Group #28481683 — subgroups

Subgroups — #28481683

Order 15104.

320 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 101 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	#21433031	C₂	1	1	yes	maximal, characteristic
2	#21422857	C₂	1	2	yes	maximal
3	#21422734	C₂	1	1	yes	maximal, characteristic
4	#21422006	C₂	1	2	yes	maximal
5	C₅₉×#1537	C₂	1	1	yes	maximal, characteristic
6	#16837533	—	2	2	no	—
7	Q₈×C₅₉⋊C₈	—	2	2	no	—
8	#16837459	C₂²	1	2	yes	—
9	#16837429	C₄	1	2	yes	—
10	#16837426	C₂²	1	1	yes	characteristic
11	#16837358	C₄	1	2	yes	—
12	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
13	#16837154	—	2	2	no	—
14	#16837152	—	2	2	no	—
15	#16837144	—	2	2	no	—
16	C₅₉×#533	C₂²	1	1	yes	characteristic
17	C₅₉×#432	C₂²	1	2	yes	—
18	C₅₉×#330	—	2	2	no	—
19	C₂×#11745664	C₂×C₄	1	2	yes	—
20	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
24	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
25	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
26	#14273638	—	2	2	no	—
27	#14273638	—	2	2	no	—
28	C₄×C₅₉⋊C₈	—	2	2	no	—
29	C₄×C₅₉⋊C₈	D₄	1	2	yes	—
30	C₄×C₅₉⋊C₈	D₄	1	2	yes	—
31	C₄⋊Q₈×C₅₉	Q₈	1	2	yes	—
32	#125×C₅₉	D₄	1	2	yes	—
33	C₂²⋊Q₈×C₅₉	—	2	2	no	—
34	C₄×Q₈×C₅₉	—	2	2	no	—
35	#117×C₅₉	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₅₉	—	2	2	no	—
38	C₂²⋊C₈×C₅₉	—	2	2	no	—
39	#97×C₅₉	—	2	2	no	—
40	C₂×Q₈×C₅₉	—	2	2	no	—
41	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₉	#41	1	2	yes	—
43	C₂×C₈×C₅₉	—	4	2	no	—
44	C₄⋊C₄×C₅₉	—	2	4	no	—
45	C₄⋊C₄×C₅₉	—	2	2	no	—
46	C₄⋊C₄×C₅₉	C₄⋊C₄	1	4	yes	—
47	C₂²⋊C₄×C₅₉	—	2	2	no	—
48	C₄²×C₅₉	—	2	2	no	—
49	C₄²×C₅₉	C₂×D₄	1	2	yes	—
50	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
52	#11745664	—	4	2	no	—
53	C₂×C₅₉⋊C₈	—	2	2	no	—
54	C₂×C₅₉⋊C₈	—	2	2	no	—
55	C₂×C₅₉⋊C₈	—	4	1	no	—
56	C₂×C₅₉⋊C₈	—	4	2	no	—
57	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	2	yes	—
59	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₂³×C₅₉	#117	1	1	yes	characteristic
61	Q₈×C₅₉	—	4	4	no	—
62	C₂×C₄×C₅₉	—	2	1	no	—
63	C₂×C₄×C₅₉	—	2	1	no	—
64	C₂×C₄×C₅₉	—	2	4	no	—
65	C₂×C₄×C₅₉	—	4	2	no	—
66	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₉	#121	1	2	yes	—
69	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
70	C₈×C₅₉	—	8	2	no	—
71	C₅₉⋊C₈	—	2	1	no	—
72	C₅₉⋊C₈	—	2	2	no	—
73	C₅₉⋊C₈	—	4	2	no	—
74	C₅₉⋊C₈	—	8	2	no	—
75	C₅₉⋊C₈	#101	1	2	yes	—
76	#8631	—	59	1	no	maximal, Sylow
77	C₂²×C₅₉	—	2	1	no	—
78	C₂²×C₅₉	—	2	2	no	—
79	C₂²×C₅₉	#380	1	1	yes	characteristic
80	C₄×C₅₉	—	2	1	no	—
81	C₄×C₅₉	—	2	2	no	—
