Subgroups — #22577841

Order 8768.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984110	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₁₀₉₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984033	C₂	1	2	yes	maximal, complemented
4	D₂₁₉₂	C₂	1	2	yes	maximal, complemented
5	M₃₂×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#14555510	—	2	1	no	—
7	C₂×D₅₄₈	—	2	1	no	—
8	Dic₅₄₈	C₂²	1	1	yes	characteristic, complemented
9	#14555499	—	2	1	no	—
10	D₁₀₉₆	—	2	1	no	—
11	D₁₀₉₆	C₂²	1	2	yes	—
12	D₁₀₉₆	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₁₆×C₁₃₇	C₂²	1	2	yes	complemented
15	C₂²×D₁₃₇	—	4	1	no	—
16	Dic₂₇₄	—	2	1	no	—
17	D₅₄₈	—	2	2	no	—
18	D₅₄₈	—	2	1	no	—
19	D₅₄₈	—	4	1	no	—
20	C₂²⋊D₁₃₇	—	4	1	no	—
21	C₄×D₁₃₇	—	4	1	no	—
22	C₂×C₄×C₁₃₇	D₄	1	1	yes	characteristic
23	C₈×C₁₃₇	D₄	1	1	yes	characteristic
24	C₈×C₁₃₇	C₂³	1	1	yes	characteristic
25	C₂²×C₁₃₇	D₈	1	1	yes	characteristic
26	C₂×D₁₃₇	—	4	1	no	—
27	C₂×D₁₃₇	—	4	2	no	—
28	C₂×D₁₃₇	—	8	2	no	—
29	C₄×C₁₃₇	D₈	1	1	yes	characteristic
30	C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
31	Dic₁₃₇	—	4	1	no	—
32	C₂×C₁₃₇	—	2	1	no	—
33	C₂×C₁₃₇	C₂×D₈	1	1	yes	characteristic
34	D₁₃₇	—	8	1	no	—
35	D₁₃₇	—	8	2	no	—
36	C₁₃₇	#509	1	1	yes	characteristic, complemented, Sylow
37	#509	—	137	1	no	maximal, Sylow
38	#135	—	137	1	no	—
39	C₂×D₈	—	137	1	no	—
40	SD₃₂	—	137	2	no	—
41	D₁₆	—	137	2	no	—
42	M₃₂	D₁₃₇	1	1	yes	characteristic, complemented
43	#41	—	274	1	no	—
44	C₂×D₄	—	274	1	no	—
45	Q₁₆	—	137	1	no	—
46	SD₁₆	—	274	1	no	—
47	D₈	—	274	1	no	—
48	D₈	—	137	2	no	—
49	D₈	—	137	1	no	—
50	C₂×C₈	C₂×D₁₃₇	1	1	yes	characteristic
51	C₁₆	C₂×D₁₃₇	1	2	yes	complemented
52	C₂³	—	548	1	no	—
53	Q₈	—	274	1	no	—
54	D₄	—	274	2	no	—
55	D₄	—	274	1	no	—
56	D₄	—	548	1	no	—
57	D₄	—	548	1	no	—
58	C₂×C₄	—	548	1	no	—
59	C₂×C₄	D₅₄₈	1	1	yes	characteristic
60	C₈	D₅₄₈	1	1	yes	characteristic
61	C₈	C₂²×D₁₃₇	1	1	yes	characteristic
62	C₂²	—	548	1	no	—
63	C₂²	—	548	2	no	—
64	C₂²	—	1096	2	no	—
65	C₂²	D₁₀₉₆	1	1	yes	characteristic
66	C₄	—	548	1	no	—
67	C₄	D₁₀₉₆	1	1	yes	characteristic
68	C₄	C₂×D₅₄₈	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	—	1096	2	no	—
71	C₂	—	1096	1	no	—
72	C₂	C₂×D₁₀₉₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₁₀₉₆
commutator	C₈×C₁₃₇	C₂³
Frattini	C₈	C₂²×D₁₃₇
Fitting	M₃₂×C₁₃₇	C₂
socle	C₂×C₁₃₇	C₂×D₈
hypercenter	M₃₂	D₁₃₇
hypocenter	C₁₃₇	#509
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
1. #17984110	C₂	semidirect
2. C₂×D₁₀₉₆	C₂	semidirect
3. #17984033	C₂	semidirect
4. D₂₁₉₂	C₂	semidirect
5. M₃₂×C₁₃₇	C₂	semidirect
8. Dic₅₄₈	C₂²	semidirect
12. D₁₀₉₆	C₂²	semidirect
14. C₁₆×C₁₃₇	C₂²	semidirect
36. C₁₃₇	#509	semidirect
42. M₃₂	D₁₃₇	semidirect
51. C₁₆	C₂×D₁₃₇	semidirect