Subgroups — C₂₃×#18609 #19166235

Order 5888.

874 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3518	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3252	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3185	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3076	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1652	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1651	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1646	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1645	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1638	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1637	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1634	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1633	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1550	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1489	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1483	C₂	1	1	yes	maximal, characteristic, complemented
16	Q₈²×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#556	—	2	1	no	—
18	C₂₃×#556	—	2	2	no	—
19	C₂₃×#556	—	2	1	no	—
20	C₂₃×#556	C₂²	1	1	yes	characteristic
21	C₂₃×#555	—	2	1	no	—
22	C₂₃×#555	C₂²	1	1	yes	characteristic
23	D₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#548	—	2	1	no	—
25	C₂₃×#542	—	2	2	no	—
26	C₂₃×#542	—	2	1	no	—
27	C₂₃×#533	—	2	1	no	—
28	C₂₃×#533	C₂²	1	1	yes	characteristic
29	C₂₃×#533	C₂²	1	1	yes	characteristic
30	C₂₃×#529	—	2	1	no	—
31	C₂₃×#529	C₂²	1	1	yes	characteristic
32	C₂₃×#519	—	2	1	no	—
33	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
34	C₂₃×#498	C₂²	1	1	yes	characteristic
35	C₂₃×#494	C₂²	1	1	yes	characteristic
36	C₂₃×#487	C₂²	1	1	yes	characteristic
37	C₂₃×#487	C₂²	1	1	yes	characteristic
38	C₂₃×#479	—	2	1	no	—
39	C₂₃×#479	—	2	1	no	—
40	C₂₃×#478	—	2	1	no	—
41	C₂₃×#477	C₂²	1	1	yes	characteristic
42	C₂₃×#477	C₂²	1	1	yes	characteristic
43	C₂₃×#476	C₂²	1	1	yes	characteristic
44	C₂₃×#476	C₂²	1	1	yes	characteristic
45	C₂₃×#475	C₂²	1	1	yes	characteristic
46	C₂₃×#474	C₂²	1	1	yes	characteristic
47	C₂₃×#470	—	2	2	no	—
48	C₂₃×#470	—	2	1	no	—
49	C₂₃×#468	—	2	1	no	—
50	C₂₃×#464	—	2	1	no	—
51	C₂₃×#462	C₂²	1	1	yes	characteristic
52	C₂₃×#461	C₂²	1	1	yes	characteristic
53	C₂₃×#441	—	2	2	no	—
54	C₂₃×#427	—	2	1	no	—
55	C₂₃×#423	C₂²	1	1	yes	characteristic
56	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
57	C₂₃×#416	—	2	1	no	—
58	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
59	C₂₃×#333	C₂²	1	1	yes	characteristic
60	C₂₃×#333	C₂²	1	1	yes	characteristic
61	C₂₃×#332	C₂²	1	1	yes	characteristic
62	C₂₃×#332	C₂²	1	1	yes	characteristic
63	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
64	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
65	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
66	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
67	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
68	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
69	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
