Group #13115384 — subgroups

Subgroups — #13115384

Order 1664.

340 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 93 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×#535	C₂	1	1	yes	maximal, characteristic
2	#11719524	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718898	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#88073	C₂	1	2	yes	maximal, complemented
5	#11718125	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
7	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
8	C₁₃×#127	—	2	1	no	—
9	C₁₃×#124	—	2	1	no	—
10	C₁₃×#121	—	2	2	no	—
11	C₁₃×#120	C₄	1	4	yes	—
12	C₁₃×#117	C₂²	1	1	yes	characteristic
13	#88202	—	2	4	no	—
14	C₂×#6099	—	2	2	no	—
15	C₂×#6099	C₂²	1	2	yes	—
16	C₂×#6099	C₂²	1	2	yes	—
17	#88076	—	2	1	no	—
18	#88073	—	2	1	no	—
19	#88073	—	2	4	no	—
20	C₂⁴×C₁₃	C₂×C₄	1	2	yes	—
21	C₁₃×#41	—	2	2	no	—
22	C₁₃×#41	—	2	2	no	—
23	C₂×Q₈×C₁₃	D₄	1	1	yes	characteristic
24	C₂×D₄×C₁₃	—	2	4	no	—
25	C₂×D₄×C₁₃	—	2	2	no	—
26	C₂×D₄×C₁₃	—	2	1	no	—
27	C₂×D₄×C₁₃	—	2	1	no	—
28	C₂×D₄×C₁₃	—	2	2	no	—
29	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
30	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	D₄	1	2	yes	—
32	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
33	C₁₃×C₄⋊C₄	—	2	1	no	—
34	C₁₃×C₂²⋊C₄	—	2	4	no	—
35	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
36	C₄²×C₁₃	—	2	1	no	—
37	#6099	—	2	4	no	—
38	#6099	—	2	2	no	—
39	#6099	—	2	2	no	—
40	#6099	—	2	2	no	—
41	#6099	D₄	1	4	yes	complemented
42	C₂×#1055	—	2	4	no	—
43	C₂×#1055	—	2	2	no	—
44	C₂×#1055	—	2	2	no	—
45	#1914	—	13	1	no	maximal, Sylow
46	C₂³×C₁₃	—	2	2	no	—
47	C₂³×C₁₃	—	2	4	no	—
48	C₂³×C₁₃	—	8	2	no	—
49	C₂³×C₁₃	C₂²⋊C₄	1	2	yes	—
50	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	Q₈×C₁₃	—	2	2	no	—
52	D₄×C₁₃	—	2	4	no	—
53	D₄×C₁₃	—	2	2	no	—
54	D₄×C₁₃	—	4	1	no	—
55	D₄×C₁₃	—	4	2	no	—
56	D₄×C₁₃	—	4	1	no	—
57	D₄×C₁₃	—	8	4	no	—
58	C₂×C₄×C₁₃	—	2	2	no	—
59	C₂×C₄×C₁₃	—	2	2	no	—
60	C₂×C₄×C₁₃	—	2	2	no	—
61	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
62	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₁₃	#41	1	2	yes	—
66	#1055	—	2	4	no	—
67	#1055	—	4	2	no	—
68	#568	—	13	1	no	—
69	#535	D₁₃	1	1	yes	characteristic
70	#413	—	13	1	no	—
71	C₂×#100	—	13	2	no	—
72	#344	—	13	2	no	—
73	C₂²×C₁₃	—	2	2	no	—
74	C₂²×C₁₃	—	4	1	no	—
75	C₂²×C₁₃	—	4	2	no	—
76	C₂²×C₁₃	—	4	4	no	—
77	C₂²×C₁₃	—	4	2	no	—
78	C₂²×C₁₃	—	8	4	no	—
79	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₁₃	C₄×D₄	1	2	yes	—
