Subgroups — C₂₃×#19637 #19167263

Order 5888.

750 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3229	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3208	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3139	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#3138	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₃×#486	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1720	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1719	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1711	C₂	1	2	yes	maximal, complemented
10	C₂₃×#1708	C₂	1	2	yes	maximal, complemented
11	C₈×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#533	C₂²	1	1	yes	characteristic
13	C₂₃×#533	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
16	C₂₃×#500	C₂²	1	1	yes	characteristic, complemented
17	C₂₃×#498	C₂²	1	2	yes	complemented
18	C₂₃×#494	C₂²	1	1	yes	characteristic, complemented
19	C₂₃×#492	C₂²	1	2	yes	—
20	C₂₃×#491	—	2	2	no	—
21	C₂₃×#490	—	2	2	no	—
22	C₂₃×#487	C₂²	1	1	yes	characteristic, complemented
23	C₂₃×#487	C₂²	1	1	yes	characteristic, complemented
24	C₂₃×#487	C₂²	1	2	yes	complemented
25	C₂₃×#487	C₂²	1	2	yes	complemented
26	C₂₃×#487	C₂²	1	2	yes	complemented
27	C₂₃×#486	—	2	2	no	—
28	C₂₃×#486	C₂²	1	2	yes	—
29	C₂₃×#486	C₂²	1	2	yes	—
30	C₂₃×#486	C₂²	1	2	yes	—
31	C₂₃×#486	C₂²	1	2	yes	—
32	C₂₃×#486	C₂²	1	2	yes	—
33	C₂₃×#483	—	2	2	no	—
34	C₂₃×#475	—	2	2	no	—
35	C₂₃×#467	—	2	2	no	—
36	C₂₃×#460	—	2	2	no	—
37	C₂₃×#450	—	2	2	no	—
38	C₄×Q₁₆×C₂₃	—	2	2	no	—
39	C₂₃×#431	C₂²	1	1	yes	characteristic
40	C₂₃×#419	—	2	2	no	—
41	C₂₃×#418	—	2	2	no	—
42	C₂×C₂₃×#102	—	2	2	no	—
43	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
45	C₂₃×#322	C₂²	1	2	yes	complemented
46	C₈²×C₂₃	C₂²	1	2	yes	complemented
47	C₂²×Q₈×C₂₃	—	2	1	no	—
48	C₂²×D₄×C₂₃	—	2	1	no	—
49	C₂²×D₄×C₂₃	—	2	1	no	—
50	C₂²×D₄×C₂₃	—	2	1	no	—
51	C₂×Q₁₆×C₂₃	—	2	2	no	—
52	C₂×SD₁₆×C₂₃	—	2	4	no	—
53	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
54	C₂²×C₈×C₂₃	D₄	1	2	yes	—
55	C₂₃×C₄⋊Q₈	—	2	1	no	—
56	C₂₃×C₄⋊Q₈	—	2	1	no	—
57	C₂₃×C₄⋊Q₈	—	2	1	no	—
58	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
59	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
60	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
61	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
62	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
63	C₂₃×#127	—	2	1	no	—
64	C₂₃×#127	—	2	1	no	—
65	C₂₃×#127	—	2	1	no	—
66	C₂₃×#127	C₂³	1	2	yes	—
67	C₂₃×#125	—	2	1	no	—
68	C₂₃×#125	—	2	1	no	—
69	C₂₃×C₂²⋊Q₈	—	2	1	no	—
70	C₂₃×C₂²⋊Q₈	—	2	1	no	—
71	C₂₃×C₂²⋊Q₈	—	2	1	no	—
72	C₂₃×C₂²⋊Q₈	—	2	1	no	—
73	C₄×Q₈×C₂₃	—	2	1	no	—
74	C₄×Q₈×C₂₃	—	2	1	no	—
75	C₄×Q₈×C₂₃	—	2	1	no	—
76	C₄×Q₈×C₂₃	—	2	1	no	—
77	C₂₃×#117	—	2	1	no	—
78	C₂₃×#117	—	2	1	no	—
79	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
80	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
81	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
82	C₂₃×#107	—	2	2	no	—
