Group #14308998 — subgroups

Subgroups — #14308998

Order 1920.

580 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 129 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	C₃×C₅×#543	C₂	1	1	yes	maximal, characteristic
2	#11751044	C₂	1	3	yes	maximal
3	#11746347	C₂	1	3	yes	maximal
4	#10602687	—	3	1	no	maximal, Hall
5	C₂³×Dic₁₅	C₂²	1	1	yes	characteristic
6	C₃×C₅×C₂²⋊Q₈	—	2	6	no	—
7	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	3	yes	—
8	C₂×#6719	C₂²	1	3	yes	—
9	#91115	—	2	6	no	—
10	#71894	—	5	1	no	maximal, Hall
11	C₅×#543	D₃	1	1	yes	characteristic
12	#65232	—	3	3	no	—
13	#64746	—	3	3	no	—
14	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
15	C₂×C₃×C₅×Q₈	—	2	2	no	—
16	C₂²×C₃×C₄×C₅	D₄	1	6	yes	—
17	C₂²×Dic₁₅	—	2	3	no	—
18	C₂²×Dic₁₅	—	2	3	no	—
19	C₂²×Dic₁₅	—	2	3	no	—
20	C₂²×Dic₁₅	Q₈	1	2	yes	complemented
21	C₃×C₅×C₄⋊C₄	—	2	6	no	—
22	C₃×C₅×C₂²⋊C₄	—	2	6	no	—
23	#6719	—	2	6	no	—
24	C₃×#543	D₅	1	1	yes	characteristic
25	#4943	—	5	3	no	—
26	#4526	—	5	3	no	—
27	C₂³×Dic₅	—	3	1	no	—
28	C₅×C₂²⋊Q₈	—	2	6	no	—
29	C₂×C₅×C₂²⋊C₄	C₂×D₃	1	3	yes	—
30	C₂×C₂²⋊Dic₅	—	3	3	no	—
31	#3963	—	6	6	no	—
32	#2030	—	15	1	no	Sylow
33	C₂³×C₃×C₅	—	4	2	no	—
34	C₂³×C₃×C₅	C₂×D₄	1	3	yes	—
35	C₂³×C₃×C₅	C₂×Q₈	1	1	yes	characteristic
36	C₂³×C₃×C₅	#41	1	3	yes	—
37	C₃×C₅×Q₈	—	8	2	no	—
38	C₂×C₃×C₄×C₅	—	2	6	no	—
39	C₂×C₃×C₄×C₅	—	4	6	no	—
40	C₂×Dic₁₅	—	2	6	no	—
41	C₂×Dic₁₅	—	4	1	no	—
42	C₂×Dic₁₅	—	4	3	no	—
43	C₂×Dic₁₅	—	4	6	no	—
44	C₂×Dic₁₅	—	4	3	no	—
45	C₂³×Dic₃	—	5	1	no	—
46	C₃×C₂²⋊Q₈	—	2	6	no	—
47	C₂×C₃×C₂²⋊C₄	C₂×D₅	1	3	yes	—
48	C₂×#217	—	5	3	no	—
49	C₃⋊#95	—	10	6	no	—
50	C₂⁴×C₅	C₂²×D₃	1	1	yes	characteristic
51	C₂×C₅×Q₈	—	2	2	no	—
52	C₂²×C₄×C₅	#67	1	6	yes	—
53	C₂²×Dic₅	—	3	2	no	—
54	C₂²×Dic₅	—	6	3	no	—
55	C₂²×Dic₅	—	6	3	no	—
56	C₂²×Dic₅	—	6	3	no	—
57	C₅×C₄⋊C₄	—	2	6	no	—
58	C₅×C₂²⋊C₄	—	2	6	no	—
59	C₂²⋊Dic₅	—	6	6	no	—
60	#543	D₁₅	1	1	yes	characteristic
61	#381	—	15	3	no	—
62	#342	—	15	3	no	—
63	C₂²×C₃×C₅	—	2	6	no	—
64	C₂²×C₃×C₅	—	4	1	no	—
65	C₂²×C₃×C₅	—	4	3	no	—
66	C₂²×C₃×C₅	#120	1	1	yes	characteristic
67	C₂²×C₃×C₅	C₂²⋊Q₈	1	3	yes	—
68	C₂²×C₃×C₅	#124	1	3	yes	—
69	C₃×C₄×C₅	—	4	6	no	—
70	Dic₁₅	—	4	2	no	—
71	Dic₁₅	—	8	3	no	—
72	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
73	C₂×C₃×Q₈	—	2	2	no	—
74	C₂²×C₃×C₄	C₂²⋊D₅	1	6	yes	—
75	C₂²×Dic₃	—	5	2	no	—
76	C₂²×Dic₃	—	10	3	no	—
77	C₂²×Dic₃	—	10	3	no	—
78	C₂²×Dic₃	—	10	3	no	—
79	C₃×C₄⋊C₄	—	2	6	no	—
80	C₃×C₂²⋊C₄	—	2	6	no	—
81	#217	—	10	6	no	—
82	C₂³×C₅	—	4	2	no	—
83	C₂³×C₅	D₃×Q₈	1	1	yes	characteristic
84	C₂³×C₅	#239	1	3	yes	—
