Subgroups — #66056

Order 320.

212 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	#4090	C₂	1	4	yes	maximal
3	#4034	C₂	1	4	yes	maximal
4	#4033	C₂	1	4	yes	maximal
5	C₄×Dic₁₀	C₂	1	2	yes	maximal
6	C₂×C₅×Q₈	C₂²	1	2	yes	—
7	C₂×Dic₁₀	C₂²	1	4	yes	—
8	C₂×Dic₁₀	C₂²	1	2	yes	—
9	C₅×C₄⋊C₄	C₂²	1	4	yes	—
10	C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₅	C₂²	1	2	yes	—
12	#683	C₂²	1	4	yes	—
13	#683	C₂²	1	8	yes	—
14	C₄×Dic₅	C₂²	1	4	yes	—
15	C₄×Dic₅	C₂²	1	4	yes	—
16	Q₈²	—	5	1	no	maximal, Sylow
17	C₅×Q₈	—	4	2	no	—
18	C₂×C₄×C₅	C₂³	1	4	yes	—
19	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₅	C₂³	1	2	yes	—
21	C₂×Dic₅	C₂³	1	4	yes	—
22	C₂×Dic₅	C₂³	1	4	yes	—
23	Dic₁₀	—	4	4	no	—
24	Dic₁₀	Q₈	1	8	yes	complemented
25	C₄⋊Q₈	—	5	4	no	—
26	C₄⋊Q₈	—	5	4	no	—
27	C₄⋊Q₈	D₅	1	1	yes	characteristic
28	C₄×Q₈	—	5	4	no	—
29	C₄×Q₈	—	5	2	no	—
30	C₂²×C₅	C₂⁴	1	1	yes	characteristic
31	C₄×C₅	—	2	4	no	—
32	C₄×C₅	—	2	1	no	—
33	C₄×C₅	C₂×Q₈	1	4	yes	—
34	Dic₅	—	2	4	no	—
35	Dic₅	C₂×Q₈	1	8	yes	—
36	C₂×Q₈	—	5	2	no	—
37	C₂×Q₈	—	5	4	no	—
38	C₂×Q₈	C₂×D₅	1	2	yes	—
39	C₄⋊C₄	—	5	8	no	—
40	C₄⋊C₄	—	5	4	no	—
41	C₄⋊C₄	—	5	2	no	—
42	C₄⋊C₄	C₂×D₅	1	4	yes	—
43	C₄²	—	5	4	no	—
44	C₄²	—	5	4	no	—
45	C₄²	C₂×D₅	1	1	yes	characteristic
46	C₂×C₅	C₂²×Q₈	1	2	yes	—
47	C₂×C₅	H(5,𝔽₂)	1	1	yes	characteristic
48	Q₈	—	4	2	no	—
49	Q₈	—	5	8	no	—
50	Q₈	—	20	4	no	—
51	C₂×C₄	—	5	4	no	—
52	C₂×C₄	—	5	4	no	—
53	C₂×C₄	C₂²×D₅	1	4	yes	—
54	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
55	C₂×C₄	C₂²×D₅	1	2	yes	—
56	C₅	Q₈²	1	1	yes	characteristic, complemented, Sylow
57	C₂²	C₂³×D₅	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	4	no	—
60	C₄	—	5	8	no	—
61	C₄	—	10	4	no	—
62	C₄	Q₈×D₅	1	4	yes	—
63	C₂	#4144	1	1	yes	characteristic
64	C₂	C₂×Q₈×D₅	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, prime radical, cosocle	C₂²×C₅	C₂⁴
Fitting	C₅×C₄⋊Q₈	C₂
hypercenter	C₄⋊Q₈	D₅
hypocenter	C₅	Q₈²
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
24. Dic₁₀	Q₈	semidirect
56. C₅	Q₈²	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
#4090	2
#4034	2
#4033	2
C₄×Dic₁₀	2
Q₈²	5