Group #13069340 — subgroups

Subgroups — #13069340

Order 1408.

248 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 121 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	C₄×C₁₁×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₁₁⋊C₈)	C₂	1	2	yes	maximal
4	C₂×#66867	C₂	1	1	yes	maximal, characteristic
5	#10622162	C₂	1	2	yes	maximal
6	C₂×C₁₁×C₄⋊C₄	C₄	1	2	yes	—
7	C₂×C₄²×C₁₁	C₂²	1	2	yes	—
8	C₂×C₄²×C₁₁	C₂²	1	1	yes	characteristic
9	C₂²×C₁₁⋊C₈	C₂²	1	2	yes	—
10	C₂²×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₁₁×#95	C₄	1	1	yes	characteristic
13	C₁₁×#95	C₄	1	1	yes	characteristic
14	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
15	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
16	#66867	—	2	2	no	—
17	C₂²×C₄×C₁₁	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₁	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
22	C₁₁×C₄⋊C₄	—	2	4	no	—
23	C₄²×C₁₁	—	2	2	no	—
24	C₄²×C₁₁	—	2	2	no	—
25	C₄²×C₁₁	D₄	1	2	yes	—
26	C₄²×C₁₁	Q₈	1	2	yes	—
27	C₂×C₁₁⋊C₈	—	2	1	no	—
28	C₂×C₁₁⋊C₈	—	2	1	no	—
29	C₂×C₁₁⋊C₈	—	2	2	no	—
30	C₂×C₁₁⋊C₈	—	2	2	no	—
31	C₂×C₁₁⋊C₈	—	2	1	no	—
32	C₂×C₁₁⋊C₈	—	2	1	no	—
33	C₂×C₁₁⋊C₈	—	2	2	no	—
34	C₂×C₁₁⋊C₈	D₄	1	2	yes	—
35	C₂×C₁₁⋊C₈	Q₈	1	2	yes	—
36	#1940	—	11	1	no	maximal, Sylow
37	C₂³×C₁₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₁	—	2	2	no	—
39	C₂×C₄×C₁₁	—	2	2	no	—
40	C₂×C₄×C₁₁	—	2	2	no	—
41	C₂×C₄×C₁₁	—	2	2	no	—
42	C₂×C₄×C₁₁	—	2	2	no	—
43	C₂×C₄×C₁₁	—	2	2	no	—
44	C₂×C₄×C₁₁	—	2	2	no	—
45	C₂×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₁₁	M₁₆	1	4	yes	—
48	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₁₁	#41	1	1	yes	characteristic
53	C₂×C₄×C₁₁	#41	1	1	yes	characteristic
54	C₁₁⋊C₈	—	2	2	no	—
55	C₁₁⋊C₈	—	2	2	no	—
56	C₁₁⋊C₈	—	4	2	no	—
57	C₂×C₄⋊C₈	—	11	2	no	—
58	C₂×C₄⋊C₈	—	11	1	no	—
59	C₂×#97	—	11	1	no	—
60	C₄×C₄⋊C₄	D₁₁	1	1	yes	characteristic
61	#336	—	11	2	no	—
62	C₂²×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₁₁	C₄×D₄	1	1	yes	characteristic
64	C₂²×C₁₁	C₄×Q₈	1	1	yes	characteristic
65	C₂²×C₁₁	C₂×M₁₆	1	2	yes	—
66	C₂²×C₁₁	#131	1	2	yes	—
67	C₄×C₁₁	—	2	2	no	—
68	C₄×C₁₁	—	2	2	no	—
69	C₄×C₁₁	—	2	4	no	—
70	C₄×C₁₁	—	4	2	no	—
71	C₄×C₁₁	#121	1	1	yes	characteristic
72	C₄×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
