Group #48008105 — subgroups

Subgroups — C₁₃₉×#13509 #48008105

Order 35584.

268 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 114 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	#2159×C₁₃₉	C₂	1	2	yes	maximal
2	#1945×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#1846×C₁₃₉	C₂	1	1	yes	maximal, characteristic
4	#1387×C₁₃₉	C₂	1	2	yes	maximal
5	#1376×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×M₃₂×C₁₃₉	C₂²	1	2	yes	—
7	C₂×#108×C₁₃₉	C₂²	1	1	yes	characteristic
8	#427×C₁₃₉	C₂²	1	1	yes	characteristic
9	C₂×#107×C₁₃₉	C₂²	1	2	yes	—
10	C₂×#106×C₁₃₉	—	2	1	no	—
11	#424×C₁₃₉	—	2	1	no	—
12	C₂×C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
13	#382×C₁₃₉	—	2	1	no	—
14	#367×C₁₃₉	—	2	2	no	—
15	#366×C₁₃₉	—	2	2	no	—
16	#339×C₁₃₉	—	2	1	no	—
17	C₂×M₁₆×C₁₃₉	—	2	1	no	—
18	C₂²×C₈×C₁₃₉	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
20	#117×C₁₃₉	—	2	1	no	—
21	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
22	C₂×C₄²×C₁₃₉	D₄	1	1	yes	characteristic
23	M₃₂×C₁₃₉	—	2	2	no	—
24	M₃₂×C₁₃₉	Q₈	1	4	yes	—
25	C₂×C₁₆×C₁₃₉	—	2	2	no	—
26	#108×C₁₃₉	—	2	2	no	—
27	#107×C₁₃₉	—	2	2	no	—
28	#107×C₁₃₉	—	2	2	no	—
29	#107×C₁₃₉	D₄	1	2	yes	—
30	#106×C₁₃₉	—	2	1	no	—
31	#106×C₁₃₉	—	4	2	no	—
32	C₄⋊C₈×C₁₃₉	—	4	2	no	—
33	C₄×C₈×C₁₃₉	—	2	2	no	—
34	#95×C₁₃₉	—	4	2	no	—
35	C₂²×C₄×C₁₃₉	—	2	1	no	—
36	C₂²×C₄×C₁₃₉	—	4	2	no	—
37	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
38	M₁₆×C₁₃₉	—	4	1	no	—
39	M₁₆×C₁₃₉	—	4	1	no	—
40	C₂×C₈×C₁₃₉	—	2	1	no	—
41	C₂×C₈×C₁₃₉	—	2	1	no	—
42	C₂×C₈×C₁₃₉	—	4	1	no	—
43	C₂×C₈×C₁₃₉	D₈	1	2	yes	—
44	C₂×C₈×C₁₃₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₁₃₉	C₂×Q₈	1	2	yes	—
46	C₂×C₈×C₁₃₉	#41	1	2	yes	—
47	C₂×C₈×C₁₃₉	#41	1	1	yes	characteristic
48	C₄⋊C₄×C₁₃₉	—	2	2	no	—
49	C₄⋊C₄×C₁₃₉	—	4	2	no	—
50	C₄⋊C₄×C₁₃₉	—	4	2	no	—
51	C₂²⋊C₄×C₁₃₉	—	4	1	no	—
52	C₄²×C₁₃₉	—	2	1	no	—
53	C₄²×C₁₃₉	—	4	1	no	—
54	C₄²×C₁₃₉	Q₁₆	1	2	yes	—
55	C₁₆×C₁₃₉	—	2	4	no	—
56	C₂³×C₁₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
57	C₂×C₄×C₁₃₉	—	2	2	no	—
58	C₂×C₄×C₁₃₉	—	4	2	no	—
59	C₂×C₄×C₁₃₉	—	4	2	no	—
60	C₂×C₄×C₁₃₉	—	8	4	no	—
61	C₂×C₄×C₁₃₉	—	8	2	no	—
62	C₂×C₄×C₁₃₉	#121	1	1	yes	characteristic
63	C₂×C₄×C₁₃₉	#123	1	1	yes	characteristic
64	C₂×C₄×C₁₃₉	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₁₃₉	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₁₃₉	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×C₄×C₁₃₉	#137	1	1	yes	characteristic
68	C₈×C₁₃₉	—	2	2	no	—
69	C₈×C₁₃₉	—	8	1	no	—
70	C₈×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
71	C₈×C₁₃₉	#125	1	2	yes	—
72	C₂²×C₁₃₉	—	2	2	no	—
73	C₂²×C₁₃₉	#459	1	1	yes	characteristic
74	C₂²×C₁₃₉	#484	1	1	yes	characteristic
75	C₂²×C₁₃₉	#501	1	1	yes	characteristic
76	C₄×C₁₃₉	—	4	2	no	—
77	C₄×C₁₃₉	—	8	2	no	—
78	C₄×C₁₃₉	—	16	2	no	—
79	C₄×C₁₃₉	#398	1	1	yes	characteristic
80	C₄×C₁₃₉	#462	1	1	yes	characteristic
81	C₄×C₁₃₉	#480	1	1	yes	characteristic
