Group #13116246 — subgroups

Subgroups — #13116246

Order 1664.

238 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 87 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×#88056	C₂	1	2	yes	maximal
2	#11718596	C₂	1	1	yes	maximal, characteristic
3	#11718590	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#336	C₂	1	1	yes	maximal, characteristic
5	#11718143	C₂	1	2	yes	maximal
6	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₁₃	C₂²	1	2	yes	—
8	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₃	—	2	1	no	—
10	C₂×C₄⋊Dic₁₃	—	2	1	no	—
11	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
13	#88071	—	2	1	no	—
14	#88071	—	2	1	no	—
15	#88056	—	2	4	no	—
16	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₁₃	—	2	1	no	—
20	C₂²×Dic₁₃	—	2	1	no	—
21	C₂²×Dic₁₃	—	2	2	no	—
22	C₂×Dic₂₆	—	2	2	no	—
23	C₂×Dic₂₆	—	2	2	no	—
24	C₂×Dic₂₆	—	2	1	no	—
25	C₂×Dic₂₆	D₄	1	2	yes	—
26	C₂×C₈×C₁₃	—	2	2	no	—
27	C₂×C₈×C₁₃	—	2	2	no	—
28	C₂×C₈×C₁₃	—	2	2	no	—
29	C₄²×C₁₃	—	2	2	no	—
30	C₄⋊Dic₁₃	—	2	2	no	—
31	C₄⋊Dic₁₃	—	2	2	no	—
32	C₄⋊Dic₁₃	—	2	2	no	—
33	C₄⋊Dic₁₃	—	4	2	no	—
34	C₄⋊Dic₁₃	—	4	2	no	—
35	#2072	—	13	1	no	maximal, Sylow
36	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₃	—	2	1	no	—
38	C₂×C₄×C₁₃	—	2	1	no	—
39	C₂×C₄×C₁₃	—	2	1	no	—
40	C₂×C₄×C₁₃	—	2	1	no	—
41	C₂×C₄×C₁₃	SD₁₆	1	2	yes	—
42	C₂×C₄×C₁₃	Q₁₆	1	2	yes	—
43	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₃	#41	1	2	yes	—
45	C₂×C₄×C₁₃	#41	1	2	yes	—
46	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
47	C₂×Dic₁₃	—	2	2	no	—
48	C₂×Dic₁₃	—	4	1	no	—
49	C₂×Dic₁₃	—	4	1	no	—
50	C₂×Dic₁₃	—	4	2	no	—
51	C₂×Dic₁₃	—	4	1	no	—
52	C₂×Dic₁₃	—	4	2	no	—
53	C₂×Dic₁₃	—	4	2	no	—
54	C₂×Dic₁₃	—	4	1	no	—
55	C₂×Dic₁₃	—	4	1	no	—
56	Dic₂₆	—	2	4	no	—
57	Dic₂₆	—	4	2	no	—
58	C₈×C₁₃	—	4	2	no	—
59	C₂×#103	—	13	2	no	—
60	#391	—	13	1	no	—
61	#384	—	13	1	no	—
62	#340	—	13	2	no	—
63	#336	D₁₃	1	1	yes	characteristic
64	C₂²×C₁₃	#121	1	1	yes	characteristic
65	C₂²×C₁₃	#123	1	1	yes	characteristic
66	C₂²×C₁₃	#124	1	1	yes	characteristic
67	C₂²×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
68	C₂²×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
69	C₂²×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₂²×C₁₃	#137	1	1	yes	characteristic
71	C₄×C₁₃	—	2	2	no	—
72	C₄×C₁₃	—	2	2	no	—
73	C₄×C₁₃	#123	1	2	yes	—
74	C₄×C₁₃	#126	1	2	yes	—
75	Dic₁₃	—	4	2	no	—
76	Dic₁₃	—	8	2	no	—
77	Dic₁₃	—	8	1	no	—
78	C₂²×Q₈	—	13	1	no	—
79	C₂²×C₈	C₂×D₁₃	1	2	yes	—
80	C₂×C₄⋊C₄	—	13	1	no	—
81	C₂×C₄⋊C₄	—	13	2	no	—
82	C₂×C₄⋊C₄	—	26	1	no	—
83	C₂×C₄⋊C₄	—	26	1	no	—
84	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
85	#103	—	26	4	no	—
86	#95	—	26	1	no	—
87	#95	—	26	1	no	—
88	C₂×C₁₃	#397	1	1	yes	characteristic
89	C₂×C₁₃	#460	1	1	yes	characteristic
90	C₂×C₁₃	#462	1	1	yes	characteristic
91	C₂×C₁₃	#483	1	1	yes	characteristic
92	C₂×C₁₃	#484	1	1	yes	characteristic
93	C₂×C₁₃	#487	1	1	yes	characteristic
94	C₂×C₁₃	#489	1	1	yes	characteristic
95	C₂×Q₈	—	13	2	no	—
96	C₂×Q₈	—	26	1	no	—
97	C₂×Q₈	—	26	2	no	—
98	C₂×Q₈	—	26	2	no	—
99	C₂²×C₄	—	26	1	no	—
100	C₂²×C₄	—	26	1	no	—
101	C₂²×C₄	—	26	2	no	—
102	C₂²×C₄	D₅₂	1	1	yes	characteristic
103	C₂²×C₄	D₅₂	1	1	yes	characteristic
104	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₄⋊C₄	—	26	2	no	—
109	C₄⋊C₄	—	26	2	no	—
110	C₄⋊C₄	—	26	2	no	—
111	C₄⋊C₄	—	52	2	no	—
112	C₄⋊C₄	—	52	2	no	—
113	C₄²	—	2	2	no	—
114	C₁₃	#2072	1	1	yes	characteristic, complemented, Sylow
115	C₂³	C₂×D₅₂	1	1	yes	characteristic
116	Q₈	—	26	4	no	—
117	Q₈	—	52	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	26	2	no	—
123	C₂×C₄	—	52	1	no	—
124	C₂×C₄	—	52	1	no	—
125	C₂×C₄	—	52	1	no	—
126	C₂×C₄	—	52	2	no	—
127	C₂×C₄	—	52	2	no	—
128	C₂×C₄	—	52	2	no	—
129	C₂×C₄	—	52	1	no	—
130	C₂×C₄	—	52	1	no	—
131	C₂×C₄	#6095	1	2	yes	—
132	C₂×C₄	Dic₅₂	1	2	yes	—
133	C₂×C₄	#6127	1	2	yes	—
134	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
135	C₂×C₄	#6129	1	2	yes	—
136	C₂×C₄	#6131	1	1	yes	characteristic
137	C₈	—	4	2	no	—
138	C₂²	#88129	1	1	yes	characteristic
139	C₂²	#88140	1	1	yes	characteristic
140	C₂²	#88149	1	1	yes	characteristic
141	C₂²	C₂×#6095	1	1	yes	characteristic
142	C₂²	C₂×Dic₅₂	1	1	yes	characteristic
143	C₂²	#88162	1	1	yes	characteristic
144	C₂²	#88163	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	52	2	no	—
148	C₄	—	104	2	no	—
149	C₄	—	104	1	no	—
150	C₄	#88137	1	2	yes	—
151	C₄	#88152	1	2	yes	—
152	C₂	#11718335	1	1	yes	characteristic
153	C₂	#11718342	1	1	yes	characteristic
154	C₂	#11718375	1	1	yes	characteristic
155	C₂	#11718382	1	1	yes	characteristic
156	C₂	#11718383	1	1	yes	characteristic
157	C₂	#11718509	1	1	yes	characteristic
158	C₂	#11718510	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88056

