Subgroups — #27163064

Order 12704.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊D₃₉₇	C₂	1	2	yes	maximal, complemented
3	C₂²⋊Dic₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#20263708	C₂	1	2	yes	maximal, complemented
5	C₂×D₄×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₉₇	—	2	2	no	—
7	C₂²×D₃₉₇	—	2	4	no	—
8	C₂²×D₃₉₇	C₂²	1	2	yes	—
9	C₂²⋊D₃₉₇	—	2	4	no	—
10	C₂×Dic₃₉₇	C₂²	1	2	yes	complemented
11	C₂³×C₃₉₇	C₂²	1	1	yes	characteristic
12	C₂³×C₃₉₇	C₂²	1	1	yes	characteristic, complemented
13	D₄×C₃₉₇	—	2	2	no	—
14	C₂×C₄×C₃₉₇	C₂²	1	1	yes	characteristic, complemented
15	C₂²×C₃₉₇	—	2	2	no	—
16	C₂²×C₃₉₇	—	2	1	no	—
17	C₂²×C₃₉₇	—	2	2	no	—
18	C₂²×C₃₉₇	D₄	1	2	yes	complemented
19	C₂²×C₃₉₇	C₂³	1	1	yes	characteristic
20	C₂×D₃₉₇	—	2	8	no	—
21	C₂×D₃₉₇	—	2	2	no	—
22	C₂×D₃₉₇	—	2	2	no	—
23	C₂×D₃₉₇	D₄	1	4	yes	complemented
24	C₄×C₃₉₇	—	2	1	no	—
25	Dic₃₉₇	—	2	2	no	—
26	C₂×C₃₉₇	—	2	2	no	—
27	C₂×C₃₉₇	—	4	1	no	—
28	C₂×C₃₉₇	C₂×D₄	1	2	yes	—
29	C₂×C₃₉₇	C₂×D₄	1	1	yes	characteristic
30	D₃₉₇	—	2	4	no	—
31	C₃₉₇	#120	1	1	yes	characteristic, complemented, Sylow
32	#120	—	397	1	no	maximal, Sylow
33	C₂⁴	—	397	1	no	—
34	C₂×D₄	—	397	2	no	—
35	C₂×D₄	D₃₉₇	1	1	yes	characteristic, complemented
36	C₂²⋊C₄	—	397	1	no	—
37	C₂²⋊C₄	—	397	2	no	—
38	C₂³	—	794	4	no	—
39	C₂³	—	794	2	no	—
40	C₂³	—	397	2	no	—
41	C₂³	C₂×D₃₉₇	1	1	yes	characteristic
42	C₂³	C₂×D₃₉₇	1	1	yes	characteristic, complemented
43	D₄	—	2	2	no	—
44	D₄	—	794	4	no	—
45	C₂×C₄	—	397	2	no	—
46	C₂×C₄	C₂×D₃₉₇	1	1	yes	characteristic, complemented
47	C₂²	—	2	2	no	—
48	C₂²	—	2	2	no	—
49	C₂²	—	2	1	no	—
50	C₂²	—	794	8	no	—
51	C₂²	—	794	2	no	—
52	C₂²	—	794	2	no	—
53	C₂²	—	397	4	no	—
54	C₂²	C₂²⋊D₃₉₇	1	2	yes	complemented
55	C₂²	C₂²×D₃₉₇	1	1	yes	characteristic
56	C₄	—	2	1	no	—
57	C₄	—	794	2	no	—
58	C₂	—	2	2	no	—
59	C₂	—	4	1	no	—
60	C₂	—	794	4	no	—
61	C₂	D₄×D₃₉₇	1	2	yes	—
62	C₂	C₂×C₂²⋊D₃₉₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃₉₇
commutator, socle, cosocle	C₂²×C₃₉₇	C₂³
Fitting	C₂×D₄×C₃₉₇	C₂
hypercenter	C₂×D₄	D₃₉₇
hypocenter	C₃₉₇	#120
LCS	G ▷ C₂²×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×D₃₉₇	C₂	semidirect
2. C₂×C₂²⋊D₃₉₇	C₂	semidirect
3. C₂²⋊Dic₃₉₇	C₂	semidirect
4. #20263708	C₂	semidirect
5. C₂×D₄×C₃₉₇	C₂	semidirect
10. C₂×Dic₃₉₇	C₂²	semidirect
12. C₂³×C₃₉₇	C₂²	semidirect
14. C₂×C₄×C₃₉₇	C₂²	semidirect
18. C₂²×C₃₉₇	D₄	semidirect
23. C₂×D₃₉₇	D₄	semidirect
31. C₃₉₇	#120	semidirect
35. C₂×D₄	D₃₉₇	semidirect
42. C₂³	C₂×D₃₉₇	semidirect
46. C₂×C₄	C₂×D₃₉₇	semidirect
54. C₂²	C₂²⋊D₃₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×D₃₉₇	2
C₂×C₂²⋊D₃₉₇	2
C₂²⋊Dic₃₉₇	2
#20263708	2
C₂×D₄×C₃₉₇	2
#120	397