82	C₄×C₁₃₉	#500	1	1	yes	characteristic
83	C₂×C₁₃₉	—	2	1	no	—
84	C₂×C₁₃₉	—	2	1	no	—
85	C₂×C₁₃₉	#2083	1	1	yes	characteristic
86	C₂×C₁₃₉	#2212	1	2	yes	—
87	#13509	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₁₃₉	#13509	1	1	yes	characteristic, direct product factor, Sylow
89	#2159	C₂×C₁₃₉	1	2	yes	—
90	#1945	C₂×C₁₃₉	1	1	yes	characteristic
91	#1846	C₂×C₁₃₉	1	1	yes	characteristic
92	#1387	C₂×C₁₃₉	1	2	yes	—
93	#1376	C₂×C₁₃₉	1	1	yes	characteristic
94	C₂×M₃₂	C₂²×C₁₃₉	1	2	yes	—
95	C₂×#108	C₂²×C₁₃₉	1	1	yes	characteristic
96	#427	C₂²×C₁₃₉	1	1	yes	characteristic
97	C₂×#107	C₂²×C₁₃₉	1	2	yes	—
98	C₂×#106	—	2	1	no	—
99	#424	—	2	1	no	—
100	C₂×C₄×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
101	#382	—	2	1	no	—
102	#367	—	2	2	no	—
103	#366	—	2	2	no	—
104	#339	—	2	1	no	—
105	C₂×M₁₆	—	2	1	no	—
106	C₂²×C₈	D₄×C₁₃₉	1	1	yes	characteristic
107	C₂²×C₈	C₂³×C₁₃₉	1	1	yes	characteristic
108	#117	—	2	1	no	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄²	D₄×C₁₃₉	1	1	yes	characteristic
111	M₃₂	—	2	2	no	—
112	M₃₂	Q₈×C₁₃₉	1	4	yes	—
113	C₂×C₁₆	—	2	2	no	—
114	#108	—	2	2	no	—
115	#107	—	2	2	no	—
116	#107	—	2	2	no	—
117	#107	D₄×C₁₃₉	1	2	yes	—
118	#106	—	2	1	no	—
119	#106	—	4	2	no	—
120	C₄⋊C₈	—	4	2	no	—
121	C₄×C₈	—	2	2	no	—
122	#95	—	4	2	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	—	4	2	no	—
125	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
126	M₁₆	—	4	1	no	—
127	M₁₆	—	4	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	4	1	no	—
131	C₂×C₈	D₈×C₁₃₉	1	2	yes	—
132	C₂×C₈	C₂×D₄×C₁₃₉	1	1	yes	characteristic
133	C₂×C₈	C₂×Q₈×C₁₃₉	1	2	yes	—
134	C₂×C₈	#41×C₁₃₉	1	2	yes	—
135	C₂×C₈	#41×C₁₃₉	1	1	yes	characteristic
136	C₄⋊C₄	—	2	2	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	4	2	no	—
139	C₂²⋊C₄	—	4	1	no	—
140	C₄²	—	2	1	no	—
141	C₄²	—	4	1	no	—
142	C₄²	Q₁₆×C₁₃₉	1	2	yes	—
143	C₁₆	—	2	4	no	—
144	C₂³	AGL(1,ℤ/8)×C₁₃₉	1	1	yes	characteristic
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	8	4	no	—
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	#121×C₁₃₉	1	1	yes	characteristic
151	C₂×C₄	#123×C₁₃₉	1	1	yes	characteristic
152	C₂×C₄	C₄⋊Q₈×C₁₃₉	1	1	yes	characteristic
153	C₂×C₄	C₂×D₈×C₁₃₉	1	1	yes	characteristic
154	C₂×C₄	C₂×Q₁₆×C₁₃₉	1	1	yes	characteristic
155	C₂×C₄	#137×C₁₃₉	1	1	yes	characteristic
156	C₈	—	2	2	no	—
157	C₈	—	8	1	no	—
158	C₈	C₂²⋊Q₈×C₁₃₉	1	2	yes	—
159	C₈	#125×C₁₃₉	1	2	yes	—
160	C₂²	—	2	2	no	—
161	C₂²	#459×C₁₃₉	1	1	yes	characteristic
162	C₂²	#484×C₁₃₉	1	1	yes	characteristic
163	C₂²	#501×C₁₃₉	1	1	yes	characteristic
164	C₄	—	4	2	no	—
165	C₄	—	8	2	no	—
166	C₄	—	16	2	no	—
167	C₄	#398×C₁₃₉	1	1	yes	characteristic
168	C₄	#462×C₁₃₉	1	1	yes	characteristic
169	C₄	#480×C₁₃₉	1	1	yes	characteristic
170	C₄	#500×C₁₃₉	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	#2083×C₁₃₉	1	1	yes	characteristic
174	C₂	#2212×C₁₃₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#2159×C₁₃₉

