Subgroups — #38762890

Order 25408.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27163088	C₂	1	1	yes	maximal, characteristic
2	#27163020	C₂	1	4	yes	maximal
3	#27163007	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₃₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#15691787	C₂²	1	2	yes	—
6	C₄⋊Dic₃₉₇	C₂²	1	2	yes	—
7	#20263710	—	2	2	no	—
8	C₂×Q₈×C₃₉₇	C₂²	1	2	yes	—
9	C₄⋊C₄×C₃₉₇	—	2	2	no	—
10	C₄²×C₃₉₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₉₇	—	2	2	no	—
12	#15691787	—	2	2	no	—
13	Q₈×C₃₉₇	—	2	4	no	—
14	C₂×C₄×C₃₉₇	—	2	2	no	—
15	C₂×C₄×C₃₉₇	D₄	1	2	yes	—
16	C₂×C₄×C₃₉₇	C₂³	1	1	yes	characteristic
17	C₂²×C₃₉₇	C₂×D₄	1	1	yes	characteristic
18	C₄×C₃₉₇	—	2	2	no	—
19	C₄×C₃₉₇	—	4	2	no	—
20	C₄×C₃₉₇	#41	1	2	yes	—
21	Dic₃₉₇	—	4	2	no	—
22	C₂×C₃₉₇	#124	1	1	yes	characteristic
23	C₂×C₃₉₇	#137	1	2	yes	—
24	C₃₉₇	#491	1	1	yes	characteristic, complemented, Sylow
25	#491	—	397	1	no	maximal, Sylow
26	C₄⋊Q₈	D₃₉₇	1	1	yes	characteristic
27	#125	—	397	1	no	—
28	#103	—	397	4	no	—
29	#97	—	397	1	no	—
30	C₂×Q₈	C₂×D₃₉₇	1	2	yes	—
31	C₂×C₈	—	397	2	no	—
32	C₄⋊C₄	—	2	2	no	—
33	C₄⋊C₄	—	794	2	no	—
34	C₄⋊C₄	—	397	2	no	—
35	C₄²	C₂×D₃₉₇	1	1	yes	characteristic
36	Q₈	—	2	4	no	—
37	C₂×C₄	—	2	2	no	—
38	C₂×C₄	—	794	2	no	—
39	C₂×C₄	C₂²⋊D₃₉₇	1	2	yes	—
40	C₂×C₄	C₂²×D₃₉₇	1	1	yes	characteristic
41	C₈	—	794	2	no	—
42	C₂²	C₂×C₂²⋊D₃₉₇	1	1	yes	characteristic
43	C₄	—	2	2	no	—
44	C₄	—	4	2	no	—
45	C₄	—	1588	2	no	—
46	C₄	#20263730	1	2	yes	—
47	C₂	#27163083	1	1	yes	characteristic
48	C₂	#27163127	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₃₉₇
commutator, cosocle	C₂×C₄×C₃₉₇	C₂³
Frattini	C₂×C₄	C₂²×D₃₉₇
Fitting	C₄⋊Q₈×C₃₉₇	C₂
socle, prime radical	C₂²×C₃₉₇	C₂×D₄
hypercenter	C₄⋊Q₈	D₃₉₇
hypocenter	C₃₉₇	#491
LCS	G ▷ C₂×C₄×C₃₉₇ ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
24. C₃₉₇	#491	semidirect

Maximal subgroups

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

subgroup	index
#27163088	2
#27163020	2
#27163007	2
C₄⋊Q₈×C₃₉₇	2
#491	397