Subgroups — C₂₃×#19657 #19167283

Order 5888.

700 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3230	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3229	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3226	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#3206	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#3131	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1719	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1718	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1710	C₂	1	2	yes	maximal, complemented
10	C₂₃×#1708	C₂	1	2	yes	maximal, complemented
11	C₂₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#533	C₂²	1	1	yes	characteristic
13	C₂₃×#532	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂₃×#500	C₂²	1	2	yes	—
17	C₂₃×#500	C₂²	1	2	yes	—
18	C₂₃×#498	C₂²	1	2	yes	—
19	C₂₃×#492	C₂²	1	2	yes	—
20	C₂₃×#488	—	2	2	no	—
21	C₂₃×#487	—	2	2	no	—
22	C₂₃×#487	C₂²	1	2	yes	—
23	C₂₃×#487	C₂²	1	2	yes	—
24	C₂₃×#487	C₂²	1	2	yes	—
25	C₂₃×#487	C₂²	1	2	yes	—
26	C₂₃×#487	C₂²	1	1	yes	characteristic
27	C₂₃×#487	C₂²	1	1	yes	characteristic
28	C₂₃×#487	C₂²	1	2	yes	—
29	C₂₃×#486	C₂²	1	2	yes	—
30	C₂₃×#486	C₂²	1	1	yes	characteristic
31	C₂₃×#486	C₂²	1	1	yes	characteristic
32	C₂₃×#484	—	2	2	no	—
33	C₂₃×#483	—	2	2	no	—
34	C₂₃×#480	—	2	2	no	—
35	C₂₃×#477	—	2	2	no	—
36	C₂₃×#475	—	2	2	no	—
37	C₂₃×#474	—	2	2	no	—
38	C₂₃×#461	—	2	2	no	—
39	C₂₃×#448	—	2	2	no	—
40	C₂₃×#431	C₂²	1	1	yes	characteristic
41	C₂₃×#431	C₂²	1	1	yes	characteristic
42	C₂₃×#416	—	2	2	no	—
43	C₂×C₂₃×#103	—	2	2	no	—
44	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
45	C₂₃×#322	C₂²	1	2	yes	—
46	C₈²×C₂₃	C₂²	1	2	yes	—
47	C₂×#41×C₂₃	—	2	1	no	—
48	C₂²×Q₈×C₂₃	—	2	1	no	—
49	C₂²×Q₈×C₂₃	—	2	1	no	—
50	C₂×SD₁₆×C₂₃	—	2	4	no	—
51	C₂²×C₈×C₂₃	D₄	1	2	yes	—
52	C₂₃×C₄⋊Q₈	—	2	1	no	—
53	C₂₃×C₄⋊Q₈	—	2	1	no	—
54	C₂₃×C₄⋊Q₈	—	2	1	no	—
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	2	yes	—
62	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
63	C₂₃×#127	C₂³	1	1	yes	characteristic
64	C₂₃×#125	—	2	1	no	—
65	C₂₃×#124	—	2	1	no	—
66	C₂₃×C₂²⋊Q₈	—	2	1	no	—
67	C₂₃×C₂²⋊Q₈	—	2	1	no	—
68	C₂₃×#121	—	2	1	no	—
69	C₂₃×#121	—	2	1	no	—
70	C₄×Q₈×C₂₃	—	2	1	no	—
71	C₄×Q₈×C₂₃	—	2	1	no	—
72	C₄×D₄×C₂₃	—	2	1	no	—
73	C₄×D₄×C₂₃	—	2	1	no	—
74	C₂₃×#117	—	2	1	no	—
75	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
76	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
77	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
78	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
79	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
80	C₂₃×#107	—	2	4	no	—
81	C₂₃×#107	—	2	4	no	—
82	C₂₃×#106	—	2	4	no	—
