Group #28488152 — subgroups

Subgroups — #28488152

Order 15104.

478 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 155 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×#16837782	C₂	1	1	yes	maximal, characteristic
2	#21424268	C₂	1	1	yes	maximal, characteristic
3	#21422767	C₂	1	4	yes	maximal
4	C₅₉×#1450	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₃₆	—	2	2	no	—
6	C₂×#14273727	C₂²	1	2	yes	—
7	#16837782	—	2	1	no	—
8	#16837782	—	2	1	no	—
9	#16837782	C₄	1	4	yes	—
10	C₂×#14273663	—	2	2	no	—
11	#16837426	C₂²	1	2	yes	—
12	#16837147	—	2	8	no	—
13	C₅₉×#431	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉×#322	—	2	1	no	—
16	C₈²×C₅₉	—	2	1	no	—
17	C₂²×Dic₁₁₈	—	2	2	no	—
18	C₂×Dic₂₃₆	—	2	4	no	—
19	C₂×Dic₂₃₆	—	2	4	no	—
20	C₂×Dic₂₃₆	—	4	4	no	—
21	C₂×Dic₂₃₆	—	4	4	no	—
22	C₂×#11745664	—	2	2	no	—
23	#14273727	—	2	4	no	—
24	#14273727	—	2	2	no	—
25	#14273727	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊Dic₅₉	—	2	4	no	—
27	#14273663	—	4	4	no	—
28	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
29	C₂²×C₈×C₅₉	D₄	1	2	yes	—
30	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₅₉	D₄	1	4	yes	—
32	C₂²⋊C₈×C₅₉	—	2	2	no	—
33	C₄×C₈×C₅₉	—	2	2	no	—
34	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
36	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₉	#41	1	2	yes	—
39	C₂²×Dic₅₉	—	4	2	no	—
40	C₂×Dic₁₁₈	—	2	4	no	—
41	C₂×Dic₁₁₈	—	4	4	no	—
42	C₂×Dic₁₁₈	—	4	4	no	—
43	C₂×Dic₁₁₈	—	4	2	no	—
44	C₂×C₈×C₅₉	—	2	4	no	—
45	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
47	C₂×C₈×C₅₉	D₈	1	4	yes	—
48	C₂×C₈×C₅₉	SD₁₆	1	4	yes	—
49	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
50	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₄²×C₅₉	C₂×D₄	1	2	yes	—
52	C₄⋊Dic₅₉	—	2	8	no	—
53	C₄⋊Dic₅₉	—	4	4	no	—
54	#11745664	—	4	4	no	—
55	C₂×C₅₉⋊C₈	—	4	2	no	—
56	Dic₂₃₆	—	4	8	no	—
57	Dic₂₃₆	—	4	8	no	—
58	C₂³×C₅₉	#124	1	1	yes	characteristic
59	C₂×C₄×C₅₉	—	2	4	no	—
60	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
62	C₂×C₄×C₅₉	#121	1	2	yes	—
63	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
65	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
66	C₂×Dic₅₉	—	4	4	no	—
67	C₂×Dic₅₉	—	8	4	no	—
68	Dic₁₁₈	—	4	8	no	—
69	Dic₁₁₈	—	8	2	no	—
70	Dic₁₁₈	—	8	2	no	—
71	C₈×C₅₉	—	4	4	no	—
72	C₈×C₅₉	#102	1	4	yes	—
73	C₈×C₅₉	#102	1	4	yes	—
74	C₅₉⋊C₈	—	8	2	no	—
75	#9772	—	59	1	no	maximal, Sylow
76	C₂²×C₅₉	—	2	2	no	—
77	C₂²×C₅₉	#390	1	1	yes	characteristic
78	C₂²×C₅₉	#486	1	2	yes	—
79	C₄×C₅₉	—	2	2	no	—
80	C₄×C₅₉	C₂×#102	1	2	yes	—
81	C₄×C₅₉	#473	1	4	yes	—
82	C₄×C₅₉	#492	1	1	yes	characteristic
83	C₄×C₅₉	#493	1	1	yes	characteristic
84	Dic₅₉	—	8	4	no	—
85	C₂×#494	—	59	1	no	—
86	#1848	—	59	1	no	—
87	#1546	—	59	4	no	—
