Group #28479129 — subgroups

Subgroups — #28479129

Order 15104.

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

normal subgroups only

This group has 354 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	#21423784	C₂	1	1	yes	maximal, characteristic
2	#21422380	C₂	1	2	yes	maximal
3	#21421927	C₂	1	1	yes	maximal, characteristic
4	#21421924	C₂	1	2	yes	maximal
5	C₅₉×#1275	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₂²	1	2	yes	—
7	C₂×#14273637	C₄	1	2	yes	—
8	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₈×Dic₅₉	C₄	1	2	yes	—
10	#16837248	C₂²	1	1	yes	characteristic
11	#16837124	C₄	1	4	yes	complemented
12	C₈×C₅₉⋊C₈	C₄	1	4	yes	complemented
13	C₅₉×#431	C₂²	1	2	yes	—
14	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
15	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
18	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
19	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
20	#14273637	—	2	4	no	—
21	C₄×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
22	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
23	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
24	C₈×Dic₅₉	C₈	1	8	yes	complemented
25	C₅₉⋊#95	—	2	1	no	—
26	C₅₉⋊#95	—	2	1	no	—
27	C₂²×C₈×C₅₉	C₂×C₄	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	C₂×C₄	1	4	yes	—
33	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
35	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
36	C₂²×Dic₅₉	—	2	2	no	—
37	C₂×C₈×C₅₉	—	2	2	no	—
38	C₂×C₈×C₅₉	—	2	2	no	—
39	C₂×C₈×C₅₉	—	2	4	no	—
40	C₂×C₈×C₅₉	C₂×C₈	1	4	yes	—
41	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
42	C₄²×C₅₉	#41	1	2	yes	—
43	C₄×Dic₅₉	—	2	2	no	—
44	C₄×Dic₅₉	C₂×C₈	1	4	yes	—
45	C₂×C₅₉⋊C₈	—	2	4	no	—
46	C₂×C₅₉⋊C₈	C₄²	1	1	yes	characteristic
47	C₂×C₅₉⋊C₈	C₄²	1	1	yes	characteristic
48	C₂×C₅₉⋊C₈	C₄²	1	2	yes	—
49	C₂×C₅₉⋊C₈	C₂×C₈	1	4	yes	—
50	C₂×C₅₉⋊C₈	M₁₆	1	4	yes	—
51	C₂³×C₅₉	#117	1	1	yes	characteristic
52	C₂×C₄×C₅₉	—	4	1	no	—
53	C₂×C₄×C₅₉	—	4	1	no	—
54	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
55	C₂×C₄×C₅₉	#117	1	2	yes	—
56	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
57	C₂×C₄×C₅₉	#117	1	2	yes	—
58	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
59	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
60	C₂×Dic₅₉	—	2	4	no	—
61	C₂×Dic₅₉	—	4	2	no	—
62	C₂×Dic₅₉	—	4	2	no	—
63	C₈×C₅₉	—	2	4	no	—
64	C₈×C₅₉	—	4	4	no	—
65	C₅₉⋊C₈	—	4	4	no	—
66	C₅₉⋊C₈	C₄×C₈	1	2	yes	—
67	C₅₉⋊C₈	C₄×C₈	1	2	yes	—
68	C₅₉⋊C₈	#97	1	4	yes	—
69	#8137	—	59	1	no	maximal, Sylow
70	C₂²×C₅₉	—	2	1	no	—
71	C₂²×C₅₉	—	2	1	no	—
72	C₂²×C₅₉	#376	1	1	yes	characteristic
73	C₂²×C₅₉	#431	1	2	yes	—
