Subgroups — #65304

Order 320.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#686	C₂	1	2	yes	maximal, complemented
3	#4020	C₂	1	1	yes	maximal, characteristic, complemented
4	#3941	C₂	1	2	yes	maximal, complemented
5	C₄⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×D₄	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₅	—	2	1	no	—
8	C₂×D₂₀	—	2	1	no	—
9	C₂×D₂₀	—	2	1	no	—
10	C₂×D₂₀	C₂²	1	2	yes	—
11	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₅×C₂²⋊C₄	—	2	1	no	—
13	C₄²×C₅	C₂²	1	1	yes	characteristic
14	#686	—	2	4	no	—
15	C₂×#168	C₂²	1	2	yes	—
16	#459	—	5	1	no	maximal, Sylow
17	C₂³×C₅	—	2	1	no	—
18	C₂²×D₅	—	2	2	no	—
19	C₅×D₄	—	2	1	no	—
20	C₅×D₄	D₄	1	2	yes	complemented
21	C₂×C₄×C₅	—	2	1	no	—
22	C₂×C₄×C₅	—	4	2	no	—
23	C₂×C₄×C₅	D₄	1	1	yes	characteristic
24	C₂×C₄×C₅	D₄	1	1	yes	characteristic
25	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
26	D₂₀	—	2	2	no	—
27	D₂₀	—	2	2	no	—
28	D₂₀	—	4	2	no	—
29	D₂₀	—	4	2	no	—
30	#168	—	2	2	no	—
31	C₂×D₈	—	5	2	no	—
32	#127	—	5	1	no	—
33	C₄×D₄	D₅	1	1	yes	characteristic, complemented
34	C₄⋊C₈	—	5	1	no	—
35	#102	—	5	2	no	—
36	C₂²×C₅	—	2	2	no	—
37	C₂²×C₅	—	4	1	no	—
38	C₂²×C₅	—	4	1	no	—
39	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
40	C₂×D₅	—	4	2	no	—
41	C₂×D₅	—	4	2	no	—
42	C₂×D₅	—	4	2	no	—
43	C₄×C₅	—	2	1	no	—
44	C₄×C₅	—	4	1	no	—
45	C₄×C₅	D₈	1	2	yes	complemented
46	C₄×C₅	C₂×D₄	1	1	yes	characteristic
47	C₄×C₅	#41	1	1	yes	characteristic
48	C₂×D₄	—	5	2	no	—
49	C₂×D₄	—	10	1	no	—
50	C₂×D₄	—	10	1	no	—
51	C₂×D₄	C₂×D₅	1	1	yes	characteristic
52	C₂²×C₄	—	2	1	no	—
53	D₈	—	10	4	no	—
54	C₂×C₈	—	5	2	no	—
55	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
56	C₂²⋊C₄	—	2	1	no	—
57	C₄²	C₂×D₅	1	1	yes	characteristic
58	C₂×C₅	—	4	2	no	—
59	C₂×C₅	#121	1	1	yes	characteristic
60	C₂×C₅	C₂×D₈	1	1	yes	characteristic
61	C₂×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
62	D₅	—	8	2	no	—
63	C₂³	—	2	1	no	—
64	C₂³	—	10	2	no	—
65	D₄	—	2	1	no	—
66	D₄	—	10	2	no	—
67	D₄	—	10	2	no	—
68	D₄	—	20	2	no	—
69	D₄	—	20	2	no	—
70	D₄	D₂₀	1	2	yes	complemented
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
74	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
75	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
76	C₈	—	10	2	no	—
77	C₅	#459	1	1	yes	characteristic, complemented, Sylow
78	C₂²	—	2	2	no	—
79	C₂²	—	4	1	no	—
80	C₂²	—	4	1	no	—
81	C₂²	—	20	2	no	—
82	C₂²	—	20	2	no	—
83	C₂²	—	20	2	no	—
84	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	—	4	1	no	—
87	C₄	#686	1	2	yes	complemented
88	C₄	C₂×D₂₀	1	1	yes	characteristic
89	C₄	#709	1	1	yes	characteristic
90	C₂	—	4	2	no	—
91	C₂	—	40	2	no	—
92	C₂	#4076	1	1	yes	characteristic
93	C₂	C₂×#686	1	1	yes	characteristic
94	C₂	#4095	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₄×C₅×D₄	C₂
socle	C₂²×C₅	C₂×D₄
hypercenter	C₄×D₄	D₅
hypocenter	C₅	#459
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₄×C₅×D₄	C₂	semidirect
2. C₂×#686	C₂	semidirect
3. #4020	C₂	semidirect
4. #3941	C₂	semidirect
5. C₄⋊#168	C₂	semidirect
20. C₅×D₄	D₄	semidirect
33. C₄×D₄	D₅	semidirect
45. C₄×C₅	D₈	semidirect
70. D₄	D₂₀	semidirect
77. C₅	#459	semidirect
87. C₄	#686	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×D₄	2
C₂×#686	2
#4020	2
#3941	2
C₄⋊#168	2
#459	5