Subgroups — #20299579

Order 7232.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813701	C₂	1	2	yes	maximal, complemented
2	#16813681	C₂	1	4	yes	maximal, complemented
3	C₂×#14268648	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#90673	—	2	2	no	—
5	C₂²⋊#90673	C₂²	1	4	yes	—
6	C₄×#90673	—	2	2	no	—
7	#14268648	—	2	2	no	—
8	#14268648	—	2	2	no	—
9	C₂×D₄₅₂	C₄	1	1	yes	characteristic, complemented
10	C₂×D₄₅₂	C₂²	1	2	yes	—
11	C₂×C₄×D₁₁₃	C₄	1	2	yes	complemented
12	C₂×C₄×D₁₁₃	C₂²	1	1	yes	characteristic
13	C₂×Q₈×C₁₁₃	C₄	1	1	yes	characteristic, complemented
14	C₂²×D₁₁₃	C₂×C₄	1	2	yes	—
15	C₂²×D₁₁₃	C₂³	1	1	yes	characteristic
16	C₂×#90673	—	2	4	no	—
17	Q₈×C₁₁₃	—	2	2	no	—
18	C₂×C₄×C₁₁₃	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
21	D₄₅₂	—	2	4	no	—
22	D₄₅₂	—	2	2	no	—
23	C₄×D₁₁₃	—	2	2	no	—
24	C₄×D₁₁₃	—	2	2	no	—
25	C₄×D₁₁₃	D₄	1	2	yes	—
26	C₄×D₁₁₃	D₄	1	2	yes	—
27	C₂²×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
28	C₂×D₁₁₃	—	2	2	no	—
29	C₂×D₁₁₃	—	4	1	no	—
30	C₂×D₁₁₃	—	4	2	no	—
31	C₂×D₁₁₃	C₂×D₄	1	2	yes	—
32	#90673	—	4	4	no	—
33	C₄×C₁₁₃	—	2	2	no	—
34	C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
35	Dic₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
36	Dic₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₁₁₃	—	2	1	no	—
38	C₂×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
39	D₁₁₃	—	2	2	no	—
40	D₁₁₃	—	4	2	no	—
41	C₁₁₃	#410	1	1	yes	characteristic, complemented, Sylow
42	#410	—	113	1	no	maximal, Sylow
43	C₂×#41	—	113	1	no	—
44	#117	—	113	2	no	—
45	#99	—	113	4	no	—
46	#41	—	226	2	no	—
47	#41	—	226	2	no	—
48	C₂×Q₈	#90673	1	1	yes	characteristic, complemented
49	C₂×D₄	—	113	2	no	—
50	C₂×D₄	—	113	1	no	—
51	C₂²×C₄	—	113	1	no	—
52	C₂²×C₄	—	113	2	no	—
53	C₄⋊C₄	—	226	2	no	—
54	C₂²⋊C₄	—	113	4	no	—
55	C₄²	—	226	2	no	—
56	C₂³	—	113	1	no	—
57	C₂³	—	113	2	no	—
58	Q₈	—	2	2	no	—
59	D₄	—	226	4	no	—
60	D₄	—	226	2	no	—
61	C₂×C₄	—	113	1	no	—
62	C₂×C₄	—	113	2	no	—
63	C₂×C₄	—	113	2	no	—
64	C₂×C₄	—	226	2	no	—
65	C₂×C₄	—	226	2	no	—
66	C₂×C₄	—	226	4	no	—
67	C₂×C₄	C₂×#90673	1	2	yes	—
68	C₂×C₄	C₂×#90673	1	1	yes	characteristic
69	C₂²	—	113	2	no	—
70	C₂²	—	452	2	no	—
71	C₂²	—	452	1	no	—
72	C₂²	—	226	2	no	—
73	C₂²	C₂²×#90673	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	—	113	1	no	—
76	C₄	—	113	1	no	—
77	C₄	—	452	4	no	—
78	C₄	C₂²⋊#90673	1	2	yes	—
79	C₂	—	2	1	no	—
80	C₂	—	452	2	no	—
81	C₂	—	226	2	no	—
82	C₂	C₂×C₂²⋊#90673	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊#90673
commutator	C₂²×C₁₁₃	C₂²×C₄
Frattini	C₂²	C₂²×#90673
Fitting	C₂×Q₈×C₁₁₃	C₄
socle	C₂×C₁₁₃	C₂×C₂²⋊C₄
hypercenter	C₂×Q₈	#90673
hypocenter	C₁₁₃	#410
LCS	G ▷ C₂²×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. #16813701	C₂	semidirect
2. #16813681	C₂	semidirect
9. C₂×D₄₅₂	C₄	semidirect
11. C₂×C₄×D₁₁₃	C₄	semidirect
13. C₂×Q₈×C₁₁₃	C₄	semidirect
41. C₁₁₃	#410	semidirect
48. C₂×Q₈	#90673	semidirect