Subgroups — #20299403

Order 7232.

268 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813668	C₂	1	2	yes	maximal, complemented
2	#16813650	C₂	1	6	yes	maximal, complemented
3	#16813644	C₂	1	3	yes	maximal, complemented
4	C₂×D₉₀₄	C₂	1	3	yes	maximal, complemented
5	#131×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#14268648	—	2	2	no	—
7	#14268647	C₂²	1	6	yes	complemented
8	C₂×D₄₅₂	C₂²	1	6	yes	complemented
9	D₄×D₁₁₃	—	2	6	no	—
10	Dic₄₅₂	C₂²	1	1	yes	characteristic, complemented
11	#14268636	C₂²	1	6	yes	complemented
12	D₉₀₄	C₂²	1	3	yes	complemented
13	D₉₀₄	C₂²	1	6	yes	complemented
14	#41×C₁₁₃	C₂²	1	1	yes	characteristic
15	M₁₆×C₁₁₃	C₂²	1	3	yes	complemented
16	C₂×C₈×C₁₁₃	C₂²	1	3	yes	complemented
17	C₂²×D₁₁₃	—	2	6	no	—
18	Q₈×C₁₁₃	D₄	1	1	yes	characteristic
19	D₄×C₁₁₃	D₄	1	3	yes	—
20	C₂×C₄×C₁₁₃	C₂³	1	3	yes	—
21	C₂²⋊D₁₁₃	—	2	6	no	—
22	D₄₅₂	—	2	6	no	—
23	D₄₅₂	C₂³	1	6	yes	—
24	C₄×D₁₁₃	—	2	6	no	—
25	Dic₂₂₆	C₂³	1	2	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	6	no	—
30	C₂×D₁₁₃	—	8	6	no	—
31	C₄×C₁₁₃	C₂×D₄	1	3	yes	—
32	C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
33	Dic₁₁₃	—	2	2	no	—
34	C₂×C₁₁₃	—	2	3	no	—
35	C₂×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
36	D₁₁₃	—	4	6	no	—
37	C₁₁₃	#576	1	1	yes	characteristic, complemented, Sylow
38	#576	—	113	1	no	maximal, Sylow
39	H(5,𝔽₂)	—	113	2	no	—
40	AGL(1,ℤ/8)	—	113	6	no	—
41	#135	—	113	3	no	—
42	C₂×D₈	—	113	3	no	—
43	#131	D₁₁₃	1	1	yes	characteristic, complemented
44	#41	—	113	6	no	—
45	#41	—	226	2	no	—
46	#41	C₂×D₁₁₃	1	1	yes	characteristic
47	C₂×D₄	—	113	6	no	—
48	C₂×D₄	—	226	6	no	—
49	Q₁₆	—	113	1	no	—
50	SD₁₆	—	113	6	no	—
51	D₈	—	113	3	no	—
52	D₈	—	113	6	no	—
53	M₁₆	C₂×D₁₁₃	1	3	yes	complemented
54	C₂×C₈	C₂×D₁₁₃	1	3	yes	complemented
55	C₂³	—	226	6	no	—
56	Q₈	—	113	2	no	—
57	Q₈	D₄₅₂	1	1	yes	characteristic
58	D₄	—	113	6	no	—
59	D₄	—	226	6	no	—
60	D₄	—	226	6	no	—
61	D₄	D₄₅₂	1	3	yes	—
62	C₂×C₄	—	226	6	no	—
63	C₂×C₄	C₂²×D₁₁₃	1	3	yes	—
64	C₈	C₂²×D₁₁₃	1	1	yes	characteristic
65	C₈	C₂²×D₁₁₃	1	3	yes	—
66	C₂²	—	904	6	no	—
67	C₂²	—	226	6	no	—
68	C₂²	C₂×D₄₅₂	1	3	yes	—
69	C₄	—	226	2	no	—
70	C₄	C₂×D₄₅₂	1	3	yes	—
71	C₄	C₂³×D₁₁₃	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	—	452	6	no	—
74	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	#576
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. #16813668	C₂	semidirect
2. #16813650	C₂	semidirect
3. #16813644	C₂	semidirect
4. C₂×D₉₀₄	C₂	semidirect
5. #131×C₁₁₃	C₂	semidirect
7. #14268647	C₂²	semidirect
8. C₂×D₄₅₂	C₂²	semidirect
10. Dic₄₅₂	C₂²	semidirect
11. #14268636	C₂²	semidirect
12. D₉₀₄	C₂²	semidirect
13. D₉₀₄	C₂²	semidirect
15. M₁₆×C₁₁₃	C₂²	semidirect
16. C₂×C₈×C₁₁₃	C₂²	semidirect
37. C₁₁₃	#576	semidirect
43. #131	D₁₁₃	semidirect
53. M₁₆	C₂×D₁₁₃	semidirect
54. C₂×C₈	C₂×D₁₁₃	semidirect