Group #13125296 — subgroups

Subgroups — #13125296

Order 1664.

552 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 157 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	C₁₃×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×#88205	C₂	1	4	yes	maximal, complemented
3	C₂×#88123	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718753	C₂	1	2	yes	maximal, complemented
5	#11718737	C₂	1	2	yes	maximal, complemented
6	#11718721	C₂	1	4	yes	maximal, complemented
7	C₄×#6099	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#41	C₂²	1	2	yes	—
9	C₂²×Dic₂₆	C₂²	1	2	yes	—
10	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₁₃×#127	C₂²	1	1	yes	characteristic
12	C₁₃×#124	C₂²	1	2	yes	—
13	C₁₃×#121	—	2	2	no	—
14	C₄×D₄×C₁₃	—	2	2	no	—
15	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
16	#88205	—	2	16	no	—
17	C₂×#6107	C₂²	1	8	yes	—
18	C₂×#6105	C₂²	1	8	yes	—
19	C₂×C₄⋊Dic₁₃	—	2	2	no	—
20	C₂×#6099	C₂²	1	2	yes	—
21	#88123	—	2	2	no	—
22	#88123	C₂²	1	2	yes	—
23	#88123	C₂²	1	2	yes	—
24	#88043	C₂²	1	2	yes	—
25	C₄×#1055	C₂²	1	2	yes	—
26	C₁₃×#41	—	2	8	no	—
27	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
28	C₂×D₄×C₁₃	—	2	2	no	—
29	C₂×D₄×C₁₃	—	2	2	no	—
30	C₂×D₄×C₁₃	C₂³	1	2	yes	—
31	C₂²×C₄×C₁₃	—	2	2	no	—
32	C₂²×C₄×C₁₃	D₄	1	2	yes	—
33	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₂²×Dic₁₃	—	2	2	no	—
35	C₂×Dic₂₆	—	2	8	no	—
36	C₂×Dic₂₆	—	2	2	no	—
37	C₂×Dic₂₆	C₂³	1	4	yes	—
38	C₁₃×C₄⋊C₄	—	2	2	no	—
39	C₁₃×C₂²⋊C₄	—	2	4	no	—
40	C₄²×C₁₃	D₄	1	2	yes	—
41	C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	#6107	—	2	16	no	—
44	#6105	—	2	16	no	—
45	C₄⋊Dic₁₃	—	2	4	no	—
46	C₄⋊Dic₁₃	—	2	4	no	—
47	#6099	D₄	1	8	yes	complemented
48	C₂×#1055	C₂³	1	4	yes	—
49	#3152	—	13	1	no	maximal, Sylow
50	C₂³×C₁₃	—	2	2	no	—
51	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
52	Q₈×C₁₃	—	2	4	no	—
53	D₄×C₁₃	—	2	4	no	—
54	D₄×C₁₃	—	4	4	no	—
55	D₄×C₁₃	—	4	4	no	—
56	C₂×C₄×C₁₃	—	2	4	no	—
57	C₂×C₄×C₁₃	—	2	2	no	—
58	C₂×C₄×C₁₃	—	4	4	no	—
59	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
62	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
64	C₂×Dic₁₃	—	2	4	no	—
65	C₂×Dic₁₃	—	4	4	no	—
66	Dic₂₆	—	2	8	no	—
67	Dic₂₆	—	4	4	no	—
68	#1055	C₂×D₄	1	8	yes	—
69	C₂×#137	—	13	4	no	—
70	#532	D₁₃	1	1	yes	characteristic
71	C₂×C₄⋊Q₈	—	13	1	no	—
72	#497	—	13	4	no	—
73	#494	—	13	2	no	—
74	#492	—	13	2	no	—
75	C₄×M₁₆	—	13	1	no	—
76	C₂²×C₁₃	—	2	2	no	—
77	C₂²×C₁₃	—	4	2	no	—
78	C₂²×C₁₃	—	4	4	no	—
79	C₂²×C₁₃	#127	1	2	yes	—
