Subgroups — #15686110

Order 2944.

372 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093136	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₄₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092776	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092775	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092766	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092713	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092710	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092695	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092630	C₂	1	1	yes	maximal, characteristic
10	#13092620	C₂	1	1	yes	maximal, characteristic, complemented
11	#13092546	C₂	1	1	yes	maximal, characteristic, complemented
12	#13092518	C₂	1	1	yes	maximal, characteristic, complemented
13	#13092517	C₂	1	1	yes	maximal, characteristic, complemented
14	D₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic
15	C₂₃×#463	C₂	1	1	yes	maximal, characteristic
16	C₂²×Dic₄₆	—	2	1	no	—
17	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#124	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
21	Q₈×Dic₂₃	C₂²	1	1	yes	characteristic
22	#10625115	C₂²	1	1	yes	characteristic
23	#10625107	C₂²	1	1	yes	characteristic
24	D₄×Dic₂₃	C₂²	1	1	yes	characteristic
25	D₄×Dic₂₃	C₂²	1	1	yes	characteristic
26	D₄×Dic₂₃	C₂²	1	1	yes	characteristic
27	C₂×#67273	C₂²	1	1	yes	characteristic
28	#10625095	—	2	1	no	—
29	#10625059	C₂²	1	1	yes	characteristic
30	#10625059	C₂²	1	1	yes	characteristic
31	#10625050	—	2	1	no	—
32	#10625050	—	2	1	no	—
33	#10625049	—	2	1	no	—
34	#10625049	—	2	1	no	—
35	#10625049	—	2	1	no	—
36	#10625048	—	2	1	no	—
37	#10625040	C₂²	1	1	yes	characteristic
38	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
39	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
40	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
41	C₂₃×#103	C₂²	1	1	yes	characteristic
42	C₂₃×#102	C₂²	1	1	yes	characteristic
43	#10625017	C₂²	1	1	yes	characteristic
44	#10625017	C₂²	1	1	yes	characteristic
45	#10625014	C₂²	1	1	yes	characteristic
46	#10625014	C₂²	1	1	yes	characteristic
47	#10625014	C₂²	1	1	yes	characteristic
48	#10625000	C₂²	1	1	yes	characteristic
49	#10625000	C₂²	1	1	yes	characteristic
50	#10624998	C₂²	1	1	yes	characteristic
51	#10624998	C₂²	1	1	yes	characteristic
52	#10624997	C₂²	1	1	yes	characteristic
53	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
54	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
55	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
56	#10624992	C₂²	1	1	yes	characteristic
57	#10624989	C₂²	1	1	yes	characteristic
58	#10624985	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₂₃	—	2	1	no	—
63	C₂²×Dic₂₃	—	2	1	no	—
64	C₂²×Dic₂₃	—	2	1	no	—
65	C₂²×Dic₂₃	—	2	1	no	—
66	C₂×Dic₄₆	—	2	1	no	—
67	C₂×Dic₄₆	—	2	1	no	—
68	C₂×Dic₄₆	—	2	1	no	—
69	C₂×Dic₄₆	—	4	4	no	—
70	C₂×Dic₄₆	D₄	1	1	yes	characteristic
71	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
72	SD₁₆×C₂₃	—	2	2	no	—
73	D₈×C₂₃	—	2	2	no	—
74	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
77	C₂²⋊C₄×C₂₃	—	2	1	no	—
78	C₂²⋊C₄×C₂₃	—	2	1	no	—
79	C₂²⋊C₄×C₂₃	—	2	1	no	—
80	C₄²×C₂₃	C₂³	1	1	yes	characteristic
81	C₂²⋊Dic₂₃	—	2	1	no	—
82	C₂²⋊Dic₂₃	—	2	1	no	—
83	C₂²⋊Dic₂₃	—	2	1	no	—
84	C₂²⋊Dic₂₃	—	2	1	no	—
85	C₂²⋊Dic₂₃	—	2	1	no	—
86	#67273	—	2	2	no	—
87	C₄⋊Dic₂₃	—	2	1	no	—
88	C₄⋊Dic₂₃	—	2	1	no	—
