Subgroups — #15682122

Order 2944.

502 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625078	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092928	C₂	1	1	yes	maximal, characteristic
3	#13092493	C₂	1	2	yes	maximal, complemented
4	#13092488	C₂	1	2	yes	maximal, complemented
5	#13092486	C₂	1	2	yes	maximal, complemented
6	C₈×C₂²⋊D₂₃	C₂	1	2	yes	maximal, complemented
7	#13092445	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092385	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂²⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092375	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×#67288	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₂₃	—	2	4	no	—
15	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
17	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
19	#10625124	C₄	1	2	yes	—
20	#10625101	C₄	1	1	yes	characteristic
21	#10625101	C₄	1	1	yes	characteristic
22	#10625100	C₄	1	2	yes	—
23	C₄×C₂²⋊D₂₃	C₂²	1	2	yes	—
24	C₄×C₂²⋊D₂₃	C₂²	1	2	yes	—
25	#10625097	C₂²	1	1	yes	characteristic
26	#10625095	C₄	1	1	yes	characteristic
27	#10625095	C₄	1	1	yes	characteristic
28	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
29	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
30	#10625078	—	2	4	no	—
31	C₂×#67262	C₂²	1	2	yes	complemented
32	C₂×C₈×D₂₃	C₂²	1	2	yes	complemented
33	C₂²⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
34	C₂²⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
35	C₂²⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
36	C₂²⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
37	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	complemented
38	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	complemented
39	#10624997	C₂²	1	2	yes	complemented
40	Dic₂₃⋊C₈	C₂²	1	2	yes	complemented
41	Dic₂₃⋊C₈	C₂²	1	2	yes	complemented
42	C₈×Dic₂₃	C₂²	1	2	yes	complemented
43	C₂⁴×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	—	2	2	no	—
45	C₂²×C₄×C₂₃	—	2	2	no	—
46	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
52	C₂×C₂²⋊D₂₃	C₂×C₄	1	2	yes	—
53	C₂×C₂²⋊D₂₃	C₂×C₄	1	2	yes	—
54	C₂×C₂²⋊D₂₃	C₂×C₄	1	2	yes	—
55	#67288	—	2	2	no	—
56	#67288	—	2	2	no	—
57	C₂×D₉₂	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×D₂₃	C₂³	1	2	yes	—
59	C₂×Dic₄₆	C₂×C₄	1	1	yes	characteristic
60	M₁₆×C₂₃	—	2	2	no	—
61	M₁₆×C₂₃	—	2	4	no	—
62	M₁₆×C₂₃	D₄	1	4	yes	complemented
63	C₂×C₈×C₂₃	—	2	4	no	—
64	C₂×C₈×C₂₃	C₂³	1	2	yes	—
65	C₂×C₈×C₂₃	C₂³	1	2	yes	—
66	C₂²⋊Dic₂₃	C₂×C₄	1	2	yes	—
67	C₂²⋊Dic₂₃	C₂×C₄	1	2	yes	—
68	C₂²⋊Dic₂₃	C₂×C₄	1	2	yes	—
69	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
70	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
71	C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
72	C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
73	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
74	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
75	C₄×Dic₂₃	C₂³	1	2	yes	—
76	C₂×C₂₃⋊C₈	—	2	2	no	—
77	C₂×C₂₃⋊C₈	—	2	2	no	—
78	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
79	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
80	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
81	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
82	#67262	—	2	4	no	—
83	C₈×D₂₃	—	2	4	no	—
84	C₂³×C₂₃	—	2	2	no	—
85	C₂³×C₂₃	—	2	2	no	—
86	C₂³×C₂₃	—	4	1	no	—
87	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
