Subgroups — C₂₃×#17417 #19165043

Order 5888.

944 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#552	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2924	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1473	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1470	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1468	C₂	1	2	yes	maximal
6	C₂₃×#1467	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1464	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1463	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1461	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1458	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×#324	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#552	—	2	2	no	—
13	C₂₃×#552	—	2	2	no	—
14	C₂₃×#552	—	2	2	no	—
15	C₂₃×#552	—	2	2	no	—
16	C₂×C₂₃×#126	C₄	1	1	yes	characteristic
17	C₂×C₂₃×#126	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#125	C₄	1	1	yes	characteristic
19	C₂×C₂₃×#125	C₄	1	1	yes	characteristic
20	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#123	C₄	1	1	yes	characteristic
22	C₂×C₂₃×#123	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×C₂²⋊Q₈	C₄	1	1	yes	characteristic
24	C₂×C₂₃×C₂²⋊Q₈	C₄	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×D₄×C₂₃	C₄	1	1	yes	characteristic
27	C₂×C₂₃×#117	C₄	1	1	yes	characteristic
28	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
30	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
31	C₂₃×#433	—	2	2	no	—
32	C₂₃×#433	—	2	2	no	—
33	C₂₃×#432	C₂²	1	2	yes	—
34	C₂₃×#432	C₂²	1	2	yes	—
35	C₂₃×#431	C₂²	1	2	yes	—
36	C₂₃×#431	C₂²	1	2	yes	—
37	C₂₃×#424	—	2	2	no	—
38	C₂₃×#407	C₂²	1	2	yes	—
39	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
40	C₂₃×#405	—	2	2	no	—
41	C₂₃×#324	C₂²	1	1	yes	characteristic
42	C₂₃×#324	C₂²	1	1	yes	characteristic
43	C₂₃×#324	C₂²	1	2	yes	—
44	C₂₃×#324	C₂²	1	1	yes	characteristic
45	C₂₃×#324	C₂²	1	1	yes	characteristic
46	C₂₃×#324	C₂²	1	2	yes	—
47	C₂₃×#324	C₂²	1	2	yes	—
48	C₂₃×#324	C₂²	1	2	yes	—
49	C₂₃×#323	C₂²	1	2	yes	—
50	C₂₃×#323	C₂²	1	2	yes	—
51	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
53	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
54	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
55	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
56	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
57	C₂×M₁₆×C₂₃	—	2	2	no	—
58	C₂²×C₈×C₂₃	—	2	2	no	—
59	C₂₃×#126	—	2	2	no	—
60	C₂₃×#126	—	2	2	no	—
61	C₂₃×#126	—	2	2	no	—
62	C₂₃×#126	—	2	2	no	—
63	C₂₃×#125	—	2	2	no	—
64	C₂₃×#125	—	2	2	no	—
65	C₂₃×#125	—	2	2	no	—
66	C₂₃×#125	—	2	2	no	—
67	C₂₃×#125	—	2	2	no	—
68	C₂₃×#125	—	2	2	no	—
69	C₂₃×#123	—	2	2	no	—
70	C₂₃×#123	—	2	2	no	—
71	C₂₃×#123	—	2	2	no	—
72	C₂₃×#123	—	2	2	no	—
73	C₂₃×C₂²⋊Q₈	—	2	2	no	—
74	C₂₃×C₂²⋊Q₈	—	2	2	no	—
75	C₂₃×C₂²⋊Q₈	—	2	2	no	—
76	C₂₃×C₂²⋊Q₈	—	2	2	no	—
77	C₄×Q₈×C₂₃	—	2	2	no	—
78	C₄×Q₈×C₂₃	—	2	2	no	—
79	C₄×D₄×C₂₃	—	2	1	no	—
80	C₄×D₄×C₂₃	—	2	1	no	—
81	C₄×D₄×C₂₃	—	2	1	no	—
82	C₄×D₄×C₂₃	—	2	1	no	—
83	C₂₃×#117	—	2	1	no	—
84	C₂₃×#117	—	2	1	no	—
85	C₂₃×#117	—	2	1	no	—
86	C₂₃×#117	—	2	1	no	—
87	C₂₃×#117	—	2	2	no	—
88	C₂₃×#117	—	2	2	no	—
89	C₂₃×#117	D₄	1	2	yes	—
90	C₂₃×#117	D₄	1	2	yes	—
