Group #13122908 — subgroups

Subgroups — #13122908

Order 1664.

456 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 133 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×C₁₃×#126	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88130	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718655	C₂	1	3	yes	maximal, complemented
4	#11718645	C₂	1	3	yes	maximal, complemented
5	#11718622	C₂	1	3	yes	maximal
6	#11718609	C₂	1	3	yes	maximal, complemented
7	#11718596	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₂²⋊D₁₃	C₂²	1	1	yes	characteristic
9	C₁₃×#126	—	2	4	no	—
10	C₂×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
11	C₂×C₁₃×C₂²⋊C₄	C₂²	1	3	yes	—
12	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊Dic₁₃	C₂²	1	3	yes	—
14	C₂×#6103	C₂²	1	3	yes	—
15	C₂×#6103	C₂²	1	3	yes	—
16	C₂×C₄⋊Dic₁₃	C₂²	1	3	yes	—
17	C₂×C₄⋊Dic₁₃	C₂²	1	3	yes	—
18	C₂×#6101	C₂²	1	3	yes	—
19	C₂×#6101	C₂²	1	3	yes	—
20	#88130	—	2	4	no	—
21	#88071	C₂²	1	3	yes	—
22	#88071	C₂²	1	3	yes	—
23	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
24	C₂³×D₁₃	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
26	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
27	C₂×C₂²⋊D₁₃	—	4	3	no	—
28	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
29	C₂²×Dic₁₃	C₂³	1	3	yes	—
30	C₂²×Dic₁₃	C₂³	1	3	yes	—
31	C₁₃×C₄⋊C₄	—	2	6	no	—
32	C₁₃×C₂²⋊C₄	—	2	6	no	—
33	C₄²×C₁₃	D₄	1	4	yes	complemented
34	C₂²⋊Dic₁₃	—	4	3	no	—
35	#6103	—	2	6	no	—
36	#6103	—	4	3	no	—
37	C₄⋊Dic₁₃	—	4	3	no	—
38	C₄⋊Dic₁₃	—	4	3	no	—
39	#6101	—	2	6	no	—
40	#6101	—	4	3	no	—
41	#2823	—	13	1	no	maximal, Sylow
42	C₂³×C₁₃	—	2	1	no	—
43	C₂³×C₁₃	—	2	3	no	—
44	C₂³×C₁₃	—	2	3	no	—
45	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
46	C₂²×D₁₃	—	2	1	no	—
47	C₂²×D₁₃	—	2	3	no	—
48	C₂²×D₁₃	—	2	3	no	—
49	C₂×C₄×C₁₃	—	2	3	no	—
50	C₂×C₄×C₁₃	—	2	3	no	—
51	C₂×C₄×C₁₃	—	2	3	no	—
52	C₂×C₄×C₁₃	—	2	3	no	—
53	C₂×C₄×C₁₃	—	2	3	no	—
54	C₂×C₄×C₁₃	C₂×D₄	1	6	yes	—
55	C₂²⋊D₁₃	—	4	4	no	—
56	C₂×Dic₁₃	—	2	3	no	—
57	C₂×Dic₁₃	—	2	3	no	—
58	C₂×Dic₁₃	—	2	3	no	—
59	C₂×Dic₁₃	—	2	3	no	—
60	C₂×Dic₁₃	—	2	3	no	—
61	C₂×Dic₁₃	—	2	3	no	—
62	C₂×Dic₁₃	—	2	3	no	—
63	C₂×#126	—	13	1	no	—
64	C₂×#126	D₁₃	1	1	yes	characteristic, complemented
65	#398	—	13	3	no	—
66	#396	—	13	3	no	—
67	#396	—	13	3	no	—
68	#394	—	13	3	no	—
69	#384	—	13	1	no	—
70	C₂²×C₁₃	—	4	3	no	—
71	C₂²×C₁₃	—	4	1	no	—
72	C₂²×C₁₃	—	4	3	no	—
