Subgroups — #27163086

Order 12704.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Dic₃₉₇	C₂	1	1	yes	maximal, characteristic
2	#20263710	C₂	1	1	yes	maximal, characteristic
3	#20263710	C₂	1	1	yes	maximal, characteristic
4	#20263710	C₂	1	1	yes	maximal, characteristic
5	#20263710	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₃₉₇	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₄×C₃₉₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₃₉₇	C₂²	1	1	yes	characteristic
9	C₂×Dic₃₉₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₃₉₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₉₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₃₉₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₃₉₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₃₉₇	C₂²	1	1	yes	characteristic
15	C₂²×C₃₉₇	C₂³	1	1	yes	characteristic
16	C₄×C₃₉₇	—	2	1	no	—
17	C₄×C₃₉₇	—	2	1	no	—
18	C₄×C₃₉₇	—	2	1	no	—
19	Dic₃₉₇	—	2	1	no	—
20	Dic₃₉₇	—	2	1	no	—
21	Dic₃₉₇	—	2	1	no	—
22	Dic₃₉₇	Q₈	1	2	yes	—
23	C₂×C₃₉₇	C₂×Q₈	1	1	yes	characteristic
24	C₂×C₃₉₇	#41	1	1	yes	characteristic
25	C₂×C₃₉₇	#41	1	1	yes	characteristic
26	C₃₉₇	#125	1	1	yes	characteristic, complemented, Sylow
27	#125	—	397	1	no	maximal, Sylow
28	C₄⋊C₄	—	397	1	no	—
29	C₄⋊C₄	—	397	1	no	—
30	C₄⋊C₄	—	397	1	no	—
31	C₄⋊C₄	—	397	1	no	—
32	C₄⋊C₄	—	397	1	no	—
33	C₄⋊C₄	D₃₉₇	1	1	yes	characteristic
34	C₄²	—	397	1	no	—
35	C₂×C₄	—	397	1	no	—
36	C₂×C₄	—	397	1	no	—
37	C₂×C₄	—	397	1	no	—
38	C₂×C₄	—	397	1	no	—
39	C₂×C₄	C₂×D₃₉₇	1	1	yes	characteristic
40	C₂×C₄	C₂×D₃₉₇	1	1	yes	characteristic
41	C₂×C₄	C₂×D₃₉₇	1	1	yes	characteristic
42	C₂²	C₂²×D₃₉₇	1	1	yes	characteristic
43	C₄	—	2	1	no	—
44	C₄	—	2	1	no	—
45	C₄	—	2	1	no	—
46	C₄	—	794	1	no	—
47	C₄	—	794	1	no	—
48	C₄	—	794	1	no	—
49	C₄	—	397	2	no	—
50	C₂	Q₈×D₃₉₇	1	1	yes	characteristic
51	C₂	#20263730	1	1	yes	characteristic
52	C₂	#20263731	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃₉₇
commutator, socle, prime radical, cosocle	C₂²×C₃₉₇	C₂³
Fitting	C₄⋊C₄×C₃₉₇	C₂
hypercenter	C₄⋊C₄	D₃₉₇
hypocenter	C₃₉₇	#125
LCS	G ▷ C₂²×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
26. C₃₉₇	#125	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Dic₃₉₇	2
#20263710	2
C₄×Dic₃₉₇	2
C₄⋊C₄×C₃₉₇	2
#125	397