Group #13131014 — subgroups

Subgroups — #13131014

Order 1664.

292 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 88 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	D₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#261	C₂	1	1	yes	maximal, characteristic
3	D₁₃⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#11718246	C₂	1	4	yes	maximal
5	C₂×Q₈×D₁₃	—	2	1	no	—
6	C₂×C₄⋊#261	C₂²	1	2	yes	—
7	C₂×C₄×#261	—	2	1	no	—
8	C₂×D₁₃⋊C₈	C₂²	1	2	yes	—
9	C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
10	#88198	—	2	1	no	—
11	C₄⋊C₄×D₁₃	—	2	1	no	—
12	C₄⋊C₄×D₁₃	C₂²	1	2	yes	—
13	#88142	—	2	1	no	—
14	#88142	C₄	1	2	yes	complemented
15	C₄²×D₁₃	C₂²	1	1	yes	characteristic
16	#88123	C₄	1	1	yes	characteristic, complemented
17	(C₂×#261)⋊C₄	—	2	1	no	—
18	#88112	—	2	1	no	—
19	C₄⋊#1057	—	2	1	no	—
20	C₂²×#261	—	2	2	no	—
21	C₂×Q₈×C₁₃	—	2	1	no	—
22	Q₈×D₁₃	—	4	4	no	—
23	C₂×C₄×D₁₃	—	2	2	no	—
24	C₂×C₄×D₁₃	D₄	1	1	yes	characteristic
25	C₂×C₄×D₁₃	D₄	1	1	yes	characteristic
26	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
27	C₂×Dic₂₆	—	2	1	no	—
28	C₂×Dic₂₆	—	2	2	no	—
29	C₂×#1057	—	2	2	no	—
30	C₄⋊#261	—	2	2	no	—
31	C₄⋊#261	Q₈	1	4	yes	complemented
32	C₄×#261	—	4	2	no	—
33	D₁₃⋊C₈	—	2	4	no	—
34	C₁₃×C₄⋊C₄	—	2	1	no	—
35	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
36	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₄⋊Dic₁₃	—	2	1	no	—
38	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
39	#6101	—	2	2	no	—
40	#6101	—	2	2	no	—
41	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₁₃	D₄	1	1	yes	characteristic
43	C₄×Dic₁₃	D₄	1	1	yes	characteristic
44	#1987	—	13	1	no	maximal, Sylow
45	C₂²×D₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂×#261	—	2	4	no	—
47	C₂×#261	—	4	2	no	—
48	C₂×#261	—	4	2	no	—
49	Q₈×C₁₃	—	4	2	no	—
50	C₂×C₄×C₁₃	—	2	2	no	—
51	C₂×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×Dic₁₃	—	2	2	no	—
55	C₂×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
58	C₄×D₁₃	—	2	2	no	—
59	C₄×D₁₃	—	4	4	no	—
60	C₄×D₁₃	SD₁₆	1	2	yes	—
61	C₄×D₁₃	Q₁₆	1	2	yes	—
62	C₄×D₁₃	C₂×Q₈	1	2	yes	—
63	C₄×D₁₃	#41	1	2	yes	—
64	Dic₂₆	—	4	4	no	—
65	Dic₂₆	—	4	2	no	—
66	#1057	—	4	2	no	—
67	C₂×C₄⋊Q₈	—	13	1	no	—
68	C₂×C₄⋊C₈	—	13	1	no	—
69	C₄×C₄⋊C₄	—	13	1	no	—
70	#340	—	13	4	no	—
71	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×D₁₃	C₂²⋊Q₈	1	2	yes	—
73	C₂×D₁₃	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×D₁₃	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×D₁₃	AGL(1,ℤ/8)	1	2	yes	—
76	#261	—	4	4	no	—
77	C₄×C₁₃	—	2	1	no	—
78	C₄×C₁₃	—	4	2	no	—
