Subgroups — #22577205

Order 8768.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₀₉₆	C₂	1	1	yes	maximal, characteristic
2	#17984022	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₅₄₈	—	2	1	no	—
5	D₁₀₉₆	—	2	1	no	—
6	D₁₀₉₆	C₄	1	2	yes	complemented
7	C₄⋊Dic₁₃₇	—	2	1	no	—
8	C₂×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₁₆×C₁₃₇	—	2	1	no	—
10	C₂²×D₁₃₇	—	4	1	no	—
11	D₅₄₈	—	2	2	no	—
12	D₅₄₈	—	4	1	no	—
13	C₂×Dic₁₃₇	—	4	1	no	—
14	C₂×C₄×C₁₃₇	D₄	1	1	yes	characteristic
15	C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
16	C₈×C₁₃₇	D₄	1	1	yes	characteristic
17	C₂²×C₁₃₇	D₈	1	1	yes	characteristic
18	C₂×D₁₃₇	—	4	2	no	—
19	C₂×D₁₃₇	—	8	1	no	—
20	C₂×D₁₃₇	—	8	1	no	—
21	C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₁₃₇	SD₁₆	1	1	yes	characteristic
23	Dic₁₃₇	—	8	1	no	—
24	C₂×C₁₃₇	#102	1	1	yes	characteristic
25	C₂×C₁₃₇	D₁₆	1	1	yes	characteristic
26	C₂×C₁₃₇	SD₃₂	1	1	yes	characteristic
27	D₁₃₇	—	8	2	no	—
28	C₁₃₇	#357	1	1	yes	characteristic, complemented, Sylow
29	#357	—	137	1	no	maximal, Sylow
30	C₂×D₈	—	137	1	no	—
31	C₂×C₁₆	D₁₃₇	1	1	yes	characteristic, complemented
32	#107	—	137	1	no	—
33	C₂×D₄	—	274	1	no	—
34	D₈	—	274	1	no	—
35	D₈	—	137	2	no	—
36	C₂×C₈	C₂×D₁₃₇	1	1	yes	characteristic
37	C₄⋊C₄	—	274	1	no	—
38	C₁₆	—	2	1	no	—
39	C₂³	—	548	1	no	—
40	D₄	—	274	2	no	—
41	D₄	—	548	1	no	—
42	C₂×C₄	—	548	1	no	—
43	C₂×C₄	D₅₄₈	1	1	yes	characteristic
44	C₈	C₄×D₁₃₇	1	1	yes	characteristic
45	C₈	C₂²⋊D₁₃₇	1	1	yes	characteristic
46	C₂²	—	548	2	no	—
47	C₂²	—	1096	1	no	—
48	C₂²	—	1096	1	no	—
49	C₂²	D₁₀₉₆	1	1	yes	characteristic
50	C₄	—	1096	1	no	—
51	C₄	#14555487	1	1	yes	characteristic
52	C₄	#14555499	1	1	yes	characteristic
53	C₂	—	1096	2	no	—
54	C₂	#17984010	1	1	yes	characteristic
55	C₂	D₂₁₉₂	1	1	yes	characteristic
56	C₂	#17984033	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₀₉₆
commutator	C₈×C₁₃₇	C₂×C₄
Frattini	C₂×C₈	C₂×D₁₃₇
Fitting	C₂×C₁₆×C₁₃₇	C₂
socle	C₂²×C₁₃₇	D₈
hypercenter	C₂×C₁₆	D₁₃₇
hypocenter	C₁₃₇	#357
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
2. #17984022	C₂	semidirect
3. C₂×C₁₆×C₁₃₇	C₂	semidirect
6. D₁₀₉₆	C₄	semidirect
28. C₁₃₇	#357	semidirect
31. C₂×C₁₆	D₁₃₇	semidirect