Group #28502873 — subgroups

Subgroups — #28502873

Order 15104.

230 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 71 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	#21426143	C₂	1	2	yes	maximal
2	C₈×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic
3	#21422166	C₂	1	1	yes	maximal, characteristic
4	#21422145	C₂	1	2	yes	maximal
5	C₅₉×#1713	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273673	—	2	2	no	—
7	C₈×Dic₁₁₈	—	2	1	no	—
8	C₄×Dic₂₃₆	C₂²	1	1	yes	characteristic
9	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
10	#16837206	—	2	2	no	—
11	#16837155	C₂²	1	1	yes	characteristic
12	#16837134	—	2	1	no	—
13	C₅₉×#500	—	2	1	no	—
14	C₅₉×#500	—	2	1	no	—
15	C₅₉×#494	C₂²	1	2	yes	—
16	C₈²×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₃₆	D₄	1	1	yes	characteristic
18	C₄×Dic₁₁₈	—	2	1	no	—
19	#14273673	—	4	4	no	—
20	C₂×C₅₉⋊C₁₆	—	2	2	no	—
21	#14273661	D₄	1	1	yes	characteristic
22	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
23	Dic₅₉⋊C₈	—	4	1	no	—
24	#14273652	—	2	1	no	—
25	C₈×Dic₅₉	—	4	1	no	—
26	C₂×Q₁₆×C₅₉	—	2	2	no	—
27	C₂×Q₁₆×C₅₉	—	2	2	no	—
28	C₄⋊Q₈×C₅₉	—	2	2	no	—
29	#107×C₅₉	—	4	2	no	—
30	#107×C₅₉	—	4	1	no	—
31	#107×C₅₉	—	4	1	no	—
32	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
33	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
34	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
35	C₂×Q₈×C₅₉	—	4	2	no	—
36	C₂×Dic₁₁₈	—	2	1	no	—
37	Q₁₆×C₅₉	—	4	2	no	—
38	Q₁₆×C₅₉	—	4	2	no	—
39	Q₁₆×C₅₉	—	8	2	no	—
40	C₂×C₈×C₅₉	—	2	1	no	—
41	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
42	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
43	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
45	C₄⋊C₄×C₅₉	—	4	2	no	—
46	C₄⋊C₄×C₅₉	—	4	2	no	—
47	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₄⋊Dic₅₉	—	2	1	no	—
49	Dic₅₉⋊C₄	—	4	1	no	—
50	C₄×Dic₅₉	—	4	1	no	—
51	C₂×C₅₉⋊C₈	—	4	1	no	—
52	Dic₂₃₆	—	2	1	no	—
53	Dic₂₃₆	D₈	1	2	yes	—
54	C₅₉⋊C₁₆	—	4	2	no	—
55	Q₈×C₅₉	—	8	2	no	—
56	Q₈×C₅₉	—	8	2	no	—
57	C₂×C₄×C₅₉	—	8	2	no	—
58	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
59	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂×Dic₅₉	—	4	1	no	—
62	C₂×Dic₅₉	—	4	1	no	—
63	Dic₁₁₈	—	2	2	no	—
64	Dic₁₁₈	—	4	1	no	—
65	C₈×C₅₉	—	2	2	no	—
66	C₈×C₅₉	—	2	1	no	—
67	C₈×C₅₉	—	2	1	no	—
68	C₈×C₅₉	Q₃₂	1	2	yes	complemented
69	C₈×C₅₉	C₂×D₈	1	1	yes	characteristic
70	C₈×C₅₉	#135	1	1	yes	characteristic
71	C₅₉⋊C₈	—	8	1	no	—
72	#12427	—	59	1	no	maximal, Sylow
73	C₂²×C₅₉	#459	1	1	yes	characteristic
74	C₄×C₅₉	—	2	1	no	—
75	C₄×C₅₉	—	16	2	no	—
76	C₄×C₅₉	#466	1	1	yes	characteristic
77	C₄×C₅₉	#472	1	1	yes	characteristic
78	C₄×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