70	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
71	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
72	C₂²×Q₈×C₂₃	—	2	1	no	—
73	C₂×Q₁₆×C₂₃	—	2	2	no	—
74	C₂×Q₁₆×C₂₃	—	2	1	no	—
75	C₂×SD₁₆×C₂₃	—	2	1	no	—
76	C₂×M₁₆×C₂₃	—	2	1	no	—
77	C₂×M₁₆×C₂₃	—	2	1	no	—
78	C₂²×C₈×C₂₃	—	2	1	no	—
79	C₂₃×C₄⋊Q₈	—	2	2	no	—
80	C₂₃×C₄⋊Q₈	—	2	1	no	—
81	C₂₃×C₄⋊Q₈	—	2	1	no	—
82	C₂₃×C₄⋊Q₈	—	2	1	no	—
83	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
84	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
85	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
86	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
87	C₂₃×#126	—	2	2	no	—
88	C₂₃×#126	—	2	2	no	—
89	C₂₃×#126	—	2	2	no	—
90	C₂₃×#126	—	2	2	no	—
91	C₂₃×#126	—	2	1	no	—
92	C₂₃×#125	—	2	1	no	—
93	C₂₃×#125	—	2	1	no	—
94	C₂₃×#125	D₄	1	1	yes	characteristic
95	C₂₃×#124	—	2	2	no	—
96	C₂₃×#124	—	2	2	no	—
97	C₂₃×#124	—	2	2	no	—
98	C₂₃×#124	—	2	2	no	—
99	C₂₃×#124	—	2	1	no	—
100	C₂₃×#124	D₄	1	1	yes	characteristic
101	C₂₃×#123	—	2	2	no	—
102	C₂₃×#123	—	2	1	no	—
103	C₂₃×C₂²⋊Q₈	—	2	2	no	—
104	C₂₃×C₂²⋊Q₈	—	2	2	no	—
105	C₂₃×C₂²⋊Q₈	—	2	1	no	—
106	C₂₃×C₂²⋊Q₈	—	2	1	no	—
107	C₂₃×C₂²⋊Q₈	—	2	1	no	—
108	C₂₃×C₂²⋊Q₈	—	2	2	no	—
109	C₂₃×C₂²⋊Q₈	—	2	1	no	—
110	C₂₃×C₂²⋊Q₈	—	2	1	no	—
111	C₂₃×C₂²⋊Q₈	D₄	1	1	yes	characteristic
112	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
113	C₂₃×#121	—	2	2	no	—
114	C₂₃×#121	—	2	1	no	—
115	C₂₃×#121	—	2	1	no	—
116	C₂₃×#121	D₄	1	1	yes	characteristic
117	C₄×Q₈×C₂₃	—	2	2	no	—
118	C₄×Q₈×C₂₃	—	2	2	no	—
119	C₄×Q₈×C₂₃	—	2	2	no	—
120	C₄×Q₈×C₂₃	—	2	1	no	—
121	C₄×Q₈×C₂₃	—	2	1	no	—
122	C₄×Q₈×C₂₃	—	2	1	no	—
123	C₄×Q₈×C₂₃	—	2	1	no	—
124	C₄×Q₈×C₂₃	—	2	1	no	—
125	C₄×Q₈×C₂₃	—	2	1	no	—
126	C₄×Q₈×C₂₃	D₄	1	2	yes	—
127	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
128	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
129	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
130	C₄×D₄×C₂₃	—	2	2	no	—
131	C₄×D₄×C₂₃	—	2	1	no	—
132	C₄×D₄×C₂₃	—	2	1	no	—
133	C₄×D₄×C₂₃	—	2	1	no	—
134	C₄×D₄×C₂₃	—	2	1	no	—
135	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
136	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
137	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
138	C₂₃×#117	—	2	2	no	—
139	C₂₃×#117	—	2	2	no	—
140	C₂₃×#117	—	2	1	no	—
141	C₂₃×#117	—	2	1	no	—
142	C₂₃×#117	—	2	1	no	—
143	C₂₃×#117	—	2	1	no	—
144	C₂₃×#117	—	2	1	no	—
145	C₂₃×#117	—	2	1	no	—
146	C₂₃×#117	—	2	1	no	—
147	C₂₃×#117	D₄	1	1	yes	characteristic
148	C₂×C₄²×C₂₃	—	2	1	no	—
149	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
150	C₂₃×#107	—	2	1	no	—
151	C₂₃×#107	—	2	1	no	—
152	C₂₃×#107	—	2	1	no	—
153	C₂₃×#106	—	2	1	no	—
154	C₂₃×#106	—	2	1	no	—
155	C₂₃×#106	—	2	2	no	—
156	C₂₃×#106	—	2	1	no	—
157	C₂₃×C₄⋊C₈	—	2	1	no	—
158	C₂₃×C₄⋊C₈	—	2	1	no	—
159	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
160	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
161	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
