Subgroups — #20298990

Order 7232.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14268637	C₂	1	2	yes	maximal, complemented
2	C₂×#14268634	C₂	1	2	yes	maximal
3	#16813610	C₂	1	1	yes	maximal, characteristic
4	#16813533	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊Q₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₂₂₆	C₂²	1	1	yes	characteristic
7	C₂×D₄₅₂	C₂²	1	1	yes	characteristic
8	#14268637	—	2	4	no	—
9	#14268634	—	2	4	no	—
10	C₂×#11744751	C₂²	1	2	yes	—
11	#14268624	—	2	2	no	—
12	C₂×Q₈×C₁₁₃	C₂²	1	2	yes	—
13	C₄⋊C₄×C₁₁₃	—	2	2	no	—
14	C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
15	C₂²×D₁₁₃	—	2	1	no	—
16	Q₈×C₁₁₃	—	2	2	no	—
17	Q₈×C₁₁₃	—	2	2	no	—
18	C₂×C₄×C₁₁₃	—	2	2	no	—
19	C₂×C₄×C₁₁₃	D₄	1	2	yes	—
20	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
21	C₂×Dic₁₁₃	—	2	1	no	—
22	D₄₅₂	—	2	2	no	—
23	Dic₂₂₆	—	2	2	no	—
24	#11744751	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	—	4	2	no	—
30	C₄×C₁₁₃	C₂×D₄	1	2	yes	—
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	D₁₁₃	1	1	yes	characteristic, complemented
40	#124	—	113	1	no	—
41	#97	—	113	1	no	—
42	C₂×Q₈	—	113	1	no	—
43	C₂×Q₈	C₂×D₁₁₃	1	2	yes	—
44	C₂×D₄	—	113	1	no	—
45	Q₁₆	—	226	4	no	—
46	SD₁₆	—	226	4	no	—
47	C₂×C₈	—	113	2	no	—
48	C₄⋊C₄	—	2	2	no	—
49	C₂²⋊C₄	—	226	2	no	—
50	C₄²	C₂×D₁₁₃	1	1	yes	characteristic
51	C₂³	—	226	1	no	—
52	Q₈	—	2	2	no	—
53	Q₈	—	2	2	no	—
54	Q₈	—	226	2	no	—
55	D₄	—	226	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	226	1	no	—
58	C₂×C₄	C₂²⋊D₁₁₃	1	2	yes	—
59	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
60	C₈	—	113	4	no	—
61	C₂²	—	452	1	no	—
62	C₂²	—	452	2	no	—
63	C₂²	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	4	2	no	—
66	C₄	—	452	1	no	—
67	C₄	D₄×D₁₁₃	1	2	yes	—
68	C₂	—	904	1	no	—
69	C₂	#16813618	1	1	yes	characteristic
70	C₂	#16813653	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₁₃
commutator	C₂×C₄×C₁₁₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₁₃
Fitting	C₄⋊Q₈×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂×D₄
hypercenter	C₄⋊Q₈	D₁₁₃
hypocenter	C₁₁₃	#497
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#14268637	C₂	semidirect
4. #16813533	C₂	semidirect
5. C₄⋊Q₈×C₁₁₃	C₂	semidirect
35. C₁₁₃	#497	semidirect
39. C₄⋊Q₈	D₁₁₃	semidirect