Subgroups — C₂₃×#18584 #19166210

Order 5888.

882 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3509	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3254	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3141	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3106	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1638	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1636	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1628	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1626	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1549	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1489	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1485	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#557	—	2	1	no	—
13	C₂₃×#557	C₂²	1	2	yes	—
14	C₂₃×#556	—	2	1	no	—
15	C₂₃×#556	C₂²	1	2	yes	—
16	C₂₃×#552	—	2	2	no	—
17	C₂₃×#552	—	2	2	no	—
18	C₂₃×#541	—	2	2	no	—
19	C₂₃×#536	—	2	1	no	—
20	C₂₃×#533	—	2	1	no	—
21	C₂₃×#533	—	2	1	no	—
22	C₂₃×#533	C₂²	1	1	yes	characteristic
23	C₂₃×#532	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
25	C₂₃×#519	—	2	1	no	—
26	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#489	—	2	2	no	—
28	C₂₃×#487	C₂²	1	2	yes	—
29	C₂₃×#486	C₂²	1	2	yes	—
30	C₂₃×#482	—	2	1	no	—
31	C₂₃×#477	C₂²	1	1	yes	characteristic
32	C₂₃×#477	C₂²	1	2	yes	—
33	C₂₃×#476	C₂²	1	1	yes	characteristic
34	C₂₃×#474	—	2	1	no	—
35	C₂₃×#468	—	2	2	no	—
36	C₂₃×#461	—	2	1	no	—
37	C₂₃×#460	C₂²	1	2	yes	—
38	C₂₃×#460	C₂²	1	1	yes	characteristic
39	C₂₃×#459	C₂²	1	1	yes	characteristic
40	C₂₃×#449	—	2	1	no	—
41	C₄×SD₁₆×C₂₃	—	2	1	no	—
42	C₄×D₈×C₂₃	—	2	1	no	—
43	C₂₃×#431	C₂²	1	1	yes	characteristic
44	C₂₃×#423	C₂²	1	1	yes	characteristic
45	C₂₃×#417	—	2	2	no	—
46	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
47	C₂₃×#332	C₂²	1	2	yes	—
48	C₂₃×#332	C₂²	1	2	yes	—
49	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
50	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
51	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
52	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
53	C₂×#41×C₂₃	—	2	1	no	—
54	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
55	C₂²×Q₈×C₂₃	—	2	1	no	—
56	C₂×SD₁₆×C₂₃	—	2	1	no	—
57	C₂×SD₁₆×C₂₃	—	2	2	no	—
58	C₂×D₈×C₂₃	—	2	1	no	—
59	C₂×M₁₆×C₂₃	—	2	1	no	—
60	C₂×M₁₆×C₂₃	—	2	1	no	—
61	C₂₃×C₄⋊Q₈	—	2	2	no	—
62	C₂₃×C₄⋊Q₈	—	2	1	no	—
63	C₂₃×C₄⋊Q₈	—	2	1	no	—
64	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
65	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
66	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
67	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
