Subgroups — #65407

Order 320.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₂×#681	C₂	1	1	yes	maximal, characteristic
4	#3946	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
6	C₂²×Dic₅	C₄	1	2	yes	—
7	C₅×M₁₆	—	2	2	no	—
8	C₂×C₅×C₈	C₂²	1	2	yes	—
9	C₄×Dic₅	C₄	1	1	yes	characteristic
10	C₄×Dic₅	C₄	1	1	yes	characteristic
11	C₄×Dic₅	C₂²	1	2	yes	—
12	#681	—	2	2	no	—
13	C₂×#168	C₂²	1	2	yes	—
14	#423	—	5	1	no	maximal, Sylow
15	C₂³×C₅	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₅	D₄	1	2	yes	—
18	C₂×C₄×C₅	Q₈	1	2	yes	—
19	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
20	C₂×Dic₅	—	2	4	no	—
21	C₂×Dic₅	C₂×C₄	1	2	yes	—
22	C₂×Dic₅	C₂×C₄	1	2	yes	—
23	C₅×C₈	—	2	2	no	—
24	#168	—	2	2	no	—
25	C₂×M₁₆	—	5	1	no	—
26	C₂×M₁₆	D₅	1	1	yes	characteristic
27	C₂×C₄²	—	5	1	no	—
28	C₄⋊C₈	—	5	4	no	—
29	C₂²×C₅	—	2	2	no	—
30	C₂²×C₅	C₄⋊C₄	1	2	yes	—
31	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₅	C₄⋊C₄	1	2	yes	—
33	C₄×C₅	C₂×D₄	1	1	yes	characteristic
34	C₄×C₅	C₂×Q₈	1	1	yes	characteristic
35	Dic₅	—	2	2	no	—
36	Dic₅	M₁₆	1	4	yes	complemented
37	C₂²×C₄	—	5	2	no	—
38	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
39	M₁₆	—	2	2	no	—
40	M₁₆	—	10	2	no	—
41	C₂×C₈	—	5	2	no	—
42	C₂×C₈	C₂×D₅	1	2	yes	—
43	C₄²	—	5	1	no	—
44	C₄²	—	5	1	no	—
45	C₄²	—	5	2	no	—
46	C₂×C₅	—	2	2	no	—
47	C₂×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₅	C₂×M₁₆	1	2	yes	—
49	C₂³	C₄×D₅	1	1	yes	characteristic
50	C₂×C₄	—	5	2	no	—
51	C₂×C₄	—	5	2	no	—
52	C₂×C₄	—	10	4	no	—
53	C₂×C₄	Dic₁₀	1	2	yes	—
54	C₂×C₄	C₄×D₅	1	1	yes	characteristic
55	C₂×C₄	C₂²⋊D₅	1	2	yes	—
56	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
57	C₈	—	2	2	no	—
58	C₈	—	10	2	no	—
59	C₅	#423	1	1	yes	characteristic, complemented, Sylow
60	C₂²	—	2	2	no	—
61	C₂²	#683	1	2	yes	—
62	C₂²	C₂×C₄×D₅	1	1	yes	characteristic
63	C₄	—	5	4	no	—
64	C₄	—	10	2	no	—
65	C₄	#683	1	2	yes	—
66	C₄	C₂×Dic₁₀	1	1	yes	characteristic
67	C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
68	C₂	—	2	2	no	—
69	C₂	D₅×M₁₆	1	2	yes	—
70	C₂	C₂×#683	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₅
commutator, socle	C₂²×C₅	C₂²×C₄
Fitting	C₂×C₅×M₁₆	C₂
prime radical	C₂³×C₅	C₂×C₄
cosocle	C₂×C₄×C₅	C₂³
hypercenter	C₂×M₁₆	D₅
hypocenter	C₅	#423
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
4. #3946	C₂	semidirect
36. Dic₅	M₁₆	semidirect
59. C₅	#423	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×M₁₆	2
C₂×C₄×Dic₅	2
C₂×#681	2
#3946	2
#423	5