Subgroups — #65903

Order 320.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₅	C₂	1	1	yes	maximal, characteristic
3	Q₈×Dic₅	C₂	1	1	yes	maximal, characteristic
4	#4036	C₂	1	3	yes	maximal
5	#4034	C₂	1	3	yes	maximal
6	#4016	C₂	1	3	yes	maximal
7	C₄×Dic₁₀	C₂	1	3	yes	maximal
8	C₂×C₅×Q₈	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₀	C₂²	1	3	yes	—
10	C₅×C₄⋊C₄	C₂²	1	3	yes	—
11	C₄²×C₅	C₂²	1	3	yes	—
12	C₄⋊Dic₅	C₂²	1	3	yes	—
13	C₄⋊Dic₅	C₂²	1	3	yes	—
14	C₄⋊Dic₅	C₂²	1	3	yes	—
15	#683	C₂²	1	1	yes	characteristic
16	#683	C₂²	1	3	yes	—
17	#683	C₂²	1	6	yes	—
18	C₄×Dic₅	C₂²	1	3	yes	—
19	C₄×Dic₅	C₂²	1	3	yes	—
20	#557	—	5	1	no	maximal, Sylow
21	C₅×Q₈	Q₈	1	4	yes	complemented
22	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂×C₄×C₅	C₂³	1	3	yes	—
24	C₂×C₄×C₅	C₂³	1	3	yes	—
25	C₂×Dic₅	C₂³	1	1	yes	characteristic
26	C₂×Dic₅	C₂³	1	3	yes	—
27	C₂×Dic₅	C₂³	1	3	yes	—
28	C₂×Dic₅	C₂³	1	1	yes	characteristic
29	Dic₁₀	—	2	6	no	—
30	C₄⋊Q₈	—	5	3	no	—
31	#125	—	5	3	no	—
32	#125	—	5	3	no	—
33	C₄×Q₈	—	5	1	no	—
34	C₄×Q₈	—	5	1	no	—
35	C₄×Q₈	—	5	3	no	—
36	C₄×Q₈	D₅	1	1	yes	characteristic
37	C₂²×C₅	C₂⁴	1	1	yes	characteristic
38	C₄×C₅	—	2	3	no	—
39	C₄×C₅	—	2	1	no	—
40	C₄×C₅	C₂×Q₈	1	6	yes	—
41	Dic₅	—	2	3	no	—
42	Dic₅	—	2	1	no	—
43	Dic₅	—	2	3	no	—
44	Dic₅	#41	1	2	yes	—
45	C₂×Q₈	—	5	3	no	—
46	C₂×Q₈	C₂×D₅	1	1	yes	characteristic
47	C₄⋊C₄	—	5	1	no	—
48	C₄⋊C₄	—	5	3	no	—
49	C₄⋊C₄	—	5	3	no	—
50	C₄⋊C₄	—	5	3	no	—
51	C₄⋊C₄	—	5	3	no	—
52	C₄⋊C₄	—	5	6	no	—
53	C₄⋊C₄	C₂×D₅	1	3	yes	—
54	C₄²	—	5	3	no	—
55	C₄²	—	5	3	no	—
56	C₄²	C₂×D₅	1	3	yes	—
57	C₂×C₅	C₂²×Q₈	1	1	yes	characteristic
58	C₂×C₅	C₂×#41	1	1	yes	characteristic
59	C₂×C₅	2₋¹⁺⁴	1	1	yes	characteristic
60	Q₈	—	10	6	no	—
61	Q₈	Dic₁₀	1	4	yes	complemented
62	C₂×C₄	—	5	1	no	—
63	C₂×C₄	—	5	1	no	—
64	C₂×C₄	—	5	3	no	—
65	C₂×C₄	—	5	3	no	—
66	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
67	C₂×C₄	C₂²×D₅	1	3	yes	—
68	C₂×C₄	C₂²×D₅	1	3	yes	—
69	C₅	#557	1	1	yes	characteristic, complemented, Sylow
70	C₂²	C₂³×D₅	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	2	3	no	—
73	C₄	—	5	2	no	—
74	C₄	—	10	1	no	—
75	C₄	—	10	3	no	—
76	C₄	—	10	3	no	—
77	C₄	C₂×Dic₁₀	1	6	yes	—
78	C₂	C₂²×Dic₁₀	1	1	yes	characteristic
79	C₂	#4147	1	1	yes	characteristic
80	C₂	D₅×#41	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, prime radical, cosocle	C₂²×C₅	C₂⁴
Fitting	C₄×C₅×Q₈	C₂
hypercenter	C₄×Q₈	D₅
hypocenter	C₅	#557
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
21. C₅×Q₈	Q₈	semidirect
61. Q₈	Dic₁₀	semidirect
69. C₅	#557	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×Q₈	2
Q₈×Dic₅	2
#4036	2
#4034	2
#4016	2
C₄×Dic₁₀	2
#557	5