68	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
69	C₂₃×#127	C₂³	1	1	yes	characteristic
70	C₂₃×#126	—	2	4	no	—
71	C₂₃×#126	—	2	4	no	—
72	C₂₃×#125	—	2	2	no	—
73	C₂₃×#125	—	2	4	no	—
74	C₂₃×#125	—	2	2	no	—
75	C₂₃×#125	—	2	4	no	—
76	C₂₃×#125	—	2	1	no	—
77	C₂₃×#124	—	2	2	no	—
78	C₂₃×#124	—	2	1	no	—
79	C₂₃×#123	—	2	4	no	—
80	C₂₃×C₂²⋊Q₈	—	2	2	no	—
81	C₂₃×C₂²⋊Q₈	—	2	2	no	—
82	C₂₃×C₂²⋊Q₈	—	2	2	no	—
83	C₂₃×C₂²⋊Q₈	—	2	2	no	—
84	C₂₃×C₂²⋊Q₈	—	2	1	no	—
85	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
86	C₂₃×#121	—	2	1	no	—
87	C₂₃×#121	—	2	1	no	—
88	C₄×Q₈×C₂₃	—	2	2	no	—
89	C₄×Q₈×C₂₃	—	2	2	no	—
90	C₄×Q₈×C₂₃	—	2	2	no	—
91	C₄×Q₈×C₂₃	—	2	2	no	—
92	C₄×Q₈×C₂₃	—	2	2	no	—
93	C₄×Q₈×C₂₃	—	2	2	no	—
94	C₄×Q₈×C₂₃	—	2	1	no	—
95	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
96	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
97	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
98	C₄×D₄×C₂₃	—	2	1	no	—
99	C₄×D₄×C₂₃	—	2	1	no	—
100	C₄×D₄×C₂₃	—	2	1	no	—
101	C₄×D₄×C₂₃	—	2	1	no	—
102	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
103	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
104	C₂₃×#117	—	2	4	no	—
105	C₂₃×#117	—	2	2	no	—
106	C₂₃×#117	—	2	1	no	—
107	C₂₃×#117	—	2	2	no	—
108	C₂₃×#117	—	2	1	no	—
109	C₂₃×#117	D₄	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
111	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
112	C₂×C₄²×C₂₃	—	2	1	no	—
113	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
114	C₂₃×#107	—	2	1	no	—
115	C₂₃×#107	—	2	2	no	—
116	C₂₃×#106	—	2	1	no	—
117	C₂₃×C₄⋊C₈	—	2	1	no	—
118	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
119	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
120	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
121	C₂₃×#103	—	2	1	no	—
122	C₂₃×#103	—	2	2	no	—
123	C₂₃×#103	—	2	4	no	—
124	C₂₃×#102	—	2	1	no	—
125	C₂₃×#102	—	2	4	no	—
126	C₂₃×#102	—	2	1	no	—
127	C₂₃×#102	—	2	1	no	—
128	C₂₃×#102	—	2	2	no	—
129	C₂₃×C₂²⋊C₈	—	2	1	no	—
130	C₂₃×#97	—	2	1	no	—
131	C₄×C₈×C₂₃	—	2	1	no	—
132	C₄×C₈×C₂₃	C₂³	1	2	yes	—
133	#41×C₂₃	—	4	1	no	—
134	#41×C₂₃	—	4	1	no	—
135	#41×C₂₃	—	4	4	no	—
136	C₂×Q₈×C₂₃	—	2	1	no	—
137	C₂×Q₈×C₂₃	—	2	2	no	—
138	C₂×Q₈×C₂₃	—	2	2	no	—
139	C₂×Q₈×C₂₃	—	2	1	no	—
140	C₂×Q₈×C₂₃	—	4	2	no	—
141	C₂×Q₈×C₂₃	—	4	2	no	—
142	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
143	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
144	C₂×D₄×C₂₃	—	2	1	no	—
145	C₂×D₄×C₂₃	—	2	1	no	—
146	C₂×D₄×C₂₃	—	2	1	no	—
147	C₂×D₄×C₂₃	—	2	1	no	—
148	C₂×D₄×C₂₃	—	4	1	no	—
149	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₂₃	—	2	2	no	—
151	C₂²×C₄×C₂₃	—	2	1	no	—
