Group #13070092 — subgroups

Subgroups — #13070092

Order 1408.

230 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₁₁×#389	C₂	1	1	yes	maximal, characteristic
2	C₂×#66872	C₂	1	2	yes	maximal
3	#10622548	C₂	1	1	yes	maximal, characteristic
4	#10622167	C₂	1	2	yes	maximal
5	#10622162	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₁×C₄⋊C₄	—	2	1	no	—
7	C₂×C₁₁×C₄⋊C₄	C₂²	1	1	yes	characteristic
8	C₂×C₁₁×C₄⋊C₄	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₁₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₁	—	2	1	no	—
11	C₂×C₄⋊Dic₁₁	—	2	1	no	—
12	C₂×C₄⋊Dic₁₁	C₂²	1	2	yes	—
13	C₂²×C₁₁⋊C₈	C₂²	1	2	yes	—
14	C₁₁×#95	—	2	1	no	—
15	#66872	—	2	4	no	—
16	C₂²×C₄×C₁₁	—	2	1	no	—
17	C₂²×C₄×C₁₁	—	2	1	no	—
18	C₂²×C₄×C₁₁	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₁	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
21	C₂²×Dic₁₁	—	2	2	no	—
22	C₁₁×C₄⋊C₄	—	2	1	no	—
23	C₁₁×C₄⋊C₄	—	2	2	no	—
24	C₁₁×C₄⋊C₄	—	4	2	no	—
25	C₁₁×C₄⋊C₄	D₄	1	2	yes	—
26	C₄²×C₁₁	—	2	2	no	—
27	C₄⋊Dic₁₁	—	2	2	no	—
28	C₄⋊Dic₁₁	—	2	2	no	—
29	C₄⋊Dic₁₁	—	4	2	no	—
30	C₄⋊Dic₁₁	—	4	2	no	—
31	C₂×C₁₁⋊C₈	—	2	2	no	—
32	C₂×C₁₁⋊C₈	—	2	2	no	—
33	C₂×C₁₁⋊C₈	Q₈	1	4	yes	—
34	#2084	—	11	1	no	maximal, Sylow
35	C₂³×C₁₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₁	—	2	1	no	—
37	C₂×C₄×C₁₁	—	2	1	no	—
38	C₂×C₄×C₁₁	—	2	1	no	—
39	C₂×C₄×C₁₁	—	2	1	no	—
40	C₂×C₄×C₁₁	—	2	2	no	—
41	C₂×C₄×C₁₁	—	4	1	no	—
42	C₂×C₄×C₁₁	—	4	1	no	—
43	C₂×C₄×C₁₁	—	4	1	no	—
44	C₂×C₄×C₁₁	—	4	1	no	—
45	C₂×C₄×C₁₁	—	4	1	no	—
46	C₂×C₄×C₁₁	SD₁₆	1	2	yes	—
47	C₂×C₄×C₁₁	SD₁₆	1	2	yes	—
48	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₁	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₁₁	#41	1	1	yes	characteristic
51	C₂×C₄×C₁₁	#41	1	2	yes	—
52	C₂×Dic₁₁	—	4	2	no	—
53	C₂×Dic₁₁	—	4	2	no	—
54	C₂×Dic₁₁	—	4	2	no	—
55	C₁₁⋊C₈	—	2	4	no	—
56	C₂×#106	—	11	2	no	—
57	#389	D₁₁	1	1	yes	characteristic
58	#384	—	11	1	no	—
59	#340	—	11	2	no	—
60	#336	—	11	1	no	—
61	C₂²×C₁₁	#121	1	1	yes	characteristic
62	C₂²×C₁₁	#123	1	1	yes	characteristic
63	C₂²×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
64	C₂²×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
65	C₂²×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
66	C₂²×C₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₂²×C₁₁	#137	1	1	yes	characteristic
68	C₄×C₁₁	—	2	2	no	—