79	C₄×C₅₉	#510	1	1	yes	characteristic
80	Dic₅₉	—	4	2	no	—
81	Dic₅₉	—	8	1	no	—
82	#2209	—	59	2	no	—
83	#1713	D₅₉	1	1	yes	characteristic
84	C₈×Q₁₆	—	59	1	no	—
85	#1370	—	59	1	no	—
86	#1362	—	59	2	no	—
87	C₂×C₅₉	#1667	1	1	yes	characteristic
88	C₂×C₅₉	#2209	1	1	yes	characteristic
89	C₂×C₅₉	#2214	1	1	yes	characteristic
90	C₂×Q₃₂	—	118	2	no	—
91	#500	—	2	1	no	—
92	#500	—	2	1	no	—
93	#494	C₂×D₅₉	1	2	yes	—
94	C₈×Q₈	—	118	1	no	—
95	C₄×Q₁₆	—	59	1	no	—
96	C₄⋊C₁₆	—	59	2	no	—
97	#358	—	118	2	no	—
98	#335	—	59	1	no	—
99	Q₈⋊C₈	—	118	1	no	—
100	C₈²	C₂×D₅₉	1	1	yes	characteristic
101	C₅₉	#12427	1	1	yes	characteristic, complemented, Sylow
102	C₂×Q₁₆	—	2	2	no	—
103	C₂×Q₁₆	—	2	2	no	—
104	C₂×Q₁₆	—	59	1	no	—
105	C₄⋊Q₈	—	2	2	no	—
106	C₄×Q₈	—	118	1	no	—
107	Q₃₂	—	236	4	no	—
108	C₂×C₁₆	—	118	2	no	—
109	#107	—	4	2	no	—
110	#107	—	4	1	no	—
111	#107	—	4	1	no	—
112	#107	—	59	1	no	—
113	C₄⋊C₈	—	118	1	no	—
114	C₄⋊C₈	—	236	1	no	—
115	#103	—	118	1	no	—
116	C₄×C₈	—	236	1	no	—
117	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
118	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
119	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
120	C₂×Q₈	—	4	2	no	—
121	C₂×Q₈	—	118	1	no	—
122	Q₁₆	—	4	2	no	—
123	Q₁₆	—	4	2	no	—
124	Q₁₆	—	8	2	no	—
125	Q₁₆	—	59	2	no	—
126	Q₁₆	—	118	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	236	1	no	—
129	C₂×C₈	#11745669	1	1	yes	characteristic
130	C₂×C₈	#11745669	1	1	yes	characteristic
131	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
132	C₂×C₈	#11745687	1	1	yes	characteristic
133	C₄⋊C₄	—	4	2	no	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	118	1	no	—
136	C₄⋊C₄	—	236	1	no	—
137	C₄²	—	236	1	no	—
138	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
139	C₁₆	—	236	2	no	—
140	Q₈	—	8	2	no	—
141	Q₈	—	8	2	no	—
142	Q₈	—	118	2	no	—
143	Q₈	—	236	1	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	—	236	1	no	—
146	C₂×C₄	—	236	1	no	—
147	C₂×C₄	#14273698	1	1	yes	characteristic
148	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
149	C₂×C₄	#14273753	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	472	1	no	—
154	C₈	#14273673	1	2	yes	complemented
155	C₈	D₈×D₅₉	1	1	yes	characteristic
156	C₈	#14273746	1	1	yes	characteristic
157	C₂²	#16837594	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	16	2	no	—
160	C₄	—	472	1	no	—
161	C₄	—	236	2	no	—
162	C₄	#16837635	1	1	yes	characteristic
163	C₄	#16837669	1	1	yes	characteristic
164	C₄	C₂×#14273673	1	1	yes	characteristic
165	C₄	#16837843	1	1	yes	characteristic
166	C₂	#21423448	1	1	yes	characteristic
167	C₂	#21426143	1	1	yes	characteristic
168	C₂	#21426174	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21426143

