Subgroups — C₂₃×#18745 #19166371

Order 5888.

818 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3555	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3249	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3233	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#3185	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1653	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1652	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1645	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1644	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1638	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1636	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1633	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1632	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1542	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1485	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1483	C₂	1	1	yes	maximal, characteristic
16	C₂₃×#563	—	2	2	no	—
17	Q₈²×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#557	—	2	1	no	—
19	C₂₃×#557	C₂²	1	1	yes	characteristic
20	C₂₃×#556	—	2	1	no	—
21	C₂₃×#556	—	2	1	no	—
22	C₂₃×#556	C₂²	1	1	yes	characteristic
23	C₂₃×#554	C₂²	1	1	yes	characteristic
24	C₂₃×#552	—	2	1	no	—
25	C₂₃×#544	—	2	1	no	—
26	C₂₃×#542	—	2	1	no	—
27	C₂₃×#541	—	2	1	no	—
28	C₂₃×#541	—	2	1	no	—
29	C₂₃×#541	—	2	2	no	—
30	C₂₃×#533	C₂²	1	1	yes	characteristic
31	C₂₃×#533	C₂²	1	1	yes	characteristic
32	C₂₃×#533	C₂²	1	1	yes	characteristic
33	C₂₃×#529	C₂²	1	1	yes	characteristic
34	C₂₃×#499	—	2	1	no	—
35	C₂₃×#498	C₂²	1	1	yes	characteristic
36	C₂₃×#494	C₂²	1	1	yes	characteristic
37	C₂₃×#488	—	2	1	no	—
38	C₂₃×#487	C₂²	1	1	yes	characteristic
39	C₂₃×#487	C₂²	1	1	yes	characteristic
40	C₂₃×#485	—	2	1	no	—
41	C₂₃×#482	—	2	1	no	—
42	C₂₃×#477	—	2	1	no	—
43	C₂₃×#477	C₂²	1	1	yes	characteristic
44	C₂₃×#477	C₂²	1	1	yes	characteristic
45	C₂₃×#477	C₂²	1	1	yes	characteristic
46	C₂₃×#476	C₂²	1	1	yes	characteristic
47	C₂₃×#475	C₂²	1	1	yes	characteristic
48	C₂₃×#474	C₂²	1	1	yes	characteristic
49	C₂₃×#470	—	2	2	no	—
50	C₂₃×#467	—	2	1	no	—
51	C₂₃×#467	—	2	1	no	—
52	C₂₃×#462	—	2	1	no	—
53	C₂₃×#462	C₂²	1	1	yes	characteristic
54	C₂₃×#462	C₂²	1	1	yes	characteristic
55	C₂₃×#441	—	2	2	no	—
56	C₂₃×#431	C₂²	1	1	yes	characteristic
57	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
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×Q₈⋊C₈	C₂²	1	1	yes	characteristic
68	C₂₃×Q₈⋊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×Q₁₆×C₂₃	—	2	2	no	—
72	C₂×Q₁₆×C₂₃	—	2	1	no	—
73	C₂×Q₁₆×C₂₃	—	2	1	no	—
74	C₂×M₁₆×C₂₃	—	2	1	no	—
75	C₂²×C₈×C₂₃	—	2	1	no	—
76	C₂₃×C₄⋊Q₈	—	2	2	no	—
77	C₂₃×C₄⋊Q₈	—	2	1	no	—
78	C₂₃×C₄⋊Q₈	—	2	2	no	—
79	C₂₃×C₄⋊Q₈	—	2	1	no	—
80	C₂₃×C₄⋊Q₈	—	2	1	no	—