162	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
163	C₂₃×#103	—	2	1	no	—
164	C₂₃×#103	—	2	1	no	—
165	C₂₃×#103	—	2	2	no	—
166	C₂₃×#103	—	2	1	no	—
167	C₂₃×#103	—	2	2	no	—
168	C₂₃×#103	—	2	1	no	—
169	C₂₃×#103	—	2	1	no	—
170	C₂₃×#102	—	2	1	no	—
171	C₂₃×#102	—	2	1	no	—
172	C₂₃×#102	—	2	1	no	—
173	C₂₃×#97	—	2	1	no	—
174	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
175	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
176	#41×C₂₃	—	2	2	no	—
177	#41×C₂₃	—	4	1	no	—
178	C₂×Q₈×C₂₃	—	2	2	no	—
179	C₂×Q₈×C₂₃	—	2	2	no	—
180	C₂×Q₈×C₂₃	—	2	1	no	—
181	C₂×Q₈×C₂₃	—	2	1	no	—
182	C₂×Q₈×C₂₃	—	2	1	no	—
183	C₂×Q₈×C₂₃	—	4	2	no	—
184	C₂×Q₈×C₂₃	—	4	2	no	—
185	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
186	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
187	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
188	C₂×D₄×C₂₃	—	2	1	no	—
189	C₂×D₄×C₂₃	—	2	2	no	—
190	C₂×D₄×C₂₃	—	2	1	no	—
191	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
192	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
193	C₂²×C₄×C₂₃	—	2	2	no	—
194	C₂²×C₄×C₂₃	—	2	1	no	—
195	C₂²×C₄×C₂₃	—	2	1	no	—
196	C₂²×C₄×C₂₃	—	2	1	no	—
197	C₂²×C₄×C₂₃	—	2	1	no	—
198	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
199	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
200	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
201	Q₁₆×C₂₃	—	4	4	no	—
202	Q₁₆×C₂₃	—	8	1	no	—
203	SD₁₆×C₂₃	—	8	1	no	—
204	M₁₆×C₂₃	—	4	2	no	—
205	M₁₆×C₂₃	—	8	1	no	—
206	C₂×C₈×C₂₃	—	2	1	no	—
207	C₂×C₈×C₂₃	—	2	1	no	—
208	C₂×C₈×C₂₃	—	2	1	no	—
209	C₂×C₈×C₂₃	—	2	1	no	—
210	C₂×C₈×C₂₃	—	2	1	no	—
211	C₂×C₈×C₂₃	—	2	1	no	—
212	C₂×C₈×C₂₃	—	4	2	no	—
213	C₄⋊C₄×C₂₃	—	2	2	no	—
214	C₄⋊C₄×C₂₃	—	2	2	no	—
215	C₄⋊C₄×C₂₃	—	2	2	no	—
216	C₄⋊C₄×C₂₃	—	2	2	no	—
217	C₄⋊C₄×C₂₃	—	2	1	no	—
218	C₄⋊C₄×C₂₃	—	2	1	no	—
219	C₄⋊C₄×C₂₃	—	2	2	no	—
220	C₄⋊C₄×C₂₃	—	2	1	no	—
221	C₄⋊C₄×C₂₃	—	2	1	no	—
222	C₄⋊C₄×C₂₃	—	2	2	no	—
223	C₄⋊C₄×C₂₃	—	2	1	no	—
224	C₄⋊C₄×C₂₃	—	2	1	no	—
225	C₄⋊C₄×C₂₃	—	2	1	no	—
226	C₄⋊C₄×C₂₃	—	2	1	no	—
227	C₄⋊C₄×C₂₃	—	2	1	no	—
228	C₄⋊C₄×C₂₃	—	2	1	no	—
229	C₄⋊C₄×C₂₃	—	2	1	no	—
230	C₄⋊C₄×C₂₃	—	2	1	no	—
231	C₄⋊C₄×C₂₃	—	2	1	no	—
232	C₄⋊C₄×C₂₃	—	4	1	no	—
233	C₄⋊C₄×C₂₃	—	4	1	no	—
234	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
235	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
236	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
237	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
238	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
239	C₂²⋊C₄×C₂₃	—	2	2	no	—
240	C₂²⋊C₄×C₂₃	—	2	2	no	—
241	C₂²⋊C₄×C₂₃	—	2	2	no	—
242	C₂²⋊C₄×C₂₃	—	2	2	no	—
243	C₂²⋊C₄×C₂₃	—	2	2	no	—
244	C₂²⋊C₄×C₂₃	—	2	2	no	—
245	C₂²⋊C₄×C₂₃	—	2	1	no	—
246	C₂²⋊C₄×C₂₃	—	2	1	no	—
247	C₂²⋊C₄×C₂₃	—	2	1	no	—
248	C₂²⋊C₄×C₂₃	—	2	1	no	—
249	C₂²⋊C₄×C₂₃	—	2	1	no	—
250	C₂²⋊C₄×C₂₃	—	2	1	no	—
251	C₂²⋊C₄×C₂₃	—	2	1	no	—
252	C₂²⋊C₄×C₂₃	—	2	1	no	—
253	C₂²⋊C₄×C₂₃	—	4	1	no	—
254	C₄²×C₂₃	—	2	2	no	—
255	C₄²×C₂₃	—	2	2	no	—
256	C₄²×C₂₃	—	2	2	no	—
257	C₄²×C₂₃	—	2	1	no	—
258	C₄²×C₂₃	—	2	1	no	—
259	C₄²×C₂₃	—	2	2	no	—
