Group #11771422 — subgroups

Subgroups — #11771422

Order 1152.

336 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 85 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	#10592828	C₂	1	1	yes	maximal, characteristic
2	C₃²×#349	C₂	1	1	yes	maximal, characteristic, complemented
3	#10589419	C₂	1	1	yes	maximal, characteristic, complemented
4	#67531	—	3	4	no	maximal
5	C₂×#3828	C₄	1	1	yes	characteristic
6	C₂×C₃²×M₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₃⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
8	C₃²×M₃₂	—	2	1	no	—
9	#63647	—	2	1	no	—
10	#63647	—	2	1	no	—
11	#63639	—	2	1	no	—
12	#5129	—	3	4	no	—
13	C₃×#349	D₃	1	4	yes	complemented
14	#4553	—	3	4	no	—
15	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
16	C₂×#638	—	2	1	no	—
17	#3828	—	4	1	no	—
18	#3828	—	4	1	no	—
19	C₂×C₄⋊(C₃⋊D₃)	C₈	1	1	yes	characteristic
20	C₂×C₄×C₃⋊D₃	—	2	1	no	—
21	C₂×#634	C₈	1	1	yes	characteristic
22	C₃²×M₁₆	—	4	1	no	—
23	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
24	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
25	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
26	C₂×C₃⋊(C₃⋊C₈)	—	4	1	no	—
27	C₃²×C₁₆	—	4	1	no	—
28	C₃⋊(C₃⋊C₁₆)	—	4	1	no	—
29	#1321	—	9	1	no	Sylow
30	C₂×#235	—	3	4	no	—
31	C₂×C₃×M₁₆	C₂×D₃	1	4	yes	—
32	C₂²×C₃⋊C₈	—	3	4	no	—
33	C₃×M₃₂	—	2	4	no	—
34	(C₂×D₃)⋊C₈	—	6	4	no	—
35	(C₂×D₃)⋊C₈	—	6	4	no	—
36	#812	—	6	4	no	—
37	C₂³×C₃²	M₁₆	1	1	yes	characteristic
38	C₂²×C₃⋊D₃	—	2	1	no	—
39	C₂×C₃²×C₄	—	2	1	no	—
40	C₂×C₃²×C₄	—	2	1	no	—
41	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
43	#638	—	4	2	no	—
44	C₂×C₃⋊Dic₃	—	2	1	no	—
45	C₄⋊(C₃⋊D₃)	—	4	1	no	—
46	C₄×C₃⋊D₃	—	4	1	no	—
47	C₄×C₃⋊D₃	—	4	1	no	—
48	#634	—	4	1	no	—
49	C₃²×C₈	—	2	1	no	—
50	C₃²×C₈	—	2	1	no	—
51	C₃⋊(C₃⋊C₈)	—	4	1	no	—
52	#408	—	9	1	no	—
53	#349	—	9	1	no	—
54	#349	C₃⋊D₃	1	1	yes	characteristic, complemented
55	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
56	C₂×#67	—	6	4	no	—
57	#235	—	12	4	no	—
58	#235	—	12	4	no	—
59	C₂×D₁₂	—	3	4	no	—
60	C₂×C₄×D₃	—	6	4	no	—
61	C₂×Dic₆	—	3	4	no	—
62	C₃×M₁₆	—	4	4	no	—
63	C₂×C₃×C₈	D₁₂	1	4	yes	—
64	C₂×C₃×C₈	#67	1	4	yes	—
65	C₂×C₃⋊C₈	—	6	4	no	—
66	C₂×C₃⋊C₈	—	12	4	no	—
67	C₃×C₁₆	—	4	4	no	—
68	C₃⋊C₁₆	—	12	4	no	—
69	C₂²×C₃²	—	2	1	no	—
70	C₂²×C₃²	—	4	1	no	—
71	C₂²×C₃²	C₂²⋊C₈	1	1	yes	characteristic
72	C₂×C₃⋊D₃	—	4	2	no	—
73	C₂×C₃⋊D₃	—	4	1	no	—
74	C₃²×C₄	—	2	1	no	—
75	C₃²×C₄	#99	1	1	yes	characteristic
76	C₃²×C₄	#100	1	1	yes	characteristic
77	C₃⋊Dic₃	—	4	1	no	—
78	C₂×#41	—	9	1	no	—
79	C₂×M₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
80	C₂²×C₈	—	9	1	no	—
81	M₃₂	—	2	1	no	—
82	M₃₂	—	18	1	no	—
83	C₂²⋊C₈	—	18	1	no	—
84	C₂²⋊C₈	—	18	1	no	—
85	C₂³×C₃	ΓL(1,𝔽₂₅)	1	4	yes	—
86	C₂²×D₃	—	6	4	no	—
87	C₂×C₃×C₄	—	2	4	no	—
88	C₂×C₃×C₄	—	2	4	no	—
89	C₂×C₃×C₄	D₃×C₈	1	4	yes	—
90	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
91	#67	—	12	8	no	—
92	C₂×Dic₃	—	6	4	no	—
93	D₁₂	—	12	4	no	—
94	C₄×D₃	—	12	4	no	—
95	C₄×D₃	—	12	4	no	—
96	Dic₆	—	12	4	no	—
97	C₃×C₈	—	2	4	no	—
98	C₃×C₈	—	2	4	no	—
99	C₃⋊C₈	—	12	4	no	—
100	C₂×C₃²	—	2	1	no	—
101	C₂×C₃²	—	4	1	no	—
102	C₂×C₃²	#323	1	1	yes	characteristic
103	C₃⋊D₃	—	8	1	no	—
104	#41	—	36	1	no	—
105	#41	—	36	1	no	—
106	C₂×Q₈	—	9	1	no	—
107	C₂×D₄	—	9	1	no	—
108	C₂×D₄	—	18	1	no	—
109	C₂²×C₄	—	18	1	no	—
110	C₂²×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
111	M₁₆	—	4	1	no	—
112	C₂×C₈	—	18	1	no	—
113	C₂×C₈	—	36	1	no	—
114	C₂×C₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
115	C₂×C₈	#638	1	1	yes	characteristic
116	C₁₆	—	4	1	no	—
117	C₁₆	—	36	1	no	—
118	C₂²×C₃	—	2	4	no	—
119	C₂²×C₃	—	4	4	no	—
120	C₂²×C₃	(C₂×D₃)⋊C₈	1	4	yes	—
121	C₂×D₃	—	12	8	no	—
122	C₂×D₃	—	12	4	no	—
123	C₃×C₄	—	2	4	no	—
124	C₃×C₄	#806	1	4	yes	—
125	C₃×C₄	#823	1	4	yes	—
126	Dic₃	—	12	4	no	—
127	C₃²	#1321	1	1	yes	characteristic, complemented, Sylow
128	C₂³	—	18	1	no	—
129	C₂³	#3743	1	1	yes	characteristic
130	Q₈	—	36	1	no	—
131	D₄	—	36	1	no	—
132	D₄	—	36	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	18	1	no	—
136	C₂×C₄	—	36	1	no	—
137	C₂×C₄	—	36	1	no	—
138	C₂×C₄	C₈×C₃⋊D₃	1	1	yes	characteristic
139	C₂×C₄	#3752	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	36	1	no	—
143	C₂×C₃	—	2	4	no	—
144	C₂×C₃	—	4	4	no	—
145	C₂×C₃	#4468	1	4	yes	—
146	D₃	—	24	4	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	4	1	no	—
149	C₂²	—	36	2	no	—
150	C₂²	—	36	1	no	—
151	C₂²	#63647	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	36	1	no	—
154	C₄	#63633	1	1	yes	characteristic
155	C₄	#63650	1	1	yes	characteristic
156	C₃	#67531	1	4	yes	complemented
157	C₂	—	2	1	no	—
158	C₂	—	4	1	no	—
159	C₂	—	72	1	no	—
160	C₂	#10589334	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10592828

