Group #13098219 — subgroups

Subgroups — #13098219

Order 1600.

327 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 327 conjugacy classes of subgroups — too many to draw. Only its 109 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×Dic₂₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊C₄×C₂₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊Dic₂₅	C₂	1	1	yes	maximal, characteristic
4	#11716379	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₂₅	C₂²	1	1	yes	characteristic
6	C₂²×C₄×C₂₅	C₂²	1	2	yes	—
7	C₂²×Dic₂₅	—	2	2	no	—
8	C₂²×Dic₂₅	—	2	2	no	—
9	C₂²×Dic₂₅	C₄	1	4	yes	complemented
10	C₂²×Dic₂₅	C₂²	1	2	yes	—
11	C₂²×Dic₂₅	C₂²	1	2	yes	—
12	C₂²⋊C₄×C₂₅	—	2	2	no	—
13	C₂²⋊Dic₂₅	—	2	2	no	—
14	#65232	—	5	1	no	maximal
15	C₂³×C₂₅	—	2	4	no	—
16	C₂³×C₂₅	C₂×C₄	1	2	yes	—
17	C₂³×C₂₅	D₄	1	2	yes	—
18	C₂³×C₂₅	D₄	1	2	yes	—
19	C₂³×C₂₅	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₂₅	—	2	2	no	—
21	C₂×C₄×C₂₅	—	2	4	no	—
22	C₂×Dic₂₅	—	2	8	no	—
23	C₂×Dic₂₅	—	2	2	no	—
24	C₂×Dic₂₅	—	2	2	no	—
25	C₂×Dic₂₅	—	2	2	no	—
26	C₂×Dic₂₅	—	2	4	no	—
27	C₂×Dic₂₅	C₂×C₄	1	4	yes	—
28	C₂³×Dic₅	—	5	1	no	—
29	C₂×C₅×C₂²⋊C₄	D₅	1	1	yes	characteristic
30	C₂×C₂²⋊Dic₅	—	5	1	no	—
31	#3963	—	5	4	no	—
32	C₂²×C₂₅	—	2	4	no	—
33	C₂²×C₂₅	—	2	4	no	—
34	C₂²×C₂₅	—	2	4	no	—
35	C₂²×C₂₅	C₂²⋊C₄	1	4	yes	—
36	C₂²×C₂₅	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₂₅	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₂₅	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₂₅	#41	1	4	yes	—
40	C₄×C₂₅	—	4	2	no	—
41	Dic₂₅	—	2	4	no	—
42	Dic₂₅	—	4	2	no	—
43	C₂⁴×C₅	C₂×D₅	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₂×D₅	1	2	yes	—
45	C₂²×Dic₅	—	5	4	no	—
46	C₂²×Dic₅	—	5	2	no	—
47	C₂²×Dic₅	—	5	2	no	—
48	C₂²×Dic₅	—	10	2	no	—
49	C₂²×Dic₅	—	10	2	no	—
50	C₅×C₂²⋊C₄	—	2	2	no	—
51	C₂²⋊Dic₅	—	10	2	no	—
52	#381	—	25	1	no	Sylow
53	C₂×C₂₅	—	2	4	no	—
54	C₂×C₂₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₂₅	#117	1	2	yes	—
56	C₂×C₂₅	#123	1	2	yes	—
57	C₂×C₂₅	#123	1	2	yes	—