C₂

yes

maximal

#1945×C₁₃₉

C₂

yes

maximal, characteristic

#1846×C₁₃₉

C₂

yes

maximal, characteristic

#1387×C₁₃₉

C₂

yes

maximal

#1376×C₁₃₉

C₂

yes

maximal, characteristic

C₂×M₃₂×C₁₃₉

C₂²

yes

—

C₂×#108×C₁₃₉

C₂²

yes

characteristic

#427×C₁₃₉

C₂²

yes

characteristic

C₂×#107×C₁₃₉

C₂²

yes

—

C₂×#106×C₁₃₉

—

#424×C₁₃₉

—

C₂×C₄×C₈×C₁₃₉

C₂²

yes

characteristic

#382×C₁₃₉

—

#367×C₁₃₉

—

#366×C₁₃₉

—

#339×C₁₃₉

—

C₂×M₁₆×C₁₃₉

—

C₂²×C₈×C₁₃₉

D₄

yes

characteristic

C₂²×C₈×C₁₃₉

C₂³

yes

characteristic

#117×C₁₃₉

—

C₂×C₄⋊C₄×C₁₃₉

—

C₂×C₄²×C₁₃₉

D₄

yes

characteristic

M₃₂×C₁₃₉

—

M₃₂×C₁₃₉

Q₈

yes

—

C₂×C₁₆×C₁₃₉

—

#108×C₁₃₉

—

#107×C₁₃₉

—

#107×C₁₃₉

—

#107×C₁₃₉

D₄

yes

—

#106×C₁₃₉

—

#106×C₁₃₉

—

C₄⋊C₈×C₁₃₉

—

C₄×C₈×C₁₃₉

—

#95×C₁₃₉

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

C₂×D₄

yes

characteristic

M₁₆×C₁₃₉

—

M₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

D₈

yes

—

C₂×C₈×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃₉

C₂×Q₈

yes

—

C₂×C₈×C₁₃₉

#41

yes

—

C₂×C₈×C₁₃₉

#41

yes

characteristic

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₂²⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