260	C₄²×C₂₃	—	2	1	no	—
261	C₄²×C₂₃	—	2	1	no	—
262	C₄²×C₂₃	—	4	1	no	—
263	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
264	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
265	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
266	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
267	#18609	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
268	C₂³×C₂₃	—	2	1	no	—
269	C₂³×C₂₃	—	2	1	no	—
270	C₂³×C₂₃	#120	1	1	yes	characteristic
271	Q₈×C₂₃	—	2	1	no	—
272	Q₈×C₂₃	—	2	1	no	—
273	Q₈×C₂₃	—	2	1	no	—
274	Q₈×C₂₃	—	4	2	no	—
275	Q₈×C₂₃	—	4	4	no	—
276	Q₈×C₂₃	—	4	2	no	—
277	Q₈×C₂₃	—	4	1	no	—
278	Q₈×C₂₃	—	4	1	no	—
279	Q₈×C₂₃	—	4	1	no	—
280	Q₈×C₂₃	—	4	1	no	—
281	Q₈×C₂₃	—	8	2	no	—
282	Q₈×C₂₃	#137	1	2	yes	—
283	D₄×C₂₃	—	2	1	no	—
284	D₄×C₂₃	—	2	1	no	—
285	D₄×C₂₃	—	2	1	no	—
286	D₄×C₂₃	—	2	1	no	—
287	D₄×C₂₃	—	4	2	no	—
288	D₄×C₂₃	—	8	1	no	—
289	D₄×C₂₃	—	8	2	no	—
290	C₂×C₄×C₂₃	—	2	1	no	—
291	C₂×C₄×C₂₃	—	2	1	no	—
292	C₂×C₄×C₂₃	—	2	1	no	—
293	C₂×C₄×C₂₃	—	2	1	no	—
294	C₂×C₄×C₂₃	—	2	1	no	—
295	C₂×C₄×C₂₃	—	2	1	no	—
296	C₂×C₄×C₂₃	—	2	1	no	—
297	C₂×C₄×C₂₃	—	2	2	no	—
298	C₂×C₄×C₂₃	—	2	1	no	—
299	C₂×C₄×C₂₃	—	2	1	no	—
300	C₂×C₄×C₂₃	—	2	2	no	—
301	C₂×C₄×C₂₃	—	2	1	no	—
302	C₂×C₄×C₂₃	—	4	1	no	—
303	C₂×C₄×C₂₃	—	4	2	no	—
304	C₂×C₄×C₂₃	—	4	4	no	—
305	C₂×C₄×C₂₃	—	4	1	no	—
306	C₂×C₄×C₂₃	—	4	1	no	—
307	C₂×C₄×C₂₃	—	4	1	no	—
308	C₂×C₄×C₂₃	—	4	1	no	—
309	C₂×C₄×C₂₃	—	4	1	no	—
310	C₂×C₄×C₂₃	—	4	1	no	—
311	C₂×C₄×C₂₃	—	8	1	no	—
312	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
313	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
314	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
315	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
316	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
317	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
318	C₈×C₂₃	—	2	2	no	—
319	C₈×C₂₃	—	4	1	no	—
320	C₈×C₂₃	—	4	1	no	—
321	C₈×C₂₃	—	4	1	no	—
322	C₈×C₂₃	—	4	1	no	—
323	C₈×C₂₃	—	4	1	no	—
324	#3518	C₂×C₂₃	1	1	yes	characteristic
325	#3252	C₂×C₂₃	1	1	yes	characteristic, complemented
326	#3185	C₂×C₂₃	1	1	yes	characteristic, complemented
327	#3076	C₂×C₂₃	1	1	yes	characteristic, complemented
328	#1652	C₂×C₂₃	1	1	yes	characteristic, complemented
329	#1651	C₂×C₂₃	1	1	yes	characteristic, complemented
330	#1646	C₂×C₂₃	1	1	yes	characteristic, complemented
331	#1645	C₂×C₂₃	1	1	yes	characteristic, complemented
332	#1638	C₂×C₂₃	1	1	yes	characteristic, complemented
333	#1637	C₂×C₂₃	1	1	yes	characteristic, complemented
334	#1634	C₂×C₂₃	1	1	yes	characteristic, complemented
335	#1633	C₂×C₂₃	1	1	yes	characteristic, complemented
336	#1550	C₂×C₂₃	1	1	yes	characteristic, complemented
337	#1489	C₂×C₂₃	1	1	yes	characteristic, complemented
338	#1483	C₂×C₂₃	1	1	yes	characteristic, complemented
339	C₂²×C₂₃	—	2	1	no	—
340	C₂²×C₂₃	—	2	1	no	—
341	C₂²×C₂₃	—	2	1	no	—
342	C₂²×C₂₃	—	4	1	no	—
343	C₂²×C₂₃	—	4	1	no	—
344	C₂²×C₂₃	—	4	1	no	—
345	C₂²×C₂₃	—	4	1	no	—
346	C₂²×C₂₃	—	4	1	no	—
347	C₂²×C₂₃	—	4	1	no	—
348	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
