Group #13111911 — subgroups

Subgroups — C₁₃×#2853 #13111911

Order 1664.

582 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 216 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×#123	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄²×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#388	C₂	1	4	yes	maximal, complemented
4	C₁₃×#387	C₂	1	4	yes	maximal, complemented
5	C₁₃×#386	C₂	1	2	yes	maximal
6	C₁₃×#385	C₂	1	2	yes	maximal, complemented
7	C₁₃×#381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₂³×C₄×C₁₃	C₂²	1	2	yes	—
10	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₁₃×#123	—	2	4	no	—
12	C₂×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
13	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
15	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×C₂²⋊C₄	C₂²	1	4	yes	—
17	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
18	C₂×C₄²×C₁₃	—	2	4	no	—
19	C₂×C₄²×C₁₃	C₂²	1	4	yes	—
20	C₁₃×#95	C₂²	1	4	yes	—
21	C₁₃×#95	C₂²	1	4	yes	—
22	C₁₃×#95	C₂²	1	4	yes	—
23	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
24	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
25	C₂×D₄×C₁₃	—	2	2	no	—
26	C₂×D₄×C₁₃	—	2	4	no	—
27	C₂²×C₄×C₁₃	—	2	4	no	—
28	C₂²×C₄×C₁₃	—	2	4	no	—
29	C₂²×C₄×C₁₃	—	2	8	no	—
30	C₂²×C₄×C₁₃	D₄	1	4	yes	—
31	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
32	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
34	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
35	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
36	C₁₃×C₄⋊C₄	—	2	4	no	—
37	C₁₃×C₄⋊C₄	—	2	8	no	—
38	C₁₃×C₂²⋊C₄	—	2	2	no	—
39	C₁₃×C₂²⋊C₄	—	2	4	no	—
40	C₁₃×C₂²⋊C₄	—	2	4	no	—
41	C₁₃×C₂²⋊C₄	—	2	8	no	—
42	C₄²×C₁₃	—	2	8	no	—
43	#2853	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂³×C₁₃	—	2	4	no	—
45	C₂³×C₁₃	—	2	1	no	—
46	C₂³×C₁₃	—	2	4	no	—
47	C₂³×C₁₃	—	2	2	no	—
48	C₂³×C₁₃	C₂×D₄	1	2	yes	—
49	C₂³×C₁₃	#41	1	4	yes	—
50	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
51	D₄×C₁₃	—	4	4	no	—
52	C₂×C₄×C₁₃	—	2	16	no	—
53	C₂×C₄×C₁₃	—	2	8	no	—
54	C₂×C₄×C₁₃	—	2	4	no	—
55	C₂×C₄×C₁₃	—	2	4	no	—
56	C₂×C₄×C₁₃	—	2	4	no	—
57	C₂×C₄×C₁₃	—	2	2	no	—
58	C₂×C₄×C₁₃	—	2	1	no	—
59	C₂×C₄×C₁₃	—	2	4	no	—
60	C₂×C₄×C₁₃	—	2	4	no	—
61	C₂×C₄×C₁₃	—	2	4	no	—
62	C₂×C₄×C₁₃	—	2	4	no	—
63	C₂×C₄×C₁₃	—	2	2	no	—
64	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
65	C₂×C₄×C₁₃	#41	1	8	yes	—
66	C₂×#123	C₂×C₁₃	1	1	yes	characteristic
67	C₂²×C₄²	C₂×C₁₃	1	1	yes	characteristic, complemented
68	#388	C₂×C₁₃	1	4	yes	complemented
69	#387	C₂×C₁₃	1	4	yes	complemented
70	#386	C₂×C₁₃	1	2	yes	—
71	#385	C₂×C₁₃	1	2	yes	complemented
72	#381	C₂×C₁₃	1	1	yes	characteristic, complemented
73	C₂²×C₁₃	—	2	4	no	—
74	C₂²×C₁₃	—	2	8	no	—
75	C₂²×C₁₃	—	4	1	no	—
76	C₂²×C₁₃	—	4	2	no	—
77	C₂²×C₁₃	—	4	4	no	—
78	C₂²×C₁₃	#123	1	4	yes	—
79	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
80	C₂²×C₁₃	C₂×#41	1	2	yes	—
81	C₂²×C₁₃	C₂×#41	1	4	yes	—
82	C₄×C₁₃	—	2	4	no	—
83	C₄×C₁₃	—	2	8	no	—
84	C₄×C₁₃	—	4	1	no	—
85	C₄×C₁₃	—	4	2	no	—
86	C₄×C₁₃	—	4	4	no	—
87	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
88	C₂³×C₄	C₂²×C₁₃	1	2	yes	—
89	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
90	#123	—	2	4	no	—
91	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
92	C₂×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
93	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
94	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
95	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
96	C₂×C₂²⋊C₄	C₂²×C₁₃	1	4	yes	—
97	C₂×C₄²	—	2	4	no	—
98	C₂×C₄²	C₂²×C₁₃	1	4	yes	—
99	#95	C₂²×C₁₃	1	4	yes	—
100	#95	C₂²×C₁₃	1	4	yes	—
101	#95	C₂²×C₁₃	1	4	yes	—
102	C₂×C₁₃	—	2	4	no	—
103	C₂×C₁₃	—	8	1	no	—
104	C₂×C₁₃	C₂×#123	1	1	yes	characteristic
105	C₂×C₁₃	#525	1	2	yes	—
106	C₂×C₁₃	#529	1	4	yes	—
107	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
108	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
109	C₂×D₄	—	2	2	no	—
110	C₂×D₄	—	2	4	no	—
111	C₂²×C₄	—	2	4	no	—
112	C₂²×C₄	—	2	4	no	—
113	C₂²×C₄	—	2	8	no	—
114	C₂²×C₄	D₄×C₁₃	1	4	yes	—
115	C₂²×C₄	C₂³×C₁₃	1	4	yes	—
116	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
117	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
118	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
119	C₂²×C₄	C₂³×C₁₃	1	4	yes	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	2	8	no	—
122	C₂²⋊C₄	—	2	2	no	—
123	C₂²⋊C₄	—	2	4	no	—
124	C₂²⋊C₄	—	2	4	no	—
125	C₂²⋊C₄	—	2	8	no	—
126	C₄²	—	2	8	no	—
127	C₁₃	#2853	1	1	yes	characteristic, direct product factor, Sylow
128	C₂³	—	2	4	no	—
129	C₂³	—	2	1	no	—
130	C₂³	—	2	4	no	—
131	C₂³	—	2	2	no	—
132	C₂³	C₂×D₄×C₁₃	1	2	yes	—
133	C₂³	C₁₃×#41	1	4	yes	—
134	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
135	D₄	—	4	4	no	—
136	C₂×C₄	—	2	16	no	—
137	C₂×C₄	—	2	8	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	C₂×D₄×C₁₃	1	4	yes	—
149	C₂×C₄	C₁₃×#41	1	8	yes	—
150	C₂²	—	2	4	no	—
151	C₂²	—	2	8	no	—
152	C₂²	—	4	1	no	—
153	C₂²	—	4	2	no	—
154	C₂²	—	4	4	no	—
155	C₂²	C₁₃×#123	1	4	yes	—
156	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
157	C₂²	C₂×C₁₃×#41	1	2	yes	—
158	C₂²	C₂×C₁₃×#41	1	4	yes	—
159	C₄	—	2	4	no	—
160	C₄	—	2	8	no	—
161	C₄	—	4	1	no	—
162	C₄	—	4	2	no	—
163	C₄	—	4	4	no	—
164	C₂	—	2	4	no	—
165	C₂	—	8	1	no	—
166	C₂	C₂×C₁₃×#123	1	1	yes	characteristic
167	C₂	C₁₃×#525	1	2	yes	—
168	C₂	C₁₃×#529	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#123

