Group #11771117 — subgroups

Subgroups — #11771117

Order 1152.

450 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 181 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×#423	C₂	1	1	yes	maximal, characteristic
2	#10592624	C₂	1	1	yes	maximal, characteristic
3	C₄×#3748	C₂	1	1	yes	maximal, characteristic
4	#67710	—	3	4	no	maximal
5	C₂×C₃²×M₁₆	C₄	1	2	yes	—
6	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
7	C₂×#3748	—	2	1	no	—
8	C₃²×C₄⋊C₈	—	2	1	no	—
9	C₃²×C₄⋊C₈	—	2	1	no	—
10	#63657	—	2	1	no	—
11	C₄⋊(C₃⋊(C₃⋊C₈))	C₄	1	2	yes	—
12	#63630	C₄	1	2	yes	—
13	C₄×C₃⋊(C₃⋊C₈)	—	2	1	no	—
14	C₄×C₃⋊(C₃⋊C₈)	—	2	1	no	—
15	C₃×#423	D₃	1	4	yes	—
16	#4925	—	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	C₃²×M₁₆	—	4	2	no	—
22	C₂×C₃²×C₈	—	2	2	no	—
23	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
24	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
25	C₃²×C₄²	D₄	1	1	yes	characteristic
26	C₃²×C₄²	D₄	1	1	yes	characteristic
27	#3748	—	2	4	no	—
28	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
29	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
30	#1299	—	9	1	no	Sylow
31	C₂×C₃×M₁₆	Dic₃	1	8	yes	—
32	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
33	C₂×#208	—	6	4	no	—
34	C₃×C₄⋊C₈	—	2	4	no	—
35	C₃×C₄⋊C₈	—	2	4	no	—
36	#830	—	6	4	no	—
37	C₄⋊(C₃⋊C₈)	—	3	8	no	—
38	#803	—	3	8	no	—
39	C₄×C₃⋊C₈	—	6	4	no	—
40	C₄×C₃⋊C₈	—	6	4	no	—
41	C₂³×C₃²	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₃²×C₄	—	2	1	no	—
43	C₂×C₃²×C₄	—	2	1	no	—
44	C₂×C₃²×C₄	—	2	2	no	—
45	C₂×C₃²×C₄	—	2	2	no	—
46	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
47	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
52	C₃²×C₈	—	4	2	no	—
53	C₃⋊(C₃⋊C₈)	—	2	4	no	—
54	C₃⋊(C₃⋊C₈)	—	4	2	no	—
55	#431	—	9	1	no	—
56	#423	C₃⋊D₃	1	1	yes	characteristic
57	C₄×M₁₆	—	9	1	no	—
58	C₂²×C₃×C₄	Dic₆	1	4	yes	—
59	C₂²×C₃×C₄	D₁₂	1	4	yes	—
60	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
61	C₃×M₁₆	—	4	8	no	—
62	C₂×C₃×C₈	—	2	8	no	—
63	C₃×C₄²	C₄×D₃	1	4	yes	—
64	C₃×C₄²	C₄×D₃	1	4	yes	—
65	C₃×C₄²	#67	1	4	yes	—
66	C₃×C₄²	#67	1	4	yes	—
67	#208	—	6	16	no	—
68	C₂×C₃⋊C₈	—	6	8	no	—
69	C₂×C₃⋊C₈	—	6	8	no	—
70	C₂²×C₃²	—	2	1	no	—
71	C₂²×C₃²	—	2	1	no	—
72	C₂²×C₃²	—	2	1	no	—
73	C₂²×C₃²	#95	1	1	yes	characteristic
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	C₃²×C₄	#104	1	2	yes	—
80	C₃²×C₄	#108	1	2	yes	—
81	C₂×M₁₆	—	18	1	no	—
82	C₂×M₁₆	C₃⋊Dic₃	1	2	yes	—
83	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
84	C₄⋊C₈	—	2	1	no	—
85	C₄⋊C₈	—	2	1	no	—
86	C₄⋊C₈	—	9	2	no	—
87	C₂²⋊C₈	—	18	1	no	—
88	#97	—	9	2	no	—
89	C₄×C₈	—	18	1	no	—
90	C₄×C₈	—	18	1	no	—
91	C₂³×C₃	C₄⋊Dic₃	1	4	yes	—
92	C₂×C₃×C₄	—	2	8	no	—
93	C₂×C₃×C₄	—	2	8	no	—
94	C₂×C₃×C₄	—	2	4	no	—
95	C₂×C₃×C₄	—	2	4	no	—
96	C₂×C₃×C₄	C₄×Dic₃	1	4	yes	—
97	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
98	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
99	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
100	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
101	C₂×C₃×C₄	#217	1	4	yes	—
102	C₃×C₈	—	4	8	no	—
103	C₃⋊C₈	—	6	16	no	—
104	C₃⋊C₈	—	12	8	no	—
105	C₂×C₃²	—	4	1	no	—
106	C₂×C₃²	#339	1	1	yes	characteristic
107	C₂×C₃²	#341	1	1	yes	characteristic
108	C₂×C₃²	#343	1	1	yes	characteristic
109	C₂²×C₄	#634	1	1	yes	characteristic
110	C₂²×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
111	C₂²×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
112	M₁₆	—	4	2	no	—
113	M₁₆	—	18	4	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	18	2	no	—
116	C₂×C₈	—	18	2	no	—
117	C₄²	C₄×C₃⋊D₃	1	1	yes	characteristic
118	C₄²	C₄×C₃⋊D₃	1	1	yes	characteristic
119	C₄²	#638	1	1	yes	characteristic
120	C₄²	#638	1	1	yes	characteristic
121	C₂²×C₃	—	2	4	no	—
122	C₂²×C₃	—	2	4	no	—
123	C₂²×C₃	—	2	4	no	—
124	C₂²×C₃	C₃⋊#95	1	4	yes	—
125	C₃×C₄	—	2	4	no	—
126	C₃×C₄	—	2	4	no	—
127	C₃×C₄	—	2	4	no	—
128	C₃×C₄	#822	1	8	yes	—
129	C₃×C₄	#825	1	8	yes	—
130	C₃×C₄	#833	1	4	yes	—
131	C₃×C₄	#836	1	4	yes	—
132	C₃²	#1299	1	1	yes	characteristic, complemented, Sylow
133	C₂³	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	C₄×C₃⋊Dic₃	1	1	yes	characteristic
139	C₂×C₄	#3750	1	1	yes	characteristic
140	C₂×C₄	#3750	1	1	yes	characteristic
141	C₂×C₄	#3752	1	1	yes	characteristic
142	C₂×C₄	#3752	1	1	yes	characteristic
143	C₂×C₄	#3757	1	1	yes	characteristic
144	C₈	—	4	2	no	—
145	C₈	—	18	4	no	—
146	C₈	—	36	2	no	—
147	C₂×C₃	—	4	4	no	—
148	C₂×C₃	#4530	1	4	yes	—
149	C₂×C₃	#4555	1	4	yes	—
150	C₂×C₃	#4558	1	4	yes	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	#63658	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	#63649	1	2	yes	—
159	C₄	#63652	1	2	yes	—
160	C₄	#63660	1	1	yes	characteristic
161	C₄	#63663	1	1	yes	characteristic
162	C₃	#67710	1	4	yes	complemented
163	C₂	—	4	1	no	—
164	C₂	#10589396	1	1	yes	characteristic
165	C₂	#10589421	1	1	yes	characteristic
166	C₂	#10589424	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#423