349	C₄×C₂₃	—	2	1	no	—
350	C₄×C₂₃	—	2	2	no	—
351	C₄×C₂₃	—	2	1	no	—
352	C₄×C₂₃	—	2	1	no	—
353	C₄×C₂₃	—	2	1	no	—
354	C₄×C₂₃	—	4	2	no	—
355	C₄×C₂₃	—	4	1	no	—
356	C₄×C₂₃	—	4	1	no	—
357	C₄×C₂₃	—	4	1	no	—
358	C₄×C₂₃	—	4	1	no	—
359	C₄×C₂₃	—	4	2	no	—
360	C₄×C₂₃	—	4	1	no	—
361	C₄×C₂₃	—	8	1	no	—
362	C₄×C₂₃	—	8	1	no	—
363	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
364	C₄×C₂₃	#575	1	1	yes	characteristic
365	C₄×C₂₃	#577	1	1	yes	characteristic
366	C₄×C₂₃	#578	1	1	yes	characteristic
367	Q₈²	C₂²×C₂₃	1	1	yes	characteristic
368	#556	—	2	1	no	—
369	#556	—	2	2	no	—
370	#556	—	2	1	no	—
371	#556	C₂²×C₂₃	1	1	yes	characteristic
372	#555	—	2	1	no	—
373	#555	C₂²×C₂₃	1	1	yes	characteristic
374	D₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
375	#548	—	2	1	no	—
376	#542	—	2	2	no	—
377	#542	—	2	1	no	—
378	#533	—	2	1	no	—
379	#533	C₂²×C₂₃	1	1	yes	characteristic
380	#533	C₂²×C₂₃	1	1	yes	characteristic
381	#529	—	2	1	no	—
382	#529	C₂²×C₂₃	1	1	yes	characteristic
383	#519	—	2	1	no	—
384	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
385	#498	C₂²×C₂₃	1	1	yes	characteristic
386	#494	C₂²×C₂₃	1	1	yes	characteristic
387	#487	C₂²×C₂₃	1	1	yes	characteristic
388	#487	C₂²×C₂₃	1	1	yes	characteristic
389	#479	—	2	1	no	—
390	#479	—	2	1	no	—
391	#478	—	2	1	no	—
392	#477	C₂²×C₂₃	1	1	yes	characteristic
393	#477	C₂²×C₂₃	1	1	yes	characteristic
394	#476	C₂²×C₂₃	1	1	yes	characteristic
395	#476	C₂²×C₂₃	1	1	yes	characteristic
396	#475	C₂²×C₂₃	1	1	yes	characteristic
397	#474	C₂²×C₂₃	1	1	yes	characteristic
398	#470	—	2	2	no	—
399	#470	—	2	1	no	—
400	#468	—	2	1	no	—
401	#464	—	2	1	no	—
402	#462	C₂²×C₂₃	1	1	yes	characteristic
403	#461	C₂²×C₂₃	1	1	yes	characteristic
404	#441	—	2	2	no	—
405	#427	—	2	1	no	—
406	#423	C₂²×C₂₃	1	1	yes	characteristic
407	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
408	#416	—	2	1	no	—
409	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
410	#333	C₂²×C₂₃	1	1	yes	characteristic
411	#333	C₂²×C₂₃	1	1	yes	characteristic
412	#332	C₂²×C₂₃	1	1	yes	characteristic
413	#332	C₂²×C₂₃	1	1	yes	characteristic
414	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
415	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
416	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
417	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
418	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
419	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
420	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
421	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
422	C₂×C₂₃	—	4	1	no	—
423	C₂×C₂₃	—	8	1	no	—
424	C₂×C₂₃	—	8	1	no	—
425	C₂×C₂₃	#3007	1	1	yes	characteristic
426	C₂×C₂₃	#3013	1	1	yes	characteristic
427	C₂×C₂₃	#3018	1	1	yes	characteristic
428	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
429	C₂²×Q₈	—	2	1	no	—
430	C₂×Q₁₆	—	2	2	no	—
431	C₂×Q₁₆	—	2	1	no	—
432	C₂×SD₁₆	—	2	1	no	—
433	C₂×M₁₆	—	2	1	no	—
434	C₂×M₁₆	—	2	1	no	—
435	C₂²×C₈	—	2	1	no	—
436	C₄⋊Q₈	—	2	2	no	—
