Group #28422744 — subgroups

Subgroups — C₅₉×#10635 #28422744

Order 15104.

294 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 98 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	C₅₉×#1960	C₂	1	1	yes	maximal, characteristic
2	C₅₉×#1838	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#1546	C₂	1	2	yes	maximal
4	C₅₉×#1286	C₂	1	2	yes	maximal
5	C₅₉×#1281	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
7	C₅₉×#431	C₂²	1	2	yes	—
8	C₅₉×#423	C₂²	1	2	yes	—
9	C₂×#103×C₅₉	—	2	2	no	—
10	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
11	C₅₉×#382	C₂²	1	1	yes	characteristic
12	C₅₉×#341	—	2	2	no	—
13	C₅₉×#332	—	2	4	no	—
14	C₂²×Q₈×C₅₉	—	2	1	no	—
15	C₂×M₁₆×C₅₉	—	2	2	no	—
16	C₂²×C₈×C₅₉	D₄	1	2	yes	—
17	C₄⋊Q₈×C₅₉	—	2	2	no	—
18	C₄⋊Q₈×C₅₉	—	2	1	no	—
19	C₄⋊Q₈×C₅₉	D₄	1	2	yes	—
20	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
21	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
22	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
23	C₄⋊C₈×C₅₉	—	2	2	no	—
24	C₄⋊C₈×C₅₉	—	2	2	no	—
25	C₄⋊C₈×C₅₉	—	2	2	no	—
26	#103×C₅₉	—	4	4	no	—
27	C₂²⋊C₈×C₅₉	—	2	2	no	—
28	#97×C₅₉	—	2	2	no	—
29	C₄×C₈×C₅₉	—	2	2	no	—
30	#95×C₅₉	—	2	2	no	—
31	C₂×Q₈×C₅₉	—	2	2	no	—
32	C₂×Q₈×C₅₉	—	4	1	no	—
33	C₂×Q₈×C₅₉	—	4	2	no	—
34	C₂×Q₈×C₅₉	—	4	2	no	—
35	C₂²×C₄×C₅₉	—	4	1	no	—
36	C₂²×C₄×C₅₉	—	4	1	no	—
37	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₉	#41	1	2	yes	—
39	M₁₆×C₅₉	—	4	2	no	—
40	M₁₆×C₅₉	—	4	2	no	—
41	C₂×C₈×C₅₉	—	2	2	no	—
42	C₂×C₈×C₅₉	—	2	2	no	—
43	C₂×C₈×C₅₉	—	2	2	no	—
44	C₂×C₈×C₅₉	—	4	2	no	—
45	C₂×C₈×C₅₉	—	4	2	no	—
46	C₄⋊C₄×C₅₉	—	2	4	no	—
47	C₄⋊C₄×C₅₉	—	4	2	no	—
48	C₄⋊C₄×C₅₉	—	4	2	no	—
49	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
50	C₄²×C₅₉	#41	1	1	yes	characteristic
51	C₄²×C₅₉	#41	1	2	yes	—
52	C₂³×C₅₉	#124	1	1	yes	characteristic
53	Q₈×C₅₉	—	4	4	no	—
54	Q₈×C₅₉	—	8	2	no	—
55	C₂×C₄×C₅₉	—	4	2	no	—
56	C₂×C₄×C₅₉	—	4	2	no	—
57	C₂×C₄×C₅₉	—	8	1	no	—
58	C₂×C₄×C₅₉	—	8	1	no	—
59	C₂×C₄×C₅₉	—	8	2	no	—
60	C₂×C₄×C₅₉	—	8	1	no	—
61	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
62	C₂×C₄×C₅₉	#123	1	2	yes	—
63	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
64	C₂×C₄×C₅₉	#126	1	2	yes	—
65	C₂×C₄×C₅₉	#135	1	2	yes	—
66	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
67	C₈×C₅₉	—	4	2	no	—
68	C₈×C₅₉	—	8	2	no	—
69	C₈×C₅₉	—	8	2	no	—
70	#10635	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂²×C₅₉	—	2	2	no	—
72	C₂²×C₅₉	#397	1	1	yes	characteristic
73	C₂²×C₅₉	#488	1	1	yes	characteristic
74	C₂²×C₅₉	#490	1	1	yes	characteristic
