Group #28488174 — subgroups

Subgroups — #28488174

Order 15104.

390 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 123 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×#16837800	C₂	1	1	yes	maximal, characteristic
2	#21424234	C₂	1	1	yes	maximal, characteristic
3	#21422780	C₂	1	2	yes	maximal
4	#21422776	C₂	1	2	yes	maximal
5	C₅₉×#1452	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273727	C₂²	1	2	yes	—
7	#16837800	—	2	2	no	—
8	#16837800	C₄	1	4	yes	—
9	C₂×#14273659	—	2	2	no	—
10	#16837426	C₂²	1	1	yes	characteristic
11	#16837426	C₂²	1	1	yes	characteristic
12	#16837167	—	2	2	no	—
13	#16837150	—	2	4	no	—
14	#16837149	—	2	4	no	—
15	C₅₉×#431	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₅₉×#322	—	2	2	no	—
18	C₂²×Dic₁₁₈	—	2	2	no	—
19	C₂×#11745664	—	2	1	no	—
20	C₂×#11745664	—	2	1	no	—
21	#14273727	—	2	2	no	—
22	#14273727	—	2	2	no	—
23	#14273727	—	2	2	no	—
24	#14273727	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊Dic₅₉	—	2	2	no	—
26	C₂×C₄⋊Dic₅₉	—	2	2	no	—
27	#14273659	—	2	8	no	—
28	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
29	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
30	C₂²×C₈×C₅₉	D₄	1	2	yes	—
31	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₅₉	D₄	1	4	yes	—
33	C₂²⋊C₈×C₅₉	—	2	2	no	—
34	C₄×C₈×C₅₉	—	2	2	no	—
35	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₅₉	D₄	1	2	yes	—
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	2	no	—
42	C₂×Dic₁₁₈	—	8	4	no	—
43	C₂×C₈×C₅₉	—	2	2	no	—
44	C₂×C₈×C₅₉	—	2	2	no	—
45	C₂×C₈×C₅₉	—	2	4	no	—
46	C₂×C₈×C₅₉	C₂²⋊C₄	1	4	yes	—
47	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₅₉	C₂×D₄	1	2	yes	—
50	C₄⋊Dic₅₉	—	2	4	no	—
51	C₄⋊Dic₅₉	—	2	4	no	—
52	C₄⋊Dic₅₉	—	4	2	no	—
53	C₄⋊Dic₅₉	—	4	2	no	—
54	#11745664	—	4	2	no	—
55	#11745664	—	4	2	no	—
56	C₂×C₅₉⋊C₈	—	4	1	no	—
57	C₂×C₅₉⋊C₈	—	4	1	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	#137	1	2	yes	—
65	C₂×C₄×C₅₉	#137	1	2	yes	—
66	C₂×Dic₅₉	—	4	4	no	—
67	C₂×Dic₅₉	—	8	4	no	—
68	Dic₁₁₈	—	8	4	no	—
69	Dic₁₁₈	—	8	4	no	—
70	C₈×C₅₉	—	2	4	no	—
71	C₈×C₅₉	—	4	4	no	—
72	C₅₉⋊C₈	—	8	1	no	—
73	C₅₉⋊C₈	—	8	1	no	—
74	#9776	—	59	1	no	maximal, Sylow
75	C₂²×C₅₉	—	2	2	no	—
76	C₂²×C₅₉	#390	1	1	yes	characteristic
77	C₂²×C₅₉	#491	1	2	yes	—
78	C₄×C₅₉	—	2	2	no	—
79	C₄×C₅₉	#419	1	2	yes	—
80	C₄×C₅₉	#471	1	4	yes	—
81	C₄×C₅₉	#497	1	1	yes	characteristic
82	C₄×C₅₉	#497	1	1	yes	characteristic
83	Dic₅₉	—	8	4	no	—
84	C₂×#498	—	59	1	no	—
