Subgroups — C₂₃×#60270 #19207896

Order 5888.

1786 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3455	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2995	C₂	1	4	yes	maximal, complemented
3	C₂₃×#2994	C₂	1	4	yes	maximal, complemented
4	C₂₃×#2993	C₂	1	4	yes	maximal, complemented
5	C₂₃×#2992	C₂	1	4	yes	maximal, complemented
6	C₂₃×#2961	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2960	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2933	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2933	C₂	1	2	yes	maximal, complemented
10	C₂₃×#2932	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#2931	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#2925	C₂	1	2	yes	maximal, complemented
13	C₂×C₂₃×#433	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#2906	C₂	1	2	yes	maximal, complemented
15	C₂₃×#2905	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×#131	C₂²	1	2	yes	—
17	C₂×C₂₃×#131	C₂²	1	2	yes	—
18	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
19	C₂₃×#542	C₄	1	2	yes	—
20	C₂₃×#542	C₄	1	2	yes	—
21	C₂₃×#541	C₄	1	2	yes	—
22	C₂₃×#540	C₄	1	2	yes	—
23	C₂₃×#539	C₄	1	2	yes	—
24	C₂₃×#539	C₄	1	2	yes	—
25	C₂₃×#538	C₄	1	2	yes	—
26	C₂₃×#538	C₄	1	2	yes	—
27	C₂₃×#533	C₂²	1	1	yes	characteristic
28	C₂₃×#532	C₂²	1	1	yes	characteristic
29	C₂₃×#529	C₂²	1	4	yes	—
30	C₂₃×#529	C₂²	1	4	yes	—
31	C₂₃×#525	C₂²	1	2	yes	—
32	C₄×#41×C₂₃	C₂²	1	2	yes	—
33	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
34	C₂₃×#446	C₂²	1	2	yes	complemented
35	C₂₃×#446	C₂²	1	2	yes	complemented
36	C₂₃×#446	C₂²	1	4	yes	complemented
37	C₂₃×#446	C₂²	1	4	yes	complemented
38	C₈×Q₈×C₂₃	C₂²	1	2	yes	complemented
39	C₈×Q₈×C₂₃	C₂²	1	2	yes	complemented
40	C₂₃×#436	C₂²	1	4	yes	—
41	C₂₃×#436	C₂²	1	4	yes	—
42	C₂₃×#436	C₂²	1	2	yes	complemented
43	C₂₃×#436	C₂²	1	2	yes	complemented
44	C₂₃×#435	C₂²	1	4	yes	—
45	C₂₃×#435	C₂²	1	4	yes	—
46	C₂₃×#435	C₂²	1	4	yes	—
47	C₂₃×#435	C₂²	1	4	yes	—
48	C₂₃×#435	C₂²	1	4	yes	complemented
49	C₂₃×#435	C₂²	1	4	yes	complemented
50	C₈×D₄×C₂₃	C₂²	1	4	yes	—
51	C₈×D₄×C₂₃	C₂²	1	4	yes	—
52	C₈×D₄×C₂₃	C₂²	1	2	yes	complemented
53	C₈×D₄×C₂₃	C₂²	1	2	yes	complemented
54	C₂₃×#433	C₂²	1	2	yes	—
55	C₂₃×#433	C₂²	1	2	yes	—
56	C₂₃×#433	C₂²	1	2	yes	—
57	C₂₃×#433	C₂²	1	2	yes	—
58	C₂₃×#433	C₂²	1	4	yes	complemented
59	C₂₃×#433	C₂²	1	4	yes	complemented
60	C₂₃×#432	C₂²	1	4	yes	complemented
61	C₂₃×#431	C₂²	1	4	yes	complemented
62	C₂₃×#424	C₂²	1	2	yes	—
63	C₂₃×#424	C₂²	1	2	yes	—
64	C₂₃×#424	C₂²	1	4	yes	—
65	C₂₃×#424	C₂²	1	4	yes	—
66	C₂₃×#424	C₂²	1	4	yes	complemented
67	C₂₃×#424	C₂²	1	4	yes	complemented
68	C₂₃×#423	C₂²	1	2	yes	—
69	C₂₃×#423	C₂²	1	1	yes	characteristic
70	C₂₃×#423	C₂²	1	1	yes	characteristic
71	C₂₃×#423	C₂²	1	4	yes	complemented
72	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
73	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
74	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	complemented
75	C₂₃×#408	C₂²	1	2	yes	—
76	C₂₃×#408	C₂²	1	2	yes	—
77	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
78	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
79	C₂₃×#405	C₂²	1	2	yes	—
80	C₂₃×#405	C₂²	1	2	yes	—
81	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
82	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
83	C₂×#41×C₂₃	C₂³	1	2	yes	—
84	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
85	C₂₃×#131	—	4	4	no	—
86	C₂₃×#131	—	4	4	no	—
87	C₂×M₁₆×C₂₃	—	4	4	no	—
88	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
89	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
90	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
91	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
92	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
93	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
94	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
95	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
96	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
97	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
98	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
