Group #13120777 — subgroups

Subgroups — #13120777

Order 1664.

500 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 149 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×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88129	C₂	1	1	yes	maximal, characteristic
3	#11718892	C₂	1	4	yes	maximal, complemented
4	#11718654	C₂	1	4	yes	maximal, complemented
5	#11718649	C₂	1	2	yes	maximal
6	#11718635	C₂	1	2	yes	maximal, complemented
7	#11718593	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
9	C₂²×D₅₂	C₂²	1	1	yes	characteristic
10	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
11	C₁₃×C₄⋊Q₈	—	2	4	no	—
12	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×#6103	C₂²	1	2	yes	—
16	C₂×#6103	C₂²	1	2	yes	—
17	C₂×#6103	C₂²	1	8	yes	—
18	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
19	#88129	—	2	4	no	—
20	#88071	C₂²	1	2	yes	—
21	#88071	C₂²	1	2	yes	—
22	#88071	C₂²	1	8	yes	—
23	C₂³×D₁₃	C₂³	1	2	yes	—
24	C₂×Q₈×C₁₃	—	2	4	no	—
25	C₂×Q₈×C₁₃	—	4	4	no	—
26	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
27	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
30	C₂²×Dic₁₃	C₂³	1	4	yes	—
31	C₂²×Dic₁₃	C₂³	1	2	yes	—
32	C₂×D₅₂	—	2	2	no	—
33	C₂×D₅₂	—	2	4	no	—
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	C₄²×C₁₃	D₄	1	4	yes	—
39	#6103	—	2	4	no	—
40	#6103	—	2	4	no	—
41	#6103	—	4	8	no	—
42	C₄×Dic₁₃	—	2	4	no	—
43	#2640	—	13	1	no	maximal, Sylow
44	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
45	C₂²×D₁₃	—	2	2	no	—
46	C₂²×D₁₃	—	2	4	no	—
47	C₂²×D₁₃	—	2	4	no	—
48	C₂²×D₁₃	—	2	4	no	—
49	Q₈×C₁₃	—	4	8	no	—
50	C₂×C₄×C₁₃	—	2	2	no	—
51	C₂×C₄×C₁₃	—	2	2	no	—
52	C₂×C₄×C₁₃	—	2	4	no	—
53	C₂×C₄×C₁₃	—	2	8	no	—
54	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
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	2	no	—
59	C₂×Dic₁₃	—	2	4	no	—
60	C₂×Dic₁₃	—	2	4	no	—
61	C₂×Dic₁₃	—	2	4	no	—
62	C₂×Dic₁₃	—	2	4	no	—
63	D₅₂	—	4	4	no	—
64	Dic₂₆	—	4	4	no	—
65	C₂×C₄⋊Q₈	D₁₃	1	1	yes	characteristic, complemented
66	C₂×#124	—	13	1	no	—
67	#397	—	13	4	no	—
68	#393	—	13	4	no	—
69	#391	—	13	2	no	—
70	#390	—	13	2	no	—
71	#376	—	13	1	no	—
72	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
73	C₂²×C₁₃	C₂×#41	1	2	yes	—
74	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
75	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
76	C₂×D₁₃	—	4	2	no	—
77	C₂×D₁₃	—	4	2	no	—
78	C₂×D₁₃	—	4	2	no	—
79	C₂×D₁₃	—	4	8	no	—
80	C₄×C₁₃	—	2	2	no	—
81	C₄×C₁₃	—	2	2	no	—
82	C₄×C₁₃	—	4	4	no	—
83	C₄×C₁₃	#124	1	4	yes	—
84	Dic₁₃	—	4	2	no	—
85	Dic₁₃	—	4	4	no	—
86	C₂²×Q₈	—	13	1	no	—
87	C₂²×Q₈	C₂×D₁₃	1	2	yes	—
88	C₂²×D₄	—	13	1	no	—
89	C₄⋊Q₈	—	2	4	no	—
90	#124	—	26	4	no	—
91	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
92	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
93	C₂×C₂²⋊C₄	—	13	2	no	—
94	C₂×C₂²⋊C₄	—	13	2	no	—
95	C₂×C₂²⋊C₄	—	13	8	no	—
96	C₂×C₄²	—	13	2	no	—
97	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
98	#95	—	13	2	no	—
99	#95	—	13	2	no	—
100	#95	—	13	8	no	—
101	C₂×C₁₃	C₂×#124	1	1	yes	characteristic
102	C₂×C₁₃	#535	1	1	yes	characteristic
103	C₂×C₁₃	#536	1	1	yes	characteristic
104	C₂×C₁₃	#542	1	4	yes	—
105	D₁₃	—	8	2	no	—
106	C₂⁴	—	13	2	no	—
107	C₂×Q₈	—	2	4	no	—
108	C₂×Q₈	—	4	4	no	—
109	C₂×Q₈	—	26	4	no	—
110	C₂×Q₈	—	26	2	no	—
111	C₂×D₄	—	26	4	no	—
112	C₂×D₄	—	26	2	no	—
113	C₂²×C₄	—	13	4	no	—
114	C₂²×C₄	—	13	2	no	—
115	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
116	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
117	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
118	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
119	C₄⋊C₄	—	2	4	no	—
120	C₄⋊C₄	—	2	4	no	—
121	C₂²⋊C₄	—	26	4	no	—
122	C₂²⋊C₄	—	26	4	no	—
123	C₂²⋊C₄	—	52	8	no	—
124	C₄²	—	26	4	no	—
125	C₄²	C₂²⋊D₁₃	1	4	yes	—
126	C₁₃	#2640	1	1	yes	characteristic, complemented, Sylow
127	C₂³	—	26	4	no	—
128	C₂³	—	26	4	no	—
129	C₂³	—	26	4	no	—
130	C₂³	—	26	2	no	—
131	C₂³	C₂³×D₁₃	1	1	yes	characteristic
132	Q₈	—	4	8	no	—
133	Q₈	—	52	4	no	—
134	D₄	—	52	4	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	8	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	26	4	no	—
140	C₂×C₄	—	26	4	no	—
141	C₂×C₄	—	26	4	no	—
142	C₂×C₄	—	26	4	no	—
143	C₂×C₄	—	26	2	no	—
144	C₂×C₄	#6131	1	4	yes	—
145	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
146	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
148	C₂²	—	52	8	no	—
149	C₂²	—	52	2	no	—
150	C₂²	—	52	2	no	—
151	C₂²	—	52	2	no	—
152	C₂²	#88251	1	2	yes	—
153	C₂²	C₂×#6131	1	2	yes	—
154	C₂²	#88254	1	2	yes	—
155	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
156	C₄	—	2	2	no	—
157	C₄	—	2	2	no	—
158	C₄	—	4	4	no	—
159	C₄	—	52	4	no	—
160	C₄	—	52	2	no	—
161	C₄	#88201	1	4	yes	—
162	C₂	—	104	2	no	—
163	C₂	#11719438	1	4	yes	—
164	C₂	#11719510	1	1	yes	characteristic
165	C₂	C₂×#88201	1	1	yes	characteristic
166	C₂	#11719521	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×C₄⋊Q₈

