Group #28507283 — subgroups

Subgroups — #28507283

Order 15104.

334 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 135 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	#21425852	C₂	1	1	yes	maximal, characteristic
2	C₄×#14273663	C₂	1	1	yes	maximal, characteristic
3	#21422204	C₂	1	4	yes	maximal
4	C₂×C₅₉×#499	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273667	C₂²	1	2	yes	—
6	C₂×#14273663	C₂²	1	2	yes	—
7	C₄×#11745664	—	2	1	no	—
8	C₄⋊(C₅₉⋊C₁₆)	—	2	2	no	—
9	#16837167	—	2	1	no	—
10	C₂×#125×C₅₉	—	2	1	no	—
11	C₅₉×#499	—	2	2	no	—
12	C₅₉×#499	C₄	1	4	yes	—
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×#11745664	—	2	2	no	—
17	#14273667	—	2	4	no	—
18	C₂×C₅₉⋊C₁₆	—	2	2	no	—
19	#14273663	—	2	2	no	—
20	#14273663	Q₈	1	4	yes	—
21	#14273638	—	4	1	no	—
22	C₄×C₅₉⋊C₈	—	4	1	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	1	no	—
27	#125×C₅₉	—	4	1	no	—
28	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₉	—	4	2	no	—
30	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
31	#107×C₅₉	—	2	2	no	—
32	#107×C₅₉	C₂×C₄	1	4	yes	—
33	#106×C₅₉	—	2	4	no	—
34	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₅₉	D₄	1	2	yes	—
36	C₂²×C₄×C₅₉	—	4	2	no	—
37	C₂²×C₄×C₅₉	SD₁₆	1	2	yes	—
38	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₅₉	—	2	2	no	—
40	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
41	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
44	C₂×C₈×C₅₉	#41	1	2	yes	—
45	C₄⋊C₄×C₅₉	—	2	4	no	—
46	C₄⋊C₄×C₅₉	—	4	4	no	—
47	C₄⋊C₄×C₅₉	—	4	4	no	—
48	C₄⋊C₄×C₅₉	—	4	2	no	—
49	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
50	C₄²×C₅₉	D₈	1	1	yes	characteristic
51	C₄²×C₅₉	D₈	1	1	yes	characteristic
52	#11745664	—	2	4	no	—
53	#11745664	—	4	2	no	—
54	C₂×C₅₉⋊C₈	—	4	2	no	—
55	C₅₉⋊C₁₆	—	4	2	no	—
56	C₂³×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₅₉	—	4	2	no	—
58	C₂×C₄×C₅₉	—	4	4	no	—
59	C₂×C₄×C₅₉	—	8	4	no	—
60	C₂×C₄×C₅₉	#102	1	2	yes	—
61	C₂×C₄×C₅₉	#102	1	2	yes	—
62	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
64	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₅₉	#137	1	2	yes	—
66	C₈×C₅₉	—	2	2	no	—
67	C₈×C₅₉	C₄×Q₈	1	2	yes	—
68	C₈×C₅₉	#124	1	1	yes	characteristic
69	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
70	C₅₉⋊C₈	—	4	4	no	—
71	#13280	—	59	1	no	maximal, Sylow
72	C₂²×C₅₉	—	2	2	no	—
73	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
74	C₂²×C₅₉	#475	1	2	yes	—
75	C₄×C₅₉	—	2	2	no	—
76	C₄×C₅₉	—	2	2	no	—
77	C₄×C₅₉	—	8	4	no	—
78	C₄×C₅₉	#391	1	1	yes	characteristic
79	C₄×C₅₉	#419	1	1	yes	characteristic