81	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
82	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
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	2	no	—
92	C₂₃×#126	—	2	2	no	—
93	C₂₃×#126	—	2	1	no	—
94	C₂₃×#125	—	2	2	no	—
95	C₂₃×#125	—	2	2	no	—
96	C₂₃×#125	—	2	2	no	—
97	C₂₃×#125	—	2	2	no	—
98	C₂₃×#125	—	2	1	no	—
99	C₂₃×#125	—	2	1	no	—
100	C₂₃×#125	—	2	2	no	—
101	C₂₃×#125	—	2	1	no	—
102	C₂₃×#125	—	2	1	no	—
103	C₂₃×#125	—	2	1	no	—
104	C₂₃×#125	D₄	1	1	yes	characteristic
105	C₂₃×#124	—	2	2	no	—
106	C₂₃×#124	—	2	1	no	—
107	C₂₃×#124	D₄	1	1	yes	characteristic
108	C₂₃×#123	—	2	2	no	—
109	C₂₃×#123	—	2	1	no	—
110	C₂₃×C₂²⋊Q₈	—	2	2	no	—
111	C₂₃×C₂²⋊Q₈	—	2	2	no	—
112	C₂₃×C₂²⋊Q₈	—	2	1	no	—
113	C₂₃×C₂²⋊Q₈	—	2	1	no	—
114	C₂₃×C₂²⋊Q₈	—	2	2	no	—
115	C₂₃×C₂²⋊Q₈	—	2	1	no	—
116	C₂₃×C₂²⋊Q₈	—	2	1	no	—
117	C₂₃×C₂²⋊Q₈	—	2	1	no	—
118	C₂₃×C₂²⋊Q₈	—	2	1	no	—
119	C₂₃×C₂²⋊Q₈	D₄	1	1	yes	characteristic
120	C₂₃×C₂²⋊Q₈	D₄	1	1	yes	characteristic
121	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
122	C₂₃×C₂²⋊Q₈	D₄	1	1	yes	characteristic
123	C₂₃×#121	D₄	1	1	yes	characteristic
124	C₄×Q₈×C₂₃	—	2	2	no	—
125	C₄×Q₈×C₂₃	—	2	1	no	—
126	C₄×Q₈×C₂₃	—	2	1	no	—
127	C₄×Q₈×C₂₃	—	2	1	no	—
128	C₄×Q₈×C₂₃	—	2	1	no	—
129	C₄×Q₈×C₂₃	—	2	1	no	—
130	C₄×Q₈×C₂₃	—	2	1	no	—
131	C₄×Q₈×C₂₃	—	2	1	no	—
132	C₄×Q₈×C₂₃	—	2	1	no	—
133	C₄×Q₈×C₂₃	—	2	1	no	—
134	C₄×Q₈×C₂₃	D₄	1	2	yes	—
135	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
136	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
137	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
138	C₄×D₄×C₂₃	—	2	1	no	—
139	C₄×D₄×C₂₃	—	2	1	no	—
140	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
141	C₂₃×#117	—	2	2	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	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
148	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
149	C₂₃×#107	—	2	1	no	—
150	C₂₃×#107	—	2	1	no	—
151	C₂₃×#107	—	2	1	no	—
152	C₂₃×#107	—	2	1	no	—
153	C₂₃×#106	—	2	2	no	—
154	C₂₃×#106	—	2	1	no	—
155	C₂₃×#106	—	2	1	no	—
156	C₂₃×C₄⋊C₈	—	2	1	no	—
157	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
158	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
159	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
160	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
161	C₂₃×#103	—	2	2	no	—
162	C₂₃×#103	—	2	1	no	—
163	C₂₃×#103	—	2	1	no	—
164	C₂₃×#103	—	2	1	no	—
165	C₂₃×#103	—	2	1	no	—
166	C₂₃×#103	—	2	2	no	—
167	C₂₃×#103	—	2	1	no	—
168	C₂₃×#103	—	2	1	no	—
169	C₂₃×#102	—	2	1	no	—
170	C₂₃×#102	—	2	1	no	—
171	C₂₃×C₂²⋊C₈	—	2	1	no	—
172	C₂₃×#97	—	2	1	no	—
173	C₄×C₈×C₂₃	—	2	1	no	—
174	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
175	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
