Subgroups — #27163054

Order 12704.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₃₉₇	C₂	1	1	yes	maximal, characteristic
2	C₄⋊Dic₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#20263708	C₂	1	2	yes	maximal
4	C₄×Dic₃₉₇	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₄×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₉₇	C₂²	1	1	yes	characteristic
7	C₂×Dic₃₉₇	C₂²	1	2	yes	—
8	C₂×Dic₃₉₇	C₂²	1	1	yes	characteristic
9	C₄×D₃₉₇	C₄	1	4	yes	complemented
10	C₂×C₄×C₃₉₇	C₂²	1	2	yes	—
11	C₂×C₄×C₃₉₇	C₂²	1	1	yes	characteristic
12	C₂²×C₃₉₇	C₂³	1	1	yes	characteristic
13	C₂×D₃₉₇	—	2	1	no	—
14	C₂×D₃₉₇	—	2	1	no	—
15	C₂×D₃₉₇	C₂×C₄	1	2	yes	—
16	C₄×C₃₉₇	—	2	2	no	—
17	C₄×C₃₉₇	C₂×C₄	1	2	yes	—
18	Dic₃₉₇	—	2	2	no	—
19	Dic₃₉₇	C₂×C₄	1	2	yes	—
20	C₂×C₃₉₇	C₂²×C₄	1	1	yes	characteristic
21	C₂×C₃₉₇	#41	1	1	yes	characteristic
22	C₂×C₃₉₇	#41	1	1	yes	characteristic
23	D₃₉₇	—	2	2	no	—
24	C₃₉₇	#117	1	1	yes	characteristic, complemented, Sylow
25	#117	—	397	1	no	maximal, Sylow
26	C₂²×C₄	—	397	1	no	—
27	C₄⋊C₄	—	397	1	no	—
28	C₄⋊C₄	D₃₉₇	1	1	yes	characteristic, complemented
29	C₂²⋊C₄	—	397	2	no	—
30	C₄²	—	397	2	no	—
31	C₂³	—	397	1	no	—
32	C₂×C₄	—	397	2	no	—
33	C₂×C₄	—	397	1	no	—
34	C₂×C₄	—	397	4	no	—
35	C₂×C₄	C₂×D₃₉₇	1	2	yes	—
36	C₂×C₄	C₂×D₃₉₇	1	1	yes	characteristic
37	C₂²	—	794	1	no	—
38	C₂²	—	794	1	no	—
39	C₂²	—	397	2	no	—
40	C₂²	C₂²×D₃₉₇	1	1	yes	characteristic
41	C₄	—	2	2	no	—
42	C₄	—	794	2	no	—
43	C₄	—	397	2	no	—
44	C₄	C₄×D₃₉₇	1	2	yes	—
45	C₂	—	794	2	no	—
46	C₂	C₂×C₄×D₃₉₇	1	1	yes	characteristic
47	C₂	#20263730	1	1	yes	characteristic
48	C₂	#20263732	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₄⋊Dic₃₉₇	C₂	semidirect
4. C₄×Dic₃₉₇	C₂	semidirect
5. C₄⋊C₄×C₃₉₇	C₂	semidirect
9. C₄×D₃₉₇	C₄	semidirect
24. C₃₉₇	#117	semidirect
28. C₄⋊C₄	D₃₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₃₉₇	2
C₄⋊Dic₃₉₇	2
#20263708	2
C₄×Dic₃₉₇	2
C₄⋊C₄×C₃₉₇	2
#117	397