80	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
81	C₄×C₁₃	—	2	2	no	—
82	C₄×C₁₃	—	4	2	no	—
83	C₄×C₁₃	#127	1	2	yes	—
84	C₄×C₁₃	#137	1	4	yes	complemented
85	C₄×C₁₃	C₂²×D₄	1	2	yes	—
86	Dic₁₃	—	4	4	no	—
87	C₂×#41	C₂×D₁₃	1	2	yes	—
88	C₂²×Q₈	—	13	2	no	—
89	#137	—	26	16	no	—
90	C₂×Q₁₆	—	13	8	no	—
91	C₂×SD₁₆	—	13	8	no	—
92	C₂×M₁₆	—	13	2	no	—
93	C₄⋊Q₈	—	13	2	no	—
94	C₄⋊Q₈	—	13	2	no	—
95	C₄⋊Q₈	—	26	2	no	—
96	C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
97	#127	C₂×D₁₃	1	1	yes	characteristic
98	#124	C₂×D₁₃	1	2	yes	—
99	#121	—	2	2	no	—
100	C₄×D₄	—	2	2	no	—
101	C₂×C₄⋊C₄	—	26	2	no	—
102	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
103	#97	—	13	2	no	—
104	C₄×C₈	—	13	2	no	—
105	C₂×C₁₃	—	2	2	no	—
106	C₂×C₁₃	—	8	2	no	—
107	C₂×C₁₃	C₂×#127	1	1	yes	characteristic
108	C₂×C₁₃	C₂×#137	1	2	yes	—
109	#41	—	2	8	no	—
110	C₂×Q₈	—	13	4	no	—
111	C₂×Q₈	—	26	2	no	—
112	C₂×Q₈	—	26	8	no	—
113	C₂×Q₈	C₂²×D₁₃	1	2	yes	—
114	C₂×D₄	—	2	2	no	—
115	C₂×D₄	—	2	2	no	—
116	C₂×D₄	C₂²×D₁₃	1	2	yes	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	26	2	no	—
119	C₂²×C₄	C₂²⋊D₁₃	1	2	yes	—
120	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
121	Q₁₆	—	26	16	no	—
122	SD₁₆	—	26	16	no	—
123	M₁₆	—	13	8	no	—
124	C₂×C₈	—	13	4	no	—
125	C₄⋊C₄	—	2	2	no	—
126	C₄⋊C₄	—	26	4	no	—
127	C₄⋊C₄	—	26	4	no	—
128	C₂²⋊C₄	—	2	4	no	—
129	C₄²	C₂²⋊D₁₃	1	2	yes	—
130	C₄²	C₂²×D₁₃	1	1	yes	characteristic
131	C₄²	C₂²×D₁₃	1	1	yes	characteristic
132	C₁₃	#3152	1	1	yes	characteristic, complemented, Sylow
133	C₂³	—	2	2	no	—
134	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
135	Q₈	—	2	4	no	—
136	Q₈	—	26	8	no	—
137	Q₈	—	52	4	no	—
138	D₄	—	2	4	no	—
139	D₄	—	4	4	no	—
140	D₄	—	4	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	4	4	no	—
144	C₂×C₄	—	26	4	no	—
145	C₂×C₄	—	52	4	no	—
146	C₂×C₄	D₄×D₁₃	1	4	yes	—
147	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
148	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
149	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
150	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
151	C₈	—	13	8	no	—
152	C₂²	—	2	2	no	—
153	C₂²	—	4	2	no	—
154	C₂²	—	4	4	no	—
155	C₂²	#88194	1	2	yes	—
156	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	4	2	no	—
159	C₄	—	52	4	no	—
160	C₄	#88194	1	2	yes	—
161	C₄	#88205	1	4	yes	complemented
162	C₄	C₂×D₄×D₁₃	1	2	yes	—
163	C₂	—	2	2	no	—
164	C₂	—	8	2	no	—
165	C₂	C₂×#88194	1	1	yes	characteristic
166	C₂	C₂×#88205	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#532

