Group #11771255 — subgroups

Subgroups — #11771255

Order 1152.

390 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 157 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×#432	C₂	1	1	yes	maximal, characteristic
2	#10592625	C₂	1	1	yes	maximal, characteristic
3	#10592623	C₂	1	1	yes	maximal, characteristic
4	#67758	—	3	4	no	maximal
5	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
6	C₂×#3748	C₄	1	2	yes	—
7	C₃²×C₄⋊C₈	—	2	1	no	—
8	C₃²×C₂²⋊C₈	C₄	1	2	yes	—
9	C₃²×#97	—	2	1	no	—
10	#63657	C₄	1	2	yes	—
11	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
12	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
13	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
14	#63630	—	2	1	no	—
15	C₃×#432	D₃	1	4	yes	—
16	#4926	—	3	4	no	—
17	#4924	—	3	4	no	—
18	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃²×C₈	—	2	2	no	—
22	C₃²×C₄²	D₄	1	1	yes	characteristic
23	C₃²×C₄²	D₄	1	1	yes	characteristic
24	C₃²×C₄²	D₄	1	1	yes	characteristic
25	C₃²×C₄²	Q₈	1	1	yes	characteristic
26	#3748	—	4	2	no	—
27	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
28	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
29	#1307	—	9	1	no	Sylow
30	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
31	C₂×#208	—	3	8	no	—
32	C₃×C₄⋊C₈	—	2	4	no	—
33	C₃×C₂²⋊C₈	Dic₃	1	8	yes	—
34	C₃×#97	—	2	4	no	—
35	#830	—	3	8	no	—
36	C₄⋊(C₃⋊C₈)	—	6	4	no	—
37	C₄⋊(C₃⋊C₈)	—	6	4	no	—
38	C₄⋊(C₃⋊C₈)	—	6	4	no	—
39	#803	—	6	4	no	—
40	C₂³×C₃²	C₄²	1	1	yes	characteristic
41	C₂×C₃²×C₄	—	2	1	no	—
42	C₂×C₃²×C₄	—	2	1	no	—
43	C₂×C₃²×C₄	—	4	1	no	—
44	C₂×C₃²×C₄	—	4	1	no	—
45	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
51	C₃²×C₈	—	4	2	no	—
52	C₃⋊(C₃⋊C₈)	—	4	2	no	—
53	C₃⋊(C₃⋊C₈)	—	4	2	no	—
54	#432	—	9	1	no	—
55	#432	C₃⋊D₃	1	1	yes	characteristic
56	#423	—	9	1	no	—
57	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
58	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
59	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
60	C₂×C₃×C₈	—	2	8	no	—
61	C₃×C₄²	Dic₆	1	4	yes	—
62	C₃×C₄²	D₁₂	1	4	yes	—
63	C₃×C₄²	#67	1	4	yes	—
64	C₃×C₄²	#67	1	4	yes	—
65	#208	—	12	8	no	—
66	C₂×C₃⋊C₈	—	6	8	no	—
67	C₂×C₃⋊C₈	—	6	8	no	—
68	C₂²×C₃²	—	2	1	no	—
69	C₂²×C₃²	—	2	1	no	—
70	C₂²×C₃²	—	2	1	no	—
71	C₂²×C₃²	#95	1	1	yes	characteristic
72	C₃²×C₄	—	2	1	no	—
73	C₃²×C₄	—	2	1	no	—
74	C₃²×C₄	—	2	1	no	—
75	C₃²×C₄	—	2	1	no	—
76	C₃²×C₄	—	2	1	no	—
77	C₃²×C₄	#100	1	1	yes	characteristic
78	C₃²×C₄	#101	1	1	yes	characteristic
79	C₂×M₁₆	—	9	2	no	—
80	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
81	C₄⋊C₈	—	2	1	no	—
82	C₄⋊C₈	—	18	1	no	—
83	C₄⋊C₈	—	18	1	no	—
84	C₄⋊C₈	—	18	1	no	—
85	C₂²⋊C₈	—	9	2	no	—
86	C₂²⋊C₈	C₃⋊Dic₃	1	2	yes	—
87	#97	—	2	1	no	—
88	#97	—	18	1	no	—
89	C₂³×C₃	C₄×Dic₃	1	4	yes	—
90	C₂×C₃×C₄	—	2	4	no	—
91	C₂×C₃×C₄	—	2	4	no	—
92	C₂×C₃×C₄	—	4	4	no	—
93	C₂×C₃×C₄	—	4	4	no	—
94	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
95	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
96	C₂×C₃×C₄	C₄⋊Dic₃	1	4	yes	—
97	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
98	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
99	C₂×C₃×C₄	#217	1	4	yes	—
100	C₃×C₈	—	4	8	no	—
101	C₃⋊C₈	—	12	8	no	—
102	C₃⋊C₈	—	12	8	no	—
103	C₂×C₃²	—	4	1	no	—
104	C₂×C₃²	#343	1	1	yes	characteristic
105	C₂×C₃²	#344	1	1	yes	characteristic
106	C₂×C₃²	#344	1	1	yes	characteristic
107	C₂²×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
108	C₂²×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
109	C₂²×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
110	M₁₆	—	36	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	18	2	no	—
113	C₂×C₈	—	18	2	no	—
114	C₄²	#634	1	1	yes	characteristic
115	C₄²	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
116	C₄²	#638	1	1	yes	characteristic
117	C₄²	#638	1	1	yes	characteristic
118	C₂²×C₃	—	2	4	no	—
119	C₂²×C₃	—	2	4	no	—
120	C₂²×C₃	—	2	4	no	—
121	C₂²×C₃	C₃⋊#95	1	4	yes	—
122	C₃×C₄	—	2	4	no	—
123	C₃×C₄	—	2	4	no	—
124	C₃×C₄	—	2	4	no	—
125	C₃×C₄	—	2	4	no	—
126	C₃×C₄	—	2	4	no	—
127	C₃×C₄	#823	1	4	yes	—
128	C₃×C₄	#824	1	4	yes	—
129	C₃²	#1307	1	1	yes	characteristic, complemented, Sylow
130	C₂³	C₄×C₃⋊Dic₃	1	1	yes	characteristic
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	#3750	1	1	yes	characteristic
136	C₂×C₄	#3750	1	1	yes	characteristic
137	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
138	C₂×C₄	#3752	1	1	yes	characteristic
139	C₂×C₄	#3752	1	1	yes	characteristic
140	C₂×C₄	#3757	1	1	yes	characteristic
141	C₈	—	4	2	no	—
142	C₈	—	36	2	no	—
143	C₈	—	36	2	no	—
144	C₂×C₃	—	4	4	no	—
145	C₂×C₃	#4533	1	4	yes	—
146	C₂×C₃	#4554	1	4	yes	—
147	C₂×C₃	#4556	1	4	yes	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	#63658	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	#63650	1	1	yes	characteristic
158	C₄	#63651	1	1	yes	characteristic
159	C₃	#67758	1	4	yes	complemented
160	C₂	—	4	1	no	—
161	C₂	#10589399	1	1	yes	characteristic
162	C₂	#10589420	1	1	yes	characteristic
163	C₂	#10589422	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#432

