Subgroups — #65322

Order 320.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#121	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
3	#3964	C₂	1	2	yes	maximal, complemented
4	C₂²⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
5	#3939	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×D₄	—	2	1	no	—
7	C₂×C₅×D₄	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₂²×Dic₅	—	2	1	no	—
10	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₅×C₂²⋊C₄	—	2	1	no	—
12	C₄⋊Dic₅	—	2	1	no	—
13	C₄⋊Dic₅	C₂²	1	2	yes	—
14	C₂×#168	C₂²	1	2	yes	—
15	#480	—	5	1	no	maximal, Sylow
16	C₂³×C₅	—	2	1	no	—
17	C₂³×C₅	D₄	1	1	yes	characteristic
18	C₅×D₄	—	2	2	no	—
19	C₅×D₄	—	4	2	no	—
20	C₂×C₄×C₅	—	2	2	no	—
21	C₂×C₄×C₅	—	2	1	no	—
22	C₂×C₄×C₅	D₄	1	1	yes	characteristic
23	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂×Dic₅	—	2	2	no	—
25	C₂×Dic₅	—	4	2	no	—
26	#168	—	2	2	no	—
27	#121	D₅	1	1	yes	characteristic
28	C₂×C₄⋊C₄	—	5	1	no	—
29	#107	—	5	2	no	—
30	#102	—	5	2	no	—
31	C₂²⋊C₈	—	5	1	no	—
32	C₂²×C₅	—	2	1	no	—
33	C₂²×C₅	—	2	1	no	—
34	C₂²×C₅	—	4	1	no	—
35	C₂²×C₅	—	4	1	no	—
36	C₂²×C₅	—	4	1	no	—
37	C₂²×C₅	D₈	1	2	yes	—
38	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
39	C₄×C₅	—	2	1	no	—
40	C₄×C₅	—	4	1	no	—
41	C₄×C₅	#41	1	2	yes	—
42	Dic₅	—	4	2	no	—
43	C₂×D₄	—	2	1	no	—
44	C₂×D₄	C₂×D₅	1	1	yes	characteristic
45	C₂²×C₄	—	10	1	no	—
46	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
47	C₂×C₈	—	5	2	no	—
48	C₄⋊C₄	—	5	2	no	—
49	C₄⋊C₄	—	10	1	no	—
50	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
51	C₂²⋊C₄	—	2	1	no	—
52	C₂×C₅	—	2	2	no	—
53	C₂×C₅	—	8	1	no	—
54	C₂×C₅	#123	1	1	yes	characteristic
55	C₂×C₅	C₂×D₈	1	1	yes	characteristic
56	C₂×C₅	#137	1	1	yes	characteristic
57	C₂³	—	2	1	no	—
58	C₂³	C₂²⋊D₅	1	1	yes	characteristic
59	D₄	—	2	2	no	—
60	D₄	—	4	2	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	10	2	no	—
64	C₂×C₄	—	20	2	no	—
65	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
66	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
67	C₈	—	10	2	no	—
68	C₅	#480	1	1	yes	characteristic, complemented, Sylow
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	4	1	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	4	1	no	—
74	C₂²	#686	1	2	yes	—
75	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	20	2	no	—
79	C₄	#711	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	—	8	1	no	—
82	C₂	C₂×#686	1	1	yes	characteristic
83	C₂	#4081	1	1	yes	characteristic
84	C₂	#4097	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting, prime radical	C₅×#121	C₂
socle	C₂²×C₅	C₂×D₄
hypercenter	#121	D₅
hypocenter	C₅	#480
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #121

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊Dic₅	C₂	semidirect
3. #3964	C₂	semidirect
4. C₂²⋊#168	C₂	semidirect
5. #3939	C₂	semidirect
68. C₅	#480	semidirect

Maximal subgroups

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

subgroup	index
C₅×#121	2
C₂×C₄⋊Dic₅	2
#3964	2
C₂²⋊#168	2
#3939	2
#480	5