80	C₄×C₅₉	#474	1	2	yes	—
81	C₂×#499	D₅₉	1	1	yes	characteristic
82	#2150	—	59	1	no	—
83	C₄×#108	—	59	1	no	—
84	#1387	—	59	4	no	—
85	C₂×C₅₉	—	2	2	no	—
86	C₂×C₅₉	#1988	1	1	yes	characteristic
87	C₂×C₅₉	#2148	1	2	yes	—
88	C₂×#125	—	2	1	no	—
89	C₂×M₃₂	—	59	2	no	—
90	#499	—	2	2	no	—
91	#499	Dic₅₉	1	4	yes	—
92	C₂×#108	—	59	2	no	—
93	C₂×#107	C₂×D₅₉	1	2	yes	—
94	C₂×#106	—	2	1	no	—
95	C₄×M₁₆	—	118	1	no	—
96	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
97	C₄⋊C₁₆	—	118	2	no	—
98	#341	—	118	1	no	—
99	C₅₉	#13280	1	1	yes	characteristic, complemented, Sylow
100	C₂×M₁₆	—	118	2	no	—
101	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
103	#125	—	2	4	no	—
104	#125	—	4	1	no	—
105	#125	—	4	1	no	—
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄⋊C₄	—	4	2	no	—
108	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
109	M₃₂	—	118	4	no	—
110	C₂×C₁₆	—	118	2	no	—
111	#108	—	59	4	no	—
112	#108	—	118	2	no	—
113	#107	—	2	2	no	—
114	#107	C₂×Dic₅₉	1	4	yes	—
115	#106	—	2	4	no	—
116	#97	—	236	1	no	—
117	C₄×C₈	—	236	1	no	—
118	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
119	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
120	C₂²×C₄	—	4	2	no	—
121	C₂²×C₄	#11745670	1	2	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
123	M₁₆	—	118	4	no	—
124	M₁₆	—	236	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	236	2	no	—
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	4	no	—
134	C₄⋊C₄	—	4	4	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄²	#11745669	1	1	yes	characteristic
137	C₄²	#11745669	1	1	yes	characteristic
138	C₄²	C₂²⋊Dic₅₉	1	2	yes	—
139	C₁₆	—	236	2	no	—
140	C₂³	C₂×#11745670	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	4	no	—
143	C₂×C₄	—	8	4	no	—
144	C₂×C₄	#14273648	1	2	yes	—
145	C₂×C₄	#14273648	1	2	yes	—
146	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
147	C₂×C₄	#14273703	1	2	yes	—
148	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
149	C₂×C₄	#14273758	1	2	yes	—
150	C₈	—	2	2	no	—
151	C₈	—	236	4	no	—
152	C₈	Q₈×Dic₅₉	1	2	yes	—
153	C₈	#14273716	1	1	yes	characteristic
154	C₈	#14273725	1	1	yes	characteristic
155	C₂²	—	2	2	no	—
156	C₂²	C₂×#14273648	1	1	yes	characteristic
157	C₂²	#16837684	1	2	yes	—
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	8	4	no	—
161	C₄	#16837309	1	1	yes	characteristic
162	C₄	#16837411	1	1	yes	characteristic
163	C₄	#16837679	1	2	yes	—
164	C₂	—	2	2	no	—
165	C₂	#21425049	1	1	yes	characteristic
166	C₂	#21425845	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21425852

