Group #28493681 — subgroups

Subgroups — #28493681

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	#21425036	C₂	1	1	yes	maximal, characteristic
2	#21424271	C₂	1	1	yes	maximal, characteristic
3	#21422867	C₂	1	2	yes	maximal
4	#21422526	C₂	1	2	yes	maximal
5	C₅₉×#1275	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
7	C₂×#14273663	—	2	2	no	—
8	#16837426	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
10	#16837156	—	2	2	no	—
11	#16837153	—	2	4	no	—
12	#16837134	—	2	4	no	—
13	C₅₉×#431	C₂²	1	2	yes	—
14	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₁₈	Q₈	1	1	yes	characteristic
16	C₂×#11745664	—	2	2	no	—
17	C₂²×C₅₉⋊C₈	—	2	1	no	—
18	C₄×Dic₁₁₈	—	2	2	no	—
19	C₄×Dic₁₁₈	—	2	2	no	—
20	C₂×C₄⋊Dic₅₉	Q₈	1	1	yes	characteristic
21	C₂×Dic₅₉⋊C₄	—	2	1	no	—
22	C₂×C₄×Dic₅₉	—	2	1	no	—
23	#14273663	—	4	4	no	—
24	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
25	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
26	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
27	C₂²×C₈×C₅₉	D₄	1	2	yes	—
28	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₅₉	—	2	2	no	—
30	C₂²⋊C₈×C₅₉	—	2	2	no	—
31	#97×C₅₉	—	2	2	no	—
32	C₄×C₈×C₅₉	D₄	1	4	yes	—
33	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
36	C₂²×Dic₅₉	—	2	1	no	—
37	C₂²×Dic₅₉	—	2	1	no	—
38	C₂×Dic₁₁₈	—	2	2	no	—
39	C₂×Dic₁₁₈	—	4	2	no	—
40	C₂×C₈×C₅₉	—	2	2	no	—
41	C₂×C₈×C₅₉	—	2	2	no	—
42	C₂×C₈×C₅₉	—	2	4	no	—
43	C₂×C₈×C₅₉	SD₁₆	1	4	yes	—
44	C₄²×C₅₉	C₂×D₄	1	2	yes	—
45	C₄²×C₅₉	#41	1	2	yes	—
46	C₄⋊Dic₅₉	—	2	2	no	—
47	Dic₅₉⋊C₄	—	4	2	no	—
48	C₄×Dic₅₉	—	4	1	no	—
49	C₄×Dic₅₉	—	4	1	no	—
50	#11745664	—	4	4	no	—
51	C₂×C₅₉⋊C₈	—	4	1	no	—
52	C₂×C₅₉⋊C₈	—	4	2	no	—
53	C₂×C₅₉⋊C₈	—	4	2	no	—
54	C₂³×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
55	C₂×C₄×C₅₉	—	4	1	no	—
56	C₂×C₄×C₅₉	—	4	1	no	—
57	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
58	C₂×C₄×C₅₉	#123	1	2	yes	—
59	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
60	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
61	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
62	C₂×C₄×C₅₉	#137	1	2	yes	—
63	C₂×Dic₅₉	—	2	2	no	—
64	C₂×Dic₅₉	—	4	1	no	—
65	C₂×Dic₅₉	—	4	1	no	—
66	C₂×Dic₅₉	—	4	2	no	—
67	C₂×Dic₅₉	—	4	1	no	—
68	Dic₁₁₈	—	2	4	no	—
69	Dic₁₁₈	—	4	2	no	—
70	C₈×C₅₉	—	2	4	no	—
71	C₈×C₅₉	—	4	4	no	—
72	C₅₉⋊C₈	—	8	2	no	—
73	C₅₉⋊C₈	—	8	1	no	—
74	#10663	—	59	1	no	maximal, Sylow
75	C₂²×C₅₉	—	2	1	no	—
76	C₂²×C₅₉	—	2	1	no	—
77	C₂²×C₅₉	#399	1	1	yes	characteristic
78	C₂²×C₅₉	#475	1	2	yes	—
79	C₄×C₅₉	—	2	2	no	—
80	C₄×C₅₉	—	2	2	no	—
81	C₄×C₅₉	#483	1	2	yes	—
82	C₄×C₅₉	#492	1	1	yes	characteristic
83	C₄×C₅₉	#497	1	1	yes	characteristic