C₂

yes

maximal

#11718596

C₂

yes

maximal, characteristic

#11718590

C₂

yes

maximal, characteristic

C₁₃×#336

C₂

yes

maximal, characteristic

#11718143

C₂

yes

maximal

C₂²×Dic₂₆

C₂²

yes

characteristic

C₂²×C₈×C₁₃

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

—

—

C₂²×C₄×C₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

D₄

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#2072

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

SD₁₆

yes

—

C₂×C₄×C₁₃

Q₁₆

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

#41

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×C₄×C₁₃

#41

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

—

—

—

—

D₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

C₂×SD₁₆

yes

characteristic

C₂²×C₁₃

C₂×Q₁₆

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#123

yes

—

C₄×C₁₃

#126

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×C₈

C₂×D₁₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₃

yes

—

—

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

D₅₂

yes

characteristic

103

C₂²×C₄

D₅₂

yes

characteristic

104

C₂²×C₄

C₂²×D₁₃

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄²

—

114

C₁₃

#2072

yes

characteristic, complemented, Sylow

115

C₂³

C₂×D₅₂

yes

characteristic

116

Q₈

—

117

Q₈

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

#6095

yes

—

132

C₂×C₄

Dic₅₂

yes

—

133

C₂×C₄

#6127

yes

—

134

C₂×C₄

D₄×D₁₃

yes

characteristic

135

C₂×C₄

#6129

yes

—

136

C₂×C₄

#6131

yes

characteristic

137

C₈

—

138

C₂²

#88129

yes

characteristic

139

C₂²

#88140

yes

characteristic

140

C₂²

#88149

yes

characteristic

141

C₂²

C₂×#6095

yes

characteristic

142

C₂²

C₂×Dic₅₂

yes

characteristic

143

C₂²

#88162

yes

characteristic

144

C₂²

#88163

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

104

—

149

C₄

—

104

—

150

C₄

#88137

yes

—

151

C₄

#88152

yes

—

152

C₂

#11718335

yes

characteristic

153

C₂

#11718342

yes

characteristic

154

C₂

#11718375

yes

characteristic

155

C₂

#11718382

yes

characteristic

156

C₂

#11718383

yes

characteristic

157

C₂

#11718509

yes

characteristic

158

C₂

#11718510

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₂
commutator, cosocle	C₂²×C₄×C₁₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₁₃×#336	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₄
hypercenter	#336	D₁₃
hypocenter	C₁₃	#2072
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₅₂

commutator, cosocle

C₂²×C₄×C₁₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₁₃×#336

C₂

socle, prime radical

C₂³×C₁₃

C₂×D₄

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
114. C₁₃	#2072	semidirect

H ≅ G/N

type

114. C₁₃

#2072

semidirect

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

subgroup	index
C₂×#88056	2
#11718596	2
#11718590	2
C₁₃×#336	2
#11718143	2
#2072	13

subgroup

index

C₁₃×#336

Subgroups — #13116246

Conjugacy classes of subgroups

Complements

Maximal subgroups