Subgroups — #20299001

Order 7232.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813622	C₂	1	1	yes	maximal, characteristic
2	#16813555	C₂	1	4	yes	maximal
3	C₄×#11744751	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₂₂₆	—	2	1	no	—
6	C₂×#11744751	C₂²	1	2	yes	—
7	C₄⋊Dic₁₁₃	—	2	1	no	—
8	C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
9	C₂×Q₈×C₁₁₃	—	2	1	no	—
10	C₄⋊C₄×C₁₁₃	—	2	1	no	—
11	C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
12	C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
13	Q₈×C₁₁₃	—	4	2	no	—
14	C₂×C₄×C₁₁₃	—	2	2	no	—
15	C₂×C₄×C₁₁₃	D₄	1	1	yes	characteristic
16	C₂×C₄×C₁₁₃	D₄	1	1	yes	characteristic
17	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
18	C₂×Dic₁₁₃	—	2	2	no	—
19	Dic₂₂₆	—	4	2	no	—
20	#11744751	Q₈	1	4	yes	complemented
21	C₂²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
22	C₄×C₁₁₃	—	4	2	no	—
23	C₄×C₁₁₃	D₈	1	2	yes	—
24	C₄×C₁₁₃	Q₁₆	1	2	yes	—
25	C₄×C₁₁₃	C₂×Q₈	1	2	yes	—
26	Dic₁₁₃	—	4	2	no	—
27	C₂×C₁₁₃	C₄⋊Q₈	1	1	yes	characteristic
28	C₂×C₁₁₃	C₂×D₈	1	1	yes	characteristic
29	C₂×C₁₁₃	C₂×Q₁₆	1	1	yes	characteristic
30	C₁₁₃	#500	1	1	yes	characteristic, complemented, Sylow
31	#500	—	113	1	no	maximal, Sylow
32	C₄⋊Q₈	—	113	1	no	—
33	C₄⋊Q₈	D₁₁₃	1	1	yes	characteristic
34	#107	—	113	4	no	—
35	C₄×C₈	—	113	1	no	—
36	C₂×Q₈	—	2	1	no	—
37	C₂×Q₈	—	226	1	no	—
38	C₂×C₈	—	113	2	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	113	2	no	—
41	C₄⋊C₄	—	226	1	no	—
42	C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
43	C₄²	C₂×D₁₁₃	1	1	yes	characteristic
44	Q₈	—	4	2	no	—
45	Q₈	—	452	2	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	226	2	no	—
48	C₂×C₄	C₂²⋊D₁₁₃	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₁₁₃	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
51	C₈	—	113	4	no	—
52	C₂²	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	—	452	2	no	—
55	C₄	#14268632	1	2	yes	—
56	C₄	#14268637	1	2	yes	—
57	C₄	Q₈×D₁₁₃	1	2	yes	—
58	C₂	#16813620	1	1	yes	characteristic
59	C₂	C₂×#14268632	1	1	yes	characteristic
60	C₂	C₂×#14268637	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	#500
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
20. #11744751	Q₈	semidirect
30. C₁₁₃	#500	semidirect