Subgroups — #20299036

Order 7232.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813644	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₉₀₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#16813567	C₂	1	2	yes	maximal, complemented
4	D₁₈₀₈	C₂	1	2	yes	maximal, complemented
5	M₃₂×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#14268647	—	2	1	no	—
7	C₂×D₄₅₂	—	2	1	no	—
8	Dic₄₅₂	C₂²	1	1	yes	characteristic, complemented
9	#14268636	—	2	1	no	—
10	D₉₀₄	—	2	1	no	—
11	D₉₀₄	C₂²	1	2	yes	—
12	D₉₀₄	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
14	C₁₆×C₁₁₃	C₂²	1	2	yes	complemented
15	C₂²×D₁₁₃	—	4	1	no	—
16	C₂×C₄×C₁₁₃	D₄	1	1	yes	characteristic
17	C₂²⋊D₁₁₃	—	4	1	no	—
18	D₄₅₂	—	2	2	no	—
19	D₄₅₂	—	2	1	no	—
20	D₄₅₂	—	4	1	no	—
21	C₄×D₁₁₃	—	4	1	no	—
22	Dic₂₂₆	—	2	1	no	—
23	C₈×C₁₁₃	D₄	1	1	yes	characteristic
24	C₈×C₁₁₃	C₂³	1	1	yes	characteristic
25	C₂²×C₁₁₃	D₈	1	1	yes	characteristic
26	C₂×D₁₁₃	—	4	1	no	—
27	C₂×D₁₁₃	—	4	2	no	—
28	C₂×D₁₁₃	—	8	2	no	—
29	C₄×C₁₁₃	D₈	1	1	yes	characteristic
30	C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
31	Dic₁₁₃	—	4	1	no	—
32	C₂×C₁₁₃	—	2	1	no	—
33	C₂×C₁₁₃	C₂×D₈	1	1	yes	characteristic
34	D₁₁₃	—	8	1	no	—
35	D₁₁₃	—	8	2	no	—
36	C₁₁₃	#509	1	1	yes	characteristic, complemented, Sylow
37	#509	—	113	1	no	maximal, Sylow
38	#135	—	113	1	no	—
39	C₂×D₈	—	113	1	no	—
40	SD₃₂	—	113	2	no	—
41	D₁₆	—	113	2	no	—
42	M₃₂	D₁₁₃	1	1	yes	characteristic, complemented
43	#41	—	226	1	no	—
44	C₂×D₄	—	226	1	no	—
45	Q₁₆	—	113	1	no	—
46	SD₁₆	—	226	1	no	—
47	D₈	—	113	1	no	—
48	D₈	—	113	2	no	—
49	D₈	—	226	1	no	—
50	C₂×C₈	C₂×D₁₁₃	1	1	yes	characteristic
51	C₁₆	C₂×D₁₁₃	1	2	yes	complemented
52	C₂³	—	452	1	no	—
53	Q₈	—	226	1	no	—
54	D₄	—	452	1	no	—
55	D₄	—	452	1	no	—
56	D₄	—	226	2	no	—
57	D₄	—	226	1	no	—
58	C₂×C₄	—	452	1	no	—
59	C₂×C₄	D₄₅₂	1	1	yes	characteristic
60	C₈	D₄₅₂	1	1	yes	characteristic
61	C₈	C₂²×D₁₁₃	1	1	yes	characteristic
62	C₂²	—	904	2	no	—
63	C₂²	—	452	1	no	—
64	C₂²	—	452	2	no	—
65	C₂²	D₉₀₄	1	1	yes	characteristic
66	C₄	—	452	1	no	—
67	C₄	D₉₀₄	1	1	yes	characteristic
68	C₄	C₂×D₄₅₂	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	—	904	2	no	—
71	C₂	—	904	1	no	—
72	C₂	C₂×D₉₀₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₉₀₄
commutator	C₈×C₁₁₃	C₂³
Frattini	C₈	C₂²×D₁₁₃
Fitting	M₃₂×C₁₁₃	C₂
socle	C₂×C₁₁₃	C₂×D₈
hypercenter	M₃₂	D₁₁₃
hypocenter	C₁₁₃	#509
LCS	G ▷ C₈×C₁₁₃ ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
1. #16813644	C₂	semidirect
2. C₂×D₉₀₄	C₂	semidirect
3. #16813567	C₂	semidirect
4. D₁₈₀₈	C₂	semidirect
5. M₃₂×C₁₁₃	C₂	semidirect
8. Dic₄₅₂	C₂²	semidirect
12. D₉₀₄	C₂²	semidirect
14. C₁₆×C₁₁₃	C₂²	semidirect
36. C₁₁₃	#509	semidirect
42. M₃₂	D₁₁₃	semidirect
51. C₁₆	C₂×D₁₁₃	semidirect