Group #13116123 — subgroups

Subgroups — #13116123

Order 1664.

266 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 65 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	#11719453	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718793	C₂	1	2	yes	maximal, complemented
4	C₂×#88045	C₂	1	2	yes	maximal, complemented
5	#11718152	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#6127	C₂²	1	2	yes	—
7	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#6099	C₂²	1	2	yes	—
11	#88163	—	2	2	no	—
12	#88162	—	2	2	no	—
13	#88158	—	2	2	no	—
14	C₁₃×C₄⋊C₈	—	2	2	no	—
15	#88064	—	2	2	no	—
16	#88063	—	2	2	no	—
17	#88045	—	2	2	no	—
18	#88045	—	2	2	no	—
19	C₂²×C₄×C₁₃	—	2	1	no	—
20	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
21	C₂×C₂²⋊D₁₃	—	2	2	no	—
22	#6127	—	2	2	no	—
23	#6127	—	2	2	no	—
24	#6127	—	2	2	no	—
25	#6127	—	2	2	no	—
26	C₂×D₅₂	D₄	1	2	yes	—
27	C₂×C₄×D₁₃	—	2	2	no	—
28	C₂×Dic₂₆	D₄	1	2	yes	—
29	C₁₃×M₁₆	—	2	2	no	—
30	C₁₃×M₁₆	—	2	2	no	—
31	C₂×C₈×C₁₃	—	2	1	no	—
32	C₂×C₈×C₁₃	D₄	1	2	yes	—
33	C₄²×C₁₃	—	2	1	no	—
34	C₄²×C₁₃	D₄	1	1	yes	characteristic
35	C₄²×C₁₃	D₄	1	1	yes	characteristic
36	#6099	—	2	2	no	—
37	#6099	—	2	2	no	—
38	C₂×#1055	—	2	2	no	—
39	Dic₅₂	—	2	2	no	—
40	D₁₀₄	—	2	2	no	—
41	#6095	—	2	4	no	—
42	#2050	—	13	1	no	maximal, Sylow
43	C₂³×C₁₃	#41	1	1	yes	characteristic
44	C₂²×D₁₃	—	2	2	no	—
45	C₂×C₄×C₁₃	—	2	1	no	—
46	C₂×C₄×C₁₃	—	2	1	no	—
47	C₂×C₄×C₁₃	—	4	2	no	—
48	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₃	#41	1	2	yes	—
52	C₂²⋊D₁₃	—	4	2	no	—
53	C₂²⋊D₁₃	—	4	2	no	—
54	C₂×Dic₁₃	—	2	2	no	—
55	D₅₂	—	2	2	no	—
56	D₅₂	—	2	2	no	—
57	C₄×D₁₃	—	4	2	no	—
58	C₄×D₁₃	—	4	2	no	—
59	Dic₂₆	—	2	2	no	—
60	Dic₂₆	—	2	2	no	—
61	C₈×C₁₃	—	2	2	no	—
62	C₈×C₁₃	—	4	1	no	—
63	#1055	—	4	2	no	—
64	#575	—	13	1	no	—
65	#423	D₁₃	1	1	yes	characteristic, complemented
66	C₂×#104	—	13	2	no	—
67	#413	—	13	2	no	—
68	#338	—	13	1	no	—
69	C₂²×C₁₃	—	4	1	no	—
70	C₂²×C₁₃	#121	1	1	yes	characteristic
71	C₂²×C₁₃	#123	1	2	yes	—
72	C₂×D₁₃	—	4	2	no	—
73	C₂×D₁₃	—	8	2	no	—
74	C₄×C₁₃	—	2	2	no	—
75	C₄×C₁₃	—	4	1	no	—
76	C₄×C₁₃	#120	1	1	yes	characteristic
77	C₄×C₁₃	#121	1	2	yes	—
78	C₄×C₁₃	#124	1	1	yes	characteristic
79	Dic₁₃	—	4	2	no	—
80	C₂×#41	—	13	2	no	—
81	#137	—	26	2	no	—
82	AGL(1,ℤ/8)	—	26	2	no	—
83	#135	—	26	2	no	—
84	C₂×M₁₆	—	13	2	no	—
85	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
86	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
87	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
88	C₄⋊C₈	—	2	2	no	—
89	#104	—	26	2	no	—
90	#104	—	26	2	no	—
91	#101	—	26	2	no	—
92	#100	—	26	2	no	—
93	C₂×C₁₃	—	2	1	no	—
94	C₂×C₁₃	—	2	2	no	—
95	C₂×C₁₃	#394	1	1	yes	characteristic
96	D₁₃	—	8	2	no	—
97	#41	—	26	2	no	—
98	#41	—	26	2	no	—
99	#41	—	26	2	no	—
100	#41	—	26	2	no	—
101	C₂×Q₈	—	13	2	no	—
102	C₂×D₄	—	13	2	no	—
103	C₂×D₄	—	26	2	no	—
104	C₂²×C₄	—	2	1	no	—
105	C₂²×C₄	—	26	2	no	—
106	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
107	Q₁₆	—	26	2	no	—
108	SD₁₆	—	26	4	no	—
109	D₈	—	26	2	no	—
110	M₁₆	—	2	2	no	—
111	M₁₆	—	2	2	no	—
112	M₁₆	—	26	2	no	—
113	M₁₆	—	26	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	26	2	no	—
116	C₂×C₈	C₂²⋊D₁₃	1	2	yes	—
117	C₄²	—	2	1	no	—
118	C₄²	D₅₂	1	1	yes	characteristic
119	C₄²	D₅₂	1	1	yes	characteristic
120	C₁₃	#2050	1	1	yes	characteristic, complemented, Sylow
121	C₂³	—	26	2	no	—
122	C₂³	#6127	1	1	yes	characteristic
123	Q₈	—	26	2	no	—
124	Q₈	—	26	2	no	—
125	D₄	—	26	2	no	—
126	D₄	—	26	2	no	—
127	D₄	—	52	2	no	—
128	D₄	—	52	2	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	26	2	no	—
133	C₂×C₄	—	52	2	no	—
134	C₂×C₄	—	52	2	no	—
135	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
136	C₂×C₄	D₄×D₁₃	1	2	yes	—
137	C₂×C₄	#6131	1	2	yes	—
138	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	4	1	no	—
141	C₈	—	52	2	no	—
142	C₂²	—	4	1	no	—
143	C₂²	—	52	2	no	—
144	C₂²	—	104	2	no	—
145	C₂²	#88148	1	2	yes	—
146	C₂²	#88184	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	4	1	no	—
149	C₄	—	52	2	no	—
150	C₄	#88149	1	2	yes	—
151	C₄	#88191	1	1	yes	characteristic
152	C₄	#88201	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	2	2	no	—
155	C₂	—	104	2	no	—
156	C₂	#11718652	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11719453

