Subgroups — #20261540

Order 6208.

268 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15690856	C₂	1	2	yes	maximal, complemented
2	#15690838	C₂	1	6	yes	maximal, complemented
3	#15690832	C₂	1	3	yes	maximal, complemented
4	C₂×D₇₇₆	C₂	1	3	yes	maximal, complemented
5	#131×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#13095512	—	2	2	no	—
7	#13095511	C₂²	1	6	yes	complemented
8	C₂×D₃₈₈	C₂²	1	6	yes	complemented
9	D₄×D₉₇	—	2	6	no	—
10	Dic₃₈₈	C₂²	1	1	yes	characteristic, complemented
11	#13095500	C₂²	1	6	yes	complemented
12	D₇₇₆	C₂²	1	3	yes	complemented
13	D₇₇₆	C₂²	1	6	yes	complemented
14	#41×C₉₇	C₂²	1	1	yes	characteristic
15	M₁₆×C₉₇	C₂²	1	3	yes	complemented
16	C₂×C₈×C₉₇	C₂²	1	3	yes	complemented
17	C₂²×D₉₇	—	2	6	no	—
18	Q₈×C₉₇	D₄	1	1	yes	characteristic
19	D₄×C₉₇	D₄	1	3	yes	—
20	C₂×C₄×C₉₇	C₂³	1	3	yes	—
21	C₂²⋊D₉₇	—	2	6	no	—
22	D₃₈₈	—	2	6	no	—
23	D₃₈₈	C₂³	1	6	yes	—
24	C₄×D₉₇	—	2	6	no	—
25	Dic₁₉₄	C₂³	1	2	yes	—
26	C₈×C₉₇	C₂³	1	1	yes	characteristic
27	C₈×C₉₇	C₂³	1	3	yes	—
28	C₂²×C₉₇	C₂×D₄	1	3	yes	—
29	C₂×D₉₇	—	2	6	no	—
30	C₂×D₉₇	—	8	6	no	—
31	C₄×C₉₇	C₂×D₄	1	3	yes	—
32	C₄×C₉₇	C₂⁴	1	1	yes	characteristic
33	Dic₉₇	—	2	2	no	—
34	C₂×C₉₇	—	2	3	no	—
35	C₂×C₉₇	C₂²×D₄	1	1	yes	characteristic
36	D₉₇	—	4	6	no	—
37	C₉₇	#576	1	1	yes	characteristic, complemented, Sylow
38	#576	—	97	1	no	maximal, Sylow
39	H(5,𝔽₂)	—	97	2	no	—
40	AGL(1,ℤ/8)	—	97	6	no	—
41	#135	—	97	3	no	—
42	C₂×D₈	—	97	3	no	—
43	#131	D₉₇	1	1	yes	characteristic, complemented
44	#41	—	97	6	no	—
45	#41	—	194	2	no	—
46	#41	C₂×D₉₇	1	1	yes	characteristic
47	C₂×D₄	—	97	6	no	—
48	C₂×D₄	—	194	6	no	—
49	Q₁₆	—	97	1	no	—
50	SD₁₆	—	97	6	no	—
51	D₈	—	97	3	no	—
52	D₈	—	97	6	no	—
53	M₁₆	C₂×D₉₇	1	3	yes	complemented
54	C₂×C₈	C₂×D₉₇	1	3	yes	complemented
55	C₂³	—	194	6	no	—
56	Q₈	—	97	2	no	—
57	Q₈	D₃₈₈	1	1	yes	characteristic
58	D₄	—	97	6	no	—
59	D₄	—	194	6	no	—
60	D₄	—	194	6	no	—
61	D₄	D₃₈₈	1	3	yes	—
62	C₂×C₄	—	194	6	no	—
63	C₂×C₄	C₂²×D₉₇	1	3	yes	—
64	C₈	C₂²×D₉₇	1	1	yes	characteristic
65	C₈	C₂²×D₉₇	1	3	yes	—
66	C₂²	—	776	6	no	—
67	C₂²	—	194	6	no	—
68	C₂²	C₂×D₃₈₈	1	3	yes	—
69	C₄	—	194	2	no	—
70	C₄	C₂×D₃₈₈	1	3	yes	—
71	C₄	C₂³×D₉₇	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	—	388	6	no	—
74	C₂	C₂²×D₃₈₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₃₈₈
commutator	C₄×C₉₇	C₂⁴
Frattini	C₄	C₂³×D₉₇
Fitting	#131×C₉₇	C₂
socle	C₂×C₉₇	C₂²×D₄
hypercenter	#131	D₉₇
hypocenter	C₉₇	#576
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #15690856	C₂	semidirect
2. #15690838	C₂	semidirect
3. #15690832	C₂	semidirect
4. C₂×D₇₇₆	C₂	semidirect
5. #131×C₉₇	C₂	semidirect
7. #13095511	C₂²	semidirect
8. C₂×D₃₈₈	C₂²	semidirect
10. Dic₃₈₈	C₂²	semidirect
11. #13095500	C₂²	semidirect
12. D₇₇₆	C₂²	semidirect
13. D₇₇₆	C₂²	semidirect
15. M₁₆×C₉₇	C₂²	semidirect
16. C₂×C₈×C₉₇	C₂²	semidirect
37. C₉₇	#576	semidirect
43. #131	D₉₇	semidirect
53. M₁₆	C₂×D₉₇	semidirect
54. C₂×C₈	C₂×D₉₇	semidirect