69	C₄×C₁₁	—	2	2	no	—
70	C₄×C₁₁	—	4	2	no	—
71	C₄×C₁₁	—	8	1	no	—
72	C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
73	C₄×C₁₁	#125	1	2	yes	—
74	Dic₁₁	—	8	2	no	—
75	C₂²×C₈	—	11	2	no	—
76	C₂×C₄⋊C₄	—	2	1	no	—
77	C₂×C₄⋊C₄	—	11	2	no	—
78	C₂×C₄⋊C₄	—	22	1	no	—
79	C₂×C₄⋊C₄	—	22	1	no	—
80	C₂×C₄⋊C₄	C₂×D₁₁	1	1	yes	characteristic
81	C₂×C₄⋊C₄	C₂×D₁₁	1	1	yes	characteristic
82	C₂×C₄²	C₂×D₁₁	1	1	yes	characteristic
83	#106	—	22	4	no	—
84	#95	—	2	1	no	—
85	C₂×C₁₁	#398	1	1	yes	characteristic
86	C₂×C₁₁	#460	1	1	yes	characteristic
87	C₂×C₁₁	#461	1	1	yes	characteristic
88	C₂×C₁₁	#481	1	1	yes	characteristic
89	C₂×C₁₁	#483	1	1	yes	characteristic
90	C₂×C₁₁	#498	1	1	yes	characteristic
91	C₂×C₁₁	#501	1	1	yes	characteristic
92	C₂²×C₄	—	2	1	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	22	2	no	—
95	C₂²×C₄	C₂²⋊D₁₁	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₁₁	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
98	C₂×C₈	—	11	4	no	—
99	C₂×C₈	—	22	2	no	—
100	C₂×C₈	—	22	2	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	4	2	no	—
104	C₄⋊C₄	—	22	2	no	—
105	C₄⋊C₄	—	22	2	no	—
106	C₄⋊C₄	—	44	2	no	—
107	C₄⋊C₄	—	44	2	no	—
108	C₄⋊C₄	D₄₄	1	2	yes	—
109	C₄²	—	2	2	no	—
110	C₁₁	#2084	1	1	yes	characteristic, complemented, Sylow
111	C₂³	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	44	2	no	—
123	C₂×C₄	—	44	2	no	—
124	C₂×C₄	—	44	2	no	—
125	C₂×C₄	#4324	1	2	yes	—
126	C₂×C₄	#4325	1	2	yes	—
127	C₂×C₄	C₂×D₄₄	1	1	yes	characteristic
128	C₂×C₄	#4339	1	1	yes	characteristic
129	C₂×C₄	#4341	1	2	yes	—
130	C₂×C₄	Q₈×D₁₁	1	2	yes	—
131	C₈	—	22	4	no	—
132	C₂²	#66983	1	1	yes	characteristic
133	C₂²	C₂×#4324	1	1	yes	characteristic
134	C₂²	#66988	1	1	yes	characteristic
135	C₂²	C₂×#4325	1	1	yes	characteristic
136	C₂²	#66997	1	1	yes	characteristic
137	C₂²	#67002	1	1	yes	characteristic
138	C₂²	#67004	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	2	2	no	—
141	C₄	—	4	2	no	—
142	C₄	—	8	1	no	—
143	C₄	—	88	2	no	—
144	C₄	#66942	1	2	yes	—
145	C₄	#66950	1	2	yes	—
146	C₂	#10622596	1	1	yes	characteristic
147	C₂	#10622632	1	1	yes	characteristic
148	C₂	#10622640	1	1	yes	characteristic
149	C₂	#10622647	1	1	yes	characteristic
150	C₂	#10622656	1	1	yes	characteristic
151	C₂	#10622678	1	1	yes	characteristic
152	C₂	#10622692	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₁×#389

