Group #28493499 — subgroups

Subgroups — #28493499

Order 15104.

294 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 87 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	#21424237	C₂	1	1	yes	maximal, characteristic
2	#21422768	C₂	1	2	yes	maximal
3	#21421954	C₂	1	2	yes	maximal
4	#21421941	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#1960	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₂²	1	2	yes	—
7	#16837426	C₂²	1	2	yes	—
8	#16837167	—	2	2	no	—
9	#16837148	—	2	4	no	—
10	C₂×C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
11	C₂×#103×C₅₉	—	2	2	no	—
12	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
13	C₅₉×#382	C₂²	1	1	yes	characteristic
14	C₂×#11745664	—	2	2	no	—
15	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
16	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
17	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
18	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
19	C₄×C₅₉⋊C₈	—	2	2	no	—
20	C₂²×Q₈×C₅₉	—	2	1	no	—
21	C₂²×C₈×C₅₉	D₄	1	2	yes	—
22	C₄⋊Q₈×C₅₉	—	2	2	no	—
23	C₄⋊Q₈×C₅₉	—	2	1	no	—
24	C₄⋊Q₈×C₅₉	D₄	1	2	yes	—
25	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
26	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
27	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
28	#103×C₅₉	—	4	4	no	—
29	#97×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	C₂×C₈×C₅₉	—	2	2	no	—
40	C₂×C₈×C₅₉	—	2	2	no	—
41	C₂×C₈×C₅₉	—	2	2	no	—
42	C₄⋊C₄×C₅₉	—	2	4	no	—
43	C₄⋊C₄×C₅₉	—	4	2	no	—
44	C₄⋊C₄×C₅₉	—	4	2	no	—
45	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₄²×C₅₉	#41	1	1	yes	characteristic
47	C₄²×C₅₉	#41	1	2	yes	—
48	#11745664	—	4	2	no	—
49	#11745664	—	4	2	no	—
50	C₂×C₅₉⋊C₈	—	4	2	no	—
51	C₂×C₅₉⋊C₈	—	4	2	no	—
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	1	yes	characteristic
63	C₂×C₄×C₅₉	#123	1	2	yes	—
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	C₅₉⋊C₈	—	8	2	no	—
69	C₅₉⋊C₈	—	8	2	no	—
70	#10635	—	59	1	no	maximal, 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	D₅₉	1	1	yes	characteristic
82	#1838	—	59	1	no	—
83	#1546	—	59	2	no	—
84	#1286	—	59	2	no	—
85	#1281	—	59	1	no	—
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	C₂×D₅₉	1	1	yes	characteristic
91	#431	—	59	2	no	—
92	#423	—	59	2	no	—
93	C₂×#103	—	2	2	no	—
94	C₂×#97	C₂×D₅₉	1	1	yes	characteristic
95	#382	C₂×D₅₉	1	1	yes	characteristic
96	#341	—	118	2	no	—
97	#332	—	118	4	no	—
98	C₅₉	#10635	1	1	yes	characteristic, complemented, Sylow
99	C₂²×Q₈	—	2	1	no	—
100	C₂×M₁₆	—	118	2	no	—
101	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
102	C₄⋊Q₈	—	2	1	no	—
103	C₄⋊Q₈	—	2	2	no	—
104	C₄⋊Q₈	D₂₃₆	1	2	yes	—
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄⋊C₄	—	2	1	no	—
107	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
108	C₄⋊C₈	—	118	2	no	—
109	C₄⋊C₈	—	118	2	no	—
110	C₄⋊C₈	—	118	2	no	—
111	#103	—	4	4	no	—
112	C₂²⋊C₈	—	118	2	no	—
113	#97	—	2	2	no	—
114	C₄×C₈	—	118	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	#11745687	1	2	yes	—
123	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
124	M₁₆	—	236	2	no	—
125	M₁₆	—	236	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	236	2	no	—
130	C₂×C₈	—	236	2	no	—
131	C₄⋊C₄	—	2	4	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
135	C₄²	#11745685	1	1	yes	characteristic
136	C₄²	#11745687	1	2	yes	—
137	C₂³	#14273714	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	#14273701	1	1	yes	characteristic
147	C₂×C₄	#14273708	1	1	yes	characteristic
148	C₂×C₄	#14273711	1	2	yes	—
149	C₂×C₄	#14273720	1	2	yes	—
150	C₂×C₄	#14273747	1	2	yes	—
151	C₂×C₄	#14273753	1	2	yes	—
152	C₈	—	4	2	no	—
153	C₈	—	472	2	no	—
154	C₈	—	472	2	no	—
155	C₂²	—	2	2	no	—
156	C₂²	#16837327	1	1	yes	characteristic
157	C₂²	#16837756	1	1	yes	characteristic
158	C₂²	#16837766	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	#16837617	1	2	yes	—
164	C₄	#16837723	1	2	yes	—
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#21425509	1	1	yes	characteristic
168	C₂	#21425560	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21424237

