Group #13130987 — subgroups

Subgroups — #13130987

Order 1664.

300 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 70 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×#88173	C₂	1	1	yes	maximal, characteristic
2	#11719159	C₂	1	2	yes	maximal, complemented
3	C₂×#88118	C₂	1	2	yes	maximal, complemented
4	C₂×#88104	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈×#261	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#6131	C₂²	1	2	yes	—
7	C₂×C₄×#261	C₂²	1	1	yes	characteristic
8	C₂×#6118	C₂²	1	2	yes	—
9	C₂×D₁₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×Q₁₆	C₄	1	1	yes	characteristic, complemented
11	C₂×#6106	C₄	1	2	yes	complemented
12	#88173	—	2	2	no	—
13	#88173	—	2	2	no	—
14	C₂×D₁₀₄	C₄	1	1	yes	characteristic, complemented
15	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
16	#88118	—	2	4	no	—
17	#88111	—	2	4	no	—
18	#88104	—	2	2	no	—
19	C₈×#261	—	2	2	no	—
20	C₂²×#261	—	2	1	no	—
21	C₂×Q₈×C₁₃	C₂×C₄	1	2	yes	—
22	#6131	—	2	2	no	—
23	#6131	—	2	2	no	—
24	#6131	—	2	2	no	—
25	#6131	—	2	2	no	—
26	C₂×D₅₂	—	2	2	no	—
27	C₂×D₅₂	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₁₃	—	2	2	no	—
29	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
30	C₂×#1057	—	2	2	no	—
31	C₂×#1057	D₄	1	2	yes	—
32	C₄×#261	—	2	1	no	—
33	C₄×#261	D₄	1	2	yes	—
34	#6118	—	2	2	no	—
35	#6118	—	2	2	no	—
36	D₁₃⋊C₈	—	2	1	no	—
37	D₁₃⋊C₈	—	2	1	no	—
38	C₁₃×Q₁₆	—	2	2	no	—
39	C₂×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	#6106	—	2	4	no	—
41	C₂×#1055	C₂×C₄	1	1	yes	characteristic
42	D₁₀₄	—	2	2	no	—
43	C₈×D₁₃	D₄	1	2	yes	—
44	C₈×D₁₃	D₄	1	2	yes	—
45	#1974	—	13	1	no	maximal, Sylow
46	C₂²×D₁₃	—	2	2	no	—
47	C₂²×D₁₃	#41	1	1	yes	characteristic
48	C₂×#261	—	2	4	no	—
49	C₂×#261	—	2	2	no	—
50	Q₈×C₁₃	—	2	2	no	—
51	Q₈×C₁₃	—	2	2	no	—
52	C₂×C₄×C₁₃	—	2	2	no	—
53	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
55	D₅₂	—	2	2	no	—
56	D₅₂	—	2	2	no	—
57	D₅₂	—	4	2	no	—
58	D₅₂	—	4	2	no	—
59	C₄×D₁₃	—	4	2	no	—
60	C₄×D₁₃	—	4	2	no	—
61	C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
62	C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
63	C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
64	C₄×D₁₃	#41	1	1	yes	characteristic
65	#1057	—	2	2	no	—
66	#1057	—	4	2	no	—
67	C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
68	#1055	C₂²⋊C₄	1	2	yes	—
69	C₂×#135	—	13	1	no	—
70	C₂×#108	—	13	1	no	—
71	C₂×#104	—	13	2	no	—
72	#408	—	13	2	no	—
73	C₂×C₄×C₈	—	13	1	no	—
