Subgroups — #65411

Order 320.

130 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	#4072	C₂	1	1	yes	maximal, characteristic
3	C₂²×#168	C₂	1	1	yes	maximal, characteristic
4	#3947	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₅	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₅×M₁₆	C₄	1	4	yes	complemented
8	C₂×C₅×C₈	C₂²	1	2	yes	—
9	C₂²⋊Dic₅	C₄	1	2	yes	—
10	C₄⋊Dic₅	C₄	1	2	yes	—
11	C₄×Dic₅	C₂²	1	2	yes	—
12	C₂×#168	C₄	1	4	yes	complemented
13	C₂×#168	C₂²	1	1	yes	characteristic
14	C₂×#168	C₂²	1	1	yes	characteristic
15	#405	—	5	1	no	maximal, Sylow
16	C₂³×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₅	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₅	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
21	C₂×Dic₅	C₂×C₄	1	4	yes	—
22	C₅×C₈	C₂×C₄	1	4	yes	—
23	#168	C₂×C₄	1	2	yes	—
24	#168	C₂×C₄	1	2	yes	—
25	C₂×M₁₆	D₅	1	1	yes	characteristic
26	C₂²×C₈	—	5	1	no	—
27	#117	—	5	1	no	—
28	#97	—	5	2	no	—
29	C₄×C₈	—	5	2	no	—
30	C₂²×C₅	—	2	2	no	—
31	C₂²×C₅	C₄²	1	2	yes	—
32	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
33	C₄×C₅	C₄²	1	2	yes	—
34	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
35	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
36	Dic₅	—	2	4	no	—
37	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
38	M₁₆	Dic₅	1	4	yes	complemented
39	C₂×C₈	—	5	4	no	—
40	C₂×C₈	—	5	1	no	—
41	C₂×C₈	—	5	1	no	—
42	C₂×C₈	C₂×D₅	1	2	yes	—
43	C₄⋊C₄	—	5	2	no	—
44	C₂²⋊C₄	—	5	2	no	—
45	C₄²	—	5	2	no	—
46	C₂×C₅	—	2	2	no	—
47	C₂×C₅	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₅	#131	1	2	yes	—
49	C₂³	C₄×D₅	1	1	yes	characteristic
50	C₂×C₄	—	5	4	no	—
51	C₂×C₄	C₄×D₅	1	1	yes	characteristic
52	C₂×C₄	C₄×D₅	1	2	yes	—
53	C₂×C₄	C₂×Dic₅	1	2	yes	—
54	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
55	C₈	—	5	2	no	—
56	C₈	—	5	2	no	—
57	C₈	C₂×Dic₅	1	4	yes	—
58	C₅	#405	1	1	yes	characteristic, complemented, Sylow
59	C₂²	—	2	2	no	—
60	C₂²	C₄×Dic₅	1	2	yes	—
61	C₂²	C₂×C₄×D₅	1	1	yes	characteristic
62	C₄	—	10	4	no	—
63	C₄	C₄×Dic₅	1	2	yes	—
64	C₄	C₂×C₄×D₅	1	1	yes	characteristic
65	C₄	C₂²×Dic₅	1	1	yes	characteristic
66	C₂	—	2	2	no	—
67	C₂	#4053	1	2	yes	—
68	C₂	C₂×C₄×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₅
commutator	C₂×C₅	C₂×C₄²
Fitting	C₂×C₅×M₁₆	C₂
socle	C₂²×C₅	C₂²×C₄
prime radical	C₂³×C₅	C₂×C₄
cosocle	C₂×C₄×C₅	C₂³
hypercenter	C₂×M₁₆	D₅
hypocenter	C₅	#405
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
4. #3947	C₂	semidirect
5. C₈×Dic₅	C₂	semidirect
7. C₅×M₁₆	C₄	semidirect
12. C₂×#168	C₄	semidirect
38. M₁₆	Dic₅	semidirect
58. C₅	#405	semidirect

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
#4072	2
C₂²×#168	2
#3947	2
C₈×Dic₅	2
#405	5