83	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
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×#103	—	2	4	no	—
91	C₂₃×#103	—	2	4	no	—
92	C₂₃×#102	—	2	2	no	—
93	C₂₃×#102	—	2	2	no	—
94	C₂₃×#102	—	2	4	no	—
95	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
96	C₂₃×C₂²⋊C₈	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	#41×C₂₃	—	4	4	no	—
103	C₂×Q₈×C₂₃	—	2	1	no	—
104	C₂×Q₈×C₂₃	—	2	1	no	—
105	C₂×Q₈×C₂₃	—	2	1	no	—
106	C₂×Q₈×C₂₃	—	2	2	no	—
107	C₂×Q₈×C₂₃	—	2	2	no	—
108	C₂×Q₈×C₂₃	—	4	1	no	—
109	C₂×Q₈×C₂₃	—	4	1	no	—
110	C₂×Q₈×C₂₃	—	4	4	no	—
111	C₂×Q₈×C₂₃	—	8	2	no	—
112	C₂×D₄×C₂₃	—	2	1	no	—
113	C₂×D₄×C₂₃	—	2	1	no	—
114	C₂×D₄×C₂₃	—	2	1	no	—
115	C₂×D₄×C₂₃	—	4	1	no	—
116	C₂²×C₄×C₂₃	—	4	1	no	—
117	C₂²×C₄×C₂₃	—	4	1	no	—
118	C₂²×C₄×C₂₃	—	4	1	no	—
119	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
122	SD₁₆×C₂₃	—	8	4	no	—
123	C₂×C₈×C₂₃	—	2	4	no	—
124	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
125	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
126	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
127	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
128	C₄⋊C₄×C₂₃	—	2	1	no	—
129	C₄⋊C₄×C₂₃	—	2	1	no	—
130	C₄⋊C₄×C₂₃	—	2	2	no	—
131	C₄⋊C₄×C₂₃	—	2	2	no	—
132	C₄⋊C₄×C₂₃	—	2	1	no	—
133	C₄⋊C₄×C₂₃	—	2	2	no	—
134	C₄⋊C₄×C₂₃	—	2	2	no	—
135	C₄⋊C₄×C₂₃	—	2	1	no	—
136	C₄⋊C₄×C₂₃	—	2	1	no	—
137	C₄⋊C₄×C₂₃	—	2	1	no	—
138	C₄⋊C₄×C₂₃	—	4	1	no	—
139	C₄⋊C₄×C₂₃	—	4	1	no	—
140	C₄⋊C₄×C₂₃	—	4	1	no	—
141	C₄⋊C₄×C₂₃	—	4	1	no	—
142	C₄⋊C₄×C₂₃	—	4	1	no	—
143	C₄⋊C₄×C₂₃	—	4	1	no	—
144	C₂²⋊C₄×C₂₃	—	4	1	no	—
145	C₂²⋊C₄×C₂₃	—	4	1	no	—
146	C₂²⋊C₄×C₂₃	—	4	1	no	—
147	C₄²×C₂₃	—	4	1	no	—
148	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
149	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
150	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
151	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
152	#19657	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂³×C₂₃	—	4	1	no	—
154	C₂³×C₂₃	#127	1	1	yes	characteristic
155	Q₈×C₂₃	—	4	4	no	—
156	Q₈×C₂₃	—	4	2	no	—
157	Q₈×C₂₃	—	4	4	no	—
158	Q₈×C₂₃	—	4	2	no	—
159	Q₈×C₂₃	—	4	2	no	—
160	Q₈×C₂₃	—	8	2	no	—
161	Q₈×C₂₃	—	8	2	no	—
162	D₄×C₂₃	—	4	2	no	—
163	D₄×C₂₃	—	4	2	no	—
164	D₄×C₂₃	—	4	2	no	—
165	D₄×C₂₃	—	8	2	no	—
166	C₂×C₄×C₂₃	—	2	2	no	—
167	C₂×C₄×C₂₃	—	2	2	no	—
168	C₂×C₄×C₂₃	—	4	1	no	—
169	C₂×C₄×C₂₃	—	4	1	no	—
170	C₂×C₄×C₂₃	—	4	1	no	—
171	C₂×C₄×C₂₃	—	4	2	no	—
172	C₂×C₄×C₂₃	—	4	2	no	—
173	C₂×C₄×C₂₃	—	8	2	no	—
174	C₂×C₄×C₂₃	—	8	2	no	—
175	C₂×C₄×C₂₃	—	8	2	no	—
176	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
177	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
178	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
179	C₂×C₄×C₂₃	#135	1	4	yes	—
