Group #14512500 — subgroups

Subgroups — #14512500

Order 1920.

236 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 50 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	Dic₅×#41⋊C₃	C₂	1	1	yes	maximal, characteristic
2	SL(2,𝔽₃)⋊#170	C₂	1	2	yes	maximal, complemented
3	#10605001	—	3	1	no	maximal, Hall
4	C₂×C₅×#41⋊C₃	C₄	1	1	yes	characteristic
5	C₄⋊(C₃⋊#170)	—	4	1	no	maximal
6	Dic₅×SL(2,𝔽₃)	C₄	1	1	yes	characteristic
7	Dic₅×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
8	#73071	—	5	1	no	maximal, Hall
9	#41×Dic₅	D₃	1	1	yes	characteristic
10	#65748	—	3	1	no	—
11	C₂×C₄⋊#170	—	3	1	no	—
12	Q₈⋊#170	—	3	2	no	—
13	D₄⋊#170	—	3	2	no	—
14	C₅×#41⋊C₃	C₈	1	2	yes	complemented
15	C₂×C₅×SL(2,𝔽₃)	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃⋊#170	—	4	2	no	—
17	C₃×C₄×Dic₅	—	4	1	no	—
18	C₄×#41⋊C₃	—	5	1	no	—
19	SL(2,𝔽₃)⋊C₈	—	5	2	no	—
20	C₂×C₅×#41	Dic₃	1	1	yes	characteristic
21	C₂²×#170	—	6	1	no	—
22	Q₈×Dic₅	Dic₃	1	1	yes	characteristic
23	Q₈×Dic₅	C₂×D₃	1	1	yes	characteristic
24	D₄×Dic₅	—	3	1	no	—
25	D₄×Dic₅	—	3	1	no	—
26	#4072	—	3	1	no	—
27	C₂×C₄×Dic₅	—	3	1	no	—
28	C₂²⋊#170	—	6	1	no	—
29	C₄⋊#170	—	3	2	no	—
30	C₄⋊#170	—	6	1	no	—
31	C₄⋊#170	—	6	1	no	—
32	C₄×#170	—	3	2	no	—
33	#1477	—	15	1	no	Sylow
34	C₂×C₃×C₄×C₅	—	4	1	no	—
35	C₂×C₃×Dic₅	—	4	1	no	—
36	C₂×C₃×Dic₅	—	4	1	no	—
37	C₅×SL(2,𝔽₃)	C₂×C₈	1	1	yes	characteristic, complemented
38	C₃⋊#170	—	8	2	no	—
39	C₂×#41⋊C₃	AGL(1,𝔽₅)	1	1	yes	characteristic
40	C₄×SL(2,𝔽₃)	—	5	1	no	—
41	C₄×SL(2,𝔽₃)	—	5	1	no	—
42	C₄⋊(C₃⋊C₈)	—	20	1	no	—
43	C₅×#41	—	3	2	no	—
44	C₅×#41	C₃⋊C₈	1	2	yes	complemented
45	C₂×C₅×Q₈	C₂×Dic₃	1	1	yes	characteristic
46	C₂×C₅×D₄	—	3	1	no	—
47	C₂²×C₄×C₅	—	3	1	no	—
48	C₂²×Dic₅	—	3	1	no	—
49	C₂²×Dic₅	—	3	1	no	—
50	C₂×#170	—	6	2	no	—
51	C₂×#170	—	6	2	no	—
52	C₂×#170	—	12	2	no	—
53	C₂²⋊Dic₅	—	6	1	no	—
54	C₄⋊Dic₅	—	3	1	no	—
55	C₄⋊Dic₅	—	3	1	no	—
56	C₄×Dic₅	—	3	1	no	—
57	C₄×Dic₅	—	3	1	no	—
58	C₄×Dic₅	—	3	1	no	—
59	C₄×Dic₅	S₄	1	1	yes	characteristic
60	C₄×#41	—	5	1	no	—
61	#431	—	15	1	no	—
62	C₂×C₄⋊C₈	—	15	1	no	—
63	Q₈⋊C₈	—	15	2	no	—
64	D₄⋊C₈	—	15	2	no	—
65	C₂²×C₃×C₅	—	4	1	no	—
66	C₃×C₄×C₅	—	4	2	no	—
67	C₃×Dic₅	—	4	1	no	—
68	C₃×Dic₅	—	4	1	no	—
69	C₃×Dic₅	—	8	1	no	—
70	#41⋊C₃	#170	1	2	yes	complemented
71	C₂×SL(2,𝔽₃)	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
72	C₃×C₄²	—	20	1	no	—
73	C₂×C₃⋊C₈	—	20	2	no	—
74	C₂³×C₅	—	3	1	no	—
75	C₅×Q₈	—	3	1	no	—
76	C₅×Q₈	C₂×C₃⋊C₈	1	1	yes	characteristic
77	C₅×D₄	—	3	2	no	—
78	C₅×D₄	—	6	1	no	—
79	C₂×C₄×C₅	—	3	1	no	—
80	C₂×C₄×C₅	—	3	2	no	—
81	C₂×C₄×C₅	—	3	2	no	—
82	C₂×C₄×C₅	SL(2,ℤ/4)	1	1	yes	characteristic
83	C₂×Dic₅	—	3	1	no	—
84	C₂×Dic₅	—	3	1	no	—
85	C₂×Dic₅	—	6	1	no	—
86	C₂×Dic₅	—	6	1	no	—
87	C₂×Dic₅	—	12	1	no	—
88	C₂×Dic₅	SL(2,ℤ/4)	1	1	yes	characteristic