85	#1837	—	59	1	no	—
86	#1549	—	59	2	no	—
87	#1548	—	59	2	no	—
88	#1452	D₅₉	1	1	yes	characteristic
89	C₂×C₅₉	—	2	2	no	—
90	C₂×C₅₉	#1960	1	1	yes	characteristic
91	C₂×C₅₉	#1977	1	2	yes	—
92	C₂×C₄⋊Q₈	—	59	2	no	—
93	#498	—	59	4	no	—
94	#498	—	118	2	no	—
95	#431	C₂×D₅₉	1	2	yes	—
96	C₂×#106	—	118	2	no	—
97	#423	—	59	1	no	—
98	#423	—	59	1	no	—
99	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
100	#341	—	118	2	no	—
101	#333	—	118	4	no	—
102	#332	—	118	4	no	—
103	#322	—	2	2	no	—
104	C₅₉	#9776	1	1	yes	characteristic, complemented, Sylow
105	C₂²×Q₈	—	118	2	no	—
106	C₂×M₁₆	—	118	1	no	—
107	C₂×M₁₆	—	118	1	no	—
108	C₂²×C₈	D₂₃₆	1	2	yes	—
109	C₄⋊Q₈	—	59	4	no	—
110	C₄⋊Q₈	—	118	2	no	—
111	C₄⋊Q₈	—	118	2	no	—
112	C₄⋊Q₈	—	118	2	no	—
113	C₂×C₄⋊C₄	—	118	2	no	—
114	C₂×C₄⋊C₄	—	118	2	no	—
115	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
116	#106	—	118	8	no	—
117	C₄⋊C₈	—	118	2	no	—
118	C₄⋊C₈	—	118	2	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	2	yes	—
124	C₂×Q₈	—	118	4	no	—
125	C₂×Q₈	—	472	4	no	—
126	C₂×Q₈	—	236	2	no	—
127	C₂²×C₄	—	236	2	no	—
128	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
129	C₂²×C₄	#11745685	1	2	yes	—
130	M₁₆	—	236	2	no	—
131	M₁₆	—	236	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	236	1	no	—
136	C₂×C₈	—	236	1	no	—
137	C₂×C₈	(C₂×D₅₉)⋊C₄	1	4	yes	—
138	C₄⋊C₄	—	118	4	no	—
139	C₄⋊C₄	—	118	4	no	—
140	C₄⋊C₄	—	236	2	no	—
141	C₄⋊C₄	—	236	2	no	—
142	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
143	C₄²	C₂×D₂₃₆	1	2	yes	—
144	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
145	C₂³	#14273715	1	1	yes	characteristic
146	Q₈	—	472	4	no	—
147	Q₈	—	472	4	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	472	4	no	—
150	C₂×C₄	—	236	4	no	—
151	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
152	C₂×C₄	C₄×D₂₃₆	1	2	yes	—
153	C₂×C₄	#14273701	1	2	yes	—
154	C₂×C₄	#14273724	1	1	yes	characteristic
155	C₂×C₄	#14273760	1	2	yes	—
156	C₂×C₄	#14273760	1	2	yes	—
157	C₈	—	2	4	no	—
158	C₈	—	4	4	no	—
159	C₈	—	472	1	no	—
160	C₈	—	472	1	no	—
161	C₂²	—	2	2	no	—
162	C₂²	#16837304	1	1	yes	characteristic
163	C₂²	#16837771	1	2	yes	—
164	C₄	—	2	2	no	—
165	C₄	—	472	4	no	—
166	C₄	#16837412	1	2	yes	—
167	C₄	#16837665	1	4	yes	—
168	C₄	#16837796	1	1	yes	characteristic
169	C₄	#16837796	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	#21424916	1	1	yes	characteristic
172	C₂	#21424999	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837800