152	C₂²×C₄×C₂₃	—	2	1	no	—
153	C₂²×C₄×C₂₃	—	2	1	no	—
154	C₂²×C₄×C₂₃	—	4	1	no	—
155	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
158	SD₁₆×C₂₃	—	4	2	no	—
159	SD₁₆×C₂₃	—	8	2	no	—
160	D₈×C₂₃	—	4	2	no	—
161	M₁₆×C₂₃	—	4	2	no	—
162	M₁₆×C₂₃	—	8	1	no	—
163	C₂×C₈×C₂₃	—	2	2	no	—
164	C₂×C₈×C₂₃	—	2	1	no	—
165	C₂×C₈×C₂₃	—	2	2	no	—
166	C₂×C₈×C₂₃	—	2	1	no	—
167	C₄⋊C₄×C₂₃	—	2	4	no	—
168	C₄⋊C₄×C₂₃	—	2	4	no	—
169	C₄⋊C₄×C₂₃	—	2	4	no	—
170	C₄⋊C₄×C₂₃	—	2	4	no	—
171	C₄⋊C₄×C₂₃	—	2	2	no	—
172	C₄⋊C₄×C₂₃	—	2	2	no	—
173	C₄⋊C₄×C₂₃	—	2	1	no	—
174	C₄⋊C₄×C₂₃	—	2	1	no	—
175	C₄⋊C₄×C₂₃	—	2	1	no	—
176	C₄⋊C₄×C₂₃	—	2	1	no	—
177	C₄⋊C₄×C₂₃	—	2	2	no	—
178	C₄⋊C₄×C₂₃	—	2	2	no	—
179	C₄⋊C₄×C₂₃	—	2	1	no	—
180	C₄⋊C₄×C₂₃	—	2	2	no	—
181	C₄⋊C₄×C₂₃	—	2	1	no	—
182	C₄⋊C₄×C₂₃	—	2	1	no	—
183	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
184	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
185	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
186	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
187	C₂²⋊C₄×C₂₃	—	2	4	no	—
188	C₂²⋊C₄×C₂₃	—	2	4	no	—
189	C₂²⋊C₄×C₂₃	—	2	1	no	—
190	C₂²⋊C₄×C₂₃	—	2	2	no	—
191	C₂²⋊C₄×C₂₃	—	2	1	no	—
192	C₂²⋊C₄×C₂₃	—	2	1	no	—
193	C₂²⋊C₄×C₂₃	—	4	1	no	—
194	C₂²⋊C₄×C₂₃	—	4	1	no	—
195	C₄²×C₂₃	—	2	4	no	—
196	C₄²×C₂₃	—	2	4	no	—
197	C₄²×C₂₃	—	2	1	no	—
198	C₄²×C₂₃	—	2	1	no	—
199	C₄²×C₂₃	—	2	1	no	—
200	C₄²×C₂₃	—	2	2	no	—
201	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
202	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
203	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
204	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
205	#18584	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
206	C₂³×C₂₃	—	2	1	no	—
207	C₂³×C₂₃	—	4	1	no	—
208	C₂³×C₂₃	#120	1	1	yes	characteristic
209	Q₈×C₂₃	—	2	1	no	—
210	Q₈×C₂₃	—	2	1	no	—
211	Q₈×C₂₃	—	2	2	no	—
212	Q₈×C₂₃	—	2	2	no	—
213	Q₈×C₂₃	—	4	4	no	—
214	Q₈×C₂₃	—	4	2	no	—
215	Q₈×C₂₃	—	4	2	no	—
216	Q₈×C₂₃	—	4	2	no	—
217	Q₈×C₂₃	—	4	2	no	—
218	D₄×C₂₃	—	2	1	no	—
219	D₄×C₂₃	—	2	1	no	—
220	D₄×C₂₃	—	4	2	no	—
221	D₄×C₂₃	—	4	2	no	—
222	D₄×C₂₃	—	4	1	no	—
223	D₄×C₂₃	—	4	1	no	—
224	D₄×C₂₃	—	4	1	no	—
225	D₄×C₂₃	—	4	1	no	—
226	D₄×C₂₃	—	8	2	no	—
227	C₂×C₄×C₂₃	—	2	1	no	—
228	C₂×C₄×C₂₃	—	2	1	no	—
229	C₂×C₄×C₂₃	—	2	1	no	—
230	C₂×C₄×C₂₃	—	2	1	no	—
231	C₂×C₄×C₂₃	—	2	2	no	—
232	C₂×C₄×C₂₃	—	2	1	no	—
233	C₂×C₄×C₂₃	—	2	1	no	—
234	C₂×C₄×C₂₃	—	2	2	no	—
235	C₂×C₄×C₂₃	—	2	2	no	—
236	C₂×C₄×C₂₃	—	2	4	no	—
237	C₂×C₄×C₂₃	—	2	1	no	—
238	C₂×C₄×C₂₃	—	4	2	no	—
239	C₂×C₄×C₂₃	—	4	4	no	—
240	C₂×C₄×C₂₃	—	4	1	no	—
241	C₂×C₄×C₂₃	—	4	1	no	—
242	C₂×C₄×C₂₃	—	4	2	no	—
243	C₂×C₄×C₂₃	—	4	1	no	—
244	C₂×C₄×C₂₃	—	8	2	no	—
