Group #13077333 — subgroups

Subgroups — #13077333

Order 1408.

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

normal subgroups only

This group has 484 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	C₂×#66960	C₂	1	2	yes	maximal, complemented
2	#10623008	C₂	1	1	yes	maximal, characteristic
3	C₁₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
4	#10622742	C₂	1	2	yes	maximal, complemented
5	C₂×#66867	C₂	1	1	yes	maximal, characteristic, complemented
6	#10622449	C₂	1	8	yes	maximal, complemented
7	C₂×#4339	C₂²	1	2	yes	—
8	C₂×C₁₁×M₁₆	C₂²	1	2	yes	—
9	C₂×C₄²×C₁₁	C₂²	1	1	yes	characteristic
10	#66983	C₄	1	4	yes	—
11	C₂²×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₂²×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
13	#66960	—	2	8	no	—
14	C₂×#4313	C₂²	1	4	yes	—
15	C₂×C₈×D₁₁	C₂²	1	4	yes	—
16	#66928	C₄	1	2	yes	—
17	#66927	C₄	1	1	yes	characteristic
18	C₄×D₄₄	C₂²	1	4	yes	—
19	#66922	C₄	1	1	yes	characteristic
20	C₁₁×C₄⋊C₈	C₂²	1	4	yes	—
21	(C₂×D₁₁)⋊C₈	C₂²	1	8	yes	—
22	#66867	C₂²	1	4	yes	—
23	C₂²×C₄×C₁₁	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
25	C₂×C₂²⋊D₁₁	C₂×C₄	1	4	yes	—
26	#4339	—	2	4	no	—
27	#4339	—	2	4	no	—
28	C₂×D₄₄	C₂×C₄	1	2	yes	—
29	C₂×D₄₄	C₂×C₄	1	4	yes	—
30	C₂×C₄×D₁₁	C₂³	1	4	yes	—
31	C₂×Dic₂₂	C₂×C₄	1	2	yes	—
32	C₁₁×M₁₆	—	2	4	no	—
33	C₂×C₈×C₁₁	C₂³	1	4	yes	—
34	C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₁₁	C₂³	1	2	yes	—
37	(C₂×D₁₁)⋊C₄	C₂×C₄	1	8	yes	—
38	C₄⋊Dic₁₁	C₂×C₄	1	4	yes	—
39	C₂×C₁₁⋊C₈	—	2	2	no	—
40	C₂×C₁₁⋊C₈	D₄	1	4	yes	complemented
41	C₂×C₁₁⋊C₈	C₂³	1	2	yes	—
42	C₂×C₁₁⋊C₈	C₂³	1	2	yes	—
43	#4313	—	2	8	no	—
44	C₈×D₁₁	—	2	8	no	—
45	#2932	—	11	1	no	maximal, Sylow
46	C₂³×C₁₁	C₂²×C₄	1	1	yes	characteristic
47	C₂²×D₁₁	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₁₁	—	4	2	no	—
49	C₂×C₄×C₁₁	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₁₁	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₁	#41	1	2	yes	—
54	C₂×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
55	C₂²⋊D₁₁	—	2	8	no	—
56	C₂×Dic₁₁	C₂²×C₄	1	4	yes	—
57	D₄₄	—	2	4	no	—
58	D₄₄	—	4	4	no	—
59	C₄×D₁₁	—	2	4	no	—
60	C₄×D₁₁	—	2	4	no	—
61	Dic₂₂	—	2	4	no	—
62	C₈×C₁₁	—	2	4	no	—
63	C₁₁⋊C₈	—	2	2	no	—
64	C₁₁⋊C₈	C₂×D₄	1	4	yes	—
65	C₂×#131	—	11	2	no	—
66	#532	—	11	1	no	—
67	#435	—	11	8	no	—
68	#423	D₁₁	1	1	yes	characteristic, complemented
69	#408	—	11	2	no	—
70	C₂×#97	—	11	1	no	—
71	C₂²×C₁₁	—	2	2	no	—
