Subgroups — #13102798

Order 1600.

616 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×C₅²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#1039	C₂	1	2	yes	maximal
3	#11716698	C₂	1	4	yes	maximal
4	C₂²×C₄×C₅²	C₂²	1	2	yes	—
5	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
6	C₂²×#1039	C₂²	1	4	yes	—
7	C₄²×C₅²	C₄	1	4	yes	complemented
8	C₄⋊#1039	—	2	4	no	—
9	#65225	—	5	6	no	maximal
10	C₂³×C₅²	C₂³	1	1	yes	characteristic
11	C₂×C₄×C₅²	—	2	4	no	—
12	C₂×C₄×C₅²	C₂×C₄	1	4	yes	—
13	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₅²	D₄	1	2	yes	—
15	C₂×C₄×C₅²	Q₈	1	2	yes	—
16	C₂×#1039	—	2	4	no	—
17	C₂×#1039	—	2	8	no	—
18	C₂×C₄²×C₅	D₅	1	6	yes	—
19	C₂×C₄⋊Dic₅	—	5	12	no	—
20	#3963	—	5	24	no	—
21	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
22	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
23	C₂²×C₅²	C₂×Q₈	1	1	yes	characteristic
24	C₂²×C₅²	#41	1	4	yes	—
25	C₄×C₅²	—	2	4	no	—
26	C₄×C₅²	C₄⋊C₄	1	4	yes	—
27	#1039	—	4	4	no	—
28	C₂²×C₄×C₅	C₂×D₅	1	12	yes	—
29	C₂²×C₄×C₅	C₂×D₅	1	6	yes	—
30	C₂²×Dic₅	—	5	24	no	—
31	C₄²×C₅	Dic₅	1	24	yes	complemented
32	C₄⋊Dic₅	—	10	24	no	—
33	#382	—	25	1	no	Sylow
34	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₅²	#117	1	2	yes	—
36	C₂×C₅²	#124	1	2	yes	—
37	C₂×C₅²	#125	1	2	yes	—
38	C₂³×C₅	C₂²×D₅	1	6	yes	—
39	C₂×C₄×C₅	—	2	24	no	—
40	C₂×C₄×C₅	Dic₁₀	1	12	yes	—
41	C₂×C₄×C₅	D₂₀	1	12	yes	—
42	C₂×C₄×C₅	C₂×Dic₅	1	24	yes	—
43	C₂×C₄×C₅	C₂×Dic₅	1	12	yes	—
44	C₂×Dic₅	—	10	24	no	—
45	C₂×Dic₅	—	10	48	no	—
46	C₂×C₄⋊C₄	—	25	2	no	—
47	C₂×C₄²	C₅⋊D₅	1	1	yes	characteristic
48	#95	—	25	4	no	—
49	C₅²	#382	1	1	yes	characteristic, complemented, Sylow
50	C₂²×C₅	C₂×Dic₁₀	1	6	yes	—
51	C₂²×C₅	C₂×D₂₀	1	6	yes	—
52	C₂²×C₅	#709	1	24	yes	—
53	C₂²×C₅	C₂²×Dic₅	1	6	yes	—
54	C₄×C₅	—	2	24	no	—
55	C₄×C₅	C₄⋊Dic₅	1	24	yes	—
56	Dic₅	—	20	24	no	—
57	C₂²×C₄	—	25	4	no	—
58	C₂²×C₄	C₂×C₅⋊D₅	1	2	yes	—
59	C₂²×C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
60	C₄⋊C₄	—	50	4	no	—
61	C₄²	#1039	1	4	yes	complemented
62	C₂×C₅	#4016	1	12	yes	—
63	C₂×C₅	#4021	1	12	yes	—
64	C₂×C₅	C₂×C₄⋊Dic₅	1	6	yes	—
65	C₂×C₅	#4072	1	12	yes	—
66	C₂³	C₂²×C₅⋊D₅	1	1	yes	characteristic
67	C₂×C₄	—	2	4	no	—
68	C₂×C₄	—	50	4	no	—
69	C₂×C₄	—	50	8	no	—
70	C₂×C₄	#6045	1	2	yes	—
71	C₂×C₄	C₄⋊(C₅⋊D₅)	1	2	yes	—
72	C₂×C₄	C₂×#1039	1	4	yes	—
73	C₂×C₄	C₂×#1039	1	2	yes	—
74	C₅	#65225	1	6	yes	complemented
75	C₂²	C₂×#6045	1	1	yes	characteristic
76	C₂²	C₂×C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
77	C₂²	#87900	1	4	yes	—
78	C₂²	C₂²×#1039	1	1	yes	characteristic
79	C₄	—	2	4	no	—
80	C₄	—	100	4	no	—
81	C₄	C₄⋊#1039	1	4	yes	—
82	C₂	#11717190	1	2	yes	—
83	C₂	#11717195	1	2	yes	—
84	C₂	C₂×C₄⋊#1039	1	1	yes	characteristic
85	C₂	#11717246	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×C₅⋊D₅
commutator	C₂²×C₅²	C₂²×C₄
Fitting	C₂×C₄²×C₅²	C₂
socle, prime radical, cosocle	C₂³×C₅²	C₂³
hypercenter	C₂×C₄²	C₅⋊D₅
hypocenter	C₅²	#382
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
7. C₄²×C₅²	C₄	semidirect
31. C₄²×C₅	Dic₅	semidirect
49. C₅²	#382	semidirect
61. C₄²	#1039	semidirect
74. C₅	#65225	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄²×C₅²	2
C₂×C₄⋊#1039	2
#11716698	2
#65225	5