245	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
246	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
247	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
248	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
249	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
250	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
251	C₈×C₂₃	—	4	2	no	—
252	C₈×C₂₃	—	4	1	no	—
253	C₈×C₂₃	—	4	1	no	—
254	C₈×C₂₃	—	4	2	no	—
255	#3509	C₂×C₂₃	1	1	yes	characteristic, complemented
256	#3254	C₂×C₂₃	1	1	yes	characteristic, complemented
257	#3141	C₂×C₂₃	1	1	yes	characteristic, complemented
258	#3106	C₂×C₂₃	1	1	yes	characteristic
259	#1638	C₂×C₂₃	1	2	yes	complemented
260	#1636	C₂×C₂₃	1	2	yes	complemented
261	#1628	C₂×C₂₃	1	2	yes	complemented
262	#1626	C₂×C₂₃	1	2	yes	complemented
263	#1549	C₂×C₂₃	1	1	yes	characteristic, complemented
264	#1489	C₂×C₂₃	1	1	yes	characteristic, complemented
265	#1485	C₂×C₂₃	1	1	yes	characteristic, complemented
266	C₂²×C₂₃	—	2	1	no	—
267	C₂²×C₂₃	—	2	1	no	—
268	C₂²×C₂₃	—	2	1	no	—
269	C₂²×C₂₃	—	4	1	no	—
270	C₂²×C₂₃	—	4	1	no	—
271	C₂²×C₂₃	—	4	1	no	—
272	C₂²×C₂₃	—	8	1	no	—
273	C₂²×C₂₃	—	8	1	no	—
274	C₂²×C₂₃	—	8	1	no	—
275	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
276	C₄×C₂₃	—	2	1	no	—
277	C₄×C₂₃	—	2	1	no	—
278	C₄×C₂₃	—	2	1	no	—
279	C₄×C₂₃	—	4	1	no	—
280	C₄×C₂₃	—	4	4	no	—
281	C₄×C₂₃	—	4	1	no	—
282	C₄×C₂₃	—	4	2	no	—
283	C₄×C₂₃	—	4	1	no	—
284	C₄×C₂₃	—	4	2	no	—
285	C₄×C₂₃	—	8	1	no	—
286	C₄×C₂₃	#454	1	2	yes	—
287	C₄×C₂₃	#575	1	1	yes	characteristic
288	C₄×C₂₃	#575	1	1	yes	characteristic
289	C₄×C₂₃	#577	1	1	yes	characteristic
290	C₄×C₂₃	#578	1	1	yes	characteristic
291	#557	—	2	1	no	—
292	#557	C₂²×C₂₃	1	2	yes	—
293	#556	—	2	1	no	—
294	#556	C₂²×C₂₃	1	2	yes	—
295	#552	—	2	2	no	—
296	#552	—	2	2	no	—
297	#541	—	2	2	no	—
298	#536	—	2	1	no	—
299	#533	—	2	1	no	—
300	#533	—	2	1	no	—
301	#533	C₂²×C₂₃	1	1	yes	characteristic
302	#532	C₂²×C₂₃	1	1	yes	characteristic
303	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
304	#519	—	2	1	no	—
305	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
306	#489	—	2	2	no	—
307	#487	C₂²×C₂₃	1	2	yes	—
308	#486	C₂²×C₂₃	1	2	yes	—
309	#482	—	2	1	no	—
310	#477	C₂²×C₂₃	1	1	yes	characteristic
311	#477	C₂²×C₂₃	1	2	yes	—
312	#476	C₂²×C₂₃	1	1	yes	characteristic
313	#474	—	2	1	no	—
314	#468	—	2	2	no	—
315	#461	—	2	1	no	—
316	#460	C₂²×C₂₃	1	1	yes	characteristic
317	#460	C₂²×C₂₃	1	2	yes	—
318	#459	C₂²×C₂₃	1	1	yes	characteristic
319	#449	—	2	1	no	—
320	C₄×SD₁₆	—	2	1	no	—
321	C₄×D₈	—	2	1	no	—
322	#431	C₂²×C₂₃	1	1	yes	characteristic
323	#423	C₂²×C₂₃	1	1	yes	characteristic
324	#417	—	2	2	no	—
