Subgroups — #22578315

Order 8768.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984161	C₂	1	1	yes	maximal, characteristic, complemented
2	#17984157	C₂	1	1	yes	maximal, characteristic, complemented
3	D₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#10587739	—	2	1	no	—
5	#14555513	—	2	1	no	—
6	D₄×D₁₃₇	—	2	1	no	—
7	D₁₀₉₆	C₄	1	1	yes	characteristic, complemented
8	#14555495	—	2	1	no	—
9	C₈×D₁₃₇	C₂²	1	1	yes	characteristic
10	D₈×C₁₃₇	C₄	1	1	yes	characteristic, complemented
11	C₂×#10587739	—	4	1	no	—
12	C₂²×D₁₃₇	—	4	1	no	—
13	D₅₄₈	—	2	1	no	—
14	C₂²⋊D₁₃₇	—	4	1	no	—
15	C₄×D₁₃₇	D₄	1	1	yes	characteristic
16	#11764040	D₄	1	1	yes	characteristic
17	D₄×C₁₃₇	—	2	1	no	—
18	C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₁₃₇	—	4	1	no	—
20	C₂×D₁₃₇	—	4	1	no	—
21	C₂×D₁₃₇	—	8	1	no	—
22	C₂×D₁₃₇	—	8	1	no	—
23	C₂×D₁₃₇	D₈	1	1	yes	characteristic
24	#10587739	—	8	1	no	—
25	C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
26	Dic₁₃₇	SD₁₆	1	1	yes	characteristic
27	C₂×C₁₃₇	—	8	1	no	—
28	C₂×C₁₃₇	#102	1	1	yes	characteristic
29	D₁₃₇	—	8	1	no	—
30	D₁₃₇	D₁₆	1	1	yes	characteristic
31	D₁₃₇	SD₃₂	1	1	yes	characteristic
32	C₁₃₇	#357	1	1	yes	characteristic, complemented, Sylow
33	#357	—	137	1	no	maximal, Sylow
34	C₂×D₈	—	137	1	no	—
35	C₂×C₁₆	—	137	1	no	—
36	#107	—	137	1	no	—
37	C₂×D₄	—	274	1	no	—
38	D₈	—	274	1	no	—
39	D₈	—	137	1	no	—
40	D₈	#10587739	1	1	yes	characteristic, complemented
41	C₂×C₈	—	137	1	no	—
42	C₄⋊C₄	—	274	1	no	—
43	C₁₆	—	274	1	no	—
44	C₂³	—	548	1	no	—
45	D₄	—	2	1	no	—
46	D₄	—	274	1	no	—
47	D₄	—	548	1	no	—
48	C₂×C₄	—	548	1	no	—
49	C₂×C₄	—	137	1	no	—
50	C₈	—	137	1	no	—
51	C₈	C₂×#10587739	1	1	yes	characteristic
52	C₂²	—	4	1	no	—
53	C₂²	—	548	1	no	—
54	C₂²	—	1096	1	no	—
55	C₂²	—	1096	1	no	—
56	C₂²	—	137	1	no	—
57	C₄	—	1096	1	no	—
58	C₄	—	137	1	no	—
59	C₄	C₂²⋊#10587739	1	1	yes	characteristic
60	C₂	—	8	1	no	—
61	C₂	—	1096	1	no	—
62	C₂	—	137	1	no	—
63	C₂	—	137	1	no	—
64	C₂	#17984150	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#17984150
commutator	C₈×C₁₃₇	C₂×C₄
Frattini	C₈	C₂×#10587739
Fitting	D₈×C₁₃₇	C₄
socle	C₂×C₁₃₇	#102
hypercenter	D₈	#10587739
hypocenter	C₁₃₇	#357
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #17984161	C₂	semidirect
2. #17984157	C₂	semidirect
7. D₁₀₉₆	C₄	semidirect
10. D₈×C₁₃₇	C₄	semidirect
32. C₁₃₇	#357	semidirect
40. D₈	#10587739	semidirect