75	C₄×C₅₉	—	2	2	no	—
76	C₄×C₅₉	—	2	2	no	—
77	C₄×C₅₉	—	8	2	no	—
78	C₄×C₅₉	—	16	1	no	—
79	C₄×C₅₉	#463	1	2	yes	—
80	C₄×C₅₉	#482	1	2	yes	—
81	#1960	C₂×C₅₉	1	1	yes	characteristic
82	#1838	C₂×C₅₉	1	1	yes	characteristic
83	#1546	C₂×C₅₉	1	2	yes	—
84	#1286	C₂×C₅₉	1	2	yes	—
85	#1281	C₂×C₅₉	1	1	yes	characteristic
86	C₂×C₅₉	—	2	1	no	—
87	C₂×C₅₉	—	2	1	no	—
88	C₂×C₅₉	#2073	1	1	yes	characteristic
89	C₂×C₅₉	#2082	1	2	yes	—
90	C₂×C₄⋊Q₈	C₂²×C₅₉	1	1	yes	characteristic
91	#431	C₂²×C₅₉	1	2	yes	—
92	#423	C₂²×C₅₉	1	2	yes	—
93	C₂×#103	—	2	2	no	—
94	C₂×#97	C₂²×C₅₉	1	1	yes	characteristic
95	#382	C₂²×C₅₉	1	1	yes	characteristic
96	#341	—	2	2	no	—
97	#332	—	2	4	no	—
98	C₅₉	#10635	1	1	yes	characteristic, direct product factor, Sylow
99	C₂²×Q₈	—	2	1	no	—
100	C₂×M₁₆	—	2	2	no	—
101	C₂²×C₈	D₄×C₅₉	1	2	yes	—
102	C₄⋊Q₈	—	2	2	no	—
103	C₄⋊Q₈	—	2	1	no	—
104	C₄⋊Q₈	D₄×C₅₉	1	2	yes	—
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄⋊C₄	—	2	1	no	—
107	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
108	C₄⋊C₈	—	2	2	no	—
109	C₄⋊C₈	—	2	2	no	—
110	C₄⋊C₈	—	2	2	no	—
111	#103	—	4	4	no	—
112	C₂²⋊C₈	—	2	2	no	—
113	#97	—	2	2	no	—
114	C₄×C₈	—	2	2	no	—
115	#95	—	2	2	no	—
116	C₂×Q₈	—	2	2	no	—
117	C₂×Q₈	—	4	1	no	—
118	C₂×Q₈	—	4	2	no	—
119	C₂×Q₈	—	4	2	no	—
120	C₂²×C₄	—	4	1	no	—
121	C₂²×C₄	—	4	1	no	—
122	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
123	C₂²×C₄	#41×C₅₉	1	2	yes	—
124	M₁₆	—	4	2	no	—
125	M₁₆	—	4	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	4	2	no	—
131	C₄⋊C₄	—	2	4	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
135	C₄²	#41×C₅₉	1	1	yes	characteristic
136	C₄²	#41×C₅₉	1	2	yes	—
137	C₂³	#124×C₅₉	1	1	yes	characteristic
138	Q₈	—	4	4	no	—
139	Q₈	—	8	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	8	1	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	—	8	1	no	—
146	C₂×C₄	#121×C₅₉	1	1	yes	characteristic
147	C₂×C₄	#123×C₅₉	1	2	yes	—
148	C₂×C₄	#123×C₅₉	1	1	yes	characteristic
149	C₂×C₄	#126×C₅₉	1	2	yes	—
150	C₂×C₄	#135×C₅₉	1	2	yes	—
151	C₂×C₄	AGL(1,ℤ/8)×C₅₉	1	2	yes	—
152	C₈	—	4	2	no	—
153	C₈	—	8	2	no	—
154	C₈	—	8	2	no	—
155	C₂²	—	2	2	no	—
156	C₂²	C₅₉×#397	1	1	yes	characteristic
157	C₂²	C₅₉×#488	1	1	yes	characteristic
158	C₂²	C₅₉×#490	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	8	2	no	—
162	C₄	—	16	1	no	—
163	C₄	C₅₉×#463	1	2	yes	—
164	C₄	C₅₉×#482	1	2	yes	—
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	C₅₉×#2073	1	1	yes	characteristic
168	C₂	C₅₉×#2082	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₉×#1960