89	C₂×Dic₅	C₂×S₄	1	1	yes	characteristic
90	#170	—	12	2	no	—
91	#170	—	12	2	no	—
92	C₂×#41	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
93	C₂²×C₈	—	30	1	no	—
94	C₄×Q₈	—	5	1	no	—
95	C₄×Q₈	—	5	1	no	—
96	C₄×D₄	—	15	1	no	—
97	C₄×D₄	—	15	1	no	—
98	#117	—	15	1	no	—
99	C₂×C₄²	—	15	1	no	—
100	C₄⋊C₈	—	15	2	no	—
101	C₄⋊C₈	—	30	1	no	—
102	C₄⋊C₈	—	30	1	no	—
103	C₂²⋊C₈	—	30	1	no	—
104	C₄×C₈	—	15	2	no	—
105	C₂×C₃×C₅	—	4	1	no	—
106	C₂×C₃×C₅	—	4	1	no	—
107	C₂×C₃×C₅	—	4	1	no	—
108	C₂×C₃×C₄	—	4	1	no	—
109	C₂×C₃×C₄	—	20	1	no	—
110	C₂×C₃×C₄	—	20	1	no	—
111	SL(2,𝔽₃)	C₂×#170	1	1	yes	characteristic, complemented
112	C₃⋊C₈	—	40	2	no	—
113	C₂²×C₅	—	3	2	no	—
114	C₂²×C₅	—	6	1	no	—
115	C₂²×C₅	—	6	1	no	—
116	C₂²×C₅	C₂×SL(2,ℤ/4)	1	1	yes	characteristic
117	C₄×C₅	—	3	1	no	—
118	C₄×C₅	—	3	1	no	—
119	C₄×C₅	A₄⋊C₈	1	2	yes	—
120	Dic₅	—	2	1	no	—
121	Dic₅	—	6	1	no	—
122	Dic₅	—	6	1	no	—
123	Dic₅	#984	1	1	yes	characteristic
124	Dic₅	#986	1	1	yes	characteristic
125	#41	—	3	2	no	—
126	#41	C₃⋊#170	1	2	yes	complemented
127	C₂×Q₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
128	C₂×D₄	—	3	1	no	—
129	C₂²×C₄	—	3	1	no	—
130	C₂²×C₄	—	15	1	no	—
131	C₂²×C₄	—	15	1	no	—
132	C₂×C₈	—	30	2	no	—
133	C₂×C₈	—	30	2	no	—
134	C₂×C₈	—	60	2	no	—
135	C₄⋊C₄	—	15	1	no	—
136	C₄⋊C₄	—	15	1	no	—
137	C₂²⋊C₄	—	30	1	no	—
138	C₄²	—	5	1	no	—
139	C₄²	—	15	1	no	—
140	C₄²	—	15	1	no	—
141	C₄²	—	15	1	no	—
142	C₃×C₅	—	4	1	no	Hall
143	C₂²×C₃	—	4	1	no	—
144	C₃×C₄	—	4	2	no	—
145	C₃×C₄	—	20	1	no	—
146	C₃×C₄	—	20	1	no	—
147	C₃×C₄	—	40	1	no	—
148	C₂×C₅	—	6	2	no	—
149	C₂×C₅	C₂×A₄⋊C₈	1	1	yes	characteristic
150	C₂×C₅	#5423	1	1	yes	characteristic
151	C₂×C₅	#5426	1	1	yes	characteristic
152	C₂³	—	3	1	no	—
153	Q₈	—	3	1	no	—
154	Q₈	C₂×C₃⋊#170	1	1	yes	characteristic
155	D₄	—	3	2	no	—
156	D₄	—	6	1	no	—
157	C₂×C₄	—	3	1	no	—
158	C₂×C₄	—	3	2	no	—
159	C₂×C₄	—	3	2	no	—
160	C₂×C₄	—	5	1	no	—
161	C₂×C₄	—	5	1	no	—
162	C₂×C₄	—	15	1	no	—
163	C₂×C₄	—	15	1	no	—
164	C₂×C₄	—	30	1	no	—
165	C₂×C₄	—	30	1	no	—
166	C₂×C₄	—	60	1	no	—
167	C₂×C₄	A₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
168	C₈	—	60	2	no	—
169	C₈	—	60	2	no	—
170	C₂×C₃	—	4	1	no	—
171	C₂×C₃	—	4	1	no	—
172	C₂×C₃	—	4	1	no	—
173	C₅	#73071	1	1	yes	characteristic, complemented, Sylow
174	C₂²	—	3	2	no	—
175	C₂²	—	6	1	no	—
176	C₂²	—	6	1	no	—
177	C₂²	C₂×A₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
178	C₄	—	3	1	no	—
179	C₄	—	3	1	no	—
180	C₄	—	5	1	no	—
181	C₄	—	5	1	no	—
182	C₄	—	10	1	no	—
183	C₄	—	30	1	no	—
184	C₄	—	30	1	no	—
185	C₄	A₄⋊#170	1	2	yes	—
186	C₃	—	4	1	no	Sylow
187	C₂	—	6	2	no	—
188	C₂	#11756724	1	1	yes	characteristic
189	C₂	#11756726	1	1	yes	characteristic
190	C₂	C₂×A₄⋊#170	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Dic₅×#41⋊C₃

