Subgroups — #22577516

Order 8768.

132 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	C₄×D₅₄₈	C₂	1	2	yes	maximal, complemented
3	#17984020	C₂	1	1	yes	maximal, characteristic, complemented
4	#17984010	C₂	1	2	yes	maximal, complemented
5	#97×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₅₄₈	C₂²	1	2	yes	—
7	C₂×C₄×D₁₃₇	—	2	2	no	—
8	D₁₀₉₆	C₄	1	4	yes	complemented
9	C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
10	#14555487	—	2	2	no	—
11	C₂×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂²×D₁₃₇	—	2	2	no	—
15	D₅₄₈	—	2	2	no	—
16	D₅₄₈	C₂×C₄	1	4	yes	complemented
17	C₂×Dic₁₃₇	—	2	2	no	—
18	C₄×D₁₃₇	—	4	4	no	—
19	C₂×C₄×C₁₃₇	D₄	1	2	yes	—
20	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
21	C₈×C₁₃₇	—	2	1	no	—
22	C₈×C₁₃₇	C₂×C₄	1	2	yes	complemented
23	C₂²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
24	C₂×D₁₃₇	—	2	4	no	—
25	C₂×D₁₃₇	—	4	2	no	—
26	C₂×D₁₃₇	—	4	2	no	—
27	C₄×C₁₃₇	—	2	2	no	—
28	C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₁₃₇	#41	1	1	yes	characteristic
30	Dic₁₃₇	—	4	2	no	—
31	C₂×C₁₃₇	C₄×D₄	1	1	yes	characteristic
32	C₂×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
33	D₁₃₇	—	4	4	no	—
34	C₁₃₇	#442	1	1	yes	characteristic, complemented, Sylow
35	#442	—	137	1	no	maximal, Sylow
36	C₂×D₈	—	137	1	no	—
37	C₄×D₄	—	137	2	no	—
38	#106	—	137	1	no	—
39	#102	—	137	2	no	—
40	#97	D₁₃₇	1	1	yes	characteristic, complemented
41	C₂×D₄	—	137	2	no	—
42	C₂²×C₄	—	274	2	no	—
43	D₈	—	137	4	no	—
44	C₂×C₈	C₂×D₁₃₇	1	1	yes	characteristic
45	C₂×C₈	C₂×D₁₃₇	1	1	yes	characteristic
46	C₄⋊C₄	—	137	2	no	—
47	C₂²⋊C₄	—	274	2	no	—
48	C₄²	C₂×D₁₃₇	1	1	yes	characteristic
49	C₂³	—	274	2	no	—
50	D₄	—	274	2	no	—
51	D₄	—	137	4	no	—
52	C₂×C₄	—	274	2	no	—
53	C₂×C₄	—	548	4	no	—
54	C₂×C₄	D₅₄₈	1	2	yes	—
55	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
56	C₈	—	2	1	no	—
57	C₈	C₄×D₁₃₇	1	2	yes	complemented
58	C₂²	—	274	4	no	—
59	C₂²	—	548	2	no	—
60	C₂²	—	548	2	no	—
61	C₂²	C₂×D₅₄₈	1	1	yes	characteristic
62	C₄	—	2	2	no	—
63	C₄	—	548	2	no	—
64	C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
65	C₄	#14555510	1	1	yes	characteristic
66	C₂	—	548	4	no	—
67	C₂	C₄×D₅₄₈	1	1	yes	characteristic
68	C₂	#17984116	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₅₄₈
commutator	C₄×C₁₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₃₇
Fitting	#97×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×D₄
hypercenter	#97	D₁₃₇
hypocenter	C₁₃₇	#442
LCS	G ▷ 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
2. C₄×D₅₄₈	C₂	semidirect
3. #17984020	C₂	semidirect
4. #17984010	C₂	semidirect
5. #97×C₁₃₇	C₂	semidirect
8. D₁₀₉₆	C₄	semidirect
16. D₅₄₈	C₂×C₄	semidirect
22. C₈×C₁₃₇	C₂×C₄	semidirect
34. C₁₃₇	#442	semidirect
40. #97	D₁₃₇	semidirect
57. C₈	C₄×D₁₃₇	semidirect