Subgroups — #65332

Order 320.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
3	#3967	C₂	1	2	yes	maximal, complemented
4	C₂²⋊#168	C₂	1	1	yes	maximal, characteristic
5	#3940	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×Q₈	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₂²×Dic₅	—	2	1	no	—
9	C₅×C₄⋊C₄	—	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	#483	—	5	1	no	maximal, Sylow
16	C₂³×C₅	D₄	1	1	yes	characteristic
17	C₅×Q₈	—	2	2	no	—
18	C₂×C₄×C₅	—	2	2	no	—
19	C₂×C₄×C₅	—	2	1	no	—
20	C₂×C₄×C₅	—	2	1	no	—
21	C₂×C₄×C₅	D₄	1	1	yes	characteristic
22	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂×Dic₅	—	2	2	no	—
24	C₂×Dic₅	—	4	2	no	—
25	#168	—	2	2	no	—
26	C₂²⋊Q₈	D₅	1	1	yes	characteristic
27	C₂×C₄⋊C₄	—	5	1	no	—
28	#106	—	5	2	no	—
29	#103	—	5	2	no	—
30	C₂²⋊C₈	—	5	1	no	—
31	C₂²×C₅	—	2	1	no	—
32	C₂²×C₅	—	2	1	no	—
33	C₂²×C₅	SD₁₆	1	2	yes	—
34	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
35	C₄×C₅	—	2	1	no	—
36	C₄×C₅	—	4	1	no	—
37	C₄×C₅	—	4	1	no	—
38	C₄×C₅	#41	1	2	yes	—
39	Dic₅	—	4	2	no	—
40	C₂×Q₈	C₂×D₅	1	1	yes	characteristic
41	C₂²×C₄	—	10	1	no	—
42	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
43	C₂×C₈	—	5	2	no	—
44	C₄⋊C₄	—	2	1	no	—
45	C₄⋊C₄	—	5	2	no	—
46	C₄⋊C₄	—	10	1	no	—
47	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
48	C₂²⋊C₄	—	2	1	no	—
49	C₂×C₅	—	2	2	no	—
50	C₂×C₅	#123	1	1	yes	characteristic
51	C₂×C₅	C₂×SD₁₆	1	1	yes	characteristic
52	C₂×C₅	#137	1	1	yes	characteristic
53	C₂³	C₂²⋊D₅	1	1	yes	characteristic
54	Q₈	—	2	2	no	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	10	2	no	—
59	C₂×C₄	—	20	2	no	—
60	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
61	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
62	C₈	—	10	2	no	—
63	C₅	#483	1	1	yes	characteristic, complemented, Sylow
64	C₂²	—	2	1	no	—
65	C₂²	—	2	1	no	—
66	C₂²	#688	1	2	yes	—
67	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	4	1	no	—
70	C₄	—	4	1	no	—
71	C₄	—	20	2	no	—
72	C₄	#711	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	#4081	1	1	yes	characteristic
75	C₂	C₂×#688	1	1	yes	characteristic
76	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	C₅×C₂²⋊Q₈	C₂
socle	C₂²×C₅	C₂×D₄
prime radical	C₂³×C₅	D₄
hypercenter	C₂²⋊Q₈	D₅
hypocenter	C₅	#483
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
3. #3967	C₂	semidirect
5. #3940	C₂	semidirect
63. C₅	#483	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₂²⋊Q₈	2
C₂×C₄⋊Dic₅	2
#3967	2
C₂²⋊#168	2
#3940	2
#483	5