Group #13098482 — subgroups

Subgroups — #13098482

Order 1600.

285 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₂₅	C₂	1	1	yes	maximal, characteristic
2	C₂×#87716	C₂	1	2	yes	maximal, complemented
3	C₂²⋊#6014	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₂₅	C₄	1	1	yes	characteristic
5	C₂²×C₄×C₂₅	—	2	4	no	—
6	C₂²×C₄×C₂₅	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₂₅	C₄	1	2	yes	—
8	C₂²×C₄×C₂₅	C₂²	1	2	yes	—
9	C₂²×C₄×C₂₅	C₂²	1	1	yes	characteristic
10	#87716	—	2	4	no	—
11	C₂×#6014	C₂²	1	4	yes	complemented
12	#65496	—	5	1	no	maximal
13	C₂³×C₂₅	—	2	2	no	—
14	C₂³×C₂₅	—	2	4	no	—
15	C₂³×C₂₅	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₂₅	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₂₅	—	2	2	no	—
18	C₂×C₄×C₂₅	—	2	4	no	—
19	C₂×C₄×C₂₅	—	2	4	no	—
20	C₂×C₄×C₂₅	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₂₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₂₅	D₄	1	4	yes	—
23	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
24	#6014	—	2	4	no	—
25	C₂³×C₄×C₅	D₅	1	1	yes	characteristic
26	C₂×#681	—	5	2	no	—
27	C₂²⋊#168	—	5	4	no	—
28	C₂²×C₂₅	—	2	8	no	—
29	C₂²×C₂₅	—	2	2	no	—
30	C₂²×C₂₅	—	2	2	no	—
31	C₂²×C₂₅	—	2	2	no	—
32	C₂²×C₂₅	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₂₅	M₁₆	1	4	yes	complemented
34	C₂²×C₂₅	C₂²×C₄	1	1	yes	characteristic
35	C₄×C₂₅	—	2	2	no	—
36	C₄×C₂₅	C₂²⋊C₄	1	2	yes	—
37	C₄×C₂₅	C₂×D₄	1	2	yes	—
38	C₂⁴×C₅	Dic₅	1	1	yes	characteristic
39	C₂²×C₄×C₅	—	2	4	no	—
40	C₂²×C₄×C₅	Dic₅	1	1	yes	characteristic
41	C₂²×C₄×C₅	Dic₅	1	2	yes	—
42	C₂²×C₄×C₅	C₂×D₅	1	2	yes	—
43	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
44	#681	—	10	4	no	—
45	C₂×#168	—	5	4	no	—
46	#407	—	25	1	no	Sylow
47	C₂×C₂₅	—	2	2	no	—
48	C₂×C₂₅	—	2	4	no	—
49	C₂×C₂₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₂₅	C₂×M₁₆	1	2	yes	—
51	C₂³×C₅	—	2	4	no	—
52	C₂³×C₅	—	2	2	no	—
53	C₂³×C₅	C₂×Dic₅	1	2	yes	—
54	C₂³×C₅	C₂×Dic₅	1	1	yes	characteristic
55	C₂×C₄×C₅	—	2	2	no	—
56	C₂×C₄×C₅	—	2	4	no	—
57	C₂×C₄×C₅	—	2	4	no	—
58	C₂×C₄×C₅	C₂×Dic₅	1	2	yes	—
59	C₂×C₄×C₅	C₂×Dic₅	1	1	yes	characteristic
60	C₂×C₄×C₅	C₂²⋊D₅	1	4	yes	—
61	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
62	#168	—	10	4	no	—
63	C₂³×C₄	D₂₅	1	1	yes	characteristic
64	C₂×M₁₆	—	25	2	no	—
65	C₂²⋊C₈	—	25	4	no	—
66	C₂₅	#407	1	1	yes	characteristic, complemented, Sylow
67	C₂²×C₅	—	2	8	no	—
68	C₂²×C₅	—	2	2	no	—
69	C₂²×C₅	—	2	2	no	—
70	C₂²×C₅	—	2	2	no	—
71	C₂²×C₅	#681	1	4	yes	—
72	C₂²×C₅	C₂²⋊Dic₅	1	2	yes	—
73	C₂²×C₅	C₂²×Dic₅	1	1	yes	characteristic
74	C₄×C₅	—	2	2	no	—
75	C₄×C₅	C₂²⋊Dic₅	1	2	yes	—
76	C₄×C₅	C₂×C₂²⋊D₅	1	2	yes	—
77	C₂⁴	Dic₂₅	1	1	yes	characteristic
78	C₂²×C₄	—	2	4	no	—
79	C₂²×C₄	Dic₂₅	1	1	yes	characteristic
80	C₂²×C₄	Dic₂₅	1	2	yes	—
81	C₂²×C₄	C₂×D₂₅	1	2	yes	—
82	C₂²×C₄	C₂×D₂₅	1	1	yes	characteristic
83	M₁₆	—	50	4	no	—
84	C₂×C₈	—	25	4	no	—
85	C₂×C₅	—	2	4	no	—
86	C₂×C₅	—	2	2	no	—
87	C₂×C₅	C₂×#681	1	2	yes	—
88	C₂×C₅	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
89	C₂³	—	2	4	no	—
90	C₂³	—	2	2	no	—
91	C₂³	C₂×Dic₂₅	1	2	yes	—
92	C₂³	C₂×Dic₂₅	1	1	yes	characteristic
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	C₂×Dic₂₅	1	2	yes	—
97	C₂×C₄	C₂×Dic₂₅	1	1	yes	characteristic
98	C₂×C₄	C₂²⋊D₂₅	1	4	yes	—
99	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
100	C₈	—	50	4	no	—
101	C₅	#65496	1	1	yes	characteristic
102	C₂²	—	2	8	no	—
103	C₂²	—	2	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	—	2	2	no	—
106	C₂²	#87716	1	4	yes	complemented
107	C₂²	C₂²⋊Dic₂₅	1	2	yes	—
108	C₂²	C₂²×Dic₂₅	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	C₂²⋊Dic₂₅	1	2	yes	—
111	C₄	C₂×C₂²⋊D₂₅	1	2	yes	—
112	C₂	—	2	4	no	—
113	C₂	—	2	2	no	—
114	C₂	C₂×#87716	1	2	yes	—
115	C₂	C₂×C₂²⋊Dic₂₅	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₄×C₂₅