73	C₄×C₁₁	#125	1	1	yes	characteristic
74	C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
75	C₂²×C₈	—	11	1	no	—
76	C₂²×C₈	—	11	1	no	—
77	C₂²×C₈	—	11	2	no	—
78	C₂×C₄⋊C₄	Dic₁₁	1	2	yes	—
79	C₂×C₄²	C₂×D₁₁	1	2	yes	—
80	C₂×C₄²	C₂×D₁₁	1	1	yes	characteristic
81	C₄⋊C₈	—	22	4	no	—
82	C₄⋊C₈	—	22	2	no	—
83	#97	—	22	2	no	—
84	#95	Dic₁₁	1	1	yes	characteristic
85	#95	Dic₁₁	1	1	yes	characteristic
86	C₂×C₁₁	#389	1	1	yes	characteristic
87	C₂×C₁₁	#423	1	1	yes	characteristic
88	C₂×C₁₁	#424	1	1	yes	characteristic
89	C₂×C₁₁	#435	1	2	yes	—
90	C₂×C₁₁	#446	1	2	yes	—
91	C₂²×C₄	C₂×Dic₁₁	1	2	yes	—
92	C₂²×C₄	C₂×Dic₁₁	1	2	yes	—
93	C₂²×C₄	C₂×Dic₁₁	1	1	yes	characteristic
94	C₂²×C₄	C₂×Dic₁₁	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
96	C₂×C₈	—	11	2	no	—
97	C₂×C₈	—	11	2	no	—
98	C₂×C₈	—	22	1	no	—
99	C₂×C₈	—	22	1	no	—
100	C₂×C₈	—	22	1	no	—
101	C₂×C₈	—	22	1	no	—
102	C₂×C₈	—	22	2	no	—
103	C₂×C₈	—	22	2	no	—
104	C₂×C₈	—	22	2	no	—
105	C₄⋊C₄	—	2	4	no	—
106	C₄²	—	2	2	no	—
107	C₄²	—	2	2	no	—
108	C₄²	Dic₂₂	1	2	yes	—
109	C₄²	D₄₄	1	2	yes	—
110	C₁₁	#1940	1	1	yes	characteristic, complemented, Sylow
111	C₂³	C₂²×Dic₁₁	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	#4318	1	4	yes	—
120	C₂×C₄	C₄⋊Dic₁₁	1	2	yes	—
121	C₂×C₄	C₄⋊Dic₁₁	1	2	yes	—
122	C₂×C₄	C₂×Dic₂₂	1	1	yes	characteristic
123	C₂×C₄	C₂×D₄₄	1	1	yes	characteristic
124	C₂×C₄	D₄×D₁₁	1	1	yes	characteristic
125	C₂×C₄	#4341	1	1	yes	characteristic
126	C₂×C₄	Q₈×D₁₁	1	1	yes	characteristic
127	C₂×C₄	#4343	1	1	yes	characteristic
128	C₈	—	22	2	no	—
129	C₈	—	22	2	no	—
130	C₈	—	44	2	no	—
131	C₂²	C₂×#4318	1	2	yes	—
132	C₂²	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
133	C₂²	D₄×Dic₁₁	1	1	yes	characteristic
134	C₂²	Q₈×Dic₁₁	1	1	yes	characteristic
135	C₂²	#67001	1	2	yes	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	2	4	no	—
139	C₄	—	4	2	no	—
140	C₄	#66941	1	1	yes	characteristic
141	C₄	#66943	1	1	yes	characteristic
142	C₄	#66948	1	1	yes	characteristic
143	C₄	#66950	1	1	yes	characteristic
144	C₂	#10622537	1	1	yes	characteristic
145	C₂	#10622598	1	1	yes	characteristic
146	C₂	#10622612	1	1	yes	characteristic
147	C₂	#10622624	1	2	yes	—
148	C₂	#10622637	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₁₁×C₄⋊C₄

