Group #13085163 — subgroups

Subgroups — D₉×#700 #13085163

Order 1440.

204 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 62 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	C₄×D₅×D₉	C₂	1	1	yes	maximal, characteristic
2	#10623650	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#700	C₂	1	1	yes	maximal, characteristic, complemented
4	#10623629	C₂	1	2	yes	maximal, complemented
5	D₉×#170	C₂	1	2	yes	maximal, complemented
6	D₃×#700	—	3	1	no	maximal
7	C₂×D₅×D₉	C₄	1	1	yes	characteristic
8	C₄×D₄₅	C₄	1	1	yes	characteristic
9	C₄×C₅×D₉	C₄	1	1	yes	characteristic
10	C₄×C₉×D₅	C₂²	1	1	yes	characteristic
11	D₅×Dic₉	C₄	1	1	yes	characteristic
12	#67079	C₂²	1	1	yes	characteristic
13	D₉×Dic₅	C₂²	1	1	yes	characteristic
14	C₉⋊#170	C₂²	1	2	yes	complemented
15	C₉×#170	C₂²	1	2	yes	complemented
16	M₁₆×D₉	—	5	1	no	maximal, Hall
17	C₄×D₃×D₅	—	3	1	no	—
18	#6758	—	3	1	no	—
19	C₃×#700	D₃	1	1	yes	characteristic
20	#6740	—	3	2	no	—
21	D₃×#170	—	3	2	no	—
22	C₂×D₄₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₅×D₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₉×D₅	C₂×C₄	1	1	yes	characteristic
25	D₅×D₉	—	2	2	no	—
26	C₄×C₅×C₉	C₂×C₄	1	1	yes	characteristic
27	Dic₄₅	C₂×C₄	1	1	yes	characteristic
28	C₉×Dic₅	C₂³	1	1	yes	characteristic
29	C₅×Dic₉	C₂×C₄	1	1	yes	characteristic
30	C₂×#700	—	9	1	no	Hall
31	C₂×C₄×D₉	—	5	1	no	—
32	C₉×M₁₆	—	5	1	no	—
33	#3667	—	5	1	no	—
34	#3663	—	5	2	no	—
35	C₈×D₉	—	5	2	no	—
36	C₂×D₃×D₅	—	3	1	no	—
37	C₄×D₁₅	—	3	1	no	—
38	C₄×C₅×D₃	—	3	1	no	—
39	C₃×C₄×D₅	C₂×D₃	1	1	yes	characteristic
40	#1200	—	3	1	no	—
41	D₃×Dic₅	—	3	1	no	—
42	D₅×Dic₃	—	3	1	no	—
43	C₃⋊#170	—	3	2	no	—
44	C₃×#170	C₂×D₃	1	2	yes	—
45	D₃×M₁₆	—	15	1	no	—
46	C₂×C₅×C₉	C₂²×C₄	1	1	yes	characteristic
47	D₄₅	—	2	1	no	—
48	C₉×D₅	—	2	1	no	—
49	C₅×D₉	M₁₆	1	2	yes	complemented
50	C₂×C₄×D₅	—	9	1	no	—
51	C₂×#170	—	9	2	no	—
52	#700	—	9	1	no	—
53	#700	—	9	2	no	—
54	#700	D₉	1	1	yes	characteristic, direct product factor
55	C₂²×D₉	—	5	1	no	—
56	C₂×C₄×C₉	—	5	1	no	—
57	C₂×Dic₉	—	5	1	no	—
58	C₄×D₉	—	5	1	no	—
59	C₄×D₉	—	5	1	no	—
60	C₄×D₉	—	5	1	no	—
61	C₄×D₉	AGL(1,𝔽₅)	1	1	yes	characteristic
62	C₈×C₉	—	5	2	no	—
63	C₉⋊C₈	—	5	2	no	—
64	C₂×D₁₅	—	3	1	no	—
65	C₂×C₅×D₃	—	3	1	no	—
66	C₂×C₃×D₅	C₄×D₃	1	1	yes	characteristic
67	D₃×D₅	—	6	2	no	—
68	C₃×C₄×C₅	C₄×D₃	1	1	yes	characteristic
69	Dic₁₅	—	3	1	no	—
70	C₃×Dic₅	C₂²×D₃	1	1	yes	characteristic
71	C₅×Dic₃	—	3	1	no	—
72	C₂×C₄×D₃	—	15	1	no	—
73	C₃×M₁₆	—	5	1	no	—
74	#208	—	15	1	no	—
75	ΓL(1,𝔽₂₅)	—	15	2	no	—
76	D₃×C₈	—	15	2	no	—
