Group #11771002 — subgroups

Subgroups — #11771002

Order 1152.

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

normal subgroups only

This group has 414 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	C₄×#3748	C₂	1	1	yes	maximal, characteristic
4	#67754	—	3	4	no	maximal
5	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
6	C₂×#3748	—	2	1	no	—
7	C₃²×C₄⋊C₈	—	2	1	no	—
8	C₃²×C₂²⋊C₈	C₄	1	2	yes	—
9	C₃²×#97	—	2	1	no	—
10	#63657	—	2	1	no	—
11	C₄⋊(C₃⋊(C₃⋊C₈))	C₄	1	2	yes	—
12	#63630	—	2	1	no	—
13	#63630	—	2	1	no	—
14	C₄×C₃⋊(C₃⋊C₈)	C₄	1	2	yes	—
15	C₃×#432	D₃	1	4	yes	—
16	#4926	—	3	4	no	—
17	C₄×#208	—	3	4	no	—
18	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
20	C₂²×C₃²×C₄	Q₈	1	1	yes	characteristic
21	C₂×C₃²×C₈	—	2	2	no	—
22	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
23	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
24	C₃²×C₄²	D₄	1	1	yes	characteristic
25	C₃²×C₄²	D₄	1	1	yes	characteristic
26	#3748	—	2	4	no	—
27	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
28	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
29	#1291	—	9	1	no	Sylow
30	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
31	C₂×#208	—	6	4	no	—
32	C₃×C₄⋊C₈	—	2	4	no	—
33	C₃×C₂²⋊C₈	Dic₃	1	8	yes	—
34	C₃×#97	—	2	4	no	—
35	#830	—	6	4	no	—
36	C₄⋊(C₃⋊C₈)	—	3	8	no	—
37	#803	—	6	4	no	—
38	#803	—	6	4	no	—
39	C₄×C₃⋊C₈	—	3	8	no	—
40	C₂³×C₃²	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₃²×C₄	—	2	2	no	—
42	C₂×C₃²×C₄	—	4	1	no	—
43	C₂×C₃²×C₄	—	4	1	no	—
44	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
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	C₃²×C₈	—	4	2	no	—
51	C₃⋊(C₃⋊C₈)	—	2	4	no	—
52	C₃⋊(C₃⋊C₈)	—	4	2	no	—
53	#432	—	9	1	no	—
54	#432	C₃⋊D₃	1	1	yes	characteristic
55	C₄×M₁₆	—	9	1	no	—
56	C₂²×C₃×C₄	Dic₆	1	4	yes	—
57	C₂²×C₃×C₄	D₁₂	1	4	yes	—
58	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
59	C₂×C₃×C₈	—	2	8	no	—
60	C₃×C₄²	C₄×D₃	1	4	yes	—
61	C₃×C₄²	C₄×D₃	1	4	yes	—
62	C₃×C₄²	#67	1	4	yes	—
63	C₃×C₄²	#67	1	4	yes	—
64	#208	—	6	16	no	—
65	C₂×C₃⋊C₈	—	6	8	no	—
66	C₂×C₃⋊C₈	—	6	8	no	—
67	C₂²×C₃²	—	2	1	no	—
68	C₂²×C₃²	—	2	1	no	—
69	C₂²×C₃²	—	2	1	no	—
70	C₂²×C₃²	#95	1	1	yes	characteristic
71	C₃²×C₄	—	2	1	no	—
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	#104	1	2	yes	—
77	C₂×M₁₆	—	18	1	no	—
78	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
79	C₄⋊C₈	—	2	1	no	—
80	C₄⋊C₈	—	9	2	no	—
81	C₂²⋊C₈	—	18	1	no	—
82	C₂²⋊C₈	C₃⋊Dic₃	1	2	yes	—
83	#97	—	2	1	no	—
84	#97	—	18	1	no	—
85	#97	—	18	1	no	—
86	C₄×C₈	—	9	2	no	—
87	C₂³×C₃	C₄⋊Dic₃	1	4	yes	—
88	C₂×C₃×C₄	—	2	8	no	—
89	C₂×C₃×C₄	—	4	4	no	—
90	C₂×C₃×C₄	—	4	4	no	—
91	C₂×C₃×C₄	C₄×Dic₃	1	4	yes	—
92	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
93	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
94	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
95	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
96	C₂×C₃×C₄	#217	1	4	yes	—
97	C₃×C₈	—	4	8	no	—
98	C₃⋊C₈	—	6	16	no	—
99	C₃⋊C₈	—	12	8	no	—
100	C₂×C₃²	—	4	1	no	—
101	C₂×C₃²	#338	1	1	yes	characteristic
102	C₂×C₃²	#339	1	1	yes	characteristic
103	C₂×C₃²	#344	1	1	yes	characteristic
104	C₂²×C₄	#634	1	1	yes	characteristic
105	C₂²×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
106	C₂²×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
107	M₁₆	—	18	4	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	18	2	no	—
110	C₂×C₈	—	18	2	no	—
111	C₄²	C₄×C₃⋊D₃	1	1	yes	characteristic
112	C₄²	C₄×C₃⋊D₃	1	1	yes	characteristic
113	C₄²	#638	1	1	yes	characteristic
114	C₄²	#638	1	1	yes	characteristic
115	C₂²×C₃	—	2	4	no	—
116	C₂²×C₃	—	2	4	no	—
117	C₂²×C₃	—	2	4	no	—
118	C₂²×C₃	C₃⋊#95	1	4	yes	—
119	C₃×C₄	—	2	4	no	—
120	C₃×C₄	—	2	4	no	—
121	C₃×C₄	—	2	4	no	—
122	C₃×C₄	—	2	4	no	—
123	C₃×C₄	—	2	4	no	—
124	C₃×C₄	#825	1	8	yes	—
125	C₃²	#1291	1	1	yes	characteristic, complemented, Sylow
126	C₂³	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	C₄×C₃⋊Dic₃	1	1	yes	characteristic
131	C₂×C₄	#3750	1	1	yes	characteristic
132	C₂×C₄	#3750	1	1	yes	characteristic
133	C₂×C₄	#3752	1	1	yes	characteristic
134	C₂×C₄	#3752	1	1	yes	characteristic
135	C₂×C₄	#3757	1	1	yes	characteristic
136	C₈	—	4	2	no	—
137	C₈	—	18	4	no	—
138	C₈	—	36	2	no	—
139	C₂×C₃	—	4	4	no	—
140	C₂×C₃	#4533	1	4	yes	—
141	C₂×C₃	#4555	1	4	yes	—
142	C₂×C₃	C₃⋊#338	1	4	yes	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	#63658	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	#63652	1	2	yes	—
153	C₃	#67754	1	4	yes	complemented
154	C₂	—	4	1	no	—
155	C₂	#10589399	1	1	yes	characteristic
156	C₂	#10589421	1	1	yes	characteristic
157	C₂	#10589426	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#432