180	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
182	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
183	C₈×C₂₃	—	2	2	no	—
184	C₈×C₂₃	—	2	2	no	—
185	C₈×C₂₃	—	2	4	no	—
186	C₈×C₂₃	—	2	4	no	—
187	#3230	C₂×C₂₃	1	1	yes	characteristic, complemented
188	#3229	C₂×C₂₃	1	1	yes	characteristic, complemented
189	#3226	C₂×C₂₃	1	1	yes	characteristic
190	#3206	C₂×C₂₃	1	1	yes	characteristic
191	#3131	C₂×C₂₃	1	1	yes	characteristic
192	C₂×#487	C₂×C₂₃	1	1	yes	characteristic, complemented
193	#1719	C₂×C₂₃	1	2	yes	complemented
194	#1718	C₂×C₂₃	1	2	yes	complemented
195	#1710	C₂×C₂₃	1	2	yes	complemented
196	#1708	C₂×C₂₃	1	2	yes	complemented
197	#1450	C₂×C₂₃	1	1	yes	characteristic, complemented
198	C₂²×C₂₃	—	2	1	no	—
199	C₂²×C₂₃	—	2	1	no	—
200	C₂²×C₂₃	—	8	1	no	—
201	C₂²×C₂₃	—	8	1	no	—
202	C₂²×C₂₃	—	8	1	no	—
203	C₂²×C₂₃	#495	1	2	yes	—
204	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
205	C₄×C₂₃	—	2	1	no	—
206	C₄×C₂₃	—	2	1	no	—
207	C₄×C₂₃	—	8	1	no	—
208	C₄×C₂₃	—	8	2	no	—
209	C₄×C₂₃	—	8	2	no	—
210	C₄×C₂₃	—	8	1	no	—
211	C₄×C₂₃	—	8	1	no	—
212	C₄×C₂₃	#495	1	2	yes	—
213	C₄×C₂₃	C₂×#135	1	2	yes	—
214	C₄×C₂₃	#577	1	2	yes	—
215	C₄×C₂₃	#578	1	2	yes	—
216	#533	C₂²×C₂₃	1	1	yes	characteristic
217	#532	C₂²×C₂₃	1	1	yes	characteristic
218	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
219	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
220	#500	C₂²×C₂₃	1	2	yes	—
221	#500	C₂²×C₂₃	1	2	yes	—
222	#498	C₂²×C₂₃	1	2	yes	—
223	#492	C₂²×C₂₃	1	2	yes	—
224	#488	—	2	2	no	—
225	#487	—	2	2	no	—
226	#487	C₂²×C₂₃	1	2	yes	—
227	#487	C₂²×C₂₃	1	2	yes	—
228	#487	C₂²×C₂₃	1	1	yes	characteristic
229	#487	C₂²×C₂₃	1	1	yes	characteristic
230	#487	C₂²×C₂₃	1	2	yes	—
231	#487	C₂²×C₂₃	1	2	yes	—
232	#487	C₂²×C₂₃	1	2	yes	—
233	#486	C₂²×C₂₃	1	1	yes	characteristic
234	#486	C₂²×C₂₃	1	1	yes	characteristic
235	#486	C₂²×C₂₃	1	2	yes	—
236	#484	—	2	2	no	—
237	#483	—	2	2	no	—
238	#480	—	2	2	no	—
239	#477	—	2	2	no	—
240	#475	—	2	2	no	—
241	#474	—	2	2	no	—
242	#461	—	2	2	no	—
243	#448	—	2	2	no	—
244	#431	C₂²×C₂₃	1	1	yes	characteristic
245	#431	C₂²×C₂₃	1	1	yes	characteristic
246	#416	—	2	2	no	—
247	C₂×#103	—	2	2	no	—
248	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
249	#322	C₂²×C₂₃	1	2	yes	—
250	C₈²	C₂²×C₂₃	1	2	yes	—
251	C₂×C₂₃	—	2	2	no	—
252	C₂×C₂₃	—	16	1	no	—
253	C₂×C₂₃	C₂×#495	1	1	yes	characteristic
254	C₂×C₂₃	#3154	1	1	yes	characteristic
255	C₂×C₂₃	#3156	1	1	yes	characteristic
256	C₂×#41	—	2	1	no	—
257	C₂²×Q₈	—	2	1	no	—
258	C₂²×Q₈	—	2	1	no	—
259	C₂×SD₁₆	—	2	4	no	—
260	C₂²×C₈	D₄×C₂₃	1	2	yes	—
261	C₄⋊Q₈	—	2	1	no	—
262	C₄⋊Q₈	—	2	1	no	—
263	C₄⋊Q₈	—	2	1	no	—
264	C₄⋊Q₈	—	2	1	no	—
265	C₄⋊Q₈	—	2	1	no	—
266	C₄⋊Q₈	—	2	1	no	—