85	C₂³×C₅	C₂×#67	1	3	yes	—
86	C₅×Q₈	—	8	2	no	—
87	C₂×C₄×C₅	—	2	6	no	—
88	C₂×C₄×C₅	—	4	6	no	—
89	C₂×Dic₅	—	6	6	no	—
90	C₂×Dic₅	—	12	3	no	—
91	C₂×Dic₅	—	12	1	no	—
92	C₂×Dic₅	—	12	6	no	—
93	C₂×Dic₅	—	12	3	no	—
94	C₂³×C₄	—	15	1	no	—
95	C₂²⋊Q₈	—	2	6	no	—
96	C₂×C₂²⋊C₄	—	15	3	no	—
97	C₂×C₂²⋊C₄	C₂×D₁₅	1	3	yes	—
98	#95	—	30	6	no	—
99	C₂×C₃×C₅	—	2	3	no	—
100	C₂×C₃×C₅	—	2	3	no	—
101	C₂×C₃×C₅	#393	1	1	yes	characteristic
102	C₂×C₃×C₅	#458	1	2	yes	—
103	C₂³×C₃	—	4	2	no	—
104	C₂³×C₃	Q₈×D₅	1	1	yes	characteristic
105	C₂³×C₃	#713	1	3	yes	—
106	C₂³×C₃	C₂×C₂²⋊D₅	1	3	yes	—
107	C₃×Q₈	—	8	2	no	—
108	C₂×C₃×C₄	—	2	6	no	—
109	C₂×C₃×C₄	—	4	6	no	—
110	C₂×Dic₃	—	10	6	no	—
111	C₂×Dic₃	—	20	1	no	—
112	C₂×Dic₃	—	20	3	no	—
113	C₂×Dic₃	—	20	6	no	—
114	C₂×Dic₃	—	20	3	no	—
115	C₂²×C₅	—	2	6	no	—
116	C₂²×C₅	—	4	1	no	—
117	C₂²×C₅	—	4	3	no	—
118	C₂²×C₅	#946	1	3	yes	—
119	C₂²×C₅	#947	1	3	yes	—
120	C₂²×C₅	#953	1	1	yes	characteristic
121	C₄×C₅	—	4	6	no	—
122	Dic₅	—	12	2	no	—
123	Dic₅	—	24	3	no	—
124	C₂⁴	C₂²×D₁₅	1	1	yes	characteristic
125	C₂×Q₈	—	2	2	no	—
126	C₂²×C₄	—	15	2	no	—
127	C₂²×C₄	—	30	3	no	—
128	C₂²×C₄	—	30	3	no	—
129	C₂²×C₄	—	30	3	no	—
130	C₂²×C₄	#1220	1	6	yes	—
131	C₄⋊C₄	—	2	6	no	—
132	C₂²⋊C₄	—	2	6	no	—
133	C₂²⋊C₄	—	30	6	no	—
134	C₃×C₅	#2030	1	1	yes	characteristic, complemented, Hall
135	C₂²×C₃	—	2	6	no	—
136	C₂²×C₃	—	4	1	no	—
137	C₂²×C₃	—	4	3	no	—
138	C₂²×C₃	#4092	1	3	yes	—
139	C₂²×C₃	#4093	1	3	yes	—
140	C₂²×C₃	#4099	1	1	yes	characteristic
141	C₃×C₄	—	4	6	no	—
142	Dic₃	—	20	2	no	—
143	Dic₃	—	40	3	no	—
144	C₂×C₅	—	2	3	no	—
145	C₂×C₅	—	2	3	no	—
146	C₂×C₅	#5231	1	1	yes	characteristic
147	C₂×C₅	#5243	1	2	yes	—
148	C₂³	—	4	2	no	—
149	C₂³	Q₈×D₁₅	1	1	yes	characteristic
150	C₂³	#6821	1	3	yes	—
151	C₂³	C₂×#1220	1	3	yes	—
152	Q₈	—	8	2	no	—
153	C₂×C₄	—	2	6	no	—
154	C₂×C₄	—	4	6	no	—
155	C₂×C₄	—	30	6	no	—
156	C₂×C₄	—	60	3	no	—
157	C₂×C₄	—	60	1	no	—
158	C₂×C₄	—	60	6	no	—
159	C₂×C₄	—	60	3	no	—
160	C₂×C₃	—	2	3	no	—
161	C₂×C₃	—	2	3	no	—
162	C₂×C₃	#65520	1	1	yes	characteristic
163	C₂×C₃	#65532	1	2	yes	—
164	C₅	#71894	1	1	yes	characteristic, complemented, Sylow
165	C₂²	—	2	6	no	—
166	C₂²	—	4	1	no	—
167	C₂²	—	4	3	no	—
168	C₂²	#91835	1	3	yes	—
169	C₂²	#91836	1	3	yes	—
170	C₂²	#91842	1	1	yes	characteristic
171	C₄	—	4	6	no	—
172	C₄	—	60	2	no	—
173	C₄	—	120	3	no	—
174	C₃	#10602687	1	1	yes	characteristic, complemented, Sylow
175	C₂	—	2	3	no	—
176	C₂	—	2	3	no	—
177	C₂	#11751332	1	1	yes	characteristic
178	C₂	#11751344	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₅×#543

