Subgroups — #38762872

Order 25408.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27163096	C₂	1	1	yes	maximal, characteristic
2	#27163020	C₂	1	4	yes	maximal
3	C₄×#15691787	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₃₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₇₉₄	—	2	1	no	—
6	C₂×#15691787	C₂²	1	2	yes	—
7	C₄⋊Dic₃₉₇	—	2	1	no	—
8	C₄⋊Dic₃₉₇	C₂²	1	2	yes	—
9	C₂×Q₈×C₃₉₇	C₂²	1	2	yes	—
10	C₄⋊C₄×C₃₉₇	—	2	1	no	—
11	C₄⋊C₄×C₃₉₇	—	2	1	no	—
12	C₄²×C₃₉₇	C₂²	1	1	yes	characteristic
13	Dic₇₉₄	—	4	2	no	—
14	C₂×Dic₃₉₇	—	2	2	no	—
15	#15691787	—	2	2	no	—
16	Q₈×C₃₉₇	—	2	4	no	—
17	C₂×C₄×C₃₉₇	—	2	2	no	—
18	C₂×C₄×C₃₉₇	D₄	1	1	yes	characteristic
19	C₂×C₄×C₃₉₇	D₄	1	1	yes	characteristic
20	C₂×C₄×C₃₉₇	C₂³	1	1	yes	characteristic
21	C₂²×C₃₉₇	C₂×D₄	1	1	yes	characteristic
22	C₄×C₃₉₇	—	4	2	no	—
23	C₄×C₃₉₇	SD₁₆	1	2	yes	—
24	C₄×C₃₉₇	Q₁₆	1	2	yes	—
25	C₄×C₃₉₇	#41	1	2	yes	—
26	Dic₃₉₇	—	4	2	no	—
27	C₂×C₃₉₇	#124	1	1	yes	characteristic
28	C₂×C₃₉₇	C₂×SD₁₆	1	1	yes	characteristic
29	C₂×C₃₉₇	C₂×Q₁₆	1	1	yes	characteristic
30	C₃₉₇	#487	1	1	yes	characteristic, complemented, Sylow
31	#487	—	397	1	no	maximal, Sylow
32	C₄⋊Q₈	—	397	1	no	—
33	C₄⋊Q₈	D₃₉₇	1	1	yes	characteristic
34	#103	—	397	4	no	—
35	C₄×C₈	—	397	1	no	—
36	C₂×Q₈	—	794	1	no	—
37	C₂×Q₈	C₂×D₃₉₇	1	2	yes	—
38	C₂×C₈	—	397	2	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	2	1	no	—
41	C₄⋊C₄	—	794	1	no	—
42	C₄⋊C₄	—	397	2	no	—
43	C₄²	C₂×D₃₉₇	1	1	yes	characteristic
44	Q₈	—	2	4	no	—
45	Q₈	—	1588	2	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	794	2	no	—
48	C₂×C₄	C₂²⋊D₃₉₇	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₃₉₇	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₃₉₇	1	1	yes	characteristic
51	C₈	—	794	2	no	—
52	C₂²	C₂×C₂²⋊D₃₉₇	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	—	1588	2	no	—
55	C₄	#20263718	1	2	yes	—
56	C₄	#20263721	1	2	yes	—
57	C₄	#20263730	1	2	yes	—
58	C₂	#27163083	1	1	yes	characteristic
59	C₂	C₂×#20263718	1	1	yes	characteristic
60	C₂	C₂×#20263721	1	1	yes	characteristic

	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	#487
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
30. C₃₉₇	#487	semidirect

Maximal subgroups

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

subgroup	index
#27163096	2
#27163020	2
C₄×#15691787	2
C₄⋊Q₈×C₃₉₇	2
#487	397