Subgroups — C₅×#344 #64811

Order 320.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₄×C₅	C₂²⋊C₄
commutator	C₂²	C₄²×C₅
Frattini, cosocle	C₂²×C₄	C₂²×C₅
socle	C₂×C₅	#95
prime radical	C₂³×C₅	C₂×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₅×M₁₆	C₄	semidirect
10. C₂×C₅×C₈	C₄	semidirect
14. #344	C₅	direct (N × H)
41. M₁₆	C₄×C₅	semidirect
43. C₂×C₈	C₄×C₅	semidirect
61. C₅	#344	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×M₁₆	2
C₅×#117	2
#344	5

Subgroups — C5×#344 #64811

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#344 #64811