Subgroups — #65901

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	#4090	C₂	1	2	yes	maximal
3	#4035	C₂	1	4	yes	maximal
4	#4033	C₂	1	2	yes	maximal
5	#4016	C₂	1	2	yes	maximal
6	#4015	C₂	1	1	yes	maximal, characteristic
7	C₄×Dic₁₀	C₂	1	2	yes	maximal
8	C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic
9	C₂×C₅×Q₈	C₂²	1	1	yes	characteristic
10	C₂×Dic₁₀	C₂²	1	2	yes	—
11	C₂×Dic₁₀	C₂²	1	1	yes	characteristic
12	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₅×C₄⋊C₄	C₂²	1	2	yes	—
14	C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₄²×C₅	C₂²	1	2	yes	—
16	C₄⋊Dic₅	C₂²	1	2	yes	—
17	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₅	C₂²	1	2	yes	—
19	#683	C₂²	1	2	yes	—
20	#683	C₂²	1	4	yes	—
21	#683	C₂²	1	4	yes	—
22	#683	C₂²	1	4	yes	—
23	C₄×Dic₅	C₂²	1	4	yes	—
24	C₄×Dic₅	C₂²	1	2	yes	—
25	#557	—	5	1	no	maximal, Sylow
26	C₅×Q₈	—	2	2	no	—
27	C₂×C₄×C₅	C₂³	1	2	yes	—
28	C₂×C₄×C₅	C₂³	1	2	yes	—
29	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂×Dic₅	C₂³	1	2	yes	—
33	C₂×Dic₅	C₂³	1	4	yes	—
34	C₂×Dic₅	C₂³	1	2	yes	—
35	Dic₁₀	—	2	4	no	—
36	Dic₁₀	Q₈	1	4	yes	complemented
37	C₄⋊Q₈	—	5	2	no	—
38	C₄⋊Q₈	—	5	1	no	—
39	#125	—	5	4	no	—
40	#125	—	5	2	no	—
41	C₄×Q₈	—	5	2	no	—
42	C₄×Q₈	—	5	2	no	—
43	C₄×Q₈	—	5	1	no	—
44	C₄×Q₈	D₅	1	1	yes	characteristic
45	C₂²×C₅	C₂⁴	1	1	yes	characteristic
46	C₄×C₅	—	2	2	no	—
47	C₄×C₅	—	2	2	no	—
48	C₄×C₅	—	2	1	no	—
49	C₄×C₅	C₂×Q₈	1	2	yes	—
50	C₄×C₅	#41	1	2	yes	—
51	Dic₅	—	2	4	no	—
52	Dic₅	—	2	2	no	—
53	Dic₅	C₂×Q₈	1	4	yes	—
54	C₂×Q₈	—	5	1	no	—
55	C₂×Q₈	—	5	2	no	—
56	C₂×Q₈	C₂×D₅	1	1	yes	characteristic
57	C₄⋊C₄	—	5	2	no	—
58	C₄⋊C₄	—	5	4	no	—
59	C₄⋊C₄	—	5	1	no	—
60	C₄⋊C₄	—	5	4	no	—
61	C₄⋊C₄	—	5	2	no	—
62	C₄⋊C₄	—	5	4	no	—
63	C₄⋊C₄	—	5	2	no	—
64	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
65	C₄⋊C₄	C₂×D₅	1	2	yes	—
66	C₄²	—	5	4	no	—
67	C₄²	—	5	2	no	—
68	C₄²	C₂×D₅	1	2	yes	—
69	C₄²	C₂×D₅	1	1	yes	characteristic
70	C₂×C₅	C₂²×Q₈	1	1	yes	characteristic
71	C₂×C₅	C₂×#41	1	1	yes	characteristic
72	C₂×C₅	2₋¹⁺⁴	1	1	yes	characteristic
73	Q₈	—	2	2	no	—
74	Q₈	—	5	4	no	—
75	Q₈	—	10	4	no	—
76	C₂×C₄	—	5	2	no	—
77	C₂×C₄	—	5	2	no	—
78	C₂×C₄	—	5	4	no	—
79	C₂×C₄	C₂²×D₅	1	2	yes	—
80	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
81	C₂×C₄	C₂²×D₅	1	2	yes	—
82	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
83	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
84	C₅	#557	1	1	yes	characteristic, complemented, Sylow
85	C₂²	C₂³×D₅	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	2	2	no	—
88	C₄	—	2	1	no	—
89	C₄	—	5	4	no	—
90	C₄	—	10	4	no	—
91	C₄	—	10	2	no	—
92	C₄	#709	1	2	yes	—
93	C₄	Q₈×D₅	1	2	yes	—
94	C₂	C₂×#709	1	1	yes	characteristic
95	C₂	C₂×Q₈×D₅	1	1	yes	characteristic
96	C₂	#4150	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
36. Dic₁₀	Q₈	semidirect
84. 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
#4090	2
#4035	2
#4033	2
#4016	2
#4015	2
C₄×Dic₁₀	2
#557	5