79	C₄×C₁₃	#103	1	2	yes	—
80	C₄×C₁₃	C₄×Q₈	1	2	yes	—
81	Dic₁₃	—	2	1	no	—
82	Dic₁₃	—	4	2	no	—
83	Dic₁₃	#103	1	2	yes	—
84	Dic₁₃	#124	1	1	yes	characteristic
85	Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
86	C₂²×Q₈	—	26	1	no	—
87	C₂²×C₈	—	13	2	no	—
88	C₄⋊Q₈	—	13	2	no	—
89	C₄⋊Q₈	—	13	1	no	—
90	C₄⋊Q₈	—	26	1	no	—
91	C₄⋊Q₈	—	26	1	no	—
92	C₄⋊Q₈	#261	1	1	yes	characteristic, complemented
93	C₂×C₄⋊C₄	—	13	2	no	—
94	C₂×C₄⋊C₄	—	13	2	no	—
95	C₂×C₄⋊C₄	—	26	1	no	—
96	C₂×C₄²	—	13	1	no	—
97	C₂×C₄²	—	26	1	no	—
98	C₄⋊C₈	—	26	1	no	—
99	C₄⋊C₈	—	26	1	no	—
100	#95	—	26	1	no	—
101	C₂×C₁₃	#391	1	1	yes	characteristic
102	C₂×C₁₃	C₂×#103	1	1	yes	characteristic
103	C₂×C₁₃	#418	1	1	yes	characteristic
104	D₁₃	#476	1	2	yes	—
105	D₁₃	#477	1	2	yes	—
106	C₂×Q₈	—	2	1	no	—
107	C₂×Q₈	—	26	2	no	—
108	C₂×Q₈	—	26	1	no	—
109	C₂×Q₈	—	52	4	no	—
110	C₂²×C₄	—	13	1	no	—
111	C₂²×C₄	—	13	1	no	—
112	C₂²×C₄	—	13	1	no	—
113	C₂²×C₄	—	26	2	no	—
114	C₂²×C₄	—	26	2	no	—
115	C₂×C₈	—	26	2	no	—
116	C₂×C₈	—	26	4	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	13	4	no	—
119	C₄⋊C₄	—	13	2	no	—
120	C₄⋊C₄	—	26	2	no	—
121	C₄⋊C₄	—	26	2	no	—
122	C₄⋊C₄	—	26	2	no	—
123	C₄⋊C₄	—	26	1	no	—
124	C₄⋊C₄	C₂×#261	1	2	yes	—
125	C₄²	—	13	1	no	—
126	C₄²	—	13	1	no	—
127	C₄²	—	13	1	no	—
128	C₄²	—	52	2	no	—
129	C₄²	C₂×#261	1	1	yes	characteristic
130	C₁₃	#1987	1	1	yes	characteristic, complemented, Sylow
131	C₂³	—	13	1	no	—
132	Q₈	—	4	2	no	—
133	Q₈	—	52	2	no	—
134	Q₈	—	52	4	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	13	1	no	—
137	C₂×C₄	—	13	1	no	—
138	C₂×C₄	—	13	2	no	—
139	C₂×C₄	—	13	2	no	—
140	C₂×C₄	—	13	2	no	—
141	C₂×C₄	—	13	2	no	—
142	C₂×C₄	—	13	1	no	—
143	C₂×C₄	—	26	2	no	—
144	C₂×C₄	—	26	4	no	—
145	C₂×C₄	—	26	2	no	—
146	C₂×C₄	—	52	4	no	—
147	C₂×C₄	—	52	2	no	—
148	C₂×C₄	—	52	2	no	—
149	C₂×C₄	C₂²⋊#261	1	1	yes	characteristic
150	C₂×C₄	C₂²⋊#261	1	1	yes	characteristic
151	C₂×C₄	C₂²×#261	1	1	yes	characteristic
152	C₈	—	52	2	no	—
153	C₂²	—	13	2	no	—
154	C₂²	—	13	1	no	—
155	C₂²	—	13	1	no	—
156	C₂²	—	13	2	no	—
157	C₂²	C₂×C₂²⋊#261	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	4	2	no	—
160	C₄	—	13	2	no	—
161	C₄	—	13	1	no	—
162	C₄	—	13	1	no	—
163	C₄	—	26	1	no	—
164	C₄	—	52	4	no	—
165	C₄	—	52	2	no	—
166	C₄	#88117	1	2	yes	—
167	C₄	Q₈×#261	1	2	yes	—
168	C₂	—	13	2	no	—
169	C₂	—	13	2	no	—
170	C₂	#11719138	1	1	yes	characteristic
171	C₂	C₂×#88117	1	1	yes	characteristic
172	C₂	(C₂×#261)⋊Q₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₁₃×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₄×C₄⋊#261