C₂

yes

maximal, characteristic

C₂²×C₄²×C₁₃

C₂

yes

maximal, characteristic, complemented

C₁₃×#388

C₂

yes

maximal, complemented

C₁₃×#387

C₂

yes

maximal, complemented

C₁₃×#386

C₂

yes

maximal

C₁₃×#385

C₂

yes

maximal, complemented

C₁₃×#381

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₁₃

C₂²

yes

characteristic

C₂³×C₄×C₁₃

C₂²

yes

—

C₂³×C₄×C₁₃

C₂²

yes

characteristic

C₁₃×#123

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₁₃×C₂²⋊C₄

C₂²

yes

—

C₂×C₁₃×C₂²⋊C₄

C₂²

yes

characteristic

C₂×C₁₃×C₂²⋊C₄

C₂²

yes

—

C₂×C₁₃×C₂²⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

—

C₂×C₄²×C₁₃

C₂²

yes

—

C₁₃×#95

C₂²

yes

—

C₁₃×#95

C₂²

yes

—

C₁₃×#95

C₂²

yes

—

C₂⁴×C₁₃

C₂³

yes

characteristic

C₂⁴×C₁₃

C₂³

yes

characteristic

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

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

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

#2853

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂×D₄

yes

—

C₂³×C₁₃

#41

yes

—

C₂³×C₁₃

C₂⁴

yes

characteristic

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×#123

C₂×C₁₃

yes

characteristic

C₂²×C₄²

C₂×C₁₃

yes

characteristic, complemented

#388

C₂×C₁₃

yes

complemented

#387

C₂×C₁₃

yes

complemented

#386

C₂×C₁₃

yes

—

#385

C₂×C₁₃

yes

complemented

#381

C₂×C₁₃

yes

characteristic, complemented

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#123

yes

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

—

C₂²×C₁₃

C₂×#41

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₂²×D₄

C₂²×C₁₃

yes

characteristic

C₂³×C₄

C₂²×C₁₃

yes

—

C₂³×C₄

C₂²×C₁₃

yes

characteristic

#123

—

C₂×C₄⋊C₄

C₂²×C₁₃

yes

—

C₂×C₄⋊C₄

C₂²×C₁₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₁₃

yes

characteristic

C₂×C₂²⋊C₄

C₂²×C₁₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₁₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₁₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×C₁₃

