Group #13119310 — subgroups

Subgroups — #13119310

Order 1664.

544 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 181 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	C₂²×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#88178	C₂	1	1	yes	maximal, characteristic
3	#11718871	C₂	1	4	yes	maximal
4	C₄×C₂²⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
5	#11718641	C₂	1	4	yes	maximal, complemented
6	#11718637	C₂	1	2	yes	maximal, complemented
7	#11718635	C₂	1	2	yes	maximal, complemented
8	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₂³×C₄×C₁₃	C₂²	1	2	yes	—
10	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
12	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₂²⋊Dic₁₃	C₂²	1	2	yes	—
15	C₂×C₂²⋊Dic₁₃	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
17	#88178	—	2	4	no	—
18	C₂×#6101	C₂²	1	2	yes	—
19	C₂×#6101	C₂²	1	8	yes	—
20	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
21	#88071	C₂²	1	2	yes	—
22	#88071	C₂²	1	8	yes	—
23	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₁₃	—	2	4	no	—
25	C₂²×C₄×C₁₃	—	2	4	no	—
26	C₂²×C₄×C₁₃	—	4	4	no	—
27	C₂²×C₄×C₁₃	D₄	1	4	yes	—
28	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
29	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×Dic₁₃	C₂³	1	2	yes	—
32	C₂²×Dic₁₃	C₂³	1	2	yes	—
33	C₂²×Dic₁₃	C₂³	1	4	yes	—
34	C₂×Dic₂₆	—	2	2	no	—
35	C₂×Dic₂₆	—	2	4	no	—
36	C₁₃×C₄⋊C₄	—	2	4	no	—
37	C₁₃×C₄⋊C₄	—	2	4	no	—
38	C₂²⋊Dic₁₃	—	2	4	no	—
39	C₂²⋊Dic₁₃	Q₈	1	8	yes	complemented
40	C₄⋊Dic₁₃	—	2	2	no	—
41	#6101	—	2	4	no	—
42	#6101	—	4	8	no	—
43	C₄×Dic₁₃	—	2	4	no	—
44	#2502	—	13	1	no	maximal, Sylow
45	C₂³×C₁₃	—	2	2	no	—
46	C₂³×C₁₃	—	2	2	no	—
47	C₂³×C₁₃	C₂×D₄	1	2	yes	—
48	C₂³×C₁₃	C₂×Q₈	1	4	yes	—
49	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
50	C₂×C₄×C₁₃	—	2	2	no	—
51	C₂×C₄×C₁₃	—	2	8	no	—
52	C₂×C₄×C₁₃	—	2	4	no	—
53	C₂×C₄×C₁₃	—	2	8	no	—
54	C₂×C₄×C₁₃	—	4	8	no	—
55	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₁₃	#41	1	4	yes	—
58	C₂×Dic₁₃	—	2	4	no	—
59	C₂×Dic₁₃	—	2	4	no	—
60	C₂×Dic₁₃	—	2	2	no	—
61	C₂×Dic₁₃	—	2	2	no	—
62	C₂×Dic₁₃	—	2	4	no	—
63	C₂×Dic₁₃	—	2	4	no	—
64	C₂×Dic₁₃	C₂×Q₈	1	8	yes	—
65	Dic₂₆	—	4	4	no	—
66	C₂×C₂²⋊Q₈	—	13	1	no	—
67	C₂²×C₄⋊C₄	D₁₃	1	1	yes	characteristic
68	#398	—	13	4	no	—
69	#391	—	13	2	no	—
70	#385	—	13	4	no	—
71	#384	—	13	2	no	—
72	C₄×C₂²⋊C₄	—	13	1	no	—
73	C₂²×C₁₃	—	2	2	no	—
74	C₂²×C₁₃	—	2	2	no	—
75	C₂²×C₁₃	—	2	8	no	—
76	C₂²×C₁₃	C₄⋊Q₈	1	4	yes	—
77	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
78	C₂²×C₁₃	C₂²×Q₈	1	2	yes	—
79	C₂²×C₁₃	C₂×#41	1	2	yes	—
80	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
