Subgroups — #27163077

Order 12704.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#20263710	C₂	1	2	yes	maximal, complemented
3	C₂²⋊Dic₃₉₇	C₂	1	2	yes	maximal, complemented
4	C₂²⋊Dic₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄×C₃₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₉₇	—	2	2	no	—
7	C₂×Dic₃₉₇	C₂²	1	2	yes	—
8	C₂×Dic₃₉₇	C₂²	1	2	yes	—
9	C₂³×C₃₉₇	C₂²	1	1	yes	characteristic
10	C₂³×C₃₉₇	C₂²	1	1	yes	characteristic
11	D₄×C₃₉₇	—	2	2	no	—
12	C₂×C₄×C₃₉₇	C₂²	1	1	yes	characteristic
13	C₂²×C₃₉₇	—	2	2	no	—
14	C₂²×C₃₉₇	—	2	1	no	—
15	C₂²×C₃₉₇	—	2	2	no	—
16	C₂²×C₃₉₇	D₄	1	2	yes	—
17	C₂²×C₃₉₇	C₂³	1	1	yes	characteristic
18	C₄×C₃₉₇	—	2	1	no	—
19	Dic₃₉₇	—	2	2	no	—
20	Dic₃₉₇	—	2	2	no	—
21	C₂×C₃₉₇	—	2	2	no	—
22	C₂×C₃₉₇	—	4	1	no	—
23	C₂×C₃₉₇	C₂×D₄	1	1	yes	characteristic
24	C₂×C₃₉₇	#41	1	2	yes	—
25	C₃₉₇	#123	1	1	yes	characteristic, complemented, Sylow
26	#123	—	397	1	no	maximal, Sylow
27	C₂×D₄	D₃₉₇	1	1	yes	characteristic
28	C₂²×C₄	—	397	1	no	—
29	C₄⋊C₄	—	397	2	no	—
30	C₂²⋊C₄	—	397	1	no	—
31	C₂²⋊C₄	—	397	2	no	—
32	C₂³	C₂×D₃₉₇	1	1	yes	characteristic
33	C₂³	C₂×D₃₉₇	1	1	yes	characteristic
34	D₄	—	2	2	no	—
35	C₂×C₄	—	794	2	no	—
36	C₂×C₄	—	397	2	no	—
37	C₂×C₄	—	397	2	no	—
38	C₂×C₄	C₂×D₃₉₇	1	1	yes	characteristic
39	C₂²	—	2	2	no	—
40	C₂²	—	2	1	no	—
41	C₂²	—	2	2	no	—
42	C₂²	C₂²⋊D₃₉₇	1	2	yes	—
43	C₂²	C₂²×D₃₉₇	1	1	yes	characteristic
44	C₄	—	2	1	no	—
45	C₄	—	794	2	no	—
46	C₄	—	794	2	no	—
47	C₂	—	2	2	no	—
48	C₂	—	4	1	no	—
49	C₂	C₂×C₂²⋊D₃₉₇	1	1	yes	characteristic
50	C₂	#20263730	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃₉₇
commutator, socle, cosocle	C₂²×C₃₉₇	C₂³
Fitting, prime radical	C₂×D₄×C₃₉₇	C₂
hypercenter	C₂×D₄	D₃₉₇
hypocenter	C₃₉₇	#123
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. C₂²×Dic₃₉₇	C₂	semidirect
2. #20263710	C₂	semidirect
3. C₂²⋊Dic₃₉₇	C₂	semidirect
4. C₂²⋊Dic₃₉₇	C₂	semidirect
25. C₃₉₇	#123	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₃₉₇	2
#20263710	2
C₂²⋊Dic₃₉₇	2
C₂×D₄×C₃₉₇	2
#123	397