Group #13070150 — subgroups

Subgroups — #13070150

Order 1408.

290 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 91 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×#384	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#66873	C₂	1	2	yes	maximal
3	C₂×#66872	C₂	1	2	yes	maximal, complemented
4	#10622552	C₂	1	1	yes	maximal, characteristic
5	#10622162	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄₄	C₂²	1	1	yes	characteristic
7	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
8	C₂×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₄²×C₁₁	C₂²	1	1	yes	characteristic
10	C₂×(C₂×D₁₁)⋊C₄	—	2	2	no	—
11	C₂×C₄⋊Dic₁₁	C₂²	1	1	yes	characteristic
12	C₂²×C₁₁⋊C₈	C₂²	1	2	yes	—
13	#66873	—	2	4	no	—
14	#66872	—	2	4	no	—
15	C₂³×D₁₁	—	2	1	no	—
16	C₂²×C₄×C₁₁	—	2	2	no	—
17	C₂²×C₄×C₁₁	D₄	1	2	yes	—
18	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
19	C₂²×Dic₁₁	—	2	1	no	—
20	C₂×D₄₄	—	2	2	no	—
21	C₂×D₄₄	—	2	1	no	—
22	C₂×D₄₄	—	2	2	no	—
23	C₂×D₄₄	Q₈	1	2	yes	—
24	C₁₁×C₄⋊C₄	—	2	2	no	—
25	C₁₁×C₄⋊C₄	—	2	2	no	—
26	C₁₁×C₄⋊C₄	—	4	4	no	—
27	C₄²×C₁₁	—	2	2	no	—
28	(C₂×D₁₁)⋊C₄	—	4	4	no	—
29	C₄⋊Dic₁₁	—	2	2	no	—
30	C₂×C₁₁⋊C₈	—	2	2	no	—
31	C₂×C₁₁⋊C₈	—	2	2	no	—
32	C₂×C₁₁⋊C₈	D₄	1	4	yes	—
33	#2094	—	11	1	no	maximal, Sylow
34	C₂³×C₁₁	C₂×D₄	1	1	yes	characteristic
35	C₂²×D₁₁	—	2	2	no	—
36	C₂²×D₁₁	—	4	1	no	—
37	C₂²×D₁₁	—	4	1	no	—
38	C₂²×D₁₁	—	4	2	no	—
39	C₂²×D₁₁	—	4	2	no	—
40	C₂×C₄×C₁₁	—	2	2	no	—
41	C₂×C₄×C₁₁	—	2	2	no	—
42	C₂×C₄×C₁₁	—	4	2	no	—
43	C₂×C₄×C₁₁	—	4	4	no	—
44	C₂×C₄×C₁₁	SD₁₆	1	4	yes	—
45	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₁₁	#41	1	2	yes	—
48	C₂×C₄×C₁₁	#41	1	1	yes	characteristic
49	C₂×Dic₁₁	—	4	1	no	—
50	C₂×Dic₁₁	—	4	2	no	—
51	D₄₄	—	2	4	no	—
52	D₄₄	—	4	2	no	—
53	C₁₁⋊C₈	—	2	4	no	—
54	C₂×#106	—	11	2	no	—
55	C₂×#102	—	11	2	no	—
56	#390	—	11	1	no	—
57	#384	D₁₁	1	1	yes	characteristic, complemented
58	#336	—	11	1	no	—
59	C₂²×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
60	C₂²×C₁₁	#123	1	1	yes	characteristic
61	C₂²×C₁₁	#127	1	1	yes	characteristic
62	C₂²×C₁₁	C₂×SD₁₆	1	2	yes	—
63	C₂²×C₁₁	AGL(1,ℤ/8)	1	2	yes	—
64	C₂×D₁₁	—	4	2	no	—
65	C₂×D₁₁	—	4	4	no	—
66	C₂×D₁₁	—	8	2	no	—