73	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
74	C₂²×C₁₃	H(5,𝔽₂)	1	3	yes	—
75	C₂²×C₁₃	2₋¹⁺⁴	1	3	yes	—
76	C₂×D₁₃	—	4	3	no	—
77	C₂×D₁₃	—	4	1	no	—
78	C₂×D₁₃	—	4	3	no	—
79	C₄×C₁₃	—	2	6	no	—
80	C₄×C₁₃	—	4	3	no	—
81	Dic₁₃	—	4	3	no	—
82	Dic₁₃	—	4	3	no	—
83	Dic₁₃	—	4	1	no	—
84	C₂²×D₄	—	13	1	no	—
85	#126	—	2	4	no	—
86	#126	—	26	4	no	—
87	C₂×C₄⋊C₄	—	13	3	no	—
88	C₂×C₄⋊C₄	—	13	3	no	—
89	C₂×C₄⋊C₄	—	13	3	no	—
90	C₂×C₄⋊C₄	—	13	3	no	—
91	C₂×C₄⋊C₄	C₂×D₁₃	1	3	yes	—
92	C₂×C₂²⋊C₄	—	13	3	no	—
93	C₂×C₂²⋊C₄	—	13	3	no	—
94	C₂×C₂²⋊C₄	—	13	3	no	—
95	C₂×C₂²⋊C₄	C₂×D₁₃	1	3	yes	—
96	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
97	#95	—	13	3	no	—
98	#95	—	13	3	no	—
99	C₂×C₁₃	—	8	1	no	—
100	C₂×C₁₃	#537	1	3	yes	—
101	C₂×C₁₃	#560	1	1	yes	characteristic
102	C₂×C₁₃	#563	1	3	yes	—
103	D₁₃	—	8	1	no	—
104	C₂⁴	—	13	1	no	—
105	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
106	C₂×D₄	—	52	3	no	—
107	C₂²×C₄	—	13	1	no	—
108	C₂²×C₄	—	13	3	no	—
109	C₂²×C₄	—	13	3	no	—
110	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
111	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
112	C₄⋊C₄	—	2	6	no	—
113	C₄⋊C₄	—	26	6	no	—
114	C₄⋊C₄	—	52	3	no	—
115	C₄⋊C₄	—	52	3	no	—
116	C₄⋊C₄	—	52	3	no	—
117	C₂²⋊C₄	—	2	6	no	—
118	C₂²⋊C₄	—	26	6	no	—
119	C₂²⋊C₄	—	52	3	no	—
120	C₂²⋊C₄	—	52	3	no	—
121	C₄²	C₂²⋊D₁₃	1	4	yes	complemented
122	C₁₃	#2823	1	1	yes	characteristic, complemented, Sylow
123	C₂³	—	2	3	no	—
124	C₂³	—	2	3	no	—
125	C₂³	—	2	1	no	—
126	C₂³	—	26	3	no	—
127	C₂³	—	26	3	no	—
128	C₂³	—	26	1	no	—
129	C₂³	C₂³×D₁₃	1	1	yes	characteristic
130	D₄	—	52	4	no	—
131	C₂×C₄	—	2	3	no	—
132	C₂×C₄	—	2	3	no	—
133	C₂×C₄	—	2	3	no	—
134	C₂×C₄	—	2	3	no	—
135	C₂×C₄	—	2	3	no	—
136	C₂×C₄	—	26	3	no	—
137	C₂×C₄	—	26	3	no	—
138	C₂×C₄	—	26	3	no	—
139	C₂×C₄	—	26	3	no	—
140	C₂×C₄	—	26	3	no	—
141	C₂×C₄	—	26	3	no	—
142	C₂×C₄	—	26	3	no	—
143	C₂×C₄	C₂×C₂²⋊D₁₃	1	6	yes	—
144	C₂²	—	4	3	no	—
145	C₂²	—	4	1	no	—
146	C₂²	—	4	3	no	—
147	C₂²	—	52	3	no	—
148	C₂²	—	52	1	no	—
149	C₂²	—	52	3	no	—
150	C₂²	#88256	1	3	yes	—
151	C₂²	#88257	1	3	yes	—
152	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
153	C₄	—	2	6	no	—
154	C₄	—	4	3	no	—
155	C₄	—	52	3	no	—
156	C₄	—	52	3	no	—
157	C₄	—	52	1	no	—
158	C₂	—	8	1	no	—
159	C₂	—	104	1	no	—
160	C₂	#11719416	1	1	yes	characteristic
161	C₂	#11719417	1	3	yes	—
162	C₂	#11719538	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#126

