Subgroups — #27161816

Order 12608.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263320	C₂	1	2	yes	maximal, complemented
2	#20263307	C₂	1	3	yes	maximal, complemented
3	#20263298	C₂	1	6	yes	maximal, complemented
4	#20263294	C₂	1	3	yes	maximal, complemented
5	#127×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₁₉₇	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊D₁₉₇	C₂²	1	6	yes	—
8	C₂²×Dic₁₉₇	C₂²	1	3	yes	complemented
9	#15691337	C₂²	1	6	yes	complemented
10	C₂²⋊Dic₁₉₇	C₂²	1	3	yes	—
11	C₂²⋊Dic₁₉₇	C₂²	1	3	yes	complemented
12	#15691335	C₂²	1	6	yes	—
13	C₂×D₄×C₁₉₇	C₂²	1	3	yes	—
14	C₂×D₄×C₁₉₇	C₂²	1	3	yes	complemented
15	C₄²×C₁₉₇	C₂²	1	1	yes	characteristic, complemented
16	C₂²×D₁₉₇	—	4	3	no	—
17	C₂²×D₁₉₇	C₂³	1	2	yes	—
18	C₂²⋊D₁₉₇	—	4	6	no	—
19	C₂×Dic₁₉₇	—	4	3	no	—
20	C₂×Dic₁₉₇	C₂³	1	6	yes	—
21	C₂³×C₁₉₇	C₂³	1	1	yes	characteristic
22	C₂³×C₁₉₇	C₂³	1	3	yes	—
23	D₄×C₁₉₇	—	4	3	no	—
24	D₄×C₁₉₇	—	4	3	no	—
25	C₂×C₄×C₁₉₇	C₂³	1	3	yes	—
26	C₂²×C₁₉₇	—	2	3	no	—
27	C₂²×C₁₉₇	—	2	6	no	—
28	C₂²×C₁₉₇	—	2	3	no	—
29	C₂²×C₁₉₇	C₂⁴	1	1	yes	characteristic
30	C₂×D₁₉₇	—	2	6	no	—
31	C₂×D₁₉₇	—	4	4	no	—
32	C₄×C₁₉₇	—	2	3	no	—
33	Dic₁₉₇	—	2	6	no	—
34	C₂×C₁₉₇	—	4	1	no	—
35	C₂×C₁₉₇	—	4	3	no	—
36	C₂×C₁₉₇	H(5,𝔽₂)	1	3	yes	—
37	D₁₉₇	—	4	2	no	—
38	C₁₉₇	#560	1	1	yes	characteristic, complemented, Sylow
39	#560	—	197	1	no	maximal, Sylow
40	#127	D₁₉₇	1	1	yes	characteristic, complemented
41	#126	—	197	2	no	—
42	#123	—	197	3	no	—
43	#121	—	197	6	no	—
44	#120	—	197	3	no	—
45	C₂⁴	—	197	1	no	—
46	C₂×D₄	—	197	6	no	—
47	C₂×D₄	C₂×D₁₉₇	1	3	yes	—
48	C₂×D₄	C₂×D₁₉₇	1	3	yes	complemented
49	C₂²×C₄	—	197	3	no	—
50	C₄⋊C₄	—	197	6	no	—
51	C₂²⋊C₄	—	197	3	no	—
52	C₂²⋊C₄	—	197	3	no	—
53	C₂²⋊C₄	—	197	6	no	—
54	C₄²	C₂×D₁₉₇	1	1	yes	characteristic, complemented
55	C₂³	—	788	3	no	—
56	C₂³	—	197	2	no	—
57	C₂³	C₂²×D₁₉₇	1	1	yes	characteristic
58	C₂³	C₂²×D₁₉₇	1	3	yes	—
59	D₄	—	4	3	no	—
60	D₄	—	4	3	no	—
61	D₄	—	788	6	no	—
62	C₂×C₄	—	788	3	no	—
63	C₂×C₄	—	197	6	no	—
64	C₂×C₄	C₂²×D₁₉₇	1	3	yes	—
65	C₂²	—	2	3	no	—
66	C₂²	—	2	6	no	—
67	C₂²	—	2	3	no	—
68	C₂²	—	788	4	no	—
69	C₂²	—	394	6	no	—
70	C₂²	C₂³×D₁₉₇	1	1	yes	characteristic
71	C₄	—	2	3	no	—
72	C₄	—	394	6	no	—
73	C₂	—	4	1	no	—
74	C₂	—	4	3	no	—
75	C₂	—	788	2	no	—
76	C₂	#20263371	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	#560
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. #20263320	C₂	semidirect
2. #20263307	C₂	semidirect
3. #20263298	C₂	semidirect
4. #20263294	C₂	semidirect
5. #127×C₁₉₇	C₂	semidirect
8. C₂²×Dic₁₉₇	C₂²	semidirect
9. #15691337	C₂²	semidirect
11. C₂²⋊Dic₁₉₇	C₂²	semidirect
14. C₂×D₄×C₁₉₇	C₂²	semidirect
15. C₄²×C₁₉₇	C₂²	semidirect
38. C₁₉₇	#560	semidirect
40. #127	D₁₉₇	semidirect
48. C₂×D₄	C₂×D₁₉₇	semidirect
54. C₄²	C₂×D₁₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20263320	2
#20263307	2
#20263298	2
#20263294	2
#127×C₁₉₇	2
#560	197