67	C₂×D₁₁	—	8	1	no	—
68	C₂×D₁₁	—	8	1	no	—
69	C₄×C₁₁	—	2	4	no	—
70	C₄×C₁₁	—	8	2	no	—
71	C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
72	C₄×C₁₁	#123	1	2	yes	—
73	Dic₁₁	—	8	1	no	—
74	C₂²×D₄	—	11	1	no	—
75	C₂²×C₈	—	11	2	no	—
76	C₂×C₄⋊C₄	—	2	2	no	—
77	C₂×C₄⋊C₄	—	11	1	no	—
78	C₂×C₄⋊C₄	C₂×D₁₁	1	2	yes	—
79	C₂×C₂²⋊C₄	—	22	2	no	—
80	C₂×C₄²	C₂×D₁₁	1	1	yes	characteristic
81	#106	—	22	4	no	—
82	#102	—	22	4	no	—
83	C₂×C₁₁	#399	1	1	yes	characteristic
84	C₂×C₁₁	#476	1	2	yes	—
85	C₂×C₁₁	#481	1	2	yes	—
86	C₂×C₁₁	#492	1	1	yes	characteristic
87	C₂×C₁₁	#496	1	1	yes	characteristic
88	D₁₁	—	8	2	no	—
89	C₂⁴	—	22	1	no	—
90	C₂×D₄	—	11	2	no	—
91	C₂×D₄	—	22	2	no	—
92	C₂×D₄	—	22	2	no	—
93	C₂×D₄	—	22	1	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	22	1	no	—
96	C₂²×C₄	C₂²⋊D₁₁	1	2	yes	—
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	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	4	4	no	—
104	C₄⋊C₄	—	22	2	no	—
105	C₂²⋊C₄	—	44	4	no	—
106	C₄²	—	2	2	no	—
107	C₁₁	#2094	1	1	yes	characteristic, complemented, Sylow
108	C₂³	—	22	2	no	—
109	C₂³	—	44	1	no	—
110	C₂³	—	44	2	no	—
111	C₂³	—	44	1	no	—
112	C₂³	—	44	2	no	—
113	C₂³	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
114	D₄	—	22	4	no	—
115	D₄	—	44	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	4	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	44	1	no	—
121	C₂×C₄	—	44	2	no	—
122	C₂×C₄	#4325	1	4	yes	—
123	C₂×C₄	#4339	1	2	yes	—
124	C₂×C₄	D₄×D₁₁	1	2	yes	—
125	C₂×C₄	Q₈×D₁₁	1	1	yes	characteristic
126	C₂×C₄	#4343	1	1	yes	characteristic
127	C₈	—	22	4	no	—
128	C₂²	—	44	2	no	—
129	C₂²	—	44	4	no	—
130	C₂²	—	88	2	no	—
131	C₂²	—	88	1	no	—
132	C₂²	—	88	1	no	—
133	C₂²	#66982	1	1	yes	characteristic
134	C₂²	#66985	1	2	yes	—
135	C₂²	#66993	1	1	yes	characteristic
136	C₂²	C₂×#4325	1	2	yes	—
137	C₂²	#66999	1	1	yes	characteristic
138	C₄	—	2	4	no	—
139	C₄	—	8	2	no	—
140	C₄	—	88	1	no	—
141	C₄	#66947	1	2	yes	—
142	C₄	#66949	1	2	yes	—
143	C₂	—	88	2	no	—
144	C₂	#10622581	1	2	yes	—
145	C₂	#10622605	1	1	yes	characteristic
146	C₂	#10622687	1	1	yes	characteristic
147	C₂	#10622697	1	2	yes	—
148	C₂	#10622698	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₁×#384

