Subgroups — C₂₃×#18812 #19166438

Order 5888.

968 conjugacy classes of subgroups.

The subgroup lattice has 762 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3522	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3251	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3207	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3037	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1645	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1644	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1641	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1640	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1623	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1622	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1621	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1620	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1560	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1495	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1487	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×#137	—	2	1	no	—
17	C₂₃×#559	—	2	1	no	—
18	C₂₃×#559	C₂²	1	1	yes	characteristic
19	Q₈²×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#554	—	2	1	no	—
21	C₂₃×#554	C₂²	1	1	yes	characteristic
22	C₂₃×#550	C₂²	1	1	yes	characteristic
23	C₂₃×#548	—	2	1	no	—
24	C₂₃×#547	—	2	2	no	—
25	C₂₃×#547	—	2	1	no	—
26	C₂₃×#542	—	2	2	no	—
27	C₂₃×#542	—	2	1	no	—
28	C₂₃×#532	—	2	1	no	—
29	C₂₃×#532	C₂²	1	1	yes	characteristic
30	C₂₃×#532	C₂²	1	1	yes	characteristic
31	C₂₃×#529	—	2	1	no	—
32	C₂₃×#529	C₂²	1	1	yes	characteristic
33	C₂₃×#520	—	2	1	no	—
34	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
35	C₂₃×#493	C₂²	1	1	yes	characteristic
36	C₂₃×#492	C₂²	1	1	yes	characteristic
37	C₂₃×#487	C₂²	1	1	yes	characteristic
38	C₂₃×#486	C₂²	1	1	yes	characteristic
39	C₂₃×#478	—	2	1	no	—
40	C₂₃×#475	C₂²	1	1	yes	characteristic
41	C₂₃×#474	C₂²	1	1	yes	characteristic
42	C₂₃×#469	—	2	1	no	—
43	C₂₃×#468	—	2	1	no	—
44	C₂₃×#464	—	2	1	no	—
45	C₂₃×#464	—	2	1	no	—
46	C₂₃×#463	—	2	2	no	—
47	C₂₃×#463	—	2	1	no	—
48	C₂₃×#462	C₂²	1	1	yes	characteristic
49	C₂₃×#462	C₂²	1	1	yes	characteristic
50	C₂₃×#460	C₂²	1	1	yes	characteristic
51	C₂₃×#460	C₂²	1	1	yes	characteristic
52	C₂₃×#460	C₂²	1	1	yes	characteristic
53	C₂₃×#459	C₂²	1	1	yes	characteristic
54	C₂₃×#449	—	2	2	no	—
55	C₂₃×#431	C₂²	1	1	yes	characteristic
56	C₂₃×#423	C₂²	1	1	yes	characteristic
57	C₂₃×#423	C₂²	1	1	yes	characteristic
58	C₂₃×#417	—	2	1	no	—
59	C₂₃×#333	C₂²	1	1	yes	characteristic
60	C₂₃×#333	C₂²	1	1	yes	characteristic
61	C₂₃×#331	C₂²	1	1	yes	characteristic
62	C₂₃×#331	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×#41×C₂₃	—	2	2	no	—
72	C₂×#41×C₂₃	—	2	1	no	—
73	C₂×#41×C₂₃	—	2	1	no	—
74	C₂×#41×C₂₃	D₄	1	2	yes	—
75	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
76	C₂₃×#137	—	4	4	no	—
77	C₂×Q₁₆×C₂₃	—	2	1	no	—
78	C₂×Q₁₆×C₂₃	—	2	1	no	—
79	C₂×SD₁₆×C₂₃	—	2	1	no	—
80	C₂×SD₁₆×C₂₃	—	2	1	no	—
81	C₂×SD₁₆×C₂₃	—	2	2	no	—
82	C₂×SD₁₆×C₂₃	—	2	1	no	—
83	C₂×D₈×C₂₃	—	2	2	no	—
84	C₂×D₈×C₂₃	—	2	1	no	—