C₂

yes

maximal, characteristic

C₂×#87716

C₂

yes

maximal, complemented

C₂²⋊#6014

C₂

yes

maximal, complemented

C₂⁴×C₂₅

C₄

yes

characteristic

C₂²×C₄×C₂₅

—

C₂²×C₄×C₂₅

C₄

yes

characteristic

C₂²×C₄×C₂₅

C₄

yes

—

C₂²×C₄×C₂₅

C₂²

yes

—

C₂²×C₄×C₂₅

C₂²

yes

characteristic

#87716

—

C₂×#6014

C₂²

yes

complemented

#65496

—

maximal

C₂³×C₂₅

—

C₂³×C₂₅

—

C₂³×C₂₅

C₂×C₄

yes

characteristic

C₂³×C₂₅

C₂×C₄

yes

—

C₂×C₄×C₂₅

—

C₂×C₄×C₂₅

—

C₂×C₄×C₂₅

—

C₂×C₄×C₂₅

C₂×C₄

yes

—

C₂×C₄×C₂₅

C₂×C₄

yes

characteristic

C₂×C₄×C₂₅

D₄

yes

—

C₂×C₄×C₂₅

C₂³

yes

characteristic

#6014

—

C₂³×C₄×C₅

D₅

yes

characteristic

C₂×#681

—

C₂²⋊#168

—

C₂²×C₂₅

—

C₂²×C₂₅

—

C₂²×C₂₅

—

C₂²×C₂₅

—

C₂²×C₂₅

C₂²⋊C₄

yes

—

C₂²×C₂₅

M₁₆

yes

complemented

C₂²×C₂₅

C₂²×C₄

yes

characteristic

C₄×C₂₅

—

C₄×C₂₅

C₂²⋊C₄

yes

—

C₄×C₂₅

C₂×D₄

yes

—

C₂⁴×C₅

Dic₅

yes

characteristic

C₂²×C₄×C₅

—

C₂²×C₄×C₅

Dic₅

yes

characteristic

C₂²×C₄×C₅

Dic₅

yes

—

C₂²×C₄×C₅

C₂×D₅

yes

—

C₂²×C₄×C₅

C₂×D₅

yes

—

—

—

C₂×C₂₅

—

C₂×C₂₅

—

C₂×C₂₅

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₂₅

C₂×M₁₆

yes

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

C₂×Dic₅

yes

—

C₂³×C₅

C₂×Dic₅

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×Dic₅

yes

—

C₂×C₄×C₅

C₂×Dic₅

yes

characteristic

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

characteristic

#168

—

C₂³×C₄

D₂₅

yes

characteristic

C₂×M₁₆

—

C₂²⋊C₈

—

C₂₅

#407

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#681

yes

—

C₂²×C₅

C₂²⋊Dic₅

yes

—

C₂²×C₅

C₂²×Dic₅

yes

characteristic

C₄×C₅

—

C₄×C₅

C₂²⋊Dic₅

yes

—

C₄×C₅

C₂×C₂²⋊D₅

yes

—

C₂⁴

Dic₂₅

yes

characteristic

C₂²×C₄

—

C₂²×C₄

Dic₂₅

yes

characteristic

C₂²×C₄

Dic₂₅

yes

—

C₂²×C₄

C₂×D₂₅

yes

—

C₂²×C₄

C₂×D₂₅

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

