Group #13120944 — subgroups

Subgroups — #13120944

Order 1664.

844 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 844 conjugacy classes of subgroups — too many to draw. Only its 165 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×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88201	C₂	1	2	yes	maximal, complemented
3	C₂×#88194	C₂	1	2	yes	maximal, complemented
4	C₂×#88128	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718621	C₂	1	8	yes	maximal, complemented
6	#11718593	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×C₂²⋊D₁₃	C₂²	1	8	yes	—
10	C₂²×D₅₂	C₂²	1	2	yes	—
11	C₂²×D₅₂	C₂²	1	4	yes	—
12	C₁₃×#124	—	2	4	no	—
13	C₂×C₁₃×C₂²⋊C₄	C₂²	1	4	yes	—
14	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
15	#88201	—	2	8	no	—
16	#88194	—	2	8	no	—
17	C₂×#6103	C₂²	1	8	yes	—
18	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
19	#88128	—	2	4	no	—
20	#88071	C₂²	1	4	yes	—
21	C₂⁴×C₁₃	C₂³	1	2	yes	—
22	C₂³×D₁₃	C₂³	1	4	yes	—
23	C₂×Q₈×C₁₃	—	2	4	no	—
24	C₂×Q₈×C₁₃	—	2	2	no	—
25	C₂×D₄×C₁₃	—	2	4	no	—
26	C₂×D₄×C₁₃	—	2	2	no	—
27	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
28	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
29	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
30	C₂×C₂²⋊D₁₃	—	2	16	no	—
31	C₂×C₂²⋊D₁₃	—	4	16	no	—
32	C₂²×Dic₁₃	C₂³	1	4	yes	—
33	C₂×D₅₂	—	2	8	no	—
34	C₂×D₅₂	—	2	4	no	—
35	C₂×D₅₂	—	2	4	no	—
36	C₂×D₅₂	—	2	4	no	—
37	C₂×D₅₂	—	4	8	no	—
38	C₁₃×C₂²⋊C₄	—	2	8	no	—
39	C₄²×C₁₃	D₄	1	4	yes	complemented
40	#6103	—	2	16	no	—
41	C₄×Dic₁₃	D₄	1	8	yes	complemented
42	#2657	—	13	1	no	maximal, Sylow
43	C₂³×C₁₃	—	2	2	no	—
44	C₂³×C₁₃	—	2	4	no	—
45	C₂³×C₁₃	—	2	8	no	—
46	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
47	C₂²×D₁₃	—	2	4	no	—
48	C₂²×D₁₃	—	2	4	no	—
49	C₂²×D₁₃	—	2	4	no	—
50	C₂²×D₁₃	—	2	16	no	—
51	Q₈×C₁₃	—	4	4	no	—
52	D₄×C₁₃	—	4	4	no	—
53	C₂×C₄×C₁₃	—	2	4	no	—
54	C₂×C₄×C₁₃	—	2	4	no	—
55	C₂×C₄×C₁₃	—	2	2	no	—
56	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
57	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
58	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
59	C₂²⋊D₁₃	—	4	32	no	—
60	C₂×Dic₁₃	—	2	8	no	—
61	C₂×Dic₁₃	C₂×D₄	1	8	yes	—
62	D₅₂	—	4	16	no	—
63	D₅₂	—	4	8	no	—
64	C₂×#127	—	13	2	no	—
65	C₂×#127	—	13	1	no	—
66	C₂×#124	—	13	2	no	—
67	C₂×#124	D₁₃	1	1	yes	characteristic, complemented
68	#392	—	13	8	no	—
69	#376	—	13	1	no	—
70	C₂²×C₁₃	—	4	2	no	—
71	C₂²×C₁₃	—	4	8	no	—
72	C₂²×C₁₃	—	4	4	no	—
73	C₂²×C₁₃	C₂²×D₄	1	2	yes	—