85	C₂×M₁₆×C₂₃	—	2	1	no	—
86	C₂×M₁₆×C₂₃	—	2	1	no	—
87	C₂₃×C₄⋊Q₈	—	2	2	no	—
88	C₂₃×C₄⋊Q₈	—	2	1	no	—
89	C₂₃×C₄⋊Q₈	—	2	1	no	—
90	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
91	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
92	C₂₃×#127	—	2	1	no	—
93	C₂₃×#127	C₂³	1	1	yes	characteristic
94	C₂₃×#127	C₂³	1	1	yes	characteristic
95	C₂₃×#126	—	2	2	no	—
96	C₂₃×#126	—	2	1	no	—
97	C₂₃×#125	—	2	1	no	—
98	C₂₃×#125	D₄	1	1	yes	characteristic
99	C₂₃×#124	—	2	2	no	—
100	C₂₃×#124	—	2	2	no	—
101	C₂₃×#124	—	2	1	no	—
102	C₂₃×#124	—	2	1	no	—
103	C₂₃×#124	—	2	1	no	—
104	C₂₃×#124	D₄	1	1	yes	characteristic
105	C₂₃×#124	D₄	1	2	yes	—
106	C₂₃×#123	—	2	2	no	—
107	C₂₃×#123	—	2	2	no	—
108	C₂₃×#123	—	2	2	no	—
109	C₂₃×#123	—	2	2	no	—
110	C₂₃×#123	—	2	1	no	—
111	C₂₃×C₂²⋊Q₈	—	2	2	no	—
112	C₂₃×C₂²⋊Q₈	—	2	2	no	—
113	C₂₃×C₂²⋊Q₈	—	2	2	no	—
114	C₂₃×C₂²⋊Q₈	—	2	2	no	—
115	C₂₃×C₂²⋊Q₈	—	2	1	no	—
116	C₂₃×C₂²⋊Q₈	D₄	1	1	yes	characteristic
117	C₂₃×#121	—	2	2	no	—
118	C₂₃×#121	—	2	2	no	—
119	C₂₃×#121	—	2	1	no	—
120	C₂₃×#121	—	2	1	no	—
121	C₂₃×#121	—	2	1	no	—
122	C₂₃×#121	—	2	1	no	—
123	C₂₃×#121	—	2	1	no	—
124	C₂₃×#121	D₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
130	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
131	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
132	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
133	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
134	C₄×D₄×C₂₃	—	2	2	no	—
135	C₄×D₄×C₂₃	—	2	2	no	—
136	C₄×D₄×C₂₃	—	2	2	no	—
137	C₄×D₄×C₂₃	—	2	1	no	—
138	C₄×D₄×C₂₃	—	2	1	no	—
139	C₄×D₄×C₂₃	—	2	1	no	—
140	C₄×D₄×C₂₃	—	2	1	no	—
141	C₄×D₄×C₂₃	—	2	1	no	—
142	C₄×D₄×C₂₃	—	2	1	no	—
143	C₄×D₄×C₂₃	—	2	1	no	—
144	C₄×D₄×C₂₃	—	2	1	no	—
145	C₄×D₄×C₂₃	—	2	1	no	—
146	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
147	C₂₃×#117	—	2	2	no	—
148	C₂₃×#117	—	2	1	no	—
149	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
150	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
151	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
152	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
153	C₂×C₄²×C₂₃	—	2	1	no	—
154	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
155	C₂₃×#107	—	2	1	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	—	2	1	no	—
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×C₄⋊C₈	C₂³	1	1	yes	characteristic
164	C₂₃×#103	—	2	2	no	—
165	C₂₃×#103	—	2	1	no	—
166	C₂₃×#103	—	2	1	no	—
167	C₂₃×#103	—	2	1	no	—
168	C₂₃×#102	—	2	2	no	—
169	C₂₃×#102	—	2	1	no	—
170	C₂₃×#102	—	2	1	no	—
171	C₂₃×#102	—	2	1	no	—
172	C₂₃×#102	—	2	1	no	—
173	C₂₃×#102	—	2	1	no	—
174	C₂₃×C₂²⋊C₈	—	2	1	no	—
175	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
176	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
177	#41×C₂₃	—	4	2	no	—
178	#41×C₂₃	—	4	2	no	—
179	#41×C₂₃	—	4	2	no	—
180	#41×C₂₃	—	4	2	no	—
181	#41×C₂₃	—	8	4	no	—
182	#41×C₂₃	—	8	2	no	—
183	C₂×Q₈×C₂₃	—	2	2	no	—
184	C₂×Q₈×C₂₃	—	2	1	no	—
185	C₂×Q₈×C₂₃	—	2	2	no	—
186	C₂×Q₈×C₂₃	—	2	1	no	—
187	C₂×Q₈×C₂₃	—	4	1	no	—
188	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
