Group #13077549 — subgroups

Subgroups — #13077549

Order 1408.

364 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 185 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	#10623048	C₂	1	1	yes	maximal, characteristic
2	C₁₁×#424	C₂	1	1	yes	maximal, characteristic
3	#10622735	C₂	1	1	yes	maximal, characteristic
4	#10622731	C₂	1	1	yes	maximal, characteristic
5	#10622715	C₂	1	1	yes	maximal, characteristic
6	C₂×C₈×Dic₁₁	C₂	1	1	yes	maximal, characteristic
7	#10622610	C₂	1	1	yes	maximal, characteristic
8	#10622443	C₂	1	4	yes	maximal, complemented
9	#10622441	C₂	1	4	yes	maximal, complemented
10	C₂×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₁₁	C₂²	1	1	yes	characteristic
12	C₁₁×#117	C₂²	1	1	yes	characteristic
13	#66979	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₁₁	C₂²	1	1	yes	characteristic
15	C₂×#4318	C₂²	1	1	yes	characteristic
16	C₂²×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
17	#66943	C₄	1	2	yes	—
18	#66941	C₄	1	2	yes	—
19	#66931	C₄	1	4	yes	—
20	C₄×Dic₂₂	C₂²	1	4	yes	—
21	C₁₁×C₄⋊C₈	C₂²	1	4	yes	—
22	#66879	C₂²	1	2	yes	—
23	Dic₁₁⋊C₈	C₂²	1	4	yes	—
24	Dic₁₁⋊C₈	C₂²	1	4	yes	—
25	C₈×Dic₁₁	C₂²	1	2	yes	—
26	C₈×Dic₁₁	C₂²	1	2	yes	—
27	C₈×Dic₁₁	C₂²	1	2	yes	—
28	#66867	C₂²	1	2	yes	—
29	C₄×C₁₁⋊C₈	C₂²	1	2	yes	—
30	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
31	C₂²×Dic₁₁	C₂×C₄	1	2	yes	—
32	C₂×Dic₂₂	C₂×C₄	1	4	yes	—
33	C₁₁×M₁₆	—	2	2	no	—
34	C₂×C₈×C₁₁	—	2	2	no	—
35	C₂×C₈×C₁₁	C₂³	1	2	yes	—
36	C₂×C₈×C₁₁	C₂³	1	2	yes	—
37	C₁₁×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₁₁×C₂²⋊C₄	C₂×C₄	1	2	yes	—
39	C₄²×C₁₁	C₂³	1	2	yes	—
40	C₂²⋊Dic₁₁	C₂×C₄	1	2	yes	—
41	C₄⋊Dic₁₁	C₂×C₄	1	4	yes	—
42	C₄⋊Dic₁₁	C₂×C₄	1	2	yes	—
43	Dic₁₁⋊C₄	C₂×C₄	1	4	yes	—
44	Dic₁₁⋊C₄	C₂×C₄	1	4	yes	—
45	C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
46	C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
47	C₄×Dic₁₁	C₂³	1	2	yes	—
48	C₄×Dic₁₁	C₂³	1	2	yes	—
49	#4318	—	2	2	no	—
50	C₂×C₁₁⋊C₈	Q₈	1	4	yes	complemented
51	C₂×C₁₁⋊C₈	C₂³	1	2	yes	—
52	C₂×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
53	C₂×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
54	#2960	—	11	1	no	maximal, Sylow
55	C₂³×C₁₁	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₁	C₂²×C₄	1	4	yes	—
57	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₁	C₂×Q₈	1	2	yes	—
59	C₂×C₄×C₁₁	#41	1	2	yes	—
60	C₂×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
61	C₂×Dic₁₁	—	2	4	no	—
62	C₂×Dic₁₁	C₂²×C₄	1	2	yes	—
63	C₂×Dic₁₁	C₂²×C₄	1	4	yes	—
64	C₂×Dic₁₁	C₂²×C₄	1	2	yes	—
65	Dic₂₂	—	2	8	no	—
66	C₈×C₁₁	—	2	2	no	—
67	C₈×C₁₁	—	2	2	no	—
68	C₁₁⋊C₈	—	2	2	no	—
69	C₁₁⋊C₈	C₂×Q₈	1	2	yes	—
