Group #11786156 — subgroups

Subgroups — #11786156

Order 1152.

243 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 243 conjugacy classes of subgroups — too many to draw. Only its 88 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	M₁₆×Dic₉	C₂	1	1	yes	maximal, characteristic
2	C₉×#338	C₂	1	1	yes	maximal, characteristic
3	#10588237	C₂	1	1	yes	maximal, characteristic
4	#10588235	C₂	1	2	yes	maximal
5	#10588232	C₂	1	2	yes	maximal
6	#68716	—	3	1	no	maximal
7	C₂×C₉×M₁₆	C₂²	1	2	yes	—
8	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
9	#63488	C₂²	1	2	yes	—
10	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
11	C₂×#3667	C₂²	1	1	yes	characteristic
12	#63375	C₄	1	2	yes	complemented
13	C₈×Dic₉	C₄	1	2	yes	complemented
14	Dic₃×M₁₆	—	3	1	no	—
15	C₃×#338	D₃	1	1	yes	characteristic
16	#4559	—	3	2	no	—
17	#4557	—	3	1	no	—
18	#4555	—	3	2	no	—
19	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
20	C₂²×Dic₉	—	2	1	no	—
21	C₉×M₁₆	—	2	2	no	—
22	C₉×M₁₆	—	2	2	no	—
23	C₉×M₁₆	—	2	2	no	—
24	C₂×C₈×C₉	—	2	2	no	—
25	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
26	#3676	—	2	2	no	—
27	C₄⋊Dic₉	—	2	2	no	—
28	C₄×Dic₉	—	2	1	no	—
29	C₄×Dic₉	—	2	2	no	—
30	C₄×Dic₉	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₉	C₂×C₄	1	1	yes	characteristic
32	#3667	—	2	2	no	—
33	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
34	#1841	—	9	1	no	Sylow
35	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
36	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
37	#926	—	3	2	no	—
38	C₂×C₄×Dic₃	—	3	1	no	—
39	C₂×#208	—	3	1	no	—
40	#815	—	3	2	no	—
41	C₈×Dic₃	—	3	2	no	—
42	C₂³×C₉	#41	1	1	yes	characteristic
43	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₉	#41	1	2	yes	—
45	C₂×C₄×C₉	#41	1	2	yes	—
46	C₂×C₄×C₉	#41	1	1	yes	characteristic
47	C₂×Dic₉	—	2	1	no	—
48	C₂×Dic₉	—	2	1	no	—
49	C₂×Dic₉	—	4	2	no	—
50	C₂×Dic₉	—	4	2	no	—
51	C₈×C₉	—	2	2	no	—
52	C₈×C₉	—	4	2	no	—
53	C₉⋊C₈	—	2	2	no	—
54	C₄×M₁₆	—	9	1	no	—
55	#344	—	9	2	no	—
56	#339	—	9	2	no	—
57	#338	D₉	1	1	yes	characteristic
58	H(3,ℤ/4)	—	9	1	no	—
59	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
60	C₂²×Dic₃	—	6	1	no	—
61	C₃×M₁₆	—	2	2	no	—
62	C₃×M₁₆	—	2	2	no	—
63	C₃×M₁₆	—	2	2	no	—
64	C₂×C₃×C₈	—	2	2	no	—
65	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
66	#217	—	6	2	no	—
67	C₄⋊Dic₃	—	6	2	no	—
68	C₄×Dic₃	—	3	1	no	—
69	C₄×Dic₃	—	3	1	no	—
70	C₄×Dic₃	—	6	1	no	—
