Subgroups — #65729

Order 320.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#700	C₂	1	1	yes	maximal, characteristic
3	D₅×M₁₆	C₂	1	1	yes	maximal, characteristic
4	#3996	C₂	1	2	yes	maximal, complemented
5	#3995	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
7	#704	—	2	1	no	—
8	C₂×#170	—	2	1	no	—
9	#700	—	2	1	no	—
10	#700	C₂²	1	2	yes	—
11	#700	C₂²	1	1	yes	characteristic
12	#700	C₂²	1	1	yes	characteristic
13	D₅⋊C₈	—	2	1	no	—
14	C₅×M₁₆	C₄	1	1	yes	characteristic
15	#681	C₄	1	1	yes	characteristic
16	#676	C₄	1	2	yes	—
17	C₈×D₅	C₂²	1	2	yes	—
18	#430	—	5	1	no	maximal, Sylow
19	C₂²×D₅	Q₈	1	1	yes	characteristic
20	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₅	D₄	1	1	yes	characteristic
22	C₄×D₅	C₂×C₄	1	1	yes	characteristic
23	C₄×D₅	D₄	1	1	yes	characteristic
24	C₄×D₅	Q₈	1	1	yes	characteristic
25	C₄×D₅	C₂³	1	1	yes	characteristic
26	#170	—	2	1	no	—
27	#170	—	2	1	no	—
28	#170	—	2	2	no	—
29	C₅×C₈	C₂×C₄	1	2	yes	—
30	#168	C₂×C₄	1	2	yes	—
31	C₂×M₁₆	—	5	1	no	—
32	C₂×M₁₆	—	5	1	no	—
33	C₂×M₁₆	—	5	1	no	—
34	#108	—	5	2	no	—
35	#108	—	5	2	no	—
36	C₂²×C₅	C₄⋊C₄	1	1	yes	characteristic
37	C₂×D₅	—	4	1	no	—
38	C₂×D₅	C₄⋊C₄	1	1	yes	characteristic
39	C₂×D₅	C₂×Q₈	1	1	yes	characteristic
40	C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
41	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
42	Dic₅	C₄⋊C₄	1	1	yes	characteristic
43	Dic₅	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄	—	5	1	no	—
45	M₁₆	—	5	1	no	—
46	M₁₆	—	5	2	no	—
47	M₁₆	—	5	1	no	—
48	M₁₆	—	5	2	no	—
49	M₁₆	—	5	1	no	—
50	M₁₆	—	10	1	no	—
51	M₁₆	—	10	1	no	—
52	M₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
53	C₂×C₈	—	5	2	no	—
54	C₂×C₈	—	10	1	no	—
55	C₂×C₈	—	10	1	no	—
56	C₂×C₅	—	2	1	no	—
57	C₂×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
58	D₅	—	2	1	no	—
59	D₅	—	2	1	no	—
60	C₂³	—	5	1	no	—
61	C₂×C₄	—	5	1	no	—
62	C₂×C₄	—	5	1	no	—
63	C₂×C₄	—	5	1	no	—
64	C₂×C₄	—	5	1	no	—
65	C₂×C₄	—	5	1	no	—
66	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
67	C₈	—	5	2	no	—
68	C₈	—	10	1	no	—
69	C₈	—	10	1	no	—
70	C₈	—	10	2	no	—
71	C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
72	C₅	#430	1	1	yes	characteristic, complemented, Sylow
73	C₂²	—	5	1	no	—
74	C₂²	—	5	1	no	—
75	C₂²	—	20	1	no	—
76	C₂²	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
77	C₄	—	5	1	no	—
78	C₄	—	5	1	no	—
79	C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
80	C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
81	C₂	—	2	1	no	—
82	C₂	—	10	1	no	—
83	C₂	—	10	1	no	—
84	C₂	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₄⋊AGL(1,𝔽₅)
commutator	C₄×C₅	C₂²×C₄
Frattini	C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×M₁₆	C₄
socle	C₂×C₅	C₂×C₄⋊C₄
prime radical	C₂²×D₅	Q₈
cosocle	C₄×D₅	C₂³
hypercenter	M₁₆	AGL(1,𝔽₅)
hypocenter	C₅	#430
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
4. #3996	C₂	semidirect
5. #3995	C₂	semidirect
72. C₅	#430	semidirect

Maximal subgroups

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

subgroup	index
D₅⋊M₁₆	2
C₂×#700	2
D₅×M₁₆	2
#3996	2
#3995	2
#430	5