C₂

yes

maximal, characteristic

#10592624

C₂

yes

maximal, characteristic

C₄×#3748

C₂

yes

maximal, characteristic

#67710

—

maximal

C₂×C₃²×M₁₆

C₄

yes

—

C₂×C₃²×C₄²

C₂²

yes

characteristic

C₂×#3748

—

C₃²×C₄⋊C₈

—

C₃²×C₄⋊C₈

—

#63657

—

C₄⋊(C₃⋊(C₃⋊C₈))

C₄

yes

—

#63630

C₄

yes

—

C₄×C₃⋊(C₃⋊C₈)

—

C₄×C₃⋊(C₃⋊C₈)

—

C₃×#423

D₃

yes

—

#4925

—

C₄×#208

—

C₂²×C₃²×C₄

C₂×C₄

yes

characteristic

C₂²×C₃²×C₄

D₄

yes

characteristic

C₂²×C₃²×C₄

Q₈

yes

characteristic

C₃²×M₁₆

—

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)

—

#1299

—

Sylow

C₂×C₃×M₁₆

Dic₃

yes

—

C₂×C₃×C₄²

C₂×D₃

yes

—

C₂×#208

—

C₃×C₄⋊C₈

—

C₃×C₄⋊C₈

—

#830

—

C₄⋊(C₃⋊C₈)

—

#803

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

C₂³×C₃²

C₄⋊C₄

yes

characteristic

C₂×C₃²×C₄

—

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)

