Group #13074737 — subgroups

Subgroups — #13074737

Order 1408.

448 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 165 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×C₁₁×#125	C₂	1	1	yes	maximal, characteristic
2	C₂×#66922	C₂	1	1	yes	maximal, characteristic
3	#10622596	C₂	1	4	yes	maximal
4	#10622592	C₂	1	4	yes	maximal
5	#10622590	C₂	1	2	yes	maximal
6	#10622589	C₂	1	2	yes	maximal
7	#10622545	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₂	C₂²	1	2	yes	—
9	C₁₁×#125	—	2	4	no	—
10	C₂×C₁₁×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₂×C₁₁×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₁	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₁₁	C₂²	1	2	yes	—
17	C₂×Dic₁₁⋊C₄	C₂²	1	8	yes	—
18	C₂×C₄×Dic₁₁	C₂²	1	2	yes	—
19	#66922	—	2	4	no	—
20	C₁₁⋊#95	C₂²	1	2	yes	—
21	C₁₁⋊#95	C₂²	1	2	yes	—
22	C₁₁⋊#95	C₂²	1	8	yes	—
23	C₂²×C₄×C₁₁	C₂³	1	2	yes	—
24	C₂²×C₄×C₁₁	C₂³	1	2	yes	—
25	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
28	C₂²×Dic₁₁	C₂³	1	4	yes	—
29	C₂²×Dic₁₁	C₂³	1	4	yes	—
30	C₂×Dic₂₂	—	2	4	no	—
31	C₂×Dic₂₂	—	4	4	no	—
32	C₁₁×C₄⋊C₄	—	2	4	no	—
33	C₁₁×C₄⋊C₄	—	2	4	no	—
34	C₁₁×C₄⋊C₄	—	2	2	no	—
35	C₁₁×C₄⋊C₄	—	2	2	no	—
36	C₄²×C₁₁	D₄	1	4	yes	—
37	C₄⋊Dic₁₁	—	2	4	no	—
38	C₄⋊Dic₁₁	—	2	4	no	—
39	Dic₁₁⋊C₄	—	4	8	no	—
40	C₄×Dic₁₁	Q₈	1	8	yes	complemented
41	#2660	—	11	1	no	maximal, Sylow
42	C₂³×C₁₁	C₂⁴	1	1	yes	characteristic
43	C₂×C₄×C₁₁	—	2	2	no	—
44	C₂×C₄×C₁₁	—	2	2	no	—
45	C₂×C₄×C₁₁	—	2	2	no	—
46	C₂×C₄×C₁₁	—	2	2	no	—
47	C₂×C₄×C₁₁	—	2	4	no	—
48	C₂×C₄×C₁₁	—	2	4	no	—
49	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₁	C₂×Q₈	1	4	yes	—
53	C₂×Dic₁₁	—	2	4	no	—
54	C₂×Dic₁₁	—	2	4	no	—
55	C₂×Dic₁₁	—	2	4	no	—
56	C₂×Dic₁₁	—	2	8	no	—
57	C₂×Dic₁₁	C₂×Q₈	1	8	yes	—
58	Dic₂₂	—	4	8	no	—
59	C₂×C₄⋊Q₈	—	11	1	no	—
60	C₂×#125	D₁₁	1	1	yes	characteristic
61	#398	—	11	4	no	—
62	#395	—	11	4	no	—
63	#384	—	11	2	no	—
64	#382	—	11	2	no	—
65	#376	—	11	1	no	—
66	C₂²×C₁₁	C₂²×D₄	1	1	yes	characteristic
67	C₂²×C₁₁	C₂²×Q₈	1	2	yes	—
68	C₂²×C₁₁	H(5,𝔽₂)	1	2	yes	—