99	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
100	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
101	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
102	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
103	C₂₃×#127	C₂×C₄	1	1	yes	characteristic
104	C₂₃×#126	C₂×C₄	1	4	yes	—
105	C₂₃×#126	C₂×C₄	1	4	yes	—
106	C₂₃×#126	C₂×C₄	1	4	yes	—
107	C₂₃×#126	C₂×C₄	1	4	yes	—
108	C₂₃×#125	C₂×C₄	1	4	yes	—
109	C₂₃×#125	C₂×C₄	1	4	yes	—
110	C₂₃×#125	C₂×C₄	1	2	yes	—
111	C₂₃×#124	C₂×C₄	1	4	yes	—
112	C₂₃×#124	C₂×C₄	1	4	yes	—
113	C₂₃×#124	C₂×C₄	1	2	yes	—
114	C₂₃×#123	C₂×C₄	1	4	yes	—
115	C₂₃×#123	C₂×C₄	1	4	yes	—
116	C₂₃×#123	C₂×C₄	1	4	yes	—
117	C₂₃×#123	C₂×C₄	1	4	yes	—
118	C₂₃×#123	C₂×C₄	1	4	yes	—
119	C₂₃×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
120	C₂₃×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
121	C₂₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
122	C₂₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
123	C₂₃×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
124	C₂₃×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
125	C₂₃×#121	C₂×C₄	1	2	yes	—
126	C₂₃×#121	C₂×C₄	1	2	yes	—
127	C₂₃×#121	C₂×C₄	1	4	yes	—
128	C₂₃×#121	C₂×C₄	1	4	yes	—
129	C₂₃×#121	C₂×C₄	1	2	yes	—
130	C₂₃×#121	C₂×C₄	1	2	yes	—
131	C₂₃×#120	C₂×C₄	1	4	yes	—
132	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
133	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
134	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
135	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
136	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
137	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
138	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
139	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
140	C₄×D₄×C₂₃	C₂³	1	4	yes	—
141	C₄×D₄×C₂₃	C₂³	1	4	yes	—
142	C₄×D₄×C₂₃	C₂³	1	2	yes	—
143	C₄×D₄×C₂₃	C₂³	1	2	yes	—
144	C₂₃×#117	C₂×C₄	1	2	yes	—
145	C₂₃×#117	C₂×C₄	1	2	yes	—
146	C₂₃×#117	C₂³	1	2	yes	—
147	C₂₃×#117	C₂³	1	2	yes	—
148	C₂₃×#117	C₂³	1	2	yes	—
149	C₂₃×#117	C₂³	1	4	yes	—
150	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
151	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
152	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
153	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
154	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
155	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
156	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
157	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
158	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
159	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
160	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
161	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
162	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
163	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
164	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
165	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
166	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
167	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
168	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
169	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
170	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
171	C₂₃×C₂²⋊C₈	C₂³	1	2	yes	—
172	C₂₃×C₂²⋊C₈	C₂³	1	2	yes	—
173	C₂₃×C₂²⋊C₈	C₂³	1	2	yes	—
174	C₂₃×C₂²⋊C₈	C₂³	1	2	yes	—
175	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
176	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
177	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
178	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
179	C₂₃×#97	C₂³	1	2	yes	—
180	C₂₃×#97	C₂³	1	2	yes	—
181	C₂₃×#97	C₂³	1	4	yes	—
182	C₂₃×#97	C₂³	1	4	yes	—
183	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
184	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
185	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
186	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
187	C₄×C₈×C₂₃	C₂³	1	4	yes	—