91	C₂₃×#117	D₄	1	2	yes	—
92	C₂₃×#117	D₄	1	2	yes	—
93	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
94	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
95	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
111	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
112	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
113	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
114	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
115	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
116	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
117	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
118	C₂×C₂²⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
120	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
121	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
122	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
123	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
124	C₂₃×C₄⋊C₈	—	2	2	no	—
125	C₂₃×C₄⋊C₈	—	2	2	no	—
126	C₂₃×C₄⋊C₈	—	2	2	no	—
127	C₂₃×C₄⋊C₈	—	2	2	no	—
128	C₂₃×C₂²⋊C₈	—	2	2	no	—
129	C₂₃×C₂²⋊C₈	—	2	2	no	—
130	C₂₃×C₂²⋊C₈	C₂³	1	2	yes	—
131	C₂₃×C₂²⋊C₈	C₂³	1	2	yes	—
132	C₂₃×C₂²⋊C₈	C₂³	1	2	yes	—
133	C₂₃×C₂²⋊C₈	C₂³	1	2	yes	—
134	C₂₃×#97	—	2	2	no	—
135	C₂₃×#97	—	2	2	no	—
136	C₄×C₈×C₂₃	—	2	2	no	—
137	C₄×C₈×C₂₃	—	2	2	no	—
138	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
139	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
140	C₂×Q₈×C₂₃	—	2	2	no	—
141	C₂×Q₈×C₂₃	—	2	2	no	—
142	C₂×Q₈×C₂₃	—	4	1	no	—
143	C₂×D₄×C₂₃	—	2	1	no	—
144	C₂×D₄×C₂₃	—	2	1	no	—
145	C₂×D₄×C₂₃	—	4	2	no	—
146	C₂²×C₄×C₂₃	—	2	1	no	—
147	C₂²×C₄×C₂₃	—	2	1	no	—
148	C₂²×C₄×C₂₃	—	2	1	no	—
149	C₂²×C₄×C₂₃	—	2	1	no	—
150	C₂²×C₄×C₂₃	—	2	1	no	—
151	C₂²×C₄×C₂₃	—	2	1	no	—
152	C₂²×C₄×C₂₃	—	2	2	no	—
153	C₂²×C₄×C₂₃	—	2	2	no	—
154	C₂²×C₄×C₂₃	—	4	1	no	—
155	C₂²×C₄×C₂₃	—	4	1	no	—
156	C₂²×C₄×C₂₃	—	4	2	no	—
157	C₂²×C₄×C₂₃	—	4	2	no	—
158	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
159	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
160	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
162	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
163	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
164	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
165	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
166	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
167	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
168	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
169	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
170	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
171	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
173	M₁₆×C₂₃	—	4	4	no	—
174	C₂×C₈×C₂₃	—	2	2	no	—
175	C₂×C₈×C₂₃	—	2	2	no	—
176	C₂×C₈×C₂₃	—	2	2	no	—
177	C₂×C₈×C₂₃	—	2	2	no	—
178	C₂×C₈×C₂₃	—	4	4	no	—
179	C₄⋊C₄×C₂₃	—	2	2	no	—
180	C₄⋊C₄×C₂₃	—	2	2	no	—
181	C₄⋊C₄×C₂₃	—	2	2	no	—
182	C₄⋊C₄×C₂₃	—	2	1	no	—
183	C₄⋊C₄×C₂₃	—	2	1	no	—
184	C₄⋊C₄×C₂₃	—	2	1	no	—
185	C₄⋊C₄×C₂₃	—	2	1	no	—
186	C₄⋊C₄×C₂₃	—	2	1	no	—
187	C₄⋊C₄×C₂₃	—	2	1	no	—
188	C₄⋊C₄×C₂₃	—	2	2	no	—
189	C₄⋊C₄×C₂₃	—	2	2	no	—
190	C₄⋊C₄×C₂₃	—	2	2	no	—
191	C₄⋊C₄×C₂₃	—	2	2	no	—