176	C₂×Q₈×C₂₃	—	2	2	no	—
177	C₂×Q₈×C₂₃	—	2	1	no	—
178	C₂×Q₈×C₂₃	—	2	1	no	—
179	C₂×Q₈×C₂₃	—	2	1	no	—
180	C₂×Q₈×C₂₃	—	2	1	no	—
181	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
182	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
183	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
184	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
185	C₂×D₄×C₂₃	—	2	1	no	—
186	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
187	C₂²×C₄×C₂₃	—	2	2	no	—
188	C₂²×C₄×C₂₃	—	2	1	no	—
189	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
190	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
191	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
192	Q₁₆×C₂₃	—	4	4	no	—
193	Q₁₆×C₂₃	—	8	1	no	—
194	Q₁₆×C₂₃	—	8	1	no	—
195	M₁₆×C₂₃	—	4	2	no	—
196	C₂×C₈×C₂₃	—	2	1	no	—
197	C₂×C₈×C₂₃	—	2	1	no	—
198	C₂×C₈×C₂₃	—	2	1	no	—
199	C₂×C₈×C₂₃	—	2	1	no	—
200	C₂×C₈×C₂₃	—	2	1	no	—
201	C₂×C₈×C₂₃	—	2	1	no	—
202	C₂×C₈×C₂₃	—	8	1	no	—
203	C₄⋊C₄×C₂₃	—	2	2	no	—
204	C₄⋊C₄×C₂₃	—	2	2	no	—
205	C₄⋊C₄×C₂₃	—	2	2	no	—
206	C₄⋊C₄×C₂₃	—	2	2	no	—
207	C₄⋊C₄×C₂₃	—	2	2	no	—
208	C₄⋊C₄×C₂₃	—	2	2	no	—
209	C₄⋊C₄×C₂₃	—	2	2	no	—
210	C₄⋊C₄×C₂₃	—	2	2	no	—
211	C₄⋊C₄×C₂₃	—	2	1	no	—
212	C₄⋊C₄×C₂₃	—	2	1	no	—
213	C₄⋊C₄×C₂₃	—	2	1	no	—
214	C₄⋊C₄×C₂₃	—	2	1	no	—
215	C₄⋊C₄×C₂₃	—	2	1	no	—
216	C₄⋊C₄×C₂₃	—	2	1	no	—
217	C₄⋊C₄×C₂₃	—	2	1	no	—
218	C₄⋊C₄×C₂₃	—	2	2	no	—
219	C₄⋊C₄×C₂₃	—	2	1	no	—
220	C₄⋊C₄×C₂₃	—	2	2	no	—
221	C₄⋊C₄×C₂₃	—	2	1	no	—
222	C₄⋊C₄×C₂₃	—	2	1	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	—	4	1	no	—
232	C₄⋊C₄×C₂₃	—	4	1	no	—
233	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
234	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
235	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
236	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
237	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
238	C₂²⋊C₄×C₂₃	—	2	2	no	—
239	C₂²⋊C₄×C₂₃	—	2	2	no	—
240	C₂²⋊C₄×C₂₃	—	2	2	no	—
241	C₂²⋊C₄×C₂₃	—	2	1	no	—
242	C₂²⋊C₄×C₂₃	—	2	2	no	—
243	C₂²⋊C₄×C₂₃	—	2	1	no	—
244	C₂²⋊C₄×C₂₃	—	2	1	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	—	4	1	no	—
250	C₄²×C₂₃	—	2	2	no	—
251	C₄²×C₂₃	—	2	2	no	—
252	C₄²×C₂₃	—	2	2	no	—
253	C₄²×C₂₃	—	2	2	no	—
254	C₄²×C₂₃	—	2	1	no	—
255	C₄²×C₂₃	—	2	1	no	—
256	C₄²×C₂₃	—	2	1	no	—
257	C₄²×C₂₃	—	4	1	no	—
258	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
259	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
260	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
261	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
262	#18745	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
263	C₂³×C₂₃	—	2	1	no	—
264	C₂³×C₂₃	#120	1	1	yes	characteristic
265	Q₈×C₂₃	—	2	1	no	—
266	Q₈×C₂₃	—	2	1	no	—
267	Q₈×C₂₃	—	2	1	no	—
268	Q₈×C₂₃	—	2	1	no	—