70	C₁₁⋊C₈	C₂×Q₈	1	2	yes	—
71	#533	—	11	1	no	—
72	#446	—	11	4	no	—
73	C₈×Q₈	—	11	4	no	—
74	#424	—	11	1	no	—
75	#424	D₁₁	1	1	yes	characteristic
76	#423	—	11	1	no	—
77	#405	—	11	1	no	—
78	#405	—	11	1	no	—
79	C₂×C₄×C₈	—	11	1	no	—
80	C₂²×C₁₁	—	2	2	no	—
81	C₂²×C₁₁	C₄×Q₈	1	2	yes	—
82	C₂²×C₁₁	C₂³×C₄	1	1	yes	characteristic
83	C₄×C₁₁	—	2	4	no	—
84	C₄×C₁₁	C₄×Q₈	1	2	yes	—
85	C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
86	C₄×C₁₁	C₂×#41	1	1	yes	characteristic
87	Dic₁₁	—	2	4	no	—
88	Dic₁₁	—	2	2	no	—
89	Dic₁₁	#131	1	4	yes	—
90	C₂×M₁₆	—	11	1	no	—
91	C₂×M₁₆	C₂×D₁₁	1	1	yes	characteristic
92	C₂²×C₈	—	11	1	no	—
93	C₂²×C₈	C₂×D₁₁	1	1	yes	characteristic
94	C₄⋊Q₈	—	11	2	no	—
95	#125	—	11	2	no	—
96	C₂²⋊Q₈	—	11	4	no	—
97	C₄×Q₈	—	11	4	no	—
98	#117	—	11	1	no	—
99	#117	C₂×D₁₁	1	1	yes	characteristic
100	C₂×C₄²	—	11	1	no	—
101	C₄⋊C₈	—	11	4	no	—
102	C₄⋊C₈	—	11	4	no	—
103	C₄⋊C₈	C₂×D₁₁	1	4	yes	—
104	#97	—	11	2	no	—
105	#97	—	11	2	no	—
106	C₄×C₈	—	11	2	no	—
107	C₄×C₈	—	11	2	no	—
108	C₄×C₈	—	11	2	no	—
109	C₄×C₈	—	11	2	no	—
110	C₂×C₁₁	—	2	2	no	—
111	C₂×C₁₁	C₂×C₄×Q₈	1	1	yes	characteristic
112	C₂×C₁₁	C₂×#131	1	2	yes	—
113	C₂×Q₈	—	11	4	no	—
114	C₂²×C₄	—	11	2	no	—
115	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
116	M₁₆	—	2	2	no	—
117	M₁₆	—	22	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	11	4	no	—
120	C₂×C₈	—	11	1	no	—
121	C₂×C₈	—	11	1	no	—
122	C₂×C₈	—	11	2	no	—
123	C₂×C₈	C₂²×D₁₁	1	2	yes	—
124	C₂×C₈	C₂²×D₁₁	1	2	yes	—
125	C₄⋊C₄	—	11	2	no	—
126	C₄⋊C₄	—	11	4	no	—
127	C₄⋊C₄	—	11	4	no	—
128	C₄⋊C₄	—	11	4	no	—
129	C₄⋊C₄	C₄×D₁₁	1	2	yes	—
130	C₂²⋊C₄	—	11	2	no	—
131	C₂²⋊C₄	C₄×D₁₁	1	2	yes	—
132	C₄²	—	11	1	no	—
133	C₄²	—	11	1	no	—
134	C₄²	—	11	2	no	—
135	C₄²	—	11	2	no	—
136	C₄²	C₂²×D₁₁	1	2	yes	—
137	C₁₁	#2960	1	1	yes	characteristic, complemented, Sylow
138	C₂³	C₂×C₄×D₁₁	1	1	yes	characteristic
139	Q₈	—	22	8	no	—
140	C₂×C₄	—	11	2	no	—
141	C₂×C₄	—	11	2	no	—
142	C₂×C₄	—	11	4	no	—
143	C₂×C₄	—	22	4	no	—
144	C₂×C₄	C₂×C₄×D₁₁	1	4	yes	—
145	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
146	C₂×C₄	#4341	1	2	yes	—
147	C₂×C₄	Q₈×D₁₁	1	2	yes	—
148	C₂×C₄	C₂³×D₁₁	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	2	2	no	—
151	C₈	—	11	2	no	—
152	C₈	—	11	2	no	—
153	C₈	—	22	2	no	—
154	C₂²	—	2	2	no	—
155	C₂²	#66940	1	2	yes	—
156	C₂²	C₂²×C₄×D₁₁	1	1	yes	characteristic
157	C₄	—	2	4	no	—
158	C₄	—	11	4	no	—
159	C₄	—	22	2	no	—
160	C₄	—	22	4	no	—
161	C₄	#66940	1	2	yes	—
162	C₄	C₂×#4341	1	1	yes	characteristic
163	C₄	C₂×Q₈×D₁₁	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	C₂×#66940	1	1	yes	characteristic
166	C₂	D₁₁×#131	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10623048

