Subgroups — #66068

Order 320.

236 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, complemented
2	#4093	C₂	1	4	yes	maximal, complemented
3	#4041	C₂	1	4	yes	maximal, complemented
4	#4038	C₂	1	4	yes	maximal, complemented
5	C₄×D₂₀	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×Q₈	C₂²	1	2	yes	—
7	C₂×D₂₀	C₂²	1	2	yes	—
8	C₂×D₂₀	C₂²	1	4	yes	—
9	C₂×C₄×D₅	C₂²	1	4	yes	—
10	C₅×C₄⋊C₄	C₂²	1	4	yes	—
11	C₄²×C₅	C₂²	1	1	yes	characteristic
12	#685	C₂²	1	4	yes	—
13	#685	C₂²	1	8	yes	—
14	C₄⋊Dic₅	C₂²	1	2	yes	—
15	C₄×Dic₅	C₂²	1	4	yes	—
16	#555	—	5	1	no	maximal, Sylow
17	C₂²×D₅	C₂³	1	4	yes	—
18	C₅×Q₈	—	4	2	no	—
19	C₂×C₄×C₅	C₂³	1	4	yes	—
20	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₅	C₂³	1	2	yes	—
22	C₂×Dic₅	C₂³	1	4	yes	—
23	D₂₀	—	2	4	no	—
24	D₂₀	—	4	4	no	—
25	C₄×D₅	—	2	8	no	—
26	C₄⋊Q₈	D₅	1	1	yes	characteristic, complemented
27	#124	—	5	4	no	—
28	#121	—	5	4	no	—
29	C₄×D₄	—	5	2	no	—
30	#117	—	5	4	no	—
31	C₂²×C₅	C₂⁴	1	1	yes	characteristic
32	C₂×D₅	—	2	4	no	—
33	C₂×D₅	—	2	4	no	—
34	C₂×D₅	—	2	4	no	—
35	C₄×C₅	—	2	4	no	—
36	C₄×C₅	—	2	1	no	—
37	C₄×C₅	#41	1	4	yes	—
38	Dic₅	—	2	4	no	—
39	C₂×Q₈	C₂×D₅	1	2	yes	—
40	C₂×D₄	—	5	2	no	—
41	C₂×D₄	—	5	4	no	—
42	C₂²×C₄	—	5	4	no	—
43	C₄⋊C₄	—	5	2	no	—
44	C₄⋊C₄	C₂×D₅	1	4	yes	—
45	C₂²⋊C₄	—	5	8	no	—
46	C₂²⋊C₄	—	5	4	no	—
47	C₄²	—	5	4	no	—
48	C₄²	C₂×D₅	1	1	yes	characteristic
49	C₂×C₅	C₂×#41	1	2	yes	—
50	C₂×C₅	H(5,𝔽₂)	1	1	yes	characteristic
51	D₅	—	4	4	no	—
52	C₂³	—	5	4	no	—
53	Q₈	—	4	2	no	—
54	D₄	—	10	4	no	—
55	D₄	—	20	4	no	—
56	C₂×C₄	—	5	4	no	—
57	C₂×C₄	—	10	8	no	—
58	C₂×C₄	C₂²×D₅	1	4	yes	—
59	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₅	1	2	yes	—
61	C₅	#555	1	1	yes	characteristic, complemented, Sylow
62	C₂²	—	10	4	no	—
63	C₂²	—	10	4	no	—
64	C₂²	—	10	4	no	—
65	C₂²	C₂³×D₅	1	1	yes	characteristic
66	C₄	—	2	4	no	—
67	C₄	—	2	1	no	—
68	C₄	—	10	4	no	—
69	C₄	#713	1	4	yes	—
70	C₂	—	20	4	no	—
71	C₂	#4144	1	1	yes	characteristic
72	C₂	C₂×#713	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting	C₅×C₄⋊Q₈	C₂
hypercenter	C₄⋊Q₈	D₅
hypocenter	C₅	#555
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
1. C₅×C₄⋊Q₈	C₂	semidirect
2. #4093	C₂	semidirect
3. #4041	C₂	semidirect
4. #4038	C₂	semidirect
5. C₄×D₂₀	C₂	semidirect
26. C₄⋊Q₈	D₅	semidirect
61. C₅	#555	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
#4093	2
#4041	2
#4038	2
C₄×D₂₀	2
#555	5