yes

—

#95

C₂²×C₁₃

yes

—

100

#95

C₂²×C₁₃

yes

—

101

#95

C₂²×C₁₃

yes

—

102

C₂×C₁₃

—

103

C₂×C₁₃

—

104

C₂×C₁₃

C₂×#123

yes

characteristic

105

C₂×C₁₃

#525

yes

—

106

C₂×C₁₃

#529

yes

—

107

C₂⁴

C₂³×C₁₃

yes

characteristic

108

C₂⁴

C₂³×C₁₃

yes

characteristic

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

D₄×C₁₃

yes

—

115

C₂²×C₄

C₂³×C₁₃

yes

—

116

C₂²×C₄

C₂³×C₁₃

yes

characteristic

117

C₂²×C₄

C₂³×C₁₃

yes

—

118

C₂²×C₄

C₂³×C₁₃

yes

—

119

C₂²×C₄

C₂³×C₁₃

yes

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₂²⋊C₄

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

125

C₂²⋊C₄

—

126

C₄²

—

127

C₁₃

#2853

yes

characteristic, direct product factor, Sylow

128

C₂³

—

129

C₂³

—

130

C₂³

—

131

C₂³

—

132

C₂³

C₂×D₄×C₁₃

yes

—

133

C₂³

C₁₃×#41

yes

—

134

C₂³

C₂⁴×C₁₃

yes

characteristic

135

D₄

—

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

C₂×D₄×C₁₃

yes

—

149

C₂×C₄

C₁₃×#41

yes

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

C₁₃×#123

yes

—

156

C₂²

C₂²×D₄×C₁₃

yes

characteristic

157

C₂²

C₂×C₁₃×#41

yes

—

158

C₂²

C₂×C₁₃×#41

yes

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₂

—

165

C₂

—

166

C₂

C₂×C₁₃×#123

yes

characteristic

167

C₂

C₁₃×#525

yes

—

168

C₂

C₁₃×#529

yes

—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃
prime radical	C₂²×D₄×C₁₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ G

subgroup

G/H

center, socle

C₂³×C₁₃

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₁₃

prime radical

C₂²×D₄×C₁₃

C₂²

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₃ ◁ G

N	H ≅ G/N	type
2. C₂²×C₄²×C₁₃	C₂	semidirect
3. C₁₃×#388	C₂	semidirect
4. C₁₃×#387	C₂	semidirect
6. C₁₃×#385	C₂	semidirect
7. C₁₃×#381	C₂	semidirect
43. #2853	C₁₃	direct (N × H)
67. C₂²×C₄²	C₂×C₁₃	semidirect
68. #388	C₂×C₁₃	semidirect
69. #387	C₂×C₁₃	semidirect
71. #385	C₂×C₁₃	semidirect
72. #381	C₂×C₁₃	semidirect
127. C₁₃	#2853	direct (N × H)

H ≅ G/N

type

2. C₂²×C₄²×C₁₃

C₂

semidirect

3. C₁₃×#388

C₂

semidirect

4. C₁₃×#387

C₂

semidirect

6. C₁₃×#385

C₂

semidirect

7. C₁₃×#381

C₂

semidirect

43. #2853

C₁₃

direct (N × H)

67. C₂²×C₄²

C₂×C₁₃

semidirect

68. #388

C₂×C₁₃

semidirect

69. #387

C₂×C₁₃

semidirect

71. #385

C₂×C₁₃

semidirect

72. #381

C₂×C₁₃

semidirect

127. C₁₃

#2853

direct (N × H)

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

subgroup	index
C₂×C₁₃×#123	2
C₂²×C₄²×C₁₃	2
C₁₃×#388	2
C₁₃×#387	2
C₁₃×#386	2
C₁₃×#385	2
C₁₃×#381	2
#2853	13

subgroup

index

C₂×C₁₃×#123

C₂²×C₄²×C₁₃

C₁₃×#388

C₁₃×#387

C₁₃×#386

C₁₃×#385

C₁₃×#381

#2853

Subgroups — C₁₃×#2853 #13111911

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C13×#2853 #13111911

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2853 #13111911