77	C₅×C₉	C₂×M₁₆	1	1	yes	characteristic, complemented, Hall
78	C₂²×D₅	—	9	1	no	—
79	C₂×C₄×C₅	—	9	1	no	—
80	C₂×Dic₅	—	9	1	no	—
81	C₄×D₅	—	9	1	no	—
82	C₄×D₅	—	9	1	no	—
83	C₄×D₅	—	9	1	no	—
84	C₄×D₅	C₂×D₉	1	1	yes	characteristic
85	#170	—	9	2	no	—
86	#170	C₂×D₉	1	2	yes	complemented
87	C₂²×C₉	—	5	1	no	—
88	C₂×D₉	—	5	1	no	—
89	C₂×D₉	—	10	2	no	—
90	C₂×D₉	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
91	C₄×C₉	—	5	1	no	—
92	C₄×C₉	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
93	Dic₉	—	5	1	no	—
94	Dic₉	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
95	C₂×M₁₆	—	45	1	no	Sylow
96	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
97	D₁₅	—	6	1	no	—
98	C₃×D₅	—	2	1	no	—
99	C₅×D₃	—	3	2	no	—
100	C₂²×D₃	—	15	1	no	—
101	C₂×C₃×C₄	—	5	1	no	—
102	C₂×Dic₃	—	15	1	no	—
103	C₄×D₃	—	3	1	no	—
104	C₄×D₃	—	15	1	no	—
105	C₄×D₃	—	15	1	no	—
106	C₄×D₃	—	15	1	no	—
107	C₃×C₈	—	5	2	no	—
108	C₃⋊C₈	—	15	2	no	—
109	C₂²×C₅	—	9	1	no	—
110	C₂×D₅	—	9	1	no	—
111	C₂×D₅	—	18	2	no	—
112	C₂×D₅	C₄×D₉	1	1	yes	characteristic
113	C₄×C₅	—	9	1	no	—
114	C₄×C₅	C₄×D₉	1	1	yes	characteristic
115	Dic₅	—	9	1	no	—
116	Dic₅	C₂²×D₉	1	1	yes	characteristic
117	C₂×C₉	—	10	1	no	—
118	C₂×C₉	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
119	D₉	—	10	1	no	—
120	D₉	#700	1	2	yes	direct product factor
121	C₂²×C₄	—	45	1	no	—
122	M₁₆	—	5	1	no	—
123	M₁₆	—	45	2	no	—
124	M₁₆	—	45	1	no	—
125	C₂×C₈	—	45	2	no	—
126	C₃×C₅	D₃×M₁₆	1	1	yes	characteristic
127	C₂²×C₃	—	5	1	no	—
128	C₂×D₃	—	3	1	no	—
129	C₂×D₃	—	15	1	no	—
130	C₂×D₃	—	30	2	no	—
131	C₃×C₄	—	5	1	no	—
132	C₃×C₄	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
133	Dic₃	—	3	1	no	—
134	Dic₃	—	15	1	no	—
135	C₂×C₅	—	9	2	no	—
136	C₂×C₅	C₂×C₄×D₉	1	1	yes	characteristic
137	D₅	—	2	1	no	—
138	D₅	—	18	1	no	—
139	C₉	C₂×#700	1	1	yes	characteristic, complemented, Sylow
140	C₂³	—	45	1	no	—
141	C₂×C₄	—	5	1	no	—
142	C₂×C₄	—	9	1	no	—
143	C₂×C₄	—	45	1	no	—
144	C₂×C₄	—	45	1	no	—
145	C₂×C₄	—	45	1	no	—
146	C₂×C₄	—	45	1	no	—
147	C₈	—	5	2	no	—
148	C₈	—	45	2	no	—
149	C₂×C₃	—	10	1	no	—
150	C₂×C₃	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
151	D₃	—	3	2	no	—
152	D₃	—	30	1	no	—
153	C₅	M₁₆×D₉	1	1	yes	characteristic, complemented, Sylow
154	C₂²	—	5	1	no	—
155	C₂²	—	9	1	no	—
156	C₂²	—	45	1	no	—
157	C₂²	—	90	2	no	—
158	C₄	—	5	1	no	—
159	C₄	—	9	1	no	—
160	C₄	—	45	1	no	—
161	C₄	D₉×AGL(1,𝔽₅)	1	1	yes	characteristic
162	C₃	D₃×#700	1	1	yes	characteristic
163	C₂	—	9	2	no	—
164	C₂	—	10	1	no	—
165	C₂	—	90	1	no	—
166	C₂	C₂×D₉×AGL(1,𝔽₅)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₅×D₉