192	C₄⋊C₄×C₂₃	—	2	2	no	—
193	C₄⋊C₄×C₂₃	—	2	2	no	—
194	C₄⋊C₄×C₂₃	—	2	2	no	—
195	C₄⋊C₄×C₂₃	—	2	2	no	—
196	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
197	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
198	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
199	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
200	C₂²⋊C₄×C₂₃	—	2	1	no	—
201	C₂²⋊C₄×C₂₃	—	2	1	no	—
202	C₂²⋊C₄×C₂₃	—	2	2	no	—
203	C₂²⋊C₄×C₂₃	—	2	1	no	—
204	C₂²⋊C₄×C₂₃	—	2	1	no	—
205	C₂²⋊C₄×C₂₃	—	2	2	no	—
206	C₂²⋊C₄×C₂₃	—	2	2	no	—
207	C₂²⋊C₄×C₂₃	—	2	2	no	—
208	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
209	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
210	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
211	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
212	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
213	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
214	C₄²×C₂₃	—	2	1	no	—
215	C₄²×C₂₃	—	2	1	no	—
216	C₄²×C₂₃	—	2	1	no	—
217	C₄²×C₂₃	—	2	1	no	—
218	C₄²×C₂₃	—	2	1	no	—
219	C₄²×C₂₃	—	2	1	no	—
220	C₄²×C₂₃	C₂×D₄	1	2	yes	—
221	C₄²×C₂₃	C₂×D₄	1	2	yes	—
222	C₄²×C₂₃	C₂×D₄	1	2	yes	—
223	C₄²×C₂₃	C₂×D₄	1	2	yes	—
224	#17417	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
225	C₂³×C₂₃	—	2	1	no	—
226	C₂³×C₂₃	—	2	1	no	—
227	C₂³×C₂₃	—	2	2	no	—
228	C₂³×C₂₃	—	2	1	no	—
229	C₂³×C₂₃	—	2	1	no	—
230	C₂³×C₂₃	—	2	1	no	—
231	C₂³×C₂₃	—	4	1	no	—
232	C₂³×C₂₃	—	4	1	no	—
233	C₂³×C₂₃	—	4	1	no	—
234	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
235	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
236	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
237	Q₈×C₂₃	—	4	4	no	—
238	D₄×C₂₃	—	4	2	no	—
239	D₄×C₂₃	—	4	2	no	—
240	C₂×C₄×C₂₃	—	2	1	no	—
241	C₂×C₄×C₂₃	—	2	1	no	—
242	C₂×C₄×C₂₃	—	2	1	no	—
243	C₂×C₄×C₂₃	—	2	1	no	—
244	C₂×C₄×C₂₃	—	2	1	no	—
245	C₂×C₄×C₂₃	—	2	1	no	—
246	C₂×C₄×C₂₃	—	2	4	no	—
247	C₂×C₄×C₂₃	—	2	1	no	—
248	C₂×C₄×C₂₃	—	2	1	no	—
249	C₂×C₄×C₂₃	—	2	1	no	—
250	C₂×C₄×C₂₃	—	2	1	no	—
251	C₂×C₄×C₂₃	—	2	1	no	—
252	C₂×C₄×C₂₃	—	2	1	no	—
253	C₂×C₄×C₂₃	—	2	1	no	—
254	C₂×C₄×C₂₃	—	2	1	no	—
255	C₂×C₄×C₂₃	—	2	1	no	—
256	C₂×C₄×C₂₃	—	2	1	no	—
257	C₂×C₄×C₂₃	—	4	1	no	—
258	C₂×C₄×C₂₃	—	4	2	no	—
259	C₂×C₄×C₂₃	—	4	1	no	—
260	C₂×C₄×C₂₃	—	4	1	no	—
261	C₂×C₄×C₂₃	—	4	1	no	—
262	C₂×C₄×C₂₃	—	4	2	no	—
263	C₂×C₄×C₂₃	—	4	2	no	—
264	C₂×C₄×C₂₃	—	4	2	no	—
265	C₂×C₄×C₂₃	—	4	2	no	—
266	C₂×C₄×C₂₃	—	4	2	no	—
267	C₂×C₄×C₂₃	—	4	2	no	—
268	C₂×C₄×C₂₃	—	4	1	no	—
269	C₂×C₄×C₂₃	—	4	1	no	—
270	C₂×C₄×C₂₃	—	4	1	no	—
271	C₂×C₄×C₂₃	—	4	1	no	—
272	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
273	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
274	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
275	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
276	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
277	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
278	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
279	C₂×C₄×C₂₃	#131	1	2	yes	—
280	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
281	C₈×C₂₃	—	4	2	no	—
282	C₈×C₂₃	—	4	2	no	—
283	C₈×C₂₃	—	4	2	no	—
284	C₈×C₂₃	—	4	2	no	—
285	C₂×#552	C₂×C₂₃	1	1	yes	characteristic
286	#2924	C₂×C₂₃	1	2	yes	complemented