325	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
326	#332	C₂²×C₂₃	1	2	yes	—
327	#332	C₂²×C₂₃	1	2	yes	—
328	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
329	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
330	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
331	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
332	C₂×C₂₃	—	4	1	no	—
333	C₂×C₂₃	—	8	1	no	—
334	C₂×C₂₃	—	16	1	no	—
335	C₂×C₂₃	#3005	1	1	yes	characteristic
336	C₂×C₂₃	#3013	1	1	yes	characteristic
337	C₂×C₂₃	C₂×#454	1	1	yes	characteristic
338	C₂×#41	—	2	1	no	—
339	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
340	C₂²×Q₈	—	2	1	no	—
341	C₂×SD₁₆	—	2	1	no	—
342	C₂×SD₁₆	—	2	2	no	—
343	C₂×D₈	—	2	1	no	—
344	C₂×M₁₆	—	2	1	no	—
345	C₂×M₁₆	—	2	1	no	—
346	C₄⋊Q₈	—	2	2	no	—
347	C₄⋊Q₈	—	2	1	no	—
348	C₄⋊Q₈	—	2	1	no	—
349	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
350	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
351	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
352	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
353	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
354	#127	C₂³×C₂₃	1	1	yes	characteristic
355	#126	—	2	4	no	—
356	#126	—	2	4	no	—
357	#125	—	2	2	no	—
358	#125	—	2	4	no	—
359	#125	—	2	2	no	—
360	#125	—	2	4	no	—
361	#125	—	2	1	no	—
362	#124	—	2	2	no	—
363	#124	—	2	1	no	—
364	#123	—	2	4	no	—
365	C₂²⋊Q₈	—	2	2	no	—
366	C₂²⋊Q₈	—	2	2	no	—
367	C₂²⋊Q₈	—	2	2	no	—
368	C₂²⋊Q₈	—	2	2	no	—
369	C₂²⋊Q₈	—	2	1	no	—
370	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
371	#121	—	2	1	no	—
372	#121	—	2	1	no	—
373	C₄×Q₈	—	2	2	no	—
374	C₄×Q₈	—	2	2	no	—
375	C₄×Q₈	—	2	2	no	—
376	C₄×Q₈	—	2	2	no	—
377	C₄×Q₈	—	2	2	no	—
378	C₄×Q₈	—	2	2	no	—
379	C₄×Q₈	—	2	1	no	—
380	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
381	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
382	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
383	C₄×D₄	—	2	1	no	—
384	C₄×D₄	—	2	1	no	—
385	C₄×D₄	—	2	1	no	—
386	C₄×D₄	—	2	1	no	—
387	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
388	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
389	#117	—	2	4	no	—
390	#117	—	2	2	no	—
391	#117	—	2	1	no	—
392	#117	—	2	2	no	—
393	#117	—	2	1	no	—
394	#117	D₄×C₂₃	1	1	yes	characteristic
395	C₂×C₄⋊C₄	—	2	1	no	—
396	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
397	C₂×C₄²	—	2	1	no	—
398	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
399	#107	—	2	1	no	—
400	#107	—	2	2	no	—
401	#106	—	2	1	no	—
402	C₄⋊C₈	—	2	1	no	—
403	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
404	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