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₁₁⋊C₈)

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₁₁⋊C₈)

C₂

yes

maximal

C₂×#66867

C₂

yes

maximal, characteristic

#10622162

C₂

yes

maximal

C₂×C₁₁×C₄⋊C₄

C₄

yes

—

C₂×C₄²×C₁₁

C₂²

yes

—

C₂×C₄²×C₁₁

C₂²

yes

characteristic

C₂²×C₁₁⋊C₈

C₂²

yes

—

C₂²×C₁₁⋊C₈

C₂²

yes

characteristic

C₂²×C₁₁⋊C₈

C₂²

yes

characteristic

C₁₁×#95

C₄

yes

characteristic

C₁₁×#95

C₄

yes

characteristic

C₄⋊(C₁₁⋊C₈)

—

C₄⋊(C₁₁⋊C₈)

—

#66867

—

C₂²×C₄×C₁₁

C₂×C₄

yes

—

C₂²×C₄×C₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₁

C₂×C₄

yes

—

C₂²×C₄×C₁₁

C₂³

yes

characteristic

C₁₁×C₄⋊C₄

—

C₄²×C₁₁

—

C₄²×C₁₁

—

C₄²×C₁₁

D₄

yes

—

C₄²×C₁₁

Q₈

yes

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

D₄

yes

—

C₂×C₁₁⋊C₈

Q₈

yes

—

#1940

—

maximal, Sylow

C₂³×C₁₁

C₂²×C₄

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

C₄⋊C₄

yes

—

C₂×C₄×C₁₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₁

M₁₆

yes

—

C₂×C₄×C₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₁

C₂×Q₈

yes

C₂×C₄×C₁₁

yes

C₂×C₄×C₁₁

yes

C₁₁⋊C₈

—

C₁₁⋊C₈

—

C₁₁⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₄×C₄⋊C₄

D₁₁

yes

characteristic

#336

—

C₂²×C₁₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₁

C₄×D₄

yes

characteristic

C₂²×C₁₁

C₄×Q₈

yes

characteristic

C₂²×C₁₁

C₂×M₁₆

yes

—

C₂²×C₁₁

#131

yes

—

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

#121

yes

characteristic

C₄×C₁₁

C₂²⋊Q₈

yes

characteristic

C₄×C₁₁

#125

yes

characteristic

C₄×C₁₁

C₄⋊Q₈

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

Dic₁₁

yes

—

C₂×C₄²

C₂×D₁₁

yes

—

C₂×C₄²

C₂×D₁₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#95

Dic₁₁

yes

characteristic

#95

Dic₁₁

yes

C₂×C₁₁

yes

C₂×C₁₁

yes

C₂×C₁₁

yes

C₂×C₁₁

yes

—

C₂×C₁₁

#446

yes

—

C₂²×C₄

C₂×Dic₁₁

yes

—

C₂²×C₄

C₂×Dic₁₁

yes

—

C₂²×C₄

C₂×Dic₁₁

yes

characteristic

C₂²×C₄

C₂×Dic₁₁

yes

characteristic

C₂²×C₄

C₂²×D₁₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₄⋊C₄

—

106

C₄²

—

107

C₄²

—

108

C₄²

Dic₂₂

yes

—

109

C₄²

D₄₄

yes

—

110

C₁₁

#1940

yes

characteristic, complemented, Sylow

111

C₂³

C₂²×Dic₁₁

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

#4318

yes

—

120

C₂×C₄

C₄⋊Dic₁₁

yes

—

121

C₂×C₄

C₄⋊Dic₁₁

yes

—

122

C₂×C₄

C₂×Dic₂₂

yes

characteristic

123

C₂×C₄

C₂×D₄₄

yes

characteristic

124

C₂×C₄

D₄×D₁₁

yes

characteristic

125

C₂×C₄

#4341

yes

characteristic

126

C₂×C₄

Q₈×D₁₁

yes

characteristic

127

C₂×C₄

#4343

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

131

C₂²

C₂×#4318

yes

—

132

C₂²

C₂×C₄⋊Dic₁₁

yes

characteristic

133

C₂²

D₄×Dic₁₁

yes

characteristic

134

C₂²

Q₈×Dic₁₁

yes

characteristic

135

C₂²

#67001

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#66941

yes

characteristic

141

C₄

#66943

yes

characteristic

142

C₄

#66948

yes

characteristic

143

C₄

#66950

yes

characteristic

144

C₂

#10622537

yes

characteristic

145

C₂

#10622598

yes

characteristic

146

C₂

#10622612

yes

characteristic

147

C₂

#10622624

yes

—

148

C₂

#10622637

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₁
commutator, socle, prime radical	C₂³×C₁₁	C₂²×C₄
Fitting	C₄×C₁₁×C₄⋊C₄	C₂
cosocle	C₂²×C₄×C₁₁	C₂³
hypercenter	C₄×C₄⋊C₄	D₁₁
hypocenter	C₁₁	#1940
LCS	G ▷ C₂³×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₁

commutator, socle, prime radical

C₂³×C₁₁

C₂²×C₄

Fitting

C₄×C₁₁×C₄⋊C₄

C₂

cosocle

C₂²×C₄×C₁₁

C₂³

hypercenter

C₄×C₄⋊C₄

D₁₁

C₁₁

G ▷ C₂³×C₁₁ ▷ C₁₁

C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
110. C₁₁	#1940	semidirect

H ≅ G/N

type

110. C₁₁

#1940

semidirect

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

subgroup	index
C₄×C₁₁×C₄⋊C₄	2
C₂×C₄⋊(C₁₁⋊C₈)	2
C₂×#66867	2
#10622162	2
#1940	11

subgroup

index

C₄×C₁₁×C₄⋊C₄

C₂×C₄⋊(C₁₁⋊C₈)

C₂×#66867

#10622162

#1940

Subgroups — #13069340

Conjugacy classes of subgroups

Complements

Maximal subgroups