Group #13130196 — subgroups

Subgroups — #13130196

Order 1664.

246 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 60 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	#11719421	C₂	1	1	yes	maximal, characteristic
2	#11719054	C₂	1	1	yes	maximal, characteristic, complemented
3	#11719049	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊#1057	C₂	1	4	yes	maximal, complemented
5	C₂×#6129	—	2	1	no	—
6	C₁₃×#127	C₄	1	1	yes	characteristic
7	#88192	—	2	1	no	—
8	#88192	C₄	1	2	yes	—
9	#88190	—	2	1	no	—
10	D₄×Dic₁₃	—	2	1	no	—
11	D₄×Dic₁₃	C₂²	1	2	yes	—
12	C₄²×D₁₃	C₂²	1	1	yes	characteristic
13	#88123	C₄	1	1	yes	characteristic
14	#88112	—	2	1	no	—
15	#88111	—	2	1	no	—
16	#88111	—	2	1	no	—
17	#88110	—	2	1	no	—
18	C₄⋊#1057	C₂²	1	2	yes	—
19	C₄×#1057	C₂²	1	2	yes	—
20	C₂×D₄×C₁₃	—	2	1	no	—
21	C₂×D₄×C₁₃	—	2	1	no	—
22	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
23	C₂×C₂²⋊D₁₃	—	2	2	no	—
24	C₂²×Dic₁₃	—	2	2	no	—
25	#6129	—	4	4	no	—
26	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×D₁₃	D₄	1	1	yes	characteristic
28	C₂×C₄×D₁₃	D₄	1	1	yes	characteristic
29	C₂×Dic₂₆	—	2	1	no	—
30	C₂×#1057	—	2	2	no	—
31	C₂×#1057	—	2	2	no	—
32	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²⋊Dic₁₃	—	2	2	no	—
34	C₂²⋊Dic₁₃	—	2	2	no	—
35	C₄⋊Dic₁₃	—	2	1	no	—
36	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
37	C₄×Dic₁₃	D₄	1	1	yes	characteristic
38	C₄×Dic₁₃	D₄	1	1	yes	characteristic
39	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
40	#1496	—	13	1	no	maximal, Sylow
41	C₂³×C₁₃	—	2	2	no	—
42	C₂²×D₁₃	C₂²⋊C₄	1	1	yes	characteristic
43	D₄×C₁₃	—	2	4	no	—
44	D₄×C₁₃	—	4	2	no	—
45	D₄×C₁₃	—	4	2	no	—
46	C₂×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²⋊D₁₃	—	4	4	no	—
50	C₂×Dic₁₃	—	2	2	no	—
51	C₂×Dic₁₃	—	4	4	no	—
52	C₂×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
55	C₄×D₁₃	—	2	2	no	—
56	C₄×D₁₃	—	2	2	no	—
57	C₄×D₁₃	—	4	1	no	—
58	Dic₂₆	—	4	2	no	—
59	#1057	—	4	2	no	—
60	#1057	—	4	2	no	—
61	#532	—	13	1	no	—
62	#432	—	13	1	no	—
63	#431	—	13	1	no	—
64	D₄⋊C₈	—	13	4	no	—
65	C₂²×C₁₃	—	4	2	no	—
66	C₂²×C₁₃	—	4	2	no	—
67	C₂²×C₁₃	—	4	2	no	—
68	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×D₁₃	—	2	1	no	—
70	C₂×D₁₃	—	2	1	no	—
