Group #28508339 — subgroups

Subgroups — #28508339

Order 15104.

324 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 105 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×#16837224	C₂	1	2	yes	maximal
2	#21424844	C₂	1	1	yes	maximal, characteristic
3	#21422203	C₂	1	2	yes	maximal
4	#21422178	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₉×#499	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273667	C₂²	1	2	yes	—
7	C₂×#14273663	C₂²	1	1	yes	characteristic
8	#16837440	C₂²	1	1	yes	characteristic
9	#16837428	—	2	1	no	—
10	#16837224	—	2	4	no	—
11	#16837162	—	2	1	no	—
12	C₂×#125×C₅₉	—	2	1	no	—
13	C₅₉×#499	—	2	4	no	—
14	C₂×#107×C₅₉	—	2	1	no	—
15	C₂×#106×C₅₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×#11745664	—	2	1	no	—
18	#14273686	—	2	1	no	—
19	#14273667	—	2	2	no	—
20	#14273667	Q₈	1	4	yes	—
21	C₂×C₅₉⋊C₁₆	—	2	2	no	—
22	#14273663	—	2	2	no	—
23	#14273661	Q₈	1	2	yes	—
24	#14273659	—	2	1	no	—
25	Dic₅₉⋊C₈	—	4	2	no	—
26	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
28	#125×C₅₉	—	2	4	no	—
29	#125×C₅₉	—	4	2	no	—
30	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
31	C₂×C₄⋊C₄×C₅₉	—	4	2	no	—
32	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
33	#107×C₅₉	—	2	4	no	—
34	#106×C₅₉	—	2	2	no	—
35	#106×C₅₉	D₄	1	4	yes	—
36	C₄×C₈×C₅₉	—	2	2	no	—
37	C₂²×C₄×C₅₉	—	2	1	no	—
38	C₂²×C₄×C₅₉	—	4	2	no	—
39	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₅₉	—	2	1	no	—
41	C₂×C₈×C₅₉	—	2	1	no	—
42	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
43	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
44	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
45	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
47	C₄⋊C₄×C₅₉	—	2	4	no	—
48	C₄⋊C₄×C₅₉	—	4	8	no	—
49	C₄⋊C₄×C₅₉	—	4	2	no	—
50	C₄²×C₅₉	—	2	1	no	—
51	C₄²×C₅₉	SD₁₆	1	2	yes	—
52	C₂²⋊Dic₅₉	—	4	1	no	—
53	C₄⋊Dic₅₉	—	2	2	no	—
54	C₄×Dic₅₉	—	4	1	no	—
55	#11745664	—	4	1	no	—
56	#11745664	—	4	1	no	—
57	C₂×C₅₉⋊C₈	—	4	1	no	—
58	C₅₉⋊C₁₆	—	2	4	no	—
59	C₂³×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂×C₄×C₅₉	—	2	2	no	—
61	C₂×C₄×C₅₉	—	4	2	no	—
62	C₂×C₄×C₅₉	—	4	4	no	—
63	C₂×C₄×C₅₉	—	8	4	no	—
64	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
65	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
70	C₂×Dic₅₉	—	4	2	no	—
71	C₈×C₅₉	—	2	2	no	—
72	C₈×C₅₉	C₂²⋊Q₈	1	2	yes	—
73	C₈×C₅₉	C₄⋊Q₈	1	2	yes	—
74	C₅₉⋊C₈	—	8	1	no	—
75	#13458	—	59	1	no	maximal, Sylow
76	C₂²×C₅₉	—	2	2	no	—
77	C₂²×C₅₉	#450	1	1	yes	characteristic
78	C₂²×C₅₉	#476	1	1	yes	characteristic
79	C₂²×C₅₉	#501	1	1	yes	characteristic
80	C₄×C₅₉	—	4	2	no	—
81	C₄×C₅₉	—	8	4	no	—
82	C₄×C₅₉	#395	1	1	yes	characteristic
83	C₄×C₅₉	#448	1	1	yes	characteristic
84	C₄×C₅₉	#475	1	1	yes	characteristic
85	C₄×C₅₉	#498	1	1	yes	characteristic
86	Dic₅₉	—	8	2	no	—
