Group #11781950 — subgroups

Subgroups — #11781950

Order 1152.

456 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 139 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	C₃²×#500	C₂	1	1	yes	maximal, characteristic
2	#10592789	C₂	1	2	yes	maximal
3	#10592378	C₂	1	1	yes	maximal, characteristic
4	#10589413	C₂	1	2	yes	maximal
5	#10589323	C₂	1	1	yes	maximal, characteristic
6	#71379	—	3	4	no	maximal
7	C₃²×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×#3756	—	2	2	no	—
9	C₄×#634	C₂²	1	1	yes	characteristic
10	C₃²×#107	—	2	1	no	—
11	C₃²×#107	—	2	1	no	—
12	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
13	#63662	—	2	2	no	—
14	#63642	—	2	1	no	—
15	#63641	—	2	1	no	—
16	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	—
17	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
18	C₃×#500	D₃	1	4	yes	—
19	#5090	—	3	8	no	—
20	#4679	—	3	4	no	—
21	#4547	—	3	8	no	—
22	#4457	—	3	4	no	—
23	C₂×C₃²×Q₈	—	2	2	no	—
24	C₂×#634	D₄	1	1	yes	characteristic
25	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
26	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
27	C₃²×C₄⋊C₄	—	2	2	no	—
28	C₃²×C₄⋊C₄	—	2	2	no	—
29	C₃²×C₄²	C₂³	1	1	yes	characteristic
30	#3756	—	4	4	no	—
31	C₄⋊(C₃⋊Dic₃)	D₄	1	1	yes	characteristic
32	#3750	—	2	1	no	—
33	C₄×C₃⋊Dic₃	—	2	1	no	—
34	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
35	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
36	#1694	—	9	1	no	Sylow
37	C₃×C₄⋊Q₈	C₂×D₃	1	8	yes	—
38	C₂×#216	—	6	8	no	—
39	C₄×Dic₆	—	3	4	no	—
40	C₃×#107	—	2	4	no	—
41	C₃×#107	—	2	4	no	—
42	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
43	#835	—	6	8	no	—
44	#815	—	6	4	no	—
45	Dic₃⋊C₈	—	6	4	no	—
46	C₄⋊(C₃⋊C₈)	—	3	4	no	—
47	C₄⋊(C₃⋊C₈)	—	3	8	no	—
48	C₃²×Q₈	—	4	2	no	—
49	C₃²×Q₈	—	4	2	no	—
50	C₂×C₃²×C₄	—	4	2	no	—
51	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
52	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
53	C₂×C₃²×C₄	#41	1	1	yes	characteristic
54	C₂×C₃⋊Dic₃	—	2	1	no	—
55	C₂×C₃⋊Dic₃	—	2	1	no	—
56	#634	—	2	2	no	—
57	C₃²×C₈	—	2	1	no	—
58	C₃²×C₈	Q₁₆	1	2	yes	complemented
59	C₃⋊(C₃⋊C₈)	—	4	2	no	—
60	C₃⋊(C₃⋊C₈)	—	4	1	no	—
61	#500	C₃⋊D₃	1	1	yes	characteristic
62	#462	—	9	2	no	—
63	#446	—	9	1	no	—
64	#334	—	9	1	no	—
65	#332	—	9	2	no	—
66	C₂×C₃×Q₈	—	2	8	no	—
67	C₂×Dic₆	—	3	4	no	—
68	C₂×C₃×C₈	#67	1	4	yes	—
69	C₂×C₃×C₈	#67	1	4	yes	—
70	C₃×C₄⋊C₄	—	2	8	no	—
71	C₃×C₄⋊C₄	—	2	8	no	—
72	C₃×C₄²	C₂²×D₃	1	4	yes	—
73	#216	—	12	16	no	—
74	C₄⋊Dic₃	—	3	4	no	—
75	Dic₃⋊C₄	—	6	4	no	—