89	C₄⋊Dic₂₃	D₄	1	1	yes	characteristic
90	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
91	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
93	Dic₂₃⋊C₄	—	2	1	no	—
94	Dic₂₃⋊C₄	—	2	1	no	—
95	Dic₂₃⋊C₄	—	2	1	no	—
96	Dic₂₃⋊C₄	—	2	1	no	—
97	Dic₂₃⋊C₄	—	2	1	no	—
98	Dic₂₃⋊C₄	D₄	1	1	yes	characteristic
99	Dic₂₃⋊C₄	D₄	1	1	yes	characteristic
100	C₄×Dic₂₃	—	2	1	no	—
101	C₄×Dic₂₃	—	2	1	no	—
102	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
103	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
104	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
106	C₂³×C₂₃	—	2	1	no	—
107	C₂³×C₂₃	—	2	1	no	—
108	Q₈×C₂₃	—	2	1	no	—
109	Q₈×C₂₃	—	2	1	no	—
110	D₄×C₂₃	—	2	1	no	—
111	D₄×C₂₃	—	2	1	no	—
112	D₄×C₂₃	—	2	1	no	—
113	D₄×C₂₃	#41	1	2	yes	—
114	C₂×C₄×C₂₃	—	2	1	no	—
115	C₂×C₄×C₂₃	—	2	1	no	—
116	C₂×C₄×C₂₃	—	4	2	no	—
117	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
120	C₂×Dic₂₃	—	2	1	no	—
121	C₂×Dic₂₃	—	2	1	no	—
122	C₂×Dic₂₃	—	2	1	no	—
123	C₂×Dic₂₃	—	2	1	no	—
124	C₂×Dic₂₃	—	4	2	no	—
125	C₂×Dic₂₃	—	4	2	no	—
126	C₂×Dic₂₃	—	4	2	no	—
127	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
131	Dic₄₆	—	2	2	no	—
132	Dic₄₆	—	2	1	no	—
133	Dic₄₆	—	2	1	no	—
134	Dic₄₆	—	4	2	no	—
135	Dic₄₆	—	4	2	no	—
136	Dic₄₆	—	4	2	no	—
137	C₈×C₂₃	—	2	1	no	—
138	C₈×C₂₃	—	2	1	no	—
139	C₂₃⋊C₈	—	2	1	no	—
140	C₂₃⋊C₈	—	2	1	no	—
141	#3367	—	23	1	no	maximal, Sylow
142	C₂²×C₂₃	—	2	2	no	—
143	C₂²×C₂₃	—	4	1	no	—
144	C₂²×C₂₃	—	4	1	no	—
145	C₂²×C₂₃	—	4	1	no	—
146	C₂²×C₂₃	—	4	1	no	—
147	C₂²×C₂₃	—	4	1	no	—
148	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
149	C₄×C₂₃	—	2	1	no	—
150	C₄×C₂₃	—	2	1	no	—
151	C₄×C₂₃	—	4	1	no	—
152	C₄×C₂₃	—	4	1	no	—
153	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
154	C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
155	Dic₂₃	—	2	1	no	—
156	Dic₂₃	—	2	1	no	—
157	Dic₂₃	—	2	1	no	—
158	Dic₂₃	—	4	1	no	—
159	Dic₂₃	—	4	1	no	—
160	Dic₂₃	—	4	1	no	—
161	Dic₂₃	—	4	1	no	—
162	Dic₂₃	AGL(1,ℤ/8)	1	2	yes	—
163	#556	—	23	1	no	—
164	D₄×Q₈	—	23	1	no	—
165	#489	—	23	1	no	—
166	#489	—	23	1	no	—
167	#487	—	23	1	no	—
168	#477	—	23	1	no	—
169	#477	—	23	1	no	—
170	#474	—	23	1	no	—
171	#463	—	23	1	no	—
172	#463	D₂₃	1	1	yes	characteristic
173	#461	—	23	1	no	—
174	#446	—	23	1	no	—
175	#440	—	23	1	no	—
176	#440	—	23	1	no	—
177	C₄×D₈	—	23	1	no	—
178	C₂×C₂₃	—	4	2	no	—
179	C₂×C₂₃	—	8	1	no	—
180	C₂×C₂₃	#548	1	1	yes	characteristic
181	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₂×C₂₃	#578	1	1	yes	characteristic
183	C₂²×Q₈	—	46	1	no	—
184	C₂×SD₁₆	—	23	1	no	—
185	C₂×SD₁₆	C₂×D₂₃	1	1	yes	characteristic
186	C₂×D₈	C₂×D₂₃	1	1	yes	characteristic
187	C₄⋊Q₈	—	23	1	no	—
188	C₄⋊Q₈	—	23	1	no	—
189	C₄⋊Q₈	—	23	1	no	—
190	C₄⋊Q₈	—	23	1	no	—
191	#126	—	46	1	no	—
192	#126	—	46	1	no	—
193	#124	—	23	1	no	—
194	#124	C₂×D₂₃	1	1	yes	characteristic
195	C₂²⋊Q₈	—	46	1	no	—
196	C₂²⋊Q₈	—	46	1	no	—
197	C₂²⋊Q₈	—	46	1	no	—
198	C₂²⋊Q₈	—	46	1	no	—
199	C₄×Q₈	—	23	1	no	—
200	C₄×Q₈	—	23	1	no	—
201	C₄×Q₈	—	23	1	no	—
202	C₄×D₄	—	23	1	no	—
203	C₄×D₄	—	23	1	no	—
204	C₄×D₄	—	23	1	no	—
205	C₄×D₄	C₂×D₂₃	1	1	yes	characteristic
206	#117	—	46	1	no	—
207	#107	—	23	1	no	—
208	#107	—	23	1	no	—
209	#107	—	23	1	no	—
210	C₄⋊C₈	—	23	1	no	—
211	C₄⋊C₈	—	23	1	no	—
212	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic
213	#103	—	23	1	no	—
214	#103	—	23	1	no	—
215	#103	—	23	1	no	—
216	#103	—	23	1	no	—
217	#103	—	23	1	no	—