87	C₂×#499	D₅₉	1	1	yes	characteristic
88	C₂×#365	—	59	2	no	—
89	#1945	—	59	1	no	—
90	#1386	—	59	2	no	—
91	#1376	—	59	1	no	—
92	C₂×C₅₉	—	2	1	no	—
93	C₂×C₅₉	—	2	1	no	—
94	C₂×C₅₉	#2058	1	1	yes	characteristic
95	C₂×C₅₉	#2231	1	2	yes	—
96	C₂×#125	—	2	1	no	—
97	C₂×M₃₂	—	59	2	no	—
98	#499	—	2	4	no	—
99	C₂×#108	—	59	1	no	—
100	#427	—	59	1	no	—
101	C₂×#107	—	2	1	no	—
102	C₂×#106	C₂×D₅₉	1	2	yes	—
103	#424	—	118	1	no	—
104	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
105	#365	—	118	4	no	—
106	#339	—	118	1	no	—
107	C₅₉	#13458	1	1	yes	characteristic, complemented, Sylow
108	C₂×M₁₆	—	118	1	no	—
109	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
110	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
111	#125	—	2	4	no	—
112	#125	—	4	2	no	—
113	#117	—	118	1	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	4	2	no	—
116	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
117	M₃₂	—	59	4	no	—
118	M₃₂	—	118	2	no	—
119	C₂×C₁₆	—	118	2	no	—
120	#108	—	118	2	no	—
121	#107	—	2	4	no	—
122	#107	—	59	2	no	—
123	#106	—	2	2	no	—
124	#106	—	118	1	no	—
125	#106	C₂²⋊D₅₉	1	4	yes	—
126	C₄⋊C₈	—	236	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
131	M₁₆	—	236	1	no	—
132	M₁₆	—	236	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	236	1	no	—
136	C₂×C₈	#11745671	1	2	yes	—
137	C₂×C₈	Q₈×D₅₉	1	2	yes	—
138	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
139	C₂×C₈	#11745689	1	1	yes	characteristic
140	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	4	8	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	118	2	no	—
145	C₂²⋊C₄	—	236	1	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	236	1	no	—
148	C₄²	#11745670	1	2	yes	—
149	C₁₆	—	118	4	no	—
150	C₂³	#14273753	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	4	no	—
154	C₂×C₄	—	8	4	no	—
155	C₂×C₄	—	236	2	no	—
156	C₂×C₄	#14273696	1	1	yes	characteristic
157	C₂×C₄	#14273703	1	1	yes	characteristic
158	C₂×C₄	#14273725	1	1	yes	characteristic
159	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
160	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
161	C₂×C₄	#14273758	1	1	yes	characteristic
162	C₈	—	2	2	no	—
163	C₈	—	472	1	no	—
164	C₈	#14273703	1	2	yes	—
165	C₈	#14273725	1	2	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	#16837553	1	1	yes	characteristic
168	C₂²	#16837689	1	1	yes	characteristic
169	C₂²	#16837808	1	1	yes	characteristic
170	C₄	—	4	2	no	—
171	C₄	—	8	4	no	—
172	C₄	—	472	2	no	—
173	C₄	#16837321	1	1	yes	characteristic
174	C₄	#16837541	1	1	yes	characteristic
175	C₄	#16837684	1	1	yes	characteristic
176	C₄	#16837798	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	#21425426	1	1	yes	characteristic
180	C₂	#21426269	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837224