C₂

yes

maximal, characteristic

C₂×#66872

C₂

yes

maximal

#10622548

C₂

yes

maximal, characteristic

#10622167

C₂

yes

maximal

#10622162

C₂

yes

maximal, characteristic

C₂×C₁₁×C₄⋊C₄

—

C₂×C₁₁×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₁₁×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄²×C₁₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₁

—

C₂×C₄⋊Dic₁₁

—

C₂×C₄⋊Dic₁₁

C₂²

yes

—

C₂²×C₁₁⋊C₈

C₂²

yes

—

C₁₁×#95

—

#66872

—

C₂²×C₄×C₁₁

—

C₂²×C₄×C₁₁

—

C₂²×C₄×C₁₁

D₄

yes

characteristic

C₂²×C₄×C₁₁

D₄

yes

characteristic

C₂²×C₄×C₁₁

C₂³

yes

characteristic

C₂²×Dic₁₁

—

C₁₁×C₄⋊C₄

—

C₁₁×C₄⋊C₄

—

C₁₁×C₄⋊C₄

—

C₁₁×C₄⋊C₄

D₄

yes

—

C₄²×C₁₁

—

C₄⋊Dic₁₁

—

C₄⋊Dic₁₁

—

C₄⋊Dic₁₁

—

C₄⋊Dic₁₁

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

Q₈

yes

—

#2084

—

maximal, Sylow

C₂³×C₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

SD₁₆

yes

—

C₂×C₄×C₁₁

SD₁₆

yes

—

C₂×C₄×C₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁

C₂×Q₈

yes

—

C₂×C₄×C₁₁

#41

yes

characteristic

C₂×C₄×C₁₁

#41

yes

—

C₂×Dic₁₁

—

C₂×Dic₁₁

—

C₂×Dic₁₁

—

C₁₁⋊C₈

—

C₂×#106

—

#389

D₁₁

yes

—

—

—

C₂²×C₁₁

yes

C₂²×C₁₁

yes

C₂²×C₁₁

C₄⋊Q₈

yes

characteristic

C₂²×C₁₁

C₂×SD₁₆

yes

characteristic

C₂²×C₁₁

C₂×SD₁₆

yes

C₂²×C₁₁

yes

C₂²×C₁₁

yes

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

C₂²⋊Q₈

yes

—

C₄×C₁₁

#125

yes

—

Dic₁₁

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₁

yes

characteristic

C₂×C₄⋊C₄

C₂×D₁₁

yes

characteristic

C₂×C₄²

C₂×D₁₁

yes

characteristic

#106

—

#95

—

C₂×C₁₁

yes

C₂×C₁₁

yes

C₂×C₁₁

yes

C₂×C₁₁

yes

C₂×C₁₁

yes

C₂×C₁₁

yes

C₂×C₁₁

yes

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₁

yes

characteristic

C₂²×C₄

C₂²×D₁₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

D₄₄

yes

—

109

C₄²

—

110

C₁₁

#2084

yes

characteristic, complemented, Sylow

111

C₂³

C₂×C₂²⋊D₁₁

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

#4324

yes

—

126

C₂×C₄

#4325

yes

—

127

C₂×C₄

C₂×D₄₄

yes

characteristic

128

C₂×C₄

#4339

yes

characteristic

129

C₂×C₄

#4341

yes

—

130

C₂×C₄

Q₈×D₁₁

yes

—

131

C₈

—

132

C₂²

#66983

yes

characteristic

133

C₂²

C₂×#4324

yes

characteristic

134

C₂²

#66988

yes

characteristic

135

C₂²

C₂×#4325

yes

characteristic

136

C₂²

#66997

yes

characteristic

137

C₂²

#67002

yes

characteristic

138

C₂²

#67004

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

#66942

yes

—

145

C₄

#66950

yes

—

146

C₂

#10622596

yes

characteristic

147

C₂

#10622632

yes

characteristic

148

C₂

#10622640

yes

characteristic

149

C₂

#10622647

yes

characteristic

150

C₂

#10622656

yes

characteristic

151

C₂

#10622678

yes

characteristic

152

C₂

#10622692

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₁
commutator, cosocle	C₂²×C₄×C₁₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₁
Fitting	C₁₁×#389	C₂
socle, prime radical	C₂³×C₁₁	C₂×D₄
hypercenter	#389	D₁₁
hypocenter	C₁₁	#2084
LCS	G ▷ C₂²×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₁

commutator, cosocle

C₂²×C₄×C₁₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₁

Fitting

C₁₁×#389

C₂

socle, prime radical

C₂³×C₁₁

C₂×D₄

D₁₁

C₁₁

G ▷ C₂²×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
110. C₁₁	#2084	semidirect

H ≅ G/N

type

110. C₁₁

#2084

semidirect

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

subgroup	index
C₁₁×#389	2
C₂×#66872	2
#10622548	2
#10622167	2
#10622162	2
#2084	11

subgroup

index

C₁₁×#389

Subgroups — #13070092

Conjugacy classes of subgroups

Complements

Maximal subgroups