71	C₂×D₁₃	—	2	1	no	—
72	C₄×C₁₃	—	2	1	no	—
73	C₄×C₁₃	—	2	1	no	—
74	C₄×C₁₃	#131	1	2	yes	—
75	Dic₁₃	—	2	1	no	—
76	Dic₁₃	—	2	1	no	—
77	Dic₁₃	—	4	2	no	—
78	Dic₁₃	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×#41	—	26	1	no	—
80	C₄⋊Q₈	—	13	1	no	—
81	#127	#261	1	1	yes	characteristic
82	#124	—	26	1	no	—
83	#121	—	13	2	no	—
84	#121	—	26	1	no	—
85	C₄×D₄	—	13	2	no	—
86	C₄×D₄	—	26	1	no	—
87	C₂×C₄²	—	13	1	no	—
88	C₄⋊C₈	—	13	2	no	—
89	C₄⋊C₈	—	26	1	no	—
90	C₂²⋊C₈	—	26	1	no	—
91	C₂²⋊C₈	—	26	1	no	—
92	#97	—	26	1	no	—
93	C₄×C₈	—	13	2	no	—
94	C₂×C₁₃	—	8	2	no	—
95	C₂×C₁₃	#408	1	1	yes	characteristic
96	C₂×C₁₃	#418	1	1	yes	characteristic
97	C₂×C₁₃	#421	1	1	yes	characteristic
98	D₁₃	—	4	1	no	—
99	#41	—	52	4	no	—
100	C₂×Q₈	—	26	1	no	—
101	C₂×D₄	—	2	1	no	—
102	C₂×D₄	—	2	1	no	—
103	C₂×D₄	—	26	2	no	—
104	C₂×D₄	C₂×#261	1	2	yes	—
105	C₂²×C₄	—	13	1	no	—
106	C₂²×C₄	—	13	1	no	—
107	C₂²×C₄	—	13	1	no	—
108	C₂²×C₄	—	26	2	no	—
109	C₂×C₈	—	26	2	no	—
110	C₂×C₈	—	26	2	no	—
111	C₄⋊C₄	—	13	2	no	—
112	C₄⋊C₄	—	26	1	no	—
113	C₂²⋊C₄	—	26	2	no	—
114	C₂²⋊C₄	—	26	2	no	—
115	C₄²	—	13	1	no	—
116	C₄²	—	13	1	no	—
117	C₄²	—	13	1	no	—
118	C₄²	C₂×#261	1	1	yes	characteristic
119	C₁₃	#1496	1	1	yes	characteristic, complemented, Sylow
120	C₂³	—	2	2	no	—
121	C₂³	—	13	1	no	—
122	Q₈	—	52	2	no	—
123	D₄	—	2	4	no	—
124	D₄	—	4	2	no	—
125	D₄	—	4	2	no	—
126	D₄	—	52	4	no	—
127	C₂×C₄	—	13	1	no	—
128	C₂×C₄	—	13	1	no	—
129	C₂×C₄	—	13	1	no	—
130	C₂×C₄	—	26	2	no	—
131	C₂×C₄	—	26	2	no	—
132	C₂×C₄	—	26	2	no	—
133	C₂×C₄	—	52	1	no	—
134	C₂×C₄	—	52	4	no	—
135	C₂×C₄	C₂²⋊#261	1	1	yes	characteristic
136	C₂×C₄	C₂²⋊#261	1	1	yes	characteristic
137	C₂×C₄	C₂²×#261	1	1	yes	characteristic
138	C₈	—	52	2	no	—
139	C₈	—	52	2	no	—
140	C₂²	—	4	2	no	—
141	C₂²	—	4	2	no	—
142	C₂²	—	4	2	no	—
143	C₂²	—	26	1	no	—
144	C₂²	—	26	1	no	—
145	C₂²	—	26	1	no	—
146	C₂²	C₂×C₂²⋊#261	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	13	2	no	—
150	C₄	—	26	1	no	—
151	C₄	—	26	1	no	—
152	C₄	—	52	2	no	—
153	C₄	#88238	1	2	yes	—
154	C₂	—	8	2	no	—
155	C₂	—	52	1	no	—
156	C₂	#11719138	1	1	yes	characteristic
157	C₂	#11719140	1	1	yes	characteristic
158	C₂	#11719147	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11719421

