Subgroups — #65082

Order 320.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4090	C₂	1	1	yes	maximal, characteristic
2	C₂×#689	C₂	1	1	yes	maximal, characteristic
3	C₂×Dic₂₀	C₂	1	1	yes	maximal, characteristic
4	#4033	C₂	1	1	yes	maximal, characteristic
5	C₅×#103	C₂	1	1	yes	maximal, characteristic
6	#3946	C₂	1	1	yes	maximal, characteristic
7	#3942	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅×Q₈	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₀	C₂²	1	1	yes	characteristic
10	C₂×Dic₁₀	C₂²	1	1	yes	characteristic
11	C₂×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	#689	—	2	2	no	—
14	#683	—	2	1	no	—
15	#683	—	2	1	no	—
16	#683	—	2	1	no	—
17	C₄×Dic₅	—	2	1	no	—
18	C₄×Dic₅	C₂²	1	1	yes	characteristic
19	C₂×#168	C₂²	1	1	yes	characteristic
20	Dic₂₀	—	2	2	no	—
21	#462	—	5	1	no	maximal, Sylow
22	C₅×Q₈	—	2	1	no	—
23	C₅×Q₈	—	2	1	no	—
24	C₂×C₄×C₅	—	2	1	no	—
25	C₂×C₄×C₅	—	2	1	no	—
26	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₂×Dic₅	—	2	1	no	—
28	C₂×Dic₅	—	2	1	no	—
29	C₂×Dic₅	D₄	1	1	yes	characteristic
30	C₂×Dic₅	D₄	1	1	yes	characteristic
31	Dic₁₀	—	2	1	no	—
32	Dic₁₀	—	2	1	no	—
33	Dic₁₀	—	2	1	no	—
34	Dic₁₀	D₄	1	2	yes	—
35	C₅×C₈	—	2	1	no	—
36	#168	—	2	1	no	—
37	C₂×Q₁₆	—	5	1	no	—
38	C₂×Q₁₆	—	5	1	no	—
39	C₄⋊Q₈	—	5	1	no	—
40	C₄×Q₈	—	5	1	no	—
41	C₄⋊C₈	—	5	1	no	—
42	#103	—	5	1	no	—
43	#103	D₅	1	1	yes	characteristic
44	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
45	C₄×C₅	—	4	1	no	—
46	C₄×C₅	—	4	1	no	—
47	C₄×C₅	C₂×D₄	1	1	yes	characteristic
48	C₄×C₅	#41	1	1	yes	characteristic
49	Dic₅	—	2	2	no	—
50	Dic₅	—	2	1	no	—
51	Dic₅	—	4	1	no	—
52	Dic₅	Q₁₆	1	2	yes	complemented
53	C₂×Q₈	—	5	1	no	—
54	C₂×Q₈	—	5	1	no	—
55	C₂×Q₈	C₂×D₅	1	1	yes	characteristic
56	Q₁₆	—	10	2	no	—
57	Q₁₆	—	10	2	no	—
58	C₂×C₈	—	5	1	no	—
59	C₂×C₈	C₂×D₅	1	1	yes	characteristic
60	C₄⋊C₄	—	10	1	no	—
61	C₄⋊C₄	—	10	1	no	—
62	C₄⋊C₄	—	10	1	no	—
63	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
64	C₄²	—	5	1	no	—
65	C₄²	—	10	1	no	—
66	C₂×C₅	#121	1	1	yes	characteristic
67	C₂×C₅	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×C₅	#137	1	1	yes	characteristic
69	Q₈	—	2	1	no	—
70	Q₈	—	2	1	no	—
71	Q₈	—	5	2	no	—
72	Q₈	—	10	1	no	—
73	Q₈	—	10	1	no	—
74	Q₈	—	10	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	5	1	no	—
78	C₂×C₄	—	5	1	no	—
79	C₂×C₄	—	10	1	no	—
80	C₂×C₄	—	10	1	no	—
81	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
82	C₈	—	2	1	no	—
83	C₈	—	10	1	no	—
84	C₅	#462	1	1	yes	characteristic, complemented, Sylow
85	C₂²	D₄×D₅	1	1	yes	characteristic
86	C₄	—	4	1	no	—
87	C₄	—	4	1	no	—
88	C₄	—	5	2	no	—
89	C₄	—	10	2	no	—
90	C₄	—	10	1	no	—
91	C₄	—	20	1	no	—
92	C₄	#709	1	1	yes	characteristic
93	C₄	D₄×D₅	1	1	yes	characteristic
94	C₂	#4030	1	1	yes	characteristic
95	C₂	#4061	1	1	yes	characteristic
96	C₂	D₅×Q₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×#103	C₂
socle, prime radical	C₂²×C₅	C₂×D₄
hypercenter	#103	D₅
hypocenter	C₅	#462
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
52. Dic₅	Q₁₆	semidirect
84. C₅	#462	semidirect

Maximal subgroups

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

subgroup	index
#4090	2
C₂×#689	2
C₂×Dic₂₀	2
#4033	2
C₅×#103	2
#3946	2
#3942	2
#462	5