84	Dic₅₉	—	4	2	no	—
85	Dic₅₉	—	8	1	no	—
86	#1985	—	59	1	no	—
87	#1849	—	59	1	no	—
88	#1570	—	59	2	no	—
89	#1487	—	59	2	no	—
90	#1275	D₅₉	1	1	yes	characteristic
91	C₂×C₅₉	—	2	2	no	—
92	C₂×C₅₉	#2091	1	1	yes	characteristic
93	C₂×C₅₉	#2096	1	1	yes	characteristic
94	C₂×C₅₉	#2098	1	1	yes	characteristic
95	C₂×C₄×Q₈	—	59	1	no	—
96	#431	C₂×D₅₉	1	2	yes	—
97	C₂×#108	—	118	2	no	—
98	#423	—	59	2	no	—
99	C₂×C₄⋊C₈	—	59	1	no	—
100	C₂×#97	C₂×D₅₉	1	1	yes	characteristic
101	#336	—	118	2	no	—
102	#334	—	118	4	no	—
103	Q₈⋊C₈	—	118	4	no	—
104	C₅₉	#10663	1	1	yes	characteristic, complemented, Sylow
105	C₂²×Q₈	—	59	1	no	—
106	C₂×M₁₆	—	118	2	no	—
107	C₂²×C₈	—	118	1	no	—
108	C₂²×C₈	D₂₃₆	1	2	yes	—
109	C₄×Q₈	—	118	2	no	—
110	C₄×Q₈	—	118	2	no	—
111	C₂×C₄⋊C₄	—	59	1	no	—
112	C₂×C₄⋊C₄	—	118	1	no	—
113	C₂×C₄²	—	118	1	no	—
114	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
115	#108	—	236	4	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	118	2	no	—
118	C₄⋊C₈	—	118	2	no	—
119	C₄⋊C₈	—	118	2	no	—
120	C₂²⋊C₈	—	2	2	no	—
121	#97	—	2	2	no	—
122	C₄×C₈	D₂₃₆	1	4	yes	—
123	C₂×Q₈	—	118	2	no	—
124	C₂×Q₈	—	236	2	no	—
125	C₂²×C₄	—	118	1	no	—
126	C₂²×C₄	—	118	1	no	—
127	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
128	C₂²×C₄	#11745687	1	1	yes	characteristic
129	C₂²×C₄	Q₈×D₅₉	1	1	yes	characteristic
130	M₁₆	—	236	4	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	—	236	1	no	—
135	C₂×C₈	—	236	2	no	—
136	C₂×C₈	—	236	2	no	—
137	C₂×C₈	#11745660	1	4	yes	—
138	C₄⋊C₄	—	118	2	no	—
139	C₄⋊C₄	—	236	2	no	—
140	C₄²	—	236	1	no	—
141	C₄²	—	236	1	no	—
142	C₄²	C₂×D₂₃₆	1	2	yes	—
143	C₄²	#11745687	1	2	yes	—
144	C₂³	#14273707	1	1	yes	characteristic
145	Q₈	—	118	4	no	—
146	Q₈	—	236	2	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	118	2	no	—
150	C₂×C₄	—	236	2	no	—
151	C₂×C₄	—	236	1	no	—
152	C₂×C₄	—	236	1	no	—
153	C₂×C₄	—	236	1	no	—
154	C₂×C₄	#14273707	1	2	yes	—
155	C₂×C₄	#14273711	1	2	yes	—
156	C₂×C₄	#14273711	1	1	yes	characteristic
157	C₂×C₄	#14273724	1	1	yes	characteristic
158	C₂×C₄	C₂×#11745660	1	2	yes	—
159	C₂×C₄	#14273760	1	2	yes	—
160	C₈	—	2	4	no	—
161	C₈	—	4	4	no	—
162	C₈	—	472	2	no	—
163	C₈	—	472	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	#16837339	1	1	yes	characteristic
167	C₂²	#16837688	1	2	yes	—
168	C₄	—	2	2	no	—
169	C₄	—	2	2	no	—
170	C₄	—	472	1	no	—
171	C₄	—	236	2	no	—
172	C₄	#16837733	1	2	yes	—
173	C₄	#16837775	1	1	yes	characteristic
174	C₄	#16837796	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	#21425613	1	1	yes	characteristic
177	C₂	#21425638	1	1	yes	characteristic
178	C₂	#21425650	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21425036

