Subgroups — #38762252

Order 25408.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₇₉₄	C₂	1	1	yes	maximal, characteristic
2	C₄×#15691787	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×C₃₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₇₉₄	C₄	1	1	yes	characteristic
5	C₂×#15691787	—	2	1	no	—
6	C₄⋊Dic₃₉₇	C₄	1	1	yes	characteristic
7	#20263710	—	2	1	no	—
8	C₄×Dic₃₉₇	—	2	1	no	—
9	C₂×C₈×C₃₉₇	—	2	1	no	—
10	C₄²×C₃₉₇	C₂²	1	1	yes	characteristic
11	Dic₇₉₄	—	2	1	no	—
12	Dic₇₉₄	C₈	1	2	yes	complemented
13	C₂×Dic₃₉₇	—	2	1	no	—
14	C₂×Dic₃₉₇	—	2	1	no	—
15	#15691787	—	2	2	no	—
16	C₂×C₄×C₃₉₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₃₉₇	D₄	1	1	yes	characteristic
18	C₂×C₄×C₃₉₇	D₄	1	1	yes	characteristic
19	C₈×C₃₉₇	—	4	1	no	—
20	C₂²×C₃₉₇	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₃₉₇	—	2	1	no	—
22	C₄×C₃₉₇	C₂×C₈	1	1	yes	characteristic
23	C₄×C₃₉₇	M₁₆	1	1	yes	characteristic
24	C₄×C₃₉₇	SD₁₆	1	1	yes	characteristic
25	C₄×C₃₉₇	Q₁₆	1	1	yes	characteristic
26	Dic₃₉₇	—	2	2	no	—
27	Dic₃₉₇	—	4	1	no	—
28	C₂×C₃₉₇	C₂²⋊C₈	1	1	yes	characteristic
29	C₂×C₃₉₇	#103	1	1	yes	characteristic
30	C₂×C₃₉₇	#104	1	1	yes	characteristic
31	C₃₉₇	Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
32	Q₈⋊C₈	—	397	1	no	maximal, Sylow
33	C₄×Q₈	—	397	1	no	—
34	C₄⋊C₈	D₃₉₇	1	1	yes	characteristic
35	C₄×C₈	—	397	1	no	—
36	C₂×Q₈	—	397	1	no	—
37	C₂×C₈	—	2	1	no	—
38	C₂×C₈	—	794	1	no	—
39	C₄⋊C₄	—	794	1	no	—
40	C₄⋊C₄	—	397	1	no	—
41	C₄²	—	794	1	no	—
42	C₄²	C₂×D₃₉₇	1	1	yes	characteristic
43	Q₈	—	794	1	no	—
44	Q₈	—	397	2	no	—
45	C₂×C₄	—	794	1	no	—
46	C₂×C₄	—	794	1	no	—
47	C₂×C₄	C₄×D₃₉₇	1	1	yes	characteristic
48	C₂×C₄	C₂²⋊D₃₉₇	1	1	yes	characteristic
49	C₂×C₄	D₁₅₈₈	1	1	yes	characteristic
50	C₈	—	4	1	no	—
51	C₈	—	794	2	no	—
52	C₂²	#20263708	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	794	2	no	—
55	C₄	—	1588	1	no	—
56	C₄	C₈×D₃₉₇	1	1	yes	characteristic
57	C₄	#20263714	1	1	yes	characteristic
58	C₄	#20263719	1	1	yes	characteristic
59	C₄	#20263721	1	1	yes	characteristic
60	C₂	#27163008	1	1	yes	characteristic
61	C₂	#27163019	1	1	yes	characteristic
62	C₂	#27163022	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₃₉₇
commutator	C₄×C₃₉₇	C₂×C₈
Frattini	C₄²	C₂×D₃₉₇
Fitting	C₄⋊C₈×C₃₉₇	C₂
socle, prime radical	C₂²×C₃₉₇	C₂²⋊C₄
cosocle	C₄²×C₃₉₇	C₂²
hypercenter	C₄⋊C₈	D₃₉₇
hypocenter	C₃₉₇	Q₈⋊C₈
LCS	G ▷ C₄×C₃₉₇ ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
12. Dic₇₉₄	C₈	semidirect
31. C₃₉₇	Q₈⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₇₉₄	2
C₄×#15691787	2
C₄⋊C₈×C₃₉₇	2
Q₈⋊C₈	397