C₂

yes

maximal, characteristic, complemented

C₂×#66873

C₂

yes

maximal

C₂×#66872

C₂

yes

maximal, complemented

#10622552

C₂

yes

maximal, characteristic

#10622162

C₂

yes

maximal, characteristic, complemented

C₂²×D₄₄

C₂²

yes

characteristic

C₂×C₁₁×C₄⋊C₄

—

C₂×C₁₁×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₁

C₂²

yes

characteristic

C₂×(C₂×D₁₁)⋊C₄

—

C₂×C₄⋊Dic₁₁

C₂²

yes

characteristic

C₂²×C₁₁⋊C₈

C₂²

yes

—

#66873

—

#66872

—

C₂³×D₁₁

—

C₂²×C₄×C₁₁

—

C₂²×C₄×C₁₁

D₄

yes

—

C₂²×C₄×C₁₁

C₂³

yes

characteristic

C₂²×Dic₁₁

—

C₂×D₄₄

—

C₂×D₄₄

—

C₂×D₄₄

—

C₂×D₄₄

Q₈

yes

—

C₁₁×C₄⋊C₄

—

C₁₁×C₄⋊C₄

—

C₁₁×C₄⋊C₄

—

C₄²×C₁₁

—

(C₂×D₁₁)⋊C₄

—

C₄⋊Dic₁₁

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

D₄

yes

—

#2094

—

maximal, Sylow

C₂³×C₁₁

C₂×D₄

yes

characteristic

C₂²×D₁₁

—

C₂²×D₁₁

—

C₂²×D₁₁

—

C₂²×D₁₁

—

C₂²×D₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

SD₁₆

yes

—

C₂×C₄×C₁₁

C₂×D₄

yes

—

C₂×C₄×C₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₁

#41

yes

—

C₂×C₄×C₁₁

#41

yes

characteristic

C₂×Dic₁₁

—

C₂×Dic₁₁

—

D₄₄

—

D₄₄

—

C₁₁⋊C₈

—

—

—

—

D₁₁

yes

characteristic, complemented

#336

—

C₂²×C₁₁

C₂²⋊Q₈

yes

C₂²×C₁₁

yes

C₂²×C₁₁

yes

C₂²×C₁₁

C₂×SD₁₆

yes

—

C₂²×C₁₁

AGL(1,ℤ/8)

yes

—

C₂×D₁₁

—

C₂×D₁₁

—

C₂×D₁₁

—

C₂×D₁₁

—

C₂×D₁₁

—

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

C₂²⋊Q₈

yes

—

C₄×C₁₁

#123

yes

—

Dic₁₁

—

C₂²×D₄

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₁

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₁₁

yes

characteristic

#106

—

#102

—

C₂×C₁₁

#399

yes

characteristic

C₂×C₁₁

#476

yes

—

C₂×C₁₁

#481

yes

—

C₂×C₁₁

yes

C₂×C₁₁

yes

D₁₁

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₁

yes

—

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

C₂²⋊C₄

—

106

C₄²

—

107

C₁₁

#2094

yes

characteristic, complemented, Sylow

108

C₂³

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

C₂×C₂²⋊D₁₁

yes

characteristic

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

#4325

yes

—

123

C₂×C₄

#4339

yes

—

124

C₂×C₄

D₄×D₁₁

yes

—

125

C₂×C₄

Q₈×D₁₁

yes

characteristic

126

C₂×C₄

#4343

yes

characteristic

127

C₈

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

#66982

yes

characteristic

134

C₂²

#66985

yes

—

135

C₂²

#66993

yes

characteristic

136

C₂²

C₂×#4325

yes

—

137

C₂²

#66999

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#66947

yes

—

142

C₄

#66949

yes

—

143

C₂

—

144

C₂

#10622581

yes

—

145

C₂

#10622605

yes

characteristic

146

C₂

#10622687

yes

characteristic

147

C₂

#10622697

yes

—

148

C₂

#10622698

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₁
commutator, cosocle	C₂²×C₄×C₁₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₁
Fitting	C₁₁×#384	C₂
socle	C₂³×C₁₁	C₂×D₄
prime radical	C₂²×D₄₄	C₂²
hypercenter	#384	D₁₁
hypocenter	C₁₁	#2094
LCS	G ▷ C₂²×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₁

commutator, cosocle

C₂²×C₄×C₁₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₁

Fitting

C₁₁×#384

C₂

socle

C₂³×C₁₁

C₂×D₄

prime radical

C₂²×D₄₄

C₂²

D₁₁

C₁₁

G ▷ C₂²×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
1. C₁₁×#384	C₂	semidirect
3. C₂×#66872	C₂	semidirect
5. #10622162	C₂	semidirect
57. #384	D₁₁	semidirect
107. C₁₁	#2094	semidirect

H ≅ G/N

type

1. C₁₁×#384

C₂

semidirect

3. C₂×#66872

C₂

semidirect

5. #10622162

C₂

semidirect

57. #384

D₁₁

semidirect

107. C₁₁

#2094

semidirect

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

subgroup	index
C₁₁×#384	2
C₂×#66873	2
C₂×#66872	2
#10622552	2
#10622162	2
#2094	11

subgroup

index

C₁₁×#384

Subgroups — #13070150

Conjugacy classes of subgroups

Complements

Maximal subgroups