C₂

yes

maximal, characteristic

#21424271

C₂

yes

maximal, characteristic

#21422867

C₂

yes

maximal

#21422526

C₂

yes

maximal

C₅₉×#1275

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₁₈

C₂²

yes

characteristic

C₂×#14273663

—

#16837426

C₂²

yes

—

C₂×C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

—

—

C₅₉×#431

C₂²

yes

—

C₂×#97×C₅₉

C₂²

yes

characteristic

C₂²×Dic₁₁₈

Q₈

yes

characteristic

C₂×#11745664

—

C₂²×C₅₉⋊C₈

—

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

—

C₂×C₄⋊Dic₅₉

Q₈

yes

characteristic

C₂×Dic₅₉⋊C₄

—

C₂×C₄×Dic₅₉

—

#14273663

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

C₂²⋊C₈×C₅₉

—

#97×C₅₉

—

C₄×C₈×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₄²×C₅₉

C₂×D₄

yes

—

C₄²×C₅₉

#41

yes

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₉

#123

yes

—

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

C₂×SD₁₆

yes

—

C₂×C₄×C₅₉

#137

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#10663

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#399

yes

characteristic

C₂²×C₅₉

#475

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#483

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

—

—

—

—

D₅₉

yes

characteristic

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₄×Q₈

—

#431

C₂×D₅₉

yes

—

C₂×#108

—

118

—

#423

—

C₂×C₄⋊C₈

—

100

C₂×#97

C₂×D₅₉

yes

characteristic

101

#336

—

118

—

102

#334

—

118

—

103

Q₈⋊C₈

—

118

—

104

C₅₉

#10663

yes

characteristic, complemented, Sylow

105

C₂²×Q₈

—

106

C₂×M₁₆

—

118

—

107

C₂²×C₈

—

118

—

108

C₂²×C₈

D₂₃₆

yes

—

109

C₄×Q₈

—

118

—

110

C₄×Q₈

—

118

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

118

—

113

C₂×C₄²

—

118

—

114

C₂×C₄²

C₂²×D₅₉

yes

characteristic

115

#108

—

236

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

—

118

C₄⋊C₈

—

118

—

119

C₄⋊C₈

—

118

—

120

C₂²⋊C₈

—

121

#97

—

122

C₄×C₈

D₂₃₆

yes

—

123

C₂×Q₈

—

118

—

124

C₂×Q₈

—

236

—

125

C₂²×C₄

—

118

—

126

C₂²×C₄

—

118

—

127

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

128

C₂²×C₄

#11745687

yes

characteristic

129

C₂²×C₄

Q₈×D₅₉

yes

characteristic

130

M₁₆

—

236

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

236

—

135

C₂×C₈

—

236

—

136

C₂×C₈

—

236

—

137

C₂×C₈

#11745660

yes

—

138

C₄⋊C₄

—

118

—

139

C₄⋊C₄

—

236

—

140

C₄²

—

236

—

141

C₄²

—

236

—

142

C₄²

C₂×D₂₃₆

yes

—

143

C₄²

#11745687

yes

—

144

C₂³

#14273707

yes

characteristic

145

Q₈

—

118

—

146

Q₈

—

236

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

118

—

150

C₂×C₄

—

236

—

151

C₂×C₄

—

236

—

152

C₂×C₄

—

236

—

153

C₂×C₄

—

236

—

154

C₂×C₄

#14273707

yes

—

155

C₂×C₄

#14273711

yes

—

156

C₂×C₄

#14273711

yes

characteristic

157

C₂×C₄

#14273724

yes

characteristic

158

C₂×C₄

C₂×#11745660

yes

—

159

C₂×C₄

#14273760

yes

—

160

C₈

—

161

C₈

—

162

C₈

—

472

—

163

C₈

—

472

—

164

C₂²

—

165

C₂²

—

166

C₂²

#16837339

yes

characteristic

167

C₂²

#16837688

yes

—

168

C₄

—

169

C₄

—

170

C₄

—

472

—

171

C₄

—

236

—

172

C₄

#16837733

yes

—

173

C₄

#16837775

yes

characteristic

174

C₄

#16837796

yes

characteristic

175

C₂

—

176

C₂

#21425613

yes

characteristic

177

C₂

#21425638

yes

characteristic

178

C₂

#21425650

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14273724
commutator, cosocle	C₂×C₄²×C₅₉	C₂³
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1275	C₂
socle	C₂²×C₅₉	#399
prime radical	C₂³×C₅₉	C₂²⋊Q₈
hypercenter	#1275	D₅₉
hypocenter	C₅₉	#10663
LCS	G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1275

subgroup

G/H

center

C₂×C₄

#14273724

commutator, cosocle

C₂×C₄²×C₅₉

C₂³

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1275

C₂

socle

C₂²×C₅₉

#399

prime radical

C₂³×C₅₉

C₂²⋊Q₈

D₅₉

C₅₉

G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1275

N	H ≅ G/N	type
104. C₅₉	#10663	semidirect

H ≅ G/N

type

104. C₅₉

#10663

semidirect

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

subgroup	index
#21425036	2
#21424271	2
#21422867	2
#21422526	2
C₅₉×#1275	2
#10663	59

subgroup

index

C₅₉×#1275

Subgroups — #28493681

Conjugacy classes of subgroups

Complements

Maximal subgroups