Group #13070159 — subgroups

Subgroups — #13070159

Order 1408.

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

normal subgroups only

This group has 254 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×#391	C₂	1	1	yes	maximal, characteristic
2	C₂×#66899	C₂	1	2	yes	maximal
3	C₂×#66872	C₂	1	2	yes	maximal
4	#10622548	C₂	1	1	yes	maximal, characteristic
5	#10622162	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₁	C₂²	1	1	yes	characteristic
7	C₂×C₁₁×C₄⋊C₄	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₁₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₁	—	2	2	no	—
10	C₂×C₄⋊Dic₁₁	C₂²	1	2	yes	—
11	C₂²×C₁₁⋊C₈	C₂²	1	2	yes	—
12	C₁₁×#95	—	2	2	no	—
13	#66899	—	2	4	no	—
14	#66872	—	2	4	no	—
15	C₂×Q₈×C₁₁	—	2	1	no	—
16	C₂×Q₈×C₁₁	—	2	2	no	—
17	C₂×Q₈×C₁₁	—	2	2	no	—
18	C₂×Q₈×C₁₁	Q₈	1	2	yes	—
19	C₂²×C₄×C₁₁	—	2	1	no	—
20	C₂²×C₄×C₁₁	—	2	1	no	—
21	C₂²×C₄×C₁₁	D₄	1	2	yes	—
22	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
23	C₂²×Dic₁₁	—	2	2	no	—
24	C₁₁×C₄⋊C₄	—	2	2	no	—
25	C₄²×C₁₁	—	2	2	no	—
26	C₄⋊Dic₁₁	—	2	2	no	—
27	C₄⋊Dic₁₁	—	2	2	no	—
28	C₄⋊Dic₁₁	—	4	4	no	—
29	C₂×C₁₁⋊C₈	—	2	2	no	—
30	C₂×C₁₁⋊C₈	—	2	2	no	—
31	C₂×C₁₁⋊C₈	D₄	1	4	yes	—
32	#2096	—	11	1	no	maximal, Sylow
33	C₂³×C₁₁	C₂×D₄	1	1	yes	characteristic
34	Q₈×C₁₁	—	2	4	no	—
35	Q₈×C₁₁	—	4	2	no	—
36	C₂×C₄×C₁₁	—	2	2	no	—
37	C₂×C₄×C₁₁	—	2	2	no	—
38	C₂×C₄×C₁₁	—	2	2	no	—
39	C₂×C₄×C₁₁	—	4	1	no	—
40	C₂×C₄×C₁₁	—	4	1	no	—
41	C₂×C₄×C₁₁	—	4	1	no	—
42	C₂×C₄×C₁₁	—	4	2	no	—
43	C₂×C₄×C₁₁	SD₁₆	1	4	yes	—
44	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₁	#41	1	1	yes	characteristic
47	C₂×C₄×C₁₁	#41	1	2	yes	—
48	C₂×Dic₁₁	—	4	2	no	—
49	C₂×Dic₁₁	—	4	4	no	—
50	C₁₁⋊C₈	—	2	4	no	—
51	C₂×#106	—	11	2	no	—
52	C₂×#103	—	11	2	no	—
53	#391	D₁₁	1	1	yes	characteristic
54	#384	—	11	1	no	—
55	#336	—	11	1	no	—
56	C₂²×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
57	C₂²×C₁₁	#123	1	1	yes	characteristic
58	C₂²×C₁₁	#127	1	1	yes	characteristic
59	C₂²×C₁₁	C₂×SD₁₆	1	2	yes	—
60	C₂²×C₁₁	#137	1	2	yes	—
61	C₄×C₁₁	—	2	4	no	—
62	C₄×C₁₁	—	4	2	no	—
63	C₄×C₁₁	—	8	1	no	—
64	C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₁₁	#123	1	2	yes	—
66	Dic₁₁	—	8	2	no	—
67	C₂²×Q₈	C₂×D₁₁	1	1	yes	characteristic
68	C₂²×C₈	—	11	2	no	—
69	C₂×C₄⋊C₄	—	11	2	no	—
70	C₂×C₄⋊C₄	—	22	2	no	—
71	C₂×C₄⋊C₄	C₂×D₁₁	1	1	yes	characteristic
72	C₂×C₄²	C₂×D₁₁	1	1	yes	characteristic
73	#106	—	22	4	no	—
74	#103	—	22	4	no	—
75	#95	—	2	2	no	—
76	C₂×C₁₁	#399	1	1	yes	characteristic
77	C₂×C₁₁	#475	1	2	yes	—
78	C₂×C₁₁	#483	1	2	yes	—
79	C₂×C₁₁	#492	1	1	yes	characteristic
80	C₂×C₁₁	#497	1	1	yes	characteristic
81	C₂×Q₈	—	2	1	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	Dic₂₂	1	2	yes	—
85	C₂²×C₄	—	2	1	no	—
86	C₂²×C₄	—	2	1	no	—
87	C₂²×C₄	—	22	2	no	—
88	C₂²×C₄	C₂²⋊D₁₁	1	2	yes	—
89	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
90	C₂×C₈	—	11	4	no	—
91	C₂×C₈	—	22	2	no	—
92	C₂×C₈	—	22	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	22	2	no	—
95	C₄⋊C₄	—	22	2	no	—
96	C₄⋊C₄	—	44	4	no	—
97	C₄²	—	2	2	no	—
98	C₁₁	#2096	1	1	yes	characteristic, complemented, Sylow
99	C₂³	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
100	Q₈	—	2	4	no	—
101	Q₈	—	4	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	44	2	no	—
110	C₂×C₄	—	44	4	no	—
111	C₂×C₄	#4325	1	4	yes	—
112	C₂×C₄	C₂×Dic₂₂	1	1	yes	characteristic
113	C₂×C₄	#4339	1	1	yes	characteristic
114	C₂×C₄	D₄×D₁₁	1	2	yes	—
115	C₂×C₄	#4341	1	2	yes	—
116	C₈	—	22	4	no	—
117	C₂²	#66977	1	1	yes	characteristic
118	C₂²	#66988	1	1	yes	characteristic
119	C₂²	#66993	1	1	yes	characteristic
120	C₂²	C₂×#4325	1	2	yes	—
121	C₂²	#67004	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	4	2	no	—
124	C₄	—	8	1	no	—
125	C₄	—	88	2	no	—
126	C₄	#66931	1	2	yes	—
127	C₄	#66936	1	2	yes	—
128	C₂	#10622562	1	1	yes	characteristic
129	C₂	#10622633	1	2	yes	—
130	C₂	#10622656	1	2	yes	—
131	C₂	#10622673	1	1	yes	characteristic
132	C₂	#10622698	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₁×#391