C₂

yes

maximal, characteristic

#10592625

C₂

yes

maximal, characteristic

C₄×#3748

C₂

yes

maximal, characteristic

#67754

—

maximal

C₂×C₃²×C₄²

C₂²

yes

characteristic

C₂×#3748

—

C₃²×C₄⋊C₈

—

C₃²×C₂²⋊C₈

C₄

yes

—

C₃²×#97

—

#63657

—

C₄⋊(C₃⋊(C₃⋊C₈))

C₄

yes

—

#63630

—

#63630

—

C₄×C₃⋊(C₃⋊C₈)

C₄

yes

—

C₃×#432

D₃

yes

—

#4926

—

C₄×#208

—

C₂²×C₃²×C₄

C₂×C₄

yes

characteristic

C₂²×C₃²×C₄

D₄

yes

characteristic

C₂²×C₃²×C₄

Q₈

yes

characteristic

C₂×C₃²×C₈

—

C₃²×C₄²

C₂×C₄

yes

characteristic

C₃²×C₄²

C₂×C₄

yes

characteristic

C₃²×C₄²

D₄

yes

characteristic

C₃²×C₄²

D₄

yes

characteristic

#3748

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂×C₃⋊(C₃⋊C₈)

—

#1291

—

Sylow

C₂×C₃×C₄²

C₂×D₃

yes

—

C₂×#208

—

C₃×C₄⋊C₈

—

C₃×C₂²⋊C₈

Dic₃

yes

—

C₃×#97

—

#830

—

C₄⋊(C₃⋊C₈)

—

#803

—

#803

—

C₄×C₃⋊C₈

—

C₂³×C₃²

C₄⋊C₄

yes

characteristic

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₄²

yes

characteristic

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

C₃²×C₈

—

C₃⋊(C₃⋊C₈)