269	Q₈×C₂₃	—	2	1	no	—
270	Q₈×C₂₃	—	2	1	no	—
271	Q₈×C₂₃	—	4	2	no	—
272	Q₈×C₂₃	—	4	2	no	—
273	Q₈×C₂₃	—	4	1	no	—
274	Q₈×C₂₃	—	4	1	no	—
275	Q₈×C₂₃	—	4	1	no	—
276	Q₈×C₂₃	—	4	1	no	—
277	Q₈×C₂₃	—	8	1	no	—
278	Q₈×C₂₃	—	8	2	no	—
279	D₄×C₂₃	—	2	1	no	—
280	D₄×C₂₃	—	2	1	no	—
281	D₄×C₂₃	—	8	1	no	—
282	C₂×C₄×C₂₃	—	2	1	no	—
283	C₂×C₄×C₂₃	—	2	1	no	—
284	C₂×C₄×C₂₃	—	2	1	no	—
285	C₂×C₄×C₂₃	—	2	1	no	—
286	C₂×C₄×C₂₃	—	2	1	no	—
287	C₂×C₄×C₂₃	—	2	1	no	—
288	C₂×C₄×C₂₃	—	2	1	no	—
289	C₂×C₄×C₂₃	—	2	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	2	no	—
294	C₂×C₄×C₂₃	—	2	1	no	—
295	C₂×C₄×C₂₃	—	4	1	no	—
296	C₂×C₄×C₂₃	—	4	1	no	—
297	C₂×C₄×C₂₃	—	4	1	no	—
298	C₂×C₄×C₂₃	—	4	1	no	—
299	C₂×C₄×C₂₃	—	4	1	no	—
300	C₂×C₄×C₂₃	—	8	2	no	—
301	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
302	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
303	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
304	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
305	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
306	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
307	C₈×C₂₃	—	2	2	no	—
308	C₈×C₂₃	—	4	1	no	—
309	C₈×C₂₃	—	4	1	no	—
310	C₈×C₂₃	—	4	1	no	—
311	C₈×C₂₃	—	4	1	no	—
312	C₈×C₂₃	—	4	1	no	—
313	#3555	C₂×C₂₃	1	1	yes	characteristic
314	#3249	C₂×C₂₃	1	1	yes	characteristic, complemented
315	#3233	C₂×C₂₃	1	1	yes	characteristic
316	#3185	C₂×C₂₃	1	1	yes	characteristic, complemented
317	#1653	C₂×C₂₃	1	1	yes	characteristic, complemented
318	#1652	C₂×C₂₃	1	1	yes	characteristic, complemented
319	#1645	C₂×C₂₃	1	1	yes	characteristic, complemented
320	#1644	C₂×C₂₃	1	1	yes	characteristic, complemented
321	#1638	C₂×C₂₃	1	1	yes	characteristic, complemented
322	#1636	C₂×C₂₃	1	1	yes	characteristic, complemented
323	#1633	C₂×C₂₃	1	1	yes	characteristic, complemented
324	#1632	C₂×C₂₃	1	1	yes	characteristic, complemented
325	#1542	C₂×C₂₃	1	1	yes	characteristic, complemented
326	#1485	C₂×C₂₃	1	1	yes	characteristic, complemented
327	#1483	C₂×C₂₃	1	1	yes	characteristic
328	C₂²×C₂₃	—	2	1	no	—
329	C₂²×C₂₃	—	2	1	no	—
330	C₂²×C₂₃	—	2	1	no	—
331	C₂²×C₂₃	—	4	1	no	—
332	C₂²×C₂₃	—	4	1	no	—
333	C₂²×C₂₃	—	4	1	no	—
334	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
335	C₄×C₂₃	—	2	1	no	—
336	C₄×C₂₃	—	2	1	no	—
337	C₄×C₂₃	—	2	1	no	—
338	C₄×C₂₃	—	2	1	no	—
339	C₄×C₂₃	—	4	1	no	—
340	C₄×C₂₃	—	4	2	no	—
341	C₄×C₂₃	—	4	1	no	—
342	C₄×C₂₃	—	4	1	no	—
343	C₄×C₂₃	—	4	1	no	—
344	C₄×C₂₃	—	4	1	no	—
345	C₄×C₂₃	—	4	2	no	—
346	C₄×C₂₃	—	4	1	no	—
347	C₄×C₂₃	—	4	1	no	—
348	C₄×C₂₃	—	8	1	no	—
349	C₄×C₂₃	—	8	1	no	—
350	C₄×C₂₃	#577	1	1	yes	characteristic
351	C₄×C₂₃	#578	1	1	yes	characteristic
352	C₄×C₂₃	#578	1	1	yes	characteristic
353	C₄×C₂₃	#578	1	1	yes	characteristic
354	#563	—	2	2	no	—