189	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
190	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
191	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
192	C₂×D₄×C₂₃	—	2	2	no	—
193	C₂×D₄×C₂₃	—	2	2	no	—
194	C₂×D₄×C₂₃	—	2	1	no	—
195	C₂×D₄×C₂₃	—	2	2	no	—
196	C₂×D₄×C₂₃	—	2	1	no	—
197	C₂×D₄×C₂₃	—	2	1	no	—
198	C₂×D₄×C₂₃	—	2	1	no	—
199	C₂×D₄×C₂₃	—	2	1	no	—
200	C₂×D₄×C₂₃	—	2	1	no	—
201	C₂×D₄×C₂₃	—	4	1	no	—
202	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
203	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
204	C₂²×C₄×C₂₃	—	2	2	no	—
205	C₂²×C₄×C₂₃	—	2	1	no	—
206	C₂²×C₄×C₂₃	—	2	1	no	—
207	C₂²×C₄×C₂₃	—	2	2	no	—
208	C₂²×C₄×C₂₃	—	2	2	no	—
209	C₂²×C₄×C₂₃	—	2	1	no	—
210	C₂²×C₄×C₂₃	—	2	2	no	—
211	C₂²×C₄×C₂₃	—	4	1	no	—
212	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
213	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
214	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
215	Q₁₆×C₂₃	—	4	2	no	—
216	Q₁₆×C₂₃	—	4	2	no	—
217	SD₁₆×C₂₃	—	4	2	no	—
218	SD₁₆×C₂₃	—	4	2	no	—
219	SD₁₆×C₂₃	—	8	1	no	—
220	SD₁₆×C₂₃	—	8	2	no	—
221	D₈×C₂₃	—	8	1	no	—
222	D₈×C₂₃	—	8	2	no	—
223	M₁₆×C₂₃	—	4	2	no	—
224	M₁₆×C₂₃	—	8	1	no	—
225	C₂×C₈×C₂₃	—	2	1	no	—
226	C₂×C₈×C₂₃	—	2	1	no	—
227	C₂×C₈×C₂₃	—	2	1	no	—
228	C₂×C₈×C₂₃	—	2	1	no	—
229	C₂×C₈×C₂₃	—	2	1	no	—
230	C₂×C₈×C₂₃	—	2	1	no	—
231	C₄⋊C₄×C₂₃	—	2	2	no	—
232	C₄⋊C₄×C₂₃	—	2	2	no	—
233	C₄⋊C₄×C₂₃	—	2	2	no	—
234	C₄⋊C₄×C₂₃	—	2	2	no	—
235	C₄⋊C₄×C₂₃	—	2	1	no	—
236	C₄⋊C₄×C₂₃	—	2	1	no	—
237	C₄⋊C₄×C₂₃	—	2	1	no	—
238	C₄⋊C₄×C₂₃	—	2	1	no	—
239	C₄⋊C₄×C₂₃	—	2	1	no	—
240	C₄⋊C₄×C₂₃	—	2	1	no	—
241	C₄⋊C₄×C₂₃	—	2	1	no	—
242	C₄⋊C₄×C₂₃	—	2	1	no	—
243	C₄⋊C₄×C₂₃	—	2	1	no	—
244	C₄⋊C₄×C₂₃	—	2	1	no	—
245	C₄⋊C₄×C₂₃	—	2	1	no	—
246	C₄⋊C₄×C₂₃	—	2	1	no	—
247	C₄⋊C₄×C₂₃	—	2	1	no	—
248	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
249	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
250	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
251	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
252	C₂²⋊C₄×C₂₃	—	2	2	no	—
253	C₂²⋊C₄×C₂₃	—	2	2	no	—
254	C₂²⋊C₄×C₂₃	—	2	2	no	—
255	C₂²⋊C₄×C₂₃	—	2	2	no	—
256	C₂²⋊C₄×C₂₃	—	2	2	no	—
257	C₂²⋊C₄×C₂₃	—	2	1	no	—
258	C₂²⋊C₄×C₂₃	—	2	2	no	—
259	C₂²⋊C₄×C₂₃	—	2	1	no	—
260	C₂²⋊C₄×C₂₃	—	2	1	no	—
261	C₂²⋊C₄×C₂₃	—	2	2	no	—
262	C₂²⋊C₄×C₂₃	—	2	1	no	—
263	C₂²⋊C₄×C₂₃	—	2	1	no	—
264	C₂²⋊C₄×C₂₃	—	4	1	no	—
265	C₂²⋊C₄×C₂₃	—	4	1	no	—
266	C₄²×C₂₃	—	2	2	no	—
267	C₄²×C₂₃	—	2	1	no	—
268	C₄²×C₂₃	—	2	1	no	—
269	C₄²×C₂₃	—	2	1	no	—
270	C₄²×C₂₃	—	2	1	no	—
271	C₄²×C₂₃	—	2	1	no	—
272	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
273	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
274	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
275	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
276	#18812	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
277	C₂³×C₂₃	—	2	1	no	—
278	C₂³×C₂₃	—	2	2	no	—
279	C₂³×C₂₃	—	4	1	no	—
280	C₂³×C₂₃	#120	1	1	yes	characteristic
281	Q₈×C₂₃	—	2	1	no	—
282	Q₈×C₂₃	—	2	1	no	—
283	Q₈×C₂₃	—	2	1	no	—
284	Q₈×C₂₃	—	2	1	no	—
285	Q₈×C₂₃	—	2	1	no	—