71	C₄×Dic₃	—	6	2	no	—
72	#208	—	6	2	no	—
73	C₂×C₃⋊C₈	—	3	2	no	—
74	C₂²×C₉	—	4	1	no	—
75	C₂²×C₉	#117	1	1	yes	characteristic
76	C₂²×C₉	#126	1	2	yes	—
77	C₄×C₉	#117	1	1	yes	characteristic
78	C₄×C₉	#117	1	1	yes	characteristic
79	C₄×C₉	#126	1	2	yes	—
80	Dic₉	—	2	2	no	—
81	Dic₉	—	4	1	no	—
82	Dic₉	—	8	2	no	—
83	C₂×M₁₆	—	9	1	no	—
84	C₂×M₁₆	C₂×D₉	1	2	yes	—
85	C₂×M₁₆	C₂×D₉	1	1	yes	characteristic
86	#117	—	9	2	no	—
87	C₂×C₄²	—	9	1	no	—
88	#97	—	9	2	no	—
89	C₄×C₈	—	9	2	no	—
90	C₂³×C₃	#239	1	1	yes	characteristic
91	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
92	C₂×C₃×C₄	#235	1	2	yes	—
93	C₂×C₃×C₄	#237	1	2	yes	—
94	C₂×C₃×C₄	#237	1	1	yes	characteristic
95	C₂×Dic₃	—	6	1	no	—
96	C₂×Dic₃	—	6	1	no	—
97	C₂×Dic₃	—	12	2	no	—
98	C₂×Dic₃	—	12	2	no	—
99	C₃×C₈	—	2	2	no	—
100	C₃×C₈	—	4	2	no	—
101	C₃⋊C₈	—	6	2	no	—
102	C₂×C₉	—	2	1	no	—
103	C₂×C₉	—	2	2	no	—
104	C₂×C₉	#383	1	1	yes	characteristic
105	C₂²×C₄	—	18	1	no	—
106	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
107	M₁₆	—	2	2	no	—
108	M₁₆	—	2	2	no	—
109	M₁₆	—	2	2	no	—
110	M₁₆	—	18	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	9	2	no	—
113	C₂×C₈	C₄×D₉	1	2	yes	—
114	C₄⋊C₄	—	18	2	no	—
115	C₂²⋊C₄	—	18	2	no	—
116	C₄²	—	9	1	no	—
117	C₄²	—	9	1	no	—
118	C₄²	—	18	1	no	—
119	C₄²	—	18	2	no	—
120	C₂²×C₃	—	4	1	no	—
121	C₂²×C₃	#892	1	1	yes	characteristic
122	C₂²×C₃	#898	1	2	yes	—
123	C₃×C₄	#872	1	1	yes	characteristic
124	C₃×C₄	#877	1	1	yes	characteristic
125	C₃×C₄	#879	1	2	yes	—
126	Dic₃	—	6	2	no	—
127	Dic₃	—	12	1	no	—
128	Dic₃	—	24	2	no	—
129	C₉	#1841	1	1	yes	characteristic, complemented, Sylow
130	C₂³	#3701	1	1	yes	characteristic
131	C₂×C₄	—	18	1	no	—
132	C₂×C₄	—	18	1	no	—
133	C₂×C₄	—	36	2	no	—
134	C₂×C₄	—	36	2	no	—
135	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
136	C₂×C₄	#3697	1	2	yes	—
137	C₂×C₄	#3699	1	2	yes	—
138	C₂×C₄	#3699	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	4	2	no	—
141	C₈	—	18	2	no	—
142	C₂×C₃	—	2	1	no	—
143	C₂×C₃	—	2	2	no	—
144	C₂×C₃	#4651	1	1	yes	characteristic
145	C₂²	—	4	1	no	—
146	C₂²	#63454	1	1	yes	characteristic
147	C₂²	#63460	1	2	yes	—
148	C₄	—	18	2	no	—
149	C₄	—	36	1	no	—
150	C₄	—	72	2	no	—
151	C₄	#63434	1	1	yes	characteristic
152	C₄	#63439	1	1	yes	characteristic
153	C₄	#63441	1	2	yes	—
154	C₃	#68716	1	1	yes	characteristic
155	C₂	—	2	1	no	—
156	C₂	—	2	2	no	—
157	C₂	#10588331	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₁₆×Dic₉

