Subgroups — #13107337

Order 1600.

1064 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅²×Q₈	C₂	1	1	yes	maximal, characteristic
2	Q₈×#1039	C₂	1	8	yes	maximal, complemented
3	#11717246	C₂	1	6	yes	maximal
4	C₂×C₅²×Q₈	C₄	1	8	yes	complemented
5	C₂×C₅²×Q₈	C₂²	1	4	yes	—
6	C₂²×C₄×C₅²	C₂²	1	3	yes	—
7	C₂²⋊#1039	C₂²	1	4	yes	—
8	C₄⋊#1039	C₂²	1	12	yes	—
9	C₄×#1039	C₂²	1	12	yes	—
10	#66146	—	5	6	no	maximal
11	C₂³×C₅²	C₂³	1	1	yes	characteristic
12	C₅²×Q₈	C₂×C₄	1	16	yes	—
13	C₂×C₄×C₅²	C₂×C₄	1	12	yes	—
14	C₂×C₄×C₅²	C₂³	1	6	yes	—
15	C₂×#1039	C₂³	1	8	yes	—
16	C₂²×C₅×Q₈	D₅	1	6	yes	—
17	Q₈×Dic₅	—	5	48	no	—
18	#4072	—	5	36	no	—
19	C₂²×C₅²	—	2	2	no	—
20	C₂²×C₅²	C₂²×C₄	1	2	yes	—
21	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
22	C₄×C₅²	C₂²×C₄	1	12	yes	—
23	#1039	—	2	8	no	—
24	C₂×C₅×Q₈	Dic₅	1	48	yes	complemented
25	C₂×C₅×Q₈	C₂×D₅	1	24	yes	—
26	C₂²×C₄×C₅	C₂×D₅	1	18	yes	—
27	C₂²⋊Dic₅	—	5	24	no	—
28	C₄⋊Dic₅	—	5	72	no	—
29	C₄×Dic₅	—	5	72	no	—
30	#519	—	25	1	no	Sylow
31	C₂×C₅²	—	2	2	no	—
32	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
33	C₂×C₅²	2₋¹⁺⁴	1	2	yes	—
34	C₂³×C₅	C₂²×D₅	1	6	yes	—
35	C₅×Q₈	C₂×Dic₅	1	96	yes	—
36	C₂×C₄×C₅	C₂×Dic₅	1	72	yes	—
37	C₂×C₄×C₅	C₂²×D₅	1	36	yes	—
38	C₂×Dic₅	—	5	48	no	—
39	C₂²×Q₈	C₅⋊D₅	1	1	yes	characteristic
40	C₄×Q₈	—	25	8	no	—
41	#117	—	25	6	no	—
42	C₅²	#519	1	1	yes	characteristic, complemented, Sylow
43	C₂²×C₅	—	2	12	no	—
44	C₂²×C₅	C₂²×Dic₅	1	12	yes	—
45	C₂²×C₅	C₂³×D₅	1	6	yes	—
46	C₄×C₅	C₂²×Dic₅	1	72	yes	—
47	Dic₅	—	10	48	no	—
48	C₂×Q₈	#1039	1	8	yes	complemented
49	C₂×Q₈	C₂×C₅⋊D₅	1	4	yes	—
50	C₂²×C₄	C₂×C₅⋊D₅	1	3	yes	—
51	C₄⋊C₄	—	25	12	no	—
52	C₂²⋊C₄	—	25	4	no	—
53	C₄²	—	25	12	no	—
54	C₂×C₅	—	2	12	no	—
55	C₂×C₅	#4147	1	12	yes	—
56	C₂×C₅	C₂³×Dic₅	1	6	yes	—
57	C₂³	C₂²×C₅⋊D₅	1	1	yes	characteristic
58	Q₈	C₂×#1039	1	16	yes	—
59	C₂×C₄	—	25	8	no	—
60	C₂×C₄	C₂×#1039	1	12	yes	—
61	C₂×C₄	C₂²×C₅⋊D₅	1	6	yes	—
62	C₅	#66146	1	6	yes	complemented
63	C₂²	—	2	2	no	—
64	C₂²	C₂²×#1039	1	2	yes	—
65	C₂²	C₂³×C₅⋊D₅	1	1	yes	characteristic
66	C₄	—	50	8	no	—
67	C₄	C₂²×#1039	1	12	yes	—
68	C₂	—	2	2	no	—
69	C₂	#11717520	1	2	yes	—
70	C₂	C₂³×#1039	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₅⋊D₅
commutator	C₂×C₅²	C₂³×C₄
Fitting	C₂²×C₅²×Q₈	C₂
socle, cosocle	C₂²×C₅²	C₂⁴
prime radical	C₂³×C₅²	C₂³
hypercenter	C₂²×Q₈	C₅⋊D₅
hypocenter	C₅²	#519
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×#1039	C₂	semidirect
4. C₂×C₅²×Q₈	C₄	semidirect
24. C₂×C₅×Q₈	Dic₅	semidirect
42. C₅²	#519	semidirect
48. C₂×Q₈	#1039	semidirect
62. C₅	#66146	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅²×Q₈	2
Q₈×#1039	2
#11717246	2
#66146	5