76	C₄×Dic₃	—	6	4	no	—
77	C₂×C₃⋊C₈	—	6	8	no	—
78	C₂×C₃⋊C₈	—	6	4	no	—
79	C₂²×C₃²	#121	1	1	yes	characteristic
80	C₃²×C₄	—	2	1	no	—
81	C₃²×C₄	—	8	2	no	—
82	C₃²×C₄	C₂×Q₁₆	1	1	yes	characteristic
83	C₃²×C₄	AGL(1,ℤ/8)	1	2	yes	—
84	C₃²×C₄	#137	1	1	yes	characteristic
85	C₃⋊Dic₃	—	4	1	no	—
86	C₃⋊Dic₃	—	4	1	no	—
87	C₂×Q₁₆	—	18	2	no	—
88	C₄⋊Q₈	C₂×C₃⋊D₃	1	2	yes	—
89	C₄×Q₈	—	9	1	no	—
90	#107	—	2	1	no	—
91	#107	—	2	1	no	—
92	C₄⋊C₈	—	9	1	no	—
93	C₄⋊C₈	—	9	2	no	—
94	C₄⋊C₈	—	18	1	no	—
95	#103	—	18	2	no	—
96	#97	—	18	1	no	—
97	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
98	C₃×Q₈	—	4	8	no	—
99	C₃×Q₈	—	4	8	no	—
100	C₂×C₃×C₄	—	4	8	no	—
101	C₂×C₃×C₄	D₃×D₄	1	4	yes	—
102	C₂×C₃×C₄	#237	1	4	yes	—
103	C₂×C₃×C₄	C₂×#67	1	4	yes	—
104	C₂×Dic₃	—	6	4	no	—
105	C₂×Dic₃	—	6	4	no	—
106	Dic₆	—	6	8	no	—
107	C₃×C₈	—	2	4	no	—
108	C₃×C₈	#216	1	8	yes	complemented
109	C₃⋊C₈	—	12	8	no	—
110	C₃⋊C₈	—	12	4	no	—
111	C₂×C₃²	#462	1	1	yes	characteristic
112	C₂×C₃²	#469	1	1	yes	characteristic
113	C₂×C₃²	#473	1	1	yes	characteristic
114	C₂×Q₈	—	2	2	no	—
115	C₂×Q₈	—	9	1	no	—
116	Q₁₆	—	36	4	no	—
117	C₂×C₈	—	18	2	no	—
118	C₂×C₈	—	18	1	no	—
119	C₂×C₈	#638	1	1	yes	characteristic
120	C₂×C₈	#638	1	1	yes	characteristic
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	9	1	no	—
124	C₄⋊C₄	—	18	1	no	—
125	C₄²	—	18	1	no	—
126	C₄²	C₂²×C₃⋊D₃	1	1	yes	characteristic
127	C₂²×C₃	#938	1	4	yes	—
128	C₃×C₄	—	2	4	no	—
129	C₃×C₄	—	8	8	no	—
130	C₃×C₄	#911	1	8	yes	—
131	C₃×C₄	#942	1	4	yes	—
132	C₃×C₄	C₂×#216	1	4	yes	—
133	Dic₃	—	12	4	no	—
134	Dic₃	—	12	4	no	—
135	C₃²	#1694	1	1	yes	characteristic, complemented, Sylow
136	Q₈	—	4	2	no	—
137	Q₈	—	4	2	no	—
138	Q₈	—	18	2	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	18	1	no	—
141	C₂×C₄	—	18	1	no	—
142	C₂×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
143	C₂×C₄	#3830	1	1	yes	characteristic
144	C₂×C₄	C₂×#638	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	36	2	no	—
147	C₈	—	36	1	no	—
148	C₈	#3756	1	2	yes	complemented
149	C₂×C₃	#5090	1	4	yes	—
150	C₂×C₃	#5159	1	4	yes	—
151	C₂×C₃	#5192	1	4	yes	—
152	C₂²	#64147	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	8	2	no	—
155	C₄	—	36	1	no	—
156	C₄	—	36	1	no	—
157	C₄	#64120	1	2	yes	—
158	C₄	#64151	1	1	yes	characteristic
159	C₄	C₂×#3756	1	1	yes	characteristic
160	C₃	#71379	1	4	yes	complemented
161	C₂	#10592789	1	1	yes	characteristic
162	C₂	#10592858	1	1	yes	characteristic
163	C₂	#10592891	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#500