C₂

yes

maximal, characteristic

SL(2,𝔽₃)⋊#170

C₂

yes

maximal, complemented

#10605001

—

maximal, Hall

C₂×C₅×#41⋊C₃

C₄

yes

—

C₄

yes

characteristic

Dic₅×SL(2,𝔽₃)

C₂²

yes

—

D₃

yes

characteristic

#65748

—

C₂×C₄⋊#170

—

Q₈⋊#170

—

D₄⋊#170

—

C₅×#41⋊C₃

C₈

yes

complemented

C₂×C₅×SL(2,𝔽₃)

C₂×C₄

yes

characteristic

C₂×C₃⋊#170

—

C₃×C₄×Dic₅

—

C₄×#41⋊C₃

—

SL(2,𝔽₃)⋊C₈

—

C₂×C₅×#41

Dic₃

yes

characteristic

C₂²×#170

—

Q₈×Dic₅

Dic₃

yes

characteristic

Q₈×Dic₅

C₂×D₃

yes

characteristic

D₄×Dic₅

—

D₄×Dic₅

—

#4072

—

C₂×C₄×Dic₅

—

C₂²⋊#170

—

—

—

—

—

—

C₂×C₃×C₄×C₅

—

C₂×C₃×Dic₅

—

C₂×C₃×Dic₅

—

C₅×SL(2,𝔽₃)

C₂×C₈

yes

characteristic, complemented

C₃⋊#170

—

C₂×#41⋊C₃

AGL(1,𝔽₅)

yes

characteristic

C₄×SL(2,𝔽₃)

—

C₄×SL(2,𝔽₃)

—

C₄⋊(C₃⋊C₈)

—

C₅×#41

—

C₅×#41

C₃⋊C₈

yes

complemented

C₂×C₅×Q₈

C₂×Dic₃

yes

characteristic

C₂×C₅×D₄

—

C₂²×C₄×C₅

—

C₂²×Dic₅

—

C₂²×Dic₅

—

C₂×#170

—

C₂×#170

—

C₂×#170

—

C₂²⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

S₄

yes

characteristic

C₄×#41

—

#431

—

C₂×C₄⋊C₈

—

Q₈⋊C₈

—

D₄⋊C₈

—

C₂²×C₃×C₅

—

C₃×C₄×C₅

—

C₃×Dic₅

—

C₃×Dic₅

—

C₃×Dic₅

—

yes

C₂×SL(2,𝔽₃)

yes

C₃×C₄²

—

C₂×C₃⋊C₈

—

C₂³×C₅

—

C₅×Q₈

—

C₅×Q₈

C₂×C₃⋊C₈

yes

characteristic

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

SL(2,ℤ/4)

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

SL(2,ℤ/4)

yes

characteristic

C₂×Dic₅

C₂×S₄

yes

characteristic

#170

—

#170

—

C₂×#41

ΓL(1,𝔽₁₆)

yes

characteristic

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

C₂²⋊C₈

—

104

C₄×C₈

—

105

C₂×C₃×C₅

—

106

C₂×C₃×C₅

—

107

C₂×C₃×C₅

—

108

C₂×C₃×C₄

—

109

C₂×C₃×C₄

—

110

C₂×C₃×C₄

—

111

SL(2,𝔽₃)