C₂

yes

maximal, characteristic

C₂×#88205

C₂

yes

maximal, complemented

C₂×#88123

C₂

yes

maximal, characteristic, complemented

#11718753

C₂

yes

maximal, complemented

#11718737

C₂

yes

maximal, complemented

#11718721

C₂

yes

maximal, complemented

C₄×#6099

C₂

yes

maximal, characteristic, complemented

C₂×C₁₃×#41

C₂²

yes

—

C₂²×Dic₂₆

C₂²

yes

—

C₁₃×C₄⋊Q₈

C₂²

yes

characteristic

C₁₃×#127

C₂²

yes

characteristic

C₁₃×#124

C₂²

yes

—

C₁₃×#121

—

C₄×D₄×C₁₃

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

#88205

—

C₂×#6107

C₂²

yes

—

C₂×#6105

C₂²

yes

—

C₂×C₄⋊Dic₁₃

—

C₂×#6099

C₂²

yes

—

#88123

—

#88123

C₂²

yes

—

#88123

C₂²

yes

—

#88043

C₂²

yes

—

C₄×#1055

C₂²

yes

—

C₁₃×#41

—

C₂×Q₈×C₁₃

C₂³

yes

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

C₂³

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

D₄

yes

—

C₄²×C₁₃

C₂³

yes

characteristic

C₄²×C₁₃

C₂³

yes

characteristic

#6107

—

#6105

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6099

D₄

yes

complemented

C₂×#1055

C₂³

yes

—

#3152

—

maximal, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

C₂×D₄

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂⁴

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

#1055

C₂×D₄

yes

—

C₂×#137

—

#532

D₁₃

yes

characteristic

C₂×C₄⋊Q₈

—

#497

—

#494

—

#492

—

C₄×M₁₆

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#127

yes

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#127

yes

—

C₄×C₁₃

#137

yes

complemented

C₄×C₁₃

C₂²×D₄

yes

—

Dic₁₃

—

C₂×#41

C₂×D₁₃

yes

—

C₂²×Q₈

—

#137

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₃

yes

characteristic

#127

C₂×D₁₃

yes

characteristic

#124

C₂×D₁₃

yes

—

#121

—

100

C₄×D₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄²

C₂×D₁₃

yes

characteristic

103

#97

—

104

C₄×C₈

—

105

C₂×C₁₃

—

106

C₂×C₁₃

—

107

C₂×C₁₃

C₂×#127

yes

characteristic

108

C₂×C₁₃

C₂×#137

yes

—

109

#41

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

C₂²×D₁₃

yes

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

C₂²×D₁₃

yes

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

C₂²⋊D₁₃

yes

—

120

C₂²×C₄

C₂²×D₁₃

yes

characteristic

121

Q₁₆

—

122

SD₁₆

—

123

M₁₆

—

124

C₂×C₈

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₂²⋊C₄

—

129

C₄²

C₂²⋊D₁₃

yes

—

130

C₄²

C₂²×D₁₃

yes

characteristic

131

C₄²

C₂²×D₁₃

yes

characteristic

132

C₁₃

#3152

yes

characteristic, complemented, Sylow

133

C₂³

—

134

C₂³

C₂×C₂²⋊D₁₃

yes

characteristic

135

Q₈

—

136

Q₈

—

137

Q₈

—

138

D₄

—

139

D₄

—

140

D₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

D₄×D₁₃

yes

—

147

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

148

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

149

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

150

C₂×C₄

C₂³×D₁₃

yes

characteristic

151

C₈

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

#88194

yes

—

156

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

#88194

yes

—

161

C₄

#88205

yes

complemented

162

C₄

C₂×D₄×D₁₃

yes

—

163

C₂

—

164

C₂

—

165

C₂

C₂×#88194

yes

characteristic

166

C₂

C₂×#88205

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting, prime radical	C₁₃×#532	C₂
socle	C₂²×C₁₃	C₂²×D₄
hypercenter	#532	D₁₃
hypocenter	C₁₃	#3152
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₃

commutator, cosocle

C₂×C₄×C₁₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃

Fitting, prime radical

C₁₃×#532

C₂

socle

C₂²×C₁₃

C₂²×D₄

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
2. C₂×#88205	C₂	semidirect
3. C₂×#88123	C₂	semidirect
4. #11718753	C₂	semidirect
5. #11718737	C₂	semidirect
6. #11718721	C₂	semidirect
7. C₄×#6099	C₂	semidirect
47. #6099	D₄	semidirect
84. C₄×C₁₃	#137	semidirect
132. C₁₃	#3152	semidirect
161. C₄	#88205	semidirect

H ≅ G/N

type

2. C₂×#88205

C₂

semidirect

3. C₂×#88123

C₂

semidirect

4. #11718753

C₂

semidirect

5. #11718737

C₂

semidirect

6. #11718721

C₂

semidirect

7. C₄×#6099

C₂

semidirect

47. #6099

D₄

semidirect

84. C₄×C₁₃

#137

semidirect

132. C₁₃

#3152

semidirect

161. C₄

#88205

semidirect

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

subgroup	index
C₁₃×#532	2
C₂×#88205	2
C₂×#88123	2
#11718753	2
#11718737	2
#11718721	2
C₄×#6099	2
#3152	13

subgroup

index

C₁₃×#532

Subgroups — #13125296

Conjugacy classes of subgroups

Complements

Maximal subgroups