C₂

yes

maximal, characteristic

C₃²×#349

C₂

yes

maximal, characteristic, complemented

#10589419

C₂

yes

maximal, characteristic, complemented

#67531

—

maximal

C₂×#3828

C₄

yes

characteristic

C₂×C₃²×M₁₆

C₂²

yes

characteristic

C₂²×C₃⋊(C₃⋊C₈)

C₄

yes

characteristic

C₃²×M₃₂

—

—

—

—

—

D₃

yes

complemented

#4553

—

C₂²×C₃²×C₄

C₂×C₄

yes

—

—

—

C₂×C₄⋊(C₃⋊D₃)

C₈

yes

characteristic

C₂×C₄×C₃⋊D₃

—

C₂×#634

C₈

yes

characteristic

C₃²×M₁₆

—

C₂×C₃²×C₈

D₄

yes

characteristic

C₂×C₃²×C₈

D₄

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

—

C₂×C₃⋊(C₃⋊C₈)

—

C₃²×C₁₆

—

C₃⋊(C₃⋊C₁₆)

—

#1321

—

Sylow

C₂×#235

—

C₂×C₃×M₁₆

C₂×D₃

yes

—

C₂²×C₃⋊C₈

—

C₃×M₃₂

—

(C₂×D₃)⋊C₈

—

(C₂×D₃)⋊C₈

—

#812

—

C₂³×C₃²

M₁₆

yes

characteristic

C₂²×C₃⋊D₃

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃²×C₄

C₂×C₈

yes

characteristic

#638

—

C₂×C₃⋊Dic₃

—

C₄⋊(C₃⋊D₃)