437	C₄⋊Q₈	—	2	1	no	—
438	C₄⋊Q₈	—	2	1	no	—
439	C₄⋊Q₈	—	2	1	no	—
440	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
441	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
442	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
443	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
444	#126	—	2	2	no	—
445	#126	—	2	2	no	—
446	#126	—	2	2	no	—
447	#126	—	2	2	no	—
448	#126	—	2	1	no	—
449	#125	—	2	1	no	—
450	#125	—	2	1	no	—
451	#125	D₄×C₂₃	1	1	yes	characteristic
452	#124	—	2	2	no	—
453	#124	—	2	2	no	—
454	#124	—	2	2	no	—
455	#124	—	2	2	no	—
456	#124	—	2	1	no	—
457	#124	D₄×C₂₃	1	1	yes	characteristic
458	#123	—	2	2	no	—
459	#123	—	2	1	no	—
460	C₂²⋊Q₈	—	2	2	no	—
461	C₂²⋊Q₈	—	2	2	no	—
462	C₂²⋊Q₈	—	2	1	no	—
463	C₂²⋊Q₈	—	2	1	no	—
464	C₂²⋊Q₈	—	2	1	no	—
465	C₂²⋊Q₈	—	2	2	no	—
466	C₂²⋊Q₈	—	2	1	no	—
467	C₂²⋊Q₈	—	2	1	no	—
468	C₂²⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
469	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
470	#121	—	2	2	no	—
471	#121	—	2	1	no	—
472	#121	—	2	1	no	—
473	#121	D₄×C₂₃	1	1	yes	characteristic
474	C₄×Q₈	—	2	2	no	—
475	C₄×Q₈	—	2	2	no	—
476	C₄×Q₈	—	2	2	no	—
477	C₄×Q₈	—	2	1	no	—
478	C₄×Q₈	—	2	1	no	—
479	C₄×Q₈	—	2	1	no	—
480	C₄×Q₈	—	2	1	no	—
481	C₄×Q₈	—	2	1	no	—
482	C₄×Q₈	—	2	1	no	—
483	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
484	C₄×Q₈	D₄×C₂₃	1	2	yes	—
485	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
486	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
487	C₄×D₄	—	2	2	no	—
488	C₄×D₄	—	2	1	no	—
489	C₄×D₄	—	2	1	no	—
490	C₄×D₄	—	2	1	no	—
491	C₄×D₄	—	2	1	no	—
492	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
493	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
494	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
495	#117	—	2	2	no	—
496	#117	—	2	2	no	—
497	#117	—	2	1	no	—
498	#117	—	2	1	no	—
499	#117	—	2	1	no	—
500	#117	—	2	1	no	—
501	#117	—	2	1	no	—
502	#117	—	2	1	no	—
503	#117	—	2	1	no	—
504	#117	D₄×C₂₃	1	1	yes	characteristic
505	C₂×C₄²	—	2	1	no	—
506	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
507	#107	—	2	1	no	—
508	#107	—	2	1	no	—
509	#107	—	2	1	no	—
510	#106	—	2	1	no	—
511	#106	—	2	1	no	—
512	#106	—	2	2	no	—
513	#106	—	2	1	no	—
514	C₄⋊C₈	—	2	1	no	—
515	C₄⋊C₈	—	2	1	no	—
516	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
517	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
518	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
519	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
520	#103	—	2	1	no	—
521	#103	—	2	1	no	—
522	#103	—	2	2	no	—
523	#103	—	2	1	no	—
524	#103	—	2	2	no	—
525	#103	—	2	1	no	—
526	#103	—	2	1	no	—
527	#102	—	2	1	no	—
528	#102	—	2	1	no	—
529	#102	—	2	1	no	—
530	#97	—	2	1	no	—
531	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
532	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
533	C₂₃	#18609	1	1	yes	characteristic, direct product factor, Sylow
534	#41	—	2	2	no	—
535	#41	—	4	1	no	—
536	C₂×Q₈	—	2	2	no	—
537	C₂×Q₈	—	2	2	no	—
538	C₂×Q₈	—	2	1	no	—
