Subgroups — #65486

Order 320.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#135	C₂	1	1	yes	maximal, characteristic
2	#4094	C₂	1	2	yes	maximal, complemented
3	#3969	C₂	1	2	yes	maximal, complemented
4	#3951	C₂	1	1	yes	maximal, characteristic, complemented
5	#3947	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#41	C₂²	1	2	yes	—
7	C₅×Q₁₆	C₄	1	1	yes	characteristic, complemented
8	C₅×SD₁₆	C₄	1	2	yes	complemented
9	C₅×D₈	C₄	1	1	yes	characteristic, complemented
10	C₂×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₄×Dic₅	C₂²	1	1	yes	characteristic
12	#681	—	2	2	no	—
13	#681	C₂²	1	2	yes	—
14	C₂×#168	—	2	2	no	—
15	C₂×#168	C₂²	1	1	yes	characteristic
16	#444	—	5	1	no	maximal, Sylow
17	C₅×Q₈	C₂×C₄	1	2	yes	—
18	C₅×D₄	—	2	2	no	—
19	C₅×D₄	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₅	—	2	2	no	—
21	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
22	C₂×Dic₅	—	2	1	no	—
23	C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	#168	—	2	2	no	—
26	#168	D₄	1	2	yes	—
27	#135	D₅	1	1	yes	characteristic
28	#131	—	5	2	no	—
29	#108	—	5	1	no	—
30	#104	—	5	2	no	—
31	#97	—	5	1	no	—
32	C₂²×C₅	—	2	2	no	—
33	C₂²×C₅	#41	1	1	yes	characteristic
34	C₄×C₅	—	2	2	no	—
35	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
36	C₄×C₅	C₂×D₄	1	1	yes	characteristic
37	Dic₅	—	4	1	no	—
38	#41	C₂×D₅	1	2	yes	—
39	Q₁₆	Dic₅	1	1	yes	characteristic, complemented
40	SD₁₆	Dic₅	1	2	yes	complemented
41	D₈	Dic₅	1	1	yes	characteristic, complemented
42	M₁₆	—	5	2	no	—
43	M₁₆	—	10	2	no	—
44	C₂×C₈	—	5	1	no	—
45	C₂×C₈	—	10	2	no	—
46	C₂×C₈	C₂×D₅	1	1	yes	characteristic
47	C₄²	—	5	1	no	—
48	C₂×C₅	—	2	1	no	—
49	C₂×C₅	—	4	2	no	—
50	C₂×C₅	C₄×D₄	1	1	yes	characteristic
51	Q₈	C₂×Dic₅	1	2	yes	—
52	D₄	—	2	2	no	—
53	D₄	C₂×Dic₅	1	2	yes	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	10	1	no	—
56	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
57	C₈	—	5	2	no	—
58	C₈	—	10	2	no	—
59	C₈	C₂×Dic₅	1	1	yes	characteristic
60	C₈	C₂×Dic₅	1	1	yes	characteristic
61	C₅	#444	1	1	yes	characteristic, complemented, Sylow
62	C₂²	—	2	2	no	—
63	C₂²	#711	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	20	1	no	—
66	C₄	D₄×D₅	1	1	yes	characteristic
67	C₄	C₂²×Dic₅	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4	2	no	—
70	C₂	D₄×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₅
commutator	C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅
Fitting, prime radical	C₅×#135	C₂
socle	C₂×C₅	C₄×D₄
cosocle	C₂×C₄×C₅	C₂³
hypercenter	#135	D₅
hypocenter	C₅	#444
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
2. #4094	C₂	semidirect
3. #3969	C₂	semidirect
4. #3951	C₂	semidirect
5. #3947	C₂	semidirect
7. C₅×Q₁₆	C₄	semidirect
8. C₅×SD₁₆	C₄	semidirect
9. C₅×D₈	C₄	semidirect
39. Q₁₆	Dic₅	semidirect
40. SD₁₆	Dic₅	semidirect
41. D₈	Dic₅	semidirect
61. C₅	#444	semidirect

Maximal subgroups

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

subgroup	index
C₅×#135	2
#4094	2
#3969	2
#3951	2
#3947	2
#444	5