Group #11785974 — subgroups

Subgroups — #11785974

Order 1152.

324 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 142 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	C₂×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#10588612	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#10588228	C₂	1	4	yes	maximal
5	#70444	—	3	1	no	maximal
6	C₂²×C₈×C₉	C₂²	1	2	yes	—
7	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₉	C₂²	1	2	yes	—
9	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
10	C₉×C₄⋊C₈	—	2	2	no	—
11	C₉⋊#95	C₄	1	2	yes	—
12	C₉⋊#95	C₄	1	2	yes	—
13	C₄⋊(C₉⋊C₈)	—	2	2	no	—
14	C₂×C₃×C₄⋊C₈	D₃	1	1	yes	characteristic
15	#4932	—	3	1	no	—
16	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
17	#4550	—	3	4	no	—
18	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
21	C₂²×Dic₉	C₂×C₄	1	4	yes	—
22	C₂×C₈×C₉	—	2	2	no	—
23	C₂×C₈×C₉	—	2	4	no	—
24	C₄²×C₉	D₄	1	2	yes	—
25	C₄²×C₉	Q₈	1	2	yes	—
26	C₄×Dic₉	—	2	2	no	—
27	C₄×Dic₉	—	2	2	no	—
28	C₂×C₉⋊C₈	—	2	2	no	—
29	C₂×C₉⋊C₈	—	2	4	no	—
30	#1832	—	9	1	no	Sylow
31	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
32	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
33	C₂×C₄×Dic₃	—	3	2	no	—
34	C₂²×C₃⋊C₈	—	3	2	no	—
35	C₃×C₄⋊C₈	—	2	2	no	—
36	C₃⋊#95	—	3	2	no	—
37	C₃⋊#95	—	3	2	no	—
38	C₄⋊(C₃⋊C₈)	—	6	2	no	—
39	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₉	—	2	2	no	—
41	C₂×C₄×C₉	—	2	2	no	—
42	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₉	#41	1	2	yes	—
47	C₂×C₄×C₉	#41	1	2	yes	—
48	C₂×Dic₉	—	2	4	no	—
49	C₂×Dic₉	—	2	4	no	—
50	C₂×Dic₉	—	2	4	no	—
51	C₈×C₉	—	4	2	no	—
52	C₉⋊C₈	—	4	2	no	—
53	C₂×C₄⋊C₈	—	9	1	no	—
54	C₂×C₄⋊C₈	D₉	1	1	yes	characteristic
55	#376	—	9	1	no	—
56	#336	—	9	4	no	—
57	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
58	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
60	C₂²×Dic₃	—	3	4	no	—
61	C₂×C₃×C₈	—	2	4	no	—
62	C₂×C₃×C₈	—	2	2	no	—
63	C₃×C₄²	Dic₆	1	2	yes	—
64	C₃×C₄²	#67	1	2	yes	—
65	C₄×Dic₃	—	6	2	no	—
66	C₄×Dic₃	—	6	2	no	—
67	C₂×C₃⋊C₈	—	6	4	no	—
68	C₂×C₃⋊C₈	—	6	2	no	—
69	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₉	#117	1	2	yes	—
71	C₂²×C₉	#131	1	2	yes	—
72	C₂²×C₉	#131	1	2	yes	—
73	C₄×C₉	—	2	2	no	—
74	C₄×C₉	—	2	2	no	—
75	C₄×C₉	#124	1	1	yes	characteristic
76	C₄×C₉	#124	1	1	yes	characteristic
77	C₄×C₉	#125	1	2	yes	—
78	Dic₉	—	4	4	no	—
79	C₂²×C₈	—	9	2	no	—
80	C₂²×C₈	C₂×D₉	1	2	yes	—
81	C₂×C₄²	—	9	2	no	—
82	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
83	C₄⋊C₈	—	2	2	no	—
84	C₄⋊C₈	—	18	2	no	—
85	#95	—	9	2	no	—
86	#95	—	9	2	no	—
87	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
88	C₂×C₃×C₄	—	2	2	no	—
89	C₂×C₃×C₄	—	2	2	no	—
90	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
91	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
92	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
93	C₂×C₃×C₄	#237	1	2	yes	—
94	C₂×C₃×C₄	#239	1	2	yes	—
95	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
96	C₂×Dic₃	—	6	4	no	—
97	C₂×Dic₃	—	6	4	no	—
98	C₂×Dic₃	—	6	4	no	—
99	C₃×C₈	—	4	2	no	—
100	C₃⋊C₈	—	12	2	no	—
101	C₂×C₉	#382	1	1	yes	characteristic
102	C₂×C₉	#424	1	1	yes	characteristic
103	C₂×C₉	#424	1	1	yes	characteristic
104	C₂×C₉	#433	1	4	yes	—
105	C₂²×C₄	—	9	4	no	—
106	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
107	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
108	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
109	C₂×C₈	—	2	4	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	18	4	no	—
112	C₂×C₈	—	18	2	no	—
113	C₄²	—	18	2	no	—
114	C₄²	—	18	2	no	—
115	C₄²	Dic₁₈	1	2	yes	—
116	C₄²	#611	1	2	yes	—
117	C₂²×C₃	#892	1	2	yes	—
118	C₂²×C₃	#901	1	2	yes	—
119	C₂²×C₃	#907	1	2	yes	—
120	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
121	C₃×C₄	—	2	2	no	—
122	C₃×C₄	—	2	2	no	—
123	C₃×C₄	#890	1	2	yes	—
124	C₃×C₄	#936	1	1	yes	characteristic
125	C₃×C₄	#947	1	1	yes	characteristic
126	Dic₃	—	12	4	no	—
127	C₉	#1832	1	1	yes	characteristic, complemented, Sylow
128	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	18	4	no	—
132	C₂×C₄	—	18	4	no	—
133	C₂×C₄	—	18	4	no	—
134	C₂×C₄	Dic₉⋊C₄	1	2	yes	—
135	C₂×C₄	Dic₉⋊C₄	1	2	yes	—
136	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
137	C₂×C₄	#3699	1	2	yes	—
138	C₂×C₄	#3701	1	2	yes	—
139	C₂×C₄	C₂×#611	1	1	yes	characteristic
140	C₈	—	4	2	no	—
141	C₈	—	36	2	no	—
142	C₂×C₃	#4840	1	4	yes	—
143	C₂×C₃	#4977	1	1	yes	characteristic
144	C₂×C₃	#5102	1	1	yes	characteristic
145	C₂×C₃	#5119	1	1	yes	characteristic
146	C₂²	#63454	1	2	yes	—
147	C₂²	#63464	1	2	yes	—
148	C₂²	#63469	1	2	yes	—
149	C₂²	C₂×Dic₉⋊C₄	1	1	yes	characteristic
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	36	4	no	—
153	C₄	#63452	1	2	yes	—
154	C₄	#63498	1	1	yes	characteristic
155	C₄	#63509	1	1	yes	characteristic
156	C₃	#70444	1	1	yes	characteristic
157	C₂	#10588524	1	4	yes	—
158	C₂	#10588657	1	1	yes	characteristic
159	C₂	#10588783	1	1	yes	characteristic
160	C₂	#10588803	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₉×C₄⋊C₈

