Subgroups — #20298992

Order 7232.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₄₅₂	C₂	1	2	yes	maximal, complemented
2	C₂×#14268636	C₂	1	2	yes	maximal
3	#16813622	C₂	1	1	yes	maximal, characteristic, complemented
4	#16813610	C₂	1	1	yes	maximal, characteristic
5	#97×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₂₂₆	C₂²	1	1	yes	characteristic
7	C₂×Dic₂₂₆	C₂²	1	2	yes	—
8	C₂×D₄₅₂	C₂²	1	1	yes	characteristic
9	Dic₄₅₂	—	2	4	no	—
10	#14268636	—	2	4	no	—
11	C₄⋊Dic₁₁₃	—	2	2	no	—
12	#14268624	—	2	2	no	—
13	C₂×C₈×C₁₁₃	C₂²	1	2	yes	—
14	C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
15	C₂²×D₁₁₃	—	2	1	no	—
16	C₂×C₄×C₁₁₃	D₄	1	2	yes	—
17	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
18	C₂×Dic₁₁₃	—	2	1	no	—
19	C₂×Dic₁₁₃	—	2	2	no	—
20	D₄₅₂	—	2	2	no	—
21	Dic₂₂₆	—	2	2	no	—
22	Dic₂₂₆	—	2	2	no	—
23	Dic₂₂₆	—	2	2	no	—
24	C₈×C₁₁₃	D₄	1	4	yes	—
25	C₂²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
26	C₂×D₁₁₃	—	4	1	no	—
27	C₂×D₁₁₃	—	4	2	no	—
28	C₄×C₁₁₃	—	2	2	no	—
29	C₄×C₁₁₃	C₂×D₄	1	2	yes	—
30	Dic₁₁₃	—	4	2	no	—
31	Dic₁₁₃	—	4	1	no	—
32	C₂×C₁₁₃	#127	1	1	yes	characteristic
33	C₂×C₁₁₃	#137	1	2	yes	—
34	D₁₁₃	—	8	1	no	—
35	C₁₁₃	#497	1	1	yes	characteristic, complemented, Sylow
36	#497	—	113	1	no	maximal, Sylow
37	C₂×Q₁₆	—	113	2	no	—
38	C₂×SD₁₆	—	113	2	no	—
39	C₄⋊Q₈	—	113	1	no	—
40	#124	—	113	1	no	—
41	#97	D₁₁₃	1	1	yes	characteristic, complemented
42	C₂×Q₈	—	113	1	no	—
43	C₂×Q₈	—	113	2	no	—
44	C₂×D₄	—	113	1	no	—
45	Q₁₆	—	226	4	no	—
46	SD₁₆	—	226	4	no	—
47	C₂×C₈	C₂×D₁₁₃	1	2	yes	—
48	C₄⋊C₄	—	226	2	no	—
49	C₂²⋊C₄	—	226	2	no	—
50	C₄²	C₂×D₁₁₃	1	1	yes	characteristic
51	C₂³	—	226	1	no	—
52	Q₈	—	226	2	no	—
53	Q₈	—	226	2	no	—
54	Q₈	—	226	2	no	—
55	D₄	—	226	2	no	—
56	C₂×C₄	—	226	1	no	—
57	C₂×C₄	—	226	2	no	—
58	C₂×C₄	D₄₅₂	1	2	yes	—
59	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
60	C₈	D₄₅₂	1	4	yes	—
61	C₂²	—	452	1	no	—
62	C₂²	—	452	2	no	—
63	C₂²	C₂×D₄₅₂	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	452	1	no	—
66	C₄	—	452	2	no	—
67	C₄	C₂×D₄₅₂	1	2	yes	—
68	C₂	—	904	1	no	—
69	C₂	#16813619	1	1	yes	characteristic
70	C₂	#16813655	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄₅₂
commutator	C₂×C₄×C₁₁₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₁₃
Fitting	#97×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂×D₄
hypercenter	#97	D₁₁₃
hypocenter	C₁₁₃	#497
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₄₅₂	C₂	semidirect
3. #16813622	C₂	semidirect
5. #97×C₁₁₃	C₂	semidirect
35. C₁₁₃	#497	semidirect
41. #97	D₁₁₃	semidirect