Subgroups — #22577776

Order 8768.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14555499	C₂	1	4	yes	maximal, complemented
2	#17984088	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984085	C₂	1	1	yes	maximal, characteristic
4	C₄×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Dic₂₇₄	C₂²	1	2	yes	—
6	C₂×D₅₄₈	—	2	2	no	—
7	C₂×D₅₄₈	C₂²	1	2	yes	—
8	#14555499	—	2	8	no	—
9	C₄⋊Dic₁₃₇	—	2	2	no	—
10	C₂×C₈×C₁₃₇	C₂²	1	2	yes	—
11	C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂²×D₁₃₇	—	2	2	no	—
13	Dic₂₇₄	—	2	4	no	—
14	D₅₄₈	—	2	4	no	—
15	D₅₄₈	—	4	4	no	—
16	C₂×Dic₁₃₇	—	2	2	no	—
17	C₂×C₄×C₁₃₇	D₄	1	2	yes	—
18	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
19	C₈×C₁₃₇	D₄	1	4	yes	complemented
20	C₂²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
21	C₂×D₁₃₇	—	4	2	no	—
22	C₂×D₁₃₇	—	4	4	no	—
23	C₄×C₁₃₇	SD₁₆	1	4	yes	complemented
24	C₄×C₁₃₇	C₂×D₄	1	2	yes	—
25	Dic₁₃₇	—	4	2	no	—
26	C₂×C₁₃₇	#127	1	1	yes	characteristic
27	C₂×C₁₃₇	C₂×SD₁₆	1	2	yes	—
28	D₁₃₇	—	8	2	no	—
29	C₁₃₇	#492	1	1	yes	characteristic, complemented, Sylow
30	#492	—	137	1	no	maximal, Sylow
31	C₂×SD₁₆	—	137	4	no	—
32	C₄⋊Q₈	—	137	1	no	—
33	#127	—	137	1	no	—
34	C₄×C₈	D₁₃₇	1	1	yes	characteristic, complemented
35	C₂×Q₈	—	137	2	no	—
36	C₂×D₄	—	274	2	no	—
37	C₂×D₄	—	137	2	no	—
38	SD₁₆	—	274	8	no	—
39	C₂×C₈	C₂×D₁₃₇	1	2	yes	—
40	C₄⋊C₄	—	274	2	no	—
41	C₄²	C₂×D₁₃₇	1	1	yes	characteristic
42	C₂³	—	274	2	no	—
43	Q₈	—	274	4	no	—
44	D₄	—	274	4	no	—
45	D₄	—	548	4	no	—
46	C₂×C₄	—	274	2	no	—
47	C₂×C₄	D₅₄₈	1	2	yes	—
48	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
49	C₈	D₅₄₈	1	4	yes	complemented
50	C₂²	—	548	2	no	—
51	C₂²	—	548	4	no	—
52	C₂²	C₂×D₅₄₈	1	1	yes	characteristic
53	C₄	—	548	2	no	—
54	C₄	#14555499	1	4	yes	complemented
55	C₄	C₂×D₅₄₈	1	2	yes	—
56	C₂	—	1096	2	no	—
57	C₂	#17984085	1	1	yes	characteristic
58	C₂	C₂×#14555499	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₅₄₈
commutator	C₂×C₄×C₁₃₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₇
Fitting	C₄×C₈×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×D₄
hypercenter	C₄×C₈	D₁₃₇
hypocenter	C₁₃₇	#492
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#14555499	C₂	semidirect
2. #17984088	C₂	semidirect
4. C₄×C₈×C₁₃₇	C₂	semidirect
19. C₈×C₁₃₇	D₄	semidirect
23. C₄×C₁₃₇	SD₁₆	semidirect
29. C₁₃₇	#492	semidirect
34. C₄×C₈	D₁₃₇	semidirect
49. C₈	D₅₄₈	semidirect
54. C₄	#14555499	semidirect