C₂

yes

maximal, characteristic

C₁₁×#424

C₂

yes

maximal, characteristic

#10622735

C₂

yes

maximal, characteristic

#10622731

C₂

yes

maximal, characteristic

#10622715

C₂

yes

maximal, characteristic

C₂×C₈×Dic₁₁

C₂

yes

maximal, characteristic

#10622610

C₂

yes

maximal, characteristic

#10622443

C₂

yes

maximal, complemented

#10622441

C₂

yes

maximal, complemented

C₂×C₁₁×M₁₆

C₂²

yes

characteristic

C₂²×C₈×C₁₁

C₂²

yes

characteristic

C₁₁×#117

C₂²

yes

characteristic

#66979

C₂²

yes

characteristic

C₂×C₄×Dic₁₁

C₂²

yes

characteristic

C₂×#4318

C₂²

yes

characteristic

C₂²×C₁₁⋊C₈

C₂²

yes

characteristic

#66943

C₄

yes

—

#66941

C₄

yes

—

#66931

C₄

yes

—

C₄×Dic₂₂

C₂²

yes

—

C₁₁×C₄⋊C₈

C₂²

yes

—

#66879

C₂²

yes

—

Dic₁₁⋊C₈

C₂²

yes

—

Dic₁₁⋊C₈

C₂²

yes

—

C₈×Dic₁₁

C₂²

yes

—

C₈×Dic₁₁

C₂²

yes

—

C₈×Dic₁₁

C₂²

yes

—

#66867

C₂²

yes

—

C₄×C₁₁⋊C₈

C₂²

yes

—

C₂²×C₄×C₁₁

C₂³

yes

characteristic

C₂²×Dic₁₁

C₂×C₄

yes

—

C₂×Dic₂₂

C₂×C₄

yes

—

C₁₁×M₁₆

—

C₂×C₈×C₁₁

—

C₂×C₈×C₁₁

C₂³

yes

—

C₂×C₈×C₁₁

C₂³

yes

—

C₁₁×C₄⋊C₄

C₂×C₄

yes

—

C₁₁×C₂²⋊C₄

C₂×C₄

yes

—

C₄²×C₁₁

C₂³

yes

—

C₂²⋊Dic₁₁

C₂×C₄

yes

—

C₄⋊Dic₁₁

C₂×C₄

yes

—

C₄⋊Dic₁₁

C₂×C₄

yes

—

Dic₁₁⋊C₄

C₂×C₄

yes

—

Dic₁₁⋊C₄

C₂×C₄

yes

—

C₄×Dic₁₁

C₂×C₄

yes

characteristic

C₄×Dic₁₁

C₂×C₄

yes

characteristic

C₄×Dic₁₁

C₂³

yes

—

C₄×Dic₁₁

C₂³

yes

—

#4318

—

C₂×C₁₁⋊C₈

Q₈

yes

complemented

C₂×C₁₁⋊C₈

C₂³

yes

—

C₂×C₁₁⋊C₈

C₂³

yes

characteristic

C₂×C₁₁⋊C₈

C₂³

yes

characteristic

#2960

—

maximal, Sylow

C₂³×C₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁

C₂²×C₄

yes

—

C₂×C₄×C₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁

C₂×Q₈

yes

—

C₂×C₄×C₁₁

#41

yes

—

C₂×C₄×C₁₁

C₂⁴

yes

characteristic

C₂×Dic₁₁

—

C₂×Dic₁₁

C₂²×C₄

yes

—

C₂×Dic₁₁

C₂²×C₄

yes

—

C₂×Dic₁₁

C₂²×C₄

yes

—

Dic₂₂

—

C₈×C₁₁

—

C₈×C₁₁

—

C₁₁⋊C₈

—

C₁₁⋊C₈

C₂×Q₈

yes

—

C₁₁⋊C₈

C₂×Q₈

yes

—

#533

—

#446

—

C₈×Q₈

—

#424

—

#424

D₁₁

yes

—

—

—

C₂×C₄×C₈

—

C₂²×C₁₁

—

C₂²×C₁₁

C₄×Q₈

yes

—

C₂²×C₁₁

C₂³×C₄

yes

characteristic

C₄×C₁₁

—

C₄×C₁₁

C₄×Q₈

yes

—

C₄×C₁₁

C₂²×Q₈

yes

characteristic

C₄×C₁₁

C₂×#41

yes

characteristic

Dic₁₁

—

Dic₁₁

—

Dic₁₁

#131

yes

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₁

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×D₁₁