C₂

yes

maximal, characteristic

#11751044

C₂

yes

maximal

#11746347

C₂

yes

maximal

#10602687

—

maximal, Hall

C₂³×Dic₁₅

C₂²

yes

characteristic

C₃×C₅×C₂²⋊Q₈

—

C₂×C₃×C₅×C₂²⋊C₄

C₂²

yes

—

C₂×#6719

C₂²

yes

—

—

—

D₃

yes

characteristic

#65232

—

#64746

—

C₂⁴×C₃×C₅

C₂³

yes

characteristic

C₂×C₃×C₅×Q₈

—

C₂²×C₃×C₄×C₅

D₄

yes

—

C₂²×Dic₁₅

—

C₂²×Dic₁₅

—

C₂²×Dic₁₅

—

C₂²×Dic₁₅

Q₈

yes

complemented

C₃×C₅×C₄⋊C₄

—

C₃×C₅×C₂²⋊C₄

—

#6719

—

C₃×#543

D₅

yes

characteristic

#4943

—

#4526

—

C₂³×Dic₅

—

C₅×C₂²⋊Q₈

—

C₂×C₅×C₂²⋊C₄

C₂×D₃

yes

—

C₂×C₂²⋊Dic₅

—

#3963

—

#2030

—

Sylow

C₂³×C₃×C₅

—

C₂³×C₃×C₅

C₂×D₄

yes

—

C₂³×C₃×C₅

C₂×Q₈

yes

characteristic

C₂³×C₃×C₅

#41

yes

—

C₃×C₅×Q₈

—

C₂×C₃×C₄×C₅

—

C₂×C₃×C₄×C₅

—

C₂×Dic₁₅

—

C₂×Dic₁₅

—

C₂×Dic₁₅

—

C₂×Dic₁₅

—

C₂×Dic₁₅

—

C₂³×Dic₃

—

C₃×C₂²⋊Q₈

—

C₂×C₃×C₂²⋊C₄

C₂×D₅

yes

—

C₂×#217

—

C₃⋊#95

—

C₂⁴×C₅

C₂²×D₃

yes

characteristic

C₂×C₅×Q₈

—

C₂²×C₄×C₅

#67

yes

—

C₂²×Dic₅

—

C₂²×Dic₅

—

C₂²×Dic₅

—

C₂²×Dic₅

—

C₅×C₄⋊C₄

—

C₅×C₂²⋊C₄

—

C₂²⋊Dic₅

—

#543

D₁₅

yes

characteristic

#381

—

#342

—

C₂²×C₃×C₅

—

C₂²×C₃×C₅

—

C₂²×C₃×C₅

—

C₂²×C₃×C₅

#120

yes

characteristic

C₂²×C₃×C₅

C₂²⋊Q₈

yes

—

C₂²×C₃×C₅

#124

yes

—

C₃×C₄×C₅

—

Dic₁₅

—

Dic₁₅

—

C₂⁴×C₃

C₂²×D₅

yes

characteristic

C₂×C₃×Q₈

—

C₂²×C₃×C₄

C₂²⋊D₅

yes

—

C₂²×Dic₃

—

C₂²×Dic₃

—

C₂²×Dic₃

—

C₂²×Dic₃

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

#217

—

C₂³×C₅

—

C₂³×C₅

D₃×Q₈

yes

characteristic

C₂³×C₅

#239

yes

—

C₂³×C₅

C₂×#67

yes

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂³×C₄

—

C₂²⋊Q₈

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₁₅

yes

—

#95

—

C₂×C₃×C₅

—

100

C₂×C₃×C₅

—

101

C₂×C₃×C₅

#393

yes

characteristic

102

C₂×C₃×C₅

#458

yes

—

103

C₂³×C₃

—

104

C₂³×C₃

Q₈×D₅

yes

characteristic

105

C₂³×C₃

#713

yes

—

106

C₂³×C₃

C₂×C₂²⋊D₅

yes

—

107

C₃×Q₈

—

108

C₂×C₃×C₄

—

109

C₂×C₃×C₄

—

110

C₂×Dic₃

—

111

C₂×Dic₃

—

112

C₂×Dic₃

—

113

C₂×Dic₃

—

114

C₂×Dic₃

—

115

C₂²×C₅

—

116

C₂²×C₅

—

117

C₂²×C₅

—

118

C₂²×C₅

#946

yes

—

119

C₂²×C₅

#947

yes

—

120

C₂²×C₅

#953

yes

characteristic