267	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
268	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
269	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
270	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
271	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
272	#127	C₂³×C₂₃	1	1	yes	characteristic
273	#125	—	2	1	no	—
274	#124	—	2	1	no	—
275	C₂²⋊Q₈	—	2	1	no	—
276	C₂²⋊Q₈	—	2	1	no	—
277	#121	—	2	1	no	—
278	#121	—	2	1	no	—
279	C₄×Q₈	—	2	1	no	—
280	C₄×Q₈	—	2	1	no	—
281	C₄×D₄	—	2	1	no	—
282	C₄×D₄	—	2	1	no	—
283	#117	—	2	1	no	—
284	C₂×C₄⋊C₄	—	2	1	no	—
285	C₂×C₄⋊C₄	—	2	1	no	—
286	C₂×C₄⋊C₄	—	2	1	no	—
287	C₂×C₄⋊C₄	—	2	1	no	—
288	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
289	#107	—	2	4	no	—
290	#107	—	2	4	no	—
291	#106	—	2	4	no	—
292	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
293	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
294	#103	—	2	4	no	—
295	#103	—	2	4	no	—
296	#103	—	2	2	no	—
297	#103	—	2	2	no	—
298	#103	—	2	4	no	—
299	#103	—	2	4	no	—
300	#103	—	2	4	no	—
301	#102	—	2	2	no	—
302	#102	—	2	2	no	—
303	#102	—	2	4	no	—
304	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
305	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
306	C₄×C₈	D₄×C₂₃	1	2	yes	—
307	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
308	C₄×C₈	C₂³×C₂₃	1	2	yes	—
309	C₄×C₈	C₂³×C₂₃	1	2	yes	—
310	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
311	C₂₃	#19657	1	1	yes	characteristic, direct product factor, Sylow
312	#41	—	4	4	no	—
313	C₂×Q₈	—	2	1	no	—
314	C₂×Q₈	—	2	1	no	—
315	C₂×Q₈	—	2	1	no	—
316	C₂×Q₈	—	2	2	no	—
317	C₂×Q₈	—	2	2	no	—
318	C₂×Q₈	—	4	1	no	—
319	C₂×Q₈	—	4	1	no	—
320	C₂×Q₈	—	4	4	no	—
321	C₂×Q₈	—	8	2	no	—
322	C₂×D₄	—	2	1	no	—
323	C₂×D₄	—	2	1	no	—
324	C₂×D₄	—	2	1	no	—
325	C₂×D₄	—	4	1	no	—
326	C₂²×C₄	—	4	1	no	—
327	C₂²×C₄	—	4	1	no	—
328	C₂²×C₄	—	4	1	no	—
329	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
330	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
331	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
332	SD₁₆	—	8	4	no	—
333	C₂×C₈	—	2	4	no	—
334	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
335	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
336	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
337	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
338	C₄⋊C₄	—	2	1	no	—
339	C₄⋊C₄	—	2	1	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	2	1	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	1	no	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	—	4	1	no	—
349	C₄⋊C₄	—	4	1	no	—
350	C₄⋊C₄	—	4	1	no	—
351	C₄⋊C₄	—	4	1	no	—
352	C₄⋊C₄	—	4	1	no	—
353	C₄⋊C₄	—	4	1	no	—
354	C₂²⋊C₄	—	4	1	no	—
355	C₂²⋊C₄	—	4	1	no	—