58	C₂³×C₅	—	2	4	no	—
59	C₂³×C₅	C₄×D₅	1	2	yes	—
60	C₂³×C₅	D₂₀	1	2	yes	—
61	C₂³×C₅	C₂²⋊D₅	1	2	yes	—
62	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
63	C₂×C₄×C₅	—	2	4	no	—
64	C₂×C₄×C₅	—	2	2	no	—
65	C₂×Dic₅	—	5	4	no	—
66	C₂×Dic₅	—	10	8	no	—
67	C₂×Dic₅	—	10	2	no	—
68	C₂×Dic₅	—	10	2	no	—
69	C₂×Dic₅	—	10	2	no	—
70	C₂×Dic₅	—	10	4	no	—
71	C₂³×C₄	—	25	1	no	—
72	C₂×C₂²⋊C₄	—	25	1	no	—
73	C₂×C₂²⋊C₄	D₂₅	1	1	yes	characteristic
74	#95	—	25	4	no	—
75	C₂₅	#381	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₅	—	2	4	no	—
77	C₂²×C₅	—	2	4	no	—
78	C₂²×C₅	—	2	4	no	—
79	C₂²×C₅	#685	1	4	yes	—
80	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
81	C₂²×C₅	C₂×D₂₀	1	1	yes	characteristic
82	C₂²×C₅	#711	1	4	yes	—
83	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
84	C₄×C₅	—	4	2	no	—
85	Dic₅	—	10	4	no	—
86	Dic₅	—	20	2	no	—
87	C₂⁴	C₂×D₂₅	1	1	yes	characteristic
88	C₂²×C₄	—	25	4	no	—
89	C₂²×C₄	—	25	2	no	—
90	C₂²×C₄	—	25	2	no	—
91	C₂²×C₄	—	50	2	no	—
92	C₂²×C₄	—	50	2	no	—
93	C₂²×C₄	C₂×D₂₅	1	2	yes	—
94	C₂²⋊C₄	—	2	2	no	—
95	C₂²⋊C₄	—	50	2	no	—
96	C₂×C₅	—	2	4	no	—
97	C₂×C₅	#4023	1	2	yes	—
98	C₂×C₅	#4032	1	2	yes	—
99	C₂×C₅	C₂×#685	1	1	yes	characteristic
100	C₂×C₅	#4081	1	2	yes	—
101	C₂³	—	2	4	no	—
102	C₂³	C₄×D₂₅	1	2	yes	—
103	C₂³	D₁₀₀	1	2	yes	—
104	C₂³	C₂²⋊D₂₅	1	2	yes	—
105	C₂³	C₂²×D₂₅	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	25	4	no	—
109	C₂×C₄	—	50	8	no	—
110	C₂×C₄	—	50	2	no	—
111	C₂×C₄	—	50	2	no	—
112	C₂×C₄	—	50	2	no	—
113	C₂×C₄	—	50	4	no	—
114	C₅	#65232	1	1	yes	characteristic
115	C₂²	—	2	4	no	—
116	C₂²	—	2	4	no	—
117	C₂²	—	2	4	no	—
118	C₂²	#87720	1	4	yes	—
119	C₂²	C₂×C₄×D₂₅	1	1	yes	characteristic
120	C₂²	C₂×D₁₀₀	1	1	yes	characteristic
121	C₂²	#87746	1	4	yes	—
122	C₂²	C₂×C₂²⋊D₂₅	1	1	yes	characteristic
123	C₄	—	4	2	no	—
124	C₄	—	50	4	no	—
125	C₄	—	100	2	no	—
126	C₂	—	2	4	no	—
127	C₂	#11716439	1	2	yes	—
128	C₂	#11716448	1	2	yes	—
129	C₂	C₂×#87720	1	1	yes	characteristic
130	C₂	#11716497	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Dic₂₅