188	C₄×C₈×C₂₃	C₂³	1	4	yes	—
189	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
190	#41×C₂₃	—	2	4	no	—
191	#41×C₂₃	—	2	4	no	—
192	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
193	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
194	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
195	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
196	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
197	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
198	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
199	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
200	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
201	C₂²×C₄×C₂₃	—	2	2	no	—
202	C₂²×C₄×C₂₃	—	2	2	no	—
203	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
204	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
205	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
206	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
207	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
208	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
209	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
210	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
211	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
212	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
213	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
214	M₁₆×C₂₃	—	2	4	no	—
215	M₁₆×C₂₃	—	2	4	no	—
216	M₁₆×C₂₃	—	4	4	no	—
217	M₁₆×C₂₃	—	4	4	no	—
218	M₁₆×C₂₃	—	4	4	no	—
219	C₂×C₈×C₂₃	—	2	2	no	—
220	C₂×C₈×C₂₃	—	2	2	no	—
221	C₂×C₈×C₂₃	—	2	2	no	—
222	C₂×C₈×C₂₃	—	2	2	no	—
223	C₂×C₈×C₂₃	—	4	4	no	—
224	C₂×C₈×C₂₃	—	4	4	no	—
225	C₂×C₈×C₂₃	—	4	4	no	—
226	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
227	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
228	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
229	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
230	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
231	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
232	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
233	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
234	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
235	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
236	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
237	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
238	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
239	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
240	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
241	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
242	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
243	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
244	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
245	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
246	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
247	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
248	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
249	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
250	C₂²⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
251	C₂²⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
252	C₂²⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
253	C₂²⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
254	C₂²⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
255	C₂²⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
256	C₂²⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
257	C₂²⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
258	C₂²⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
259	C₂²⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
260	C₂²⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
261	C₂²⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
262	C₂²⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
263	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
264	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