539	C₂×Q₈	—	2	1	no	—
540	C₂×Q₈	—	2	1	no	—
541	C₂×Q₈	—	4	2	no	—
542	C₂×Q₈	—	4	2	no	—
543	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
544	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
545	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
546	C₂×D₄	—	2	1	no	—
547	C₂×D₄	—	2	2	no	—
548	C₂×D₄	—	2	1	no	—
549	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
550	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
551	C₂²×C₄	—	2	2	no	—
552	C₂²×C₄	—	2	1	no	—
553	C₂²×C₄	—	2	1	no	—
554	C₂²×C₄	—	2	1	no	—
555	C₂²×C₄	—	2	1	no	—
556	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
557	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
558	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
559	Q₁₆	—	4	4	no	—
560	Q₁₆	—	8	1	no	—
561	SD₁₆	—	8	1	no	—
562	M₁₆	—	4	2	no	—
563	M₁₆	—	8	1	no	—
564	C₂×C₈	—	2	1	no	—
565	C₂×C₈	—	2	1	no	—
566	C₂×C₈	—	2	1	no	—
567	C₂×C₈	—	2	1	no	—
568	C₂×C₈	—	2	1	no	—
569	C₂×C₈	—	2	1	no	—
570	C₂×C₈	—	4	2	no	—
571	C₄⋊C₄	—	2	2	no	—
572	C₄⋊C₄	—	2	2	no	—
573	C₄⋊C₄	—	2	2	no	—
574	C₄⋊C₄	—	2	2	no	—
575	C₄⋊C₄	—	2	1	no	—
576	C₄⋊C₄	—	2	1	no	—
577	C₄⋊C₄	—	2	2	no	—
578	C₄⋊C₄	—	2	1	no	—
579	C₄⋊C₄	—	2	1	no	—
580	C₄⋊C₄	—	2	2	no	—
581	C₄⋊C₄	—	2	1	no	—
582	C₄⋊C₄	—	2	1	no	—
583	C₄⋊C₄	—	2	1	no	—
584	C₄⋊C₄	—	2	1	no	—
585	C₄⋊C₄	—	2	1	no	—
586	C₄⋊C₄	—	2	1	no	—
587	C₄⋊C₄	—	2	1	no	—
588	C₄⋊C₄	—	2	1	no	—
589	C₄⋊C₄	—	2	1	no	—
590	C₄⋊C₄	—	4	1	no	—
591	C₄⋊C₄	—	4	1	no	—
592	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
593	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
594	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
595	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
596	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
597	C₂²⋊C₄	—	2	2	no	—
598	C₂²⋊C₄	—	2	2	no	—
599	C₂²⋊C₄	—	2	2	no	—
600	C₂²⋊C₄	—	2	2	no	—
601	C₂²⋊C₄	—	2	2	no	—
602	C₂²⋊C₄	—	2	2	no	—
603	C₂²⋊C₄	—	2	1	no	—
604	C₂²⋊C₄	—	2	1	no	—
605	C₂²⋊C₄	—	2	1	no	—
606	C₂²⋊C₄	—	2	1	no	—
607	C₂²⋊C₄	—	2	1	no	—
608	C₂²⋊C₄	—	2	1	no	—
609	C₂²⋊C₄	—	2	1	no	—
610	C₂²⋊C₄	—	2	1	no	—
611	C₂²⋊C₄	—	4	1	no	—
612	C₄²	—	2	2	no	—
613	C₄²	—	2	2	no	—
614	C₄²	—	2	2	no	—
615	C₄²	—	2	1	no	—
616	C₄²	—	2	1	no	—
617	C₄²	—	2	2	no	—
618	C₄²	—	2	1	no	—
619	C₄²	—	2	1	no	—
620	C₄²	—	4	1	no	—
621	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
622	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
623	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
624	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
625	C₂³	—	2	1	no	—
626	C₂³	—	2	1	no	—
627	C₂³	C₂₃×#120	1	1	yes	characteristic
628	Q₈	—	2	1	no	—
629	Q₈	—	2	1	no	—
630	Q₈	—	2	1	no	—
631	Q₈	—	4	2	no	—
632	Q₈	—	4	4	no	—
633	Q₈	—	4	2	no	—
634	Q₈	—	4	1	no	—
635	Q₈	—	4	1	no	—
636	Q₈	—	4	1	no	—
637	Q₈	—	4	1	no	—
638	Q₈	—	8	2	no	—
639	Q₈	C₂₃×#137	1	2	yes	—
640	D₄	—	2	1	no	—
641	D₄	—	2	1	no	—
642	D₄	—	2	1	no	—
643	D₄	—	2	1	no	—
644	D₄	—	4	2	no	—
645	D₄	—	8	1	no	—
646	D₄	—	8	2	no	—
647	C₂×C₄	—	2	1	no	—
648	C₂×C₄	—	2	1	no	—
649	C₂×C₄	—	2	1	no	—
650	C₂×C₄	—	2	1	no	—
