Subgroups — #65381

Order 320.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#689	C₂	1	4	yes	maximal
3	#4015	C₂	1	1	yes	maximal, characteristic
4	C₄×#168	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×Q₈	C₂²	1	2	yes	—
6	C₂×Dic₁₀	C₂²	1	2	yes	—
7	C₅×C₄⋊C₄	—	2	2	no	—
8	C₄²×C₅	C₂²	1	1	yes	characteristic
9	#689	—	2	8	no	—
10	C₄⋊Dic₅	—	2	2	no	—
11	C₂×#168	C₂²	1	2	yes	—
12	#494	—	5	1	no	maximal, Sylow
13	C₅×Q₈	—	2	4	no	—
14	C₂×C₄×C₅	—	2	2	no	—
15	C₂×C₄×C₅	D₄	1	2	yes	—
16	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
17	C₂×Dic₅	—	2	2	no	—
18	Dic₁₀	—	2	4	no	—
19	#168	D₄	1	4	yes	—
20	C₂×Q₁₆	—	5	4	no	—
21	C₄⋊Q₈	—	5	1	no	—
22	C₄⋊Q₈	D₅	1	1	yes	characteristic
23	C₄×C₈	—	5	1	no	—
24	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
25	C₄×C₅	—	4	2	no	—
26	C₄×C₅	Q₁₆	1	4	yes	complemented
27	C₄×C₅	C₂×D₄	1	2	yes	—
28	Dic₅	—	4	2	no	—
29	C₂×Q₈	—	5	2	no	—
30	C₂×Q₈	C₂×D₅	1	2	yes	—
31	Q₁₆	—	10	8	no	—
32	C₂×C₈	—	5	2	no	—
33	C₄⋊C₄	—	2	2	no	—
34	C₄⋊C₄	—	10	2	no	—
35	C₄²	C₂×D₅	1	1	yes	characteristic
36	C₂×C₅	#127	1	1	yes	characteristic
37	C₂×C₅	C₂×Q₁₆	1	2	yes	—
38	Q₈	—	2	4	no	—
39	Q₈	—	10	4	no	—
40	C₂×C₄	—	2	2	no	—
41	C₂×C₄	—	10	2	no	—
42	C₂×C₄	C₂²⋊D₅	1	2	yes	—
43	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
44	C₈	—	5	4	no	—
45	C₅	#494	1	1	yes	characteristic, complemented, Sylow
46	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
47	C₄	—	4	2	no	—
48	C₄	—	20	2	no	—
49	C₄	#689	1	4	yes	complemented
50	C₄	D₄×D₅	1	2	yes	—
51	C₂	#4086	1	1	yes	characteristic
52	C₂	C₂×#689	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₅	C₂×D₄
hypercenter	C₄⋊Q₈	D₅
hypocenter	C₅	#494
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
26. C₄×C₅	Q₁₆	semidirect
45. C₅	#494	semidirect
49. C₄	#689	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₄⋊Q₈	2
C₂×#689	2
#4015	2
C₄×#168	2
#494	5