Group #13125239 — subgroups

Subgroups — #13125239

Order 1664.

392 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 133 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	C₂×C₁₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719182	C₂	1	1	yes	maximal, characteristic
3	#11718730	C₂	1	2	yes	maximal, complemented
4	#11718729	C₂	1	4	yes	maximal, complemented
5	#11718728	C₂	1	2	yes	maximal, complemented
6	#11718627	C₂	1	4	yes	maximal, complemented
7	C₄×#6099	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#6127	C₂²	1	2	yes	—
9	C₁₃×#125	—	2	2	no	—
10	C₁₃×#125	C₂²	1	2	yes	—
11	C₁₃×#125	C₂²	1	2	yes	—
12	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
13	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
15	#88184	—	2	2	no	—
16	C₂×#6099	C₂²	1	2	yes	—
17	#88129	C₂²	1	2	yes	—
18	#88128	C₂²	1	1	yes	characteristic
19	C₄×D₅₂	—	2	2	no	—
20	#88123	C₂²	1	1	yes	characteristic
21	#88050	C₂²	1	8	yes	—
22	#88049	C₂²	1	8	yes	—
23	#88043	C₂²	1	2	yes	—
24	C₄×#1055	C₂²	1	2	yes	—
25	C₂²×C₄×C₁₃	—	2	2	no	—
26	C₂²×C₄×C₁₃	D₄	1	2	yes	—
27	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₂²⋊D₁₃	—	2	2	no	—
29	#6127	—	2	8	no	—
30	C₂×D₅₂	—	2	2	no	—
31	C₂×D₅₂	C₂³	1	2	yes	—
32	C₂×C₄×D₁₃	—	2	2	no	—
33	C₂×Dic₂₆	C₂³	1	2	yes	—
34	C₁₃×C₄⋊C₄	—	2	4	no	—
35	C₁₃×C₄⋊C₄	—	2	4	no	—
36	C₁₃×C₄⋊C₄	—	2	2	no	—
37	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
38	C₄²×C₁₃	D₄	1	2	yes	—
39	C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	C₄²×C₁₃	C₂³	1	1	yes	characteristic
41	#6103	—	2	4	no	—
42	C₄⋊Dic₁₃	—	2	2	no	—
43	#6099	—	2	4	no	—
44	C₂×#1055	C₂³	1	4	yes	—
45	#3142	—	13	1	no	maximal, Sylow
46	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×D₁₃	—	2	2	no	—
48	C₂×C₄×C₁₃	—	2	4	no	—
49	C₂×C₄×C₁₃	—	4	2	no	—
50	C₂×C₄×C₁₃	—	4	4	no	—
51	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃	#41	1	4	yes	—
55	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
56	C₂²⋊D₁₃	—	4	4	no	—
57	C₂×Dic₁₃	—	2	2	no	—
58	D₅₂	—	2	4	no	—
59	D₅₂	—	4	4	no	—
60	C₄×D₁₃	—	4	4	no	—
61	Dic₂₆	—	2	4	no	—
62	#1055	—	2	4	no	—
63	#532	—	13	1	no	—
64	C₂×#125	D₁₃	1	1	yes	characteristic, complemented
65	#491	—	13	2	no	—
66	#490	—	13	2	no	—
67	#488	—	13	4	no	—
68	#417	—	13	4	no	—
69	C₄×M₁₆	—	13	1	no	—
70	C₂²×C₁₃	—	2	2	no	—
71	C₂²×C₁₃	#124	1	2	yes	—
72	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
73	C₂×D₁₃	—	4	2	no	—
74	C₂×D₁₃	—	4	4	no	—
75	C₄×C₁₃	—	2	2	no	—
76	C₄×C₁₃	—	2	2	no	—
77	C₄×C₁₃	—	4	4	no	—
78	C₄×C₁₃	#124	1	2	yes	—
79	C₄×C₁₃	C₂×#41	1	2	yes	—
80	Dic₁₃	—	4	2	no	—
81	C₂×#41	—	13	2	no	—
82	C₂×M₁₆	—	13	2	no	—
83	C₄⋊Q₈	—	13	1	no	—
84	#127	—	13	1	no	—
85	#125	—	2	2	no	—
86	#125	C₂×D₁₃	1	2	yes	—
87	#125	C₂×D₁₃	1	2	yes	—
88	#124	—	13	2	no	—
89	#121	—	26	2	no	—
90	C₄×D₄	—	26	2	no	—
91	C₂×C₄⋊C₄	—	2	2	no	—
92	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
93	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
94	#103	—	13	8	no	—
95	#102	—	13	8	no	—
96	#97	—	13	2	no	—
97	C₄×C₈	—	13	2	no	—
98	C₂×C₁₃	—	2	2	no	—
99	C₂×C₁₃	C₂×#124	1	1	yes	characteristic
100	C₂×C₁₃	#575	1	2	yes	—
101	D₁₃	—	8	2	no	—
102	#41	—	26	8	no	—
103	C₂×Q₈	—	13	2	no	—
104	C₂×D₄	—	13	2	no	—
105	C₂×D₄	—	26	2	no	—
106	C₂×D₄	—	26	2	no	—
107	C₂²×C₄	—	2	2	no	—
108	C₂²×C₄	—	26	2	no	—
109	C₂²×C₄	C₂²⋊D₁₃	1	2	yes	—
110	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
111	M₁₆	—	26	4	no	—
112	C₂×C₈	—	13	4	no	—
113	C₄⋊C₄	—	2	4	no	—
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	26	2	no	—
117	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
118	C₂²⋊C₄	—	26	4	no	—
119	C₄²	C₂²⋊D₁₃	1	2	yes	—
120	C₄²	C₂²×D₁₃	1	1	yes	characteristic
121	C₄²	C₂²×D₁₃	1	1	yes	characteristic
122	C₁₃	#3142	1	1	yes	characteristic, complemented, Sylow
123	C₂³	—	26	2	no	—
124	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
125	Q₈	—	26	4	no	—
126	D₄	—	26	4	no	—
127	D₄	—	52	4	no	—
128	D₄	—	52	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	4	no	—
132	C₂×C₄	—	26	2	no	—
133	C₂×C₄	—	52	4	no	—
134	C₂×C₄	#6131	1	4	yes	—
135	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
136	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
137	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
138	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
139	C₈	—	26	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	52	2	no	—
142	C₂²	—	52	4	no	—
143	C₂²	#88201	1	2	yes	—
144	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	4	4	no	—
148	C₄	—	52	2	no	—
149	C₄	#88201	1	2	yes	—
150	C₄	C₂×#6131	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	104	2	no	—
153	C₂	C₂×#88201	1	1	yes	characteristic
154	C₂	#11719527	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#125