286	Q₈×C₂₃	—	4	2	no	—
287	Q₈×C₂₃	—	4	2	no	—
288	Q₈×C₂₃	—	4	1	no	—
289	Q₈×C₂₃	—	4	1	no	—
290	Q₈×C₂₃	—	8	2	no	—
291	Q₈×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
292	D₄×C₂₃	—	2	1	no	—
293	D₄×C₂₃	—	2	1	no	—
294	D₄×C₂₃	—	4	2	no	—
295	D₄×C₂₃	—	4	4	no	—
296	D₄×C₂₃	—	4	2	no	—
297	D₄×C₂₃	—	4	4	no	—
298	D₄×C₂₃	—	4	2	no	—
299	D₄×C₂₃	—	4	1	no	—
300	D₄×C₂₃	—	4	1	no	—
301	D₄×C₂₃	—	4	1	no	—
302	D₄×C₂₃	—	4	1	no	—
303	D₄×C₂₃	—	4	1	no	—
304	D₄×C₂₃	—	4	1	no	—
305	D₄×C₂₃	—	4	1	no	—
306	D₄×C₂₃	—	4	1	no	—
307	D₄×C₂₃	—	8	2	no	—
308	D₄×C₂₃	—	8	2	no	—
309	C₂×C₄×C₂₃	—	2	1	no	—
310	C₂×C₄×C₂₃	—	2	1	no	—
311	C₂×C₄×C₂₃	—	2	1	no	—
312	C₂×C₄×C₂₃	—	2	1	no	—
313	C₂×C₄×C₂₃	—	2	1	no	—
314	C₂×C₄×C₂₃	—	2	1	no	—
315	C₂×C₄×C₂₃	—	2	1	no	—
316	C₂×C₄×C₂₃	—	2	1	no	—
317	C₂×C₄×C₂₃	—	2	1	no	—
318	C₂×C₄×C₂₃	—	2	2	no	—
319	C₂×C₄×C₂₃	—	2	1	no	—
320	C₂×C₄×C₂₃	—	2	1	no	—
321	C₂×C₄×C₂₃	—	4	1	no	—
322	C₂×C₄×C₂₃	—	4	2	no	—
323	C₂×C₄×C₂₃	—	4	4	no	—
324	C₂×C₄×C₂₃	—	4	2	no	—
325	C₂×C₄×C₂₃	—	4	1	no	—
326	C₂×C₄×C₂₃	—	4	1	no	—
327	C₂×C₄×C₂₃	—	4	1	no	—
328	C₂×C₄×C₂₃	—	8	2	no	—
329	C₂×C₄×C₂₃	—	8	2	no	—
330	C₂×C₄×C₂₃	—	8	2	no	—
331	C₂×C₄×C₂₃	—	8	2	no	—
332	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
333	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
334	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
335	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
336	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
337	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
338	C₈×C₂₃	—	4	1	no	—
339	C₈×C₂₃	—	4	1	no	—
340	C₈×C₂₃	—	4	1	no	—
341	C₈×C₂₃	—	4	1	no	—
342	C₈×C₂₃	—	4	1	no	—
343	C₈×C₂₃	—	4	1	no	—
344	#3522	C₂×C₂₃	1	1	yes	characteristic, complemented
345	#3251	C₂×C₂₃	1	1	yes	characteristic, complemented
346	#3207	C₂×C₂₃	1	1	yes	characteristic, complemented
347	#3037	C₂×C₂₃	1	1	yes	characteristic, complemented
348	#1645	C₂×C₂₃	1	1	yes	characteristic, complemented
349	#1644	C₂×C₂₃	1	1	yes	characteristic, complemented
350	#1641	C₂×C₂₃	1	1	yes	characteristic, complemented
351	#1640	C₂×C₂₃	1	1	yes	characteristic, complemented
352	#1623	C₂×C₂₃	1	1	yes	characteristic, complemented
353	#1622	C₂×C₂₃	1	1	yes	characteristic, complemented
354	#1621	C₂×C₂₃	1	1	yes	characteristic, complemented
355	#1620	C₂×C₂₃	1	1	yes	characteristic, complemented
356	#1560	C₂×C₂₃	1	1	yes	characteristic, complemented
357	#1495	C₂×C₂₃	1	1	yes	characteristic, complemented
358	#1487	C₂×C₂₃	1	1	yes	characteristic, complemented
359	C₂²×C₂₃	—	2	1	no	—
360	C₂²×C₂₃	—	2	1	no	—
361	C₂²×C₂₃	—	2	1	no	—
362	C₂²×C₂₃	—	4	2	no	—
363	C₂²×C₂₃	—	4	1	no	—
364	C₂²×C₂₃	—	4	1	no	—
365	C₂²×C₂₃	—	4	2	no	—
366	C₂²×C₂₃	—	4	2	no	—
367	C₂²×C₂₃	—	4	1	no	—
368	C₂²×C₂₃	—	8	1	no	—
369	C₂²×C₂₃	—	8	1	no	—
370	C₂²×C₂₃	—	8	1	no	—
371	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
372	C₄×C₂₃	—	2	1	no	—
373	C₄×C₂₃	—	2	2	no	—
374	C₄×C₂₃	—	2	1	no	—
375	C₄×C₂₃	—	2	1	no	—
376	C₄×C₂₃	—	2	1	no	—
377	C₄×C₂₃	—	4	1	no	—
378	C₄×C₂₃	—	4	1	no	—
379	C₄×C₂₃	—	4	1	no	—
380	C₄×C₂₃	—	4	1	no	—
381	C₄×C₂₃	—	4	1	no	—
382	C₄×C₂₃	—	4	2	no	—