C₂

yes

maximal

C₈×Dic₂₃₆

C₂

yes

maximal, characteristic

#21422166

C₂

yes

maximal, characteristic

#21422145

C₂

yes

maximal

C₅₉×#1713

C₂

yes

maximal, characteristic

C₂×#14273673

—

C₈×Dic₁₁₈

—

C₄×Dic₂₃₆

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₁₆)

C₂²

yes

—

#16837206

—

#16837155

C₂²

yes

characteristic

#16837134

—

C₅₉×#500

—

C₅₉×#500

—

C₅₉×#494

C₂²

yes

—

C₈²×C₅₉

C₂²

yes

characteristic

C₂×Dic₂₃₆

D₄

yes

characteristic

C₄×Dic₁₁₈

—

#14273673

—

C₂×C₅₉⋊C₁₆

—

#14273661

D₄

yes

characteristic

C₄⋊(C₅₉⋊C₈)

—

Dic₅₉⋊C₈

—

#14273652

—

C₈×Dic₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₄⋊Q₈×C₅₉

—

#107×C₅₉

—

#107×C₅₉

—

#107×C₅₉

—

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Dic₁₁₈

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₈

yes

characteristic

C₂×C₈×C₅₉

D₈

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

#41

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

Dic₂₃₆

—

Dic₂₃₆

D₈

yes

—

C₅₉⋊C₁₆

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#121

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

Q₃₂

yes

complemented

C₈×C₅₉

C₂×D₈

yes

characteristic

C₈×C₅₉

#135

yes

characteristic

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

C₂×Q₃₂

yes

characteristic

C₄×C₅₉

#510

yes

characteristic

Dic₅₉

—

Dic₅₉

—

#2209

—

#1713

D₅₉

yes

characteristic

C₈×Q₁₆

—

#1370

—

#1362

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×Q₃₂

—

118

—

#500

—

#500

—

#494

C₂×D₅₉

yes

—

C₈×Q₈

—

118

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

#358

—

118

—

#335

—

Q₈⋊C₈

—

118

—

100

C₈²

C₂×D₅₉

yes

characteristic

101

C₅₉

#12427

yes

characteristic, complemented, Sylow

102

C₂×Q₁₆

—

103

C₂×Q₁₆

—

104

C₂×Q₁₆

—

105

C₄⋊Q₈

—

106

C₄×Q₈

—

118

—

107

Q₃₂

—

236

—

108

C₂×C₁₆

—

118

—

109

#107

—

110

#107

—

111

#107

—

112

#107

—

113

C₄⋊C₈

—

118

—

114

C₄⋊C₈

—

236

—

115

#103

—

118

—

116

C₄×C₈

—

236

—

117

C₄×C₈

C₂²⋊D₅₉

yes

characteristic

118

C₄×C₈

C₂²⋊D₅₉

yes

characteristic

119

C₄×C₈

C₂²×D₅₉

yes

characteristic

120

C₂×Q₈

—

121

C₂×Q₈

—

118

—

122

Q₁₆

—

123

Q₁₆

—

124

Q₁₆

—

125

Q₁₆

—

126

Q₁₆

—

118

—

127

C₂×C₈

—

128

C₂×C₈

—

236

—

129

C₂×C₈

#11745669

yes

characteristic

130

C₂×C₈

#11745669

yes

characteristic

131

C₂×C₈

D₄×D₅₉

yes

characteristic

132

C₂×C₈

#11745687

yes

characteristic

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

118

—

136

C₄⋊C₄

—

236

—

137

C₄²

—

236

—

138

C₄²

C₂×C₂²⋊D₅₉

yes

characteristic

139

C₁₆

—

236

—

140

Q₈

—

141

Q₈

—

142

Q₈

—

118

—

143

Q₈

—

236

—

144

C₂×C₄

—

145

C₂×C₄

—

236

—

146

C₂×C₄

—

236

—

147

C₂×C₄

#14273698

yes

characteristic

148

C₂×C₄

C₂×#11745669

yes

characteristic

149

C₂×C₄

#14273753

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

472

—

154

C₈

#14273673

yes

complemented

155

C₈

D₈×D₅₉

yes

characteristic

156

C₈

#14273746

yes

characteristic

157

C₂²

#16837594

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

472

—

161

C₄

—

236

—

162

C₄

#16837635

yes

characteristic

163

C₄

#16837669

yes

characteristic

164

C₄

C₂×#14273673

yes

characteristic

165

C₄

#16837843

yes

characteristic

166

C₂

#21423448

yes

characteristic

167

C₂

#21426143

yes

characteristic

168

C₂

#21426174

yes

characteristic

	subgroup	G/H
center	C₂²	#16837594
commutator, cosocle	C₄×C₈×C₅₉	C₂³
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#1713	C₂
socle, prime radical	C₂²×C₅₉	#459
hypercenter	#1713	D₅₉
hypocenter	C₅₉	#12427
LCS	G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#16837594

commutator, cosocle

C₄×C₈×C₅₉

C₂³

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#1713

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

N	H ≅ G/N	type
68. C₈×C₅₉	Q₃₂	semidirect
101. C₅₉	#12427	semidirect
154. C₈	#14273673	semidirect

H ≅ G/N

type

68. C₈×C₅₉

Q₃₂

semidirect

101. C₅₉

#12427

semidirect

154. C₈

#14273673

semidirect

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

subgroup	index
#21426143	2
C₈×Dic₂₃₆	2
#21422166	2
#21422145	2
C₅₉×#1713	2
#12427	59

subgroup

index

C₈×Dic₂₃₆

C₅₉×#1713

Subgroups — #28502873

Conjugacy classes of subgroups

Complements

Maximal subgroups