74	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
75	C₂²×C₁₃	H(5,𝔽₂)	1	4	yes	—
76	C₂×D₁₃	—	4	4	no	—
77	C₂×D₁₃	—	4	8	no	—
78	C₂×D₁₃	—	4	8	no	—
79	C₂×D₁₃	—	4	8	no	—
80	C₄×C₁₃	—	2	4	no	—
81	C₄×C₁₃	—	4	2	no	—
82	C₄×C₁₃	#127	1	4	yes	—
83	Dic₁₃	—	2	8	no	—
84	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
85	C₂²×D₄	—	13	8	no	—
86	C₂²×D₄	—	13	2	no	—
87	C₂²×D₄	—	13	4	no	—
88	C₂²×D₄	C₂×D₁₃	1	1	yes	characteristic
89	#127	—	26	8	no	—
90	#127	—	26	4	no	—
91	#124	—	2	4	no	—
92	#124	—	26	8	no	—
93	C₂×C₂²⋊C₄	—	13	8	no	—
94	C₂×C₂²⋊C₄	C₂×D₁₃	1	4	yes	—
95	C₂×C₄²	—	13	2	no	—
96	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
97	#95	—	13	4	no	—
98	C₂×C₁₃	—	8	2	no	—
99	C₂×C₁₃	C₂×#127	1	1	yes	characteristic
100	C₂×C₁₃	#535	1	2	yes	—
101	C₂×C₁₃	#535	1	4	yes	—
102	D₁₃	—	8	4	no	—
103	C₂⁴	—	13	4	no	—
104	C₂⁴	C₂²×D₁₃	1	2	yes	—
105	C₂×Q₈	—	2	4	no	—
106	C₂×Q₈	—	2	2	no	—
107	C₂×D₄	—	2	4	no	—
108	C₂×D₄	—	2	2	no	—
109	C₂×D₄	—	26	16	no	—
110	C₂×D₄	—	26	4	no	—
111	C₂×D₄	—	26	4	no	—
112	C₂×D₄	—	26	8	no	—
113	C₂×D₄	—	26	4	no	—
114	C₂×D₄	—	52	8	no	—
115	C₂×D₄	—	52	16	no	—
116	C₂²×C₄	—	13	4	no	—
117	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
118	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
119	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
120	C₂²⋊C₄	—	2	8	no	—
121	C₂²⋊C₄	—	26	16	no	—
122	C₄²	—	13	8	no	—
123	C₄²	C₂²⋊D₁₃	1	4	yes	complemented
124	C₁₃	#2657	1	1	yes	characteristic, complemented, Sylow
125	C₂³	—	2	4	no	—
126	C₂³	—	2	8	no	—
127	C₂³	—	2	2	no	—
128	C₂³	—	26	4	no	—
129	C₂³	—	26	4	no	—
130	C₂³	—	26	16	no	—
131	C₂³	—	26	4	no	—
132	C₂³	C₂³×D₁₃	1	1	yes	characteristic
133	Q₈	—	4	4	no	—
134	D₄	—	4	4	no	—
135	D₄	—	52	16	no	—
136	D₄	—	52	32	no	—
137	D₄	—	52	8	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	13	8	no	—
142	C₂×C₄	—	26	8	no	—
143	C₂×C₄	D₄×D₁₃	1	4	yes	—
144	C₂×C₄	C₂×C₂²⋊D₁₃	1	4	yes	—
145	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
146	C₂²	—	4	2	no	—
147	C₂²	—	4	4	no	—
148	C₂²	—	4	8	no	—
149	C₂²	—	52	4	no	—
150	C₂²	—	52	8	no	—
151	C₂²	—	52	8	no	—
152	C₂²	—	52	8	no	—
153	C₂²	C₂×D₄×D₁₃	1	2	yes	—
154	C₂²	#88256	1	4	yes	—
155	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
156	C₄	—	2	4	no	—
157	C₄	—	4	2	no	—
158	C₄	—	26	8	no	—
159	C₄	#88194	1	4	yes	—
160	C₂	—	8	2	no	—
161	C₂	—	104	4	no	—
162	C₂	#11719379	1	4	yes	—
163	C₂	C₂×#88194	1	1	yes	characteristic
164	C₂	#11719535	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#124