383	C₄×C₂₃	—	4	1	no	—
384	C₄×C₂₃	—	8	1	no	—
385	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
386	C₄×C₂₃	#575	1	1	yes	characteristic
387	C₄×C₂₃	#577	1	1	yes	characteristic
388	C₄×C₂₃	#578	1	1	yes	characteristic
389	C₂×#137	—	2	1	no	—
390	#559	—	2	1	no	—
391	#559	C₂²×C₂₃	1	1	yes	characteristic
392	Q₈²	C₂²×C₂₃	1	1	yes	characteristic
393	#554	—	2	1	no	—
394	#554	C₂²×C₂₃	1	1	yes	characteristic
395	#550	C₂²×C₂₃	1	1	yes	characteristic
396	#548	—	2	1	no	—
397	#547	—	2	2	no	—
398	#547	—	2	1	no	—
399	#542	—	2	2	no	—
400	#542	—	2	1	no	—
401	#532	—	2	1	no	—
402	#532	C₂²×C₂₃	1	1	yes	characteristic
403	#532	C₂²×C₂₃	1	1	yes	characteristic
404	#529	—	2	1	no	—
405	#529	C₂²×C₂₃	1	1	yes	characteristic
406	#520	—	2	1	no	—
407	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
408	#493	C₂²×C₂₃	1	1	yes	characteristic
409	#492	C₂²×C₂₃	1	1	yes	characteristic
410	#487	C₂²×C₂₃	1	1	yes	characteristic
411	#486	C₂²×C₂₃	1	1	yes	characteristic
412	#478	—	2	1	no	—
413	#475	C₂²×C₂₃	1	1	yes	characteristic
414	#474	C₂²×C₂₃	1	1	yes	characteristic
415	#469	—	2	1	no	—
416	#468	—	2	1	no	—
417	#464	—	2	1	no	—
418	#464	—	2	1	no	—
419	#463	—	2	2	no	—
420	#463	—	2	1	no	—
421	#462	C₂²×C₂₃	1	1	yes	characteristic
422	#462	C₂²×C₂₃	1	1	yes	characteristic
423	#460	C₂²×C₂₃	1	1	yes	characteristic
424	#460	C₂²×C₂₃	1	1	yes	characteristic
425	#460	C₂²×C₂₃	1	1	yes	characteristic
426	#459	C₂²×C₂₃	1	1	yes	characteristic
427	#449	—	2	2	no	—
428	#431	C₂²×C₂₃	1	1	yes	characteristic
429	#423	C₂²×C₂₃	1	1	yes	characteristic
430	#423	C₂²×C₂₃	1	1	yes	characteristic
431	#417	—	2	1	no	—
432	#333	C₂²×C₂₃	1	1	yes	characteristic
433	#333	C₂²×C₂₃	1	1	yes	characteristic
434	#331	C₂²×C₂₃	1	1	yes	characteristic
435	#331	C₂²×C₂₃	1	1	yes	characteristic
436	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
437	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
438	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
439	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
440	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
441	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
442	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
443	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
444	C₂×C₂₃	—	4	1	no	—
445	C₂×C₂₃	—	8	1	no	—
446	C₂×C₂₃	—	8	2	no	—
447	C₂×C₂₃	—	16	1	no	—
448	C₂×C₂₃	#3006	1	1	yes	characteristic
449	C₂×C₂₃	#3013	1	1	yes	characteristic
450	C₂×C₂₃	#3019	1	1	yes	characteristic
451	C₂×#41	—	2	2	no	—
452	C₂×#41	—	2	1	no	—
453	C₂×#41	—	2	1	no	—
454	C₂×#41	D₄×C₂₃	1	2	yes	—
455	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
456	#137	—	4	4	no	—
457	C₂×Q₁₆	—	2	1	no	—
458	C₂×Q₁₆	—	2	1	no	—
459	C₂×SD₁₆	—	2	1	no	—
460	C₂×SD₁₆	—	2	1	no	—
461	C₂×SD₁₆	—	2	2	no	—
462	C₂×SD₁₆	—	2	1	no	—
463	C₂×D₈	—	2	2	no	—
464	C₂×D₈	—	2	1	no	—
465	C₂×M₁₆	—	2	1	no	—
466	C₂×M₁₆	—	2	1	no	—
467	C₄⋊Q₈	—	2	2	no	—
468	C₄⋊Q₈	—	2	1	no	—
469	C₄⋊Q₈	—	2	1	no	—
470	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
471	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
472	#127	—	2	1	no	—