C₂

yes

maximal

#21424844

C₂

yes

maximal, characteristic

#21422203

C₂

yes

maximal

#21422178

C₂

yes

maximal, characteristic

C₂×C₅₉×#499

C₂

yes

maximal, characteristic

C₂×#14273667

C₂²

yes

—

C₂×#14273663

C₂²

yes

characteristic

#16837440

C₂²

yes

—

—

—

C₂×#125×C₅₉

—

C₅₉×#499

—

C₂×#107×C₅₉

—

C₂×#106×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

—

—

—

Q₈

yes

—

C₂×C₅₉⋊C₁₆

—

#14273663

—

#14273661

Q₈

yes

—

#14273659

—

Dic₅₉⋊C₈

—

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

—

#106×C₅₉

—

#106×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₂×C₈×C₅₉

C₂×Q₈

yes

—

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₉

#41

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

SD₁₆

yes

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#120

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊Q₈

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

#137

yes

characteristic

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₂²⋊Q₈

yes

—

C₈×C₅₉

C₄⋊Q₈

yes

—

C₅₉⋊C₈

—

#13458

—

maximal, Sylow

C₂²×C₅₉

—

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

—

—

—

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#2058

yes

characteristic

C₂×C₅₉

#2231

yes

—

C₂×#125

—

C₂×M₃₂

—

#499

—

C₂×#108

—

100

#427

—

101

C₂×#107

—

102

C₂×#106

C₂×D₅₉

yes

—

103

#424

—

118

—

104

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

105

#365

—

118

—

106

#339

—

118

—

107

C₅₉

#13458

yes

characteristic, complemented, Sylow

108

C₂×M₁₆

—

118

—

109

C₂²×C₈

C₂²⋊D₅₉

yes

characteristic

110

C₂²×C₈

C₂²×D₅₉

yes

characteristic

111

#125

—

112

#125

—

113

#117

—

118

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄²

C₂²⋊D₅₉

yes

characteristic

117

M₃₂

—

118

M₃₂

—

118

—

119

C₂×C₁₆

—

118

—

120

#108

—

118

—

121

#107

—

122

#107

—

123

#106

—

124

#106

—

118

—

125

#106

C₂²⋊D₅₉

yes

—

126

C₄⋊C₈

—

236

—

127

C₄×C₈

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

131

M₁₆

—

236

—

132

M₁₆

—

236

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

236

—

136

C₂×C₈

#11745671

yes

—

137

C₂×C₈

Q₈×D₅₉

yes

—

138

C₂×C₈

Q₈×D₅₉

yes

characteristic

139

C₂×C₈

#11745689

yes

characteristic

140

C₂×C₈

C₂×C₂²⋊D₅₉

yes

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

118

—

145

C₂²⋊C₄

—

236

—

146

C₄²

—

147

C₄²

—

236

—

148

C₄²

#11745670

yes

—

149

C₁₆

—

118

—

150

C₂³

#14273753

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

236

—

156

C₂×C₄

#14273696

yes

characteristic

157

C₂×C₄

#14273703

yes

characteristic

158

C₂×C₄

#14273725

yes

characteristic

159

C₂×C₄

C₂×#11745671

yes

characteristic

160

C₂×C₄

C₂×#11745670

yes

characteristic

161

C₂×C₄

#14273758

yes

characteristic

162

C₈

—

163

C₈

—

472

—

164

C₈

#14273703

yes

—

165

C₈

#14273725

yes

—

166

C₂²

—

167

C₂²

#16837553

yes

characteristic

168

C₂²

#16837689

yes

characteristic

169

C₂²

#16837808

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

472

—

173

C₄

#16837321

yes

characteristic

174

C₄

#16837541

yes

characteristic

175

C₄

#16837684

yes

characteristic

176

C₄

#16837798

yes

characteristic

177

C₂

—

178

C₂

—

179

C₂

#21425426

yes

characteristic

180

C₂

#21426269

yes

—

	subgroup	G/H
center	C₂²	#16837689
commutator, cosocle	C₂²×C₈×C₅₉	C₂³
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#499	C₂
socle	C₂²×C₅₉	#476
prime radical	C₂³×C₅₉	AGL(1,ℤ/8)
hypercenter	C₂×#499	D₅₉
hypocenter	C₅₉	#13458
LCS	G ▷ C₂²×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#499

subgroup

G/H

center

C₂²

#16837689

commutator, cosocle

C₂²×C₈×C₅₉

C₂³

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₂×C₅₉×#499

C₂

C₂²×C₅₉

C₂³×C₅₉

D₅₉

C₅₉

G ▷ C₂²×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

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

N	H ≅ G/N	type
107. C₅₉	#13458	semidirect

H ≅ G/N

type

107. C₅₉

#13458

semidirect

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

subgroup	index
C₂×#16837224	2
#21424844	2
#21422203	2
#21422178	2
C₂×C₅₉×#499	2
#13458	59

subgroup

index

C₂×C₅₉×#499

Subgroups — #28508339

Conjugacy classes of subgroups

Complements

Maximal subgroups