265	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
266	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
267	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
268	C₄²×C₂₃	C₂⁴	1	2	yes	—
269	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
270	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
271	C₄²×C₂₃	C₂⁴	1	4	yes	—
272	#60270	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
273	C₂³×C₂₃	—	2	2	no	—
274	C₂³×C₂₃	—	4	1	no	—
275	C₂³×C₂₃	—	4	1	no	—
276	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
277	C₂³×C₂₃	C₂³×C₄	1	4	yes	—
278	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
279	Q₈×C₂₃	—	2	4	no	—
280	Q₈×C₂₃	—	4	2	no	—
281	Q₈×C₂₃	—	4	2	no	—
282	D₄×C₂₃	—	2	8	no	—
283	D₄×C₂₃	—	2	4	no	—
284	D₄×C₂₃	—	4	2	no	—
285	D₄×C₂₃	—	4	2	no	—
286	D₄×C₂₃	—	4	4	no	—
287	D₄×C₂₃	—	4	4	no	—
288	C₂×C₄×C₂₃	—	2	2	no	—
289	C₂×C₄×C₂₃	—	2	4	no	—
290	C₂×C₄×C₂₃	—	2	2	no	—
291	C₂×C₄×C₂₃	—	2	4	no	—
292	C₂×C₄×C₂₃	—	2	2	no	—
293	C₂×C₄×C₂₃	—	2	4	no	—
294	C₂×C₄×C₂₃	—	4	1	no	—
295	C₂×C₄×C₂₃	—	4	1	no	—
296	C₂×C₄×C₂₃	—	4	4	no	—
297	C₂×C₄×C₂₃	—	4	4	no	—
298	C₂×C₄×C₂₃	—	4	1	no	—
299	C₂×C₄×C₂₃	—	4	1	no	—
300	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
301	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
302	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
303	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
304	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
305	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
306	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
307	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
308	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
309	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
310	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
311	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
312	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
313	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
314	C₈×C₂₃	—	2	1	no	—
315	C₈×C₂₃	—	2	2	no	—
316	C₈×C₂₃	—	2	1	no	—
317	C₈×C₂₃	—	2	4	no	—
318	C₈×C₂₃	—	2	4	no	—
319	C₈×C₂₃	—	2	1	no	—
320	C₈×C₂₃	—	2	1	no	—
321	C₈×C₂₃	—	2	2	no	—
322	#3455	C₂×C₂₃	1	1	yes	characteristic
323	#2995	C₂×C₂₃	1	4	yes	complemented
324	#2994	C₂×C₂₃	1	4	yes	complemented
325	#2993	C₂×C₂₃	1	4	yes	complemented
326	#2992	C₂×C₂₃	1	4	yes	complemented
327	#2961	C₂×C₂₃	1	1	yes	characteristic, complemented
328	#2960	C₂×C₂₃	1	1	yes	characteristic, complemented
329	#2933	C₂×C₂₃	1	2	yes	complemented
330	#2933	C₂×C₂₃	1	2	yes	complemented
331	#2932	C₂×C₂₃	1	1	yes	characteristic, complemented
332	#2931	C₂×C₂₃	1	1	yes	characteristic, complemented
333	#2925	C₂×C₂₃	1	2	yes	complemented
334	C₂×#433	C₂×C₂₃	1	1	yes	characteristic, complemented
335	#2906	C₂×C₂₃	1	2	yes	complemented
336	#2905	C₂×C₂₃	1	1	yes	characteristic, complemented
337	C₂²×C₂₃	—	2	1	no	—
338	C₂²×C₂₃	—	2	1	no	—
339	C₂²×C₂₃	—	2	2	no	—
340	C₂²×C₂₃	—	2	4	no	—
341	C₂²×C₂₃	—	2	4	no	—
342	C₂²×C₂₃	—	2	4	no	—
343	C₂²×C₂₃	—	2	2	no	—
344	C₂²×C₂₃	—	2	2	no	—
345	C₂²×C₂₃	—	8	2	no	—
346	C₂²×C₂₃	#520	1	2	yes	—
347	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
348	C₄×C₂₃	—	2	2	no	—
349	C₄×C₂₃	—	2	4	no	—
350	C₄×C₂₃	—	2	2	no	—
351	C₄×C₂₃	—	2	2	no	—
352	C₄×C₂₃	—	2	1	no	—
353	C₄×C₂₃	—	2	1	no	—
354	C₄×C₂₃	—	2	4	no	—
355	C₄×C₂₃	—	2	4	no	—
356	C₄×C₂₃	—	2	1	no	—
357	C₄×C₂₃	—	2	1	no	—
358	C₄×C₂₃	#520	1	2	yes	—
359	C₄×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
360	C₄×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
361	C₂×#131	C₂²×C₂₃	1	2	yes	—
362	C₂×#131	C₂²×C₂₃	1	2	yes	—
363	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
364	#542	C₄×C₂₃	1	2	yes	—
365	#542	C₄×C₂₃	1	2	yes	—
366	#541	C₄×C₂₃	1	2	yes	—
367	#540	C₄×C₂₃	1	2	yes	—
368	#539	C₄×C₂₃	1	2	yes	—
369	#539	C₄×C₂₃	1	2	yes	—
370	#538	C₄×C₂₃	1	2	yes	—
371	#538	C₄×C₂₃	1	2	yes	—
372	#533	C₂²×C₂₃	1	1	yes	characteristic
373	#532	C₂²×C₂₃	1	1	yes	characteristic
374	#529	C₂²×C₂₃	1	4	yes	—
375	#529	C₂²×C₂₃	1	4	yes	—
376	#525	C₂²×C₂₃	1	2	yes	—
377	C₄×#41	C₂²×C₂₃	1	2	yes	—