C₂

yes

maximal, characteristic

C₅₉×#1838

C₂

yes

maximal, characteristic

C₅₉×#1546

C₂

yes

maximal

C₅₉×#1286

C₂

yes

maximal

C₅₉×#1281

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

C₅₉×#431

C₂²

yes

—

C₅₉×#423

C₂²

yes

—

C₂×#103×C₅₉

—

C₂×#97×C₅₉

C₂²

yes

characteristic

C₅₉×#382

C₂²

yes

characteristic

C₅₉×#341

—

C₅₉×#332

—

C₂²×Q₈×C₅₉

—

C₂×M₁₆×C₅₉

—

C₂²×C₈×C₅₉

D₄

yes

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

D₄

yes

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄⋊C₈×C₅₉

—

C₄⋊C₈×C₅₉

—

#103×C₅₉

—

C₂²⋊C₈×C₅₉

—

#97×C₅₉

—

C₄×C₈×C₅₉

—

#95×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

C₄²×C₅₉

yes

C₄²×C₅₉

yes

—

C₂³×C₅₉

#124

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#121

yes

characteristic

C₂×C₄×C₅₉

#123

yes

—

C₂×C₄×C₅₉

#123

yes

characteristic

C₂×C₄×C₅₉

#126

yes

—

C₂×C₄×C₅₉

#135

yes

—

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

#10635

C₅₉

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#463

yes

—

C₄×C₅₉

#482

yes

—

#1960

C₂×C₅₉

yes

characteristic

#1838

C₂×C₅₉

yes

characteristic

#1546

C₂×C₅₉

yes

—

#1286

C₂×C₅₉

yes

—

#1281

C₂×C₅₉

yes

characteristic

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#2073

yes

characteristic

C₂×C₅₉

#2082

yes

—

C₂×C₄⋊Q₈

C₂²×C₅₉

yes

characteristic

#431

C₂²×C₅₉

yes

—

#423

C₂²×C₅₉

yes

—

C₂×#103

—

C₂×#97

C₂²×C₅₉

yes

characteristic

#382

C₂²×C₅₉

yes

characteristic

#341

—

#332

—

C₅₉

#10635

yes

characteristic, direct product factor, Sylow

C₂²×Q₈

—

100

C₂×M₁₆

—

101

C₂²×C₈

D₄×C₅₉

yes

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

D₄×C₅₉

yes

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄²

C₂³×C₅₉

yes

characteristic

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

C₄⋊C₈

—

111

#103

—

112

C₂²⋊C₈

—

113

#97

—

114

C₄×C₈

—

115

#95

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

C₂×D₄×C₅₉

yes

characteristic

123

C₂²×C₄

#41×C₅₉

yes

—

124

M₁₆

—

125

M₁₆

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄²

C₂×D₄×C₅₉

yes

characteristic

135

C₄²

#41×C₅₉

yes

characteristic

136

C₄²

#41×C₅₉

yes

—

137

C₂³

#124×C₅₉

yes

characteristic

138

Q₈

—

139

Q₈

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

#121×C₅₉

yes

characteristic

147

C₂×C₄

#123×C₅₉

yes

—

148

C₂×C₄

#123×C₅₉

yes

characteristic

149

C₂×C₄

#126×C₅₉

yes

—

150

C₂×C₄

#135×C₅₉

yes

—

151

C₂×C₄

AGL(1,ℤ/8)×C₅₉

yes

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₂²

—

156

C₂²

C₅₉×#397

yes

characteristic

157

C₂²

C₅₉×#488

yes

characteristic

158

C₂²

C₅₉×#490

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

C₅₉×#463

yes

—

164

C₄

C₅₉×#482

yes

—

165

C₂

—

166

C₂

—

167

C₂

C₅₉×#2073

yes

characteristic

168

C₂

C₅₉×#2082

yes

—

	subgroup	G/H
center, socle	C₂²×C₅₉	#397
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₉
prime radical	C₂³×C₅₉	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₂×C₄²×C₅₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₉

#397

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₅₉

prime radical

C₂³×C₅₉

#124

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₅₉ ◁ C₂×C₄²×C₅₉ ◁ G

N	H ≅ G/N	type
70. #10635	C₅₉	direct (N × H)
98. C₅₉	#10635	direct (N × H)

H ≅ G/N

type

70. #10635

C₅₉

direct (N × H)

98. C₅₉

#10635

direct (N × H)

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

subgroup	index
C₅₉×#1960	2
C₅₉×#1838	2
C₅₉×#1546	2
C₅₉×#1286	2
C₅₉×#1281	2
#10635	59

subgroup

index

C₅₉×#1960

C₅₉×#1838

C₅₉×#1546

C₅₉×#1286

C₅₉×#1281

#10635

Subgroups — C₅₉×#10635 #28422744

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C59×#10635 #28422744

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#10635 #28422744