C₂

yes

maximal, characteristic

#21422768

C₂

yes

maximal

#21421954

C₂

yes

maximal

#21421941

C₂

yes

maximal, characteristic

C₅₉×#1960

C₂

yes

maximal, characteristic

#16837455

C₂²

yes

—

#16837426

C₂²

yes

—

#16837167

—

#16837148

—

C₂×C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

C₂×#103×C₅₉

—

C₂×#97×C₅₉

C₂²

yes

characteristic

C₅₉×#382

C₂²

yes

characteristic

C₂×#11745664

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

—

C₄×C₅₉⋊C₈

—

C₂²×Q₈×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

#103×C₅₉

—

#97×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

—

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

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

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

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

—

C₂×C₄×C₅₉

#126

yes

—

C₂×C₄×C₅₉

#135

yes

—

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#10635

—

maximal, 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

D₅₉

yes

—

—

—

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#2073

yes

characteristic

C₂×C₅₉

#2082

yes

—

C₂×C₄⋊Q₈

C₂×D₅₉

yes

—

—

—

C₂×#97

C₂×D₅₉

yes

characteristic

#382

C₂×D₅₉

yes

characteristic

#341

—

118

—

#332

—

118

—

C₅₉

#10635

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

100

C₂×M₁₆

—

118

—

101

C₂²×C₈

C₂²⋊D₅₉

yes

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

D₂₃₆

yes

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄²

C₂²×D₅₉

yes

characteristic

108

C₄⋊C₈

—

118

—

109

C₄⋊C₈

—

118

—

110

C₄⋊C₈

—

118

—

111

#103

—

112

C₂²⋊C₈

—

118

—

113

#97

—

114

C₄×C₈

—

118

—

115

#95

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

#11745687

yes

—

123

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

124

M₁₆

—

236

—

125

M₁₆

—

236

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

236

—

130

C₂×C₈

—

236

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄²

C₂×D₂₃₆

yes

characteristic

135

C₄²

#11745685

yes

characteristic

136

C₄²

#11745687

yes

—

137

C₂³

#14273714

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

#14273701

yes

characteristic

147

C₂×C₄

#14273708

yes

characteristic

148

C₂×C₄

#14273711

yes

—

149

C₂×C₄

#14273720

yes

—

150

C₂×C₄

#14273747

yes

—

151

C₂×C₄

#14273753

yes

—

152

C₈

—

153

C₈

—

472

—

154

C₈

—

472

—

155

C₂²

—

156

C₂²

#16837327

yes

characteristic

157

C₂²

#16837756

yes

characteristic

158

C₂²

#16837766

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#16837617

yes

—

164

C₄

#16837723

yes

—

165

C₂

—

166

C₂

—

167

C₂

#21425509

yes

characteristic

168

C₂

#21425560

yes

—

	subgroup	G/H
center	C₂²	#16837327
commutator, cosocle	C₂×C₄²×C₅₉	C₂³
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1960	C₂
socle	C₂²×C₅₉	#397
prime radical	C₂³×C₅₉	#124
hypercenter	#1960	D₅₉
hypocenter	C₅₉	#10635
LCS	G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1960

subgroup

G/H

center

C₂²

#16837327

commutator, cosocle

C₂×C₄²×C₅₉

C₂³

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1960

C₂

C₂²×C₅₉

C₂³×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1960

N	H ≅ G/N	type
98. C₅₉	#10635	semidirect

H ≅ G/N

type

98. C₅₉

#10635

semidirect

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

subgroup	index
#21424237	2
#21422768	2
#21421954	2
#21421941	2
C₅₉×#1960	2
#10635	59

subgroup

index

C₅₉×#1960

Subgroups — #28493499

Conjugacy classes of subgroups

Complements

Maximal subgroups