Subgroups — #27163087

Order 12704.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Dic₃₉₇	C₂	1	2	yes	maximal
2	C₄⋊Dic₃₉₇	C₂	1	1	yes	maximal, characteristic
3	#20263710	C₂	1	2	yes	maximal
4	C₄×Dic₃₉₇	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₄×C₃₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₉₇	C₂²	1	1	yes	characteristic
7	C₂×Dic₃₉₇	C₂²	1	1	yes	characteristic
8	C₂×Dic₃₉₇	C₂²	1	2	yes	—
9	C₂×C₄×C₃₉₇	C₂²	1	2	yes	—
10	C₂×C₄×C₃₉₇	C₂²	1	1	yes	characteristic
11	C₂²×C₃₉₇	C₂³	1	1	yes	characteristic
12	C₄×C₃₉₇	—	2	2	no	—
13	C₄×C₃₉₇	Q₈	1	2	yes	—
14	Dic₃₉₇	—	2	1	no	—
15	Dic₃₉₇	—	2	1	no	—
16	Dic₃₉₇	—	2	2	no	—
17	C₂×C₃₉₇	C₂×Q₈	1	1	yes	characteristic
18	C₂×C₃₉₇	#41	1	1	yes	characteristic
19	C₂×C₃₉₇	#41	1	1	yes	characteristic
20	C₃₉₇	#125	1	1	yes	characteristic, complemented, Sylow
21	#125	—	397	1	no	maximal, Sylow
22	C₄⋊C₄	—	397	2	no	—
23	C₄⋊C₄	—	397	2	no	—
24	C₄⋊C₄	—	397	1	no	—
25	C₄⋊C₄	D₃₉₇	1	1	yes	characteristic
26	C₄²	—	397	1	no	—
27	C₂×C₄	—	397	1	no	—
28	C₂×C₄	—	397	1	no	—
29	C₂×C₄	—	397	2	no	—
30	C₂×C₄	C₂×D₃₉₇	1	2	yes	—
31	C₂×C₄	C₂×D₃₉₇	1	1	yes	characteristic
32	C₂²	C₂²×D₃₉₇	1	1	yes	characteristic
33	C₄	—	2	2	no	—
34	C₄	—	794	1	no	—
35	C₄	—	794	1	no	—
36	C₄	—	794	2	no	—
37	C₄	Dic₇₉₄	1	2	yes	—
38	C₂	C₂×Dic₇₉₄	1	1	yes	characteristic
39	C₂	#20263730	1	1	yes	characteristic
40	C₂	#20263732	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

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

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

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