Subgroups — #22577684

Order 8768.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984088	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₅₄₈	C₂	1	1	yes	maximal, characteristic
3	#17984022	C₂	1	2	yes	maximal, complemented
4	#17984010	C₂	1	2	yes	maximal
5	C₄⋊C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₂₇₄	—	2	1	no	—
7	C₂×D₅₄₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₁₃₇	—	2	1	no	—
9	C₄⋊Dic₁₃₇	—	2	1	no	—
10	C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
11	C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
12	#14555487	—	2	1	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	—	4	2	no	—
17	D₅₄₈	—	2	1	no	—
18	D₅₄₈	Q₈	1	2	yes	complemented
19	C₂×Dic₁₃₇	—	2	1	no	—
20	C₂×Dic₁₃₇	—	2	2	no	—
21	C₄×D₁₃₇	—	4	2	no	—
22	C₂×C₄×C₁₃₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₁₃₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
25	C₈×C₁₃₇	—	2	2	no	—
26	C₂²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
27	C₂×D₁₃₇	—	2	2	no	—
28	C₂×D₁₃₇	—	4	1	no	—
29	C₂×D₁₃₇	—	4	1	no	—
30	C₄×C₁₃₇	—	2	1	no	—
31	C₄×C₁₃₇	D₈	1	2	yes	—
32	C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
33	C₄×C₁₃₇	#41	1	1	yes	characteristic
34	Dic₁₃₇	—	4	1	no	—
35	Dic₁₃₇	—	4	2	no	—
36	C₂×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
37	C₂×C₁₃₇	C₂×D₈	1	1	yes	characteristic
38	C₂×C₁₃₇	#137	1	1	yes	characteristic
39	D₁₃₇	—	4	2	no	—
40	C₁₃₇	#474	1	1	yes	characteristic, complemented, Sylow
41	#474	—	137	1	no	maximal, Sylow
42	C₄⋊Q₈	—	137	1	no	—
43	C₄×D₄	—	137	1	no	—
44	#107	—	137	2	no	—
45	C₄⋊C₈	D₁₃₇	1	1	yes	characteristic, complemented
46	#102	—	137	2	no	—
47	C₂×Q₈	—	274	1	no	—
48	C₂×D₄	—	137	1	no	—
49	C₂²×C₄	—	274	1	no	—
50	C₂×C₈	C₂×D₁₃₇	1	2	yes	—
51	C₄⋊C₄	—	274	1	no	—
52	C₄⋊C₄	—	137	2	no	—
53	C₄⋊C₄	—	137	1	no	—
54	C₂²⋊C₄	—	274	1	no	—
55	C₄²	C₂×D₁₃₇	1	1	yes	characteristic
56	C₂³	—	274	1	no	—
57	Q₈	—	548	2	no	—
58	D₄	—	274	1	no	—
59	D₄	—	137	2	no	—
60	C₂×C₄	—	274	1	no	—
61	C₂×C₄	—	274	2	no	—
62	C₂×C₄	—	548	2	no	—
63	C₂×C₄	D₅₄₈	1	1	yes	characteristic
64	C₂×C₄	D₅₄₈	1	1	yes	characteristic
65	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
66	C₈	—	2	2	no	—
67	C₂²	—	274	2	no	—
68	C₂²	—	548	1	no	—
69	C₂²	—	548	1	no	—
70	C₂²	C₂×D₅₄₈	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	548	1	no	—
73	C₄	—	548	2	no	—
74	C₄	D₁₀₉₆	1	2	yes	—
75	C₄	Q₈×D₁₃₇	1	1	yes	characteristic
76	C₄	#14555509	1	1	yes	characteristic
77	C₂	—	548	2	no	—
78	C₂	#17984068	1	1	yes	characteristic
79	C₂	C₂×D₁₀₉₆	1	1	yes	characteristic
80	C₂	#17984121	1	1	yes	characteristic

	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	#474
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. #17984088	C₂	semidirect
3. #17984022	C₂	semidirect
5. C₄⋊C₈×C₁₃₇	C₂	semidirect
18. D₅₄₈	Q₈	semidirect
40. C₁₃₇	#474	semidirect
45. C₄⋊C₈	D₁₃₇	semidirect