81	C₄×C₁₃	—	2	2	no	—
82	C₄×C₁₃	—	4	4	no	—
83	C₄×C₁₃	#123	1	4	yes	—
84	Dic₁₃	—	2	8	no	—
85	Dic₁₃	—	4	2	no	—
86	Dic₁₃	—	4	2	no	—
87	C₂²×Q₈	—	13	1	no	—
88	C₂³×C₄	C₂×D₁₃	1	1	yes	characteristic
89	C₂³×C₄	C₂×D₁₃	1	2	yes	—
90	C₂²⋊Q₈	—	26	4	no	—
91	C₂×C₄⋊C₄	—	2	4	no	—
92	C₂×C₄⋊C₄	—	13	2	no	—
93	C₂×C₄⋊C₄	—	13	1	no	—
94	C₂×C₄⋊C₄	—	13	8	no	—
95	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
96	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
97	C₂×C₂²⋊C₄	—	13	2	no	—
98	C₂×C₂²⋊C₄	—	13	2	no	—
99	C₂×C₄²	—	13	2	no	—
100	#95	—	13	2	no	—
101	#95	—	13	8	no	—
102	C₂×C₁₃	—	2	4	no	—
103	C₂×C₁₃	C₂×#123	1	1	yes	characteristic
104	C₂×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
105	C₂×C₁₃	#535	1	1	yes	characteristic
106	C₂×C₁₃	#554	1	4	yes	—
107	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
108	C₂×Q₈	—	26	4	no	—
109	C₂×Q₈	—	26	2	no	—
110	C₂²×C₄	—	2	4	no	—
111	C₂²×C₄	—	2	4	no	—
112	C₂²×C₄	—	4	4	no	—
113	C₂²×C₄	—	13	4	no	—
114	C₂²×C₄	—	13	2	no	—
115	C₂²×C₄	—	13	2	no	—
116	C₂²×C₄	C₂²⋊D₁₃	1	4	yes	—
117	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
118	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
119	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	2	4	no	—
122	C₄⋊C₄	—	26	4	no	—
123	C₄⋊C₄	—	26	2	no	—
124	C₄⋊C₄	—	52	8	no	—
125	C₂²⋊C₄	—	13	8	no	—
126	C₂²⋊C₄	—	26	4	no	—
127	C₄²	—	26	4	no	—
128	C₁₃	#2502	1	1	yes	characteristic, complemented, Sylow
129	C₂³	—	2	2	no	—
130	C₂³	—	2	2	no	—
131	C₂³	Q₈×D₁₃	1	4	yes	—
132	C₂³	C₂×C₂²⋊D₁₃	1	2	yes	—
133	C₂³	C₂³×D₁₃	1	1	yes	characteristic
134	Q₈	—	52	4	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	8	no	—
137	C₂×C₄	—	2	8	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	4	8	no	—
140	C₂×C₄	—	13	8	no	—
141	C₂×C₄	—	26	4	no	—
142	C₂×C₄	—	26	4	no	—
143	C₂×C₄	—	26	4	no	—
144	C₂×C₄	—	26	4	no	—
145	C₂×C₄	—	26	2	no	—
146	C₂×C₄	—	26	2	no	—
147	C₂×C₄	#6127	1	4	yes	—
148	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
149	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
150	C₂²	—	2	8	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	2	2	no	—
153	C₂²	#88198	1	4	yes	—
154	C₂²	C₂×#6127	1	2	yes	—
155	C₂²	#88251	1	2	yes	—
156	C₂²	C₂×Q₈×D₁₃	1	2	yes	—
157	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	4	4	no	—
160	C₄	—	26	8	no	—
161	C₄	—	52	2	no	—
162	C₄	—	52	2	no	—
163	C₄	#88183	1	4	yes	—
164	C₂	—	2	4	no	—
165	C₂	#11719216	1	4	yes	—
166	C₂	C₂×#88183	1	1	yes	characteristic
167	C₂	#11719510	1	1	yes	characteristic
168	C₂	C₂×#88198	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₃×C₄⋊C₄

