Group #28507287 — subgroups

Subgroups — #28507287

Order 15104.

366 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 167 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×C₄×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#21424260	C₂	1	1	yes	maximal, characteristic
3	#21422187	C₂	1	4	yes	maximal
4	C₂×C₅₉×#499	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14273661	C₂²	1	2	yes	—
7	C₂×#14273659	—	2	1	no	—
8	#16837268	—	2	1	no	—
9	C₄×C₅₉⋊C₁₆	—	2	2	no	—
10	C₂×#125×C₅₉	—	2	1	no	—
11	C₅₉×#499	—	2	2	no	—
12	C₅₉×#499	C₄	1	4	yes	complemented
13	C₂×#107×C₅₉	C₂²	1	2	yes	—
14	C₂×#106×C₅₉	—	2	1	no	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₅₉	—	2	2	no	—
17	C₂×C₅₉⋊C₁₆	—	2	2	no	—
18	C₂×C₅₉⋊C₁₆	—	2	2	no	—
19	C₂×C₅₉⋊C₁₆	Q₈	1	4	yes	—
20	#14273661	—	2	4	no	—
21	#14273659	—	4	2	no	—
22	C₅₉⋊#95	—	4	2	no	—
23	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
25	#125×C₅₉	—	2	4	no	—
26	#125×C₅₉	—	4	2	no	—
27	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
28	C₂×C₄⋊C₄×C₅₉	—	4	2	no	—
29	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
30	#107×C₅₉	—	2	2	no	—
31	#107×C₅₉	C₂×C₄	1	4	yes	—
32	#106×C₅₉	—	2	4	no	—
33	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₅₉	D₄	1	2	yes	—
35	C₂²×C₄×C₅₉	—	4	2	no	—
36	C₂²×C₄×C₅₉	D₈	1	2	yes	—
37	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₅₉	—	4	2	no	—
39	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
40	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
41	C₂×C₈×C₅₉	Q₁₆	1	2	yes	—
42	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
45	C₂×C₈×C₅₉	#41	1	2	yes	—
46	C₄⋊C₄×C₅₉	—	2	4	no	—
47	C₄⋊C₄×C₅₉	—	4	8	no	—
48	C₄⋊C₄×C₅₉	—	4	2	no	—
49	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
50	C₄²×C₅₉	SD₁₆	1	2	yes	—
51	C₄⋊Dic₅₉	—	4	2	no	—
52	C₄⋊Dic₅₉	—	4	2	no	—
53	C₅₉⋊C₁₆	—	2	4	no	—
54	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₅₉	—	2	4	no	—
56	C₂×C₄×C₅₉	—	4	4	no	—
57	C₂×C₄×C₅₉	—	8	4	no	—
58	C₂×C₄×C₅₉	#102	1	4	yes	—
59	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
61	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×Dic₅₉	—	8	2	no	—
66	C₂×Dic₅₉	—	8	4	no	—
67	C₈×C₅₉	#103	1	4	yes	—
68	C₈×C₅₉	C₄×Q₈	1	2	yes	—
69	C₈×C₅₉	C₂²⋊Q₈	1	2	yes	—
70	#13281	—	59	1	no	maximal, Sylow
71	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
72	C₂²×C₅₉	#486	1	1	yes	characteristic
73	C₂²×C₅₉	#500	1	1	yes	characteristic
74	C₂²×C₅₉	#508	1	4	yes	—
75	C₄×C₅₉	—	2	4	no	—
76	C₄×C₅₉	—	8	4	no	—
77	C₄×C₅₉	#391	1	1	yes	characteristic
78	C₄×C₅₉	C₂×#103	1	1	yes	characteristic
79	C₄×C₅₉	#487	1	1	yes	characteristic
80	C₄×C₅₉	#498	1	1	yes	characteristic
81	Dic₅₉	—	16	2	no	—
82	C₂×#499	D₅₉	1	1	yes	characteristic
83	C₂×C₄×C₁₆	—	59	1	no	—
84	#1846	—	59	1	no	—
85	#1380	—	59	4	no	—
86	C₂×C₅₉	#1990	1	1	yes	characteristic