72	C₂²×C₁₁	C₄×D₄	1	2	yes	—
73	C₂²×C₁₁	C₂³×C₄	1	1	yes	characteristic
74	C₂×D₁₁	—	2	4	no	—
75	C₂×D₁₁	—	2	8	no	—
76	C₄×C₁₁	—	2	2	no	—
77	C₄×C₁₁	—	2	2	no	—
78	C₄×C₁₁	C₄×D₄	1	2	yes	—
79	C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
80	C₄×C₁₁	C₂×#41	1	1	yes	characteristic
81	Dic₁₁	—	2	4	no	—
82	C₂×#41	—	11	2	no	—
83	#131	—	22	8	no	—
84	C₂×M₁₆	—	11	4	no	—
85	C₂×M₁₆	C₂×D₁₁	1	2	yes	—
86	C₂²×C₈	—	11	1	no	—
87	C₂²×C₈	—	11	1	no	—
88	C₂²×C₈	—	11	4	no	—
89	C₄⋊Q₈	—	11	1	no	—
90	#127	—	11	1	no	—
91	#124	—	11	2	no	—
92	#121	—	11	4	no	—
93	C₄×D₄	—	11	4	no	—
94	C₂×C₄²	C₂×D₁₁	1	1	yes	characteristic
95	C₄⋊C₈	C₂×D₁₁	1	4	yes	—
96	C₂²⋊C₈	—	11	8	no	—
97	#97	—	11	4	no	—
98	C₂×C₁₁	—	2	2	no	—
99	C₂×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
100	C₂×C₁₁	#568	1	2	yes	—
101	D₁₁	—	4	4	no	—
102	#41	—	22	4	no	—
103	#41	—	22	4	no	—
104	C₂×Q₈	—	11	2	no	—
105	C₂×D₄	—	11	4	no	—
106	C₂×D₄	—	11	2	no	—
107	C₂×D₄	—	11	4	no	—
108	C₂²×C₄	—	11	4	no	—
109	C₂²×C₄	C₄×D₁₁	1	2	yes	—
110	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
111	M₁₆	—	2	4	no	—
112	M₁₆	—	22	8	no	—
113	C₂×C₈	—	11	4	no	—
114	C₂×C₈	—	11	2	no	—
115	C₂×C₈	—	11	2	no	—
116	C₂×C₈	—	22	8	no	—
117	C₂×C₈	—	22	2	no	—
118	C₂×C₈	C₂²×D₁₁	1	4	yes	—
119	C₄⋊C₄	—	11	4	no	—
120	C₂²⋊C₄	—	11	8	no	—
121	C₄²	C₄×D₁₁	1	1	yes	characteristic
122	C₄²	C₄×D₁₁	1	1	yes	characteristic
123	C₄²	C₂²×D₁₁	1	2	yes	—
124	C₁₁	#2932	1	1	yes	characteristic, complemented, Sylow
125	C₂³	—	11	4	no	—
126	C₂³	C₂×C₄×D₁₁	1	1	yes	characteristic
127	Q₈	—	22	4	no	—
128	D₄	—	22	8	no	—
129	D₄	—	22	4	no	—
130	D₄	—	44	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	11	4	no	—
133	C₂×C₄	—	22	4	no	—
134	C₂×C₄	—	22	4	no	—
135	C₂×C₄	C₂×C₄×D₁₁	1	2	yes	—
136	C₂×C₄	C₂×C₄×D₁₁	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
138	C₂×C₄	D₄×D₁₁	1	2	yes	—
139	C₂×C₄	#4343	1	2	yes	—
140	C₂×C₄	C₂³×D₁₁	1	1	yes	characteristic
141	C₈	—	2	4	no	—
142	C₈	—	11	4	no	—
143	C₈	—	22	2	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	22	8	no	—
146	C₂²	—	22	4	no	—
147	C₂²	#66946	1	2	yes	—
148	C₂²	C₂²×C₄×D₁₁	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	22	4	no	—
152	C₄	#66946	1	2	yes	—
153	C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
154	C₄	C₂×#4343	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	44	4	no	—
157	C₂	C₂×#66946	1	1	yes	characteristic
158	C₂	#10623276	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#66960