C₂

yes

maximal, characteristic

#10592789

C₂

yes

maximal

#10592378

C₂

yes

maximal, characteristic

#10589413

C₂

yes

maximal

#10589323

C₂

yes

maximal, characteristic

#71379

—

maximal

C₃²×C₄⋊Q₈

C₂²

yes

—

C₂×#3756

—

C₄×#634

C₂²

yes

characteristic

C₃²×#107

—

C₃²×#107

—

C₃²×C₄×C₈

C₂²

yes

—

—

—

C₄⋊(C₃⋊(C₃⋊C₈))

C₂²

yes

—

C₄⋊(C₃⋊(C₃⋊C₈))

C₂²

yes

characteristic

C₃×#500

D₃

yes

—

—

—

—

—

C₂×C₃²×Q₈

—

C₂×#634

D₄

yes

characteristic

C₂×C₃²×C₈

D₄

yes

characteristic

C₂×C₃²×C₈

D₄

yes

characteristic

C₃²×C₄⋊C₄

—

C₃²×C₄⋊C₄

—

C₃²×C₄²

C₂³

yes

characteristic

#3756

—

C₄⋊(C₃⋊Dic₃)

D₄

yes

characteristic

#3750

—

C₄×C₃⋊Dic₃

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂×C₃⋊(C₃⋊C₈)

—

#1694

—

Sylow

C₃×C₄⋊Q₈

C₂×D₃

yes

—

C₂×#216

—

C₄×Dic₆

—

C₃×#107

—

C₃×#107

—

C₃×C₄×C₈

C₂×D₃

yes

—

#835

—

#815

—

Dic₃⋊C₈

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

—

C₃²×Q₈

—

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃²×C₄

#41

yes

characteristic

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

#634

—

C₃²×C₈

—

C₃²×C₈

Q₁₆

yes

complemented

C₃⋊(C₃⋊C₈)

—

C₃⋊(C₃⋊C₈)

—

#500

C₃⋊D₃

yes

—

—

—

—

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₂×C₃×C₈

#67

yes

—

C₂×C₃×C₈

#67

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂²×D₃

yes

—

#216

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₃²

#121

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₂×Q₁₆

yes

characteristic

C₃²×C₄

AGL(1,ℤ/8)

yes

—

C₃²×C₄

#137

yes

characteristic

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₂×Q₁₆

—

C₄⋊Q₈

C₂×C₃⋊D₃

yes

—

C₄×Q₈

—

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#97

—

C₄×C₈

C₂×C₃⋊D₃

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

100

C₂×C₃×C₄

—

101

C₂×C₃×C₄

D₃×D₄

yes

—

102

C₂×C₃×C₄

#237

yes

—

103

C₂×C₃×C₄

C₂×#67

yes

—

104

C₂×Dic₃

—

105

C₂×Dic₃

—

106

Dic₆

—

107

C₃×C₈

—

108

C₃×C₈

#216

yes

complemented

109

C₃⋊C₈

—

110

C₃⋊C₈

—

111

C₂×C₃²

#462

yes

characteristic

112

C₂×C₃²

#469

yes

characteristic

113

C₂×C₃²

#473

yes

characteristic

114

C₂×Q₈

—

115

C₂×Q₈

—

116

Q₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

#638

yes

characteristic

120

C₂×C₈

#638

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄²

—

126

C₄²

C₂²×C₃⋊D₃

yes

characteristic

127

C₂²×C₃

#938

yes

—

128

C₃×C₄

—

129

C₃×C₄

—

130

C₃×C₄

#911

yes

—

131

C₃×C₄

#942

yes

—

132

C₃×C₄

C₂×#216

yes

—

133

Dic₃

—

134

Dic₃

—

135

C₃²

#1694

yes

characteristic, complemented, Sylow

136

Q₈

—

137

Q₈

—

138

Q₈

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

D₄×C₃⋊D₃

yes

characteristic

143

C₂×C₄

#3830

yes

characteristic

144

C₂×C₄

C₂×#638

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

#3756

yes

complemented

149

C₂×C₃

#5090

yes

—

150

C₂×C₃

#5159

yes

—

151

C₂×C₃

#5192

yes

—

152

C₂²

#64147

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

#64120

yes

—

158

C₄

#64151

yes

characteristic

159

C₄

C₂×#3756

yes

characteristic

160

C₃

#71379

yes

complemented

161

C₂

#10592789

yes

characteristic

162

C₂

#10592858

yes

characteristic

163

C₂

#10592891

yes

characteristic

	subgroup	G/H
center	C₂²	#64147
commutator, cosocle	C₃²×C₄²	C₂³
Frattini	C₄²	C₂²×C₃⋊D₃
Fitting	C₃²×#500	C₂
socle, prime radical	C₂²×C₃²	#121
hypercenter	#500	C₃⋊D₃
hypocenter	C₃²	#1694
LCS	G ▷ C₃²×C₄² ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#64147

commutator, cosocle

C₃²×C₄²

C₂³

Frattini

C₄²

C₂²×C₃⋊D₃

Fitting

C₃²×#500

C₂

C₂²×C₃²

C₃⋊D₃

C₃²

G ▷ C₃²×C₄² ▷ C₂²×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
58. C₃²×C₈	Q₁₆	semidirect
108. C₃×C₈	#216	semidirect
135. C₃²	#1694	semidirect
148. C₈	#3756	semidirect
160. C₃	#71379	semidirect

H ≅ G/N

type

58. C₃²×C₈

Q₁₆

semidirect

108. C₃×C₈

#216

semidirect

135. C₃²

#1694

semidirect

148. C₈

#3756

semidirect

160. C₃

#71379

semidirect

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

subgroup	index
C₃²×#500	2
#10592789	2
#10592378	2
#10589413	2
#10589323	2
#71379	3

subgroup

index

C₃²×#500

Subgroups — #11781950

Conjugacy classes of subgroups

Complements

Maximal subgroups