Subgroups — #22578444

Order 8768.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×#10587739	C₂	1	2	yes	maximal, complemented
2	#17984156	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984150	C₂	1	2	yes	maximal, complemented
4	#17984142	C₂	1	1	yes	maximal, characteristic, complemented
5	D₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10587739	—	2	2	no	—
7	#14555518	C₂²	1	1	yes	characteristic
8	C₄⋊#10587739	C₂²	1	2	yes	—
9	C₂²⋊#10587739	—	2	2	no	—
10	C₄×#10587739	C₂²	1	1	yes	characteristic
11	D₄×D₁₃₇	C₂²	1	2	yes	—
12	D₁₀₉₆	C₄	1	1	yes	characteristic, complemented
13	#14555495	C₄	1	2	yes	complemented
14	C₈×D₁₃₇	C₂²	1	1	yes	characteristic
15	D₈×C₁₃₇	C₄	1	1	yes	characteristic, complemented
16	C₂×#10587739	—	2	2	no	—
17	C₂×#10587739	—	4	4	no	—
18	C₂×#10587739	D₄	1	2	yes	—
19	#11764047	—	2	1	no	—
20	C₂²×D₁₃₇	—	2	2	no	—
21	D₅₄₈	C₂×C₄	1	2	yes	complemented
22	C₂²⋊D₁₃₇	—	2	2	no	—
23	C₄×D₁₃₇	C₂³	1	1	yes	characteristic
24	#11764040	C₂×C₄	1	1	yes	characteristic, complemented
25	D₄×C₁₃₇	C₂×C₄	1	2	yes	complemented
26	C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic, complemented
27	C₂²×C₁₃₇	—	2	2	no	—
28	C₂×D₁₃₇	—	2	2	no	—
29	C₂×D₁₃₇	—	4	2	no	—
30	C₂×D₁₃₇	—	4	2	no	—
31	C₂×D₁₃₇	C₂×D₄	1	1	yes	characteristic
32	#10587739	—	2	2	no	—
33	#10587739	—	4	2	no	—
34	C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
35	Dic₁₃₇	#41	1	1	yes	characteristic
36	C₂×C₁₃₇	—	4	2	no	—
37	C₂×C₁₃₇	C₄×D₄	1	1	yes	characteristic
38	D₁₃₇	—	4	2	no	—
39	D₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
40	C₁₃₇	#442	1	1	yes	characteristic, complemented, Sylow
41	#442	—	137	1	no	maximal, Sylow
42	C₂×D₈	—	137	1	no	—
43	C₄×D₄	—	137	2	no	—
44	#106	—	137	1	no	—
45	#102	—	137	2	no	—
46	#97	—	137	1	no	—
47	C₂×D₄	—	137	2	no	—
48	C₂²×C₄	—	274	2	no	—
49	D₈	—	137	2	no	—
50	D₈	—	137	1	no	—
51	D₈	#10587739	1	1	yes	characteristic, complemented
52	C₂×C₈	—	137	1	no	—
53	C₂×C₈	—	137	1	no	—
54	C₄⋊C₄	—	137	2	no	—
55	C₂²⋊C₄	—	274	2	no	—
56	C₄²	—	137	1	no	—
57	C₂³	—	274	2	no	—
58	D₄	—	274	2	no	—
59	D₄	—	137	2	no	—
60	D₄	C₂×#10587739	1	2	yes	complemented
61	C₂×C₄	—	274	2	no	—
62	C₂×C₄	—	548	4	no	—
63	C₂×C₄	—	137	2	no	—
64	C₂×C₄	—	137	1	no	—
65	C₈	—	274	1	no	—
66	C₈	—	137	1	no	—
67	C₈	C₂×#10587739	1	1	yes	characteristic, complemented
68	C₂²	—	2	2	no	—
69	C₂²	—	274	2	no	—
70	C₂²	—	548	2	no	—
71	C₂²	—	548	2	no	—
72	C₂²	—	137	1	no	—
73	C₄	—	274	2	no	—
74	C₄	—	548	2	no	—
75	C₄	—	137	1	no	—
76	C₄	C₂²×#10587739	1	1	yes	characteristic
77	C₂	—	4	2	no	—
78	C₂	—	548	2	no	—
79	C₂	—	137	2	no	—
80	C₂	D₄×#10587739	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#10587739
commutator	C₄×C₁₃₇	C₂²×C₄
Frattini	C₄	C₂²×#10587739
Fitting	D₈×C₁₃₇	C₄
socle	C₂×C₁₃₇	C₄×D₄
hypercenter	D₈	#10587739
hypocenter	C₁₃₇	#442
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. D₄×#10587739	C₂	semidirect
2. #17984156	C₂	semidirect
3. #17984150	C₂	semidirect
4. #17984142	C₂	semidirect
12. D₁₀₉₆	C₄	semidirect
13. #14555495	C₄	semidirect
15. D₈×C₁₃₇	C₄	semidirect
21. D₅₄₈	C₂×C₄	semidirect
24. #11764040	C₂×C₄	semidirect
25. D₄×C₁₃₇	C₂×C₄	semidirect
26. C₈×C₁₃₇	C₂×C₄	semidirect
40. C₁₃₇	#442	semidirect
51. D₈	#10587739	semidirect
60. D₄	C₂×#10587739	semidirect
67. C₈	C₂×#10587739	semidirect