C₂

yes

maximal, characteristic

#21424234

C₂

yes

maximal, characteristic

#21422780

C₂

yes

maximal

#21422776

C₂

yes

maximal

C₅₉×#1452

C₂

yes

maximal, characteristic

C₂×#14273727

C₂²

yes

—

#16837800

—

#16837800

C₄

yes

—

C₂×#14273659

—

#16837426

C₂²

yes

characteristic

#16837426

C₂²

yes

—

—

—

C₅₉×#431

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#322

—

C₂²×Dic₁₁₈

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

#14273659

—

C₄⋊(C₅₉⋊C₈)

—

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

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

—

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

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

#137

yes

—

C₂×C₄×C₅₉

#137

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#9776

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#390

yes

characteristic

C₂²×C₅₉

#491

yes

—

C₄×C₅₉

—

C₄×C₅₉

#419

yes

—

C₄×C₅₉

#471

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

—

—

—

—

D₅₉

yes

characteristic

C₂×C₅₉

—

C₂×C₅₉

#1960

yes

characteristic

C₂×C₅₉

#1977

yes

—

C₂×C₄⋊Q₈

—

#498

—

#498

—

118

—

#431

C₂×D₅₉

yes

—

C₂×#106

—

118

—

#423

—

#423

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

100

#341

—

118

—

101

#333

—

118

—

102

#332

—

118

—

103

#322

—

104

C₅₉

#9776

yes

characteristic, complemented, Sylow

105

C₂²×Q₈

—

118

—

106

C₂×M₁₆

—

118

—

107

C₂×M₁₆

—

118

—

108

C₂²×C₈

D₂₃₆

yes

—

109

C₄⋊Q₈

—

110

C₄⋊Q₈

—

118

—

111

C₄⋊Q₈

—

118

—

112

C₄⋊Q₈

—

118

—

113

C₂×C₄⋊C₄

—

118

—

114

C₂×C₄⋊C₄

—

118

—

115

C₂×C₄²

C₂²×D₅₉

yes

characteristic

116

#106

—

118

—

117

C₄⋊C₈

—

118

—

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

—

124

C₂×Q₈

—

118

—

125

C₂×Q₈

—

472

—

126

C₂×Q₈

—

236

—

127

C₂²×C₄

—

236

—

128

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

129

C₂²×C₄

#11745685

yes

—

130

M₁₆

—

236

—

131

M₁₆

—

236

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

236

—

136

C₂×C₈

—

236

—

137

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

—

138

C₄⋊C₄

—

118

—

139

C₄⋊C₄

—

118

—

140

C₄⋊C₄

—

236

—

141

C₄⋊C₄

—

236

—

142

C₄²

C₂×C₄×D₅₉

yes

characteristic

143

C₄²

C₂×D₂₃₆

yes

—

144

C₄²

C₂×C₂²⋊D₅₉

yes

characteristic

145

C₂³

#14273715

yes

characteristic

146

Q₈

—

472

—

147

Q₈

—

472

—

148

C₂×C₄

—

149

C₂×C₄

—

472

—

150

C₂×C₄

—

236

—

151

C₂×C₄

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

152

C₂×C₄

C₄×D₂₃₆

yes

—

153

C₂×C₄

#14273701

yes

—

154

C₂×C₄

#14273724

yes

characteristic

155

C₂×C₄

#14273760

yes

—

156

C₂×C₄

#14273760

yes

—

157

C₈

—

158

C₈

—

159

C₈

—

472

—

160

C₈

—

472

—

161

C₂²

—

162

C₂²

#16837304

yes

characteristic

163

C₂²

#16837771

yes

—

164

C₄

—

165

C₄

—

472

—

166

C₄

#16837412

yes

—

167

C₄

#16837665

yes

—

168

C₄

#16837796

yes

characteristic

169

C₄

#16837796

yes

characteristic

170

C₂

—

171

C₂

#21424916

yes

characteristic

172

C₂

#21424999

yes

—

	subgroup	G/H
center	C₂²	#16837304
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1452	C₂
socle	C₂²×C₅₉	#390
prime radical	C₂³×C₅₉	#124
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1452	D₅₉
hypocenter	C₅₉	#9776
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂²

#16837304

commutator

C₄²×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1452

C₂

C₂²×C₅₉

C₂³×C₅₉

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

N	H ≅ G/N	type
104. C₅₉	#9776	semidirect

H ≅ G/N

type

104. C₅₉

#9776

semidirect

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

subgroup	index
C₂×#16837800	2
#21424234	2
#21422780	2
#21422776	2
C₅₉×#1452	2
#9776	59

subgroup

index

C₅₉×#1452

Subgroups — #28488174

Conjugacy classes of subgroups

Complements

Maximal subgroups