405	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
406	#103	—	2	1	no	—
407	#103	—	2	2	no	—
408	#103	—	2	4	no	—
409	#102	—	2	1	no	—
410	#102	—	2	4	no	—
411	#102	—	2	1	no	—
412	#102	—	2	1	no	—
413	#102	—	2	2	no	—
414	C₂²⋊C₈	—	2	1	no	—
415	#97	—	2	1	no	—
416	C₄×C₈	—	2	1	no	—
417	C₄×C₈	C₂³×C₂₃	1	2	yes	—
418	C₂₃	#18584	1	1	yes	characteristic, direct product factor, Sylow
419	#41	—	4	1	no	—
420	#41	—	4	1	no	—
421	#41	—	4	4	no	—
422	C₂×Q₈	—	2	1	no	—
423	C₂×Q₈	—	2	2	no	—
424	C₂×Q₈	—	2	2	no	—
425	C₂×Q₈	—	2	1	no	—
426	C₂×Q₈	—	4	2	no	—
427	C₂×Q₈	—	4	2	no	—
428	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
429	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
430	C₂×D₄	—	2	1	no	—
431	C₂×D₄	—	2	1	no	—
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	—	2	1	no	—
434	C₂×D₄	—	4	1	no	—
435	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
436	C₂²×C₄	—	2	2	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	4	1	no	—
441	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
442	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
443	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
444	SD₁₆	—	4	2	no	—
445	SD₁₆	—	8	2	no	—
446	D₈	—	4	2	no	—
447	M₁₆	—	4	2	no	—
448	M₁₆	—	8	1	no	—
449	C₂×C₈	—	2	1	no	—
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	2	no	—
452	C₂×C₈	—	2	2	no	—
453	C₄⋊C₄	—	2	4	no	—
454	C₄⋊C₄	—	2	4	no	—
455	C₄⋊C₄	—	2	4	no	—
456	C₄⋊C₄	—	2	4	no	—
457	C₄⋊C₄	—	2	2	no	—
458	C₄⋊C₄	—	2	2	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	—	2	1	no	—
463	C₄⋊C₄	—	2	2	no	—
464	C₄⋊C₄	—	2	2	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	2	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
470	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
471	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
472	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
473	C₂²⋊C₄	—	2	4	no	—
474	C₂²⋊C₄	—	2	4	no	—
475	C₂²⋊C₄	—	2	1	no	—
476	C₂²⋊C₄	—	2	2	no	—
477	C₂²⋊C₄	—	2	1	no	—
478	C₂²⋊C₄	—	2	1	no	—
479	C₂²⋊C₄	—	4	1	no	—
480	C₂²⋊C₄	—	4	1	no	—
481	C₄²	—	2	4	no	—
482	C₄²	—	2	4	no	—
483	C₄²	—	2	1	no	—
484	C₄²	—	2	1	no	—
485	C₄²	—	2	1	no	—
486	C₄²	—	2	2	no	—
487	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
488	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
489	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
490	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
491	C₂³	—	2	1	no	—
492	C₂³	—	4	1	no	—
493	C₂³	C₂₃×#120	1	1	yes	characteristic
494	Q₈	—	2	1	no	—
495	Q₈	—	2	1	no	—
496	Q₈	—	2	2	no	—
497	Q₈	—	2	2	no	—
498	Q₈	—	4	4	no	—
499	Q₈	—	4	2	no	—
500	Q₈	—	4	2	no	—
501	Q₈	—	4	2	no	—