87	C₂×C₅₉	#2138	1	2	yes	—
88	C₂×C₅₉	#2242	1	2	yes	—
89	C₂×C₅₉	#2253	1	2	yes	—
90	C₂×#125	—	2	1	no	—
91	C₂²×C₁₆	—	59	2	no	—
92	#499	—	2	2	no	—
93	#499	Dic₅₉	1	4	yes	complemented
94	C₂×#107	—	59	2	no	—
95	C₂×#107	C₂×D₅₉	1	2	yes	—
96	C₂×#106	—	2	1	no	—
97	C₂×#106	—	118	1	no	—
98	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
99	#382	—	118	1	no	—
100	C₄×C₁₆	—	118	2	no	—
101	C₅₉	#13281	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
104	#125	—	2	4	no	—
105	#125	—	4	2	no	—
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄⋊C₄	—	4	2	no	—
108	C₂×C₄⋊C₄	—	118	2	no	—
109	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
110	C₂×C₁₆	—	59	4	no	—
111	C₂×C₁₆	—	118	2	no	—
112	C₂×C₁₆	—	118	2	no	—
113	#107	—	2	2	no	—
114	#107	—	118	4	no	—
115	#107	C₂×Dic₅₉	1	4	yes	—
116	#106	—	2	4	no	—
117	#106	—	236	2	no	—
118	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
119	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
120	#95	—	236	2	no	—
121	C₂²×C₄	—	4	2	no	—
122	C₂²×C₄	—	236	2	no	—
123	C₂²×C₄	#11745669	1	2	yes	—
124	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
125	C₂×C₈	#11745671	1	2	yes	—
126	C₂×C₈	#11745672	1	2	yes	—
127	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
128	C₂×C₈	Q₈×D₅₉	1	2	yes	—
129	C₂×C₈	#11745689	1	2	yes	—
130	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
131	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
132	C₄⋊C₄	—	2	4	no	—
133	C₄⋊C₄	—	4	8	no	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	236	2	no	—
136	C₄⋊C₄	—	236	2	no	—
137	C₄²	#11745670	1	2	yes	—
138	C₄²	C₂²⋊Dic₅₉	1	2	yes	—
139	C₁₆	—	118	4	no	—
140	C₂³	C₂×#11745669	1	1	yes	characteristic
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	4	4	no	—
143	C₂×C₄	—	8	4	no	—
144	C₂×C₄	—	472	2	no	—
145	C₂×C₄	—	472	4	no	—
146	C₂×C₄	#14273648	1	4	yes	—
147	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
148	C₂×C₄	#14273716	1	1	yes	characteristic
149	C₂×C₄	#14273725	1	1	yes	characteristic
150	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
151	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
152	C₂×C₄	C₂×#11745672	1	1	yes	characteristic
153	C₈	#14273651	1	4	yes	—
154	C₈	Q₈×Dic₅₉	1	2	yes	—
155	C₈	#14273703	1	2	yes	—
156	C₂²	C₂×#14273648	1	1	yes	characteristic
157	C₂²	#16837750	1	1	yes	characteristic
158	C₂²	#16837805	1	1	yes	characteristic
159	C₂²	#16837832	1	4	yes	—
160	C₄	—	2	4	no	—
161	C₄	—	8	4	no	—
162	C₄	—	944	2	no	—
163	C₄	#16837309	1	1	yes	characteristic
164	C₄	C₂×#14273651	1	1	yes	characteristic
165	C₄	#16837754	1	1	yes	characteristic
166	C₄	#16837798	1	1	yes	characteristic
167	C₂	#21425058	1	1	yes	characteristic
168	C₂	#21425803	1	2	yes	—
169	C₂	#21426330	1	2	yes	—
170	C₂	#21426372	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₅₉⋊C₁₆

