Subgroups — #65369

Order 320.

90 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	#4016	C₂	1	1	yes	maximal, characteristic
3	#3940	C₂	1	2	yes	maximal
4	#3939	C₂	1	2	yes	maximal
5	#3935	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×Q₈	—	2	1	no	—
7	C₅×C₄⋊C₄	—	2	1	no	—
8	C₅×C₄⋊C₄	C₂²	1	2	yes	—
9	C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
12	#683	—	2	1	no	—
13	#683	—	2	1	no	—
14	C₂×#168	C₂²	1	2	yes	—
15	#501	—	5	1	no	maximal, Sylow
16	C₅×Q₈	—	4	2	no	—
17	C₂×C₄×C₅	—	2	2	no	—
18	C₂×C₄×C₅	D₄	1	1	yes	characteristic
19	C₂×C₄×C₅	D₄	1	1	yes	characteristic
20	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
21	C₂×Dic₅	—	2	1	no	—
22	C₂×Dic₅	—	2	1	no	—
23	#168	Q₈	1	4	yes	complemented
24	C₄⋊Q₈	D₅	1	1	yes	characteristic
25	#125	—	5	1	no	—
26	#107	—	5	2	no	—
27	#106	—	5	2	no	—
28	#97	—	5	1	no	—
29	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
30	C₄×C₅	—	2	1	no	—
31	C₄×C₅	—	2	1	no	—
32	C₄×C₅	—	4	2	no	—
33	C₄×C₅	C₂×Q₈	1	2	yes	—
34	Dic₅	—	4	1	no	—
35	Dic₅	—	4	1	no	—
36	C₂×Q₈	—	2	1	no	—
37	C₂×C₈	—	5	2	no	—
38	C₄⋊C₄	—	2	1	no	—
39	C₄⋊C₄	—	5	1	no	—
40	C₄⋊C₄	—	5	1	no	—
41	C₄⋊C₄	—	10	1	no	—
42	C₄⋊C₄	—	10	1	no	—
43	C₄⋊C₄	C₂×D₅	1	2	yes	—
44	C₄²	C₂×D₅	1	1	yes	characteristic
45	C₂×C₅	C₄⋊Q₈	1	1	yes	characteristic
46	C₂×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
47	C₂×C₅	#137	1	1	yes	characteristic
48	Q₈	—	4	2	no	—
49	C₂×C₄	—	2	2	no	—
50	C₂×C₄	—	10	1	no	—
51	C₂×C₄	—	10	1	no	—
52	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
53	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
54	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
55	C₈	—	5	4	no	—
56	C₅	#501	1	1	yes	characteristic, complemented, Sylow
57	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	—	4	2	no	—
61	C₄	—	20	1	no	—
62	C₄	—	20	1	no	—
63	C₄	Q₈×D₅	1	2	yes	—
64	C₂	#4078	1	1	yes	characteristic
65	C₂	#4088	1	1	yes	characteristic
66	C₂	#4090	1	1	yes	characteristic

	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	#501
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
23. #168	Q₈	semidirect
56. C₅	#501	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
#4016	2
#3940	2
#3939	2
#3935	2
#501	5