378	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
379	#446	C₂²×C₂₃	1	2	yes	complemented
380	#446	C₂²×C₂₃	1	2	yes	complemented
381	#446	C₂²×C₂₃	1	4	yes	complemented
382	#446	C₂²×C₂₃	1	4	yes	complemented
383	C₈×Q₈	C₂²×C₂₃	1	2	yes	complemented
384	C₈×Q₈	C₂²×C₂₃	1	2	yes	complemented
385	#436	C₂²×C₂₃	1	4	yes	—
386	#436	C₂²×C₂₃	1	4	yes	—
387	#436	C₂²×C₂₃	1	2	yes	complemented
388	#436	C₂²×C₂₃	1	2	yes	complemented
389	#435	C₂²×C₂₃	1	4	yes	—
390	#435	C₂²×C₂₃	1	4	yes	—
391	#435	C₂²×C₂₃	1	4	yes	—
392	#435	C₂²×C₂₃	1	4	yes	—
393	#435	C₂²×C₂₃	1	4	yes	complemented
394	#435	C₂²×C₂₃	1	4	yes	complemented
395	C₈×D₄	C₂²×C₂₃	1	4	yes	—
396	C₈×D₄	C₂²×C₂₃	1	4	yes	—
397	C₈×D₄	C₂²×C₂₃	1	2	yes	complemented
398	C₈×D₄	C₂²×C₂₃	1	2	yes	complemented
399	#433	C₂²×C₂₃	1	2	yes	—
400	#433	C₂²×C₂₃	1	2	yes	—
401	#433	C₂²×C₂₃	1	2	yes	—
402	#433	C₂²×C₂₃	1	2	yes	—
403	#433	C₂²×C₂₃	1	4	yes	complemented
404	#433	C₂²×C₂₃	1	4	yes	complemented
405	#432	C₂²×C₂₃	1	4	yes	complemented
406	#431	C₂²×C₂₃	1	4	yes	complemented
407	#424	C₂²×C₂₃	1	4	yes	—
408	#424	C₂²×C₂₃	1	4	yes	—
409	#424	C₂²×C₂₃	1	2	yes	—
410	#424	C₂²×C₂₃	1	2	yes	—
411	#424	C₂²×C₂₃	1	4	yes	complemented
412	#424	C₂²×C₂₃	1	4	yes	complemented
413	#423	C₂²×C₂₃	1	2	yes	—
414	#423	C₂²×C₂₃	1	1	yes	characteristic
415	#423	C₂²×C₂₃	1	1	yes	characteristic
416	#423	C₂²×C₂₃	1	4	yes	complemented
417	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
418	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
419	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	complemented
420	#408	C₂²×C₂₃	1	2	yes	—
421	#408	C₂²×C₂₃	1	2	yes	—
422	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
423	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
424	#405	C₂²×C₂₃	1	2	yes	—
425	#405	C₂²×C₂₃	1	2	yes	—
426	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
427	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
428	C₂×C₂₃	—	2	2	no	—
429	C₂×C₂₃	—	4	4	no	—
430	C₂×C₂₃	—	4	2	no	—
431	C₂×C₂₃	C₂×#520	1	1	yes	characteristic
432	C₂×C₂₃	#3573	1	1	yes	characteristic
433	C₂×C₂₃	#3573	1	1	yes	characteristic
434	C₂×#41	C₂³×C₂₃	1	2	yes	—
435	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
436	#131	—	4	4	no	—
437	#131	—	4	4	no	—
438	C₂×M₁₆	—	4	4	no	—
439	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
440	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
441	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
442	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
443	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
444	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
445	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
446	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
447	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
448	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
449	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
450	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
451	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
452	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
453	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
454	#127	C₂×C₄×C₂₃	1	1	yes	characteristic
455	#126	C₂×C₄×C₂₃	1	4	yes	—
456	#126	C₂×C₄×C₂₃	1	4	yes	—
457	#126	C₂×C₄×C₂₃	1	4	yes	—
458	#126	C₂×C₄×C₂₃	1	4	yes	—
459	#125	C₂×C₄×C₂₃	1	4	yes	—
460	#125	C₂×C₄×C₂₃	1	4	yes	—
461	#125	C₂×C₄×C₂₃	1	2	yes	—
462	#124	C₂×C₄×C₂₃	1	4	yes	—
463	#124	C₂×C₄×C₂₃	1	4	yes	—
464	#124	C₂×C₄×C₂₃	1	2	yes	—
465	#123	C₂×C₄×C₂₃	1	4	yes	—
466	#123	C₂×C₄×C₂₃	1	4	yes	—
467	#123	C₂×C₄×C₂₃	1	4	yes	—
468	#123	C₂×C₄×C₂₃	1	4	yes	—
469	#123	C₂×C₄×C₂₃	1	4	yes	—
470	C₂²⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
471	C₂²⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
472	C₂²⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
473	C₂²⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
474	C₂²⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
475	C₂²⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
476	#121	C₂×C₄×C₂₃	1	2	yes	—
477	#121	C₂×C₄×C₂₃	1	2	yes	—
478	#121	C₂×C₄×C₂₃	1	4	yes	—