C₂×#170

yes

characteristic, complemented

112

C₃⋊C₈

—

113

C₂²×C₅

—

114

C₂²×C₅

—

115

C₂²×C₅

—

116

C₂²×C₅

C₂×SL(2,ℤ/4)

yes

characteristic

117

C₄×C₅

—

118

C₄×C₅

—

119

C₄×C₅

A₄⋊C₈

yes

—

120

Dic₅

—

121

Dic₅

—

122

Dic₅

—

123

Dic₅

#984

yes

characteristic

124

Dic₅

#986

yes

characteristic

125

#41

—

126

#41

C₃⋊#170

yes

complemented

127

C₂×Q₈

C₂×ΓL(1,𝔽₁₆)

yes

characteristic

128

C₂×D₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₂²⋊C₄

—

138

C₄²

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₃×C₅

—

Hall

143

C₂²×C₃

—

144

C₃×C₄

—

145

C₃×C₄

—

146

C₃×C₄

—

147

C₃×C₄

—

148

C₂×C₅

—

149

C₂×C₅

C₂×A₄⋊C₈

yes

characteristic

150

C₂×C₅

#5423

yes

characteristic

151

C₂×C₅

#5426

yes

characteristic

152

C₂³

—

153

Q₈

—

154

Q₈

C₂×C₃⋊#170

yes

characteristic

155

D₄

—

156

D₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

A₄⋊AGL(1,𝔽₅)

yes

characteristic

168

C₈

—

169

C₈

—

170

C₂×C₃

—

171

C₂×C₃

—

172

C₂×C₃

—

173

C₅

#73071

yes

characteristic, complemented, Sylow

174

C₂²

—

175

C₂²

—

176

C₂²

—

177

C₂²

C₂×A₄⋊AGL(1,𝔽₅)

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

A₄⋊#170

yes

—

186

C₃

—

Sylow

187

C₂

—

188

C₂

#11756724

yes

characteristic

189

C₂

#11756726

yes

characteristic

190

C₂

C₂×A₄⋊#170

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×A₄⋊AGL(1,𝔽₅)
commutator, hypocenter	C₅×SL(2,𝔽₃)	C₂×C₈
Fitting	C₂×C₅×#41	Dic₃
socle	C₂²×C₅	C₂×SL(2,ℤ/4)
prime radical	C₂×C₅×#41⋊C₃	C₄
cosocle	Dic₅×SL(2,𝔽₃)	C₂²
hypercenter	C₂×C₄	A₄⋊AGL(1,𝔽₅)
LCS	G ▷ C₅×SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂×C₄

subgroup

G/H

center, Frattini

C₂²

C₂×A₄⋊AGL(1,𝔽₅)

commutator, hypocenter

C₅×SL(2,𝔽₃)

C₂×C₈

Fitting

C₂×C₅×#41

Dic₃

socle

C₂²×C₅

C₂×SL(2,ℤ/4)

prime radical

C₂×C₅×#41⋊C₃

C₄

C₂²

C₂×C₄

G ▷ C₅×SL(2,𝔽₃)

C₁ ◁ C₂² ◁ C₂×C₄

N	H ≅ G/N	type
2. SL(2,𝔽₃)⋊#170	C₂	semidirect
14. C₅×#41⋊C₃	C₈	semidirect
37. C₅×SL(2,𝔽₃)	C₂×C₈	semidirect
44. C₅×#41	C₃⋊C₈	semidirect
70. #41⋊C₃	#170	semidirect
111. SL(2,𝔽₃)	C₂×#170	semidirect
126. #41	C₃⋊#170	semidirect
173. C₅	#73071	semidirect

H ≅ G/N

type

2. SL(2,𝔽₃)⋊#170

C₂

semidirect

14. C₅×#41⋊C₃

C₈

semidirect

37. C₅×SL(2,𝔽₃)

C₂×C₈

semidirect

44. C₅×#41

C₃⋊C₈

semidirect

70. #41⋊C₃

#170

semidirect

111. SL(2,𝔽₃)

C₂×#170

semidirect

126. #41

C₃⋊#170

semidirect

173. C₅

#73071

semidirect

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

subgroup	index
Dic₅×#41⋊C₃	2
SL(2,𝔽₃)⋊#170	2
#10605001	3
C₄⋊(C₃⋊#170)	4
#73071	5

subgroup

index

Dic₅×#41⋊C₃

Subgroups — #14512500

Conjugacy classes of subgroups

Complements

Maximal subgroups