74	C₄×C₅₉	—	2	2	no	—
75	C₄×C₅₉	—	2	2	no	—
76	C₄×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
77	C₄×C₅₉	C₂×#97	1	1	yes	characteristic
78	C₄×C₅₉	#431	1	2	yes	—
79	Dic₅₉	—	4	4	no	—
80	#1726	—	59	1	no	—
81	#1450	—	59	2	no	—
82	#1275	—	59	1	no	—
83	#1275	D₅₉	1	1	yes	characteristic
84	#1274	—	59	2	no	—
85	C₂×C₅₉	—	2	2	no	—
86	C₂×C₅₉	#1744	1	1	yes	characteristic
87	C₂×C₅₉	#1747	1	1	yes	characteristic
88	C₂×C₅₉	#1747	1	1	yes	characteristic
89	#431	—	59	2	no	—
90	#431	C₂×D₅₉	1	2	yes	—
91	C₂×#97	—	59	2	no	—
92	C₂×#97	C₂×D₅₉	1	1	yes	characteristic
93	C₂×C₄×C₈	—	59	2	no	—
94	C₂×C₄×C₈	—	59	1	no	—
95	#376	—	59	1	no	—
96	#322	—	59	4	no	—
97	C₈²	—	59	4	no	—
98	C₅₉	#8137	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₈	—	59	1	no	—
100	C₂²×C₈	—	59	1	no	—
101	C₂²×C₈	C₄×D₅₉	1	2	yes	—
102	C₂×C₄²	—	59	2	no	—
103	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	118	2	no	—
106	C₂²⋊C₈	—	2	2	no	—
107	C₂²⋊C₈	—	118	2	no	—
108	#97	—	2	2	no	—
109	#97	—	118	4	no	—
110	C₄×C₈	—	59	8	no	—
111	C₄×C₈	—	59	2	no	—
112	C₄×C₈	—	59	2	no	—
113	C₄×C₈	—	59	4	no	—
114	C₄×C₈	C₄×D₅₉	1	4	yes	—
115	#95	—	118	1	no	—
116	#95	—	118	1	no	—
117	C₂²×C₄	—	118	2	no	—
118	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
119	C₂²×C₄	#11745687	1	1	yes	characteristic
120	C₂²×C₄	#11745689	1	1	yes	characteristic
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	4	no	—
124	C₂×C₈	—	59	1	no	—
125	C₂×C₈	—	59	1	no	—
126	C₂×C₈	—	59	2	no	—
127	C₂×C₈	—	59	4	no	—
128	C₂×C₈	—	59	4	no	—
129	C₂×C₈	—	118	4	no	—
130	C₂×C₈	C₈×D₅₉	1	4	yes	—
131	C₄²	—	59	4	no	—
132	C₄²	—	118	2	no	—
133	C₄²	C₂×C₄×D₅₉	1	2	yes	—
134	C₄²	#11745687	1	2	yes	—
135	C₂³	#14273685	1	1	yes	characteristic
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	118	4	no	—
139	C₂×C₄	—	236	2	no	—
140	C₂×C₄	—	236	2	no	—
141	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
142	C₂×C₄	#14273683	1	2	yes	—
143	C₂×C₄	#14273683	1	1	yes	characteristic
144	C₂×C₄	#14273685	1	2	yes	—
145	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
146	C₂×C₄	M₁₆×D₅₉	1	2	yes	—
147	C₈	—	2	4	no	—
148	C₈	—	4	4	no	—
149	C₈	—	59	2	no	—
150	C₈	—	59	2	no	—
151	C₈	—	59	4	no	—
152	C₈	—	236	4	no	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	#16837247	1	1	yes	characteristic
156	C₂²	#16837454	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	236	4	no	—
160	C₄	C₄×C₈×D₅₉	1	1	yes	characteristic
161	C₄	#97×D₅₉	1	1	yes	characteristic
162	C₄	#16837452	1	2	yes	—
163	C₂	—	2	2	no	—
164	C₂	#21423840	1	1	yes	characteristic
165	C₂	#21423850	1	1	yes	characteristic
166	C₂	#21423852	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21423784