C₂

yes

maximal, characteristic

#10623650

C₂

yes

maximal, characteristic, complemented

C₉×#700

C₂

yes

maximal, characteristic, complemented

#10623629

C₂

yes

maximal, complemented

D₉×#170

C₂

yes

maximal, complemented

D₃×#700

—

maximal

C₂×D₅×D₉

C₄

yes

characteristic

C₄×D₄₅

C₄

yes

characteristic

C₄×C₅×D₉

C₄

yes

characteristic

C₄×C₉×D₅

C₂²

yes

characteristic

D₅×Dic₉

C₄

yes

characteristic

#67079

C₂²

yes

characteristic

D₉×Dic₅

C₂²

yes

characteristic

C₉⋊#170

C₂²

yes

complemented

C₉×#170

C₂²

yes

complemented

M₁₆×D₉

—

maximal, Hall

C₄×D₃×D₅

—

#6758

—

C₃×#700

D₃

yes

characteristic

#6740

—

D₃×#170

—

C₂×D₄₅

C₂×C₄

yes

characteristic

C₂×C₅×D₉

C₂×C₄

yes

characteristic

C₂×C₉×D₅

C₂×C₄

yes

characteristic

D₅×D₉

—

C₄×C₅×C₉

C₂×C₄

yes

characteristic

Dic₄₅

C₂×C₄

yes

characteristic

C₉×Dic₅

C₂³

yes

characteristic

C₅×Dic₉

C₂×C₄

yes

characteristic

C₂×#700

—

Hall

C₂×C₄×D₉

—

C₉×M₁₆

—

#3667

—

#3663

—

C₈×D₉

—

C₂×D₃×D₅

—

C₄×D₁₅

—

C₄×C₅×D₃

—

C₃×C₄×D₅

C₂×D₃

yes

characteristic

#1200

—

D₃×Dic₅

—

D₅×Dic₃

—

C₃⋊#170

—

C₃×#170

C₂×D₃

yes

—

D₃×M₁₆

—

C₂×C₅×C₉

C₂²×C₄

yes

characteristic

D₄₅

—

C₉×D₅

—

C₅×D₉

M₁₆

yes

complemented

C₂×C₄×D₅

—

—

—

—

D₉

yes

characteristic, direct product factor

C₂²×D₉

—

C₂×C₄×C₉

—

C₂×Dic₉

—

C₄×D₉

—

C₄×D₉

—

C₄×D₉

—

C₄×D₉

AGL(1,𝔽₅)

yes

characteristic

C₈×C₉

—

C₉⋊C₈

—

C₂×D₁₅

—

C₂×C₅×D₃

—

C₂×C₃×D₅

C₄×D₃

yes

characteristic

D₃×D₅

—

C₃×C₄×C₅

C₄×D₃

yes

characteristic

Dic₁₅

—

C₃×Dic₅

C₂²×D₃

yes

characteristic

C₅×Dic₃

—

C₂×C₄×D₃

—

C₃×M₁₆

—

#208

—

ΓL(1,𝔽₂₅)

—

D₃×C₈

—

C₅×C₉

C₂×M₁₆

yes

characteristic, complemented, Hall

C₂²×D₅

—

C₂×C₄×C₅

—

C₂×Dic₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

C₂×D₉

yes

characteristic

#170

—

#170

C₂×D₉

yes

complemented

C₂²×C₉

—

C₂×D₉

—

C₂×D₉

—

C₂×D₉

C₂×AGL(1,𝔽₅)

yes

characteristic

C₄×C₉

—

C₄×C₉

C₂×AGL(1,𝔽₅)

yes

characteristic

Dic₉

—

Dic₉

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂×M₁₆

—

Sylow

C₂×C₃×C₅

C₂×C₄×D₃

yes

characteristic

D₁₅

—

C₃×D₅

—

C₅×D₃

—

100

C₂²×D₃

—

101

C₂×C₃×C₄

—

102

C₂×Dic₃

—

103

C₄×D₃

—

104

C₄×D₃

—

105

C₄×D₃

—

106

C₄×D₃

—

107

C₃×C₈

—

108

C₃⋊C₈

—

109

C₂²×C₅

—

110

C₂×D₅

—

111

C₂×D₅

—

112

C₂×D₅

C₄×D₉

yes

characteristic

113

C₄×C₅

—

114

C₄×C₅

C₄×D₉

yes

characteristic

115

Dic₅

—

116

Dic₅

C₂²×D₉

yes

characteristic

117

C₂×C₉

—

118

C₂×C₉

C₂²×AGL(1,𝔽₅)

