Subgroups — #20299415

Order 7232.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813669	C₂	1	2	yes	maximal, complemented
2	#16813655	C₂	1	6	yes	maximal, complemented
3	#16813644	C₂	1	3	yes	maximal, complemented
4	C₂×Dic₄₅₂	C₂	1	3	yes	maximal, complemented
5	#131×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#14268647	C₂²	1	6	yes	—
7	#14268646	—	2	6	no	—
8	C₂×Dic₂₂₆	C₂²	1	6	yes	—
9	Q₈×D₁₁₃	—	2	2	no	—
10	Dic₄₅₂	C₂²	1	3	yes	—
11	Dic₄₅₂	C₂²	1	6	yes	—
12	#14268636	C₂²	1	6	yes	—
13	D₉₀₄	C₂²	1	1	yes	characteristic
14	#41×C₁₁₃	C₂²	1	1	yes	characteristic
15	M₁₆×C₁₁₃	C₂²	1	3	yes	—
16	C₂×C₈×C₁₁₃	C₂²	1	3	yes	—
17	Q₈×C₁₁₃	D₄	1	1	yes	characteristic
18	D₄×C₁₁₃	D₄	1	3	yes	—
19	C₂×C₄×C₁₁₃	C₂³	1	3	yes	—
20	C₂²⋊D₁₁₃	—	2	6	no	—
21	C₂×Dic₁₁₃	—	2	6	no	—
22	D₄₅₂	C₂³	1	2	yes	—
23	C₄×D₁₁₃	—	2	6	no	—
24	Dic₂₂₆	—	2	6	no	—
25	Dic₂₂₆	C₂³	1	6	yes	—
26	C₈×C₁₁₃	C₂³	1	1	yes	characteristic
27	C₈×C₁₁₃	C₂³	1	3	yes	—
28	C₂²×C₁₁₃	C₂×D₄	1	3	yes	—
29	C₂×D₁₁₃	—	2	2	no	—
30	C₄×C₁₁₃	C₂×D₄	1	3	yes	—
31	C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
32	Dic₁₁₃	—	2	6	no	—
33	C₂×C₁₁₃	—	2	3	no	—
34	C₂×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
35	D₁₁₃	—	4	2	no	—
36	C₁₁₃	#578	1	1	yes	characteristic, complemented, Sylow
37	#578	—	113	1	no	maximal, Sylow
38	2₋¹⁺⁴	—	113	2	no	—
39	#137	—	113	6	no	—
40	#135	—	113	3	no	—
41	C₂×Q₁₆	—	113	3	no	—
42	#131	D₁₁₃	1	1	yes	characteristic, complemented
43	#41	—	113	6	no	—
44	#41	—	226	6	no	—
45	#41	C₂×D₁₁₃	1	1	yes	characteristic
46	C₂×Q₈	—	113	6	no	—
47	C₂×Q₈	—	226	2	no	—
48	Q₁₆	—	113	3	no	—
49	Q₁₆	—	113	6	no	—
50	SD₁₆	—	113	6	no	—
51	D₈	—	113	1	no	—
52	M₁₆	C₂×D₁₁₃	1	3	yes	—
53	C₂×C₈	C₂×D₁₁₃	1	3	yes	—
54	Q₈	—	113	6	no	—
55	Q₈	—	226	6	no	—
56	Q₈	D₄₅₂	1	1	yes	characteristic
57	D₄	—	113	2	no	—
58	D₄	—	226	6	no	—
59	D₄	D₄₅₂	1	3	yes	—
60	C₂×C₄	—	226	6	no	—
61	C₂×C₄	—	226	6	no	—
62	C₂×C₄	C₂²×D₁₁₃	1	3	yes	—
63	C₈	C₂²×D₁₁₃	1	1	yes	characteristic
64	C₈	C₂²×D₁₁₃	1	3	yes	—
65	C₂²	—	226	2	no	—
66	C₂²	C₂×D₄₅₂	1	3	yes	—
67	C₄	—	226	6	no	—
68	C₄	C₂×D₄₅₂	1	3	yes	—
69	C₄	C₂³×D₁₁₃	1	1	yes	characteristic
70	C₂	—	2	3	no	—
71	C₂	—	452	2	no	—
72	C₂	C₂²×D₄₅₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₄₅₂
commutator	C₄×C₁₁₃	C₂⁴
Frattini	C₄	C₂³×D₁₁₃
Fitting	#131×C₁₁₃	C₂
socle	C₂×C₁₁₃	C₂²×D₄
hypercenter	#131	D₁₁₃
hypocenter	C₁₁₃	#578
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #16813669	C₂	semidirect
2. #16813655	C₂	semidirect
3. #16813644	C₂	semidirect
4. C₂×Dic₄₅₂	C₂	semidirect
5. #131×C₁₁₃	C₂	semidirect
36. C₁₁₃	#578	semidirect
42. #131	D₁₁₃	semidirect