287	#1473	C₂×C₂₃	1	2	yes	complemented
288	#1470	C₂×C₂₃	1	2	yes	complemented
289	#1468	C₂×C₂₃	1	2	yes	—
290	#1467	C₂×C₂₃	1	1	yes	characteristic, complemented
291	#1464	C₂×C₂₃	1	1	yes	characteristic, complemented
292	#1463	C₂×C₂₃	1	1	yes	characteristic, complemented
293	#1461	C₂×C₂₃	1	1	yes	characteristic, complemented
294	#1458	C₂×C₂₃	1	1	yes	characteristic, complemented
295	C₂×#324	C₂×C₂₃	1	1	yes	characteristic, complemented
296	C₂²×C₂₃	—	2	1	no	—
297	C₂²×C₂₃	—	2	1	no	—
298	C₂²×C₂₃	—	2	1	no	—
299	C₂²×C₂₃	—	2	1	no	—
300	C₂²×C₂₃	—	2	1	no	—
301	C₂²×C₂₃	—	2	1	no	—
302	C₂²×C₂₃	—	2	1	no	—
303	C₂²×C₂₃	—	4	2	no	—
304	C₂²×C₂₃	—	4	2	no	—
305	C₂²×C₂₃	—	4	1	no	—
306	C₂²×C₂₃	—	4	1	no	—
307	C₂²×C₂₃	—	4	1	no	—
308	C₂²×C₂₃	—	8	1	no	—
309	C₂²×C₂₃	—	8	1	no	—
310	C₂²×C₂₃	#410	1	2	yes	—
311	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
312	C₂²×C₂₃	C₂×#131	1	1	yes	characteristic
313	C₂²×C₂₃	#568	1	1	yes	characteristic
314	C₄×C₂₃	—	2	2	no	—
315	C₄×C₂₃	—	2	1	no	—
316	C₄×C₂₃	—	2	1	no	—
317	C₄×C₂₃	—	2	2	no	—
318	C₄×C₂₃	—	2	2	no	—
319	C₄×C₂₃	—	2	2	no	—
320	C₄×C₂₃	—	2	1	no	—
321	C₄×C₂₃	—	2	1	no	—
322	C₄×C₂₃	—	2	2	no	—
323	C₄×C₂₃	—	4	1	no	—
324	C₄×C₂₃	—	4	1	no	—
325	C₄×C₂₃	—	4	1	no	—
326	C₄×C₂₃	—	4	1	no	—
327	C₄×C₂₃	—	4	1	no	—
328	C₄×C₂₃	—	4	1	no	—
329	#552	—	2	2	no	—
330	#552	—	2	2	no	—
331	#552	—	2	2	no	—
332	#552	—	2	2	no	—
333	C₂×#126	C₄×C₂₃	1	1	yes	characteristic
334	C₂×#126	C₂²×C₂₃	1	1	yes	characteristic
335	C₂×#125	C₄×C₂₃	1	1	yes	characteristic
336	C₂×#125	C₄×C₂₃	1	1	yes	characteristic
337	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
338	C₂×#123	C₄×C₂₃	1	1	yes	characteristic
339	C₂×#123	C₂²×C₂₃	1	1	yes	characteristic
340	C₂×C₂²⋊Q₈	C₄×C₂₃	1	1	yes	characteristic
341	C₂×C₂²⋊Q₈	C₄×C₂₃	1	1	yes	characteristic
342	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
343	C₂×C₄×D₄	C₄×C₂₃	1	1	yes	characteristic
344	C₂×#117	C₄×C₂₃	1	1	yes	characteristic
345	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
346	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
347	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
348	#433	—	2	2	no	—
349	#433	—	2	2	no	—
350	#432	C₂²×C₂₃	1	2	yes	—
351	#432	C₂²×C₂₃	1	2	yes	—
352	#431	C₂²×C₂₃	1	2	yes	—
353	#431	C₂²×C₂₃	1	2	yes	—
354	#424	—	2	2	no	—
355	#407	C₂²×C₂₃	1	2	yes	—
356	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
357	#405	—	2	2	no	—
358	#324	C₂²×C₂₃	1	1	yes	characteristic
359	#324	C₂²×C₂₃	1	1	yes	characteristic
360	#324	C₂²×C₂₃	1	2	yes	—
361	#324	C₂²×C₂₃	1	1	yes	characteristic
362	#324	C₂²×C₂₃	1	1	yes	characteristic
363	#324	C₂²×C₂₃	1	2	yes	—
364	#324	C₂²×C₂₃	1	2	yes	—
365	#324	C₂²×C₂₃	1	2	yes	—
366	#323	C₂²×C₂₃	1	2	yes	—
367	#323	C₂²×C₂₃	1	2	yes	—
368	C₂×C₂₃	—	2	1	no	—
369	C₂×C₂₃	—	2	1	no	—
370	C₂×C₂₃	—	2	2	no	—
371	C₂×C₂₃	—	4	1	no	—
372	C₂×C₂₃	—	8	1	no	—
373	C₂×C₂₃	#2880	1	1	yes	characteristic
374	C₂×C₂₃	C₂×#410	1	1	yes	characteristic
375	C₂×C₂₃	#2889	1	1	yes	characteristic
376	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
377	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
378	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
379	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