355	Q₈²	C₂²×C₂₃	1	1	yes	characteristic
356	#557	—	2	1	no	—
357	#557	C₂²×C₂₃	1	1	yes	characteristic
358	#556	—	2	1	no	—
359	#556	—	2	1	no	—
360	#556	C₂²×C₂₃	1	1	yes	characteristic
361	#554	C₂²×C₂₃	1	1	yes	characteristic
362	#552	—	2	1	no	—
363	#544	—	2	1	no	—
364	#542	—	2	1	no	—
365	#541	—	2	1	no	—
366	#541	—	2	1	no	—
367	#541	—	2	2	no	—
368	#533	C₂²×C₂₃	1	1	yes	characteristic
369	#533	C₂²×C₂₃	1	1	yes	characteristic
370	#533	C₂²×C₂₃	1	1	yes	characteristic
371	#529	C₂²×C₂₃	1	1	yes	characteristic
372	#499	—	2	1	no	—
373	#498	C₂²×C₂₃	1	1	yes	characteristic
374	#494	C₂²×C₂₃	1	1	yes	characteristic
375	#488	—	2	1	no	—
376	#487	C₂²×C₂₃	1	1	yes	characteristic
377	#487	C₂²×C₂₃	1	1	yes	characteristic
378	#485	—	2	1	no	—
379	#482	—	2	1	no	—
380	#477	—	2	1	no	—
381	#477	C₂²×C₂₃	1	1	yes	characteristic
382	#477	C₂²×C₂₃	1	1	yes	characteristic
383	#477	C₂²×C₂₃	1	1	yes	characteristic
384	#476	C₂²×C₂₃	1	1	yes	characteristic
385	#475	C₂²×C₂₃	1	1	yes	characteristic
386	#474	C₂²×C₂₃	1	1	yes	characteristic
387	#470	—	2	2	no	—
388	#467	—	2	1	no	—
389	#467	—	2	1	no	—
390	#462	—	2	1	no	—
391	#462	C₂²×C₂₃	1	1	yes	characteristic
392	#462	C₂²×C₂₃	1	1	yes	characteristic
393	#441	—	2	2	no	—
394	#431	C₂²×C₂₃	1	1	yes	characteristic
395	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
396	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
397	#333	C₂²×C₂₃	1	1	yes	characteristic
398	#333	C₂²×C₂₃	1	1	yes	characteristic
399	#332	C₂²×C₂₃	1	1	yes	characteristic
400	#332	C₂²×C₂₃	1	1	yes	characteristic
401	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
402	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
403	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
404	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
405	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
406	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
407	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
408	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
409	C₂×C₂₃	—	4	1	no	—
410	C₂×C₂₃	—	8	1	no	—
411	C₂×C₂₃	#3011	1	1	yes	characteristic
412	C₂×C₂₃	#3013	1	1	yes	characteristic
413	C₂×C₂₃	#3018	1	1	yes	characteristic
414	C₂×Q₁₆	—	2	2	no	—
415	C₂×Q₁₆	—	2	1	no	—
416	C₂×Q₁₆	—	2	1	no	—
417	C₂×M₁₆	—	2	1	no	—
418	C₂²×C₈	—	2	1	no	—
419	C₄⋊Q₈	—	2	2	no	—
420	C₄⋊Q₈	—	2	1	no	—
421	C₄⋊Q₈	—	2	2	no	—
422	C₄⋊Q₈	—	2	1	no	—
423	C₄⋊Q₈	—	2	1	no	—
424	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
425	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
426	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
427	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
428	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
429	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
430	#126	—	2	2	no	—