yes

characteristic

119

D₉

—

120

D₉

#700

yes

direct product factor

121

C₂²×C₄

—

122

M₁₆

—

123

M₁₆

—

124

M₁₆

—

125

C₂×C₈

—

126

C₃×C₅

D₃×M₁₆

yes

characteristic

127

C₂²×C₃

—

128

C₂×D₃

—

129

C₂×D₃

—

130

C₂×D₃

—

131

C₃×C₄

—

132

C₃×C₄

D₃×AGL(1,𝔽₅)

yes

characteristic

133

Dic₃

—

134

Dic₃

—

135

C₂×C₅

—

136

C₂×C₅

C₂×C₄×D₉

yes

characteristic

137

D₅

—

138

D₅

—

139

C₉

C₂×#700

yes

characteristic, complemented, Sylow

140

C₂³

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₈

—

148

C₈

—

149

C₂×C₃

—

150

C₂×C₃

C₂×D₃×AGL(1,𝔽₅)

yes

characteristic

151

D₃

—

152

D₃

—

153

C₅

M₁₆×D₉

yes

characteristic, complemented, Sylow

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

D₉×AGL(1,𝔽₅)

yes

characteristic

162

C₃

D₃×#700

yes

characteristic

163

C₂

—

164

C₂

—

165

C₂

—

166

C₂

C₂×D₉×AGL(1,𝔽₅)

yes

characteristic

	subgroup	G/H
center	C₂	C₂×D₉×AGL(1,𝔽₅)
commutator	C₂×C₅×C₉	C₂²×C₄
Frattini	C₂×C₃	C₂×D₃×AGL(1,𝔽₅)
Fitting	C₄×C₅×C₉	C₂×C₄
socle	C₂×C₃×C₅	C₂×C₄×D₃
prime radical	C₂×D₅×D₉	C₄
cosocle	C₉×Dic₅	C₂³
hypercenter	C₄	D₉×AGL(1,𝔽₅)
hypocenter	C₅×C₉	C₂×M₁₆
LCS	G ▷ C₂×C₅×C₉ ▷ C₅×C₉
UCS	C₁ ◁ C₂ ◁ C₄

subgroup

G/H

center

C₂

C₂×D₉×AGL(1,𝔽₅)

commutator

C₂×C₅×C₉

C₂²×C₄

Frattini

C₂×C₃

C₂×D₃×AGL(1,𝔽₅)

Fitting

C₄×C₅×C₉

C₂×C₄

socle

C₂×C₃×C₅

C₂×C₄×D₃

prime radical

C₂×D₅×D₉

C₄

cosocle

C₉×Dic₅

C₂³

hypercenter

C₄

D₉×AGL(1,𝔽₅)

hypocenter

C₅×C₉

C₂×M₁₆

LCS

G ▷ C₂×C₅×C₉ ▷ C₅×C₉

UCS

C₁ ◁ C₂ ◁ C₄

N	H ≅ G/N	type
2. #10623650	C₂	semidirect
3. C₉×#700	C₂	semidirect
4. #10623629	C₂	semidirect
5. D₉×#170	C₂	semidirect
14. C₉⋊#170	C₂²	semidirect
15. C₉×#170	C₂²	semidirect
49. C₅×D₉	M₁₆	semidirect
54. #700	D₉	direct (N × H)
77. C₅×C₉	C₂×M₁₆	semidirect
86. #170	C₂×D₉	semidirect
120. D₉	#700	direct (N × H)
139. C₉	C₂×#700	semidirect
153. C₅	M₁₆×D₉	semidirect

H ≅ G/N

type

2. #10623650

C₂

semidirect

3. C₉×#700

C₂

semidirect

4. #10623629

C₂

semidirect

5. D₉×#170

C₂

semidirect

14. C₉⋊#170

C₂²

semidirect

15. C₉×#170

C₂²

semidirect

49. C₅×D₉

M₁₆

semidirect

54. #700

D₉

direct (N × H)

77. C₅×C₉

C₂×M₁₆

semidirect

86. #170

C₂×D₉

semidirect

120. D₉

#700

direct (N × H)

139. C₉

C₂×#700

semidirect

153. C₅

M₁₆×D₉

semidirect

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

subgroup	index
C₄×D₅×D₉	2
#10623650	2
C₉×#700	2
#10623629	2
D₉×#170	2
D₃×#700	3
M₁₆×D₉	5

subgroup

index

C₄×D₅×D₉

M₁₆×D₉

Subgroups — D₉×#700 #13085163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D9×#700 #13085163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₉×#700 #13085163