69	C₂²×C₁₁	2₋¹⁺⁴	1	2	yes	—
70	C₄×C₁₁	—	2	2	no	—
71	C₄×C₁₁	—	2	2	no	—
72	C₄×C₁₁	—	4	2	no	—
73	C₄×C₁₁	—	4	2	no	—
74	C₄×C₁₁	C₄⋊Q₈	1	4	yes	—
75	Dic₁₁	—	2	8	no	—
76	Dic₁₁	—	4	4	no	—
77	C₂²×Q₈	—	11	2	no	—
78	C₄⋊Q₈	—	22	4	no	—
79	#125	—	2	4	no	—
80	C₂×C₄⋊C₄	—	11	2	no	—
81	C₂×C₄⋊C₄	—	11	2	no	—
82	C₂×C₄⋊C₄	—	11	8	no	—
83	C₂×C₄⋊C₄	C₂×D₁₁	1	1	yes	characteristic
84	C₂×C₄⋊C₄	C₂×D₁₁	1	1	yes	characteristic
85	C₂×C₄⋊C₄	C₂×D₁₁	1	2	yes	—
86	C₂×C₄⋊C₄	C₂×D₁₁	1	2	yes	—
87	C₂×C₄²	—	11	2	no	—
88	C₂×C₄²	C₂×D₁₁	1	1	yes	characteristic
89	#95	—	11	2	no	—
90	#95	—	11	2	no	—
91	#95	—	11	8	no	—
92	C₂×C₁₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
93	C₂×C₁₁	#535	1	1	yes	characteristic
94	C₂×C₁₁	#536	1	1	yes	characteristic
95	C₂×C₁₁	#544	1	4	yes	—
96	C₂×Q₈	—	22	4	no	—
97	C₂×Q₈	—	44	4	no	—
98	C₂²×C₄	—	11	4	no	—
99	C₂²×C₄	—	11	4	no	—
100	C₂²×C₄	C₂²×D₁₁	1	2	yes	—
101	C₂²×C₄	C₂²×D₁₁	1	2	yes	—
102	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
104	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
105	C₄⋊C₄	—	2	4	no	—
106	C₄⋊C₄	—	2	4	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	22	4	no	—
110	C₄⋊C₄	—	22	4	no	—
111	C₄⋊C₄	—	44	8	no	—
112	C₄²	—	11	8	no	—
113	C₄²	C₂²⋊D₁₁	1	4	yes	—
114	C₁₁	#2660	1	1	yes	characteristic, complemented, Sylow
115	C₂³	C₂³×D₁₁	1	1	yes	characteristic
116	Q₈	—	44	8	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	11	8	no	—
124	C₂×C₄	—	22	4	no	—
125	C₂×C₄	—	22	4	no	—
126	C₂×C₄	—	22	8	no	—
127	C₂×C₄	—	22	4	no	—
128	C₂×C₄	Q₈×D₁₁	1	4	yes	—
129	C₂×C₄	C₂×C₂²⋊D₁₁	1	2	yes	—
130	C₂×C₄	C₂×C₂²⋊D₁₁	1	2	yes	—
131	C₂×C₄	C₂×C₂²⋊D₁₁	1	2	yes	—
132	C₂²	C₂×Q₈×D₁₁	1	2	yes	—
133	C₂²	#67042	1	2	yes	—
134	C₂²	#67043	1	2	yes	—
135	C₂²	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	4	2	no	—
139	C₄	—	4	2	no	—
140	C₄	—	22	8	no	—
141	C₄	—	44	4	no	—
142	C₄	#66997	1	4	yes	—
143	C₂	#10623220	1	4	yes	—
144	C₂	C₂×#66997	1	1	yes	characteristic
145	C₂	#10623356	1	1	yes	characteristic
146	C₂	#10623361	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₁×#125

