Subgroups — #127×D₁₉₇ #27161617

Order 12608.

498 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×D₁₉₇	C₂	1	6	yes	maximal, complemented
2	#20263323	C₂	1	1	yes	maximal, characteristic, complemented
3	#20263322	C₂	1	6	yes	maximal, complemented
4	C₄²×D₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#127×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₁₉₇	C₂²	1	4	yes	—
7	C₂×D₇₈₈	C₂²	1	6	yes	complemented
8	C₂×C₂²⋊D₁₉₇	C₂²	1	12	yes	complemented
9	D₄×D₁₉₇	—	2	24	no	—
10	C₂×C₄×D₁₉₇	C₂²	1	3	yes	—
11	C₄×Dic₁₉₇	C₂²	1	3	yes	complemented
12	C₂×D₄×C₁₉₇	C₂²	1	6	yes	complemented
13	C₄²×C₁₉₇	C₂²	1	1	yes	characteristic, complemented
14	C₂²×D₁₉₇	—	2	24	no	—
15	C₂²×D₁₉₇	C₂³	1	4	yes	—
16	C₂²×D₁₉₇	C₂³	1	1	yes	characteristic
17	D₇₈₈	—	2	12	no	—
18	C₂²⋊D₁₉₇	—	2	24	no	—
19	C₂×Dic₁₉₇	C₂³	1	3	yes	—
20	C₄×D₁₉₇	D₄	1	12	yes	complemented
21	C₂³×C₁₉₇	C₂³	1	4	yes	—
22	D₄×C₁₉₇	—	2	12	no	—
23	C₂×C₄×C₁₉₇	C₂³	1	3	yes	—
24	C₂²×C₁₉₇	—	2	12	no	—
25	C₂²×C₁₉₇	C₂⁴	1	1	yes	characteristic
26	C₂×D₁₉₇	—	2	12	no	—
27	C₂×D₁₉₇	—	4	16	no	—
28	C₂×D₁₉₇	C₂×D₄	1	6	yes	—
29	C₄×C₁₉₇	C₂×D₄	1	6	yes	complemented
30	Dic₁₉₇	C₂×D₄	1	6	yes	complemented
31	C₂×C₁₉₇	—	4	4	no	—
32	C₂×C₁₉₇	C₂²×D₄	1	3	yes	—
33	D₁₉₇	—	4	4	no	—
34	D₁₉₇	#127	1	4	yes	direct product factor
35	C₁₉₇	C₂×#127	1	1	yes	characteristic, complemented, Sylow
36	C₂×#127	—	197	1	no	maximal, Sylow
37	C₂²×D₄	—	197	6	no	—
38	#127	—	197	6	no	—
39	#127	—	197	1	no	—
40	#127	D₁₉₇	1	1	yes	characteristic, direct product factor
41	C₂×C₄²	—	197	1	no	—
42	C₂⁴	—	197	4	no	—
43	C₂×D₄	—	394	24	no	—
44	C₂×D₄	—	197	12	no	—
45	C₂×D₄	—	197	6	no	—
46	C₂×D₄	C₂×D₁₉₇	1	6	yes	complemented
47	C₂²×C₄	—	197	3	no	—
48	C₄²	—	197	3	no	—
49	C₄²	C₂×D₁₉₇	1	1	yes	characteristic, complemented
50	C₂³	—	394	24	no	—
51	C₂³	—	197	4	no	—
52	C₂³	—	197	1	no	—
53	C₂³	C₂²×D₁₉₇	1	4	yes	—
54	D₄	—	2	12	no	—
55	D₄	—	394	24	no	—
56	D₄	—	394	12	no	—
57	C₂×C₄	—	197	3	no	—
58	C₂×C₄	—	197	12	no	—
59	C₂×C₄	C₂²×D₁₉₇	1	3	yes	—
60	C₂²	—	2	12	no	—
61	C₂²	—	788	16	no	—
62	C₂²	—	394	12	no	—
63	C₂²	—	197	6	no	—
64	C₂²	C₂³×D₁₉₇	1	1	yes	characteristic
65	C₄	—	197	6	no	—
66	C₄	D₄×D₁₉₇	1	6	yes	complemented
67	C₂	—	4	4	no	—
68	C₂	—	788	4	no	—
69	C₂	—	197	4	no	—
70	C₂	C₂×D₄×D₁₉₇	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₉₇
commutator, socle, cosocle	C₂²×C₁₉₇	C₂⁴
Fitting	#127×C₁₉₇	C₂
hypercenter	#127	D₁₉₇
hypocenter	C₁₉₇	C₂×#127
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. #20263323	C₂	semidirect
3. #20263322	C₂	semidirect
4. C₄²×D₁₉₇	C₂	semidirect
5. #127×C₁₉₇	C₂	semidirect
7. C₂×D₇₈₈	C₂²	semidirect
8. C₂×C₂²⋊D₁₉₇	C₂²	semidirect
11. C₄×Dic₁₉₇	C₂²	semidirect
12. C₂×D₄×C₁₉₇	C₂²	semidirect
13. C₄²×C₁₉₇	C₂²	semidirect
20. C₄×D₁₉₇	D₄	semidirect
29. C₄×C₁₉₇	C₂×D₄	semidirect
30. Dic₁₉₇	C₂×D₄	semidirect
34. D₁₉₇	#127	direct (N × H)
35. C₁₉₇	C₂×#127	semidirect
40. #127	D₁₉₇	direct (N × H)
46. C₂×D₄	C₂×D₁₉₇	semidirect
49. C₄²	C₂×D₁₉₇	semidirect
66. 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
#20263323	2
#20263322	2
C₄²×D₁₉₇	2
#127×C₁₉₇	2
C₂×#127	197

Subgroups — #127×D197 #27161617

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #127×D₁₉₇ #27161617