—

C₄²×C₁₃₉

—

C₄²×C₁₃₉

Q₁₆

yes

—

C₁₆×C₁₃₉

—

C₂³×C₁₃₉

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

yes

C₂×C₄×C₁₃₉

yes

C₂×C₄×C₁₃₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₃₉

#137

yes

characteristic

C₈×C₁₃₉

—

C₈×C₁₃₉

—

C₈×C₁₃₉

C₂²⋊Q₈

yes

—

C₈×C₁₃₉

#125

yes

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#2083

yes

characteristic

C₂×C₁₃₉

#2212

yes

—

#13509

C₁₃₉

yes

maximal, characteristic, direct product factor, Sylow

C₁₃₉

#13509

yes

characteristic, direct product factor, Sylow

#2159

C₂×C₁₃₉

yes

—

#1945

C₂×C₁₃₉

yes

characteristic

#1846

C₂×C₁₃₉

yes

characteristic

#1387

C₂×C₁₃₉

yes

—

#1376

C₂×C₁₃₉

yes

characteristic

C₂×M₃₂

C₂²×C₁₃₉

yes

—

C₂×#108

C₂²×C₁₃₉

yes

characteristic

#427

C₂²×C₁₃₉

yes

characteristic

C₂×#107

C₂²×C₁₃₉

yes

—

C₂×#106

—

#424

—

100

C₂×C₄×C₈

C₂²×C₁₃₉

yes

characteristic

101

#382

—

102

#367

—

103

#366

—

104

#339

—

105

C₂×M₁₆

—

106

C₂²×C₈

D₄×C₁₃₉

yes

characteristic

107

C₂²×C₈

C₂³×C₁₃₉

yes

characteristic

108

#117

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄²

D₄×C₁₃₉

yes

characteristic

111

M₃₂

—

112

M₃₂

Q₈×C₁₃₉

yes

—

113

C₂×C₁₆

—

114

#108

—

115

#107

—

116

#107

—

117

#107

D₄×C₁₃₉

yes

—

118

#106

—

119

#106

—

120

C₄⋊C₈

—

121

C₄×C₈

—

122

#95

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

C₂×D₄×C₁₃₉

yes

characteristic

126

M₁₆

—

127

M₁₆

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

D₈×C₁₃₉

yes

—

132

C₂×C₈

C₂×D₄×C₁₃₉

yes

characteristic

133

C₂×C₈

C₂×Q₈×C₁₃₉

yes

—

134

C₂×C₈

#41×C₁₃₉

yes

—

135

C₂×C₈

#41×C₁₃₉

yes

characteristic

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₂²⋊C₄

—

140

C₄²

—

141

C₄²

—

142

C₄²

Q₁₆×C₁₃₉

yes

—

143

C₁₆

—

144

C₂³

AGL(1,ℤ/8)×C₁₃₉

yes

characteristic

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

#121×C₁₃₉

yes

characteristic

151

C₂×C₄

#123×C₁₃₉

yes

characteristic

152

C₂×C₄

C₄⋊Q₈×C₁₃₉

yes

characteristic

153

C₂×C₄

C₂×D₈×C₁₃₉

yes

characteristic

154

C₂×C₄

C₂×Q₁₆×C₁₃₉

yes

characteristic

155

C₂×C₄

#137×C₁₃₉

yes

characteristic

156

C₈

—

157

C₈

—

158

C₈

C₂²⋊Q₈×C₁₃₉

yes

—

159

C₈

#125×C₁₃₉

yes

—

160

C₂²

—

161

C₂²

#459×C₁₃₉

yes

characteristic

162

C₂²

#484×C₁₃₉

yes

characteristic

163

C₂²

#501×C₁₃₉

yes

characteristic

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

#398×C₁₃₉

yes

characteristic

168

C₄

#462×C₁₃₉

yes

characteristic

169

C₄

#480×C₁₃₉

yes

characteristic

170

C₄

#500×C₁₃₉

yes

characteristic

171

C₂

—

172

C₂

—

173

C₂

#2083×C₁₃₉

yes

characteristic

174

C₂

#2212×C₁₃₉

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₃₉	#459
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₉
prime radical	C₂³×C₁₃₉	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₉ ◁ C₂²×C₄×C₁₃₉ ◁ C₂×C₄×C₈×C₁₃₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₃₉

#459

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₁₃₉

prime radical

C₂³×C₁₃₉

AGL(1,ℤ/8)

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₃₉ ◁ C₂²×C₄×C₁₃₉ ◁ C₂×C₄×C₈×C₁₃₉ ◁ G

N	H ≅ G/N	type
87. #13509	C₁₃₉	direct (N × H)
88. C₁₃₉	#13509	direct (N × H)

H ≅ G/N

type

87. #13509

C₁₃₉

direct (N × H)

88. C₁₃₉

#13509

direct (N × H)

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

subgroup	index
#2159×C₁₃₉	2
#1945×C₁₃₉	2
#1846×C₁₃₉	2
#1387×C₁₃₉	2
#1376×C₁₃₉	2
#13509	139

subgroup

index

#2159×C₁₃₉

#1945×C₁₃₉

#1846×C₁₃₉

#1387×C₁₃₉

#1376×C₁₃₉

#13509

139

Subgroups — C₁₃₉×#13509 #48008105

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C139×#13509 #48008105

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#13509 #48008105