C₂

yes

maximal, characteristic

C₉×#338

C₂

yes

maximal, characteristic

#10588237

C₂

yes

maximal, characteristic

#10588235

C₂

yes

maximal

#10588232

C₂

yes

maximal

#68716

—

maximal

C₂×C₉×M₁₆

C₂²

yes

—

C₂×C₉×M₁₆

C₂²

yes

characteristic

#63488

C₂²

yes

—

C₂×C₄×Dic₉

C₂²

yes

characteristic

C₂×#3667

C₂²

yes

characteristic

#63375

C₄

yes

complemented

C₈×Dic₉

C₄

yes

complemented

Dic₃×M₁₆

—

C₃×#338

D₃

yes

—

—

—

C₂²×C₄×C₉

C₂³

yes

characteristic

C₂²×Dic₉

—

C₉×M₁₆

—

C₉×M₁₆

—

C₉×M₁₆

—

C₂×C₈×C₉

—

C₂×C₈×C₉

C₂×C₄

yes

—

#3676

—

C₄⋊Dic₉

—

C₄×Dic₉

—

C₄×Dic₉

—

C₄×Dic₉

C₂×C₄

yes

characteristic

C₄×Dic₉

C₂×C₄

yes

characteristic

#3667

—

C₂×C₉⋊C₈

C₂×C₄

yes

—

#1841

—

Sylow

C₂×C₃×M₁₆

C₂×D₃

yes

—

C₂×C₃×M₁₆

C₂×D₃

yes

characteristic

#926

—

C₂×C₄×Dic₃

—

C₂×#208

—

#815

—

C₈×Dic₃

—

C₂³×C₉

#41

yes

characteristic

C₂×C₄×C₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₉

#41

yes

—

C₂×C₄×C₉

#41

yes

—

C₂×C₄×C₉

#41

yes

characteristic

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

—

C₈×C₉

—

C₈×C₉

—

C₉⋊C₈

—

C₄×M₁₆

—

#344

—

#339

—

#338

D₉

yes

characteristic

H(3,ℤ/4)

—

C₂²×C₃×C₄

C₂²×D₃

yes

characteristic

C₂²×Dic₃

—

C₃×M₁₆

—

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₄×D₃

yes

—

#217

—

C₄⋊Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

#208

—

C₂×C₃⋊C₈

—

C₂²×C₉

—

C₂²×C₉

#117

yes

characteristic

C₂²×C₉

#126

yes

—

C₄×C₉

yes

C₄×C₉

yes

C₄×C₉

yes

—

Dic₉

—

Dic₉

—

Dic₉

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₉

yes

—

C₂×M₁₆

C₂×D₉

yes

characteristic

#117

—

C₂×C₄²

—

#97

—

C₄×C₈

—

C₂³×C₃

#239

yes

characteristic

C₂×C₃×C₄

C₂×C₄×D₃

yes

characteristic

C₂×C₃×C₄

#235

yes

—

C₂×C₃×C₄

#237

yes

—

C₂×C₃×C₄

#237

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₃×C₈

—

100

C₃×C₈

—

101

C₃⋊C₈

—

102

C₂×C₉

—

103

C₂×C₉

—

104

C₂×C₉

#383

yes

characteristic

105

C₂²×C₄

—

106

C₂²×C₄

C₂²×D₉

yes

characteristic

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₄×D₉

yes

—

114

C₄⋊C₄

—

115

C₂²⋊C₄

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₂²×C₃

—

121

C₂²×C₃

#892

yes

characteristic

122

C₂²×C₃

#898

yes

—

123

C₃×C₄

#872

yes

characteristic

124

C₃×C₄

#877

yes

characteristic

125

C₃×C₄

#879

yes

—

126

Dic₃

—

127

Dic₃

—

128

Dic₃

—

129

C₉

#1841

yes

characteristic, complemented, Sylow

130

C₂³

#3701

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₂×C₄×D₉

yes

characteristic

136

C₂×C₄

#3697

yes

—

137

C₂×C₄

#3699

yes

—

138

C₂×C₄

#3699

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

142

C₂×C₃

—

143

C₂×C₃

—

144

C₂×C₃

#4651

yes

characteristic

145

C₂²

—

146

C₂²

#63454

yes

characteristic

147

C₂²

#63460

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

#63434

yes

characteristic

152

C₄

#63439

yes

characteristic

153

C₄

#63441

yes

—

154

C₃

#68716

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

#10588331

yes

characteristic

	subgroup	G/H
center	C₄	#63434
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₉×#338	C₂
socle	C₂×C₃	#4651
prime radical	C₂³×C₃	#239
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	#338	D₉
hypocenter	C₉	#1841
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#63434

commutator

C₂×C₄×C₉

C₂²×C₄

Frattini

C₂²×C₃×C₄

C₂²×D₃

C₂

C₂×C₃

C₂³×C₃

C₂²×C₄×C₉

C₂³

D₉

C₉

G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
12. #63375	C₄	semidirect
13. C₈×Dic₉	C₄	semidirect
129. C₉	#1841	semidirect

H ≅ G/N

type

12. #63375

C₄

semidirect

13. C₈×Dic₉

C₄

semidirect

129. C₉

#1841

semidirect

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

subgroup	index
M₁₆×Dic₉	2
C₉×#338	2
#10588237	2
#10588235	2
#10588232	2
#68716	3

subgroup

index

M₁₆×Dic₉

Subgroups — #11786156

Conjugacy classes of subgroups

Complements

Maximal subgroups