218	#103	C₂×D₂₃	1	1	yes	characteristic
219	#102	—	23	1	no	—
220	#102	—	23	1	no	—
221	#102	—	23	1	no	—
222	#102	C₂×D₂₃	1	1	yes	characteristic
223	#97	—	23	1	no	—
224	#97	—	23	1	no	—
225	C₄×C₈	—	23	1	no	—
226	C₂₃	#3367	1	1	yes	characteristic, complemented, Sylow
227	C₂×Q₈	—	23	1	no	—
228	C₂×Q₈	—	23	1	no	—
229	C₂×Q₈	—	46	1	no	—
230	C₂×Q₈	—	46	1	no	—
231	C₂×Q₈	—	46	1	no	—
232	C₂×Q₈	—	92	4	no	—
233	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
234	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
235	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	46	1	no	—
238	C₂²×C₄	—	46	1	no	—
239	C₂²×C₄	—	46	1	no	—
240	SD₁₆	—	2	2	no	—
241	SD₁₆	—	46	2	no	—
242	D₈	—	2	2	no	—
243	C₂×C₈	—	23	1	no	—
244	C₂×C₈	—	23	1	no	—
245	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
247	C₄⋊C₄	—	23	1	no	—
248	C₄⋊C₄	—	23	1	no	—
249	C₄⋊C₄	—	23	1	no	—
250	C₄⋊C₄	—	23	1	no	—
251	C₄⋊C₄	—	23	1	no	—
252	C₄⋊C₄	—	23	1	no	—
253	C₄⋊C₄	—	46	1	no	—
254	C₄⋊C₄	—	46	1	no	—
255	C₄⋊C₄	—	46	1	no	—
256	C₄⋊C₄	—	46	1	no	—
257	C₄⋊C₄	—	46	1	no	—
258	C₄⋊C₄	—	46	1	no	—
259	C₄⋊C₄	—	46	1	no	—
260	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	46	1	no	—
265	C₂²⋊C₄	—	46	1	no	—
266	C₂²⋊C₄	—	46	1	no	—
267	C₂²⋊C₄	—	46	1	no	—
268	C₂²⋊C₄	—	46	1	no	—
269	C₄²	—	23	1	no	—
270	C₄²	—	23	1	no	—
271	C₄²	—	46	1	no	—
272	C₄²	—	46	1	no	—
273	C₄²	C₂²×D₂₃	1	1	yes	characteristic
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	46	2	no	—
279	Q₈	—	46	1	no	—
280	Q₈	—	46	1	no	—
281	Q₈	—	92	2	no	—
282	Q₈	—	92	2	no	—
283	Q₈	—	92	2	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	2	1	no	—
287	D₄	#67290	1	2	yes	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	23	1	no	—
292	C₂×C₄	—	23	1	no	—
293	C₂×C₄	—	23	1	no	—
294	C₂×C₄	—	23	1	no	—
295	C₂×C₄	—	46	1	no	—
296	C₂×C₄	—	46	1	no	—
297	C₂×C₄	—	46	1	no	—
298	C₂×C₄	—	46	1	no	—
299	C₂×C₄	—	92	2	no	—
300	C₂×C₄	—	92	2	no	—
301	C₂×C₄	—	92	2	no	—
302	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
303	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
304	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	46	1	no	—
308	C₈	—	46	1	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	23	2	no	—
321	C₄	—	46	1	no	—
322	C₄	—	46	1	no	—
323	C₄	—	46	1	no	—
324	C₄	—	92	1	no	—
325	C₄	—	92	1	no	—
326	C₄	—	92	1	no	—
327	C₄	—	92	1	no	—
328	C₄	C₂×#67290	1	1	yes	characteristic
329	C₄	#10625161	1	1	yes	characteristic
330	C₂	—	4	2	no	—
331	C₂	—	8	1	no	—
332	C₂	#13093073	1	1	yes	characteristic
333	C₂	AGL(1,ℤ/8)×D₂₃	1	1	yes	characteristic
334	C₂	#13093231	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#463	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×D₈×C₂₃	C₂²
hypercenter	#463	D₂₃
hypocenter	C₂₃	#3367
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #463

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #13093136	C₂	semidirect
2. D₄×Dic₄₆	C₂	semidirect
3. #13092776	C₂	semidirect
4. #13092775	C₂	semidirect
5. #13092766	C₂	semidirect
6. #13092713	C₂	semidirect
7. #13092710	C₂	semidirect
8. #13092695	C₂	semidirect
10. #13092620	C₂	semidirect
11. #13092546	C₂	semidirect
12. #13092518	C₂	semidirect
13. #13092517	C₂	semidirect
226. C₂₃	#3367	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13093136	2
D₄×Dic₄₆	2
#13092776	2
#13092775	2
#13092766	2
#13092713	2
#13092710	2
#13092695	2
#13092630	2
#13092620	2
#13092546	2
#13092518	2
#13092517	2
D₈×Dic₂₃	2
C₂₃×#463	2
#3367	23