502	Q₈	—	4	2	no	—
503	D₄	—	2	1	no	—
504	D₄	—	2	1	no	—
505	D₄	—	4	2	no	—
506	D₄	—	4	2	no	—
507	D₄	—	4	1	no	—
508	D₄	—	4	1	no	—
509	D₄	—	4	1	no	—
510	D₄	—	4	1	no	—
511	D₄	—	8	2	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	4	2	no	—
524	C₂×C₄	—	4	4	no	—
525	C₂×C₄	—	4	1	no	—
526	C₂×C₄	—	4	1	no	—
527	C₂×C₄	—	4	2	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	—	8	2	no	—
530	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
531	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
532	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
533	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
534	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
535	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
536	C₈	—	4	2	no	—
537	C₈	—	4	1	no	—
538	C₈	—	4	1	no	—
539	C₈	—	4	2	no	—
540	C₂²	—	2	1	no	—
541	C₂²	—	2	1	no	—
542	C₂²	—	2	1	no	—
543	C₂²	—	4	1	no	—
544	C₂²	—	4	1	no	—
545	C₂²	—	4	1	no	—
546	C₂²	—	8	1	no	—
547	C₂²	—	8	1	no	—
548	C₂²	—	8	1	no	—
549	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
550	C₄	—	2	1	no	—
551	C₄	—	2	1	no	—
552	C₄	—	2	1	no	—
553	C₄	—	4	1	no	—
554	C₄	—	4	4	no	—
555	C₄	—	4	1	no	—
556	C₄	—	4	2	no	—
557	C₄	—	4	1	no	—
558	C₄	—	4	2	no	—
559	C₄	—	8	1	no	—
560	C₄	C₂₃×#454	1	2	yes	—
561	C₄	C₂₃×#575	1	1	yes	characteristic
562	C₄	C₂₃×#575	1	1	yes	characteristic
563	C₄	C₂₃×#577	1	1	yes	characteristic
564	C₄	C₂₃×#578	1	1	yes	characteristic
565	C₂	—	4	1	no	—
566	C₂	—	8	1	no	—
567	C₂	—	16	1	no	—
568	C₂	C₂₃×#3005	1	1	yes	characteristic
569	C₂	C₂₃×#3013	1	1	yes	characteristic
570	C₂	C₂×C₂₃×#454	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3106	C₂
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×#3509	C₂	semidirect
2. C₂₃×#3254	C₂	semidirect
3. C₂₃×#3141	C₂	semidirect
5. C₂₃×#1638	C₂	semidirect
6. C₂₃×#1636	C₂	semidirect
7. C₂₃×#1628	C₂	semidirect
8. C₂₃×#1626	C₂	semidirect
9. C₂₃×#1549	C₂	semidirect
10. C₂₃×#1489	C₂	semidirect
11. C₂₃×#1485	C₂	semidirect
205. #18584	C₂₃	direct (N × H)
255. #3509	C₂×C₂₃	semidirect
256. #3254	C₂×C₂₃	semidirect
257. #3141	C₂×C₂₃	semidirect
259. #1638	C₂×C₂₃	semidirect
260. #1636	C₂×C₂₃	semidirect
261. #1628	C₂×C₂₃	semidirect
262. #1626	C₂×C₂₃	semidirect
263. #1549	C₂×C₂₃	semidirect
264. #1489	C₂×C₂₃	semidirect
265. #1485	C₂×C₂₃	semidirect
418. C₂₃	#18584	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×#3509	2
C₂₃×#3254	2
C₂₃×#3141	2
C₂₃×#3106	2
C₂₃×#1638	2
C₂₃×#1636	2
C₂₃×#1628	2
C₂₃×#1626	2
C₂₃×#1549	2
C₂₃×#1489	2
C₂₃×#1485	2
#18584	23

Subgroups — C23×#18584 #19166210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18584 #19166210