C₂

yes

maximal, characteristic, complemented

C₂×#88201

C₂

yes

maximal, complemented

C₂×#88194

C₂

yes

maximal, complemented

C₂×#88128

C₂

yes

maximal, characteristic, complemented

#11718621

C₂

yes

maximal, complemented

#11718593

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₁₃

C₂²

yes

characteristic

C₂²×D₄×C₁₃

C₂²

yes

characteristic

C₂²×C₂²⋊D₁₃

C₂²

yes

—

C₂²×D₅₂

C₂²

yes

—

C₂²×D₅₂

C₂²

yes

—

C₁₃×#124

—

C₂×C₁₃×C₂²⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

—

—

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

#88128

—

#88071

C₂²

yes

—

C₂⁴×C₁₃

C₂³

yes

—

C₂³×D₁₃

C₂³

yes

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂×C₂²⋊D₁₃

—

C₂×C₂²⋊D₁₃

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

D₄

yes

complemented

#6103

—

C₄×Dic₁₃

D₄

yes

complemented

#2657

—

maximal, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

Q₈×C₁₃

—

D₄×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

C₂×D₄

yes

—

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×D₄

yes

—

D₅₂

—

D₅₂

—

—

—

—

D₁₃

yes

characteristic, complemented

#392

—

#376

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂²×D₄

yes

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

H(5,𝔽₂)

yes

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#127

yes

—

Dic₁₃

—

C₂²×Q₈

C₂×D₁₃

yes

characteristic

C₂²×D₄

—

C₂²×D₄

—

C₂²×D₄

—

C₂²×D₄

C₂×D₁₃

yes

—

—

—

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

#95

—

C₂×C₁₃

—

C₂×C₁₃

C₂×#127

yes

characteristic

100

C₂×C₁₃

#535

yes

—

101

C₂×C₁₃

#535

yes

—

102

D₁₃

—

103

C₂⁴

—

104

C₂⁴

C₂²×D₁₃

yes

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₂²×D₁₃

yes

—

118

C₂²×C₄

C₂²×D₁₃

yes

—

119

C₂²×C₄

C₂²×D₁₃

yes

characteristic

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

122

C₄²

—

123

C₄²

C₂²⋊D₁₃

yes

complemented

124

C₁₃

#2657

yes

characteristic, complemented, Sylow

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

—

130

C₂³

—

131

C₂³

—

132

C₂³

C₂³×D₁₃

yes

characteristic

133

Q₈

—

134

D₄

—

135

D₄

—

136

D₄

—

137

D₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

D₄×D₁₃

yes

—

144

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

145

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

C₂×D₄×D₁₃

yes

—

154

C₂²

#88256

yes

—

155

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

#88194

yes

—

160

C₂

—

161

C₂

—

104

—

162

C₂

#11719379

yes

—

163

C₂

C₂×#88194

yes

characteristic

164

C₂

#11719535

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₂×C₁₃×#124	C₂
hypercenter	C₂×#124	D₁₃
hypocenter	C₁₃	#2657
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#124

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₂×C₁₃×#124

C₂

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₂×#124

N	H ≅ G/N	type
1. C₂×C₁₃×#124	C₂	semidirect
2. C₂×#88201	C₂	semidirect
3. C₂×#88194	C₂	semidirect
4. C₂×#88128	C₂	semidirect
5. #11718621	C₂	semidirect
6. #11718593	C₂	semidirect
39. C₄²×C₁₃	D₄	semidirect
41. C₄×Dic₁₃	D₄	semidirect
67. C₂×#124	D₁₃	semidirect
123. C₄²	C₂²⋊D₁₃	semidirect
124. C₁₃	#2657	semidirect

H ≅ G/N

type

1. C₂×C₁₃×#124

C₂

semidirect

2. C₂×#88201

C₂

semidirect

3. C₂×#88194

C₂

semidirect

4. C₂×#88128

C₂

semidirect

5. #11718621

C₂

semidirect

6. #11718593

C₂

semidirect

39. C₄²×C₁₃

D₄

semidirect

41. C₄×Dic₁₃

D₄

semidirect

67. C₂×#124

D₁₃

semidirect

123. C₄²

C₂²⋊D₁₃

semidirect

124. C₁₃

#2657

semidirect

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

subgroup	index
C₂×C₁₃×#124	2
C₂×#88201	2
C₂×#88194	2
C₂×#88128	2
#11718621	2
#11718593	2
#2657	13

subgroup

index

C₂×C₁₃×#124

Subgroups — #13120944

Conjugacy classes of subgroups

Complements

Maximal subgroups