C₂

yes

maximal, characteristic

#10592625

C₂

yes

maximal, characteristic

#10592623

C₂

yes

maximal, characteristic

#67758

—

maximal

C₂×C₃²×C₄²

C₂²

yes

characteristic

C₂×#3748

C₄

yes

—

C₃²×C₄⋊C₈

—

C₃²×C₂²⋊C₈

C₄

yes

—

C₃²×#97

—

#63657

C₄

yes

—

C₄⋊(C₃⋊(C₃⋊C₈))

—

C₄⋊(C₃⋊(C₃⋊C₈))

—

C₄⋊(C₃⋊(C₃⋊C₈))

—

#63630

—

C₃×#432

D₃

yes

—

#4926

—

#4924

—

C₂²×C₃²×C₄

C₂×C₄

yes

characteristic

C₂²×C₃²×C₄

C₂×C₄

yes

characteristic

C₂²×C₃²×C₄

C₂×C₄

yes

characteristic

C₂×C₃²×C₈

—

C₃²×C₄²

D₄

yes

characteristic

C₃²×C₄²

D₄

yes

characteristic

C₃²×C₄²

D₄

yes

characteristic

C₃²×C₄²

Q₈

yes

characteristic

#3748

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂×C₃⋊(C₃⋊C₈)

—

#1307

—

Sylow

C₂×C₃×C₄²

C₂×D₃

yes

—

C₂×#208

—

C₃×C₄⋊C₈

—

C₃×C₂²⋊C₈

Dic₃

yes

—

C₃×#97

—

#830

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

—

#803

—

C₂³×C₃²

C₄²

yes

characteristic

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃²×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃²×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃²×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃²×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃²×C₄

C₄⋊C₄

yes

characteristic

C₃²×C₈

—

C₃⋊(C₃⋊C₈)

—

C₃⋊(C₃⋊C₈)