C₂

yes

maximal, characteristic

C₂×#66922

C₂

yes

maximal, characteristic

#10622596

C₂

yes

maximal

#10622592

C₂

yes

maximal

#10622590

C₂

yes

maximal

#10622589

C₂

yes

maximal

#10622545

C₂

yes

maximal, characteristic

C₂²×Dic₂₂

C₂²

yes

—

C₁₁×#125

—

C₂×C₁₁×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₁₁×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₁₁×C₄⋊C₄

C₂²

yes

—

C₂×C₁₁×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₁

C₂²

yes

—

C₂×C₄⋊Dic₁₁

C₂²

yes

—

C₂×Dic₁₁⋊C₄

C₂²

yes

—

C₂×C₄×Dic₁₁

C₂²

yes

—

#66922

—

C₁₁⋊#95

C₂²

yes

—

C₁₁⋊#95

C₂²

yes

—

C₁₁⋊#95

C₂²

yes

—

C₂²×C₄×C₁₁

C₂³

yes

—

C₂²×C₄×C₁₁

C₂³

yes

—

C₂²×C₄×C₁₁

C₂³

yes

characteristic

C₂²×C₄×C₁₁

C₂³

yes

characteristic

C₂²×C₄×C₁₁

C₂³

yes

characteristic

C₂²×Dic₁₁

C₂³

yes

—

C₂²×Dic₁₁

C₂³

yes

—

C₂×Dic₂₂

—

C₂×Dic₂₂

—

C₁₁×C₄⋊C₄

—

C₁₁×C₄⋊C₄

—

C₁₁×C₄⋊C₄

—

C₁₁×C₄⋊C₄

—

C₄²×C₁₁

D₄

yes

—

C₄⋊Dic₁₁

—

C₄⋊Dic₁₁

—

Dic₁₁⋊C₄

—

C₄×Dic₁₁

Q₈

yes

complemented

#2660

—

maximal, Sylow

C₂³×C₁₁

C₂⁴

yes

characteristic

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

C₂×D₄

yes

—

C₂×C₄×C₁₁

C₂×D₄

yes

—

C₂×C₄×C₁₁

C₂×D₄

yes

—

C₂×C₄×C₁₁

C₂×Q₈

yes

—

C₂×Dic₁₁

—

C₂×Dic₁₁

—

C₂×Dic₁₁

—

C₂×Dic₁₁

—

C₂×Dic₁₁

C₂×Q₈

yes

—

Dic₂₂

—

C₂×C₄⋊Q₈

—

C₂×#125

D₁₁

yes

—

—

—

—

—

C₂²×C₁₁

C₂²×D₄

yes

characteristic

C₂²×C₁₁

C₂²×Q₈

yes

—

C₂²×C₁₁

H(5,𝔽₂)

yes

—

C₂²×C₁₁

2₋¹⁺⁴

yes

—

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

C₄⋊Q₈

yes

—

Dic₁₁

—

Dic₁₁

—

C₂²×Q₈

—

C₄⋊Q₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₁

yes

characteristic

C₂×C₄⋊C₄

C₂×D₁₁

yes

characteristic

C₂×C₄⋊C₄

C₂×D₁₁

yes

—

C₂×C₄⋊C₄

C₂×D₁₁

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₁

yes

—

—

—

C₂×C₁₁

C₂×C₄⋊Q₈

yes

C₂×C₁₁

yes

C₂×C₁₁

yes

C₂×C₁₁

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

C₂²×D₁₁

yes

—

101

C₂²×C₄

C₂²×D₁₁

yes

—

102

C₂²×C₄

C₂²×D₁₁

yes

characteristic

103

C₂²×C₄

C₂²×D₁₁

yes

characteristic

104

C₂²×C₄

C₂²×D₁₁

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

—

113

C₄²

C₂²⋊D₁₁

yes

—

114

C₁₁

#2660

yes

characteristic, complemented, Sylow

115

C₂³

C₂³×D₁₁

yes

characteristic

116

Q₈

—

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

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

Q₈×D₁₁

yes

—

129

C₂×C₄

C₂×C₂²⋊D₁₁

yes

—

130

C₂×C₄

C₂×C₂²⋊D₁₁

yes

—

131

C₂×C₄

C₂×C₂²⋊D₁₁

yes

—

132

C₂²

C₂×Q₈×D₁₁

yes

—

133

C₂²

#67042

yes

—

134

C₂²

#67043

yes

—

135

C₂²

C₂²×C₂²⋊D₁₁

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

#66997

yes

—

143

C₂

#10623220

yes

—

144

C₂

C₂×#66997

yes

characteristic

145

C₂

#10623356

yes

characteristic

146

C₂

#10623361

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₁
commutator, socle, prime radical, cosocle	C₂³×C₁₁	C₂⁴
Fitting	C₂×C₁₁×#125	C₂
hypercenter	C₂×#125	D₁₁
hypocenter	C₁₁	#2660
LCS	G ▷ C₂³×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₁

commutator, socle, prime radical, cosocle

C₂³×C₁₁

C₂⁴

Fitting

C₂×C₁₁×#125

C₂

D₁₁

C₁₁

G ▷ C₂³×C₁₁ ▷ C₁₁

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
40. C₄×Dic₁₁	Q₈	semidirect
114. C₁₁	#2660	semidirect

H ≅ G/N

type

40. C₄×Dic₁₁

Q₈

semidirect

114. C₁₁

#2660

semidirect

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

subgroup	index
C₂×C₁₁×#125	2
C₂×#66922	2
#10622596	2
#10622592	2
#10622590	2
#10622589	2
#10622545	2
#2660	11

subgroup

index

C₂×C₁₁×#125

Subgroups — #13074737

Conjugacy classes of subgroups

Complements

Maximal subgroups