431	#126	—	2	2	no	—
432	#126	—	2	2	no	—
433	#126	—	2	2	no	—
434	#126	—	2	2	no	—
435	#126	—	2	2	no	—
436	#126	—	2	1	no	—
437	#125	—	2	2	no	—
438	#125	—	2	2	no	—
439	#125	—	2	2	no	—
440	#125	—	2	2	no	—
441	#125	—	2	1	no	—
442	#125	—	2	2	no	—
443	#125	—	2	1	no	—
444	#125	—	2	1	no	—
445	#125	—	2	1	no	—
446	#125	—	2	1	no	—
447	#125	D₄×C₂₃	1	1	yes	characteristic
448	#124	—	2	2	no	—
449	#124	—	2	1	no	—
450	#124	D₄×C₂₃	1	1	yes	characteristic
451	#123	—	2	2	no	—
452	#123	—	2	1	no	—
453	C₂²⋊Q₈	—	2	2	no	—
454	C₂²⋊Q₈	—	2	2	no	—
455	C₂²⋊Q₈	—	2	1	no	—
456	C₂²⋊Q₈	—	2	1	no	—
457	C₂²⋊Q₈	—	2	2	no	—
458	C₂²⋊Q₈	—	2	1	no	—
459	C₂²⋊Q₈	—	2	1	no	—
460	C₂²⋊Q₈	—	2	1	no	—
461	C₂²⋊Q₈	—	2	1	no	—
462	C₂²⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
463	C₂²⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
464	C₂²⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
465	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
466	#121	D₄×C₂₃	1	1	yes	characteristic
467	C₄×Q₈	—	2	2	no	—
468	C₄×Q₈	—	2	1	no	—
469	C₄×Q₈	—	2	1	no	—
470	C₄×Q₈	—	2	1	no	—
471	C₄×Q₈	—	2	1	no	—
472	C₄×Q₈	—	2	1	no	—
473	C₄×Q₈	—	2	1	no	—
474	C₄×Q₈	—	2	1	no	—
475	C₄×Q₈	—	2	1	no	—
476	C₄×Q₈	—	2	1	no	—
477	C₄×Q₈	D₄×C₂₃	1	2	yes	—
478	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
479	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
480	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
481	C₄×D₄	—	2	1	no	—
482	C₄×D₄	—	2	1	no	—
483	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
484	#117	—	2	2	no	—
485	#117	—	2	1	no	—
486	#117	—	2	1	no	—
487	#117	—	2	1	no	—
488	#117	—	2	1	no	—
489	#117	—	2	1	no	—
490	C₂×C₄⋊C₄	—	2	1	no	—
491	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
492	#107	—	2	1	no	—
493	#107	—	2	1	no	—
494	#107	—	2	1	no	—
495	#107	—	2	1	no	—
496	#106	—	2	2	no	—
497	#106	—	2	1	no	—
498	#106	—	2	1	no	—
499	C₄⋊C₈	—	2	1	no	—
500	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
501	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
502	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
503	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
504	#103	—	2	2	no	—
505	#103	—	2	1	no	—
506	#103	—	2	1	no	—
507	#103	—	2	1	no	—
508	#103	—	2	1	no	—
509	#103	—	2	2	no	—
510	#103	—	2	1	no	—
511	#103	—	2	1	no	—
512	#102	—	2	1	no	—
513	#102	—	2	1	no	—
514	C₂²⋊C₈	—	2	1	no	—
515	#97	—	2	1	no	—
516	C₄×C₈	—	2	1	no	—
517	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
518	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
519	C₂₃	#18745	1	1	yes	characteristic, direct product factor, Sylow
520	C₂×Q₈	—	2	2	no	—
521	C₂×Q₈	—	2	1	no	—
522	C₂×Q₈	—	2	1	no	—
523	C₂×Q₈	—	2	1	no	—
524	C₂×Q₈	—	2	1	no	—
525	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