88	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
89	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×D₂₃	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₂₃	—	2	2	no	—
92	C₂×C₄×C₂₃	—	2	2	no	—
93	C₂×C₄×C₂₃	—	2	2	no	—
94	C₂×C₄×C₂₃	—	2	1	no	—
95	C₂×C₄×C₂₃	—	2	1	no	—
96	C₂×C₄×C₂₃	—	2	1	no	—
97	C₂×C₄×C₂₃	—	2	1	no	—
98	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₂₃	#41	1	2	yes	—
103	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
104	C₂²⋊D₂₃	—	2	4	no	—
105	C₂²⋊D₂₃	—	2	4	no	—
106	C₂²⋊D₂₃	—	4	2	no	—
107	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
108	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
109	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
110	D₉₂	—	2	2	no	—
111	C₄×D₂₃	—	2	2	no	—
112	C₄×D₂₃	—	2	2	no	—
113	Dic₄₆	—	2	2	no	—
114	C₈×C₂₃	—	2	2	no	—
115	C₈×C₂₃	C₂×D₄	1	4	yes	—
116	C₂₃⋊C₈	—	2	1	no	—
117	C₂₃⋊C₈	—	2	1	no	—
118	C₂₃⋊C₈	—	2	1	no	—
119	C₂₃⋊C₈	—	2	1	no	—
120	#2933	—	23	1	no	maximal, Sylow
121	C₂²×C₂₃	—	2	1	no	—
122	C₂²×C₂₃	—	2	1	no	—
123	C₂²×C₂₃	—	2	1	no	—
124	C₂²×C₂₃	—	2	1	no	—
125	C₂²×C₂₃	—	2	1	no	—
126	C₂²×C₂₃	—	2	1	no	—
127	C₂²×C₂₃	—	2	1	no	—
128	C₂²×C₂₃	—	4	4	no	—
129	C₂²×C₂₃	C₄×D₄	1	2	yes	—
130	C₂²×C₂₃	#131	1	2	yes	complemented
131	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
132	C₂×D₂₃	—	2	2	no	—
133	C₂×D₂₃	—	2	2	no	—
134	C₂×D₂₃	—	2	2	no	—
135	C₄×C₂₃	—	2	1	no	—
136	C₄×C₂₃	—	2	1	no	—
137	C₄×C₂₃	C₄×D₄	1	2	yes	—
138	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
139	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
140	Dic₂₃	—	2	2	no	—
141	Dic₂₃	—	2	2	no	—
142	Dic₂₃	—	2	2	no	—
143	C₂×#131	—	23	1	no	—
144	C₂²×M₁₆	D₂₃	1	1	yes	characteristic, complemented
145	#525	—	23	1	no	—
146	#436	—	23	2	no	—
147	#435	—	23	2	no	—
148	#435	—	23	2	no	—
149	C₈×D₄	—	23	2	no	—
150	#424	—	23	1	no	—
151	#408	—	23	1	no	—
152	C₂×C₂²⋊C₈	—	23	1	no	—
153	#405	—	23	1	no	—
154	C₂×C₂₃	—	2	2	no	—
155	C₂×C₂₃	—	2	2	no	—
156	C₂×C₂₃	—	4	1	no	—
157	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
158	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
159	C₂×C₂₃	#568	1	1	yes	characteristic
160	D₂₃	—	4	2	no	—
161	C₂×#41	—	23	1	no	—
162	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
163	#131	—	46	4	no	—
164	C₂×M₁₆	—	2	4	no	—
165	C₂×M₁₆	—	23	2	no	—
166	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
167	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
168	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
169	C₂²×C₈	—	23	1	no	—
170	C₂²×C₈	—	23	1	no	—
171	C₂²×C₈	—	23	2	no	—
172	C₂²×C₈	C₂×D₂₃	1	2	yes	—
173	#123	—	23	2	no	—
174	C₂²⋊Q₈	—	23	1	no	—
175	C₂²⋊Q₈	—	23	1	no	—
176	#121	—	23	1	no	—
177	#121	—	23	1	no	—
178	#120	—	23	2	no	—
179	C₄×D₄	—	23	2	no	—
180	C₄×D₄	—	23	2	no	—
181	#117	—	23	1	no	—
182	C₄⋊C₈	—	23	2	no	—
183	C₄⋊C₈	—	23	2	no	—
184	C₂²⋊C₈	—	23	1	no	—
185	C₂²⋊C₈	—	23	1	no	—
186	C₂²⋊C₈	—	23	1	no	—
187	C₂²⋊C₈	—	23	1	no	—
188	C₂²⋊C₈	—	23	2	no	—
189	C₂²⋊C₈	—	23	2	no	—
190	#97	—	23	2	no	—
191	C₄×C₈	—	23	2	no	—
192	C₂₃	#2933	1	1	yes	characteristic, complemented, Sylow
193	C₂⁴	C₄×D₂₃	1	1	yes	characteristic
194	#41	—	46	2	no	—
195	#41	—	46	2	no	—
196	C₂×Q₈	—	23	1	no	—
197	C₂×D₄	—	23	2	no	—
198	C₂×D₄	—	23	2	no	—
199	C₂×D₄	—	23	2	no	—
200	C₂×D₄	—	23	1	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	23	2	no	—
204	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
205	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