81	C₄×C₁₃	—	2	2	no	—
82	C₄×C₁₃	—	4	2	no	—
83	C₄×C₁₃	#120	1	1	yes	characteristic
84	C₄×C₁₃	#121	1	2	yes	—
85	C₄×C₁₃	#123	1	1	yes	characteristic
86	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
87	C₂²×D₄	C₂×D₁₃	1	1	yes	characteristic
88	#131	—	26	4	no	—
89	C₂×M₁₆	—	13	2	no	—
90	C₂×M₁₆	—	13	2	no	—
91	C₂×M₁₆	—	26	2	no	—
92	#127	—	2	1	no	—
93	#124	—	2	1	no	—
94	#121	—	2	2	no	—
95	#120	Dic₁₃	1	4	yes	—
96	#117	C₂×D₁₃	1	1	yes	characteristic
97	#101	—	26	1	no	—
98	#100	—	26	4	no	—
99	#100	—	26	1	no	—
100	C₂×C₁₃	—	2	1	no	—
101	C₂×C₁₃	—	2	2	no	—
102	C₂×C₁₃	—	4	2	no	—
103	C₂×C₁₃	—	8	2	no	—
104	C₂×C₁₃	#386	1	1	yes	characteristic
105	C₂⁴	C₂×Dic₁₃	1	2	yes	—
106	#41	—	2	2	no	—
107	#41	—	2	2	no	—
108	C₂×Q₈	C₂²⋊D₁₃	1	1	yes	characteristic
109	C₂×D₄	—	2	4	no	—
110	C₂×D₄	—	2	2	no	—
111	C₂×D₄	—	2	2	no	—
112	C₂×D₄	—	2	1	no	—
113	C₂×D₄	—	2	1	no	—
114	C₂×D₄	C₂×Dic₁₃	1	2	yes	—
115	C₂×D₄	C₂²⋊D₁₃	1	1	yes	characteristic
116	C₂²×C₄	C₂²⋊D₁₃	1	2	yes	—
117	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
118	M₁₆	—	13	4	no	—
119	M₁₆	—	26	4	no	—
120	M₁₆	—	26	2	no	—
121	M₁₆	—	26	2	no	—
122	M₁₆	—	26	2	no	—
123	C₂×C₈	—	26	4	no	—
124	C₂×C₈	—	26	2	no	—
125	C₂×C₈	—	26	2	no	—
126	C₄⋊C₄	—	2	1	no	—
127	C₂²⋊C₄	—	2	4	no	—
128	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
129	C₄²	—	2	1	no	—
130	C₁₃	#1914	1	1	yes	characteristic, complemented, Sylow
131	C₂³	—	2	2	no	—
132	C₂³	—	2	4	no	—
133	C₂³	—	8	2	no	—
134	C₂³	C₂²⋊Dic₁₃	1	2	yes	—
135	C₂³	C₂²×Dic₁₃	1	1	yes	characteristic
136	Q₈	—	2	2	no	—
137	D₄	—	2	4	no	—
138	D₄	—	2	2	no	—
139	D₄	—	4	1	no	—
140	D₄	—	4	1	no	—
141	D₄	—	4	2	no	—
142	D₄	—	8	4	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	C₂²⋊Dic₁₃	1	2	yes	—
147	C₂×C₄	D₄×D₁₃	1	2	yes	—
148	C₂×C₄	#6129	1	2	yes	—
149	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
150	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
151	C₈	—	26	4	no	—
152	C₈	—	52	2	no	—
153	C₂²	—	2	2	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	4	2	no	—
156	C₂²	—	4	4	no	—
157	C₂²	—	4	2	no	—
158	C₂²	—	8	4	no	—
159	C₂²	D₄×Dic₁₃	1	2	yes	—
160	C₂²	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
161	C₄	—	2	2	no	—
162	C₄	—	4	2	no	—
163	C₄	#88189	1	1	yes	characteristic
164	C₄	#88191	1	1	yes	characteristic
165	C₄	#88193	1	2	yes	—
166	C₂	—	2	1	no	—
167	C₂	—	2	2	no	—
168	C₂	—	4	2	no	—
169	C₂	—	8	2	no	—
170	C₂	#11718881	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#535