C₂

yes

maximal, characteristic

#11719054

C₂

yes

maximal, characteristic, complemented

#11719049

C₂

yes

maximal, characteristic, complemented

D₄⋊#1057

C₂

yes

maximal, complemented

C₂×#6129

—

C₁₃×#127

C₄

yes

characteristic

#88192

—

#88192

C₄

yes

—

#88190

—

D₄×Dic₁₃

—

D₄×Dic₁₃

C₂²

yes

—

C₄²×D₁₃

C₂²

yes

characteristic

#88123

C₄

yes

—

—

—

—

C₂²

yes

—

C₄×#1057

C₂²

yes

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

C₂×C₄

yes

—

C₂×C₂²⋊D₁₃

—

C₂²×Dic₁₃

—

#6129

—

C₂×C₄×D₁₃

C₂×C₄

yes

characteristic

C₂×C₄×D₁₃

D₄

yes

characteristic

C₂×C₄×D₁₃

D₄

yes

characteristic

C₂×Dic₂₆

—

C₂×#1057

—

C₂×#1057

—

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂²⋊Dic₁₃

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

C₂×C₄

yes

—

C₄×Dic₁₃

D₄

yes

characteristic

C₄×Dic₁₃

D₄

yes

characteristic

C₄×Dic₁₃

C₂³

yes

characteristic

#1496

—

maximal, Sylow

C₂³×C₁₃

—

C₂²×D₁₃

C₂²⋊C₄

yes

characteristic

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₃

C₂×D₄

yes

characteristic

C₂×Dic₁₃

C₂×D₄

yes

characteristic

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

—

Dic₂₆

—

—

—

—

—

—

D₄⋊C₈

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#131

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

AGL(1,ℤ/8)

yes

—

C₂×#41

—

C₄⋊Q₈

—

yes

—

—

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

D₁₃

—

#41

—

100

C₂×Q₈

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

C₂×#261

yes

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₂²⋊C₄

—

114

C₂²⋊C₄

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

C₂×#261

yes

characteristic

119

C₁₃

#1496

yes

characteristic, complemented, Sylow

120

C₂³

—

121

C₂³

—

122

Q₈

—

123

D₄

—

124

D₄

—

125

D₄

—

126

D₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₂²⋊#261

yes

characteristic

136

C₂×C₄

C₂²⋊#261

yes

characteristic

137

C₂×C₄

C₂²×#261

yes

characteristic

138

C₈

—

139

C₈

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×C₂²⋊#261

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

#88238

yes

—

154

C₂

—

155

C₂

—

156

C₂

#11719138

yes

characteristic

157

C₂

#11719140

yes

characteristic

158

C₂

#11719147

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#261
Fitting	C₁₃×#127	C₄
socle	C₂²×C₁₃	C₂×C₂²⋊C₄
prime radical	#11719421	C₂
cosocle	C₄×Dic₁₃	C₂³
hypercenter	#127	#261
hypocenter	C₁₃	#1496
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

subgroup

G/H

center

C₂²

C₂×C₂²⋊#261

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#261

Fitting

C₁₃×#127

C₄

socle

C₂²×C₁₃

C₂×C₂²⋊C₄

prime radical

#11719421

C₂

cosocle

C₄×Dic₁₃

C₂³

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #127

N	H ≅ G/N	type
2. #11719054	C₂	semidirect
3. #11719049	C₂	semidirect
4. D₄⋊#1057	C₂	semidirect
119. C₁₃	#1496	semidirect

H ≅ G/N

type

2. #11719054

C₂

semidirect

3. #11719049

C₂

semidirect

4. D₄⋊#1057

C₂

semidirect

119. C₁₃

#1496

semidirect

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

subgroup	index
#11719421	2
#11719054	2
#11719049	2
D₄⋊#1057	2
#1496	13

subgroup

index

Subgroups — #13130196

Conjugacy classes of subgroups

Complements

Maximal subgroups