Subgroups — #22577797

Order 8768.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₅₄₈	C₂	1	2	yes	maximal, complemented
2	C₂×#14555499	C₂	1	2	yes	maximal
3	#17984088	C₂	1	1	yes	maximal, characteristic, complemented
4	#17984076	C₂	1	1	yes	maximal, characteristic
5	#97×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₂₇₄	C₂²	1	1	yes	characteristic
7	C₂×Dic₂₇₄	C₂²	1	2	yes	—
8	C₂×D₅₄₈	C₂²	1	1	yes	characteristic
9	Dic₅₄₈	—	2	4	no	—
10	#14555499	—	2	4	no	—
11	C₄⋊Dic₁₃₇	—	2	2	no	—
12	#14555487	—	2	2	no	—
13	C₂×C₈×C₁₃₇	C₂²	1	2	yes	—
14	C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂²×D₁₃₇	—	2	1	no	—
16	Dic₂₇₄	—	2	2	no	—
17	Dic₂₇₄	—	2	2	no	—
18	Dic₂₇₄	—	2	2	no	—
19	D₅₄₈	—	2	2	no	—
20	C₂×Dic₁₃₇	—	2	1	no	—
21	C₂×Dic₁₃₇	—	2	2	no	—
22	C₂×C₄×C₁₃₇	D₄	1	2	yes	—
23	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
24	C₈×C₁₃₇	D₄	1	4	yes	—
25	C₂²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
26	C₂×D₁₃₇	—	4	1	no	—
27	C₂×D₁₃₇	—	4	2	no	—
28	C₄×C₁₃₇	—	2	2	no	—
29	C₄×C₁₃₇	C₂×D₄	1	2	yes	—
30	Dic₁₃₇	—	4	1	no	—
31	Dic₁₃₇	—	4	2	no	—
32	C₂×C₁₃₇	#127	1	1	yes	characteristic
33	C₂×C₁₃₇	#137	1	2	yes	—
34	D₁₃₇	—	8	1	no	—
35	C₁₃₇	#497	1	1	yes	characteristic, complemented, Sylow
36	#497	—	137	1	no	maximal, Sylow
37	C₂×Q₁₆	—	137	2	no	—
38	C₂×SD₁₆	—	137	2	no	—
39	C₄⋊Q₈	—	137	1	no	—
40	#124	—	137	1	no	—
41	#97	D₁₃₇	1	1	yes	characteristic, complemented
42	C₂×Q₈	—	137	1	no	—
43	C₂×Q₈	—	137	2	no	—
44	C₂×D₄	—	137	1	no	—
45	Q₁₆	—	274	4	no	—
46	SD₁₆	—	274	4	no	—
47	C₂×C₈	C₂×D₁₃₇	1	2	yes	—
48	C₄⋊C₄	—	274	2	no	—
49	C₂²⋊C₄	—	274	2	no	—
50	C₄²	C₂×D₁₃₇	1	1	yes	characteristic
51	C₂³	—	274	1	no	—
52	Q₈	—	274	2	no	—
53	Q₈	—	274	2	no	—
54	Q₈	—	274	2	no	—
55	D₄	—	274	2	no	—
56	C₂×C₄	—	274	1	no	—
57	C₂×C₄	—	274	2	no	—
58	C₂×C₄	D₅₄₈	1	2	yes	—
59	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
60	C₈	D₅₄₈	1	4	yes	—
61	C₂²	—	548	1	no	—
62	C₂²	—	548	2	no	—
63	C₂²	C₂×D₅₄₈	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	548	1	no	—
66	C₄	—	548	2	no	—
67	C₄	C₂×D₅₄₈	1	2	yes	—
68	C₂	—	1096	1	no	—
69	C₂	#17984085	1	1	yes	characteristic
70	C₂	#17984121	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₅₄₈
commutator	C₂×C₄×C₁₃₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₇
Fitting	#97×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×D₄
hypercenter	#97	D₁₃₇
hypocenter	C₁₃₇	#497
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₅₄₈	C₂	semidirect
3. #17984088	C₂	semidirect
5. #97×C₁₃₇	C₂	semidirect
35. C₁₃₇	#497	semidirect
41. #97	D₁₃₇	semidirect