C₂

yes

maximal, characteristic

#21422380

C₂

yes

maximal

#21421927

C₂

yes

maximal, characteristic

#21421924

C₂

yes

maximal

C₅₉×#1275

C₂

yes

maximal, characteristic

#16837455

C₂²

yes

—

C₂×#14273637

C₄

yes

—

C₂×C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₂×C₈×Dic₅₉

C₄

yes

—

#16837248

C₂²

yes

characteristic

#16837124

C₄

yes

complemented

C₈×C₅₉⋊C₈

C₄

yes

complemented

C₅₉×#431

C₂²

yes

—

C₂×#97×C₅₉

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₉

C₂×C₄

yes

—

C₄⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

—

#14273637

—

C₄×C₅₉⋊C₈

C₂×C₄

yes

—

C₄×C₅₉⋊C₈

C₂×C₄

yes

—

C₄×C₅₉⋊C₈

C₂×C₄

yes

—

C₈×Dic₅₉

C₈

yes

complemented

C₅₉⋊#95

—

C₅₉⋊#95

—

C₂²×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

C₂²⋊C₈×C₅₉

—

#97×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

C₂²×C₄×C₅₉

yes

C₂²×C₄×C₅₉

yes

C₂²×Dic₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₈

yes

—

C₄²×C₅₉

C₂²×C₄

yes

—

C₄²×C₅₉

#41

yes

—

C₄×Dic₅₉

—

C₄×Dic₅₉

C₂×C₈

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₄²

yes

characteristic

C₂×C₅₉⋊C₈

C₄²

yes

characteristic

C₂×C₅₉⋊C₈

C₄²

yes

—

C₂×C₅₉⋊C₈

C₂×C₈

yes

—

C₂×C₅₉⋊C₈

M₁₆

yes

—

C₂³×C₅₉

#117

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₉

#117

yes

—

C₂×C₄×C₅₉

#117

yes

characteristic

C₂×C₄×C₅₉

#117

yes

—

C₂×C₄×C₅₉

C₂²×C₈

yes

—

C₂×C₄×C₅₉

C₂×M₁₆

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

C₄×C₈

yes

—

C₅₉⋊C₈

C₄×C₈

yes

—

C₅₉⋊C₈

#97

yes

—

#8137

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#376

yes

characteristic

C₂²×C₅₉

#431

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×C₄×C₈

yes

characteristic

C₄×C₅₉

C₂×#97

yes

characteristic

C₄×C₅₉

#431

yes

—

Dic₅₉

—

—

—

—

D₅₉

yes

characteristic

#1274

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×D₅₉

yes

—

C₂×#97

—

C₂×#97

C₂×D₅₉

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#376

—

#322

—

C₈²

—

C₅₉

#8137

yes

characteristic, complemented, Sylow

C₂²×C₈

—

100

C₂²×C₈

—

101

C₂²×C₈

C₄×D₅₉

yes

—

102

C₂×C₄²

—

103

C₂×C₄²

C₂²×D₅₉

yes

characteristic

104

C₄⋊C₈

—

105

C₄⋊C₈

—

118

—

106

C₂²⋊C₈

—

107

C₂²⋊C₈

—

118

—

108

#97

—

109

#97

—

118

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

C₄×D₅₉

yes

—

115

#95

—

118

—

116

#95

—

118

—

117

C₂²×C₄

—

118

—

118

C₂²×C₄

C₂×C₄×D₅₉

yes

characteristic

119

C₂²×C₄

#11745687

yes

characteristic

120

C₂²×C₄

#11745689

yes

characteristic

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

118

—

130

C₂×C₈

C₈×D₅₉

yes

—

131

C₄²

—

132

C₄²

—

118

—

133

C₄²

C₂×C₄×D₅₉

yes

—

134

C₄²

#11745687

yes

—

135

C₂³

#14273685

yes

characteristic

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

118

—

139

C₂×C₄

—

236

—

140

C₂×C₄

—

236

—

141

C₂×C₄

C₄²×D₅₉

yes

characteristic

142

C₂×C₄

#14273683

yes

—

143

C₂×C₄

#14273683

yes

characteristic

144

C₂×C₄

#14273685

yes

—

145

C₂×C₄

C₂×C₈×D₅₉

yes

—

146

C₂×C₄

M₁₆×D₅₉

yes

—

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

—

236

—

153

C₂²

—

154

C₂²

—

155

C₂²

#16837247

yes

characteristic

156

C₂²

#16837454

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

236

—

160

C₄

C₄×C₈×D₅₉

yes

characteristic

161

C₄

#97×D₅₉

yes

characteristic

162

C₄

#16837452

yes

—

163

C₂

—

164

C₂

#21423840

yes

characteristic

165

C₂

#21423850

yes

characteristic

166

C₂

#21423852

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄²×D₅₉
commutator	C₄×C₅₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1275	C₂
socle	C₂²×C₅₉	#376
prime radical	C₂³×C₅₉	#117
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1275	D₅₉
hypocenter	C₅₉	#8137
LCS	G ▷ C₄×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ #1275

subgroup

G/H

center

C₂×C₄

C₄²×D₅₉

commutator

C₄×C₅₉

C₂×C₄×C₈

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1275

C₂

C₂²×C₅₉

C₂³×C₅₉

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₄×C₅₉ ▷ C₅₉

C₁ ◁ C₂×C₄ ◁ #1275

N	H ≅ G/N	type
11. #16837124	C₄	semidirect
12. C₈×C₅₉⋊C₈	C₄	semidirect
24. C₈×Dic₅₉	C₈	semidirect
98. C₅₉	#8137	semidirect

H ≅ G/N

type

11. #16837124

C₄

semidirect

12. C₈×C₅₉⋊C₈

C₄

semidirect

24. C₈×Dic₅₉

C₈

semidirect

98. C₅₉

#8137

semidirect

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

subgroup	index
#21423784	2
#21422380	2
#21421927	2
#21421924	2
C₅₉×#1275	2
#8137	59

subgroup

index

C₅₉×#1275

Subgroups — #28479129

Conjugacy classes of subgroups

Complements

Maximal subgroups