C₂

yes

maximal, characteristic

#21424260

C₂

yes

maximal, characteristic

#21422187

C₂

yes

maximal

C₂×C₅₉×#499

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₁₆

C₂²

yes

—

C₂×#14273661

C₂²

yes

—

C₂×#14273659

—

#16837268

—

C₄×C₅₉⋊C₁₆

—

C₂×#125×C₅₉

—

C₅₉×#499

—

C₅₉×#499

C₄

yes

complemented

C₂×#107×C₅₉

C₂²

yes

—

C₂×#106×C₅₉

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

Q₈

yes

—

#14273661

—

#14273659

—

C₅₉⋊#95

—

C₂²×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

#125×C₅₉

—

#125×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄²×C₅₉

D₄

yes

characteristic

#107×C₅₉

—

#107×C₅₉

C₂×C₄

yes

—

#106×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₈

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×Dic₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₂×C₈×C₅₉

Q₁₆

yes

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

—

C₂×C₈×C₅₉

#41

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂²⋊C₄

yes

—

C₄²×C₅₉

SD₁₆

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#102

yes

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#124

yes

characteristic

C₂×C₄×C₅₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₁₆

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

#103

yes

—

C₈×C₅₉

C₄×Q₈

yes

—

C₈×C₅₉

C₂²⋊Q₈

yes

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

C₂×#499

D₅₉

yes

characteristic

C₂×C₄×C₁₆

—

#1846

—

#1380

—

C₂×C₅₉

#1990

yes

characteristic

C₂×C₅₉

#2138

yes

—

C₂×C₅₉

#2242

yes

—

C₂×C₅₉

#2253

yes

—

C₂×#125

—

C₂²×C₁₆

—

#499

—

#499

Dic₅₉

yes

complemented

C₂×#107

—

C₂×#107

C₂×D₅₉

yes

—

C₂×#106

—

C₂×#106

—

118

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

#382

—

118

—

100

C₄×C₁₆

—

118

—

101

C₅₉

#13281

yes

characteristic, complemented, Sylow

102

C₂²×C₈

C₂²⋊D₅₉

yes

characteristic

103

C₂²×C₈

C₂²×D₅₉

yes

characteristic

104

#125

—

105

#125

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

118

—

109

C₂×C₄²

C₂²⋊D₅₉

yes

characteristic

110

C₂×C₁₆

—

111

C₂×C₁₆

—

118

—

112

C₂×C₁₆

—

118

—

113

#107

—

114

#107

—

118

—

115

#107

C₂×Dic₅₉

yes

—

116

#106

—

117

#106

—

236

—

118

C₄×C₈

C₂×Dic₅₉

yes

—

119

C₄×C₈

C₂²⋊D₅₉

yes

—

120

#95

—

236

—

121

C₂²×C₄

—

122

C₂²×C₄

—

236

—

123

C₂²×C₄

#11745669

yes

—

124

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

125

C₂×C₈

#11745671

yes

—

126

C₂×C₈

#11745672

yes

—

127

C₂×C₈

C₂²⋊Dic₅₉

yes

—

128

C₂×C₈

Q₈×D₅₉

yes

—

129

C₂×C₈

#11745689

yes

—

130

C₂×C₈

C₂²×Dic₅₉

yes

characteristic

131

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

236

—

136

C₄⋊C₄

—

236

—

137

C₄²

#11745670

yes

—

138

C₄²

C₂²⋊Dic₅₉

yes

—

139

C₁₆

—

118

—

140

C₂³

C₂×#11745669

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

472

—

145

C₂×C₄

—

472

—

146

C₂×C₄

#14273648

yes

—

147

C₂×C₄

C₂×C₂²⋊Dic₅₉

yes

characteristic

148

C₂×C₄

#14273716

yes

characteristic

149

C₂×C₄

#14273725

yes

characteristic

150

C₂×C₄

C₂×#11745671

yes

characteristic

151

C₂×C₄

C₂×#11745670

yes

characteristic

152

C₂×C₄

C₂×#11745672

yes

characteristic

153

C₈

#14273651

yes

—

154

C₈

Q₈×Dic₅₉

yes

—

155

C₈

#14273703

yes

—

156

C₂²

C₂×#14273648

yes

characteristic

157

C₂²

#16837750

yes

characteristic

158

C₂²

#16837805

yes

characteristic

159

C₂²

#16837832

yes

—

160

C₄

—

161

C₄

—

162

C₄

—

944

—

163

C₄

#16837309

yes

characteristic

164

C₄

C₂×#14273651

yes

characteristic

165

C₄

#16837754

yes

characteristic

166

C₄

#16837798

yes

characteristic

167

C₂

#21425058

yes

characteristic

168

C₂

#21425803

yes

—

169

C₂

#21426330

yes

—

170

C₂

#21426372

yes

—

	subgroup	G/H
center	C₂³	C₂×#11745669
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#499	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#499	D₅₉
hypocenter	C₅₉	#13281
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

subgroup

G/H

center

C₂³

C₂×#11745669

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₂×C₅₉×#499

C₂

socle, prime radical

C₂³×C₅₉

C₂×D₈

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

N	H ≅ G/N	type
12. C₅₉×#499	C₄	semidirect
93. #499	Dic₅₉	semidirect
101. C₅₉	#13281	semidirect

H ≅ G/N

type

12. C₅₉×#499

C₄

semidirect

93. #499

Dic₅₉

semidirect

101. C₅₉

#13281

semidirect

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

subgroup	index
C₂×C₄×C₅₉⋊C₁₆	2
#21424260	2
#21422187	2
C₂×C₅₉×#499	2
#13281	59

subgroup

index

C₂×C₄×C₅₉⋊C₁₆

#21424260

#21422187

C₂×C₅₉×#499

#13281

Subgroups — #28507287

Conjugacy classes of subgroups

Complements

Maximal subgroups