Subgroups — #64867

Order 320.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#704	C₂	1	1	yes	maximal, characteristic
2	C₂×#704	C₂	1	1	yes	maximal, characteristic
3	#4023	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₅	C₂²	1	1	yes	characteristic
5	#704	—	2	1	no	—
6	#704	—	2	1	no	—
7	#704	—	2	1	no	—
8	#704	C₄	1	2	yes	complemented
9	C₂×#170	—	2	1	no	—
10	C₂×#170	C₄	1	2	yes	complemented
11	C₅×C₂²⋊C₄	C₄	1	1	yes	characteristic
12	C₂²⋊Dic₅	C₄	1	1	yes	characteristic
13	#683	—	2	1	no	—
14	C₄×Dic₅	—	2	1	no	—
15	#344	—	5	1	no	maximal, Sylow
16	C₂³×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₅	—	2	1	no	—
18	C₂×Dic₅	—	2	1	no	—
19	C₂×Dic₅	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₅	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₅	D₄	1	1	yes	characteristic
22	C₂×Dic₅	D₄	1	1	yes	characteristic
23	C₂×Dic₅	D₄	1	1	yes	characteristic
24	C₂×Dic₅	Q₈	1	1	yes	characteristic
25	#170	—	2	2	no	—
26	#170	—	2	2	no	—
27	C₂×M₁₆	—	5	1	no	—
28	C₂×M₁₆	—	5	1	no	—
29	#117	—	5	1	no	—
30	C₂²×C₅	—	4	1	no	—
31	C₂²×C₅	C₄²	1	1	yes	characteristic
32	C₂²×C₅	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₅	C₄⋊C₄	1	1	yes	characteristic
34	C₄×C₅	—	4	1	no	—
35	Dic₅	—	4	1	no	—
36	Dic₅	C₂²⋊C₄	1	1	yes	characteristic
37	Dic₅	C₂²⋊C₄	1	1	yes	characteristic
38	Dic₅	C₄⋊C₄	1	1	yes	characteristic
39	Dic₅	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₄	—	5	1	no	—
41	M₁₆	—	5	2	no	—
42	M₁₆	—	10	1	no	—
43	M₁₆	—	10	1	no	—
44	M₁₆	—	10	1	no	—
45	C₂×C₈	—	5	2	no	—
46	C₂×C₈	—	10	1	no	—
47	C₄⋊C₄	—	10	1	no	—
48	C₂²⋊C₄	—	5	1	no	—
49	C₂²⋊C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
50	C₄²	—	10	1	no	—
51	C₂×C₅	—	2	1	no	—
52	C₂×C₅	—	2	1	no	—
53	C₂×C₅	—	2	1	no	—
54	C₂×C₅	#95	1	1	yes	characteristic
55	C₂³	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	5	1	no	—
58	C₂×C₄	—	5	1	no	—
59	C₂×C₄	—	5	1	no	—
60	C₂×C₄	—	5	1	no	—
61	C₂×C₄	—	5	1	no	—
62	C₂×C₄	—	5	1	no	—
63	C₂×C₄	—	10	1	no	—
64	C₈	—	10	2	no	—
65	C₈	—	10	2	no	—
66	C₅	#344	1	1	yes	characteristic, complemented, Sylow
67	C₂²	—	4	1	no	—
68	C₂²	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
69	C₂²	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
70	C₂²	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
71	C₄	—	4	1	no	—
72	C₄	—	5	1	no	—
73	C₄	—	5	1	no	—
74	C₄	—	5	1	no	—
75	C₄	—	5	1	no	—
76	C₄	—	20	1	no	—
77	C₂	—	2	1	no	—
78	C₂	—	2	1	no	—
79	C₂	—	2	1	no	—
80	C₂	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	(C₂×AGL(1,𝔽₅))⋊C₄
commutator	C₂²×C₅	C₄²
Frattini	C₂³	C₂×AGL(1,𝔽₅)
Fitting	C₅×C₂²⋊C₄	C₄
socle	C₂×C₅	#95
prime radical	C₂³×C₅	C₂×C₄
cosocle	C₂²×Dic₅	C₂²
hypercenter	C₂²⋊C₄	AGL(1,𝔽₅)
hypocenter	C₅	#344
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
8. #704	C₄	semidirect
10. C₂×#170	C₄	semidirect
66. C₅	#344	semidirect

Maximal subgroups

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

subgroup	index
C₂×#704	2
#4023	2
#344	5