83	C₂₃×#106	—	2	4	no	—
84	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
85	C₂₃×#103	—	2	4	no	—
86	C₂₃×#103	—	2	4	no	—
87	C₂₃×#103	—	2	2	no	—
88	C₂₃×#103	—	2	2	no	—
89	C₂₃×#103	—	2	4	no	—
90	C₂₃×#102	—	2	4	no	—
91	C₂₃×#102	—	2	4	no	—
92	C₂₃×#102	—	2	4	no	—
93	C₂₃×#102	—	2	4	no	—
94	C₂₃×#102	—	2	4	no	—
95	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
96	C₂₃×#97	D₄	1	2	yes	—
97	C₄×C₈×C₂₃	D₄	1	2	yes	—
98	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
99	C₄×C₈×C₂₃	C₂³	1	2	yes	—
100	C₄×C₈×C₂₃	C₂³	1	2	yes	—
101	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
102	C₂⁴×C₂₃	—	4	1	no	—
103	C₂×Q₈×C₂₃	—	2	1	no	—
104	C₂×Q₈×C₂₃	—	2	2	no	—
105	C₂×Q₈×C₂₃	—	2	1	no	—
106	C₂×Q₈×C₂₃	—	2	1	no	—
107	C₂×Q₈×C₂₃	—	2	1	no	—
108	C₂×Q₈×C₂₃	—	4	1	no	—
109	C₂×Q₈×C₂₃	—	4	4	no	—
110	C₂×D₄×C₂₃	—	2	2	no	—
111	C₂×D₄×C₂₃	—	2	2	no	—
112	C₂×D₄×C₂₃	—	2	2	no	—
113	C₂×D₄×C₂₃	—	4	4	no	—
114	C₂×D₄×C₂₃	—	4	1	no	—
115	C₂×D₄×C₂₃	—	4	1	no	—
116	C₂×D₄×C₂₃	—	4	1	no	—
117	C₂×D₄×C₂₃	—	4	4	no	—
118	C₂×D₄×C₂₃	—	8	2	no	—
119	C₂²×C₄×C₂₃	—	4	1	no	—
120	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
123	Q₁₆×C₂₃	—	4	4	no	—
124	SD₁₆×C₂₃	—	8	4	no	—
125	M₁₆×C₂₃	—	2	4	no	—
126	C₂×C₈×C₂₃	—	4	2	no	—
127	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
128	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
129	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
130	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
131	C₄⋊C₄×C₂₃	—	2	2	no	—
132	C₄⋊C₄×C₂₃	—	2	1	no	—
133	C₄⋊C₄×C₂₃	—	2	1	no	—
134	C₄⋊C₄×C₂₃	—	2	1	no	—
135	C₄⋊C₄×C₂₃	—	2	1	no	—
136	C₄⋊C₄×C₂₃	—	2	1	no	—
137	C₄⋊C₄×C₂₃	—	2	2	no	—
138	C₄⋊C₄×C₂₃	—	2	1	no	—
139	C₄⋊C₄×C₂₃	—	2	1	no	—
140	C₄⋊C₄×C₂₃	—	2	1	no	—
141	C₄⋊C₄×C₂₃	—	4	1	no	—
142	C₄⋊C₄×C₂₃	—	4	1	no	—
143	C₄⋊C₄×C₂₃	—	4	1	no	—
144	C₄⋊C₄×C₂₃	—	4	1	no	—
145	C₄⋊C₄×C₂₃	—	4	1	no	—
146	C₄⋊C₄×C₂₃	—	4	1	no	—
147	C₂²⋊C₄×C₂₃	—	4	1	no	—
148	C₂²⋊C₄×C₂₃	—	4	1	no	—
149	C₄²×C₂₃	—	4	1	no	—
150	C₄²×C₂₃	—	4	1	no	—
151	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
152	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
153	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
154	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
155	#19637	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂³×C₂₃	—	4	2	no	—
157	C₂³×C₂₃	—	8	2	no	—
158	C₂³×C₂₃	—	8	2	no	—
159	C₂³×C₂₃	—	8	2	no	—
160	C₂³×C₂₃	#127	1	1	yes	characteristic
161	Q₈×C₂₃	—	4	4	no	—
162	Q₈×C₂₃	—	4	2	no	—
163	Q₈×C₂₃	—	4	2	no	—
164	Q₈×C₂₃	—	4	2	no	—
165	Q₈×C₂₃	—	4	2	no	—
166	Q₈×C₂₃	—	8	2	no	—
167	D₄×C₂₃	—	4	4	no	—
168	D₄×C₂₃	—	4	4	no	—
169	D₄×C₂₃	—	4	4	no	—
170	D₄×C₂₃	—	8	2	no	—
171	D₄×C₂₃	—	8	2	no	—
172	D₄×C₂₃	—	8	2	no	—
173	C₂×C₄×C₂₃	—	2	2	no	—
174	C₂×C₄×C₂₃	—	2	2	no	—
175	C₂×C₄×C₂₃	—	4	1	no	—
176	C₂×C₄×C₂₃	—	4	1	no	—
177	C₂×C₄×C₂₃	—	4	1	no	—
178	C₂×C₄×C₂₃	—	4	1	no	—
179	C₂×C₄×C₂₃	—	4	2	no	—
180	C₂×C₄×C₂₃	—	8	2	no	—
181	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
182	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