479	#121	C₂×C₄×C₂₃	1	4	yes	—
480	#121	C₂×C₄×C₂₃	1	2	yes	—
481	#121	C₂×C₄×C₂₃	1	2	yes	—
482	#120	C₂×C₄×C₂₃	1	4	yes	—
483	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
484	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
485	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
486	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
487	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
488	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
489	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
490	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
491	C₄×D₄	C₂³×C₂₃	1	4	yes	—
492	C₄×D₄	C₂³×C₂₃	1	4	yes	—
493	C₄×D₄	C₂³×C₂₃	1	2	yes	—
494	C₄×D₄	C₂³×C₂₃	1	2	yes	—
495	#117	C₂×C₄×C₂₃	1	2	yes	—
496	#117	C₂×C₄×C₂₃	1	2	yes	—
497	#117	C₂³×C₂₃	1	2	yes	—
498	#117	C₂³×C₂₃	1	2	yes	—
499	#117	C₂³×C₂₃	1	2	yes	—
500	#117	C₂³×C₂₃	1	4	yes	—
501	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
502	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
503	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
504	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
505	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
506	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
507	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
508	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
509	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
510	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
511	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
512	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
513	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
514	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
515	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
516	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
517	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
518	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
519	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
520	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
521	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
522	C₂²⋊C₈	C₂³×C₂₃	1	2	yes	—
523	C₂²⋊C₈	C₂³×C₂₃	1	2	yes	—
524	C₂²⋊C₈	C₂³×C₂₃	1	2	yes	—
525	C₂²⋊C₈	C₂³×C₂₃	1	2	yes	—
526	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
527	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
528	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
529	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
530	#97	C₂³×C₂₃	1	2	yes	—
531	#97	C₂³×C₂₃	1	2	yes	—
532	#97	C₂³×C₂₃	1	4	yes	—
533	#97	C₂³×C₂₃	1	4	yes	—
534	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
535	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
536	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
537	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
538	C₄×C₈	C₂³×C₂₃	1	4	yes	—
539	C₄×C₈	C₂³×C₂₃	1	4	yes	—
540	C₂₃	#60270	1	1	yes	characteristic, direct product factor, Sylow
541	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
542	#41	—	2	4	no	—
543	#41	—	2	4	no	—
544	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
545	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
546	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
547	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
548	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
549	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
550	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
551	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
552	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
553	C₂²×C₄	—	2	2	no	—
554	C₂²×C₄	—	2	2	no	—
555	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
556	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
557	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
558	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
559	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
560	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
561	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
