Subgroups — #20298934

Order 7232.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813601	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₁₁₃	C₂	1	1	yes	maximal, characteristic
3	#16813556	C₂	1	2	yes	maximal, complemented
4	#16813547	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₂₆	C₂²	1	1	yes	characteristic
7	C₂²×Dic₁₁₃	—	2	1	no	—
8	C₄⋊Dic₁₁₃	—	2	1	no	—
9	C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
10	C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
11	#14268626	—	2	1	no	—
12	C₂²⋊Dic₁₁₃	—	2	1	no	—
13	C₂²×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₁₁₃	C₂²	1	2	yes	—
15	C₂³×C₁₁₃	D₄	1	1	yes	characteristic
16	C₂×C₄×C₁₁₃	—	2	2	no	—
17	C₂×C₄×C₁₁₃	D₄	1	1	yes	characteristic
18	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
19	C₂×Dic₁₁₃	—	2	1	no	—
20	C₂×Dic₁₁₃	—	2	1	no	—
21	C₂×Dic₁₁₃	—	2	2	no	—
22	C₂×Dic₁₁₃	—	4	2	no	—
23	Dic₂₂₆	—	2	2	no	—
24	C₈×C₁₁₃	—	2	2	no	—
25	C₂²×C₁₁₃	—	2	1	no	—
26	C₂²×C₁₁₃	—	2	1	no	—
27	C₂²×C₁₁₃	Q₁₆	1	2	yes	—
28	C₂²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
29	C₄×C₁₁₃	—	2	1	no	—
30	C₄×C₁₁₃	#41	1	2	yes	—
31	Dic₁₁₃	—	4	1	no	—
32	Dic₁₁₃	—	4	1	no	—
33	Dic₁₁₃	—	4	2	no	—
34	C₂×C₁₁₃	—	2	2	no	—
35	C₂×C₁₁₃	#123	1	1	yes	characteristic
36	C₂×C₁₁₃	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₁₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
38	C₁₁₃	#484	1	1	yes	characteristic, complemented, Sylow
39	#484	—	113	1	no	maximal, Sylow
40	C₂²⋊Q₈	—	113	1	no	—
41	C₂×C₄⋊C₄	—	113	1	no	—
42	#107	—	113	2	no	—
43	#103	—	113	2	no	—
44	C₂²⋊C₈	D₁₁₃	1	1	yes	characteristic
45	C₂×Q₈	—	113	1	no	—
46	C₂²×C₄	—	226	1	no	—
47	C₂²×C₄	C₂×D₁₁₃	1	1	yes	characteristic
48	C₂×C₈	C₂×D₁₁₃	1	2	yes	—
49	C₄⋊C₄	—	113	2	no	—
50	C₄⋊C₄	—	113	1	no	—
51	C₄⋊C₄	—	226	1	no	—
52	C₄⋊C₄	—	226	1	no	—
53	C₂²⋊C₄	—	226	1	no	—
54	C₂³	D₄₅₂	1	1	yes	characteristic
55	Q₈	—	226	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	452	2	no	—
58	C₂×C₄	—	226	1	no	—
59	C₂×C₄	—	226	1	no	—
60	C₂×C₄	—	226	2	no	—
61	C₂×C₄	D₄₅₂	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
63	C₈	—	2	2	no	—
64	C₂²	—	2	1	no	—
65	C₂²	—	2	1	no	—
66	C₂²	Dic₄₅₂	1	2	yes	—
67	C₂²	C₂×D₄₅₂	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	452	1	no	—
70	C₄	—	452	2	no	—
71	C₄	—	452	1	no	—
72	C₄	#14268646	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	#16813606	1	1	yes	characteristic
75	C₂	C₂×Dic₄₅₂	1	1	yes	characteristic
76	C₂	#16813650	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄₅₂
commutator	C₂×C₄×C₁₁₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₁₃
Fitting	C₂²⋊C₈×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂×D₄
hypercenter	C₂²⋊C₈	D₁₁₃
hypocenter	C₁₁₃	#484
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
3. #16813556	C₂	semidirect
4. #16813547	C₂	semidirect
38. C₁₁₃	#484	semidirect