82	C₄×C₅₉	—	2	2	no	—
83	C₄×C₅₉	—	4	4	no	—
84	C₄×C₅₉	C₂×#101	1	1	yes	characteristic
85	C₄×C₅₉	#413	1	1	yes	characteristic
86	#2960	—	59	1	no	—
87	#1568	—	59	2	no	—
88	#1537	—	59	1	no	—
89	#1537	D₅₉	1	1	yes	characteristic
90	#1307	—	59	2	no	—
91	C₂×C₅₉	—	4	1	no	—
92	C₂×C₅₉	#1801	1	1	yes	characteristic
93	C₂×C₅₉	#1814	1	2	yes	—
94	#533	C₂×D₅₉	1	1	yes	characteristic
95	#446	—	118	2	no	—
96	C₈×Q₈	—	118	2	no	—
97	#432	—	59	2	no	—
98	#432	C₂×D₅₉	1	2	yes	—
99	#424	—	59	2	no	—
100	#423	—	59	1	no	—
101	#405	—	59	2	no	—
102	C₂×C₄×C₈	—	59	1	no	—
103	#335	—	118	2	no	—
104	#334	—	118	2	no	—
105	#330	—	2	2	no	—
106	#330	—	118	2	no	—
107	C₅₉	#8631	1	1	yes	characteristic, complemented, Sylow
108	C₂×M₁₆	—	59	2	no	—
109	C₂²×C₈	—	59	1	no	—
110	C₂²×C₈	—	59	1	no	—
111	C₄⋊Q₈	Dic₁₁₈	1	2	yes	—
112	#125	C₂²⋊D₅₉	1	2	yes	—
113	C₂²⋊Q₈	—	2	2	no	—
114	C₄×Q₈	—	2	2	no	—
115	#117	C₄×D₅₉	1	2	yes	—
116	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	118	4	no	—
119	C₄⋊C₈	—	118	2	no	—
120	C₄⋊C₈	—	118	2	no	—
121	C₂²⋊C₈	—	2	2	no	—
122	C₂²⋊C₈	—	118	2	no	—
123	#97	—	2	2	no	—
124	#97	—	118	2	no	—
125	#97	—	118	2	no	—
126	C₄×C₈	—	59	2	no	—
127	C₄×C₈	—	59	2	no	—
128	C₄×C₈	—	118	2	no	—
129	C₂×Q₈	—	2	2	no	—
130	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
131	C₂²×C₄	#11745687	1	2	yes	—
132	M₁₆	—	236	2	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	—	59	1	no	—
135	C₂×C₈	—	59	1	no	—
136	C₂×C₈	—	59	2	no	—
137	C₂×C₈	—	118	2	no	—
138	C₂×C₈	—	118	2	no	—
139	C₂×C₈	—	236	1	no	—
140	C₂×C₈	—	236	2	no	—
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	Dic₅₉⋊C₄	1	4	yes	—
144	C₂²⋊C₄	—	2	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
147	C₄²	D₄×D₅₉	1	2	yes	—
148	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
149	C₂³	#14273683	1	1	yes	characteristic
150	Q₈	—	4	4	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
156	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
157	C₂×C₄	#14273699	1	2	yes	—
158	C₂×C₄	#14273704	1	2	yes	—
159	C₈	—	8	2	no	—
160	C₈	—	59	2	no	—
161	C₈	—	118	1	no	—
162	C₈	—	118	2	no	—
163	C₈	—	472	2	no	—
164	C₈	—	236	2	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	2	1	no	—
167	C₂²	#16837260	1	1	yes	characteristic
168	C₄	—	2	2	no	—
169	C₄	—	2	2	no	—
170	C₄	—	2	1	no	—
171	C₄	—	4	4	no	—
172	C₄	#101×D₅₉	1	1	yes	characteristic
173	C₄	#16837385	1	1	yes	characteristic
174	C₂	—	4	1	no	—
175	C₂	#21424090	1	1	yes	characteristic
176	C₂	#21424148	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21433031

