Subgroups — #20299647

Order 7232.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813689	C₂	1	2	yes	maximal
2	C₄⋊#11744753	C₂	1	1	yes	maximal, characteristic
3	C₄×#11744753	C₂	1	1	yes	maximal, characteristic
4	C₄×#11744753	C₂	1	2	yes	maximal
5	#16813587	C₂	1	1	yes	maximal, characteristic
6	C₂×#11744753	C₂²	1	2	yes	—
7	C₂×#11744753	C₂²	1	1	yes	characteristic
8	C₂×#11744753	C₂²	1	1	yes	characteristic
9	C₂×Dic₂₂₆	C₄	1	1	yes	characteristic
10	#14268626	C₄	1	2	yes	—
11	C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
12	C₄×Dic₁₁₃	C₂²	1	2	yes	—
13	C₄⋊C₄×C₁₁₃	C₄	1	1	yes	characteristic
14	C₂×C₄×C₁₁₃	C₂×C₄	1	2	yes	—
15	C₂×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₁₁₃	C₂×C₄	1	2	yes	—
18	C₂×Dic₁₁₃	C₂³	1	1	yes	characteristic
19	Dic₂₂₆	C₈	1	4	yes	complemented
20	#11744753	—	2	2	no	—
21	#11744753	—	2	1	no	—
22	#11744753	Q₈	1	2	yes	complemented
23	C₂²×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₁₁₃	—	2	2	no	—
25	C₄×C₁₁₃	C₂×C₈	1	2	yes	—
26	Dic₁₁₃	—	2	1	no	—
27	Dic₁₁₃	C₂×C₈	1	4	yes	—
28	Dic₁₁₃	C₂×Q₈	1	1	yes	characteristic
29	Dic₁₁₃	#41	1	1	yes	characteristic
30	C₂×C₁₁₃	C₄×Q₈	1	1	yes	characteristic
31	C₂×C₁₁₃	C₂²×C₈	1	1	yes	characteristic
32	C₂×C₁₁₃	#131	1	1	yes	characteristic
33	C₁₁₃	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
34	C₈×Q₈	—	113	1	no	maximal, Sylow
35	C₄×Q₈	—	113	1	no	—
36	C₄⋊C₈	—	113	1	no	—
37	C₄⋊C₈	—	113	2	no	—
38	C₄×C₈	—	113	2	no	—
39	C₄×C₈	—	113	1	no	—
40	C₂×Q₈	—	113	1	no	—
41	C₂×C₈	—	113	1	no	—
42	C₂×C₈	—	113	2	no	—
43	C₂×C₈	—	113	1	no	—
44	C₄⋊C₄	—	113	2	no	—
45	C₄⋊C₄	#90673	1	1	yes	characteristic
46	C₄²	—	113	2	no	—
47	C₄²	—	113	1	no	—
48	Q₈	—	113	4	no	—
49	C₂×C₄	—	113	1	no	—
50	C₂×C₄	—	113	2	no	—
51	C₂×C₄	—	113	1	no	—
52	C₂×C₄	C₂×#90673	1	2	yes	—
53	C₂×C₄	C₂×#90673	1	1	yes	characteristic
54	C₈	—	113	2	no	—
55	C₈	—	226	2	no	—
56	C₈	—	226	1	no	—
57	C₂²	C₂²×#90673	1	1	yes	characteristic
58	C₄	—	2	2	no	—
59	C₄	—	113	1	no	—
60	C₄	—	113	1	no	—
61	C₄	—	113	4	no	—
62	C₄	—	226	1	no	—
63	C₄	#14268655	1	2	yes	—
64	C₂	Q₈×#90673	1	1	yes	characteristic
65	C₂	C₂×#14268655	1	1	yes	characteristic
66	C₂	#16813709	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#90673
commutator	C₂×C₁₁₃	C₂²×C₈
Fitting	C₄⋊C₄×C₁₁₃	C₄
socle	C₂²×C₁₁₃	C₂²×C₄
hypercenter	C₄⋊C₄	#90673
hypocenter	C₁₁₃	C₈×Q₈
LCS	G ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
19. Dic₂₂₆	C₈	semidirect
22. #11744753	Q₈	semidirect
33. C₁₁₃	C₈×Q₈	semidirect