Subgroups — #65758

Order 320.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#704	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₅	C₂	1	1	yes	maximal, characteristic
3	#4004	C₂	1	2	yes	maximal, complemented
4	C₄⋊#170	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₅	C₄	1	1	yes	characteristic
6	C₂²×Dic₅	C₄	1	1	yes	characteristic
7	C₂²×Dic₅	C₂²	1	1	yes	characteristic
8	#704	—	2	4	no	—
9	C₂×#170	C₂²	1	4	yes	—
10	C₄×Dic₅	C₄	1	1	yes	characteristic
11	C₄×Dic₅	C₄	1	1	yes	characteristic
12	C₄×Dic₅	C₂²	1	1	yes	characteristic
13	C₄×Dic₅	C₂²	1	1	yes	characteristic
14	#423	—	5	1	no	maximal, Sylow
15	C₂³×C₅	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₅	—	2	2	no	—
17	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₅	—	2	2	no	—
20	C₂×Dic₅	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₅	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₅	C₂×C₄	1	1	yes	characteristic
23	C₂×Dic₅	D₄	1	2	yes	—
24	C₂×Dic₅	Q₈	1	2	yes	—
25	C₂×Dic₅	C₂³	1	1	yes	characteristic
26	#170	—	2	4	no	—
27	C₂×M₁₆	—	5	2	no	—
28	C₂×C₄²	—	5	1	no	—
29	C₄⋊C₈	—	5	2	no	—
30	C₄⋊C₈	—	5	2	no	—
31	C₂²×C₅	—	2	1	no	—
32	C₂²×C₅	—	2	1	no	—
33	C₂²×C₅	C₄⋊C₄	1	2	yes	—
34	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
35	C₄×C₅	—	2	1	no	—
36	C₄×C₅	M₁₆	1	2	yes	complemented
37	Dic₅	—	2	1	no	—
38	Dic₅	C₄⋊C₄	1	2	yes	—
39	Dic₅	M₁₆	1	2	yes	complemented
40	Dic₅	C₂×D₄	1	1	yes	characteristic
41	Dic₅	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄	—	5	1	no	—
43	C₂²×C₄	—	5	1	no	—
44	C₂²×C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
45	M₁₆	—	10	4	no	—
46	C₂×C₈	—	5	4	no	—
47	C₄²	—	5	1	no	—
48	C₄²	—	5	1	no	—
49	C₄²	—	5	1	no	—
50	C₄²	—	5	1	no	—
51	C₂×C₅	—	2	2	no	—
52	C₂×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₅	C₂×M₁₆	1	1	yes	characteristic
54	C₂×C₅	C₂×M₁₆	1	1	yes	characteristic
55	C₂³	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	5	2	no	—
58	C₂×C₄	—	5	2	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	2	no	—
64	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
65	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
66	C₈	—	10	4	no	—
67	C₅	#423	1	1	yes	characteristic, complemented, Sylow
68	C₂²	—	2	1	no	—
69	C₂²	—	2	1	no	—
70	C₂²	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
71	C₂²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
72	C₄	—	2	1	no	—
73	C₄	—	5	2	no	—
74	C₄	—	5	1	no	—
75	C₄	—	5	2	no	—
76	C₄	—	5	1	no	—
77	C₄	—	10	1	no	—
78	C₄	#704	1	2	yes	complemented
79	C₂	—	2	2	no	—
80	C₂	D₅⋊M₁₆	1	1	yes	characteristic
81	C₂	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
82	C₂	C₂×#704	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂²×C₄
Fitting	C₂²×C₄×C₅	C₄
prime radical	C₂³×C₅	C₂×C₄
cosocle	C₂×Dic₅	C₂³
hypercenter	C₂²×C₄	AGL(1,𝔽₅)
hypocenter	C₅	#423
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
3. #4004	C₂	semidirect
4. C₄⋊#170	C₂	semidirect
36. C₄×C₅	M₁₆	semidirect
39. Dic₅	M₁₆	semidirect
67. C₅	#423	semidirect
78. C₄	#704	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
C₂×C₄×Dic₅	2
#4004	2
C₄⋊#170	2
#423	5