183	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
184	C₂×C₄×C₂₃	#135	1	4	yes	—
185	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
188	C₈×C₂₃	—	2	4	no	—
189	C₈×C₂₃	—	2	2	no	—
190	C₈×C₂₃	—	2	2	no	—
191	C₈×C₂₃	—	2	4	no	—
192	#3229	C₂×C₂₃	1	1	yes	characteristic, complemented
193	#3208	C₂×C₂₃	1	1	yes	characteristic
194	#3170	C₂×C₂₃	1	1	yes	characteristic, complemented
195	#3139	C₂×C₂₃	1	1	yes	characteristic, complemented
196	#3138	C₂×C₂₃	1	1	yes	characteristic
197	C₂×#486	C₂×C₂₃	1	1	yes	characteristic
198	#1720	C₂×C₂₃	1	2	yes	complemented
199	#1719	C₂×C₂₃	1	2	yes	complemented
200	#1711	C₂×C₂₃	1	2	yes	complemented
201	#1708	C₂×C₂₃	1	2	yes	complemented
202	C₈×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
203	C₂²×C₂₃	—	2	1	no	—
204	C₂²×C₂₃	—	2	1	no	—
205	C₂²×C₂₃	—	8	2	no	—
206	C₂²×C₂₃	—	8	2	no	—
207	C₂²×C₂₃	—	8	2	no	—
208	C₂²×C₂₃	—	16	4	no	—
209	C₂²×C₂₃	#495	1	2	yes	—
210	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
211	C₄×C₂₃	—	2	1	no	—
212	C₄×C₂₃	—	2	1	no	—
213	C₄×C₂₃	—	8	2	no	—
214	C₄×C₂₃	—	8	1	no	—
215	C₄×C₂₃	—	8	1	no	—
216	C₄×C₂₃	—	8	1	no	—
217	C₄×C₂₃	—	8	1	no	—
218	C₄×C₂₃	#495	1	2	yes	—
219	C₄×C₂₃	C₂×#135	1	2	yes	—
220	C₄×C₂₃	#575	1	2	yes	—
221	C₄×C₂₃	#577	1	2	yes	—
222	#533	C₂²×C₂₃	1	1	yes	characteristic
223	#533	C₂²×C₂₃	1	1	yes	characteristic
224	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
225	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
226	#500	C₂²×C₂₃	1	1	yes	characteristic, complemented
227	#498	C₂²×C₂₃	1	2	yes	complemented
228	#494	C₂²×C₂₃	1	1	yes	characteristic, complemented
229	#492	C₂²×C₂₃	1	2	yes	—
230	#491	—	2	2	no	—
231	#490	—	2	2	no	—
232	#487	C₂²×C₂₃	1	2	yes	complemented
233	#487	C₂²×C₂₃	1	2	yes	complemented
234	#487	C₂²×C₂₃	1	1	yes	characteristic, complemented
235	#487	C₂²×C₂₃	1	1	yes	characteristic, complemented
236	#487	C₂²×C₂₃	1	2	yes	complemented
237	#486	—	2	2	no	—
238	#486	C₂²×C₂₃	1	2	yes	—
239	#486	C₂²×C₂₃	1	2	yes	—
240	#486	C₂²×C₂₃	1	2	yes	—
241	#486	C₂²×C₂₃	1	2	yes	—
242	#486	C₂²×C₂₃	1	2	yes	—
243	#483	—	2	2	no	—
244	#475	—	2	2	no	—
245	#467	—	2	2	no	—
246	#460	—	2	2	no	—
247	#450	—	2	2	no	—
248	C₄×Q₁₆	—	2	2	no	—
249	#431	C₂²×C₂₃	1	1	yes	characteristic
250	#419	—	2	2	no	—
251	#418	—	2	2	no	—
252	C₂×#102	—	2	2	no	—
253	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
254	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
255	#322	C₂²×C₂₃	1	2	yes	complemented
256	C₈²	C₂²×C₂₃	1	2	yes	complemented
257	C₂×C₂₃	—	2	2	no	—
258	C₂×C₂₃	—	16	2	no	—
259	C₂×C₂₃	C₂×#495	1	1	yes	characteristic
260	C₂×C₂₃	#3153	1	1	yes	characteristic
261	C₂×C₂₃	#3154	1	1	yes	characteristic
262	C₂²×Q₈	—	2	1	no	—
263	C₂²×D₄	—	2	1	no	—
264	C₂²×D₄	—	2	1	no	—
265	C₂²×D₄	—	2	1	no	—
266	C₂×Q₁₆	—	2	2	no	—
267	C₂×SD₁₆	—	2	4	no	—
268	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
269	C₂²×C₈	D₄×C₂₃	1	2	yes	—
270	C₄⋊Q₈	—	2	1	no	—
271	C₄⋊Q₈	—	2	1	no	—
272	C₄⋊Q₈	—	2	1	no	—