473	#127	C₂³×C₂₃	1	1	yes	characteristic
474	#127	C₂³×C₂₃	1	1	yes	characteristic
475	#126	—	2	2	no	—
476	#126	—	2	1	no	—
477	#125	—	2	1	no	—
478	#125	D₄×C₂₃	1	1	yes	characteristic
479	#124	—	2	2	no	—
480	#124	—	2	2	no	—
481	#124	—	2	1	no	—
482	#124	—	2	1	no	—
483	#124	—	2	1	no	—
484	#124	D₄×C₂₃	1	1	yes	characteristic
485	#124	D₄×C₂₃	1	2	yes	—
486	#123	—	2	2	no	—
487	#123	—	2	2	no	—
488	#123	—	2	2	no	—
489	#123	—	2	2	no	—
490	#123	—	2	1	no	—
491	C₂²⋊Q₈	—	2	2	no	—
492	C₂²⋊Q₈	—	2	2	no	—
493	C₂²⋊Q₈	—	2	2	no	—
494	C₂²⋊Q₈	—	2	2	no	—
495	C₂²⋊Q₈	—	2	1	no	—
496	C₂²⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
497	#121	—	2	2	no	—
498	#121	—	2	2	no	—
499	#121	—	2	1	no	—
500	#121	—	2	1	no	—
501	#121	—	2	1	no	—
502	#121	—	2	1	no	—
503	#121	—	2	1	no	—
504	#121	D₄×C₂₃	1	1	yes	characteristic
505	C₄×Q₈	—	2	1	no	—
506	C₄×Q₈	—	2	1	no	—
507	C₄×Q₈	—	2	1	no	—
508	C₄×Q₈	—	2	1	no	—
509	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
510	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
511	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
512	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
513	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
514	C₄×D₄	—	2	2	no	—
515	C₄×D₄	—	2	2	no	—
516	C₄×D₄	—	2	2	no	—
517	C₄×D₄	—	2	1	no	—
518	C₄×D₄	—	2	1	no	—
519	C₄×D₄	—	2	1	no	—
520	C₄×D₄	—	2	1	no	—
521	C₄×D₄	—	2	1	no	—
522	C₄×D₄	—	2	1	no	—
523	C₄×D₄	—	2	1	no	—
524	C₄×D₄	—	2	1	no	—
525	C₄×D₄	—	2	1	no	—
526	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
527	#117	—	2	2	no	—
528	#117	—	2	1	no	—
529	C₂×C₄⋊C₄	—	2	2	no	—
530	C₂×C₄⋊C₄	—	2	1	no	—
531	C₂×C₄⋊C₄	—	2	1	no	—
532	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
533	C₂×C₄²	—	2	1	no	—
534	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
535	#107	—	2	1	no	—
536	#106	—	2	1	no	—
537	C₄⋊C₈	—	2	1	no	—
538	C₄⋊C₈	—	2	1	no	—
539	C₄⋊C₈	—	2	1	no	—
540	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
541	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
542	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
543	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
544	#103	—	2	2	no	—
545	#103	—	2	1	no	—
546	#103	—	2	1	no	—
547	#103	—	2	1	no	—
548	#102	—	2	2	no	—
549	#102	—	2	1	no	—
550	#102	—	2	1	no	—
551	#102	—	2	1	no	—
552	#102	—	2	1	no	—
553	#102	—	2	1	no	—
554	C₂²⋊C₈	—	2	1	no	—
555	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
556	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
557	C₂₃	#18812	1	1	yes	characteristic, direct product factor, Sylow
558	#41	—	4	2	no	—
559	#41	—	4	2	no	—
560	#41	—	4	2	no	—
561	#41	—	4	2	no	—
562	#41	—	8	4	no	—
563	#41	—	8	2	no	—
564	C₂×Q₈	—	2	2	no	—
565	C₂×Q₈	—	2	1	no	—
566	C₂×Q₈	—	2	2	no	—
567	C₂×Q₈	—	2	1	no	—
568	C₂×Q₈	—	4	1	no	—
569	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
570	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
571	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
572	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
573	C₂×D₄	—	2	2	no	—
574	C₂×D₄	—	2	2	no	—
575	C₂×D₄	—	2	1	no	—