C₂

yes

maximal, characteristic

#11719524

C₂

yes

maximal, characteristic, complemented

#11718898

C₂

yes

maximal, characteristic, complemented

C₂×#88073

C₂

yes

maximal, complemented

#11718125

C₂

yes

maximal, complemented

C₂×C₁₃×#41

C₂²

yes

characteristic

C₂²×D₄×C₁₃

C₂²

yes

characteristic

C₁₃×#127

—

C₁₃×#124

—

C₁₃×#121

—

C₁₃×#120

C₄

yes

—

C₁₃×#117

C₂²

yes

—

—

C₂²

yes

—

C₂×#6099

C₂²

yes

—

#88076

—

#88073

—

#88073

—

C₂⁴×C₁₃

C₂×C₄

yes

—

C₁₃×#41

—

C₁₃×#41

—

C₂×Q₈×C₁₃

D₄

yes

characteristic

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

C₂×C₄

yes

—

C₂×D₄×C₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₁₃×C₂²⋊C₄

C₂×C₄

yes

—

C₄²×C₁₃

—

—

—

—

—

D₄

yes

—

—

—

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂²⋊C₄

yes

—

C₂³×C₁₃

C₂²×C₄

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

#41

yes

—

—

—

—

D₁₃

yes

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃

C₄×D₄

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#120

yes

characteristic

C₄×C₁₃

#121

yes

—

C₄×C₁₃

#123

yes

characteristic

C₂×#41

C₂×D₁₃

yes

characteristic

C₂²×D₄

C₂×D₁₃

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

—

—

—

Dic₁₃

yes

—

#117

C₂×D₁₃

yes

—

—

—

100

C₂×C₁₃

—

101

C₂×C₁₃

—

102

C₂×C₁₃

—

103

C₂×C₁₃

—

104

C₂×C₁₃

#386

yes

characteristic

105

C₂⁴

C₂×Dic₁₃

yes

—

106

#41

—

107

#41

—

108

C₂×Q₈

C₂²⋊D₁₃

yes

characteristic

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

C₂×Dic₁₃

yes

—

115

C₂×D₄

C₂²⋊D₁₃

yes

characteristic

116

C₂²×C₄

C₂²⋊D₁₃

yes

—

117

C₂²×C₄

C₂²×D₁₃

yes

characteristic

118

M₁₆

—

119

M₁₆

—

120

M₁₆

—

121

M₁₆

—

122

M₁₆

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₄⋊C₄

—

127

C₂²⋊C₄

—

128

C₂²⋊C₄

C₂×Dic₁₃

yes

—

129

C₄²

—

130

C₁₃

#1914

yes

characteristic, complemented, Sylow

131

C₂³

—

132

C₂³

—

133

C₂³

—

134

C₂³

C₂²⋊Dic₁₃

yes

—

135

C₂³

C₂²×Dic₁₃

yes

characteristic

136

Q₈

—

137

D₄

—

138

D₄

—

139

D₄

—

140

D₄

—

141

D₄

—

142

D₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂²⋊Dic₁₃

yes

—

147

C₂×C₄

D₄×D₁₃

yes

—

148

C₂×C₄

#6129

yes

—

149

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

150

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

151

C₈

—

152

C₈

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

D₄×Dic₁₃

yes

—

160

C₂²

C₂×C₂²⋊Dic₁₃

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

#88189

yes

characteristic

164

C₄

#88191

yes

characteristic

165

C₄

#88193

yes

—

166

C₂

—

167

C₂

—

168

C₂

—

169

C₂

—

170

C₂

#11718881

yes

characteristic

	subgroup	G/H
center	C₂	#11718881
commutator	C₂³×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting, prime radical	C₁₃×#535	C₂
socle	C₂×C₁₃	#386
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	#535	D₁₃
hypocenter	C₁₃	#1914
LCS	G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #535

subgroup

G/H

center

C₂

#11718881

commutator

C₂³×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting, prime radical

C₁₃×#535

C₂

socle

C₂×C₁₃

#386

cosocle

C₂²×C₄×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #535

N	H ≅ G/N	type
2. #11719524	C₂	semidirect
3. #11718898	C₂	semidirect
4. C₂×#88073	C₂	semidirect
5. #11718125	C₂	semidirect
41. #6099	D₄	semidirect
130. C₁₃	#1914	semidirect

H ≅ G/N

type

2. #11719524

C₂

semidirect

3. #11718898

C₂

semidirect

4. C₂×#88073

C₂

semidirect

5. #11718125

C₂

semidirect

41. #6099

D₄

semidirect

130. C₁₃

#1914

semidirect

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

subgroup	index
C₁₃×#535	2
#11719524	2
#11718898	2
C₂×#88073	2
#11718125	2
#1914	13

subgroup

index

C₁₃×#535

Subgroups — #13115384

Conjugacy classes of subgroups

Complements

Maximal subgroups