C₂

yes

maximal, characteristic

C₂×C₂²⋊C₄×C₂₅

C₂

yes

maximal, characteristic

C₂×C₂²⋊Dic₂₅

C₂

yes

maximal, characteristic

#11716379

C₂

yes

maximal, complemented

C₂⁴×C₂₅

C₂²

yes

characteristic

C₂²×C₄×C₂₅

C₂²

yes

—

C₂²×Dic₂₅

—

C₂²×Dic₂₅

—

C₂²×Dic₂₅

C₄

yes

complemented

C₂²×Dic₂₅

C₂²

yes

—

C₂²×Dic₂₅

C₂²

yes

—

C₂²⋊C₄×C₂₅

—

C₂²⋊Dic₂₅

—

#65232

—

maximal

C₂³×C₂₅

—

C₂³×C₂₅

C₂×C₄

yes

—

C₂³×C₂₅

D₄

yes

—

C₂³×C₂₅

D₄

yes

—

C₂³×C₂₅

C₂³

yes

characteristic

C₂×C₄×C₂₅

—

C₂×C₄×C₂₅

—

C₂×Dic₂₅

—

C₂×Dic₂₅

—

C₂×Dic₂₅

—

C₂×Dic₂₅

—

C₂×Dic₂₅

—

C₂×Dic₂₅

C₂×C₄

yes

—

C₂³×Dic₅

—

C₂×C₅×C₂²⋊C₄

D₅

yes

characteristic

C₂×C₂²⋊Dic₅

—

#3963

—

C₂²×C₂₅

—

C₂²×C₂₅

—

C₂²×C₂₅

—

C₂²×C₂₅

C₂²⋊C₄

yes

—

C₂²×C₂₅

C₂²×C₄

yes

characteristic

C₂²×C₂₅

C₂×D₄

yes

characteristic

C₂²×C₂₅

C₂×D₄

yes

characteristic

C₂²×C₂₅

#41

yes

—

C₄×C₂₅

—

Dic₂₅

—

Dic₂₅

—

C₂⁴×C₅

C₂×D₅

yes

characteristic

C₂²×C₄×C₅

C₂×D₅

yes

—

C₂²×Dic₅

—

C₂²×Dic₅

—

C₂²×Dic₅

—

C₂²×Dic₅

—

C₂²×Dic₅

—

C₅×C₂²⋊C₄

—

C₂²⋊Dic₅

—

#381

—

Sylow

C₂×C₂₅

—

C₂×C₂₅

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₂₅

#117

yes

—

C₂×C₂₅

#123

yes

—

C₂×C₂₅

#123

yes

—

C₂³×C₅

—

C₂³×C₅

C₄×D₅

yes

—

C₂³×C₅

D₂₀

yes

—

C₂³×C₅

C₂²⋊D₅

yes

—

C₂³×C₅

C₂²×D₅

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂³×C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

D₂₅

yes

characteristic

#95

—

C₂₅

#381

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#685

yes

—

C₂²×C₅

C₂×C₄×D₅

yes

characteristic

C₂²×C₅

C₂×D₂₀

yes

characteristic

C₂²×C₅

#711

yes

—

C₂²×C₅

C₂×C₂²⋊D₅

yes

characteristic

C₄×C₅

—

Dic₅

—

Dic₅

—

C₂⁴

C₂×D₂₅

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₂₅

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂×C₅

—

C₂×C₅

#4023

yes

—

C₂×C₅

#4032

yes

—

C₂×C₅

C₂×#685

yes

characteristic

100

C₂×C₅

#4081

yes

—

101

C₂³

—

102

C₂³

C₄×D₂₅

yes

—

103

C₂³

D₁₀₀

yes

—

104

C₂³

C₂²⋊D₂₅

yes

—

105

C₂³

C₂²×D₂₅

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₅

#65232

yes

characteristic

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

#87720

yes

—

119

C₂²

C₂×C₄×D₂₅

yes

characteristic

120

C₂²

C₂×D₁₀₀

yes

characteristic

121

C₂²

#87746

yes

—

122

C₂²

C₂×C₂²⋊D₂₅

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

100

—

126

C₂

—

127

C₂

#11716439

yes

—

128

C₂

#11716448

yes

—

129

C₂

C₂×#87720

yes

characteristic

130

C₂

#11716497

yes

—

	subgroup	G/H
center	C₂³	C₂²×D₂₅
commutator	C₂²×C₂₅	C₂²×C₄
Frattini, socle	C₂³×C₅	C₂²×D₅
Fitting	C₂×C₂²⋊C₄×C₂₅	C₂
prime radical	C₂⁴×C₅	C₂×D₅
cosocle	C₂³×C₂₅	C₂³
hypercenter	C₂×C₂²⋊C₄	D₂₅
hypocenter	C₂₅	#381
LCS	G ▷ C₂²×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

subgroup

G/H

center

C₂³

C₂²×D₂₅

commutator

C₂²×C₂₅

C₂²×C₄

Frattini, socle

C₂³×C₅

C₂²×D₅

Fitting

C₂×C₂²⋊C₄×C₂₅

C₂

prime radical

C₂⁴×C₅

C₂×D₅

cosocle

C₂³×C₂₅

C₂³

hypercenter

C₂×C₂²⋊C₄

D₂₅

C₂₅

G ▷ C₂²×C₂₅ ▷ C₂₅

C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

N	H ≅ G/N	type
4. #11716379	C₂	semidirect
9. C₂²×Dic₂₅	C₄	semidirect
75. C₂₅	#381	semidirect

H ≅ G/N

type

4. #11716379

C₂

semidirect

9. C₂²×Dic₂₅

C₄

semidirect

75. C₂₅

#381

semidirect

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

subgroup	index
C₂³×Dic₂₅	2
C₂×C₂²⋊C₄×C₂₅	2
C₂×C₂²⋊Dic₂₅	2
#11716379	2
#65232	5

subgroup

index

C₂³×Dic₂₅

C₂×C₂²⋊C₄×C₂₅

C₂×C₂²⋊Dic₂₅

#11716379

#65232

Subgroups — #13098219

Conjugacy classes of subgroups

Complements

Maximal subgroups