121

C₄×C₅

—

122

Dic₅

—

123

Dic₅

—

124

C₂⁴

C₂²×D₁₅

yes

characteristic

125

C₂×Q₈

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

#1220

yes

—

131

C₄⋊C₄

—

132

C₂²⋊C₄

—

133

C₂²⋊C₄

—

134

C₃×C₅

#2030

yes

characteristic, complemented, Hall

135

C₂²×C₃

—

136

C₂²×C₃

—

137

C₂²×C₃

—

138

C₂²×C₃

#4092

yes

—

139

C₂²×C₃

#4093

yes

—

140

C₂²×C₃

#4099

yes

characteristic

141

C₃×C₄

—

142

Dic₃

—

143

Dic₃

—

144

C₂×C₅

—

145

C₂×C₅

—

146

C₂×C₅

#5231

yes

characteristic

147

C₂×C₅

#5243

yes

—

148

C₂³

—

149

C₂³

Q₈×D₁₅

yes

characteristic

150

C₂³

#6821

yes

—

151

C₂³

C₂×#1220

yes

—

152

Q₈

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₃

—

161

C₂×C₃

—

162

C₂×C₃

#65520

yes

characteristic

163

C₂×C₃

#65532

yes

—

164

C₅

#71894

yes

characteristic, complemented, Sylow

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

#91835

yes

—

169

C₂²

#91836

yes

—

170

C₂²

#91842

yes

characteristic

171

C₄

—

172

C₄

—

173

C₄

—

120

—

174

C₃

#10602687

yes

characteristic, complemented, Sylow

175

C₂

—

176

C₂

—

177

C₂

#11751332

yes

characteristic

178

C₂

#11751344

yes

—

	subgroup	G/H
center	C₂²	#91842
commutator, prime radical, cosocle	C₂⁴×C₃×C₅	C₂³
Frattini	C₂⁴	C₂²×D₁₅
Fitting	C₃×C₅×#543	C₂
socle	C₂²×C₃×C₅	#120
hypercenter	#543	D₁₅
hypocenter	C₃×C₅	#2030
LCS	G ▷ C₂⁴×C₃×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #543

subgroup

G/H

center

C₂²

#91842

commutator, prime radical, cosocle

C₂⁴×C₃×C₅

C₂³

Frattini

C₂⁴

C₂²×D₁₅

Fitting

C₃×C₅×#543

C₂

C₂²×C₃×C₅

D₁₅

C₃×C₅

G ▷ C₂⁴×C₃×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂² ◁ C₂⁴ ◁ #543

N	H ≅ G/N	type
20. C₂²×Dic₁₅	Q₈	semidirect
134. C₃×C₅	#2030	semidirect
164. C₅	#71894	semidirect
174. C₃	#10602687	semidirect

H ≅ G/N

type

20. C₂²×Dic₁₅

Q₈

semidirect

134. C₃×C₅

#2030

semidirect

164. C₅

#71894

semidirect

174. C₃

#10602687

semidirect

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

subgroup	index
C₃×C₅×#543	2
#11751044	2
#11746347	2
#10602687	3
#71894	5

subgroup

index

C₃×C₅×#543

Subgroups — #14308998

Conjugacy classes of subgroups

Complements

Maximal subgroups