—

C₄×C₃⋊D₃

—

C₄×C₃⋊D₃

—

#634

—

C₃²×C₈

—

C₃²×C₈

—

C₃⋊(C₃⋊C₈)

—

#408

—

#349

—

#349

C₃⋊D₃

yes

characteristic, complemented

C₂²×C₃×C₄

C₄×D₃

yes

—

C₂×#67

—

#235

—

#235

—

C₂×D₁₂

—

C₂×C₄×D₃

—

C₂×Dic₆

—

C₃×M₁₆

—

C₂×C₃×C₈

D₁₂

yes

—

C₂×C₃×C₈

#67

yes

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₂²⋊C₈

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₃²×C₄

—

C₃²×C₄

yes

C₃²×C₄

yes

C₃⋊Dic₃

—

C₂×#41

—

C₂×M₁₆

C₂×C₃⋊D₃

yes

characteristic

C₂²×C₈

—

M₃₂

—

M₃₂

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂³×C₃

ΓL(1,𝔽₂₅)

yes

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

D₃×C₈

yes

—

C₂×C₃×C₄

(C₂×D₃)⋊C₄

yes

—

#67

—

C₂×Dic₃

—

D₁₂

—

C₄×D₃

—

C₄×D₃

—

Dic₆

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

100

C₂×C₃²

—

101

C₂×C₃²

—

102

C₂×C₃²

#323

yes

characteristic

103

C₃⋊D₃

—

104

#41

—

105

#41

—

106

C₂×Q₈

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₄×C₃⋊D₃

yes

characteristic

111

M₁₆

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

C₄⋊(C₃⋊D₃)

yes

characteristic

115

C₂×C₈

#638

yes

characteristic

116

C₁₆

—

117

C₁₆

—

118

C₂²×C₃

—

119

C₂²×C₃

—

120

C₂²×C₃

(C₂×D₃)⋊C₈

yes

—

121

C₂×D₃

—

122

C₂×D₃

—

123

C₃×C₄

—

124

C₃×C₄

#806

yes

—

125

C₃×C₄

#823

yes

—

126

Dic₃

—

127

C₃²

#1321

yes

characteristic, complemented, Sylow

128

C₂³

—

129

C₂³

#3743

yes

characteristic

130

Q₈

—

131

D₄

—

132

D₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

C₈×C₃⋊D₃

yes

characteristic

139

C₂×C₄

#3752

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

143

C₂×C₃

—

144

C₂×C₃

—

145

C₂×C₃

#4468

yes

—

146

D₃

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

#63647

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

#63633

yes

characteristic

155

C₄

#63650

yes

characteristic

156

C₃

#67531

yes

complemented

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#10589334

yes

characteristic

	subgroup	G/H
center	C₄	#63633
commutator	C₂×C₃²×C₄	C₂×C₈
Frattini	C₂×M₁₆	C₂×C₃⋊D₃
Fitting	C₃²×#349	C₂
socle	C₂×C₃²	#323
prime radical	C₂×#3828	C₄
cosocle	C₂×C₃²×M₁₆	C₂²
hypercenter	#349	C₃⋊D₃
hypocenter	C₃²	#1321
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂²×C₃² ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

subgroup

G/H

center

C₄

#63633

commutator

C₂×C₃²×C₄

C₂×C₈

Frattini

C₂×M₁₆

C₂×C₃⋊D₃

Fitting

C₃²×#349

C₂

C₂×C₃²

C₄

C₂×C₃²×M₁₆

C₂²

C₃⋊D₃

C₃²

G ▷ C₂×C₃²×C₄ ▷ C₂²×C₃² ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

N	H ≅ G/N	type
2. C₃²×#349	C₂	semidirect
3. #10589419	C₂	semidirect
13. C₃×#349	D₃	semidirect
54. #349	C₃⋊D₃	semidirect
127. C₃²	#1321	semidirect
156. C₃	#67531	semidirect

H ≅ G/N

type

2. C₃²×#349

C₂

semidirect

3. #10589419

C₂

semidirect

13. C₃×#349

D₃

semidirect

54. #349

C₃⋊D₃

semidirect

127. C₃²

#1321

semidirect

156. C₃

#67531

semidirect

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

subgroup	index
#10592828	2
C₃²×#349	2
#10589419	2
#67531	3

subgroup

index

#10592828

C₃²×#349

#10589419

#67531

Subgroups — #11771422

Conjugacy classes of subgroups

Complements

Maximal subgroups