C₂×#681

yes

—

C₂×C₅

C₂×C₂²⋊Dic₅

yes

characteristic

C₂³

—

C₂³

—

C₂³

C₂×Dic₂₅

yes

—

C₂³

C₂×Dic₂₅

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×Dic₂₅

yes

—

C₂×C₄

C₂×Dic₂₅

yes

characteristic

C₂×C₄

C₂²⋊D₂₅

yes

—

C₂×C₄

C₂²×D₂₅

yes

characteristic

100

C₈

—

101

C₅

#65496

yes

characteristic

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

#87716

yes

complemented

107

C₂²

C₂²⋊Dic₂₅

yes

—

108

C₂²

C₂²×Dic₂₅

yes

characteristic

109

C₄

—

110

C₄

C₂²⋊Dic₂₅

yes

—

111

C₄

C₂×C₂²⋊D₂₅

yes

—

112

C₂

—

113

C₂

—

114

C₂

C₂×#87716

yes

—

115

C₂

C₂×C₂²⋊Dic₂₅

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₅
commutator	C₂²×C₂₅	C₂²×C₄
Frattini	C₂×C₄×C₅	C₂²×D₅
Fitting	C₂³×C₄×C₂₅	C₂
socle	C₂²×C₅	C₂²×Dic₅
prime radical	C₂⁴×C₅	Dic₅
cosocle	C₂×C₄×C₂₅	C₂³
hypercenter	C₂³×C₄	D₂₅
hypocenter	C₂₅	#407
LCS	G ▷ C₂²×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄

subgroup

G/H

center

C₂×C₄

C₂²×D₂₅

commutator

C₂²×C₂₅

C₂²×C₄

Frattini

C₂×C₄×C₅

C₂²×D₅

Fitting

C₂³×C₄×C₂₅

C₂

socle

C₂²×C₅

C₂²×Dic₅

prime radical

C₂⁴×C₅

Dic₅

cosocle

C₂×C₄×C₂₅

C₂³

hypercenter

C₂³×C₄

D₂₅

C₂₅

G ▷ C₂²×C₂₅ ▷ C₂₅

C₁ ◁ C₂×C₄ ◁ C₂³×C₄

N	H ≅ G/N	type
2. C₂×#87716	C₂	semidirect
3. C₂²⋊#6014	C₂	semidirect
11. C₂×#6014	C₂²	semidirect
33. C₂²×C₂₅	M₁₆	semidirect
66. C₂₅	#407	semidirect
106. C₂²	#87716	semidirect

H ≅ G/N

type

2. C₂×#87716

C₂

semidirect

3. C₂²⋊#6014

C₂

semidirect

11. C₂×#6014

C₂²

semidirect

33. C₂²×C₂₅

M₁₆

semidirect

66. C₂₅

#407

semidirect

106. C₂²

#87716

semidirect

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

subgroup	index
C₂³×C₄×C₂₅	2
C₂×#87716	2
C₂²⋊#6014	2
#65496	5

subgroup

index

C₂³×C₄×C₂₅

C₂×#87716

C₂²⋊#6014

#65496

Subgroups — #13098482

Conjugacy classes of subgroups

Complements

Maximal subgroups