Subgroups — #13107347

Order 1600.

1032 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×#41	C₂	1	1	yes	maximal, characteristic
2	#11717268	C₂	1	8	yes	maximal, complemented
3	C₂×#87804	C₂	1	6	yes	maximal, complemented
4	C₅²×#41	C₄	1	4	yes	—
5	C₅²×#41	C₂²	1	4	yes	—
6	C₂×C₅²×Q₈	C₄	1	1	yes	characteristic
7	C₂×C₅²×D₄	C₄	1	3	yes	—
8	C₂²×C₄×C₅²	C₂²	1	3	yes	—
9	#87804	C₂²	1	4	yes	—
10	#87804	C₂²	1	12	yes	complemented
11	C₂×#6029	C₂²	1	12	yes	complemented
12	#66153	—	5	6	no	maximal
13	C₂³×C₅²	C₂×C₄	1	3	yes	—
14	C₅²×Q₈	C₂×C₄	1	4	yes	—
15	C₅²×D₄	C₂×C₄	1	12	yes	—
16	C₂×C₄×C₅²	C₂×C₄	1	3	yes	—
17	C₂×C₄×C₅²	C₂×C₄	1	6	yes	—
18	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
19	C₂×C₄×C₅²	C₂³	1	6	yes	—
20	#6029	C₂³	1	8	yes	—
21	C₂×C₅×#41	D₅	1	6	yes	—
22	#4094	—	5	48	no	—
23	C₂×#681	—	5	36	no	—
24	C₂²×C₅²	—	4	3	no	—
25	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₅²	C₂²×C₄	1	6	yes	—
27	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
28	C₄×C₅²	C₂²×C₄	1	6	yes	—
29	C₄×C₅²	C₂⁴	1	1	yes	characteristic
30	C₅×#41	Dic₅	1	24	yes	—
31	C₅×#41	C₂×D₅	1	24	yes	—
32	C₂×C₅×Q₈	Dic₅	1	6	yes	—
33	C₂×C₅×D₄	Dic₅	1	18	yes	—
34	C₂²×C₄×C₅	C₂×D₅	1	18	yes	—
35	#681	—	5	24	no	—
36	#681	—	5	72	no	—
37	C₂×#168	—	5	72	no	—
38	#568	—	25	1	no	Sylow
39	C₂×C₅²	—	2	1	no	—
40	C₂×C₅²	—	2	6	no	—
41	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
42	C₂³×C₅	C₂×Dic₅	1	18	yes	—
43	C₅×Q₈	C₂×Dic₅	1	24	yes	—
44	C₅×D₄	C₂×Dic₅	1	72	yes	—
45	C₂×C₄×C₅	C₂×Dic₅	1	18	yes	—
46	C₂×C₄×C₅	C₂×Dic₅	1	36	yes	—
47	C₂×C₄×C₅	C₂²×D₅	1	36	yes	—
48	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
49	#168	—	5	48	no	—
50	C₂×#41	C₅⋊D₅	1	1	yes	characteristic
51	#131	—	25	8	no	—
52	C₂×M₁₆	—	25	6	no	—
53	C₅²	#568	1	1	yes	characteristic, complemented, Sylow
54	C₂²×C₅	—	4	18	no	—
55	C₂²×C₅	C₂²×Dic₅	1	6	yes	—
56	C₂²×C₅	C₂²×Dic₅	1	36	yes	—
57	C₄×C₅	C₂²×Dic₅	1	6	yes	—
58	C₄×C₅	C₂²×Dic₅	1	36	yes	—
59	C₄×C₅	C₂³×D₅	1	6	yes	—
60	#41	#1039	1	4	yes	—
61	#41	C₂×C₅⋊D₅	1	4	yes	—
62	C₂×Q₈	#1039	1	1	yes	characteristic
63	C₂×D₄	#1039	1	3	yes	—
64	C₂²×C₄	C₂×C₅⋊D₅	1	3	yes	—
65	M₁₆	—	25	4	no	—
66	M₁₆	—	25	12	no	—
67	C₂×C₈	—	25	12	no	—
68	C₂×C₅	—	2	6	no	—
69	C₂×C₅	—	2	36	no	—
70	C₂×C₅	C₂³×Dic₅	1	6	yes	—
71	C₂³	C₂×#1039	1	3	yes	—
72	Q₈	C₂×#1039	1	4	yes	—
73	D₄	C₂×#1039	1	12	yes	—
74	C₂×C₄	C₂×#1039	1	3	yes	—
75	C₂×C₄	C₂×#1039	1	6	yes	—
76	C₂×C₄	C₂²×C₅⋊D₅	1	6	yes	—
77	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
78	C₈	—	25	8	no	—
79	C₅	#66153	1	6	yes	complemented
80	C₂²	—	4	3	no	—
81	C₂²	C₂²×#1039	1	1	yes	characteristic
82	C₂²	C₂²×#1039	1	6	yes	—
83	C₄	C₂²×#1039	1	1	yes	characteristic
84	C₄	C₂²×#1039	1	6	yes	—
85	C₄	C₂³×C₅⋊D₅	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	2	6	no	—
88	C₂	C₂³×#1039	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂³×C₅⋊D₅
commutator, socle	C₂×C₅²	C₂³×C₄
Fitting, prime radical	C₂×C₅²×#41	C₂
cosocle	C₄×C₅²	C₂⁴
hypercenter	C₂×#41	C₅⋊D₅
hypocenter	C₅²	#568
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
2. #11717268	C₂	semidirect
3. C₂×#87804	C₂	semidirect
10. #87804	C₂²	semidirect
11. C₂×#6029	C₂²	semidirect
53. C₅²	#568	semidirect
79. C₅	#66153	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅²×#41	2
#11717268	2
C₂×#87804	2
#66153	5