Subgroups — #13102540

Order 1600.

744 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#87808	C₂	1	3	yes	maximal
2	C₂×C₄⋊#1039	C₂	1	3	yes	maximal, complemented
3	C₅²×#95	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×C₅⋊D₅	C₂²	1	1	yes	characteristic
5	C₂²×C₄×C₅²	C₂²	1	3	yes	—
6	C₂²×#1039	C₂²	1	3	yes	—
7	#87808	—	2	6	no	—
8	C₄⋊#1039	—	2	6	no	—
9	#64966	—	5	6	no	maximal
10	C₂³×C₅²	C₂³	1	1	yes	characteristic
11	C₂²×C₅⋊D₅	—	2	3	no	—
12	C₂²×C₅⋊D₅	—	2	3	no	—
13	C₂²×C₅⋊D₅	Q₈	1	2	yes	—
14	C₂×C₄×C₅²	—	2	6	no	—
15	C₂×C₄×C₅²	D₄	1	6	yes	—
16	C₂×#1039	—	2	3	no	—
17	C₂×#1039	—	2	6	no	—
18	C₂×#685	—	5	18	no	—
19	C₂×C₄⋊Dic₅	—	5	18	no	—
20	C₅×#95	D₅	1	6	yes	complemented
21	C₂²×C₅²	C₂×D₄	1	3	yes	—
22	C₂²×C₅²	C₂×Q₈	1	1	yes	characteristic
23	C₂²×C₅²	#41	1	3	yes	—
24	C₂×C₅⋊D₅	—	2	6	no	—
25	C₂×C₅⋊D₅	—	4	3	no	—
26	C₂×C₅⋊D₅	—	4	1	no	—
27	C₄×C₅²	—	2	6	no	—
28	#1039	—	4	3	no	—
29	C₂³×D₅	—	5	6	no	—
30	C₂²×C₄×C₅	C₂×D₅	1	18	yes	—
31	C₂²×Dic₅	—	5	18	no	—
32	#685	—	10	36	no	—
33	C₄⋊Dic₅	—	10	36	no	—
34	#399	—	25	1	no	Sylow
35	C₂×C₅²	C₂²⋊Q₈	1	3	yes	—
36	C₂×C₅²	#123	1	3	yes	—
37	C₂×C₅²	#127	1	1	yes	characteristic
38	C₅⋊D₅	—	4	2	no	—
39	C₂³×C₅	C₂²×D₅	1	6	yes	—
40	C₂²×D₅	—	5	12	no	—
41	C₂²×D₅	—	10	18	no	—
42	C₂²×D₅	—	10	18	no	—
43	C₂×C₄×C₅	—	2	36	no	—
44	C₂×C₄×C₅	D₂₀	1	36	yes	—
45	C₂×Dic₅	—	10	36	no	—
46	C₂×Dic₅	—	10	18	no	—
47	C₂×C₄⋊C₄	—	25	3	no	—
48	C₂×C₂²⋊C₄	—	25	3	no	—
49	#95	C₅⋊D₅	1	1	yes	characteristic, complemented
50	C₅²	#399	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₅	C₂×D₂₀	1	18	yes	—
52	C₂²×C₅	#711	1	18	yes	—
53	C₂²×C₅	Q₈×D₅	1	6	yes	—
54	C₂×D₅	—	10	36	no	—
55	C₂×D₅	—	20	6	no	—
56	C₂×D₅	—	20	18	no	—
57	C₄×C₅	—	2	36	no	—
58	Dic₅	—	20	18	no	—
59	C₂⁴	—	25	1	no	—
60	C₂²×C₄	—	25	3	no	—
61	C₂²×C₄	C₂×C₅⋊D₅	1	3	yes	—
62	C₄⋊C₄	—	50	6	no	—
63	C₂²⋊C₄	—	50	6	no	—
64	C₂×C₅	#4020	1	6	yes	—
65	C₂×C₅	#4032	1	18	yes	—
66	C₂×C₅	#4043	1	18	yes	—
67	D₅	—	20	12	no	—
68	C₂³	—	25	2	no	—
69	C₂³	—	50	3	no	—
70	C₂³	—	50	3	no	—
71	C₂³	C₂²×C₅⋊D₅	1	1	yes	characteristic
72	C₂×C₄	—	2	6	no	—
73	C₂×C₄	—	50	3	no	—
74	C₂×C₄	—	50	6	no	—
75	C₂×C₄	C₄⋊(C₅⋊D₅)	1	6	yes	—
76	C₅	#64966	1	6	yes	complemented
77	C₂²	—	50	6	no	—
78	C₂²	—	100	1	no	—
79	C₂²	—	100	3	no	—
80	C₂²	C₂×C₄⋊(C₅⋊D₅)	1	3	yes	—
81	C₂²	#87902	1	3	yes	—
82	C₂²	Q₈×C₅⋊D₅	1	1	yes	characteristic
83	C₄	—	2	6	no	—
84	C₄	—	100	3	no	—
85	C₂	—	100	2	no	—
86	C₂	#11717194	1	1	yes	characteristic
87	C₂	#11717206	1	3	yes	—
88	C₂	#11717217	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×C₅⋊D₅
commutator, socle, cosocle	C₂³×C₅²	C₂³
Fitting	C₅²×#95	C₂
prime radical	C₂³×C₅⋊D₅	C₂²
hypercenter	#95	C₅⋊D₅
hypocenter	C₅²	#399
LCS	G ▷ C₂³×C₅² ▷ C₅²
UCS	C₁ ◁ C₂³ ◁ #95

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊#1039	C₂	semidirect
3. C₅²×#95	C₂	semidirect
20. C₅×#95	D₅	semidirect
49. #95	C₅⋊D₅	semidirect
50. C₅²	#399	semidirect
76. C₅	#64966	semidirect

Maximal subgroups

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

subgroup	index
C₂×#87808	2
C₂×C₄⋊#1039	2
C₅²×#95	2
#64966	5