Subgroups — #65413

Order 320.

102 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×#681	C₂	1	1	yes	maximal, characteristic
3	C₂×#681	C₂	1	1	yes	maximal, characteristic
4	#3954	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
6	C₅×M₁₆	—	2	1	no	—
7	C₅×M₁₆	C₂²	1	2	yes	—
8	C₂×C₅×C₈	—	2	1	no	—
9	#681	—	2	1	no	—
10	#681	C₄	1	2	yes	—
11	#681	C₄	1	2	yes	—
12	#681	C₂²	1	2	yes	—
13	C₂×#168	—	2	1	no	—
14	C₂×#168	C₂²	1	2	yes	—
15	#430	—	5	1	no	maximal, Sylow
16	C₂³×C₅	Q₈	1	1	yes	characteristic
17	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₅	D₄	1	1	yes	characteristic
20	C₂×C₄×C₅	D₄	1	1	yes	characteristic
21	C₂×C₄×C₅	Q₈	1	1	yes	characteristic
22	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₅×C₈	—	2	2	no	—
24	#168	—	2	2	no	—
25	#168	C₂×C₄	1	4	yes	—
26	C₂×M₁₆	—	5	1	no	—
27	C₂×M₁₆	—	5	1	no	—
28	C₂×M₁₆	D₅	1	1	yes	characteristic
29	#108	—	5	4	no	—
30	C₂²×C₅	—	4	1	no	—
31	C₂²×C₅	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₅	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₅	C₂×Q₈	1	1	yes	characteristic
34	C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
35	C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
36	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
37	C₄×C₅	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
39	M₁₆	—	2	1	no	—
40	M₁₆	—	5	2	no	—
41	M₁₆	—	5	2	no	—
42	M₁₆	—	5	2	no	—
43	M₁₆	—	10	1	no	—
44	M₁₆	C₂×D₅	1	2	yes	—
45	C₂×C₈	—	2	1	no	—
46	C₂×C₈	—	5	2	no	—
47	C₂×C₈	—	10	1	no	—
48	C₂×C₅	—	2	1	no	—
49	C₂×C₅	—	2	1	no	—
50	C₂×C₅	—	2	1	no	—
51	C₂×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂³	Dic₁₀	1	1	yes	characteristic
53	C₂×C₄	Dic₁₀	1	1	yes	characteristic
54	C₂×C₄	C₄×D₅	1	1	yes	characteristic
55	C₂×C₄	C₄×D₅	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
57	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
58	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
59	C₈	—	2	2	no	—
60	C₈	—	5	4	no	—
61	C₈	—	10	2	no	—
62	C₅	#430	1	1	yes	characteristic, complemented, Sylow
63	C₂²	—	4	1	no	—
64	C₂²	#683	1	1	yes	characteristic
65	C₂²	#683	1	1	yes	characteristic
66	C₂²	C₂×Dic₁₀	1	1	yes	characteristic
67	C₄	#683	1	1	yes	characteristic
68	C₄	#683	1	1	yes	characteristic
69	C₄	C₂×C₄×D₅	1	1	yes	characteristic
70	C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	—	2	1	no	—
73	C₂	—	2	1	no	—
74	C₂	C₂×#683	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₂²⋊D₅
commutator	C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₂×C₅×M₁₆	C₂
socle	C₂×C₅	C₂×C₄⋊C₄
prime radical	C₂³×C₅	Q₈
cosocle	C₂×C₄×C₅	C₂³
hypercenter	C₂×M₁₆	D₅
hypocenter	C₅	#430
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
4. #3954	C₂	semidirect
62. C₅	#430	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×#681	2
#3954	2
#430	5