Subgroups — #13103074

Order 1600.

872 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₅²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊#1039	C₂	1	2	yes	maximal
3	#11716698	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₅²	C₂²	1	1	yes	characteristic
5	C₂²×C₄×C₅²	—	2	4	no	—
6	C₂²×C₄×C₅²	C₄	1	4	yes	complemented
7	C₂²×C₄×C₅²	C₂²	1	2	yes	—
8	C₂²×#1039	C₂²	1	4	yes	—
9	C₂²⋊#1039	—	2	4	no	—
10	#65500	—	5	6	no	maximal
11	C₂³×C₅²	—	2	4	no	—
12	C₂³×C₅²	C₂×C₄	1	2	yes	—
13	C₂³×C₅²	D₄	1	4	yes	—
14	C₂³×C₅²	C₂³	1	1	yes	characteristic
15	C₂×C₄×C₅²	—	2	8	no	—
16	C₂×C₄×C₅²	—	2	4	no	—
17	C₂×C₄×C₅²	C₂×C₄	1	4	yes	—
18	C₂×#1039	—	2	4	no	—
19	C₂×#1039	—	2	8	no	—
20	C₂³×C₄×C₅	D₅	1	6	yes	—
21	C₂×C₂²⋊Dic₅	—	5	12	no	—
22	#3963	—	5	24	no	—
23	C₂²×C₅²	—	2	4	no	—
24	C₂²×C₅²	—	2	8	no	—
25	C₂²×C₅²	C₂²⋊C₄	1	4	yes	—
26	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
27	C₂²×C₅²	C₂×D₄	1	2	yes	—
28	C₂²×C₅²	#41	1	4	yes	—
29	C₄×C₅²	—	2	4	no	—
30	#1039	—	4	4	no	—
31	C₂⁴×C₅	C₂×D₅	1	6	yes	—
32	C₂²×C₄×C₅	—	2	24	no	—
33	C₂²×C₄×C₅	Dic₅	1	24	yes	complemented
34	C₂²×C₄×C₅	C₂×D₅	1	12	yes	—
35	C₂²×Dic₅	—	5	24	no	—
36	C₂²⋊Dic₅	—	10	24	no	—
37	#381	—	25	1	no	Sylow
38	C₂×C₅²	—	2	4	no	—
39	C₂×C₅²	C₂×C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₅²	#117	1	2	yes	—
41	C₂×C₅²	#123	1	4	yes	—
42	C₂³×C₅	—	2	24	no	—
43	C₂³×C₅	C₂×Dic₅	1	12	yes	—
44	C₂³×C₅	C₂²⋊D₅	1	24	yes	—
45	C₂³×C₅	C₂²×D₅	1	6	yes	—
46	C₂×C₄×C₅	—	2	48	no	—
47	C₂×C₄×C₅	—	2	24	no	—
48	C₂×C₄×C₅	C₂×Dic₅	1	24	yes	—
49	C₂×Dic₅	—	10	48	no	—
50	C₂×Dic₅	—	10	24	no	—
51	C₂³×C₄	C₅⋊D₅	1	1	yes	characteristic
52	C₂×C₂²⋊C₄	—	25	2	no	—
53	#95	—	25	4	no	—
54	C₅²	#381	1	1	yes	characteristic, complemented, Sylow
55	C₂²×C₅	—	2	24	no	—
56	C₂²×C₅	—	2	48	no	—
57	C₂²×C₅	C₂²⋊Dic₅	1	24	yes	—
58	C₂²×C₅	#709	1	24	yes	—
59	C₂²×C₅	C₂²×Dic₅	1	6	yes	—
60	C₂²×C₅	C₂×C₂²⋊D₅	1	12	yes	—
61	C₄×C₅	—	2	24	no	—
62	Dic₅	—	20	24	no	—
63	C₂⁴	C₂×C₅⋊D₅	1	1	yes	characteristic
64	C₂²×C₄	—	2	4	no	—
65	C₂²×C₄	—	25	4	no	—
66	C₂²×C₄	#1039	1	4	yes	complemented
67	C₂²×C₄	C₂×C₅⋊D₅	1	2	yes	—
68	C₂²⋊C₄	—	50	4	no	—
69	C₂×C₅	—	2	24	no	—
70	C₂×C₅	#4072	1	12	yes	—
71	C₂×C₅	#4075	1	24	yes	—
72	C₂×C₅	C₂×C₂²⋊Dic₅	1	6	yes	—
73	C₂³	—	2	4	no	—
74	C₂³	C₂×#1039	1	2	yes	—
75	C₂³	C₂²⋊(C₅⋊D₅)	1	4	yes	—
76	C₂³	C₂²×C₅⋊D₅	1	1	yes	characteristic
77	C₂×C₄	—	2	8	no	—
78	C₂×C₄	—	2	4	no	—
79	C₂×C₄	—	50	8	no	—
80	C₂×C₄	—	50	4	no	—
81	C₂×C₄	C₂×#1039	1	4	yes	—
82	C₅	#65500	1	6	yes	complemented
83	C₂²	—	2	4	no	—
84	C₂²	—	2	8	no	—
85	C₂²	C₂²⋊#1039	1	4	yes	—
86	C₂²	#87900	1	4	yes	—
87	C₂²	C₂²×#1039	1	1	yes	characteristic
88	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	2	yes	—
89	C₄	—	2	4	no	—
90	C₄	—	100	4	no	—
91	C₂	—	2	4	no	—
92	C₂	#11717246	1	2	yes	—
93	C₂	#11717249	1	4	yes	—
94	C₂	C₂×C₂²⋊#1039	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×C₅⋊D₅
commutator	C₂²×C₅²	C₂²×C₄
Fitting	C₂³×C₄×C₅²	C₂
socle, cosocle	C₂³×C₅²	C₂³
prime radical	C₂⁴×C₅²	C₂²
hypercenter	C₂³×C₄	C₅⋊D₅
hypocenter	C₅²	#381
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
3. #11716698	C₂	semidirect
6. C₂²×C₄×C₅²	C₄	semidirect
33. C₂²×C₄×C₅	Dic₅	semidirect
54. C₅²	#381	semidirect
66. C₂²×C₄	#1039	semidirect
82. C₅	#65500	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄×C₅²	2
C₂×C₂²⋊#1039	2
#11716698	2
#65500	5