74	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
75	C₂×D₁₃	—	2	2	no	—
76	C₂×D₁₃	—	4	2	no	—
77	C₂×D₁₃	—	4	4	no	—
78	C₂×D₁₃	#121	1	1	yes	characteristic
79	C₂×D₁₃	#124	1	1	yes	characteristic
80	#261	—	2	4	no	—
81	C₄×C₁₃	—	4	2	no	—
82	C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
84	Dic₁₃	#121	1	1	yes	characteristic
85	Dic₁₃	#127	1	1	yes	characteristic
86	C₂×#41	—	13	2	no	—
87	#135	—	26	2	no	—
88	#135	—	26	2	no	—
89	C₂×Q₁₆	#261	1	1	yes	characteristic, complemented
90	C₂×SD₁₆	—	13	2	no	—
91	C₂×D₈	—	13	1	no	—
92	C₂×M₁₆	—	13	2	no	—
93	C₂²×C₈	—	13	1	no	—
94	C₂²×C₈	—	13	1	no	—
95	C₂×C₄²	—	13	1	no	—
96	#108	—	26	2	no	—
97	#104	—	26	4	no	—
98	C₂²⋊C₈	—	26	4	no	—
99	C₄×C₈	—	26	2	no	—
100	C₂×C₁₃	#390	1	1	yes	characteristic
101	C₂×C₁₃	#443	1	2	yes	—
102	D₁₃	—	2	1	no	—
103	D₁₃	—	2	1	no	—
104	D₁₃	—	8	2	no	—
105	#41	—	26	2	no	—
106	#41	—	26	2	no	—
107	#41	—	26	2	no	—
108	#41	—	26	2	no	—
109	C₂×Q₈	C₂×#261	1	2	yes	—
110	C₂×D₄	—	13	2	no	—
111	C₂×D₄	—	26	2	no	—
112	C₂²×C₄	—	13	1	no	—
113	C₂²×C₄	—	26	2	no	—
114	C₂²×C₄	—	26	1	no	—
115	Q₁₆	—	2	2	no	—
116	SD₁₆	—	26	4	no	—
117	D₈	—	26	2	no	—
118	M₁₆	—	26	2	no	—
119	M₁₆	—	26	2	no	—
120	C₂×C₈	—	13	1	no	—
121	C₂×C₈	—	13	2	no	—
122	C₂×C₈	—	13	2	no	—
123	C₂×C₈	—	13	2	no	—
124	C₂×C₈	—	26	1	no	—
125	C₂×C₈	—	26	1	no	—
126	C₂×C₈	—	26	2	no	—
127	C₂×C₈	C₂×#261	1	1	yes	characteristic
128	C₄²	—	13	2	no	—
129	C₄²	—	26	1	no	—
130	C₁₃	#1974	1	1	yes	characteristic, complemented, Sylow
131	C₂³	—	13	1	no	—
132	C₂³	—	26	2	no	—
133	Q₈	—	2	2	no	—
134	Q₈	—	2	2	no	—
135	D₄	—	26	2	no	—
136	D₄	—	26	2	no	—
137	D₄	—	52	2	no	—
138	D₄	—	52	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	13	1	no	—
141	C₂×C₄	—	13	1	no	—
142	C₂×C₄	—	13	1	no	—
143	C₂×C₄	—	13	1	no	—
144	C₂×C₄	—	13	1	no	—
145	C₂×C₄	—	26	2	no	—
146	C₂×C₄	—	26	4	no	—
147	C₂×C₄	—	52	2	no	—
148	C₂×C₄	—	52	2	no	—
149	C₂×C₄	C₂²×#261	1	1	yes	characteristic
150	C₈	—	13	2	no	—
151	C₈	—	26	2	no	—
152	C₈	—	52	2	no	—
153	C₈	C₂²⋊#261	1	2	yes	—
154	C₂²	—	13	1	no	—
155	C₂²	—	13	1	no	—
156	C₂²	—	26	2	no	—
157	C₂²	—	52	4	no	—
158	C₂²	—	52	2	no	—
159	C₂²	D₄×#261	1	1	yes	characteristic
160	C₄	—	4	2	no	—
161	C₄	—	13	1	no	—
162	C₄	—	13	1	no	—
163	C₄	—	26	4	no	—
164	C₄	D₄×#261	1	1	yes	characteristic
165	C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
166	C₂	—	26	1	no	—
167	C₂	—	26	1	no	—
168	C₂	—	104	2	no	—
169	C₂	#11719111	1	2	yes	—
170	C₂	#11719151	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88173