526	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
527	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
528	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
529	C₂×D₄	—	2	1	no	—
530	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
531	C₂²×C₄	—	2	2	no	—
532	C₂²×C₄	—	2	1	no	—
533	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
534	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
535	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
536	Q₁₆	—	4	4	no	—
537	Q₁₆	—	8	1	no	—
538	Q₁₆	—	8	1	no	—
539	M₁₆	—	4	2	no	—
540	C₂×C₈	—	2	1	no	—
541	C₂×C₈	—	2	1	no	—
542	C₂×C₈	—	2	1	no	—
543	C₂×C₈	—	2	1	no	—
544	C₂×C₈	—	2	1	no	—
545	C₂×C₈	—	2	1	no	—
546	C₂×C₈	—	8	1	no	—
547	C₄⋊C₄	—	2	2	no	—
548	C₄⋊C₄	—	2	2	no	—
549	C₄⋊C₄	—	2	2	no	—
550	C₄⋊C₄	—	2	2	no	—
551	C₄⋊C₄	—	2	2	no	—
552	C₄⋊C₄	—	2	2	no	—
553	C₄⋊C₄	—	2	2	no	—
554	C₄⋊C₄	—	2	2	no	—
555	C₄⋊C₄	—	2	1	no	—
556	C₄⋊C₄	—	2	1	no	—
557	C₄⋊C₄	—	2	1	no	—
558	C₄⋊C₄	—	2	1	no	—
559	C₄⋊C₄	—	2	1	no	—
560	C₄⋊C₄	—	2	1	no	—
561	C₄⋊C₄	—	2	1	no	—
562	C₄⋊C₄	—	2	2	no	—
563	C₄⋊C₄	—	2	1	no	—
564	C₄⋊C₄	—	2	2	no	—
565	C₄⋊C₄	—	2	1	no	—
566	C₄⋊C₄	—	2	1	no	—
567	C₄⋊C₄	—	2	1	no	—
568	C₄⋊C₄	—	2	1	no	—
569	C₄⋊C₄	—	2	1	no	—
570	C₄⋊C₄	—	2	1	no	—
571	C₄⋊C₄	—	2	1	no	—
572	C₄⋊C₄	—	2	1	no	—
573	C₄⋊C₄	—	2	1	no	—
574	C₄⋊C₄	—	2	1	no	—
575	C₄⋊C₄	—	4	1	no	—
576	C₄⋊C₄	—	4	1	no	—
577	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
578	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
579	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
580	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
581	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
582	C₂²⋊C₄	—	2	2	no	—
583	C₂²⋊C₄	—	2	2	no	—
584	C₂²⋊C₄	—	2	2	no	—
585	C₂²⋊C₄	—	2	1	no	—
586	C₂²⋊C₄	—	2	2	no	—
587	C₂²⋊C₄	—	2	1	no	—
588	C₂²⋊C₄	—	2	1	no	—
589	C₂²⋊C₄	—	2	1	no	—
590	C₂²⋊C₄	—	2	1	no	—
591	C₂²⋊C₄	—	2	1	no	—
592	C₂²⋊C₄	—	2	1	no	—
593	C₂²⋊C₄	—	4	1	no	—
594	C₄²	—	2	2	no	—
595	C₄²	—	2	2	no	—
596	C₄²	—	2	2	no	—
597	C₄²	—	2	2	no	—
598	C₄²	—	2	1	no	—
599	C₄²	—	2	1	no	—
600	C₄²	—	2	1	no	—
601	C₄²	—	4	1	no	—
602	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
603	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
604	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
605	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
606	C₂³	—	2	1	no	—
607	C₂³	C₂₃×#120	1	1	yes	characteristic
608	Q₈	—	2	1	no	—
609	Q₈	—	2	1	no	—
610	Q₈	—	2	1	no	—
611	Q₈	—	2	1	no	—
612	Q₈	—	2	1	no	—
613	Q₈	—	2	1	no	—
614	Q₈	—	4	2	no	—
615	Q₈	—	4	2	no	—
616	Q₈	—	4	1	no	—
617	Q₈	—	4	1	no	—
618	Q₈	—	4	1	no	—
619	Q₈	—	4	1	no	—
620	Q₈	—	8	1	no	—
621	Q₈	—	8	2	no	—
622	D₄	—	2	1	no	—
623	D₄	—	2	1	no	—
624	D₄	—	8	1	no	—
625	C₂×C₄	—	2	1	no	—
626	C₂×C₄	—	2	1	no	—
627	C₂×C₄	—	2	1	no	—