562	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
563	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
564	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
565	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
566	M₁₆	—	2	4	no	—
567	M₁₆	—	2	4	no	—
568	M₁₆	—	4	4	no	—
569	M₁₆	—	4	4	no	—
570	M₁₆	—	4	4	no	—
571	C₂×C₈	—	2	2	no	—
572	C₂×C₈	—	2	2	no	—
573	C₂×C₈	—	2	2	no	—
574	C₂×C₈	—	2	2	no	—
575	C₂×C₈	—	4	4	no	—
576	C₂×C₈	—	4	4	no	—
577	C₂×C₈	—	4	4	no	—
578	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
579	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
580	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
581	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
582	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
583	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
584	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
585	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
586	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
587	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
588	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
589	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
590	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
591	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
592	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
593	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
594	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
595	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
596	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
597	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
598	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
599	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
600	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
601	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
602	C₂²⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
603	C₂²⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
604	C₂²⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
605	C₂²⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
606	C₂²⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
607	C₂²⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
608	C₂²⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
609	C₂²⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
610	C₂²⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
611	C₂²⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
612	C₂²⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
613	C₂²⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
614	C₂²⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
615	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
616	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
617	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
618	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
619	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
620	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
621	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
622	C₄²	C₂⁴×C₂₃	1	4	yes	—
623	C₄²	C₂⁴×C₂₃	1	2	yes	—
624	C₂³	—	2	2	no	—
625	C₂³	—	4	1	no	—
626	C₂³	—	4	1	no	—
627	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
628	C₂³	C₂³×C₄×C₂₃	1	4	yes	—
629	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
630	Q₈	—	2	4	no	—
631	Q₈	—	4	2	no	—
632	Q₈	—	4	2	no	—
633	D₄	—	2	8	no	—
634	D₄	—	2	4	no	—
635	D₄	—	4	2	no	—
636	D₄	—	4	2	no	—
637	D₄	—	4	4	no	—
638	D₄	—	4	4	no	—
639	C₂×C₄	—	2	2	no	—
640	C₂×C₄	—	2	2	no	—
641	C₂×C₄	—	2	4	no	—
642	C₂×C₄	—	2	2	no	—
643	C₂×C₄	—	2	4	no	—
644	C₂×C₄	—	2	4	no	—
645	C₂×C₄	—	4	1	no	—
646	C₂×C₄	—	4	1	no	—
647	C₂×C₄	—	4	4	no	—
648	C₂×C₄	—	4	4	no	—
649	C₂×C₄	—	4	1	no	—
650	C₂×C₄	—	4	1	no	—
651	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
652	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
653	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
654	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
