Subgroups — C₂₃×#60208 #19207834

Order 5888.

1950 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3444	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2971	C₂	1	4	yes	maximal, complemented
3	C₂₃×#2970	C₂	1	4	yes	maximal, complemented
4	C₂₃×#2969	C₂	1	8	yes	maximal, complemented
5	C₂₃×#2960	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2931	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2930	C₂	1	4	yes	maximal, complemented
8	C₂₃×#2923	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2920	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#2875	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#131	C₂²	1	4	yes	—
13	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂₃×#533	C₄	1	2	yes	—
15	C₂₃×#533	C₂²	1	1	yes	characteristic
16	C₂₃×#532	C₄	1	2	yes	—
17	C₂₃×#532	C₂²	1	1	yes	characteristic
18	C₂₃×#529	C₄	1	4	yes	—
19	C₂₃×#529	C₄	1	4	yes	—
20	C₂₃×#529	C₂²	1	4	yes	—
21	C₂₃×#525	C₄	1	4	yes	—
22	C₂₃×#525	C₂²	1	2	yes	—
23	C₄×#41×C₂₃	C₂²	1	4	yes	—
24	C₄×#41×C₂₃	C₂²	1	2	yes	—
25	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#446	C₂²	1	8	yes	complemented
27	C₈×Q₈×C₂₃	C₂²	1	8	yes	complemented
28	C₂₃×#436	C₂²	1	16	yes	—
29	C₂₃×#436	C₂²	1	8	yes	complemented
30	C₂₃×#435	C₂²	1	16	yes	—
31	C₈×D₄×C₂₃	C₂²	1	8	yes	complemented
32	C₂₃×#433	C₂²	1	8	yes	complemented
33	C₂₃×#432	C₂²	1	4	yes	—
34	C₂₃×#432	C₂²	1	8	yes	complemented
35	C₂₃×#431	C₂²	1	4	yes	—
36	C₂₃×#424	C₂²	1	4	yes	—
37	C₂₃×#423	C₂²	1	2	yes	—
38	C₂₃×#408	C₂²	1	4	yes	—
39	C₂₃×#407	C₂²	1	2	yes	—
40	C₂₃×#405	C₂²	1	4	yes	—
41	C₂₃×#405	C₂²	1	8	yes	complemented
42	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
43	C₄×M₁₆×C₂₃	C₂²	1	4	yes	—
44	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
45	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
46	C₄×M₁₆×C₂₃	C₂²	1	4	yes	complemented
47	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
48	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
49	C₂×C₂₃×#97	C₂²	1	4	yes	complemented
50	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
51	C₂×#41×C₂₃	C₂×C₄	1	4	yes	—
52	C₂×#41×C₂₃	C₂³	1	2	yes	—
53	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
54	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
55	C₂₃×#131	—	2	16	no	—
56	C₂×M₁₆×C₂₃	—	2	8	no	—
57	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
58	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
59	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
60	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
61	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
62	C₂²×C₈×C₂₃	C₂³	1	8	yes	—
63	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
64	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
65	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
66	C₂₃×#127	C₂×C₄	1	1	yes	characteristic
67	C₂₃×#126	C₂×C₄	1	8	yes	—
68	C₂₃×#125	C₂×C₄	1	2	yes	—
69	C₂₃×#125	C₂×C₄	1	4	yes	—
70	C₂₃×#124	C₂×C₄	1	2	yes	—
71	C₂₃×#124	C₂×C₄	1	4	yes	—
72	C₂₃×#123	C₂×C₄	1	8	yes	—
73	C₂₃×#123	C₂×C₄	1	4	yes	—
74	C₂₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
75	C₂₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
76	C₂₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
77	C₂₃×#121	C₂×C₄	1	4	yes	—
78	C₂₃×#121	C₂×C₄	1	4	yes	—
79	C₂₃×#121	C₂×C₄	1	4	yes	—
80	C₂₃×#120	C₂×C₄	1	4	yes	—
81	C₄×Q₈×C₂₃	C₂×C₄	1	4	yes	—
82	C₄×Q₈×C₂₃	C₂×C₄	1	4	yes	—
83	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
84	C₄×D₄×C₂₃	C₂×C₄	1	8	yes	—
85	C₄×D₄×C₂₃	C₂×C₄	1	8	yes	—
86	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
87	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
88	C₄×D₄×C₂₃	C₂³	1	4	yes	—
89	C₄×D₄×C₂₃	C₂³	1	8	yes	—
90	C₂₃×#117	C₂×C₄	1	4	yes	—
91	C₂₃×#117	C₂×C₄	1	8	yes	—
92	C₂₃×#117	C₂³	1	2	yes	—
93	C₂₃×#117	C₂³	1	4	yes	—
94	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
95	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
96	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
97	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
100	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
