Subgroups — #27161714

Order 12608.

428 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×D₁₉₇	C₂	1	4	yes	maximal, complemented
2	#20263297	C₂	1	4	yes	maximal, complemented
3	#20263294	C₂	1	4	yes	maximal, complemented
4	C₄×D₇₈₈	C₂	1	2	yes	maximal, complemented
5	#127×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₁₉₇	C₂²	1	4	yes	complemented
7	C₂×D₇₈₈	C₂²	1	2	yes	—
8	C₂×C₂²⋊D₁₉₇	C₂²	1	8	yes	complemented
9	D₄×D₁₉₇	—	2	16	no	—
10	C₂×C₄×D₁₉₇	C₂²	1	4	yes	complemented
11	C₄⋊Dic₁₉₇	C₂²	1	2	yes	complemented
12	C₂²⋊Dic₁₉₇	C₂²	1	4	yes	complemented
13	#15691335	C₂²	1	4	yes	complemented
14	C₂×D₄×C₁₉₇	C₂²	1	4	yes	complemented
15	C₂×D₄×C₁₉₇	C₂²	1	2	yes	complemented
16	C₄²×C₁₉₇	C₂²	1	1	yes	characteristic, complemented
17	C₂²×D₁₉₇	—	2	16	no	—
18	C₂²×D₁₉₇	—	4	4	no	—
19	C₂²×D₁₉₇	C₂³	1	4	yes	—
20	D₇₈₈	D₄	1	8	yes	complemented
21	C₂²⋊D₁₉₇	—	2	16	no	—
22	C₂×Dic₁₉₇	C₂³	1	4	yes	—
23	C₄×D₁₉₇	—	2	8	no	—
24	C₂³×C₁₉₇	C₂³	1	4	yes	—
25	D₄×C₁₉₇	—	2	8	no	—
26	D₄×C₁₉₇	—	4	2	no	—
27	C₂×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₁₉₇	C₂³	1	2	yes	—
29	C₂²×C₁₉₇	—	2	8	no	—
30	C₂²×C₁₉₇	—	2	4	no	—
31	C₂²×C₁₉₇	C₂⁴	1	1	yes	characteristic
32	C₂×D₁₉₇	—	2	4	no	—
33	C₂×D₁₉₇	—	2	4	no	—
34	C₂×D₁₉₇	—	4	16	no	—
35	C₂×D₁₉₇	C₂×D₄	1	8	yes	complemented
36	C₄×C₁₉₇	—	2	1	no	—
37	C₄×C₁₉₇	C₂×D₄	1	4	yes	complemented
38	Dic₁₉₇	—	2	4	no	—
39	C₂×C₁₉₇	—	4	4	no	—
40	C₂×C₁₉₇	C₂²×D₄	1	2	yes	—
41	C₂×C₁₉₇	H(5,𝔽₂)	1	1	yes	characteristic
42	D₁₉₇	—	2	8	no	—
43	C₁₉₇	D₄²	1	1	yes	characteristic, complemented, Sylow
44	D₄²	—	197	1	no	maximal, Sylow
45	C₂²×D₄	—	197	4	no	—
46	#127	D₁₉₇	1	1	yes	characteristic, complemented
47	#121	—	197	4	no	—
48	#120	—	197	4	no	—
49	C₄×D₄	—	197	2	no	—
50	C₂⁴	—	197	4	no	—
51	C₂×D₄	—	394	16	no	—
52	C₂×D₄	—	197	8	no	—
53	C₂×D₄	—	197	2	no	—
54	C₂×D₄	C₂×D₁₉₇	1	4	yes	complemented
55	C₂×D₄	C₂×D₁₉₇	1	2	yes	complemented
56	C₂²×C₄	—	197	4	no	—
57	C₄⋊C₄	—	197	2	no	—
58	C₂²⋊C₄	—	197	4	no	—
59	C₂²⋊C₄	—	197	4	no	—
60	C₄²	C₂×D₁₉₇	1	1	yes	characteristic, complemented
61	C₂³	—	788	4	no	—
62	C₂³	—	394	16	no	—
63	C₂³	—	197	4	no	—
64	C₂³	C₂²×D₁₉₇	1	4	yes	—
65	D₄	—	2	8	no	—
66	D₄	—	4	2	no	—
67	D₄	—	394	16	no	—
68	D₄	—	197	8	no	—
69	C₂×C₄	—	394	8	no	—
70	C₂×C₄	—	197	4	no	—
71	C₂×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
72	C₂×C₄	C₂²×D₁₉₇	1	2	yes	—
73	C₂²	—	2	8	no	—
74	C₂²	—	2	4	no	—
75	C₂²	—	788	16	no	—
76	C₂²	—	394	4	no	—
77	C₂²	—	394	4	no	—
78	C₂²	—	197	8	no	—
79	C₂²	C₂³×D₁₉₇	1	1	yes	characteristic
80	C₄	—	2	1	no	—
81	C₄	—	394	4	no	—
82	C₄	D₄×D₁₉₇	1	4	yes	complemented
83	C₂	—	4	4	no	—
84	C₂	—	394	8	no	—
85	C₂	C₂×D₄×D₁₉₇	1	2	yes	—
86	C₂	#20263371	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₉₇
commutator, socle, cosocle	C₂²×C₁₉₇	C₂⁴
Fitting	#127×C₁₉₇	C₂
hypercenter	#127	D₁₉₇
hypocenter	C₁₉₇	D₄²
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×D₁₉₇	C₂	semidirect
2. #20263297	C₂	semidirect
3. #20263294	C₂	semidirect
4. C₄×D₇₈₈	C₂	semidirect
5. #127×C₁₉₇	C₂	semidirect
6. C₂³×D₁₉₇	C₂²	semidirect
8. C₂×C₂²⋊D₁₉₇	C₂²	semidirect
10. C₂×C₄×D₁₉₇	C₂²	semidirect
11. C₄⋊Dic₁₉₇	C₂²	semidirect
12. C₂²⋊Dic₁₉₇	C₂²	semidirect
13. #15691335	C₂²	semidirect
14. C₂×D₄×C₁₉₇	C₂²	semidirect
15. C₂×D₄×C₁₉₇	C₂²	semidirect
16. C₄²×C₁₉₇	C₂²	semidirect
20. D₇₈₈	D₄	semidirect
35. C₂×D₁₉₇	C₂×D₄	semidirect
37. C₄×C₁₉₇	C₂×D₄	semidirect
43. C₁₉₇	D₄²	semidirect
46. #127	D₁₉₇	semidirect
54. C₂×D₄	C₂×D₁₉₇	semidirect
55. C₂×D₄	C₂×D₁₉₇	semidirect
60. C₄²	C₂×D₁₉₇	semidirect
82. C₄	D₄×D₁₉₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×D₄×D₁₉₇	2
#20263297	2
#20263294	2
C₄×D₇₈₈	2
#127×C₁₉₇	2
D₄²	197