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

#117

C₂×D₁₁

yes

characteristic

100

C₂×C₄²

—

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

C₂×D₁₁

yes

—

104

#97

—

105

#97

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₂×C₁₁

—

111

C₂×C₁₁

C₂×C₄×Q₈

yes

characteristic

112

C₂×C₁₁

C₂×#131

yes

—

113

C₂×Q₈

—

114

C₂²×C₄

—

115

C₂²×C₄

C₂²×D₁₁

yes

characteristic

116

M₁₆

—

117

M₁₆

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

C₂²×D₁₁

yes

—

124

C₂×C₈

C₂²×D₁₁

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

C₄×D₁₁

yes

—

130

C₂²⋊C₄

—

131

C₂²⋊C₄

C₄×D₁₁

yes

—

132

C₄²

—

133

C₄²

—

134

C₄²

—

135

C₄²

—

136

C₄²

C₂²×D₁₁

yes

—

137

C₁₁

#2960

yes

characteristic, complemented, Sylow

138

C₂³

C₂×C₄×D₁₁

yes

characteristic

139

Q₈

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

C₂×C₄×D₁₁

yes

—

145

C₂×C₄

C₂×C₄×D₁₁

yes

characteristic

146

C₂×C₄

#4341

yes

—

147

C₂×C₄

Q₈×D₁₁

yes

—

148

C₂×C₄

C₂³×D₁₁

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₂²

—

155

C₂²

#66940

yes

—

156

C₂²

C₂²×C₄×D₁₁

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

#66940

yes

—

162

C₄

C₂×#4341

yes

characteristic

163

C₄

C₂×Q₈×D₁₁

yes

characteristic

164

C₂

—

165

C₂

C₂×#66940

yes

characteristic

166

C₂

D₁₁×#131

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₄
Fitting	C₁₁×#424	C₂
prime radical	C₂³×C₁₁	C₂²×C₄
cosocle	C₂×C₄×C₁₁	C₂⁴
hypercenter	#424	D₁₁
hypocenter	C₁₁	#2960
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ #424

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₁

commutator, socle

C₂²×C₁₁

C₂³×C₄

Fitting

C₁₁×#424

C₂

prime radical

C₂³×C₁₁

C₂²×C₄

cosocle

C₂×C₄×C₁₁

C₂⁴

D₁₁

C₁₁

G ▷ C₂²×C₁₁ ▷ C₁₁

C₁ ◁ C₂×C₄ ◁ #424

N	H ≅ G/N	type
8. #10622443	C₂	semidirect
9. #10622441	C₂	semidirect
50. C₂×C₁₁⋊C₈	Q₈	semidirect
137. C₁₁	#2960	semidirect

H ≅ G/N

type

8. #10622443

C₂

semidirect

9. #10622441

C₂

semidirect

50. C₂×C₁₁⋊C₈

Q₈

semidirect

137. C₁₁

#2960

semidirect

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

subgroup	index
#10623048	2
C₁₁×#424	2
#10622735	2
#10622731	2
#10622715	2
C₂×C₈×Dic₁₁	2
#10622610	2
#10622443	2
#10622441	2
#2960	11

subgroup

index

C₁₁×#424

C₂×C₈×Dic₁₁

Subgroups — #13077549

Conjugacy classes of subgroups

Complements

Maximal subgroups