C₂

yes

maximal, characteristic

#11719159

C₂

yes

maximal, complemented

C₂×#88118

C₂

yes

maximal, complemented

C₂×#88104

C₂

yes

maximal, characteristic, complemented

C₂×C₈×#261

C₂

yes

maximal, characteristic, complemented

C₂×#6131

C₂²

yes

—

C₂×C₄×#261

C₂²

yes

characteristic

C₂×#6118

C₂²

yes

—

C₂×D₁₃⋊C₈

C₂²

yes

characteristic

C₂×C₁₃×Q₁₆

C₄

yes

characteristic, complemented

C₂×#6106

C₄

yes

complemented

#88173

—

#88173

—

C₂×D₁₀₄

C₄

yes

characteristic, complemented

C₂×C₈×D₁₃

C₂²

yes

—

—

—

—

C₂²×#261

—

C₂×Q₈×C₁₃

C₂×C₄

yes

—

—

—

—

—

C₂×D₅₂

—

C₂×D₅₂

C₂×C₄

yes

—

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

C₂³

yes

characteristic

C₂×#1057

—

C₂×#1057

D₄

yes

—

C₄×#261

—

C₄×#261

D₄

yes

—

#6118

—

#6118

—

D₁₃⋊C₈

—

D₁₃⋊C₈

—

C₁₃×Q₁₆

—

C₂×C₈×C₁₃

C₂×C₄

yes

characteristic

#6106

—

C₂×#1055

C₂×C₄

yes

characteristic

D₁₀₄

—

C₈×D₁₃

D₄

yes

—

C₈×D₁₃

D₄

yes

—

#1974

—

maximal, Sylow

C₂²×D₁₃

—

C₂²×D₁₃

yes

—

—

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×Dic₁₃

C₂×D₄

yes

characteristic

D₅₂

—

D₅₂

—

D₅₂

—

D₅₂

—

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

C₂×D₄

yes

characteristic

C₄×D₁₃

C₂×D₄

yes

characteristic

C₄×D₁₃

C₂×D₄

yes

C₄×D₁₃

yes

—

—

C₈×C₁₃

C₂²⋊C₄

yes

—

#1055

C₂²⋊C₄

yes

—

—

—

—

—

C₂×C₄×C₈

—

C₂²×C₁₃

C₄×D₄

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

yes

C₂×D₁₃

yes

—

C₄×C₁₃

—

C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₃

C₄×D₄

yes

Dic₁₃

yes

Dic₁₃

yes

C₂×#41

—

#135

—

#135

—

C₂×Q₁₆

#261

yes

characteristic, complemented

C₂×SD₁₆

—

C₂×D₈

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

#108

—

#104

—

C₂²⋊C₈

—

C₄×C₈

—

100

C₂×C₁₃

#390

yes

characteristic

101

C₂×C₁₃

#443

yes

—

102

D₁₃

—

103

D₁₃

—

104

D₁₃

—

105

#41

—

106

#41

—

107

#41

—

108

#41

—

109

C₂×Q₈

C₂×#261

yes

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

Q₁₆

—

116

SD₁₆

—

117

D₈

—

118

M₁₆

—

119

M₁₆

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

C₂×#261

yes

characteristic

128

C₄²

—

129

C₄²

—

130

C₁₃

#1974

yes

characteristic, complemented, Sylow

131

C₂³

—

132

C₂³

—

133

Q₈

—

134

Q₈

—

135

D₄

—

136

D₄

—

137

D₄

—

138

D₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

C₂²×#261

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

C₂²⋊#261

yes

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

D₄×#261

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

D₄×#261

yes

characteristic

165

C₄

C₂×C₂²⋊#261

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

104

—

169

C₂

#11719111

yes

—

170

C₂

#11719151

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#261
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#261
Fitting	C₂×C₁₃×Q₁₆	C₄
socle	C₂²×C₁₃	C₄×D₄
prime radical	C₂×#88173	C₂
cosocle	C₂×C₄×D₁₃	C₂³
hypercenter	C₂×Q₁₆	#261
hypocenter	C₁₃	#1974
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂²

D₄×#261

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#261

Fitting

C₂×C₁₃×Q₁₆

C₄

socle

C₂²×C₁₃

C₄×D₄

prime radical

C₂×#88173

C₂

cosocle

C₂×C₄×D₁₃

C₂³

C₂×Q₁₆

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

N	H ≅ G/N	type
2. #11719159	C₂	semidirect
3. C₂×#88118	C₂	semidirect
4. C₂×#88104	C₂	semidirect
5. C₂×C₈×#261	C₂	semidirect
10. C₂×C₁₃×Q₁₆	C₄	semidirect
11. C₂×#6106	C₄	semidirect
14. C₂×D₁₀₄	C₄	semidirect
89. C₂×Q₁₆	#261	semidirect
130. C₁₃	#1974	semidirect

H ≅ G/N

type

2. #11719159

C₂

semidirect

3. C₂×#88118

C₂

semidirect

4. C₂×#88104

C₂

semidirect

5. C₂×C₈×#261

C₂

semidirect

10. C₂×C₁₃×Q₁₆

C₄

semidirect

11. C₂×#6106

C₄

semidirect

14. C₂×D₁₀₄

C₄

semidirect

89. C₂×Q₁₆

#261

semidirect

130. C₁₃

#1974

semidirect

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

subgroup	index
C₂×#88173	2
#11719159	2
C₂×#88118	2
C₂×#88104	2
C₂×C₈×#261	2
#1974	13

subgroup

index

C₂×C₈×#261

Subgroups — #13130987

Conjugacy classes of subgroups

Complements

Maximal subgroups