C₂

yes

maximal, characteristic

D₁₃⋊(C₄⋊C₈)

C₂

yes

maximal, characteristic

#11718246

C₂

yes

maximal

C₂×Q₈×D₁₃

—

C₂×C₄⋊#261

C₂²

yes

—

C₂×C₄×#261

—

C₂×D₁₃⋊C₈

C₂²

yes

—

C₁₃×C₄⋊Q₈

C₄

yes

characteristic, complemented

#88198

—

C₄⋊C₄×D₁₃

—

C₄⋊C₄×D₁₃

C₂²

yes

—

#88142

—

#88142

C₄

yes

complemented

C₄²×D₁₃

C₂²

yes

characteristic

#88123

C₄

yes

characteristic, complemented

(C₂×#261)⋊C₄

—

#88112

—

C₄⋊#1057

—

C₂²×#261

—

C₂×Q₈×C₁₃

—

Q₈×D₁₃

—

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

D₄

yes

characteristic

C₂×C₄×D₁₃

D₄

yes

characteristic

C₂×C₄×D₁₃

C₂³

yes

characteristic

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×#1057

—

C₄⋊#261

—

C₄⋊#261

Q₈

yes

complemented

C₄×#261

—

D₁₃⋊C₈

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

C₂×C₄

yes

—

#6101

—

#6101

—

C₄×Dic₁₃

C₂×C₄

yes

characteristic

C₄×Dic₁₃

D₄

yes

characteristic

C₄×Dic₁₃

D₄

yes

characteristic

#1987

—

maximal, Sylow

C₂²×D₁₃

C₂×D₄

yes

—

—

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₃

C₂×D₄

yes

characteristic

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

SD₁₆

yes

—

C₄×D₁₃

Q₁₆

yes

—

C₄×D₁₃

C₂×Q₈

yes

—

C₄×D₁₃

#41

yes

—

Dic₂₆

—

Dic₂₆

—

#1057

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₄×C₄⋊C₄

—

#340

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₃

C₂²⋊Q₈

yes

—

C₂×D₁₃

C₂×SD₁₆

yes

characteristic

C₂×D₁₃

C₂×Q₁₆

yes

characteristic

C₂×D₁₃

AGL(1,ℤ/8)

yes

—

#261

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#103

yes

—

C₄×C₁₃

C₄×Q₈

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

#103

yes

—

Dic₁₃

#124

yes

characteristic

Dic₁₃

C₄⋊Q₈

yes

characteristic

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

#261

yes

characteristic, complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

100

#95

—

101

C₂×C₁₃

#391

yes

characteristic

102

C₂×C₁₃

C₂×#103

yes

characteristic

103

C₂×C₁₃

#418

yes

characteristic

104

D₁₃

#476

yes

—

105

D₁₃

#477

yes

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

C₂×#261

yes

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

—

129

C₄²

C₂×#261

yes

characteristic

130

C₁₃

#1987

yes

characteristic, complemented, Sylow

131

C₂³

—

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

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

C₂×C₄

C₂²⋊#261

yes

characteristic

151

C₂×C₄

C₂²×#261

yes

characteristic

152

C₈

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

C₂×C₂²⋊#261

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

#88117

yes

—

167

C₄

Q₈×#261

yes

—

168

C₂

—

169

C₂

—

170

C₂

#11719138

yes

characteristic

171

C₂

C₂×#88117

yes

characteristic

172

C₂

(C₂×#261)⋊Q₈

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#261
Fitting	C₁₃×C₄⋊Q₈	C₄
socle	C₂²×C₁₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₃	C₂×D₄
cosocle	C₂×C₄×D₁₃	C₂³
hypercenter	C₄⋊Q₈	#261
hypocenter	C₁₃	#1987
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#261

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#261

Fitting

C₁₃×C₄⋊Q₈

C₄

socle

C₂²×C₁₃

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₃

C₂×D₄

cosocle

C₂×C₄×D₁₃

C₂³

C₄⋊Q₈

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
9. C₁₃×C₄⋊Q₈	C₄	semidirect
14. #88142	C₄	semidirect
16. #88123	C₄	semidirect
31. C₄⋊#261	Q₈	semidirect
92. C₄⋊Q₈	#261	semidirect
130. C₁₃	#1987	semidirect

H ≅ G/N

type

9. C₁₃×C₄⋊Q₈

C₄

semidirect

14. #88142

C₄

semidirect

16. #88123

C₄

semidirect

31. C₄⋊#261

Q₈

semidirect

92. C₄⋊Q₈

#261

semidirect

130. C₁₃

#1987

semidirect

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

subgroup	index
D₁₃×C₄⋊Q₈	2
C₄×C₄⋊#261	2
D₁₃⋊(C₄⋊C₈)	2
#11718246	2
#1987	13

subgroup

index

D₁₃×C₄⋊Q₈

C₄×C₄⋊#261

D₁₃⋊(C₄⋊C₈)

#11718246

#1987

Subgroups — #13131014

Conjugacy classes of subgroups

Complements

Maximal subgroups