Subgroups — #27163052

Order 12704.

76 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
2	#20263710	C₂	1	2	yes	maximal, complemented
3	C₂²⋊Dic₃₉₇	C₂	1	1	yes	maximal, characteristic
4	C₄×Dic₃₉₇	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₄×C₃₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₉₇	C₄	1	4	yes	complemented
7	C₂×Dic₃₉₇	C₂²	1	2	yes	—
8	C₂×Dic₃₉₇	C₂²	1	1	yes	characteristic
9	C₂×Dic₃₉₇	C₂²	1	1	yes	characteristic
10	C₂³×C₃₉₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₃₉₇	C₂²	1	2	yes	—
12	C₂²×C₃₉₇	—	2	2	no	—
13	C₂²×C₃₉₇	C₂×C₄	1	2	yes	—
14	C₂²×C₃₉₇	C₂³	1	1	yes	characteristic
15	C₄×C₃₉₇	—	2	2	no	—
16	Dic₃₉₇	—	2	2	no	—
17	Dic₃₉₇	C₂×C₄	1	2	yes	—
18	Dic₃₉₇	C₂×C₄	1	2	yes	—
19	C₂×C₃₉₇	—	2	2	no	—
20	C₂×C₃₉₇	C₂²×C₄	1	1	yes	characteristic
21	C₂×C₃₉₇	#41	1	2	yes	—
22	C₃₉₇	#117	1	1	yes	characteristic, complemented, Sylow
23	#117	—	397	1	no	maximal, Sylow
24	C₂²×C₄	—	397	1	no	—
25	C₄⋊C₄	—	397	2	no	—
26	C₂²⋊C₄	—	397	1	no	—
27	C₂²⋊C₄	D₃₉₇	1	1	yes	characteristic
28	C₄²	—	397	2	no	—
29	C₂³	C₂×D₃₉₇	1	1	yes	characteristic
30	C₂×C₄	—	397	2	no	—
31	C₂×C₄	—	397	4	no	—
32	C₂×C₄	—	397	1	no	—
33	C₂×C₄	—	397	1	no	—
34	C₂×C₄	C₂×D₃₉₇	1	2	yes	—
35	C₂²	—	2	2	no	—
36	C₂²	C₄×D₃₉₇	1	2	yes	—
37	C₂²	C₂²×D₃₉₇	1	1	yes	characteristic
38	C₄	—	2	2	no	—
39	C₄	—	794	2	no	—
40	C₄	—	397	2	no	—
41	C₄	—	397	2	no	—
42	C₂	—	2	2	no	—
43	C₂	C₂×C₄×D₃₉₇	1	1	yes	characteristic
44	C₂	#20263730	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃₉₇
commutator	C₂×C₃₉₇	C₂²×C₄
Fitting	C₂²⋊C₄×C₃₉₇	C₂
socle, cosocle	C₂²×C₃₉₇	C₂³
prime radical	C₂³×C₃₉₇	C₂²
hypercenter	C₂²⋊C₄	D₃₉₇
hypocenter	C₃₉₇	#117
LCS	G ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. #20263710	C₂	semidirect
4. C₄×Dic₃₉₇	C₂	semidirect
6. C₂×Dic₃₉₇	C₄	semidirect
22. C₃₉₇	#117	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
C₄×Dic₃₉₇	2
C₂²⋊C₄×C₃₉₇	2
#117	397