Subgroups — #20261370

Order 6208.

428 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×D₉₇	C₂	1	4	yes	maximal, complemented
2	#15690781	C₂	1	4	yes	maximal, complemented
3	#15690778	C₂	1	4	yes	maximal, complemented
4	C₄×D₃₈₈	C₂	1	2	yes	maximal, complemented
5	#127×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₉₇	C₂²	1	4	yes	complemented
7	C₂×D₃₈₈	C₂²	1	2	yes	—
8	C₂×C₂²⋊D₉₇	C₂²	1	8	yes	complemented
9	D₄×D₉₇	—	2	16	no	—
10	C₂×C₄×D₉₇	C₂²	1	4	yes	complemented
11	C₄⋊Dic₉₇	C₂²	1	2	yes	complemented
12	C₂²⋊Dic₉₇	C₂²	1	4	yes	complemented
13	#13095488	C₂²	1	4	yes	complemented
14	C₂×D₄×C₉₇	C₂²	1	4	yes	complemented
15	C₂×D₄×C₉₇	C₂²	1	2	yes	complemented
16	C₄²×C₉₇	C₂²	1	1	yes	characteristic, complemented
17	C₂³×C₉₇	C₂³	1	4	yes	—
18	C₂²×D₉₇	—	2	16	no	—
19	C₂²×D₉₇	—	4	4	no	—
20	C₂²×D₉₇	C₂³	1	4	yes	—
21	D₄×C₉₇	—	2	8	no	—
22	D₄×C₉₇	—	4	2	no	—
23	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₉₇	C₂³	1	2	yes	—
25	C₂²⋊D₉₇	—	2	16	no	—
26	C₂×Dic₉₇	C₂³	1	4	yes	—
27	D₃₈₈	D₄	1	8	yes	complemented
28	C₄×D₉₇	—	2	8	no	—
29	C₂²×C₉₇	—	2	8	no	—
30	C₂²×C₉₇	—	2	4	no	—
31	C₂²×C₉₇	C₂⁴	1	1	yes	characteristic
32	C₂×D₉₇	—	2	4	no	—
33	C₂×D₉₇	—	2	4	no	—
34	C₂×D₉₇	—	4	16	no	—
35	C₂×D₉₇	C₂×D₄	1	8	yes	complemented
36	C₄×C₉₇	—	2	1	no	—
37	C₄×C₉₇	C₂×D₄	1	4	yes	complemented
38	Dic₉₇	—	2	4	no	—
39	C₂×C₉₇	—	4	4	no	—
40	C₂×C₉₇	C₂²×D₄	1	2	yes	—
41	C₂×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
42	D₉₇	—	2	8	no	—
43	C₉₇	D₄²	1	1	yes	characteristic, complemented, Sylow
44	D₄²	—	97	1	no	maximal, Sylow
45	C₂²×D₄	—	97	4	no	—
46	#127	D₉₇	1	1	yes	characteristic, complemented
47	#121	—	97	4	no	—
48	#120	—	97	4	no	—
49	C₄×D₄	—	97	2	no	—
50	C₂⁴	—	97	4	no	—
51	C₂×D₄	—	97	8	no	—
52	C₂×D₄	—	97	2	no	—
53	C₂×D₄	—	194	16	no	—
54	C₂×D₄	C₂×D₉₇	1	4	yes	complemented
55	C₂×D₄	C₂×D₉₇	1	2	yes	complemented
56	C₂²×C₄	—	97	4	no	—
57	C₄⋊C₄	—	97	2	no	—
58	C₂²⋊C₄	—	97	4	no	—
59	C₂²⋊C₄	—	97	4	no	—
60	C₄²	C₂×D₉₇	1	1	yes	characteristic, complemented
61	C₂³	—	97	4	no	—
62	C₂³	—	388	4	no	—
63	C₂³	—	194	16	no	—
64	C₂³	C₂²×D₉₇	1	4	yes	—
65	D₄	—	2	8	no	—
66	D₄	—	4	2	no	—
67	D₄	—	97	8	no	—
68	D₄	—	194	16	no	—
69	C₂×C₄	—	97	4	no	—
70	C₂×C₄	—	194	8	no	—
71	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
72	C₂×C₄	C₂²×D₉₇	1	2	yes	—
73	C₂²	—	2	8	no	—
74	C₂²	—	2	4	no	—
75	C₂²	—	97	8	no	—
76	C₂²	—	388	16	no	—
77	C₂²	—	194	4	no	—
78	C₂²	—	194	4	no	—
79	C₂²	C₂³×D₉₇	1	1	yes	characteristic
80	C₄	—	2	1	no	—
81	C₄	—	194	4	no	—
82	C₄	D₄×D₉₇	1	4	yes	complemented
83	C₂	—	4	4	no	—
84	C₂	—	194	8	no	—
85	C₂	C₂×D₄×D₉₇	1	2	yes	—
86	C₂	#15690855	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂⁴
Fitting	#127×C₉₇	C₂
hypercenter	#127	D₉₇
hypocenter	C₉₇	D₄²
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×D₉₇	C₂	semidirect
2. #15690781	C₂	semidirect
3. #15690778	C₂	semidirect
4. C₄×D₃₈₈	C₂	semidirect
5. #127×C₉₇	C₂	semidirect
6. C₂³×D₉₇	C₂²	semidirect
8. C₂×C₂²⋊D₉₇	C₂²	semidirect
10. C₂×C₄×D₉₇	C₂²	semidirect
11. C₄⋊Dic₉₇	C₂²	semidirect
12. C₂²⋊Dic₉₇	C₂²	semidirect
13. #13095488	C₂²	semidirect
14. C₂×D₄×C₉₇	C₂²	semidirect
15. C₂×D₄×C₉₇	C₂²	semidirect
16. C₄²×C₉₇	C₂²	semidirect
27. D₃₈₈	D₄	semidirect
35. C₂×D₉₇	C₂×D₄	semidirect
37. C₄×C₉₇	C₂×D₄	semidirect
43. C₉₇	D₄²	semidirect
46. #127	D₉₇	semidirect
54. C₂×D₄	C₂×D₉₇	semidirect
55. C₂×D₄	C₂×D₉₇	semidirect
60. C₄²	C₂×D₉₇	semidirect
82. C₄	D₄×D₉₇	semidirect