C₂

yes

maximal, characteristic, complemented

#11719182

C₂

yes

maximal, characteristic

#11718730

C₂

yes

maximal, complemented

#11718729

C₂

yes

maximal, complemented

#11718728

C₂

yes

maximal, complemented

#11718627

C₂

yes

maximal, complemented

C₄×#6099

C₂

yes

maximal, characteristic, complemented

C₂×#6127

C₂²

yes

—

C₁₃×#125

—

C₁₃×#125

C₂²

yes

—

C₁₃×#125

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

#88184

—

C₂×#6099

C₂²

yes

—

#88129

C₂²

yes

—

#88128

C₂²

yes

characteristic

C₄×D₅₂

—

#88123

C₂²

yes

characteristic

#88050

C₂²

yes

—

#88049

C₂²

yes

—

#88043

C₂²

yes

—

C₄×#1055

C₂²

yes

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂×C₂²⋊D₁₃

—

#6127

—

C₂×D₅₂

—

C₂×D₅₂

C₂³

yes

—

C₂×C₄×D₁₃

—

C₂×Dic₂₆

C₂³

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

C₂³

yes

—

C₄²×C₁₃

D₄

yes

—

C₄²×C₁₃

C₂³

yes

characteristic

C₄²×C₁₃

C₂³

yes

characteristic

#6103

—

C₄⋊Dic₁₃

—

#6099

—

C₂×#1055

C₂³

yes

—

#3142

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

C₂²×D₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

#41

yes

—

C₂×C₄×C₁₃

C₂⁴

yes

characteristic

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

C₄×D₁₃

—

Dic₂₆

—

#1055

—

#532

—

C₂×#125

D₁₃

yes

characteristic, complemented

—

—

—

—

C₄×M₁₆

—

C₂²×C₁₃

—

C₂²×C₁₃

#124

yes

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#124

yes

—

C₄×C₁₃

C₂×#41

yes

—

Dic₁₃

—

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

—

#125

—

#125

C₂×D₁₃

yes

—

#125

C₂×D₁₃

yes

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

C₂×D₁₃

yes

—

—

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

C₂×#124

yes

characteristic

100

C₂×C₁₃

#575

yes

—

101

D₁₃

—

102

#41

—

103

C₂×Q₈

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂²⋊D₁₃

yes

—

110

C₂²×C₄

C₂²×D₁₃

yes

characteristic

111

M₁₆

—

112

C₂×C₈

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

C₂²×D₁₃

yes

—

118

C₂²⋊C₄

—

119

C₄²

C₂²⋊D₁₃

yes

—

120

C₄²

C₂²×D₁₃

yes

characteristic

121

C₄²

C₂²×D₁₃

yes

characteristic

122

C₁₃

#3142

yes

characteristic, complemented, Sylow

123

C₂³

—

124

C₂³

C₂×C₂²⋊D₁₃

yes

characteristic

125

Q₈

—

126

D₄

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

#6131

yes

—

135

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

136

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

137

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

138

C₂×C₄

C₂³×D₁₃

yes

characteristic

139

C₈

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

#88201

yes

—

144

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#88201

yes

—

150

C₄

C₂×#6131

yes

—

151

C₂

—

152

C₂

—

104

—

153

C₂

C₂×#88201

yes

characteristic

154

C₂

#11719527

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#125	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	#11719182	C₂
hypercenter	C₂×#125	D₁₃
hypocenter	C₁₃	#3142
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₃

commutator, cosocle

C₂×C₄×C₁₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃

Fitting

C₂×C₁₃×#125

C₂

socle

C₂²×C₁₃

C₂²×D₄

C₂

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
1. C₂×C₁₃×#125	C₂	semidirect
3. #11718730	C₂	semidirect
4. #11718729	C₂	semidirect
5. #11718728	C₂	semidirect
6. #11718627	C₂	semidirect
7. C₄×#6099	C₂	semidirect
64. C₂×#125	D₁₃	semidirect
122. C₁₃	#3142	semidirect

H ≅ G/N

type

1. C₂×C₁₃×#125

C₂

semidirect

3. #11718730

C₂

semidirect

4. #11718729

C₂

semidirect

5. #11718728

C₂

semidirect

6. #11718627

C₂

semidirect

7. C₄×#6099

C₂

semidirect

64. C₂×#125

D₁₃

semidirect

122. C₁₃

#3142

semidirect

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

subgroup	index
C₂×C₁₃×#125	2
#11719182	2
#11718730	2
#11718729	2
#11718728	2
#11718627	2
C₄×#6099	2
#3142	13

subgroup

index

C₂×C₁₃×#125

Subgroups — #13125239

Conjugacy classes of subgroups

Complements

Maximal subgroups