380	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
381	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
382	C₂×M₁₆	—	2	2	no	—
383	C₂²×C₈	—	2	2	no	—
384	#126	—	2	2	no	—
385	#126	—	2	2	no	—
386	#126	—	2	2	no	—
387	#126	—	2	2	no	—
388	#125	—	2	2	no	—
389	#125	—	2	2	no	—
390	#125	—	2	2	no	—
391	#125	—	2	2	no	—
392	#125	—	2	2	no	—
393	#125	—	2	2	no	—
394	#123	—	2	2	no	—
395	#123	—	2	2	no	—
396	#123	—	2	2	no	—
397	#123	—	2	2	no	—
398	C₂²⋊Q₈	—	2	2	no	—
399	C₂²⋊Q₈	—	2	2	no	—
400	C₂²⋊Q₈	—	2	2	no	—
401	C₂²⋊Q₈	—	2	2	no	—
402	C₄×Q₈	—	2	2	no	—
403	C₄×Q₈	—	2	2	no	—
404	C₄×D₄	—	2	1	no	—
405	C₄×D₄	—	2	1	no	—
406	C₄×D₄	—	2	1	no	—
407	C₄×D₄	—	2	1	no	—
408	#117	—	2	1	no	—
409	#117	—	2	1	no	—
410	#117	—	2	1	no	—
411	#117	—	2	1	no	—
412	#117	—	2	2	no	—
413	#117	—	2	2	no	—
414	#117	D₄×C₂₃	1	2	yes	—
415	#117	D₄×C₂₃	1	2	yes	—
416	#117	D₄×C₂₃	1	2	yes	—
417	#117	D₄×C₂₃	1	2	yes	—
418	C₂×C₄⋊C₄	—	2	2	no	—
419	C₂×C₄⋊C₄	—	2	2	no	—
420	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
421	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
422	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
423	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
424	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
425	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
426	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
427	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
428	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
429	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
430	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
431	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
432	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
433	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
434	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
435	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
436	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
437	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
438	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
439	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
440	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
441	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
442	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
443	C₂×C₂²⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
444	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
445	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
446	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
447	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
448	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
449	C₄⋊C₈	—	2	2	no	—
450	C₄⋊C₈	—	2	2	no	—
451	C₄⋊C₈	—	2	2	no	—
452	C₄⋊C₈	—	2	2	no	—
453	C₂²⋊C₈	—	2	2	no	—
454	C₂²⋊C₈	—	2	2	no	—
455	C₂²⋊C₈	C₂³×C₂₃	1	2	yes	—
456	C₂²⋊C₈	C₂³×C₂₃	1	2	yes	—
457	C₂²⋊C₈	C₂³×C₂₃	1	2	yes	—
458	C₂²⋊C₈	C₂³×C₂₃	1	2	yes	—
459	#97	—	2	2	no	—
460	#97	—	2	2	no	—
461	C₄×C₈	—	2	2	no	—
462	C₄×C₈	—	2	2	no	—
463	C₂₃	#17417	1	1	yes	characteristic, direct product factor, Sylow
464	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