628	C₂×C₄	—	2	1	no	—
629	C₂×C₄	—	2	1	no	—
630	C₂×C₄	—	2	1	no	—
631	C₂×C₄	—	2	1	no	—
632	C₂×C₄	—	2	2	no	—
633	C₂×C₄	—	2	1	no	—
634	C₂×C₄	—	2	1	no	—
635	C₂×C₄	—	2	1	no	—
636	C₂×C₄	—	2	2	no	—
637	C₂×C₄	—	2	1	no	—
638	C₂×C₄	—	4	1	no	—
639	C₂×C₄	—	4	1	no	—
640	C₂×C₄	—	4	1	no	—
641	C₂×C₄	—	4	1	no	—
642	C₂×C₄	—	4	1	no	—
643	C₂×C₄	—	8	2	no	—
644	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
645	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
646	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
647	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
648	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
649	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
650	C₈	—	2	2	no	—
651	C₈	—	4	1	no	—
652	C₈	—	4	1	no	—
653	C₈	—	4	1	no	—
654	C₈	—	4	1	no	—
655	C₈	—	4	1	no	—
656	C₂²	—	2	1	no	—
657	C₂²	—	2	1	no	—
658	C₂²	—	2	1	no	—
659	C₂²	—	4	1	no	—
660	C₂²	—	4	1	no	—
661	C₂²	—	4	1	no	—
662	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
663	C₄	—	2	1	no	—
664	C₄	—	2	1	no	—
665	C₄	—	2	1	no	—
666	C₄	—	2	1	no	—
667	C₄	—	4	1	no	—
668	C₄	—	4	2	no	—
669	C₄	—	4	1	no	—
670	C₄	—	4	1	no	—
671	C₄	—	4	1	no	—
672	C₄	—	4	1	no	—
673	C₄	—	4	2	no	—
674	C₄	—	4	1	no	—
675	C₄	—	4	1	no	—
676	C₄	—	8	1	no	—
677	C₄	—	8	1	no	—
678	C₄	C₂₃×#577	1	1	yes	characteristic
679	C₄	C₂₃×#578	1	1	yes	characteristic
680	C₄	C₂₃×#578	1	1	yes	characteristic
681	C₄	C₂₃×#578	1	1	yes	characteristic
682	C₂	—	4	1	no	—
683	C₂	—	8	1	no	—
684	C₂	C₂₃×#3011	1	1	yes	characteristic
685	C₂	C₂₃×#3013	1	1	yes	characteristic
686	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×#121	D₄
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×#3249	C₂	semidirect
4. C₂₃×#3185	C₂	semidirect
5. C₂₃×#1653	C₂	semidirect
6. C₂₃×#1652	C₂	semidirect
7. C₂₃×#1645	C₂	semidirect
8. C₂₃×#1644	C₂	semidirect
9. C₂₃×#1638	C₂	semidirect
10. C₂₃×#1636	C₂	semidirect
11. C₂₃×#1633	C₂	semidirect
12. C₂₃×#1632	C₂	semidirect
13. C₂₃×#1542	C₂	semidirect
14. C₂₃×#1485	C₂	semidirect
262. #18745	C₂₃	direct (N × H)
314. #3249	C₂×C₂₃	semidirect
316. #3185	C₂×C₂₃	semidirect
317. #1653	C₂×C₂₃	semidirect
318. #1652	C₂×C₂₃	semidirect
319. #1645	C₂×C₂₃	semidirect
320. #1644	C₂×C₂₃	semidirect
321. #1638	C₂×C₂₃	semidirect
322. #1636	C₂×C₂₃	semidirect
323. #1633	C₂×C₂₃	semidirect
324. #1632	C₂×C₂₃	semidirect
325. #1542	C₂×C₂₃	semidirect
326. #1485	C₂×C₂₃	semidirect
519. C₂₃	#18745	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×#3555	2
C₂₃×#3249	2
C₂₃×#3233	2
C₂₃×#3185	2
C₂₃×#1653	2
C₂₃×#1652	2
C₂₃×#1645	2
C₂₃×#1644	2
C₂₃×#1638	2
C₂₃×#1636	2
C₂₃×#1633	2
C₂₃×#1632	2
C₂₃×#1542	2
C₂₃×#1485	2
C₂₃×#1483	2
#18745	23

Subgroups — C23×#18745 #19166371

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18745 #19166371