101	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
102	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
103	C₂₃×C₄⋊C₈	C₂³	1	16	yes	—
104	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
105	C₂₃×C₂²⋊C₈	C₂³	1	16	yes	—
106	C₂₃×#97	C₂³	1	1	yes	characteristic
107	C₂₃×#97	C₂³	1	1	yes	characteristic
108	C₂₃×#97	C₂³	1	2	yes	—
109	C₂₃×#97	C₂³	1	2	yes	—
110	C₂₃×#97	C₂³	1	2	yes	—
111	C₂₃×#97	C₂³	1	4	yes	—
112	C₂₃×#97	C₂³	1	8	yes	—
113	C₄×C₈×C₂₃	C₂³	1	4	yes	—
114	C₄×C₈×C₂₃	C₂³	1	4	yes	—
115	C₄×C₈×C₂₃	C₂³	1	4	yes	—
116	C₄×C₈×C₂₃	C₂³	1	8	yes	—
117	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
118	#41×C₂₃	—	2	8	no	—
119	#41×C₂₃	—	4	8	no	—
120	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
121	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
122	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
123	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
124	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
125	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
126	C₂²×C₄×C₂₃	—	2	4	no	—
127	C₂²×C₄×C₂₃	—	4	4	no	—
128	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
129	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
131	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
132	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
133	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
134	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
135	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
136	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
137	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
138	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
139	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
140	M₁₆×C₂₃	—	2	8	no	—
141	M₁₆×C₂₃	—	2	8	no	—
142	M₁₆×C₂₃	—	2	8	no	—
143	M₁₆×C₂₃	—	2	16	no	—
144	C₂×C₈×C₂₃	—	2	4	no	—
145	C₂×C₈×C₂₃	—	2	16	no	—
146	C₂×C₈×C₂₃	#41	1	8	yes	complemented
147	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
148	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
149	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
150	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
151	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
152	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
153	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
154	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
155	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
156	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
157	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
158	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
159	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
160	C₂²⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
161	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
162	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
163	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
164	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
165	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
166	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
167	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
168	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
169	C₄²×C₂₃	C₂²×C₄	1	8	yes	—
170	C₄²×C₂₃	C₂⁴	1	2	yes	—
171	C₄²×C₂₃	C₂⁴	1	4	yes	—
172	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
173	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
174	#60208	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
175	C₂³×C₂₃	—	2	2	no	—
176	C₂³×C₂₃	—	4	2	no	—
177	C₂³×C₂₃	C₂³×C₄	1	4	yes	—
178	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
179	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
180	Q₈×C₂₃	—	2	4	no	—
181	Q₈×C₂₃	—	2	8	no	—
182	D₄×C₂₃	—	2	8	no	—
183	D₄×C₂₃	—	2	4	no	—
184	D₄×C₂₃	—	2	8	no	—
185	D₄×C₂₃	—	4	8	no	—
186	C₂×C₄×C₂₃	—	2	2	no	—
187	C₂×C₄×C₂₃	—	2	4	no	—
188	C₂×C₄×C₂₃	—	2	4	no	—
189	C₂×C₄×C₂₃	—	2	4	no	—
190	C₂×C₄×C₂₃	—	2	4	no	—
191	C₂×C₄×C₂₃	—	2	8	no	—
192	C₂×C₄×C₂₃	—	2	4	no	—
193	C₂×C₄×C₂₃	—	4	2	no	—
194	C₂×C₄×C₂₃	—	4	2	no	—
195	C₂×C₄×C₂₃	—	4	2	no	—
196	C₂×C₄×C₂₃	—	4	8	no	—