651	C₂×C₄	—	2	1	no	—
652	C₂×C₄	—	2	1	no	—
653	C₂×C₄	—	2	1	no	—
654	C₂×C₄	—	2	2	no	—
655	C₂×C₄	—	2	1	no	—
656	C₂×C₄	—	2	1	no	—
657	C₂×C₄	—	2	2	no	—
658	C₂×C₄	—	2	1	no	—
659	C₂×C₄	—	4	1	no	—
660	C₂×C₄	—	4	2	no	—
661	C₂×C₄	—	4	4	no	—
662	C₂×C₄	—	4	1	no	—
663	C₂×C₄	—	4	1	no	—
664	C₂×C₄	—	4	1	no	—
665	C₂×C₄	—	4	1	no	—
666	C₂×C₄	—	4	1	no	—
667	C₂×C₄	—	4	1	no	—
668	C₂×C₄	—	8	1	no	—
669	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
670	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
671	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
672	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
673	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
674	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
675	C₈	—	2	2	no	—
676	C₈	—	4	1	no	—
677	C₈	—	4	1	no	—
678	C₈	—	4	1	no	—
679	C₈	—	4	1	no	—
680	C₈	—	4	1	no	—
681	C₂²	—	2	1	no	—
682	C₂²	—	2	1	no	—
683	C₂²	—	2	1	no	—
684	C₂²	—	4	1	no	—
685	C₂²	—	4	1	no	—
686	C₂²	—	4	1	no	—
687	C₂²	—	4	1	no	—
688	C₂²	—	4	1	no	—
689	C₂²	—	4	1	no	—
690	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
691	C₄	—	2	1	no	—
692	C₄	—	2	2	no	—
693	C₄	—	2	1	no	—
694	C₄	—	2	1	no	—
695	C₄	—	2	1	no	—
696	C₄	—	4	2	no	—
697	C₄	—	4	1	no	—
698	C₄	—	4	1	no	—
699	C₄	—	4	1	no	—
700	C₄	—	4	1	no	—
701	C₄	—	4	2	no	—
702	C₄	—	4	1	no	—
703	C₄	—	8	1	no	—
704	C₄	—	8	1	no	—
705	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
706	C₄	C₂₃×#575	1	1	yes	characteristic
707	C₄	C₂₃×#577	1	1	yes	characteristic
708	C₄	C₂₃×#578	1	1	yes	characteristic
709	C₂	—	4	1	no	—
710	C₂	—	8	1	no	—
711	C₂	—	8	1	no	—
712	C₂	C₂₃×#3007	1	1	yes	characteristic
713	C₂	C₂₃×#3013	1	1	yes	characteristic
714	C₂	C₂₃×#3018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3518	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
2. C₂₃×#3252	C₂	semidirect
3. C₂₃×#3185	C₂	semidirect
4. C₂₃×#3076	C₂	semidirect
5. C₂₃×#1652	C₂	semidirect
6. C₂₃×#1651	C₂	semidirect
7. C₂₃×#1646	C₂	semidirect
8. C₂₃×#1645	C₂	semidirect
9. C₂₃×#1638	C₂	semidirect
10. C₂₃×#1637	C₂	semidirect
11. C₂₃×#1634	C₂	semidirect
12. C₂₃×#1633	C₂	semidirect
13. C₂₃×#1550	C₂	semidirect
14. C₂₃×#1489	C₂	semidirect
15. C₂₃×#1483	C₂	semidirect
267. #18609	C₂₃	direct (N × H)
325. #3252	C₂×C₂₃	semidirect
326. #3185	C₂×C₂₃	semidirect
327. #3076	C₂×C₂₃	semidirect
328. #1652	C₂×C₂₃	semidirect
329. #1651	C₂×C₂₃	semidirect
330. #1646	C₂×C₂₃	semidirect
331. #1645	C₂×C₂₃	semidirect
332. #1638	C₂×C₂₃	semidirect
333. #1637	C₂×C₂₃	semidirect
334. #1634	C₂×C₂₃	semidirect
335. #1633	C₂×C₂₃	semidirect
336. #1550	C₂×C₂₃	semidirect
337. #1489	C₂×C₂₃	semidirect
338. #1483	C₂×C₂₃	semidirect
533. C₂₃	#18609	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×#3518	2
C₂₃×#3252	2
C₂₃×#3185	2
C₂₃×#3076	2
C₂₃×#1652	2
C₂₃×#1651	2
C₂₃×#1646	2
C₂₃×#1645	2
C₂₃×#1638	2
C₂₃×#1637	2
C₂₃×#1634	2
C₂₃×#1633	2
C₂₃×#1550	2
C₂₃×#1489	2
C₂₃×#1483	2
#18609	23

Subgroups — C23×#18609 #19166235

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18609 #19166235