—

#431

—

#423

C₃⋊D₃

yes

characteristic

C₄×M₁₆

—

C₂²×C₃×C₄

Dic₆

yes

—

C₂²×C₃×C₄

D₁₂

yes

—

C₂²×C₃×C₄

C₂×Dic₃

yes

—

C₃×M₁₆

—

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

yes

C₃²×C₄

yes

C₃²×C₄

yes

—

C₃²×C₄

#108

yes

—

C₂×M₁₆

—

C₂×M₁₆

C₃⋊Dic₃

yes

—

C₂×C₄²

C₂×C₃⋊D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂³×C₃

C₄⋊Dic₃

yes

—

C₂×C₃×C₄

—

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

—

100

C₂×C₃×C₄

(C₂×D₃)⋊C₄

yes

—

101

C₂×C₃×C₄

#217

yes

—

102

C₃×C₈

—

103

C₃⋊C₈

—

104

C₃⋊C₈

—

105

C₂×C₃²

—

106

C₂×C₃²

#339

yes

characteristic

107

C₂×C₃²

#341

yes

characteristic

108

C₂×C₃²

#343

yes

characteristic

109

C₂²×C₄

#634

yes

characteristic

110

C₂²×C₄

C₄⋊(C₃⋊D₃)

yes

characteristic

111

C₂²×C₄

C₂×C₃⋊Dic₃

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄²

C₄×C₃⋊D₃

yes

characteristic

118

C₄²

C₄×C₃⋊D₃

yes

characteristic

119

C₄²

#638

yes

characteristic

120

C₄²

#638

yes

characteristic

121

C₂²×C₃

—

122

C₂²×C₃

—

123

C₂²×C₃

—

124

C₂²×C₃

C₃⋊#95

yes

—

125

C₃×C₄

—

126

C₃×C₄

—

127

C₃×C₄

—

128

C₃×C₄

#822

yes

—

129

C₃×C₄

#825

yes

—

130

C₃×C₄

#833

yes

—

131

C₃×C₄

#836

yes

—

132

C₃²

#1299

yes

characteristic, complemented, Sylow

133

C₂³

C₄⋊(C₃⋊Dic₃)

yes

characteristic

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

C₄×C₃⋊Dic₃

yes

characteristic

139

C₂×C₄

#3750

yes

characteristic

140

C₂×C₄

#3750

yes

characteristic

141

C₂×C₄

#3752

yes

characteristic

142

C₂×C₄

#3752

yes

characteristic

143

C₂×C₄

#3757

yes

characteristic

144

C₈

—

145

C₈

—

146

C₈

—

147

C₂×C₃

—

148

C₂×C₃

#4530

yes

—

149

C₂×C₃

#4555

yes

—

150

C₂×C₃

#4558

yes

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

#63658

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

#63649

yes

—

159

C₄

#63652

yes

—

160

C₄

#63660

yes

characteristic

161

C₄

#63663

yes

characteristic

162

C₃

#67710

yes

complemented

163

C₂

—

164

C₂

#10589396

yes

characteristic

165

C₂

#10589421

yes

characteristic

166

C₂

#10589424

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#3757
commutator	C₂×C₃²×C₄	C₄²
Frattini	C₂×C₄²	C₂×C₃⋊D₃
Fitting	C₃²×#423	C₂
socle	C₂²×C₃²	#95
prime radical	C₂³×C₃²	C₄⋊C₄
cosocle	C₂×C₃²×C₄²	C₂²
hypercenter	#423	C₃⋊D₃
hypocenter	C₃²	#1299
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

#3757

commutator

C₂×C₃²×C₄

C₄²

Frattini

C₂×C₄²

C₂×C₃⋊D₃

Fitting

C₃²×#423

C₂

socle

C₂²×C₃²

#95

prime radical

C₂³×C₃²

C₄⋊C₄

cosocle

C₂×C₃²×C₄²

C₂²

C₃⋊D₃

C₃²

G ▷ C₂×C₃²×C₄ ▷ C₂²×C₃² ▷ C₃²

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
132. C₃²	#1299	semidirect
162. C₃	#67710	semidirect

H ≅ G/N

type

132. C₃²

#1299

semidirect

162. C₃

#67710

semidirect

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

subgroup	index
C₃²×#423	2
#10592624	2
C₄×#3748	2
#67710	3

subgroup

index

C₃²×#423

#10592624

C₄×#3748

#67710

Subgroups — #11771117

Conjugacy classes of subgroups

Complements

Maximal subgroups