197	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
198	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
199	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
200	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
201	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
202	C₂×C₄×C₂₃	C₂³×C₄	1	8	yes	—
203	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
204	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
205	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
206	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
207	C₈×C₂₃	—	2	4	no	—
208	C₈×C₂₃	—	2	8	no	—
209	C₈×C₂₃	C₂×#41	1	4	yes	—
210	C₈×C₂₃	C₂×#41	1	4	yes	—
211	#3444	C₂×C₂₃	1	1	yes	characteristic
212	#2971	C₂×C₂₃	1	4	yes	complemented
213	#2970	C₂×C₂₃	1	4	yes	complemented
214	#2969	C₂×C₂₃	1	8	yes	complemented
215	#2960	C₂×C₂₃	1	2	yes	complemented
216	#2931	C₂×C₂₃	1	2	yes	complemented
217	#2930	C₂×C₂₃	1	4	yes	complemented
218	#2923	C₂×C₂₃	1	2	yes	complemented
219	#2920	C₂×C₂₃	1	1	yes	characteristic, complemented
220	#2875	C₂×C₂₃	1	2	yes	complemented
221	C₂×C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
222	C₂²×C₂₃	—	2	2	no	—
223	C₂²×C₂₃	—	2	2	no	—
224	C₂²×C₂₃	—	2	4	no	—
225	C₂²×C₂₃	—	2	2	no	—
226	C₂²×C₂₃	—	2	2	no	—
227	C₂²×C₂₃	—	2	8	no	—
228	C₂²×C₂₃	—	4	4	no	—
229	C₂²×C₂₃	C₄×#41	1	2	yes	—
230	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
231	C₄×C₂₃	—	2	2	no	—
232	C₄×C₂₃	—	2	4	no	—
233	C₄×C₂₃	—	2	2	no	—
234	C₄×C₂₃	—	2	2	no	—
235	C₄×C₂₃	—	2	2	no	—
236	C₄×C₂₃	—	2	8	no	—
237	C₄×C₂₃	C₄×#41	1	2	yes	—
238	C₄×C₂₃	#568	1	4	yes	—
239	C₄×C₂₃	C₂²×#41	1	2	yes	—
240	C₂×#131	C₂²×C₂₃	1	4	yes	—
241	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
242	#533	C₄×C₂₃	1	2	yes	—
243	#533	C₂²×C₂₃	1	1	yes	characteristic
244	#532	C₄×C₂₃	1	2	yes	—
245	#532	C₂²×C₂₃	1	1	yes	characteristic
246	#529	C₄×C₂₃	1	4	yes	—
247	#529	C₄×C₂₃	1	4	yes	—
248	#529	C₂²×C₂₃	1	4	yes	—
249	#525	C₄×C₂₃	1	4	yes	—
250	#525	C₂²×C₂₃	1	2	yes	—
251	C₄×#41	C₂²×C₂₃	1	4	yes	—
252	C₄×#41	C₂²×C₂₃	1	2	yes	—
253	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
254	#446	C₂²×C₂₃	1	8	yes	complemented
255	C₈×Q₈	C₂²×C₂₃	1	8	yes	complemented
256	#436	C₂²×C₂₃	1	16	yes	—
257	#436	C₂²×C₂₃	1	8	yes	complemented
258	#435	C₂²×C₂₃	1	16	yes	—
259	C₈×D₄	C₂²×C₂₃	1	8	yes	complemented
260	#433	C₂²×C₂₃	1	8	yes	complemented
261	#432	C₂²×C₂₃	1	4	yes	—
262	#432	C₂²×C₂₃	1	8	yes	complemented
263	#431	C₂²×C₂₃	1	4	yes	—
264	#424	C₂²×C₂₃	1	4	yes	—
265	#423	C₂²×C₂₃	1	2	yes	—
266	#408	C₂²×C₂₃	1	4	yes	—
267	#407	C₂²×C₂₃	1	2	yes	—
268	#405	C₂²×C₂₃	1	4	yes	—
269	#405	C₂²×C₂₃	1	8	yes	complemented
270	C₄×M₁₆	C₂²×C₂₃	1	4	yes	—
271	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
272	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
273	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
274	C₄×M₁₆	C₂²×C₂₃	1	4	yes	complemented
275	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
276	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
277	C₂×#97	C₂²×C₂₃	1	4	yes	complemented
278	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
279	C₂×C₂₃	—	2	2	no	—
280	C₂×C₂₃	—	4	4	no	—
281	C₂×C₂₃	—	4	2	no	—
282	C₂×C₂₃	C₂×C₄×#41	1	1	yes	characteristic
283	C₂×C₂₃	C₂×#568	1	2	yes	—
284	C₂×#41	C₂×C₄×C₂₃	1	4	yes	—
285	C₂×#41	C₂³×C₂₃	1	2	yes	—
286	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
287	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
288	#131	—	2	16	no	—
289	C₂×M₁₆	—	2	8	no	—
290	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
291	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
292	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
293	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
294	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
295	C₂²×C₈	C₂³×C₂₃	1	8	yes	—
296	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
297	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
298	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
299	#127	C₂×C₄×C₂₃	1	1	yes	characteristic
300	#126	C₂×C₄×C₂₃	1	8	yes	—
301	#125	C₂×C₄×C₂₃	1	2	yes	—
302	#125	C₂×C₄×C₂₃	1	4	yes	—
303	#124	C₂×C₄×C₂₃	1	2	yes	—