C₂

yes

maximal, characteristic

#10588612

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₉⋊C₈)

C₂

yes

maximal, characteristic

#10588228

C₂

yes

maximal

#70444

—

maximal

C₂²×C₈×C₉

C₂²

yes

—

C₂×C₄²×C₉

C₂²

yes

characteristic

C₂×C₄×Dic₉

C₂²

yes

—

C₂²×C₉⋊C₈

C₂²

yes

—

C₉×C₄⋊C₈

—

C₉⋊#95

C₄

yes

—

C₉⋊#95

C₄

yes

—

C₄⋊(C₉⋊C₈)

—

C₂×C₃×C₄⋊C₈

D₃

yes

characteristic

#4932

—

C₂×C₄⋊(C₃⋊C₈)

—

#4550

—

C₂²×C₄×C₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₉

C₂³

yes

characteristic

C₂²×Dic₉

C₂×C₄

yes

—

C₂×C₈×C₉

—

C₂×C₈×C₉

—

C₄²×C₉

D₄

yes

—

C₄²×C₉

Q₈

yes

—

C₄×Dic₉

—

C₄×Dic₉

—

C₂×C₉⋊C₈

—

C₂×C₉⋊C₈

—

#1832

—

Sylow

C₂²×C₃×C₈

C₂×D₃

yes

—

C₂×C₃×C₄²

C₂×D₃

yes

characteristic

C₂×C₄×Dic₃

—

C₂²×C₃⋊C₈

—

C₃×C₄⋊C₈

—

C₃⋊#95

—

C₃⋊#95

—

C₄⋊(C₃⋊C₈)

—

C₂³×C₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₄⋊C₄

yes

—

C₂×C₄×C₉

C₄⋊C₄

yes

—

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×C₄×C₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₉

