Subgroups — #20298808

Order 7232.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14268637	C₂	1	2	yes	maximal
2	#16813622	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11744751	C₂	1	1	yes	maximal, characteristic
4	#16813545	C₂	1	2	yes	maximal
5	C₄×Q₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₂₆	C₂²	1	2	yes	—
7	#14268637	—	2	4	no	—
8	C₂×#11744751	C₂²	1	2	yes	—
9	C₄⋊Dic₁₁₃	—	2	1	no	—
10	C₄⋊Dic₁₁₃	—	2	1	no	—
11	C₂×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₁₁₃	—	2	1	no	—
13	C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
14	C₄²×C₁₁₃	—	2	1	no	—
15	C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
16	Q₈×C₁₁₃	—	2	1	no	—
17	Q₈×C₁₁₃	D₄	1	2	yes	—
18	C₂×C₄×C₁₁₃	—	2	1	no	—
19	C₂×C₄×C₁₁₃	—	2	1	no	—
20	C₂×C₄×C₁₁₃	D₄	1	1	yes	characteristic
21	C₂×C₄×C₁₁₃	D₄	1	1	yes	characteristic
22	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
23	C₂×Dic₁₁₃	—	2	2	no	—
24	Dic₂₂₆	—	2	2	no	—
25	Dic₂₂₆	—	2	2	no	—
26	#11744751	—	2	2	no	—
27	C₂²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
28	C₄×C₁₁₃	—	2	2	no	—
29	C₄×C₁₁₃	—	2	1	no	—
30	C₄×C₁₁₃	—	4	1	no	—
31	C₄×C₁₁₃	Q₁₆	1	2	yes	complemented
32	C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
33	C₄×C₁₁₃	#41	1	1	yes	characteristic
34	Dic₁₁₃	—	4	2	no	—
35	C₂×C₁₁₃	#121	1	1	yes	characteristic
36	C₂×C₁₁₃	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₁₁₃	#137	1	1	yes	characteristic
38	C₁₁₃	#462	1	1	yes	characteristic, complemented, Sylow
39	#462	—	113	1	no	maximal, Sylow
40	C₂×Q₁₆	—	113	2	no	—
41	C₄⋊Q₈	—	113	1	no	—
42	C₄×Q₈	D₁₁₃	1	1	yes	characteristic
43	C₄⋊C₈	—	113	1	no	—
44	#103	—	113	2	no	—
45	C₂×Q₈	—	113	2	no	—
46	C₂×Q₈	C₂×D₁₁₃	1	1	yes	characteristic
47	Q₁₆	—	226	4	no	—
48	C₂×C₈	—	113	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	—	226	1	no	—
51	C₄⋊C₄	—	226	1	no	—
52	C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
53	C₄²	—	2	1	no	—
54	C₄²	C₂×D₁₁₃	1	1	yes	characteristic
55	Q₈	—	2	1	no	—
56	Q₈	—	226	2	no	—
57	Q₈	—	226	2	no	—
58	Q₈	D₄₅₂	1	2	yes	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	226	2	no	—
62	C₂×C₄	C₂²⋊D₁₁₃	1	1	yes	characteristic
63	C₂×C₄	C₂²⋊D₁₁₃	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
65	C₈	—	226	2	no	—
66	C₂²	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
67	C₄	—	2	2	no	—
68	C₄	—	2	1	no	—
69	C₄	—	4	1	no	—
70	C₄	—	452	2	no	—
71	C₄	#14268637	1	2	yes	complemented
72	C₄	C₂×D₄₅₂	1	1	yes	characteristic
73	C₄	#14268647	1	1	yes	characteristic
74	C₂	#16813596	1	1	yes	characteristic
75	C₂	C₂×#14268637	1	1	yes	characteristic
76	C₂	#16813652	1	1	yes	characteristic

	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	#462
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
31. C₄×C₁₁₃	Q₁₆	semidirect
38. C₁₁₃	#462	semidirect
71. C₄	#14268637	semidirect