—

#432

—

#432

C₃⋊D₃

yes

characteristic

#423

—

C₂²×C₃×C₄

C₄×D₃

yes

—

C₂²×C₃×C₄

C₄×D₃

yes

—

C₂²×C₃×C₄

C₂×Dic₃

yes

—

C₂×C₃×C₈

—

C₃×C₄²

Dic₆

yes

—

C₃×C₄²

D₁₂

yes

—

C₃×C₄²

#67

yes

—

C₃×C₄²

#67

yes

—

#208

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

#95

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

yes

C₃²×C₄

yes

C₂×M₁₆

—

C₂×C₄²

C₂×C₃⋊D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

C₃⋊Dic₃

yes

—

#97

—

#97

—

C₂³×C₃

C₄×Dic₃

yes

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

Dic₃⋊C₄

yes

—

C₂×C₃×C₄

Dic₃⋊C₄

yes

—

C₂×C₃×C₄

C₄⋊Dic₃

yes

—

C₂×C₃×C₄

(C₂×D₃)⋊C₄

yes

—

C₂×C₃×C₄

(C₂×D₃)⋊C₄

yes

—

C₂×C₃×C₄

#217

yes

—

100

C₃×C₈

—

101

C₃⋊C₈

—

102

C₃⋊C₈

—

103

C₂×C₃²

—

104

C₂×C₃²

#343

yes

characteristic

105

C₂×C₃²

#344

yes

characteristic

106

C₂×C₃²

#344

yes

characteristic

107

C₂²×C₄

C₄×C₃⋊D₃

yes

characteristic

108

C₂²×C₄

C₄×C₃⋊D₃

yes

characteristic

109

C₂²×C₄

C₂×C₃⋊Dic₃

yes

characteristic

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₄²

#634

yes

characteristic

115

C₄²

C₄⋊(C₃⋊D₃)

yes

characteristic

116

C₄²

#638

yes

characteristic

117

C₄²

#638

yes

characteristic

118

C₂²×C₃

—

119

C₂²×C₃

—

120

C₂²×C₃

—

121

C₂²×C₃

C₃⋊#95

yes

—

122

C₃×C₄

—

123

C₃×C₄

—

124

C₃×C₄

—

125

C₃×C₄

—

126

C₃×C₄

—

127

C₃×C₄

#823

yes

—

128

C₃×C₄

#824

yes

—

129

C₃²

#1307

yes

characteristic, complemented, Sylow

130

C₂³

C₄×C₃⋊Dic₃

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

#3750

yes

characteristic

136

C₂×C₄

#3750

yes

characteristic

137

C₂×C₄

C₄⋊(C₃⋊Dic₃)

yes

characteristic

138

C₂×C₄

#3752

yes

characteristic

139

C₂×C₄

#3752

yes

characteristic

140

C₂×C₄

#3757

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₂×C₃

—

145

C₂×C₃

#4533

yes

—

146

C₂×C₃

#4554

yes

—

147

C₂×C₃

#4556

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

#63658

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

#63650

yes

characteristic

158

C₄

#63651

yes

characteristic

159

C₃

#67758

yes

complemented

160

C₂

—

161

C₂

#10589399

yes

characteristic

162

C₂

#10589420

yes

characteristic

163

C₂

#10589422

yes

characteristic

	subgroup	G/H
center	C₂²	#63658
commutator, prime radical	C₂³×C₃²	C₄²
Frattini	C₂×C₄²	C₂×C₃⋊D₃
Fitting	C₃²×#432	C₂
socle	C₂²×C₃²	#95
cosocle	C₂×C₃²×C₄²	C₂²
hypercenter	#432	C₃⋊D₃
hypocenter	C₃²	#1307
LCS	G ▷ C₂³×C₃² ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂²

#63658

commutator, prime radical

C₂³×C₃²

C₄²

Frattini

C₂×C₄²

C₂×C₃⋊D₃

Fitting

C₃²×#432

C₂

socle

C₂²×C₃²

#95

cosocle

C₂×C₃²×C₄²

C₂²

C₃⋊D₃

C₃²

G ▷ C₂³×C₃² ▷ C₂²×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
129. C₃²	#1307	semidirect
159. C₃	#67758	semidirect

H ≅ G/N

type

129. C₃²

#1307

semidirect

159. C₃

#67758

semidirect

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

subgroup	index
C₃²×#432	2
#10592625	2
#10592623	2
#67758	3

subgroup

index

C₃²×#432

#10592625

#10592623

#67758

Subgroups — #11771255

Conjugacy classes of subgroups

Complements

Maximal subgroups