#41

yes

—

C₂×C₄×C₉

#41

yes

—

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

—

C₈×C₉

—

C₉⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₉

yes

characteristic

#376

—

#336

—

C₂²×C₃×C₄

C₄×D₃

yes

characteristic

C₂²×C₃×C₄

C₄×D₃

yes

characteristic

C₂²×C₃×C₄

C₂²×D₃

yes

characteristic

C₂²×Dic₃

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄²

Dic₆

yes

—

C₃×C₄²

#67

yes

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉

#117

yes

—

C₂²×C₉

#131

yes

—

C₂²×C₉

#131

yes

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

yes

C₄×C₉

yes

C₄×C₉

yes

—

Dic₉

—

C₂²×C₈

—

C₂²×C₈

C₂×D₉

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#95

—

#95

—

C₂³×C₃

C₂×C₄×D₃

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

Dic₃⋊C₄

yes

—

C₂×C₃×C₄

Dic₃⋊C₄

yes

—

C₂×C₃×C₄

C₂×Dic₆

yes

characteristic

C₂×C₃×C₄

#237

yes

—

C₂×C₃×C₄

#239

yes

—

C₂×C₃×C₄

C₂×#67

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₃×C₈

—

100

C₃⋊C₈

—

101

C₂×C₉

#382

yes

characteristic

102

C₂×C₉

#424

yes

characteristic

103

C₂×C₉

#424

yes

characteristic

104

C₂×C₉

#433

yes

—

105

C₂²×C₄

—

106

C₂²×C₄

C₄×D₉

yes

characteristic

107

C₂²×C₄

C₄×D₉

yes

characteristic

108

C₂²×C₄

C₂²×D₉

yes

characteristic

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₄²

—

114

C₄²

—

115

C₄²

Dic₁₈

yes

—

116

C₄²

#611

yes

—

117

C₂²×C₃

#892

yes

—

118

C₂²×C₃

#901

yes

—

119

C₂²×C₃

#907

yes

—

120

C₂²×C₃

C₂×Dic₃⋊C₄

yes

characteristic

121

C₃×C₄

—

122

C₃×C₄

—

123

C₃×C₄

#890

yes

—

124

C₃×C₄

#936

yes

characteristic

125

C₃×C₄

#947

yes

characteristic

126

Dic₃

—

127

C₉

#1832

yes

characteristic, complemented, Sylow

128

C₂³

C₂×C₄×D₉

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

Dic₉⋊C₄

yes

—

135

C₂×C₄

Dic₉⋊C₄

yes

—

136

C₂×C₄

C₂×Dic₁₈

yes

characteristic

137

C₂×C₄

#3699

yes

—

138

C₂×C₄

#3701

yes

—

139

C₂×C₄

C₂×#611

yes

characteristic

140

C₈

—

141

C₈

—

142

C₂×C₃

#4840

yes

—

143

C₂×C₃

#4977

yes

characteristic

144

C₂×C₃

#5102

yes

characteristic

145

C₂×C₃

#5119

yes

characteristic

146

C₂²

#63454

yes

—

147

C₂²

#63464

yes

—

148

C₂²

#63469

yes

—

149

C₂²

C₂×Dic₉⋊C₄

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

#63452

yes

—

154

C₄

#63498

yes

characteristic

155

C₄

#63509

yes

characteristic

156

C₃

#70444

yes

characteristic

157

C₂

#10588524

yes

—

158

C₂

#10588657

yes

characteristic

159

C₂

#10588783

yes

characteristic

160

C₂

#10588803

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉
commutator	C₂³×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₃	C₂×C₄×D₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×C₄⋊C₈	D₉
hypocenter	C₉	#1832
LCS	G ▷ C₂³×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂²×C₄

C₂²×D₉

commutator

C₂³×C₉

C₂²×C₄

Frattini

C₂²×C₃×C₄

C₂²×D₃

Fitting

C₂×C₉×C₄⋊C₈

C₂

socle, prime radical

C₂³×C₃

C₂×C₄×D₃

cosocle

C₂²×C₄×C₉

C₂³

hypercenter

C₂×C₄⋊C₈

D₉

C₉

G ▷ C₂³×C₉ ▷ C₉

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
127. C₉	#1832	semidirect

H ≅ G/N

type

127. C₉

#1832

semidirect

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

subgroup	index
C₂×C₉×C₄⋊C₈	2
#10588612	2
C₂×C₄⋊(C₉⋊C₈)	2
#10588228	2
#70444	3

subgroup

index

C₂×C₉×C₄⋊C₈

#10588612

C₂×C₄⋊(C₉⋊C₈)

#10588228

#70444

Subgroups — #11785974

Conjugacy classes of subgroups

Complements

Maximal subgroups