C₂

yes

maximal, characteristic

C₂×#66899

C₂

yes

maximal

C₂×#66872

C₂

yes

maximal

#10622548

C₂

yes

maximal, characteristic

#10622162

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₁

C₂²

yes

characteristic

C₂×C₁₁×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄²×C₁₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₁

—

C₂×C₄⋊Dic₁₁

C₂²

yes

—

C₂²×C₁₁⋊C₈

C₂²

yes

—

C₁₁×#95

—

#66899

—

#66872

—

C₂×Q₈×C₁₁

—

C₂×Q₈×C₁₁

—

C₂×Q₈×C₁₁

—

C₂×Q₈×C₁₁

Q₈

yes

—

C₂²×C₄×C₁₁

—

C₂²×C₄×C₁₁

—

C₂²×C₄×C₁₁

D₄

yes

—

C₂²×C₄×C₁₁

C₂³

yes

characteristic

C₂²×Dic₁₁

—

C₁₁×C₄⋊C₄

—

C₄²×C₁₁

—

C₄⋊Dic₁₁

—

C₄⋊Dic₁₁

—

C₄⋊Dic₁₁

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

D₄

yes

—

#2096

—

maximal, Sylow

C₂³×C₁₁

C₂×D₄

yes

characteristic

Q₈×C₁₁

—

Q₈×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

C₂×D₄

yes

—

C₂×C₄×C₁₁

C₂×Q₈

yes

C₂×C₄×C₁₁

yes

C₂×C₄×C₁₁

yes

—

C₂×Dic₁₁

—

C₂×Dic₁₁

—

C₁₁⋊C₈

—

C₂×#106

—

C₂×#103

—

#391

D₁₁

yes

characteristic

#384

—

#336

—

C₂²×C₁₁

C₂²⋊Q₈

yes

C₂²×C₁₁

yes

C₂²×C₁₁

yes

C₂²×C₁₁

C₂×SD₁₆

yes

—

C₂²×C₁₁

#137

yes

—

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

C₂²⋊Q₈

yes

—

C₄×C₁₁

#123

yes

—

Dic₁₁

—

C₂²×Q₈

C₂×D₁₁

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₁

yes

characteristic

C₂×C₄²

C₂×D₁₁

yes

—

—

—

C₂×C₁₁

#399

yes

characteristic

C₂×C₁₁

#475

yes

—

C₂×C₁₁

#483

yes

—

C₂×C₁₁

yes

C₂×C₁₁

yes

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₂₂

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₁

yes

—

C₂²×C₄

C₂²×D₁₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₁₁

#2096

yes

characteristic, complemented, Sylow

C₂³

C₂×C₂²⋊D₁₁

yes

characteristic

100

Q₈

—

101

Q₈

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

#4325

yes

—

112

C₂×C₄

C₂×Dic₂₂

yes

characteristic

113

C₂×C₄

#4339

yes

characteristic

114

C₂×C₄

D₄×D₁₁

yes

—

115

C₂×C₄

#4341

yes

—

116

C₈

—

117

C₂²

#66977

yes

characteristic

118

C₂²

#66988

yes

characteristic

119

C₂²

#66993

yes

characteristic

120

C₂²

C₂×#4325

yes

—

121

C₂²

#67004

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#66931

yes

—

127

C₄

#66936

yes

—

128

C₂

#10622562

yes

characteristic

129

C₂

#10622633

yes

—

130

C₂

#10622656

yes

—

131

C₂

#10622673

yes

characteristic

132

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×#391	C₂
socle, prime radical	C₂³×C₁₁	C₂×D₄
hypercenter	#391	D₁₁
hypocenter	C₁₁	#2096
LCS	G ▷ C₂²×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₁

commutator, cosocle

C₂²×C₄×C₁₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₁

Fitting

C₁₁×#391

C₂

socle, prime radical

C₂³×C₁₁

C₂×D₄

D₁₁

C₁₁

G ▷ C₂²×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
98. C₁₁	#2096	semidirect

H ≅ G/N

type

98. C₁₁

#2096

semidirect

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

subgroup	index
C₁₁×#391	2
C₂×#66899	2
C₂×#66872	2
#10622548	2
#10622162	2
#2096	11

subgroup

index

C₁₁×#391

Subgroups — #13070159

Conjugacy classes of subgroups

Complements

Maximal subgroups