C₂

yes

maximal, characteristic

#21422857

C₂

yes

maximal

#21422734

C₂

yes

maximal, characteristic

#21422006

C₂

yes

maximal

C₅₉×#1537

C₂

yes

maximal, characteristic

#16837533

—

Q₈×C₅₉⋊C₈

—

#16837459

C₂²

yes

—

#16837429

C₄

yes

—

#16837426

C₂²

yes

characteristic

#16837358

C₄

yes

—

C₂×C₄×C₅₉⋊C₈

C₂²

yes

—

—

—

C₅₉×#533

C₂²

yes

characteristic

C₅₉×#432

C₂²

yes

—

C₅₉×#330

—

C₂×#11745664

C₂×C₄

yes

—

C₂²×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

—

#14273638

—

#14273638

—

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

D₄

yes

—

C₄×C₅₉⋊C₈

D₄

yes

—

C₄⋊Q₈×C₅₉

Q₈

yes

—

#125×C₅₉

D₄

yes

—

C₂²⋊Q₈×C₅₉

—

C₄×Q₈×C₅₉

—

#117×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

C₂²⋊C₈×C₅₉

—

#97×C₅₉

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₄⋊C₄

yes

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×Q₈

yes

characteristic

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₅₉⋊C₈

C₂²⋊C₄

yes

—

C₂×C₅₉⋊C₈

C₂²⋊C₄

yes

characteristic

C₂³×C₅₉

#117

yes

characteristic

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#121

yes

—

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

#101

yes

—

#8631

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#380

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

—

—

—

D₅₉

yes

characteristic

#1307

—

C₂×C₅₉

—

C₂×C₅₉

#1801

yes

characteristic

C₂×C₅₉

#1814

yes

—

#533

C₂×D₅₉

yes

characteristic

#446

—

118

—

C₈×Q₈

—

118

—

#432

—

#432

C₂×D₅₉

yes

—

#424

—

100

#423

—

101

#405

—

102

C₂×C₄×C₈

—

103

#335

—

118

—

104

#334

—

118

—

105

#330

—

106

#330

—

118

—

107

C₅₉

#8631

yes

characteristic, complemented, Sylow

108

C₂×M₁₆

—

109

C₂²×C₈

—

110

C₂²×C₈

—

111

C₄⋊Q₈

Dic₁₁₈

yes

—

112

#125

C₂²⋊D₅₉

yes

—

113

C₂²⋊Q₈

—

114

C₄×Q₈

—

115

#117

C₄×D₅₉

yes

—

116

C₂×C₄²

C₂²×D₅₉

yes

characteristic

117

C₄⋊C₈

—

118

C₄⋊C₈

—

118

—

119

C₄⋊C₈

—

118

—

120

C₄⋊C₈

—

118

—

121

C₂²⋊C₈

—

122

C₂²⋊C₈

—

118

—

123

#97

—

124

#97

—

118

—

125

#97

—

118

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₄×C₈

—

118

—

129

C₂×Q₈

—

130

C₂²×C₄

C₂×C₄×D₅₉

yes

characteristic

131

C₂²×C₄

#11745687

yes

—

132

M₁₆

—

236

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

118

—

138

C₂×C₈

—

118

—

139

C₂×C₈

—

236

—

140

C₂×C₈

—

236

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

Dic₅₉⋊C₄

yes

—

144

C₂²⋊C₄

—

145

C₄²

—

146

C₄²

C₂×Dic₁₁₈

yes

characteristic

147

C₄²

D₄×D₅₉

yes

—

148

C₄²

C₂×C₂²⋊D₅₉

yes

characteristic

149

C₂³

#14273683

yes

characteristic

150

Q₈

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

C₂²⋊C₄×D₅₉

yes

characteristic

156

C₂×C₄

C₂×Dic₅₉⋊C₄

yes

characteristic

157

C₂×C₄

#14273699

yes

—

158

C₂×C₄

#14273704

yes

—

159

C₈

—

160

C₈

—

161

C₈

—

118

—

162

C₈

—

118

—

163

C₈

—

472

—

164

C₈

—

236

—

165

C₂²

—

166

C₂²

—

167

C₂²

#16837260

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

#101×D₅₉

yes

characteristic

173

C₄

#16837385

yes

characteristic

174

C₂

—

175

C₂

#21424090

yes

characteristic

176

C₂

#21424148

yes

—

	subgroup	G/H
center	C₂²	#16837260
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1537	C₂
socle	C₂²×C₅₉	#380
prime radical	C₂³×C₅₉	#117
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1537	D₅₉
hypocenter	C₅₉	#8631
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1537

subgroup

G/H

center

C₂²

#16837260

commutator

C₂²×C₄×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1537

C₂

C₂²×C₅₉

C₂³×C₅₉

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1537

N	H ≅ G/N	type
107. C₅₉	#8631	semidirect

H ≅ G/N

type

107. C₅₉

#8631

semidirect

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

subgroup	index
#21433031	2
#21422857	2
#21422734	2
#21422006	2
C₅₉×#1537	2
#8631	59

subgroup

index

C₅₉×#1537

Subgroups — #28481683

Conjugacy classes of subgroups

Complements

Maximal subgroups