465	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
466	C₂×Q₈	—	2	2	no	—
467	C₂×Q₈	—	2	2	no	—
468	C₂×Q₈	—	4	1	no	—
469	C₂×D₄	—	2	1	no	—
470	C₂×D₄	—	2	1	no	—
471	C₂×D₄	—	4	2	no	—
472	C₂²×C₄	—	2	1	no	—
473	C₂²×C₄	—	2	1	no	—
474	C₂²×C₄	—	2	1	no	—
475	C₂²×C₄	—	2	1	no	—
476	C₂²×C₄	—	2	1	no	—
477	C₂²×C₄	—	2	1	no	—
478	C₂²×C₄	—	2	2	no	—
479	C₂²×C₄	—	2	2	no	—
480	C₂²×C₄	—	4	1	no	—
481	C₂²×C₄	—	4	1	no	—
482	C₂²×C₄	—	4	2	no	—
483	C₂²×C₄	—	4	2	no	—
484	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
485	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
486	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
487	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
488	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
489	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
490	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
491	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
492	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
493	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
494	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
495	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
496	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
497	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
498	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
499	M₁₆	—	4	4	no	—
500	C₂×C₈	—	2	2	no	—
501	C₂×C₈	—	2	2	no	—
502	C₂×C₈	—	2	2	no	—
503	C₂×C₈	—	2	2	no	—
504	C₂×C₈	—	4	4	no	—
505	C₄⋊C₄	—	2	2	no	—
506	C₄⋊C₄	—	2	2	no	—
507	C₄⋊C₄	—	2	2	no	—
508	C₄⋊C₄	—	2	1	no	—
509	C₄⋊C₄	—	2	1	no	—
510	C₄⋊C₄	—	2	1	no	—
511	C₄⋊C₄	—	2	1	no	—
512	C₄⋊C₄	—	2	1	no	—
513	C₄⋊C₄	—	2	1	no	—
514	C₄⋊C₄	—	2	2	no	—
515	C₄⋊C₄	—	2	2	no	—
516	C₄⋊C₄	—	2	2	no	—
517	C₄⋊C₄	—	2	2	no	—
518	C₄⋊C₄	—	2	2	no	—
519	C₄⋊C₄	—	2	2	no	—
520	C₄⋊C₄	—	2	2	no	—
521	C₄⋊C₄	—	2	2	no	—
522	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
523	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
524	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
525	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
526	C₂²⋊C₄	—	2	1	no	—
527	C₂²⋊C₄	—	2	1	no	—
528	C₂²⋊C₄	—	2	2	no	—
529	C₂²⋊C₄	—	2	1	no	—
530	C₂²⋊C₄	—	2	1	no	—
531	C₂²⋊C₄	—	2	2	no	—
532	C₂²⋊C₄	—	2	2	no	—
533	C₂²⋊C₄	—	2	2	no	—
534	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
535	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
536	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
537	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
538	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
539	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
540	C₄²	—	2	1	no	—
541	C₄²	—	2	1	no	—
542	C₄²	—	2	1	no	—
543	C₄²	—	2	1	no	—
544	C₄²	—	2	1	no	—
545	C₄²	—	2	1	no	—
546	C₄²	C₂×D₄×C₂₃	1	2	yes	—
547	C₄²	C₂×D₄×C₂₃	1	2	yes	—
548	C₄²	C₂×D₄×C₂₃	1	2	yes	—
549	C₄²	C₂×D₄×C₂₃	1	2	yes	—
550	C₂³	—	2	1	no	—
551	C₂³	—	2	1	no	—
552	C₂³	—	2	2	no	—
553	C₂³	—	2	1	no	—
554	C₂³	—	2	1	no	—
555	C₂³	—	2	1	no	—
556	C₂³	—	4	1	no	—
557	C₂³	—	4	1	no	—
558	C₂³	—	4	1	no	—
559	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
560	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
561	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