655	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
656	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
657	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
658	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
659	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
660	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
661	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
662	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
663	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
664	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
665	C₈	—	2	1	no	—
666	C₈	—	2	2	no	—
667	C₈	—	2	1	no	—
668	C₈	—	2	4	no	—
669	C₈	—	2	4	no	—
670	C₈	—	2	1	no	—
671	C₈	—	2	1	no	—
672	C₈	—	2	2	no	—
673	C₂²	—	2	1	no	—
674	C₂²	—	2	1	no	—
675	C₂²	—	2	2	no	—
676	C₂²	—	2	4	no	—
677	C₂²	—	2	4	no	—
678	C₂²	—	2	4	no	—
679	C₂²	—	2	2	no	—
680	C₂²	—	2	2	no	—
681	C₂²	—	8	2	no	—
682	C₂²	C₂₃×#520	1	2	yes	—
683	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
684	C₄	—	2	2	no	—
685	C₄	—	2	4	no	—
686	C₄	—	2	2	no	—
687	C₄	—	2	2	no	—
688	C₄	—	2	1	no	—
689	C₄	—	2	1	no	—
690	C₄	—	2	4	no	—
691	C₄	—	2	4	no	—
692	C₄	—	2	1	no	—
693	C₄	—	2	1	no	—
694	C₄	C₂₃×#520	1	2	yes	—
695	C₄	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
696	C₄	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
697	C₂	—	2	2	no	—
698	C₂	—	4	2	no	—
699	C₂	—	4	4	no	—
700	C₂	C₂×C₂₃×#520	1	1	yes	characteristic
701	C₂	C₂₃×#3573	1	1	yes	characteristic
702	C₂	C₂₃×#3573	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
socle	C₂²×C₂₃	C₂⁴×C₄
prime radical	C₂₃×#3455	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂₃×#2995	C₂	semidirect
3. C₂₃×#2994	C₂	semidirect
4. C₂₃×#2993	C₂	semidirect
5. C₂₃×#2992	C₂	semidirect
6. C₂₃×#2961	C₂	semidirect
7. C₂₃×#2960	C₂	semidirect
8. C₂₃×#2933	C₂	semidirect
9. C₂₃×#2933	C₂	semidirect
10. C₂₃×#2932	C₂	semidirect
11. C₂₃×#2931	C₂	semidirect
12. C₂₃×#2925	C₂	semidirect
13. C₂×C₂₃×#433	C₂	semidirect
14. C₂₃×#2906	C₂	semidirect
15. C₂₃×#2905	C₂	semidirect
34. C₂₃×#446	C₂²	semidirect
35. C₂₃×#446	C₂²	semidirect
36. C₂₃×#446	C₂²	semidirect
37. C₂₃×#446	C₂²	semidirect
38. C₈×Q₈×C₂₃	C₂²	semidirect
39. C₈×Q₈×C₂₃	C₂²	semidirect
42. C₂₃×#436	C₂²	semidirect
43. C₂₃×#436	C₂²	semidirect
48. C₂₃×#435	C₂²	semidirect
49. C₂₃×#435	C₂²	semidirect
52. C₈×D₄×C₂₃	C₂²	semidirect
53. C₈×D₄×C₂₃	C₂²	semidirect
58. C₂₃×#433	C₂²	semidirect
59. C₂₃×#433	C₂²	semidirect
60. C₂₃×#432	C₂²	semidirect
61. C₂₃×#431	C₂²	semidirect
66. C₂₃×#424	C₂²	semidirect
67. C₂₃×#424	C₂²	semidirect
71. C₂₃×#423	C₂²	semidirect
74. C₂×C₂₃×C₄⋊C₈	C₂²	semidirect
272. #60270	C₂₃	direct (N × H)
323. #2995	C₂×C₂₃	semidirect
324. #2994	C₂×C₂₃	semidirect
325. #2993	C₂×C₂₃	semidirect
326. #2992	C₂×C₂₃	semidirect
327. #2961	C₂×C₂₃	semidirect
328. #2960	C₂×C₂₃	semidirect
329. #2933	C₂×C₂₃	semidirect
330. #2933	C₂×C₂₃	semidirect
331. #2932	C₂×C₂₃	semidirect
332. #2931	C₂×C₂₃	semidirect
333. #2925	C₂×C₂₃	semidirect
334. C₂×#433	C₂×C₂₃	semidirect
335. #2906	C₂×C₂₃	semidirect
336. #2905	C₂×C₂₃	semidirect
379. #446	C₂²×C₂₃	semidirect
380. #446	C₂²×C₂₃	semidirect
381. #446	C₂²×C₂₃	semidirect
382. #446	C₂²×C₂₃	semidirect
383. C₈×Q₈	C₂²×C₂₃	semidirect
384. C₈×Q₈	C₂²×C₂₃	semidirect
387. #436	C₂²×C₂₃	semidirect
388. #436	C₂²×C₂₃	semidirect
393. #435	C₂²×C₂₃	semidirect
394. #435	C₂²×C₂₃	semidirect
397. C₈×D₄	C₂²×C₂₃	semidirect
398. C₈×D₄	C₂²×C₂₃	semidirect
403. #433	C₂²×C₂₃	semidirect
404. #433	C₂²×C₂₃	semidirect
405. #432	C₂²×C₂₃	semidirect
406. #431	C₂²×C₂₃	semidirect
411. #424	C₂²×C₂₃	semidirect
412. #424	C₂²×C₂₃	semidirect
416. #423	C₂²×C₂₃	semidirect
419. C₂×C₄⋊C₈	C₂²×C₂₃	semidirect
540. C₂₃	#60270	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂₃×#3455	2
C₂₃×#2995	2
C₂₃×#2994	2
C₂₃×#2993	2
C₂₃×#2992	2
C₂₃×#2961	2
C₂₃×#2960	2
C₂₃×#2933	2
C₂₃×#2932	2
C₂₃×#2931	2
C₂₃×#2925	2
C₂×C₂₃×#433	2
C₂₃×#2906	2
C₂₃×#2905	2
#60270	23

Subgroups — C23×#60270 #19207896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60270 #19207896