206	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
208	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
209	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
210	M₁₆	—	2	2	no	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	46	4	no	—
213	M₁₆	C₂²⋊D₂₃	1	4	yes	complemented
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	23	1	no	—
216	C₂×C₈	—	23	1	no	—
217	C₂×C₈	—	23	1	no	—
218	C₂×C₈	—	23	1	no	—
219	C₂×C₈	—	46	2	no	—
220	C₂×C₈	—	46	2	no	—
221	C₂×C₈	—	46	4	no	—
222	C₂×C₈	C₂²×D₂₃	1	2	yes	—
223	C₂×C₈	C₂²×D₂₃	1	2	yes	—
224	C₄⋊C₄	—	23	1	no	—
225	C₄⋊C₄	—	23	1	no	—
226	C₄⋊C₄	—	23	2	no	—
227	C₄⋊C₄	—	23	2	no	—
228	C₂²⋊C₄	—	23	2	no	—
229	C₂²⋊C₄	—	23	2	no	—
230	C₂²⋊C₄	—	23	2	no	—
231	C₂²⋊C₄	—	23	2	no	—
232	C₂²⋊C₄	—	23	2	no	—
233	C₄²	—	23	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	4	1	no	—
237	C₂³	—	23	2	no	—
238	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
239	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
240	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
241	Q₈	—	46	2	no	—
242	D₄	—	46	4	no	—
243	D₄	—	46	4	no	—
244	D₄	—	46	2	no	—
245	D₄	—	92	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	23	2	no	—
254	C₂×C₄	—	23	2	no	—
255	C₂×C₄	—	23	2	no	—
256	C₂×C₄	—	46	2	no	—
257	C₂×C₄	—	46	2	no	—
258	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
259	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
260	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
261	C₂×C₄	#67288	1	2	yes	—
262	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
263	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	46	1	no	—
266	C₈	—	46	1	no	—
267	C₈	—	46	1	no	—
268	C₈	—	46	1	no	—
269	C₈	C₂×C₂²⋊D₂₃	1	4	yes	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	46	2	no	—
279	C₂²	—	46	2	no	—
280	C₂²	—	46	2	no	—
281	C₂²	#10625078	1	2	yes	complemented
282	C₂²	C₄×C₂²⋊D₂₃	1	2	yes	—
283	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	46	2	no	—
287	C₄	—	46	2	no	—
288	C₄	—	46	2	no	—
289	C₄	C₄×C₂²⋊D₂₃	1	2	yes	—
290	C₄	C₂×#67288	1	1	yes	characteristic
291	C₄	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
292	C₂	—	2	2	no	—
293	C₂	—	2	2	no	—
294	C₂	—	4	1	no	—
295	C₂	—	92	2	no	—
296	C₂	C₂×C₄×C₂²⋊D₂₃	1	1	yes	characteristic
297	C₂	C₂×#10625078	1	1	yes	characteristic
298	C₂	#13093183	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting	C₂²×M₁₆×C₂₃	C₂
prime radical	#13092928	C₂
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₂²×M₁₆	D₂₃
hypocenter	C₂₃	#2933
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#10625078	C₂	semidirect
3. #13092493	C₂	semidirect
4. #13092488	C₂	semidirect
5. #13092486	C₂	semidirect
6. C₈×C₂²⋊D₂₃	C₂	semidirect
7. #13092445	C₂	semidirect
8. #13092385	C₂	semidirect
9. C₂×C₂²⋊(C₂₃⋊C₈)	C₂	semidirect
10. #13092375	C₂	semidirect
11. C₂²×M₁₆×C₂₃	C₂	semidirect
31. C₂×#67262	C₂²	semidirect
32. C₂×C₈×D₂₃	C₂²	semidirect
37. (C₂×D₂₃)⋊C₈	C₂²	semidirect
38. (C₂×D₂₃)⋊C₈	C₂²	semidirect
39. #10624997	C₂²	semidirect
40. Dic₂₃⋊C₈	C₂²	semidirect
41. Dic₂₃⋊C₈	C₂²	semidirect
42. C₈×Dic₂₃	C₂²	semidirect
62. M₁₆×C₂₃	D₄	semidirect
130. C₂²×C₂₃	#131	semidirect
144. C₂²×M₁₆	D₂₃	semidirect
192. C₂₃	#2933	semidirect
213. M₁₆	C₂²⋊D₂₃	semidirect
281. C₂²	#10625078	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625078	2
#13092928	2
#13092493	2
#13092488	2
#13092486	2
C₈×C₂²⋊D₂₃	2
#13092445	2
#13092385	2
C₂×C₂²⋊(C₂₃⋊C₈)	2
#13092375	2
C₂²×M₁₆×C₂₃	2
#2933	23