576	C₂×D₄	—	2	2	no	—
577	C₂×D₄	—	2	1	no	—
578	C₂×D₄	—	2	1	no	—
579	C₂×D₄	—	2	1	no	—
580	C₂×D₄	—	2	1	no	—
581	C₂×D₄	—	2	1	no	—
582	C₂×D₄	—	4	1	no	—
583	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
584	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
585	C₂²×C₄	—	2	2	no	—
586	C₂²×C₄	—	2	1	no	—
587	C₂²×C₄	—	2	1	no	—
588	C₂²×C₄	—	2	2	no	—
589	C₂²×C₄	—	2	2	no	—
590	C₂²×C₄	—	2	1	no	—
591	C₂²×C₄	—	2	2	no	—
592	C₂²×C₄	—	4	1	no	—
593	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
594	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
595	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
596	Q₁₆	—	4	2	no	—
597	Q₁₆	—	4	2	no	—
598	SD₁₆	—	4	2	no	—
599	SD₁₆	—	4	2	no	—
600	SD₁₆	—	8	1	no	—
601	SD₁₆	—	8	2	no	—
602	D₈	—	8	1	no	—
603	D₈	—	8	2	no	—
604	M₁₆	—	4	2	no	—
605	M₁₆	—	8	1	no	—
606	C₂×C₈	—	2	1	no	—
607	C₂×C₈	—	2	1	no	—
608	C₂×C₈	—	2	1	no	—
609	C₂×C₈	—	2	1	no	—
610	C₂×C₈	—	2	1	no	—
611	C₂×C₈	—	2	1	no	—
612	C₄⋊C₄	—	2	2	no	—
613	C₄⋊C₄	—	2	2	no	—
614	C₄⋊C₄	—	2	2	no	—
615	C₄⋊C₄	—	2	2	no	—
616	C₄⋊C₄	—	2	1	no	—
617	C₄⋊C₄	—	2	1	no	—
618	C₄⋊C₄	—	2	1	no	—
619	C₄⋊C₄	—	2	1	no	—
620	C₄⋊C₄	—	2	1	no	—
621	C₄⋊C₄	—	2	1	no	—
622	C₄⋊C₄	—	2	1	no	—
623	C₄⋊C₄	—	2	1	no	—
624	C₄⋊C₄	—	2	1	no	—
625	C₄⋊C₄	—	2	1	no	—
626	C₄⋊C₄	—	2	1	no	—
627	C₄⋊C₄	—	2	1	no	—
628	C₄⋊C₄	—	2	1	no	—
629	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
630	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
631	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
632	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
633	C₂²⋊C₄	—	2	2	no	—
634	C₂²⋊C₄	—	2	2	no	—
635	C₂²⋊C₄	—	2	2	no	—
636	C₂²⋊C₄	—	2	2	no	—
637	C₂²⋊C₄	—	2	2	no	—
638	C₂²⋊C₄	—	2	1	no	—
639	C₂²⋊C₄	—	2	2	no	—
640	C₂²⋊C₄	—	2	1	no	—
641	C₂²⋊C₄	—	2	1	no	—
642	C₂²⋊C₄	—	2	2	no	—
643	C₂²⋊C₄	—	2	1	no	—
644	C₂²⋊C₄	—	2	1	no	—
645	C₂²⋊C₄	—	4	1	no	—
646	C₂²⋊C₄	—	4	1	no	—
647	C₄²	—	2	2	no	—
648	C₄²	—	2	1	no	—
649	C₄²	—	2	1	no	—
650	C₄²	—	2	1	no	—
651	C₄²	—	2	1	no	—
652	C₄²	—	2	1	no	—
653	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
654	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
655	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
656	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
657	C₂³	—	2	1	no	—
658	C₂³	—	2	2	no	—
659	C₂³	—	4	1	no	—
660	C₂³	C₂₃×#120	1	1	yes	characteristic
661	Q₈	—	2	1	no	—
662	Q₈	—	2	1	no	—
663	Q₈	—	2	1	no	—
664	Q₈	—	2	1	no	—
665	Q₈	—	2	1	no	—
666	Q₈	—	4	2	no	—
667	Q₈	—	4	2	no	—
668	Q₈	—	4	1	no	—
669	Q₈	—	4	1	no	—
670	Q₈	—	8	2	no	—
671	Q₈	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
672	D₄	—	2	1	no	—
673	D₄	—	2	1	no	—
674	D₄	—	4	2	no	—
675	D₄	—	4	4	no	—
676	D₄	—	4	2	no	—
677	D₄	—	4	4	no	—
678	D₄	—	4	2	no	—
679	D₄	—	4	1	no	—
680	D₄	—	4	1	no	—
681	D₄	—	4	1	no	—
682	D₄	—	4	1	no	—
683	D₄	—	4	1	no	—
684	D₄	—	4	1	no	—
685	D₄	—	4	1	no	—
686	D₄	—	4	1	no	—
687	D₄	—	8	2	no	—
688	D₄	—	8	2	no	—
689	C₂×C₄	—	2	1	no	—
690	C₂×C₄	—	2	1	no	—
691	C₂×C₄	—	2	1	no	—
692	C₂×C₄	—	2	1	no	—
693	C₂×C₄	—	2	1	no	—
694	C₂×C₄	—	2	1	no	—
695	C₂×C₄	—	2	1	no	—
696	C₂×C₄	—	2	1	no	—