356	C₂²⋊C₄	—	4	1	no	—
357	C₄²	—	4	1	no	—
358	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
359	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
360	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
361	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
362	C₂³	—	4	1	no	—
363	C₂³	C₂₃×#127	1	1	yes	characteristic
364	Q₈	—	4	4	no	—
365	Q₈	—	4	2	no	—
366	Q₈	—	4	4	no	—
367	Q₈	—	4	2	no	—
368	Q₈	—	4	2	no	—
369	Q₈	—	8	2	no	—
370	Q₈	—	8	2	no	—
371	D₄	—	4	2	no	—
372	D₄	—	4	2	no	—
373	D₄	—	4	2	no	—
374	D₄	—	8	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	4	1	no	—
378	C₂×C₄	—	4	1	no	—
379	C₂×C₄	—	4	1	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	8	2	no	—
383	C₂×C₄	—	8	2	no	—
384	C₂×C₄	—	8	2	no	—
385	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
386	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
387	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
388	C₂×C₄	C₂₃×#135	1	4	yes	—
389	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
390	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
391	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
392	C₈	—	2	2	no	—
393	C₈	—	2	2	no	—
394	C₈	—	2	4	no	—
395	C₈	—	2	4	no	—
396	C₂²	—	2	1	no	—
397	C₂²	—	2	1	no	—
398	C₂²	—	8	1	no	—
399	C₂²	—	8	1	no	—
400	C₂²	—	8	1	no	—
401	C₂²	C₂₃×#495	1	2	yes	—
402	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
403	C₄	—	2	1	no	—
404	C₄	—	2	1	no	—
405	C₄	—	8	1	no	—
406	C₄	—	8	2	no	—
407	C₄	—	8	2	no	—
408	C₄	—	8	1	no	—
409	C₄	—	8	1	no	—
410	C₄	C₂₃×#495	1	2	yes	—
411	C₄	C₂×C₂₃×#135	1	2	yes	—
412	C₄	C₂₃×#577	1	2	yes	—
413	C₄	C₂₃×#578	1	2	yes	—
414	C₂	—	2	2	no	—
415	C₂	—	16	1	no	—
416	C₂	C₂×C₂₃×#495	1	1	yes	characteristic
417	C₂	C₂₃×#3154	1	1	yes	characteristic
418	C₂	C₂₃×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#532	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×#3230	C₂	semidirect
2. C₂₃×#3229	C₂	semidirect
6. C₂×C₂₃×#487	C₂	semidirect
7. C₂₃×#1719	C₂	semidirect
8. C₂₃×#1718	C₂	semidirect
9. C₂₃×#1710	C₂	semidirect
10. C₂₃×#1708	C₂	semidirect
11. C₂₃×#1450	C₂	semidirect
152. #19657	C₂₃	direct (N × H)
187. #3230	C₂×C₂₃	semidirect
188. #3229	C₂×C₂₃	semidirect
192. C₂×#487	C₂×C₂₃	semidirect
193. #1719	C₂×C₂₃	semidirect
194. #1718	C₂×C₂₃	semidirect
195. #1710	C₂×C₂₃	semidirect
196. #1708	C₂×C₂₃	semidirect
197. #1450	C₂×C₂₃	semidirect
311. C₂₃	#19657	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×#3230	2
C₂₃×#3229	2
C₂₃×#3226	2
C₂₃×#3206	2
C₂₃×#3131	2
C₂×C₂₃×#487	2
C₂₃×#1719	2
C₂₃×#1718	2
C₂₃×#1710	2
C₂₃×#1708	2
C₂₃×#1450	2
#19657	23

Subgroups — C23×#19657 #19167283

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19657 #19167283