C₂

yes

maximal, characteristic

C₂×#88178

C₂

yes

maximal, characteristic

#11718871

C₂

yes

maximal

C₄×C₂²⋊Dic₁₃

C₂

yes

maximal, characteristic

#11718641

C₂

yes

maximal, complemented

#11718637

C₂

yes

maximal, complemented

#11718635

C₂

yes

maximal, complemented

C₂³×C₄×C₁₃

C₂²

yes

characteristic

C₂³×C₄×C₁₃

C₂²

yes

—

C₂²×Dic₂₆

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₁₃

C₂²

yes

—

C₂×C₂²⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

#88178

—

C₂×#6101

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂⁴×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₂²⋊Dic₁₃

—

C₂²⋊Dic₁₃

Q₈

yes

complemented

C₄⋊Dic₁₃

—

#6101

—

#6101

—

C₄×Dic₁₃

—

#2502

—

maximal, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂×D₄

yes

—

C₂³×C₁₃

C₂×Q₈

yes

—

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×Q₈

yes

—

Dic₂₆

—

C₂×C₂²⋊Q₈

—

C₂²×C₄⋊C₄

D₁₃

yes

—

—

—

—

C₄×C₂²⋊C₄

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₄⋊Q₈

yes

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

—

C₂²×C₁₃

C₂×#41

yes

—

C₂²×C₁₃

H(5,𝔽₂)

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#123

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂³×C₄

C₂×D₁₃

yes

characteristic

C₂³×C₄

C₂×D₁₃

yes

—

C₂²⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

100

#95

—

101

#95

—

102

C₂×C₁₃

—

103

C₂×C₁₃

C₂×#123

yes

characteristic

104

C₂×C₁₃

C₂×C₄⋊Q₈

yes

characteristic

105

C₂×C₁₃

#535

yes

characteristic

106

C₂×C₁₃

#554

yes

—

107

C₂⁴

C₂²×D₁₃

yes

characteristic

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

C₂²⋊D₁₃

yes

—

117

C₂²×C₄

C₂²×D₁₃

yes

—

118

C₂²×C₄

C₂²×D₁₃

yes

characteristic

119

C₂²×C₄

C₂²×D₁₃

yes

characteristic

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₂²⋊C₄

—

126

C₂²⋊C₄

—

127

C₄²

—

128

C₁₃

#2502

yes

characteristic, complemented, Sylow

129

C₂³

—

130

C₂³

—

131

C₂³

Q₈×D₁₃

yes

—

132

C₂³

C₂×C₂²⋊D₁₃

yes

—

133

C₂³

C₂³×D₁₃

yes

characteristic

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

#6127

yes

—

148

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

149

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

#88198

yes

—

154

C₂²

C₂×#6127

yes

—

155

C₂²

#88251

yes

—

156

C₂²

C₂×Q₈×D₁₃

yes

—

157

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#88183

yes

—

164

C₂

—

165

C₂

#11719216

yes

—

166

C₂

C₂×#88183

yes

characteristic

167

C₂

#11719510

yes

characteristic

168

C₂

C₂×#88198

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₂²×C₁₃×C₄⋊C₄	C₂
prime radical	C₂⁴×C₁₃	C₂³
hypercenter	C₂²×C₄⋊C₄	D₁₃
hypocenter	C₁₃	#2502
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₂²×C₁₃×C₄⋊C₄

C₂

prime radical

C₂⁴×C₁₃

C₂³

hypercenter

C₂²×C₄⋊C₄

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
5. #11718641	C₂	semidirect
6. #11718637	C₂	semidirect
7. #11718635	C₂	semidirect
39. C₂²⋊Dic₁₃	Q₈	semidirect
128. C₁₃	#2502	semidirect

H ≅ G/N

type

5. #11718641

C₂

semidirect

6. #11718637

C₂

semidirect

7. #11718635

C₂

semidirect

39. C₂²⋊Dic₁₃

Q₈

semidirect

128. C₁₃

#2502

semidirect

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

subgroup	index
C₂²×C₁₃×C₄⋊C₄	2
C₂×#88178	2
#11718871	2
C₄×C₂²⋊Dic₁₃	2
#11718641	2
#11718637	2
#11718635	2
#2502	13

subgroup

index

C₂²×C₁₃×C₄⋊C₄

C₄×C₂²⋊Dic₁₃

Subgroups — #13119310

Conjugacy classes of subgroups

Complements

Maximal subgroups