562	Q₈	—	4	4	no	—
563	D₄	—	4	2	no	—
564	D₄	—	4	2	no	—
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	1	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	4	no	—
571	C₂×C₄	—	2	1	no	—
572	C₂×C₄	—	2	1	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	1	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	1	no	—
581	C₂×C₄	—	2	1	no	—
582	C₂×C₄	—	4	1	no	—
583	C₂×C₄	—	4	2	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	1	no	—
586	C₂×C₄	—	4	1	no	—
587	C₂×C₄	—	4	2	no	—
588	C₂×C₄	—	4	2	no	—
589	C₂×C₄	—	4	2	no	—
590	C₂×C₄	—	4	2	no	—
591	C₂×C₄	—	4	2	no	—
592	C₂×C₄	—	4	2	no	—
593	C₂×C₄	—	4	1	no	—
594	C₂×C₄	—	4	1	no	—
595	C₂×C₄	—	4	1	no	—
596	C₂×C₄	—	4	1	no	—
597	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
598	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
599	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
600	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
601	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
602	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
603	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
604	C₂×C₄	C₂₃×#131	1	2	yes	—
605	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
606	C₈	—	4	2	no	—
607	C₈	—	4	2	no	—
608	C₈	—	4	2	no	—
609	C₈	—	4	2	no	—
610	C₂²	—	2	1	no	—
611	C₂²	—	2	1	no	—
612	C₂²	—	2	1	no	—
613	C₂²	—	2	1	no	—
614	C₂²	—	2	1	no	—
615	C₂²	—	2	1	no	—
616	C₂²	—	2	1	no	—
617	C₂²	—	4	2	no	—
618	C₂²	—	4	2	no	—
619	C₂²	—	4	1	no	—
620	C₂²	—	4	1	no	—
621	C₂²	—	4	1	no	—
622	C₂²	—	8	1	no	—
623	C₂²	—	8	1	no	—
624	C₂²	C₂₃×#410	1	2	yes	—
625	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
626	C₂²	C₂×C₂₃×#131	1	1	yes	characteristic
627	C₂²	C₂₃×#568	1	1	yes	characteristic
628	C₄	—	2	2	no	—
629	C₄	—	2	1	no	—
630	C₄	—	2	1	no	—
631	C₄	—	2	2	no	—
632	C₄	—	2	2	no	—
633	C₄	—	2	2	no	—
634	C₄	—	2	1	no	—
635	C₄	—	2	1	no	—
636	C₄	—	2	2	no	—
637	C₄	—	4	1	no	—
638	C₄	—	4	1	no	—
639	C₄	—	4	1	no	—
640	C₄	—	4	1	no	—
641	C₄	—	4	1	no	—
642	C₄	—	4	1	no	—
643	C₂	—	2	1	no	—
644	C₂	—	2	1	no	—
645	C₂	—	2	2	no	—
646	C₂	—	4	1	no	—
647	C₂	—	8	1	no	—
648	C₂	C₂₃×#2880	1	1	yes	characteristic
649	C₂	C₂×C₂₃×#410	1	1	yes	characteristic
650	C₂	C₂₃×#2889	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂²×D₄×C₂₃	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2924	C₂	semidirect
3. C₂₃×#1473	C₂	semidirect
4. C₂₃×#1470	C₂	semidirect
6. C₂₃×#1467	C₂	semidirect
7. C₂₃×#1464	C₂	semidirect
8. C₂₃×#1463	C₂	semidirect
9. C₂₃×#1461	C₂	semidirect
10. C₂₃×#1458	C₂	semidirect
11. C₂×C₂₃×#324	C₂	semidirect
224. #17417	C₂₃	direct (N × H)
286. #2924	C₂×C₂₃	semidirect
287. #1473	C₂×C₂₃	semidirect
288. #1470	C₂×C₂₃	semidirect
290. #1467	C₂×C₂₃	semidirect
291. #1464	C₂×C₂₃	semidirect
292. #1463	C₂×C₂₃	semidirect
293. #1461	C₂×C₂₃	semidirect
294. #1458	C₂×C₂₃	semidirect
295. C₂×#324	C₂×C₂₃	semidirect
463. C₂₃	#17417	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#552	2
C₂₃×#2924	2
C₂₃×#1473	2
C₂₃×#1470	2
C₂₃×#1468	2
C₂₃×#1467	2
C₂₃×#1464	2
C₂₃×#1463	2
C₂₃×#1461	2
C₂₃×#1458	2
C₂×C₂₃×#324	2
#17417	23

Subgroups — C23×#17417 #19165043

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17417 #19165043