697	C₂×C₄	—	2	1	no	—
698	C₂×C₄	—	2	2	no	—
699	C₂×C₄	—	2	1	no	—
700	C₂×C₄	—	2	1	no	—
701	C₂×C₄	—	4	1	no	—
702	C₂×C₄	—	4	2	no	—
703	C₂×C₄	—	4	4	no	—
704	C₂×C₄	—	4	2	no	—
705	C₂×C₄	—	4	1	no	—
706	C₂×C₄	—	4	1	no	—
707	C₂×C₄	—	4	1	no	—
708	C₂×C₄	—	8	2	no	—
709	C₂×C₄	—	8	2	no	—
710	C₂×C₄	—	8	2	no	—
711	C₂×C₄	—	8	2	no	—
712	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
713	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
714	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
715	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
716	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
717	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
718	C₈	—	4	1	no	—
719	C₈	—	4	1	no	—
720	C₈	—	4	1	no	—
721	C₈	—	4	1	no	—
722	C₈	—	4	1	no	—
723	C₈	—	4	1	no	—
724	C₂²	—	2	1	no	—
725	C₂²	—	2	1	no	—
726	C₂²	—	2	1	no	—
727	C₂²	—	4	2	no	—
728	C₂²	—	4	1	no	—
729	C₂²	—	4	1	no	—
730	C₂²	—	4	2	no	—
731	C₂²	—	4	2	no	—
732	C₂²	—	4	1	no	—
733	C₂²	—	8	1	no	—
734	C₂²	—	8	1	no	—
735	C₂²	—	8	1	no	—
736	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
737	C₄	—	2	1	no	—
738	C₄	—	2	2	no	—
739	C₄	—	2	1	no	—
740	C₄	—	2	1	no	—
741	C₄	—	2	1	no	—
742	C₄	—	4	1	no	—
743	C₄	—	4	1	no	—
744	C₄	—	4	1	no	—
745	C₄	—	4	1	no	—
746	C₄	—	4	1	no	—
747	C₄	—	4	2	no	—
748	C₄	—	4	1	no	—
749	C₄	—	8	1	no	—
750	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
751	C₄	C₂₃×#575	1	1	yes	characteristic
752	C₄	C₂₃×#577	1	1	yes	characteristic
753	C₄	C₂₃×#578	1	1	yes	characteristic
754	C₂	—	4	1	no	—
755	C₂	—	8	1	no	—
756	C₂	—	8	2	no	—
757	C₂	—	16	1	no	—
758	C₂	C₂₃×#3006	1	1	yes	characteristic
759	C₂	C₂₃×#3013	1	1	yes	characteristic
760	C₂	C₂₃×#3019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
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×#3522	C₂	semidirect
2. C₂₃×#3251	C₂	semidirect
3. C₂₃×#3207	C₂	semidirect
4. C₂₃×#3037	C₂	semidirect
5. C₂₃×#1645	C₂	semidirect
6. C₂₃×#1644	C₂	semidirect
7. C₂₃×#1641	C₂	semidirect
8. C₂₃×#1640	C₂	semidirect
9. C₂₃×#1623	C₂	semidirect
10. C₂₃×#1622	C₂	semidirect
11. C₂₃×#1621	C₂	semidirect
12. C₂₃×#1620	C₂	semidirect
13. C₂₃×#1560	C₂	semidirect
14. C₂₃×#1495	C₂	semidirect
15. C₂₃×#1487	C₂	semidirect
276. #18812	C₂₃	direct (N × H)
344. #3522	C₂×C₂₃	semidirect
345. #3251	C₂×C₂₃	semidirect
346. #3207	C₂×C₂₃	semidirect
347. #3037	C₂×C₂₃	semidirect
348. #1645	C₂×C₂₃	semidirect
349. #1644	C₂×C₂₃	semidirect
350. #1641	C₂×C₂₃	semidirect
351. #1640	C₂×C₂₃	semidirect
352. #1623	C₂×C₂₃	semidirect
353. #1622	C₂×C₂₃	semidirect
354. #1621	C₂×C₂₃	semidirect
355. #1620	C₂×C₂₃	semidirect
356. #1560	C₂×C₂₃	semidirect
357. #1495	C₂×C₂₃	semidirect
358. #1487	C₂×C₂₃	semidirect
557. C₂₃	#18812	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×#3522	2
C₂₃×#3251	2
C₂₃×#3207	2
C₂₃×#3037	2
C₂₃×#1645	2
C₂₃×#1644	2
C₂₃×#1641	2
C₂₃×#1640	2
C₂₃×#1623	2
C₂₃×#1622	2
C₂₃×#1621	2
C₂₃×#1620	2
C₂₃×#1560	2
C₂₃×#1495	2
C₂₃×#1487	2
#18812	23

Subgroups — C23×#18812 #19166438

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18812 #19166438