273	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
274	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
275	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
276	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
277	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
278	#127	—	2	1	no	—
279	#127	—	2	1	no	—
280	#127	—	2	1	no	—
281	#127	C₂³×C₂₃	1	2	yes	—
282	#125	—	2	1	no	—
283	#125	—	2	1	no	—
284	C₂²⋊Q₈	—	2	1	no	—
285	C₂²⋊Q₈	—	2	1	no	—
286	C₂²⋊Q₈	—	2	1	no	—
287	C₂²⋊Q₈	—	2	1	no	—
288	C₄×Q₈	—	2	1	no	—
289	C₄×Q₈	—	2	1	no	—
290	C₄×Q₈	—	2	1	no	—
291	C₄×Q₈	—	2	1	no	—
292	#117	—	2	1	no	—
293	#117	—	2	1	no	—
294	C₂×C₄⋊C₄	—	2	1	no	—
295	C₂×C₄⋊C₄	—	2	1	no	—
296	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
297	#107	—	2	2	no	—
298	#106	—	2	4	no	—
299	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
300	#103	—	2	4	no	—
301	#103	—	2	4	no	—
302	#103	—	2	2	no	—
303	#103	—	2	2	no	—
304	#103	—	2	4	no	—
305	#102	—	2	4	no	—
306	#102	—	2	4	no	—
307	#102	—	2	4	no	—
308	#102	—	2	4	no	—
309	#102	—	2	4	no	—
310	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
311	#97	D₄×C₂₃	1	2	yes	—
312	C₄×C₈	D₄×C₂₃	1	2	yes	—
313	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
314	C₄×C₈	C₂³×C₂₃	1	2	yes	—
315	C₄×C₈	C₂³×C₂₃	1	2	yes	—
316	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
317	C₂₃	#19637	1	1	yes	characteristic, direct product factor, Sylow
318	C₂⁴	—	4	1	no	—
319	C₂×Q₈	—	2	1	no	—
320	C₂×Q₈	—	2	2	no	—
321	C₂×Q₈	—	2	1	no	—
322	C₂×Q₈	—	2	1	no	—
323	C₂×Q₈	—	2	1	no	—
324	C₂×Q₈	—	4	1	no	—
325	C₂×Q₈	—	4	4	no	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	—	2	2	no	—
329	C₂×D₄	—	4	4	no	—
330	C₂×D₄	—	4	1	no	—
331	C₂×D₄	—	4	1	no	—
332	C₂×D₄	—	4	1	no	—
333	C₂×D₄	—	4	4	no	—
334	C₂×D₄	—	8	2	no	—
335	C₂²×C₄	—	4	1	no	—
336	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
337	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
338	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
339	Q₁₆	—	4	4	no	—
340	SD₁₆	—	8	4	no	—
341	M₁₆	—	2	4	no	—
342	C₂×C₈	—	4	2	no	—
343	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
344	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
345	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
346	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	1	no	—
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	2	1	no	—
352	C₄⋊C₄	—	2	1	no	—
353	C₄⋊C₄	—	2	2	no	—
354	C₄⋊C₄	—	2	1	no	—
355	C₄⋊C₄	—	2	1	no	—
356	C₄⋊C₄	—	2	1	no	—
357	C₄⋊C₄	—	4	1	no	—
358	C₄⋊C₄	—	4	1	no	—
359	C₄⋊C₄	—	4	1	no	—
360	C₄⋊C₄	—	4	1	no	—
361	C₄⋊C₄	—	4	1	no	—
362	C₄⋊C₄	—	4	1	no	—
363	C₂²⋊C₄	—	4	1	no	—
364	C₂²⋊C₄	—	4	1	no	—
365	C₄²	—	4	1	no	—
366	C₄²	—	4	1	no	—
367	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
368	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
369	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
370	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
371	C₂³	—	4	2	no	—
372	C₂³	—	8	2	no	—
373	C₂³	—	8	2	no	—