88	#1450	D₅₉	1	1	yes	characteristic
89	C₂×C₅₉	—	2	2	no	—
90	C₂×C₅₉	#1957	1	1	yes	characteristic
91	C₂×C₅₉	#1979	1	2	yes	—
92	C₂²×Q₁₆	—	118	2	no	—
93	C₂×C₄⋊Q₈	—	59	2	no	—
94	#494	—	59	4	no	—
95	#494	—	118	1	no	—
96	#494	—	118	1	no	—
97	#431	C₂×D₅₉	1	2	yes	—
98	C₂×#108	—	118	2	no	—
99	#423	—	59	2	no	—
100	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
101	#332	—	118	8	no	—
102	#322	—	2	1	no	—
103	C₈²	—	2	1	no	—
104	C₅₉	#9772	1	1	yes	characteristic, complemented, Sylow
105	C₂²×Q₈	—	118	2	no	—
106	C₂×Q₁₆	—	118	4	no	—
107	C₂×Q₁₆	—	118	4	no	—
108	C₂×Q₁₆	—	236	4	no	—
109	C₂×Q₁₆	—	236	4	no	—
110	C₂×M₁₆	—	118	2	no	—
111	C₂²×C₈	D₂₃₆	1	2	yes	—
112	C₄⋊Q₈	—	59	4	no	—
113	C₄⋊Q₈	—	118	4	no	—
114	C₄⋊Q₈	—	118	2	no	—
115	C₂×C₄⋊C₄	—	118	4	no	—
116	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
117	#108	—	236	4	no	—
118	C₄⋊C₈	—	118	4	no	—
119	C₄⋊C₈	D₂₃₆	1	4	yes	—
120	C₂²⋊C₈	—	2	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	C₄×D₅₉	1	2	yes	—
123	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
124	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
125	C₂×Q₈	—	118	4	no	—
126	C₂×Q₈	—	236	4	no	—
127	C₂×Q₈	—	236	4	no	—
128	C₂×Q₈	—	236	2	no	—
129	C₂²×C₄	—	236	2	no	—
130	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
131	C₂²×C₄	#11745685	1	2	yes	—
132	Q₁₆	—	236	8	no	—
133	Q₁₆	—	236	8	no	—
134	M₁₆	—	236	4	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	236	2	no	—
137	C₂×C₈	#11745660	1	4	yes	—
138	C₂×C₈	D₄₇₂	1	4	yes	—
139	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
140	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
141	C₄⋊C₄	—	118	8	no	—
142	C₄⋊C₄	—	236	4	no	—
143	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
144	C₄²	C₂×D₂₃₆	1	2	yes	—
145	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
146	C₂³	#14273715	1	1	yes	characteristic
147	Q₈	—	472	2	no	—
148	Q₈	—	472	2	no	—
149	Q₈	—	236	8	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	472	4	no	—
152	C₂×C₄	—	236	4	no	—
153	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
154	C₂×C₄	C₄×D₂₃₆	1	2	yes	—
155	C₂×C₄	#14273701	1	2	yes	—
156	C₂×C₄	#14273724	1	1	yes	characteristic
157	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
158	C₂×C₄	C₂×#11745660	1	2	yes	—
159	C₈	—	4	4	no	—
160	C₈	—	472	2	no	—
161	C₈	#14273649	1	4	yes	—
162	C₈	#14273649	1	4	yes	—
163	C₂²	—	2	2	no	—
164	C₂²	#16837304	1	1	yes	characteristic
165	C₂²	#16837749	1	2	yes	—
166	C₄	—	2	2	no	—
167	C₄	—	472	4	no	—
168	C₄	C₂×#14273649	1	2	yes	—
169	C₄	#16837677	1	4	yes	—
170	C₄	#16837775	1	1	yes	characteristic
171	C₄	#16837779	1	1	yes	characteristic
172	C₂	—	2	2	no	—
173	C₂	#21424902	1	1	yes	characteristic
174	C₂	#21425011	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837782