C₂

yes

maximal, characteristic

C₄×#14273663

C₂

yes

maximal, characteristic

#21422204

C₂

yes

maximal

C₂×C₅₉×#499

C₂

yes

maximal, characteristic

C₂×#14273667

C₂²

yes

—

C₂×#14273663

C₂²

yes

—

C₄×#11745664

—

C₄⋊(C₅₉⋊C₁₆)

—

#16837167

—

C₂×#125×C₅₉

—

C₅₉×#499

—

C₅₉×#499

C₄

yes

—

C₂×#107×C₅₉

C₂²

yes

—

C₂×#106×C₅₉

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×#11745664

—

#14273667

—

C₂×C₅₉⋊C₁₆

—

#14273663

—

#14273663

Q₈

yes

—

#14273638

—

C₄×C₅₉⋊C₈

—

C₂²×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

#125×C₅₉

—

#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

SD₁₆

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

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

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂²⋊C₄

yes

—

C₄²×C₅₉

D₈

yes

characteristic

C₄²×C₅₉

D₈

yes

characteristic

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#102

yes

—

C₂×C₄×C₅₉

#102

yes

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

#137

yes

—

C₈×C₅₉

—

C₈×C₅₉

C₄×Q₈

yes

—

C₈×C₅₉

#124

yes

characteristic

C₈×C₅₉

C₄⋊Q₈

yes

characteristic

C₅₉⋊C₈

—

#13280

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

C₂×#102

yes

characteristic

C₂²×C₅₉

#475

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

C₂×#499

D₅₉

yes

—

—

—

C₂×C₅₉

—

C₂×C₅₉

#1988

yes

characteristic

C₂×C₅₉

#2148

yes

—

C₂×#125

—

C₂×M₃₂

—

#499

—

#499

Dic₅₉

yes

—

C₂×#108

—

C₂×#107

C₂×D₅₉

yes

—

C₂×#106

—

C₄×M₁₆

—

118

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

C₄⋊C₁₆

—

118

—

#341

—

118

—

C₅₉

#13280

yes

characteristic, complemented, Sylow

100

C₂×M₁₆

—

118

—

101

C₂²×C₈

C₂²⋊D₅₉

yes

characteristic

102

C₂²×C₈

C₂²×D₅₉

yes

characteristic

103

#125

—

104

#125

—

105

#125

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄²

C₂²⋊D₅₉

yes

characteristic

109

M₃₂

—

118

—

110

C₂×C₁₆

—

118

—

111

#108

—

112

#108

—

118

—

113

#107

—

114

#107

C₂×Dic₅₉

yes

—

115

#106

—

116

#97

—

236

—

117

C₄×C₈

—

236

—

118

C₄×C₈

C₂×Dic₅₉

yes

—

119

C₄×C₈

C₂²⋊D₅₉

yes

—

120

C₂²×C₄

—

121

C₂²×C₄

#11745670

yes

—

122

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

123

M₁₆

—

118

—

124

M₁₆

—

236

—

125

C₂×C₈

—

126

C₂×C₈

—

236

—

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

—

136

C₄²

#11745669

yes

characteristic

137

C₄²

#11745669

yes

characteristic

138

C₄²

C₂²⋊Dic₅₉

yes

—

139

C₁₆

—

236

—

140

C₂³

C₂×#11745670

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

#14273648

yes

—

145

C₂×C₄

#14273648

yes

—

146

C₂×C₄

C₂×C₂²⋊Dic₅₉

yes

characteristic

147

C₂×C₄

#14273703

yes

—

148

C₂×C₄

C₂×#11745669

yes

characteristic

149

C₂×C₄

#14273758

yes

—

150

C₈

—

151

C₈

—

236

—

152

C₈

Q₈×Dic₅₉

yes

—

153

C₈

#14273716

yes

characteristic

154

C₈

#14273725

yes

characteristic

155

C₂²

—

156

C₂²

C₂×#14273648

yes

characteristic

157

C₂²

#16837684

yes

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

#16837309

yes

characteristic

162

C₄

#16837411

yes

characteristic

163

C₄

#16837679

yes

—

164

C₂

—

165

C₂

#21425049

yes

characteristic

166

C₂

#21425845

yes

—

	subgroup	G/H
center	C₂²	C₂×#14273648
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#499	C₂
socle	C₂²×C₅₉	C₂×#102
prime radical	C₂³×C₅₉	C₂×SD₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#499	D₅₉
hypocenter	C₅₉	#13280
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

subgroup

G/H

center

C₂²

C₂×#14273648

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₂×C₅₉×#499

C₂

socle

C₂²×C₅₉

C₂×#102

prime radical

C₂³×C₅₉

C₂×SD₁₆

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

N	H ≅ G/N	type
99. C₅₉	#13280	semidirect

H ≅ G/N

type

99. C₅₉

#13280

semidirect

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

subgroup	index
#21425852	2
C₄×#14273663	2
#21422204	2
C₂×C₅₉×#499	2
#13280	59

subgroup

index

C₂×C₅₉×#499

Subgroups — #28507283

Conjugacy classes of subgroups

Complements

Maximal subgroups