Subgroups — #65266

Order 320.

154 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₅	C₂	1	1	yes	maximal, characteristic
4	#3963	C₂	1	4	yes	maximal
5	C₂²×C₄×C₅	C₂²	1	2	yes	—
6	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₂²×Dic₅	C₂²	1	2	yes	—
8	C₂²×Dic₅	C₂²	1	2	yes	—
9	C₅×C₄⋊C₄	—	2	2	no	—
10	C₄⋊Dic₅	—	2	2	no	—
11	C₄×Dic₅	C₄	1	4	yes	complemented
12	#382	—	5	1	no	maximal, Sylow
13	C₂³×C₅	C₂³	1	1	yes	characteristic
14	C₂×C₄×C₅	—	2	2	no	—
15	C₂×C₄×C₅	—	2	4	no	—
16	C₂×C₄×C₅	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₅	D₄	1	2	yes	—
18	C₂×C₄×C₅	Q₈	1	2	yes	—
19	C₂×Dic₅	—	2	2	no	—
20	C₂×Dic₅	—	2	2	no	—
21	C₂×Dic₅	—	2	2	no	—
22	C₂×Dic₅	—	2	4	no	—
23	C₂×Dic₅	C₂×C₄	1	4	yes	—
24	C₂×C₄⋊C₄	—	5	1	no	—
25	C₂×C₄⋊C₄	D₅	1	1	yes	characteristic
26	C₂×C₄²	—	5	1	no	—
27	#95	—	5	4	no	—
28	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₅	C₂×Q₈	1	1	yes	characteristic
31	C₂²×C₅	#41	1	2	yes	—
32	C₂²×C₅	#41	1	2	yes	—
33	C₄×C₅	—	4	2	no	—
34	C₄×C₅	C₄⋊C₄	1	4	yes	—
35	Dic₅	—	2	4	no	—
36	Dic₅	—	4	2	no	—
37	C₂²×C₄	—	5	2	no	—
38	C₂²×C₄	—	5	2	no	—
39	C₂²×C₄	C₂×D₅	1	2	yes	—
40	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
41	C₄⋊C₄	—	2	2	no	—
42	C₄⋊C₄	—	10	2	no	—
43	C₄²	—	5	4	no	—
44	C₂×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₅	#117	1	2	yes	—
46	C₂×C₅	#124	1	1	yes	characteristic
47	C₂×C₅	#124	1	1	yes	characteristic
48	C₂×C₅	#125	1	2	yes	—
49	C₂³	C₂²×D₅	1	1	yes	characteristic
50	C₂×C₄	—	2	4	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	5	4	no	—
53	C₂×C₄	—	10	2	no	—
54	C₂×C₄	—	10	2	no	—
55	C₂×C₄	—	10	4	no	—
56	C₂×C₄	—	10	2	no	—
57	C₂×C₄	Dic₁₀	1	2	yes	—
58	C₂×C₄	C₄×D₅	1	2	yes	—
59	C₂×C₄	C₂²⋊D₅	1	2	yes	—
60	C₅	#382	1	1	yes	characteristic, complemented, Sylow
61	C₂²	C₂×Dic₁₀	1	1	yes	characteristic
62	C₂²	C₂×C₄×D₅	1	1	yes	characteristic
63	C₂²	#711	1	2	yes	—
64	C₂²	#713	1	2	yes	—
65	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
66	C₄	—	4	2	no	—
67	C₄	—	10	4	no	—
68	C₄	—	20	2	no	—
69	C₄	#683	1	4	yes	—
70	C₂	#4036	1	2	yes	—
71	C₂	#4038	1	2	yes	—
72	C₂	C₂×#683	1	1	yes	characteristic
73	C₂	#4082	1	1	yes	characteristic
74	C₂	#4093	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₅
commutator	C₂²×C₅	C₂²×C₄
Fitting	C₂×C₅×C₄⋊C₄	C₂
socle, prime radical, cosocle	C₂³×C₅	C₂³
hypercenter	C₂×C₄⋊C₄	D₅
hypocenter	C₅	#382
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
11. C₄×Dic₅	C₄	semidirect
60. C₅	#382	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×C₄⋊C₄	2
C₂×C₄⋊Dic₅	2
C₂×C₄×Dic₅	2
#3963	2
#382	5