304	#124	C₂×C₄×C₂₃	1	4	yes	—
305	#123	C₂×C₄×C₂₃	1	8	yes	—
306	#123	C₂×C₄×C₂₃	1	4	yes	—
307	C₂²⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
308	C₂²⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
309	C₂²⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
310	#121	C₂×C₄×C₂₃	1	4	yes	—
311	#121	C₂×C₄×C₂₃	1	4	yes	—
312	#121	C₂×C₄×C₂₃	1	4	yes	—
313	#120	C₂×C₄×C₂₃	1	4	yes	—
314	C₄×Q₈	C₂×C₄×C₂₃	1	4	yes	—
315	C₄×Q₈	C₂×C₄×C₂₃	1	4	yes	—
316	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
317	C₄×D₄	C₂×C₄×C₂₃	1	8	yes	—
318	C₄×D₄	C₂×C₄×C₂₃	1	8	yes	—
319	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
320	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
321	C₄×D₄	C₂³×C₂₃	1	8	yes	—
322	C₄×D₄	C₂³×C₂₃	1	4	yes	—
323	#117	C₂×C₄×C₂₃	1	4	yes	—
324	#117	C₂×C₄×C₂₃	1	8	yes	—
325	#117	C₂³×C₂₃	1	2	yes	—
326	#117	C₂³×C₂₃	1	4	yes	—
327	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
328	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
329	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
330	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
331	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
332	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
333	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
334	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
335	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
336	C₄⋊C₈	C₂³×C₂₃	1	16	yes	—
337	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
338	C₂²⋊C₈	C₂³×C₂₃	1	16	yes	—
339	#97	C₂³×C₂₃	1	1	yes	characteristic
340	#97	C₂³×C₂₃	1	1	yes	characteristic
341	#97	C₂³×C₂₃	1	2	yes	—
342	#97	C₂³×C₂₃	1	2	yes	—
343	#97	C₂³×C₂₃	1	2	yes	—
344	#97	C₂³×C₂₃	1	4	yes	—
345	#97	C₂³×C₂₃	1	8	yes	—
346	C₄×C₈	C₂³×C₂₃	1	4	yes	—
347	C₄×C₈	C₂³×C₂₃	1	4	yes	—
348	C₄×C₈	C₂³×C₂₃	1	4	yes	—
349	C₄×C₈	C₂³×C₂₃	1	8	yes	—
350	C₂₃	#60208	1	1	yes	characteristic, direct product factor, Sylow
351	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
352	#41	—	2	8	no	—
353	#41	—	4	8	no	—
354	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
355	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
356	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
357	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
358	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
359	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
360	C₂²×C₄	—	2	4	no	—
361	C₂²×C₄	—	4	4	no	—
362	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
363	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
364	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
365	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
366	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
367	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
368	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
369	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
370	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
371	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
372	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
373	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
374	M₁₆	—	2	8	no	—
375	M₁₆	—	2	8	no	—
376	M₁₆	—	2	8	no	—
377	M₁₆	—	2	16	no	—
378	C₂×C₈	—	2	4	no	—
379	C₂×C₈	—	2	16	no	—
380	C₂×C₈	#41×C₂₃	1	8	yes	complemented
381	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
382	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
383	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
384	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
385	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
386	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
387	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
388	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
389	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
390	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
391	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
392	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
393	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