C₂

yes

maximal, characteristic

#21424268

C₂

yes

maximal, characteristic

#21422767

C₂

yes

maximal

C₅₉×#1450

C₂

yes

maximal, characteristic

C₂²×Dic₂₃₆

—

C₂×#14273727

C₂²

yes

—

#16837782

—

#16837782

—

#16837782

C₄

yes

—

C₂×#14273663

—

#16837426

C₂²

yes

—

#16837147

—

C₅₉×#431

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#322

—

C₈²×C₅₉

—

C₂²×Dic₁₁₈

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊Dic₅₉

—

#14273663

—

C₄⋊(C₅₉⋊C₈)

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

D₄

yes

—

C₂²⋊C₈×C₅₉

—

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

—

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

D₈

yes

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

#11745664

—

C₂×C₅₉⋊C₈

—

Dic₂₃₆

—

Dic₂₃₆

—

C₂³×C₅₉

#124

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

—

C₂×C₄×C₅₉

#121

yes

—

C₂×C₄×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

—

C₂×C₄×C₅₉

C₂×SD₁₆

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

#102

yes

—

C₈×C₅₉

#102

yes

—

C₅₉⋊C₈

—

#9772

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#390

yes

characteristic

C₂²×C₅₉

#486

yes

—

C₄×C₅₉

—

C₄×C₅₉

C₂×#102

yes

—

C₄×C₅₉

#473

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

—

—

—

D₅₉

yes

characteristic

C₂×C₅₉

—

C₂×C₅₉

#1957

yes

characteristic

C₂×C₅₉

#1979

yes

—

C₂²×Q₁₆

—

118

—

C₂×C₄⋊Q₈

—

#494

—

#494

—

118

—

#494

—

118

—

#431

C₂×D₅₉

yes

—

C₂×#108

—

118

—

#423

—

100

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

101

#332

—

118

—

102

#322

—

103

C₈²

—

104

C₅₉

#9772

yes

characteristic, complemented, Sylow

105

C₂²×Q₈

—

118

—

106

C₂×Q₁₆

—

118

—

107

C₂×Q₁₆

—

118

—

108

C₂×Q₁₆

—

236

—

109

C₂×Q₁₆

—

236

—

110

C₂×M₁₆

—

118

—

111

C₂²×C₈

D₂₃₆

yes

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

118

—

114

C₄⋊Q₈

—

118

—

115

C₂×C₄⋊C₄

—

118

—

116

C₂×C₄²

C₂²×D₅₉

yes

characteristic

117

#108

—

236

—

118

C₄⋊C₈

—

118

—

119

C₄⋊C₈

D₂₃₆

yes

—

120

C₂²⋊C₈

—

121

C₄×C₈

—

122

C₄×C₈

C₄×D₅₉

yes

—

123

C₄×C₈

C₂²⋊D₅₉

yes

characteristic

124

C₄×C₈

C₂²⋊D₅₉

yes

characteristic

125

C₂×Q₈

—

118

—

126

C₂×Q₈

—

236

—

127

C₂×Q₈

—

236

—

128

C₂×Q₈

—

236

—

129

C₂²×C₄

—

236

—

130

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

131

C₂²×C₄

#11745685

yes

—

132

Q₁₆

—

236

—

133

Q₁₆

—

236

—

134

M₁₆

—

236

—

135

C₂×C₈

—

136

C₂×C₈

—

236

—

137

C₂×C₈

#11745660

yes

—

138

C₂×C₈

D₄₇₂

yes

—

139

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

—

140

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

—

141

C₄⋊C₄

—

118

—

142

C₄⋊C₄

—

236

—

143

C₄²

C₂×C₄×D₅₉

yes

characteristic

144

C₄²

C₂×D₂₃₆

yes

—

145

C₄²

C₂×C₂²⋊D₅₉

yes

characteristic

146

C₂³

#14273715

yes

characteristic

147

Q₈

—

472

—

148

Q₈

—

472

—

149

Q₈

—

236

—

150

C₂×C₄

—

151

C₂×C₄

—

472

—

152

C₂×C₄

—

236

—

153

C₂×C₄

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

154

C₂×C₄

C₄×D₂₃₆

yes

—

155

C₂×C₄

#14273701

yes

—

156

C₂×C₄

#14273724

yes

characteristic

157

C₂×C₄

C₂×D₄₇₂

yes

—

158

C₂×C₄

C₂×#11745660

yes

—

159

C₈

—

160

C₈

—

472

—

161

C₈

#14273649

yes

—

162

C₈

#14273649

yes

—

163

C₂²

—

164

C₂²

#16837304

yes

characteristic

165

C₂²

#16837749

yes

—

166

C₄

—

167

C₄

—

472

—

168

C₄

C₂×#14273649

yes

—

169

C₄

#16837677

yes

—

170

C₄

#16837775

yes

characteristic

171

C₄

#16837779

yes

characteristic

172

C₂

—

173

C₂

#21424902

yes

characteristic

174

C₂

#21425011

yes

—

	subgroup	G/H
center	C₂²	#16837304
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1450	C₂
socle	C₂²×C₅₉	#390
prime radical	C₂³×C₅₉	#124
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1450	D₅₉
hypocenter	C₅₉	#9772
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

subgroup

G/H

center

C₂²

#16837304

commutator

C₄²×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1450

C₂

C₂²×C₅₉

C₂³×C₅₉

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

N	H ≅ G/N	type
104. C₅₉	#9772	semidirect

H ≅ G/N

type

104. C₅₉

#9772

semidirect

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

subgroup	index
C₂×#16837782	2
#21424268	2
#21422767	2
C₅₉×#1450	2
#9772	59

subgroup

index

C₅₉×#1450

Subgroups — #28488152

Conjugacy classes of subgroups

Complements

Maximal subgroups