C₂

yes

maximal, characteristic, complemented

C₂×#88130

C₂

yes

maximal, characteristic, complemented

#11718655

C₂

yes

maximal, complemented

#11718645

C₂

yes

maximal, complemented

#11718622

C₂

yes

maximal

#11718609

C₂

yes

maximal, complemented

#11718596

C₂

yes

maximal, characteristic, complemented

C₂²×C₂²⋊D₁₃

C₂²

yes

characteristic

C₁₃×#126

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₁₃×C₂²⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₂²⋊Dic₁₃

C₂²

yes

—

C₂×#6103

C₂²

yes

—

C₂×#6103

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×#6101

C₂²

yes

—

#88130

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂⁴×C₁₃

C₂³

yes

characteristic

C₂³×D₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂×C₂²⋊D₁₃

—

C₂²×Dic₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

D₄

yes

complemented

C₂²⋊Dic₁₃

—

#6103

—

#6103

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

—

—

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×#126

—

C₂×#126

D₁₃

yes

characteristic, complemented

—

—

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

H(5,𝔽₂)

yes

—

C₂²×C₁₃

2₋¹⁺⁴

yes

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×D₄

—

#126

—

#126

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

#95

—

#95

—

C₂×C₁₃

—

100

C₂×C₁₃

#537

yes

—

101

C₂×C₁₃

#560

yes

characteristic

102

C₂×C₁₃

#563

yes

—

103

D₁₃

—

104

C₂⁴

—

105

C₂⁴

C₂²×D₁₃

yes

characteristic

106

C₂×D₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂²×D₁₃

yes

—

111

C₂²×C₄

C₂²×D₁₃

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₂²⋊C₄

—

118

C₂²⋊C₄

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

C₂²⋊D₁₃

yes

complemented

122

C₁₃

#2823

yes

characteristic, complemented, Sylow

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

C₂³×D₁₃

yes

characteristic

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

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

C₂×C₂²⋊D₁₃

yes

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

#88256

yes

—

151

C₂²

#88257

yes

—

152

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₂

—

159

C₂

—

104

—

160

C₂

#11719416

yes

characteristic

161

C₂

#11719417

yes

—

162

C₂

#11719538

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₂×C₁₃×#126	C₂
prime radical	C₂²×C₂²⋊D₁₃	C₂²
hypercenter	C₂×#126	D₁₃
hypocenter	C₁₃	#2823
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#126

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₂×C₁₃×#126

C₂

prime radical

C₂²×C₂²⋊D₁₃

C₂²

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₂×#126

N	H ≅ G/N	type
1. C₂×C₁₃×#126	C₂	semidirect
2. C₂×#88130	C₂	semidirect
3. #11718655	C₂	semidirect
4. #11718645	C₂	semidirect
6. #11718609	C₂	semidirect
7. #11718596	C₂	semidirect
33. C₄²×C₁₃	D₄	semidirect
64. C₂×#126	D₁₃	semidirect
121. C₄²	C₂²⋊D₁₃	semidirect
122. C₁₃	#2823	semidirect

H ≅ G/N

type

1. C₂×C₁₃×#126

C₂

semidirect

2. C₂×#88130

C₂

semidirect

3. #11718655

C₂

semidirect

4. #11718645

C₂

semidirect

6. #11718609

C₂

semidirect

7. #11718596

C₂

semidirect

33. C₄²×C₁₃

D₄

semidirect

64. C₂×#126

D₁₃

semidirect

121. C₄²

C₂²⋊D₁₃

semidirect

122. C₁₃

#2823

semidirect

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

subgroup	index
C₂×C₁₃×#126	2
C₂×#88130	2
#11718655	2
#11718645	2
#11718622	2
#11718609	2
#11718596	2
#2823	13

subgroup

index

C₂×C₁₃×#126

Subgroups — #13122908

Conjugacy classes of subgroups

Complements

Maximal subgroups