394	C₂²⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
395	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
396	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
397	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
398	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
399	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
400	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
401	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
402	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
403	C₄²	C₂²×C₄×C₂₃	1	8	yes	—
404	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
405	C₄²	C₂⁴×C₂₃	1	2	yes	—
406	C₄²	C₂⁴×C₂₃	1	4	yes	—
407	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
408	C₂³	—	2	2	no	—
409	C₂³	—	4	2	no	—
410	C₂³	C₂³×C₄×C₂₃	1	4	yes	—
411	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
412	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
413	Q₈	—	2	8	no	—
414	Q₈	—	2	4	no	—
415	D₄	—	2	8	no	—
416	D₄	—	2	8	no	—
417	D₄	—	2	4	no	—
418	D₄	—	4	8	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	4	no	—
421	C₂×C₄	—	2	4	no	—
422	C₂×C₄	—	2	4	no	—
423	C₂×C₄	—	2	8	no	—
424	C₂×C₄	—	2	4	no	—
425	C₂×C₄	—	2	4	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	8	no	—
430	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
431	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
432	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
433	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
434	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
435	C₂×C₄	C₂³×C₄×C₂₃	1	8	yes	—
436	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
437	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
438	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
439	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
440	C₈	—	2	4	no	—
441	C₈	—	2	8	no	—
442	C₈	C₂×#41×C₂₃	1	4	yes	—
443	C₈	C₂×#41×C₂₃	1	4	yes	—
444	C₂²	—	2	2	no	—
445	C₂²	—	2	2	no	—
446	C₂²	—	2	4	no	—
447	C₂²	—	2	2	no	—
448	C₂²	—	2	2	no	—
449	C₂²	—	2	8	no	—
450	C₂²	—	4	4	no	—
451	C₂²	C₄×#41×C₂₃	1	2	yes	—
452	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
453	C₄	—	2	2	no	—
454	C₄	—	2	4	no	—
455	C₄	—	2	2	no	—
456	C₄	—	2	2	no	—
457	C₄	—	2	2	no	—
458	C₄	—	2	8	no	—
459	C₄	C₄×#41×C₂₃	1	2	yes	—
460	C₄	C₂₃×#568	1	4	yes	—
461	C₄	C₂²×#41×C₂₃	1	2	yes	—
462	C₂	—	2	2	no	—
463	C₂	—	4	2	no	—
464	C₂	—	4	4	no	—
465	C₂	C₂×C₄×#41×C₂₃	1	1	yes	characteristic
466	C₂	C₂×C₂₃×#568	1	2	yes	—

	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×#3444	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×#2971	C₂	semidirect
3. C₂₃×#2970	C₂	semidirect
4. C₂₃×#2969	C₂	semidirect
5. C₂₃×#2960	C₂	semidirect
6. C₂₃×#2931	C₂	semidirect
7. C₂₃×#2930	C₂	semidirect
8. C₂₃×#2923	C₂	semidirect
9. C₂₃×#2920	C₂	semidirect
10. C₂₃×#2875	C₂	semidirect
11. C₂×C₄×M₁₆×C₂₃	C₂	semidirect
26. C₂₃×#446	C₂²	semidirect
27. C₈×Q₈×C₂₃	C₂²	semidirect
29. C₂₃×#436	C₂²	semidirect
31. C₈×D₄×C₂₃	C₂²	semidirect
32. C₂₃×#433	C₂²	semidirect
34. C₂₃×#432	C₂²	semidirect
41. C₂₃×#405	C₂²	semidirect
46. C₄×M₁₆×C₂₃	C₂²	semidirect
49. C₂×C₂₃×#97	C₂²	semidirect
146. C₂×C₈×C₂₃	#41	semidirect
174. #60208	C₂₃	direct (N × H)
212. #2971	C₂×C₂₃	semidirect
213. #2970	C₂×C₂₃	semidirect
214. #2969	C₂×C₂₃	semidirect
215. #2960	C₂×C₂₃	semidirect
216. #2931	C₂×C₂₃	semidirect
217. #2930	C₂×C₂₃	semidirect
218. #2923	C₂×C₂₃	semidirect
219. #2920	C₂×C₂₃	semidirect
220. #2875	C₂×C₂₃	semidirect
221. C₂×C₄×M₁₆	C₂×C₂₃	semidirect
254. #446	C₂²×C₂₃	semidirect
255. C₈×Q₈	C₂²×C₂₃	semidirect
257. #436	C₂²×C₂₃	semidirect
259. C₈×D₄	C₂²×C₂₃	semidirect
260. #433	C₂²×C₂₃	semidirect
262. #432	C₂²×C₂₃	semidirect
269. #405	C₂²×C₂₃	semidirect
274. C₄×M₁₆	C₂²×C₂₃	semidirect
277. C₂×#97	C₂²×C₂₃	semidirect
350. C₂₃	#60208	direct (N × H)
380. C₂×C₈	#41×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3444	2
C₂₃×#2971	2
C₂₃×#2970	2
C₂₃×#2969	2
C₂₃×#2960	2
C₂₃×#2931	2
C₂₃×#2930	2
C₂₃×#2923	2
C₂₃×#2920	2
C₂₃×#2875	2
C₂×C₄×M₁₆×C₂₃	2
#60208	23

Subgroups — C23×#60208 #19207834

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60208 #19207834