C₂

yes

maximal, characteristic, complemented

C₂×#88129

C₂

yes

maximal, characteristic

#11718892

C₂

yes

maximal, complemented

#11718654

C₂

yes

maximal, complemented

#11718649

C₂

yes

maximal

#11718635

C₂

yes

maximal, complemented

#11718593

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₁₃

C₂²

yes

—

C₂²×D₅₂

C₂²

yes

characteristic

C₂²×Dic₂₆

C₂²

yes

characteristic

C₁₃×C₄⋊Q₈

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×#6103

C₂²

yes

—

C₂×#6103

C₂²

yes

—

C₂×#6103

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

#88129

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂³×D₁₃

C₂³

yes

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

D₄

yes

—

#6103

—

#6103

—

#6103

—

C₄×Dic₁₃

—

#2640

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

Q₈×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

C₂×D₄

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

D₅₂

—

Dic₂₆

—

C₂×C₄⋊Q₈

D₁₃

yes

characteristic, complemented

—

—

—

—

—

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

—

C₂²×C₁₃

H(5,𝔽₂)

yes

—

C₂²×C₁₃

2₋¹⁺⁴

yes

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#124

yes

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₁₃

yes

—

C₂²×D₄

—

C₄⋊Q₈

—

#124

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

#95

—

#95

—

100

#95

—

101

C₂×C₁₃

C₂×#124

yes

characteristic

102

C₂×C₁₃

#535

yes

characteristic

103

C₂×C₁₃

#536

yes

characteristic

104

C₂×C₁₃

#542

yes

—

105

D₁₃

—

106

C₂⁴

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

C₂²×D₁₃

yes

—

116

C₂²×C₄

C₂²×D₁₃

yes

characteristic

117

C₂²×C₄

C₂²×D₁₃

yes

characteristic

118

C₂²×C₄

C₂²×D₁₃

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₂²⋊C₄

—

122

C₂²⋊C₄

—

123

C₂²⋊C₄

—

124

C₄²

—

125

C₄²

C₂²⋊D₁₃

yes

—

126

C₁₃

#2640

yes

characteristic, complemented, Sylow

127

C₂³

—

128

C₂³

—

129

C₂³

—

130

C₂³

—

131

C₂³

C₂³×D₁₃

yes

characteristic

132

Q₈

—

133

Q₈

—

134

D₄

—

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

#6131

yes

—

145

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

146

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

147

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

#88251

yes

—

153

C₂²

C₂×#6131

yes

—

154

C₂²

#88254

yes

—

155

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

#88201

yes

—

162

C₂

—

104

—

163

C₂

#11719438

yes

—

164

C₂

#11719510

yes

characteristic

165

C₂

C₂×#88201

yes

characteristic

166

C₂

#11719521

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₂×C₁₃×C₄⋊Q₈	C₂
prime radical	C₂²×D₅₂	C₂²
hypercenter	C₂×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	#2640
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₂×C₁₃×C₄⋊Q₈

C₂

prime radical

C₂²×D₅₂

C₂²

hypercenter

C₂×C₄⋊Q₈

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
1. C₂×C₁₃×C₄⋊Q₈	C₂	semidirect
3. #11718892	C₂	semidirect
4. #11718654	C₂	semidirect
6. #11718635	C₂	semidirect
7. #11718593	C₂	semidirect
65. C₂×C₄⋊Q₈	D₁₃	semidirect
126. C₁₃	#2640	semidirect

H ≅ G/N

type

1. C₂×C₁₃×C₄⋊Q₈

C₂

semidirect

3. #11718892

C₂

semidirect

4. #11718654

C₂

semidirect

6. #11718635

C₂

semidirect

7. #11718593

C₂

semidirect

65. C₂×C₄⋊Q₈

D₁₃

semidirect

126. C₁₃

#2640

semidirect

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

subgroup	index
C₂×C₁₃×C₄⋊Q₈	2
C₂×#88129	2
#11718892	2
#11718654	2
#11718649	2
#11718635	2
#11718593	2
#2640	13

subgroup

index

C₂×C₁₃×C₄⋊Q₈

Subgroups — #13120777

Conjugacy classes of subgroups

Complements

Maximal subgroups