—

C₃⋊(C₃⋊C₈)

—

#432

—

#432

C₃⋊D₃

yes

characteristic

C₄×M₁₆

—

C₂²×C₃×C₄

Dic₆

yes

—

C₂²×C₃×C₄

D₁₂

yes

—

C₂²×C₃×C₄

C₂×Dic₃

yes

—

C₂×C₃×C₈

—

C₃×C₄²

C₄×D₃

yes

—

C₃×C₄²

C₄×D₃

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

#104

yes

—

C₂×M₁₆

—

C₂×C₄²

C₂×C₃⋊D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

C₃⋊Dic₃

yes

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₂³×C₃

C₄⋊Dic₃

yes

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄×Dic₃

yes

—

C₂×C₃×C₄

Dic₃⋊C₄

yes

—

C₂×C₃×C₄

Dic₃⋊C₄

yes

—

C₂×C₃×C₄

(C₂×D₃)⋊C₄

yes

—

C₂×C₃×C₄

(C₂×D₃)⋊C₄

yes

—

C₂×C₃×C₄

#217

yes

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

100

C₂×C₃²

—

101

C₂×C₃²

#338

yes

characteristic

102

C₂×C₃²

#339

yes

characteristic

103

C₂×C₃²

#344

yes

characteristic

104

C₂²×C₄

#634

yes

characteristic

105

C₂²×C₄

C₄⋊(C₃⋊D₃)

yes

characteristic

106

C₂²×C₄

C₂×C₃⋊Dic₃

yes

characteristic

107

M₁₆

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₄²

C₄×C₃⋊D₃

yes

characteristic

112

C₄²

C₄×C₃⋊D₃

yes

characteristic

113

C₄²

#638

yes

characteristic

114

C₄²

#638

yes

characteristic

115

C₂²×C₃

—

116

C₂²×C₃

—

117

C₂²×C₃

—

118

C₂²×C₃

C₃⋊#95

yes

—

119

C₃×C₄

—

120

C₃×C₄

—

121

C₃×C₄

—

122

C₃×C₄

—

123

C₃×C₄

—

124

C₃×C₄

#825

yes

—

125

C₃²

#1291

yes

characteristic, complemented, Sylow

126

C₂³

C₄⋊(C₃⋊Dic₃)

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₄×C₃⋊Dic₃

yes

characteristic

131

C₂×C₄

#3750

yes

characteristic

132

C₂×C₄

#3750

yes

characteristic

133

C₂×C₄

#3752

yes

characteristic

134

C₂×C₄

#3752

yes

characteristic

135

C₂×C₄

#3757

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

—

139

C₂×C₃

—

140

C₂×C₃

#4533

yes

—

141

C₂×C₃

#4555

yes

—

142

C₂×C₃

C₃⋊#338

yes

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

#63658

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

#63652

yes

—

153

C₃

#67754

yes

complemented

154

C₂

—

155

C₂

#10589399

yes

characteristic

156

C₂

#10589421

yes

characteristic

157

C₂

#10589426

yes

characteristic

	subgroup	G/H
center	C₂²	#63658
commutator	C₂×C₃²×C₄	C₄²
Frattini	C₂×C₄²	C₂×C₃⋊D₃
Fitting	C₃²×#432	C₂
socle	C₂²×C₃²	#95
prime radical	C₂³×C₃²	C₄⋊C₄
cosocle	C₂×C₃²×C₄²	C₂²
hypercenter	#432	C₃⋊D₃
hypocenter	C₃²	#1291
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂²

#63658

commutator

C₂×C₃²×C₄

C₄²

Frattini

C₂×C₄²

C₂×C₃⋊D₃

Fitting

C₃²×#432

C₂

socle

C₂²×C₃²

#95

prime radical

C₂³×C₃²

C₄⋊C₄

cosocle

C₂×C₃²×C₄²

C₂²

C₃⋊D₃

C₃²

G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
125. C₃²	#1291	semidirect
153. C₃	#67754	semidirect

H ≅ G/N

type

125. C₃²

#1291

semidirect

153. C₃

#67754

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
C₄×#3748	2
#67754	3

subgroup

index

C₃²×#432

#10592625

C₄×#3748

#67754

Subgroups — #11771002

Conjugacy classes of subgroups

Complements

Maximal subgroups