C₂

yes

maximal, complemented

#10623008

C₂

yes

maximal, characteristic

C₁₁×#423

C₂

yes

maximal, characteristic, complemented

#10622742

C₂

yes

maximal, complemented

C₂×#66867

C₂

yes

maximal, characteristic, complemented

#10622449

C₂

yes

maximal, complemented

C₂×#4339

C₂²

yes

—

C₂×C₁₁×M₁₆

C₂²

yes

—

C₂×C₄²×C₁₁

C₂²

yes

characteristic

#66983

C₄

yes

—

C₂²×C₁₁⋊C₈

C₂²

yes

characteristic

C₂²×C₁₁⋊C₈

C₂²

yes

characteristic

#66960

—

C₂×#4313

C₂²

yes

—

C₂×C₈×D₁₁

C₂²

yes

—

#66928

C₄

yes

—

#66927

C₄

yes

characteristic

C₄×D₄₄

C₂²

yes

—

#66922

C₄

yes

characteristic

C₁₁×C₄⋊C₈

C₂²

yes

—

(C₂×D₁₁)⋊C₈

C₂²

yes

—

#66867

C₂²

yes

—

C₂²×C₄×C₁₁

C₂×C₄

yes

—

C₂²×C₄×C₁₁

C₂³

yes

characteristic

C₂×C₂²⋊D₁₁

C₂×C₄

yes

—

#4339

—

#4339

—

C₂×D₄₄

C₂×C₄

yes

—

C₂×D₄₄

C₂×C₄

yes

—

C₂×C₄×D₁₁

C₂³

yes

—

C₂×Dic₂₂

C₂×C₄

yes

—

C₁₁×M₁₆

—

C₂×C₈×C₁₁

C₂³

yes

—

C₄²×C₁₁

C₂×C₄

yes

characteristic

C₄²×C₁₁

C₂×C₄

yes

characteristic

C₄²×C₁₁

C₂³

yes

—

(C₂×D₁₁)⋊C₄

C₂×C₄

yes

—

C₄⋊Dic₁₁

C₂×C₄

yes

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

D₄

yes

complemented

C₂×C₁₁⋊C₈

C₂³

yes

—

C₂×C₁₁⋊C₈

C₂³

yes

—

#4313

—

C₈×D₁₁

—

#2932

—

maximal, Sylow

C₂³×C₁₁

C₂²×C₄

yes

characteristic

C₂²×D₁₁

C₂²×C₄

yes

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

C₂²×C₄

yes

—

C₂×C₄×C₁₁

C₂²×C₄

yes

—

C₂×C₄×C₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁

C₂×D₄

yes

—

C₂×C₄×C₁₁

#41

yes

—

C₂×C₄×C₁₁

C₂⁴

yes

characteristic

C₂²⋊D₁₁

—

C₂×Dic₁₁

C₂²×C₄

yes

—

D₄₄

—

D₄₄

—

C₄×D₁₁

—

C₄×D₁₁

—

Dic₂₂

—

C₈×C₁₁

—

C₁₁⋊C₈

—

C₁₁⋊C₈

C₂×D₄

yes

—

—

—

—

D₁₁

yes

characteristic, complemented

#408

—

C₂×#97

—

C₂²×C₁₁

—

C₂²×C₁₁

C₄×D₄

yes

—

C₂²×C₁₁

C₂³×C₄

yes

characteristic

C₂×D₁₁

—

C₂×D₁₁

—

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

C₄×D₄

yes

—

C₄×C₁₁

C₂²×D₄

yes

characteristic

C₄×C₁₁

C₂×#41

yes

characteristic

Dic₁₁

—

C₂×#41

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₁

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₁₁

yes

characteristic

C₄⋊C₈

C₂×D₁₁

yes

—

C₂²⋊C₈

—

#97

—

C₂×C₁₁

—

C₂×C₁₁

C₂×C₄×D₄

yes

characteristic

100

C₂×C₁₁

#568

yes

—

101

D₁₁

—

102

#41

—

103

#41

—

104

C₂×Q₈

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₄×D₁₁

yes

—

110

C₂²×C₄

C₂²×D₁₁

yes

characteristic

111

M₁₆

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

C₂²×D₁₁

yes

—

119

C₄⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

C₄×D₁₁

yes

characteristic

122

C₄²

C₄×D₁₁

yes

characteristic

123

C₄²

C₂²×D₁₁

yes

—

124

C₁₁

#2932

yes

characteristic, complemented, Sylow

125

C₂³

—

126

C₂³

C₂×C₄×D₁₁

yes

characteristic

127

Q₈

—

128

D₄

—

129

D₄

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₂×C₄×D₁₁

yes

—

136

C₂×C₄

C₂×C₄×D₁₁

yes

—

137

C₂×C₄

C₂×C₄×D₁₁

yes

characteristic

138

C₂×C₄

D₄×D₁₁

yes

—

139

C₂×C₄

#4343

yes

—

140

C₂×C₄

C₂³×D₁₁

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

#66946

yes

—

148

C₂²

C₂²×C₄×D₁₁

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

#66946

yes

—

153

C₄

C₂×D₄×D₁₁

yes

characteristic

154

C₄

C₂×#4343

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

C₂×#66946

yes

characteristic

158

C₂

#10623276

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₄
Fitting	C₁₁×#423	C₂
prime radical	#10623008	C₂
cosocle	C₂×C₄×C₁₁	C₂⁴
hypercenter	#423	D₁₁
hypocenter	C₁₁	#2932
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₁

commutator, socle

C₂²×C₁₁

C₂³×C₄

Fitting

C₁₁×#423

C₂

prime radical

#10623008

C₂

cosocle

C₂×C₄×C₁₁

C₂⁴

D₁₁

C₁₁

G ▷ C₂²×C₁₁ ▷ C₁₁

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
1. C₂×#66960	C₂	semidirect
3. C₁₁×#423	C₂	semidirect
4. #10622742	C₂	semidirect
5. C₂×#66867	C₂	semidirect
6. #10622449	C₂	semidirect
40. C₂×C₁₁⋊C₈	D₄	semidirect
68. #423	D₁₁	semidirect
124. C₁₁	#2932	semidirect

H ≅ G/N

type

1. C₂×#66960

C₂

semidirect

3. C₁₁×#423

C₂

semidirect

4. #10622742

C₂

semidirect

5. C₂×#66867

C₂

semidirect

6. #10622449

C₂

semidirect

40. C₂×C₁₁⋊C₈

D₄

semidirect

68. #423

D₁₁

semidirect

124. C₁₁

#2932

semidirect

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

subgroup	index
C₂×#66960	2
#10623008	2
C₁₁×#423	2
#10622742	2
C₂×#66867	2
#10622449	2
#2932	11

subgroup

index

C₁₁×#423

Subgroups — #13077333

Conjugacy classes of subgroups

Complements

Maximal subgroups