374	C₂³	—	8	2	no	—
375	C₂³	C₂₃×#127	1	1	yes	characteristic
376	Q₈	—	4	4	no	—
377	Q₈	—	4	2	no	—
378	Q₈	—	4	2	no	—
379	Q₈	—	4	2	no	—
380	Q₈	—	4	2	no	—
381	Q₈	—	8	2	no	—
382	D₄	—	4	4	no	—
383	D₄	—	4	4	no	—
384	D₄	—	4	4	no	—
385	D₄	—	8	2	no	—
386	D₄	—	8	2	no	—
387	D₄	—	8	2	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	4	1	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	4	1	no	—
393	C₂×C₄	—	4	1	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	8	2	no	—
396	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
397	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
398	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
399	C₂×C₄	C₂₃×#135	1	4	yes	—
400	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
401	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
402	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
403	C₈	—	2	4	no	—
404	C₈	—	2	2	no	—
405	C₈	—	2	2	no	—
406	C₈	—	2	4	no	—
407	C₂²	—	2	1	no	—
408	C₂²	—	2	1	no	—
409	C₂²	—	8	2	no	—
410	C₂²	—	8	2	no	—
411	C₂²	—	8	2	no	—
412	C₂²	—	16	4	no	—
413	C₂²	C₂₃×#495	1	2	yes	—
414	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
415	C₄	—	2	1	no	—
416	C₄	—	2	1	no	—
417	C₄	—	8	2	no	—
418	C₄	—	8	1	no	—
419	C₄	—	8	1	no	—
420	C₄	—	8	1	no	—
421	C₄	—	8	1	no	—
422	C₄	C₂₃×#495	1	2	yes	—
423	C₄	C₂×C₂₃×#135	1	2	yes	—
424	C₄	C₂₃×#575	1	2	yes	—
425	C₄	C₂₃×#577	1	2	yes	—
426	C₂	—	2	2	no	—
427	C₂	—	16	2	no	—
428	C₂	C₂×C₂₃×#495	1	1	yes	characteristic
429	C₂	C₂₃×#3153	1	1	yes	characteristic
430	C₂	C₂₃×#3154	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3229	C₂	semidirect
3. C₂₃×#3170	C₂	semidirect
4. C₂₃×#3139	C₂	semidirect
7. C₂₃×#1720	C₂	semidirect
8. C₂₃×#1719	C₂	semidirect
9. C₂₃×#1711	C₂	semidirect
10. C₂₃×#1708	C₂	semidirect
11. C₈×M₁₆×C₂₃	C₂	semidirect
16. C₂₃×#500	C₂²	semidirect
17. C₂₃×#498	C₂²	semidirect
18. C₂₃×#494	C₂²	semidirect
22. C₂₃×#487	C₂²	semidirect
23. C₂₃×#487	C₂²	semidirect
24. C₂₃×#487	C₂²	semidirect
25. C₂₃×#487	C₂²	semidirect
26. C₂₃×#487	C₂²	semidirect
45. C₂₃×#322	C₂²	semidirect
46. C₈²×C₂₃	C₂²	semidirect
155. #19637	C₂₃	direct (N × H)
192. #3229	C₂×C₂₃	semidirect
194. #3170	C₂×C₂₃	semidirect
195. #3139	C₂×C₂₃	semidirect
198. #1720	C₂×C₂₃	semidirect
199. #1719	C₂×C₂₃	semidirect
200. #1711	C₂×C₂₃	semidirect
201. #1708	C₂×C₂₃	semidirect
202. C₈×M₁₆	C₂×C₂₃	semidirect
226. #500	C₂²×C₂₃	semidirect
227. #498	C₂²×C₂₃	semidirect
228. #494	C₂²×C₂₃	semidirect
232. #487	C₂²×C₂₃	semidirect
233. #487	C₂²×C₂₃	semidirect
234. #487	C₂²×C₂₃	semidirect
235. #487	C₂²×C₂₃	semidirect
236. #487	C₂²×C₂₃	semidirect
255. #322	C₂²×C₂₃	semidirect
256. C₈²	C₂²×C₂₃	semidirect
317. C₂₃	#19637	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×#3229	2
C₂₃×#3208	2
C₂₃×#3170	2
C₂₃×#3139	2
C₂₃×#3138	2
C₂×C₂₃×#486	2
C₂₃×#1720	2
C₂₃×#1719	2
C₂₃×#1711	2
C₂₃×#1708	2
C₈×M₁₆×C₂₃	2
#19637	23

Subgroups — C23×#19637 #19167263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19637 #19167263