C₂

yes

maximal, characteristic

C₁₃×#423

C₂

yes

maximal, characteristic, complemented

#11718793

C₂

yes

maximal, complemented

C₂×#88045

C₂

yes

maximal, complemented

#11718152

C₂

yes

maximal, characteristic, complemented

C₂×#6127

C₂²

yes

—

C₂×C₁₃×M₁₆

C₂²

yes

characteristic

C₂×C₁₃×M₁₆

C₂²

yes

characteristic

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×#6099

C₂²

yes

—

#88163

—

#88162

—

#88158

—

C₁₃×C₄⋊C₈

—

—

—

—

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂×C₂²⋊D₁₃

—

—

—

—

—

C₂×D₅₂

D₄

yes

—

C₂×C₄×D₁₃

—

C₂×Dic₂₆

D₄

yes

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

D₄

yes

—

C₄²×C₁₃

—

C₄²×C₁₃

D₄

yes

characteristic

C₄²×C₁₃

D₄

yes

—

—

—

Dic₅₂

—

D₁₀₄

—

—

—

C₂³×C₁₃

yes

C₂²×D₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

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

—

C₂²⋊D₁₃

—

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

C₄×D₁₃

—

C₄×D₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

#1055

—

#575

—

#423

D₁₃

yes

characteristic, complemented

C₂×#104

—

#413

—

#338

—

C₂²×C₁₃

—

C₂²×C₁₃

#121

yes

characteristic

C₂²×C₁₃

#123

yes

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#120

yes

characteristic

C₄×C₁₃

#121

yes

—

C₄×C₁₃

#124

yes

characteristic

Dic₁₃

—

C₂×#41

—

#137

—

AGL(1,ℤ/8)

—

#135

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₃

yes

characteristic

C₂×M₁₆

C₂×D₁₃

yes

characteristic

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₄⋊C₈

—

—

—

—

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

#394

yes

characteristic

D₁₃

—

#41

—

#41

—

#41

—

100

#41

—

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

C₂²×D₁₃

yes

characteristic

107

Q₁₆

—

108

SD₁₆

—

109

D₈

—

110

M₁₆

—

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

C₂²⋊D₁₃

yes

—

117

C₄²

—

118

C₄²

D₅₂

yes

characteristic

119

C₄²

D₅₂

yes

characteristic

120

C₁₃

#2050

yes

characteristic, complemented, Sylow

121

C₂³

—

122

C₂³

#6127

yes

characteristic

123

Q₈

—

124

Q₈

—

125

D₄

—

126

D₄

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₂×D₅₂

yes

characteristic

136

C₂×C₄

D₄×D₁₃

yes

—

137

C₂×C₄

#6131

yes

—

138

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

104

—

145

C₂²

#88148

yes

—

146

C₂²

#88184

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#88149

yes

—

151

C₄

#88191

yes

characteristic

152

C₄

#88201

yes

characteristic

153

C₂

—

154

C₂

—

155

C₂

—

104

—

156

C₂

#11718652

yes

characteristic

	subgroup	G/H
center	C₄	#88201
commutator, cosocle	C₂²×C₄×C₁₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₁₃×#423	C₂
socle	C₂×C₁₃	#394
prime radical	#11719453	C₂
hypercenter	#423	D₁₃
hypocenter	C₁₃	#2050
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #423

subgroup

G/H

center

C₄

#88201

commutator, cosocle

C₂²×C₄×C₁₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₁₃×#423

C₂

C₂×C₁₃

C₂

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #423

N	H ≅ G/N	type
2. C₁₃×#423	C₂	semidirect
3. #11718793	C₂	semidirect
4. C₂×#88045	C₂	semidirect
5. #11718152	C₂	semidirect
65. #423	D₁₃	semidirect
120. C₁₃	#2050	semidirect

H ≅ G/N

type

2. C₁₃×#423

C₂

semidirect

3. #11718793

C₂

semidirect

4. C₂×#88045

C₂

semidirect

5. #11718152

C₂

semidirect

65. #423

D₁₃

semidirect

120. C₁₃

#2050

semidirect

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

subgroup	index
#11719453	2
C₁₃×#423	2
#11718793	2
C₂×#88045	2
#11718152	2
#2050	13

subgroup

index

C₁₃×#423

Subgroups — #13116123

Conjugacy classes of subgroups

Complements

Maximal subgroups