Subgroups — #65368

Order 320.

98 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	#4015	C₂	1	1	yes	maximal, characteristic
3	#3940	C₂	1	4	yes	maximal
4	C₄×#168	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×Q₈	—	2	1	no	—
6	C₂×Dic₁₀	—	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	—	2	1	no	—
11	C₄⋊Dic₅	C₂²	1	2	yes	—
12	C₂×#168	C₂²	1	2	yes	—
13	#498	—	5	1	no	maximal, Sylow
14	C₅×Q₈	—	4	2	no	—
15	C₂×C₄×C₅	—	2	2	no	—
16	C₂×C₄×C₅	D₄	1	1	yes	characteristic
17	C₂×C₄×C₅	D₄	1	1	yes	characteristic
18	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
19	C₂×Dic₅	—	2	2	no	—
20	Dic₁₀	—	4	2	no	—
21	#168	Q₈	1	4	yes	complemented
22	C₄⋊Q₈	—	5	1	no	—
23	C₄⋊Q₈	D₅	1	1	yes	characteristic
24	#106	—	5	4	no	—
25	C₄×C₈	—	5	1	no	—
26	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
27	C₄×C₅	—	4	2	no	—
28	C₄×C₅	SD₁₆	1	2	yes	—
29	C₄×C₅	SD₁₆	1	2	yes	—
30	C₄×C₅	C₂×Q₈	1	2	yes	—
31	Dic₅	—	4	2	no	—
32	C₂×Q₈	—	2	1	no	—
33	C₂×Q₈	—	10	1	no	—
34	C₂×C₈	—	5	2	no	—
35	C₄⋊C₄	—	2	1	no	—
36	C₄⋊C₄	—	5	2	no	—
37	C₄⋊C₄	—	10	1	no	—
38	C₄⋊C₄	C₂×D₅	1	2	yes	—
39	C₄²	C₂×D₅	1	1	yes	characteristic
40	C₂×C₅	C₄⋊Q₈	1	1	yes	characteristic
41	C₂×C₅	C₂×SD₁₆	1	1	yes	characteristic
42	C₂×C₅	C₂×SD₁₆	1	1	yes	characteristic
43	Q₈	—	4	2	no	—
44	Q₈	—	20	2	no	—
45	C₂×C₄	—	2	2	no	—
46	C₂×C₄	—	10	2	no	—
47	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
48	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
49	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
50	C₈	—	5	4	no	—
51	C₅	#498	1	1	yes	characteristic, complemented, Sylow
52	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	—	20	2	no	—
55	C₄	#687	1	2	yes	—
56	C₄	#688	1	2	yes	—
57	C₄	Q₈×D₅	1	2	yes	—
58	C₂	C₂×#687	1	1	yes	characteristic
59	C₂	C₂×#688	1	1	yes	characteristic
60	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	#498
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
21. #168	Q₈	semidirect
51. C₅	#498	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
#4015	2
#3940	2
C₄×#168	2
#498	5