Group #48004374 — subgroups

Subgroups — C₁₃₉×#9778 #48004374

Order 35584.

390 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 142 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	C₂×#500×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#1837×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#1549×C₁₃₉	C₂	1	2	yes	maximal
4	#1548×C₁₃₉	C₂	1	2	yes	maximal
5	#1452×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₁₃₉	C₂²	1	2	yes	—
7	#500×C₁₃₉	—	2	2	no	—
8	#500×C₁₃₉	C₄	1	4	yes	—
9	#431×C₁₃₉	C₂²	1	1	yes	characteristic
10	#431×C₁₃₉	C₂²	1	1	yes	characteristic
11	C₂×#107×C₁₃₉	—	2	2	no	—
12	#423×C₁₃₉	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
14	#341×C₁₃₉	—	2	2	no	—
15	#333×C₁₃₉	—	2	4	no	—
16	#332×C₁₃₉	—	2	4	no	—
17	#322×C₁₃₉	—	2	2	no	—
18	C₂²×Q₈×C₁₃₉	—	2	2	no	—
19	C₂×M₁₆×C₁₃₉	—	2	2	no	—
20	C₂²×C₈×C₁₃₉	D₄	1	2	yes	—
21	C₄⋊Q₈×C₁₃₉	—	2	2	no	—
22	C₄⋊Q₈×C₁₃₉	—	2	2	no	—
23	C₄⋊Q₈×C₁₃₉	—	2	2	no	—
24	C₄⋊Q₈×C₁₃₉	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
26	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
27	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
28	#107×C₁₃₉	—	2	8	no	—
29	C₄⋊C₈×C₁₃₉	—	2	2	no	—
30	C₄⋊C₈×C₁₃₉	—	2	2	no	—
31	C₄⋊C₈×C₁₃₉	D₄	1	2	yes	—
32	C₄⋊C₈×C₁₃₉	D₄	1	2	yes	—
33	C₂²⋊C₈×C₁₃₉	—	2	1	no	—
34	C₂²⋊C₈×C₁₃₉	—	2	1	no	—
35	C₄×C₈×C₁₃₉	—	2	1	no	—
36	C₄×C₈×C₁₃₉	—	2	1	no	—
37	C₄×C₈×C₁₃₉	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₁₃₉	D₄	1	2	yes	—
39	C₂×Q₈×C₁₃₉	—	2	4	no	—
40	C₂×Q₈×C₁₃₉	—	4	2	no	—
41	C₂×Q₈×C₁₃₉	—	8	4	no	—
42	C₂²×C₄×C₁₃₉	—	4	2	no	—
43	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃₉	#41	1	1	yes	characteristic
45	C₂²×C₄×C₁₃₉	#41	1	1	yes	characteristic
46	M₁₆×C₁₃₉	—	4	4	no	—
47	C₂×C₈×C₁₃₉	—	2	4	no	—
48	C₂×C₈×C₁₃₉	—	2	2	no	—
49	C₂×C₈×C₁₃₉	—	2	2	no	—
50	C₂×C₈×C₁₃₉	—	4	2	no	—
51	C₂×C₈×C₁₃₉	C₂²⋊C₄	1	4	yes	—
52	C₄⋊C₄×C₁₃₉	—	2	4	no	—
53	C₄⋊C₄×C₁₃₉	—	2	4	no	—
54	C₄⋊C₄×C₁₃₉	—	4	2	no	—
55	C₄⋊C₄×C₁₃₉	—	4	2	no	—
56	C₄²×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
60	C₂³×C₁₃₉	#124	1	1	yes	characteristic
61	Q₈×C₁₃₉	—	8	4	no	—
62	Q₈×C₁₃₉	—	8	4	no	—
63	C₂×C₄×C₁₃₉	—	2	2	no	—
64	C₂×C₄×C₁₃₉	—	2	2	no	—
65	C₂×C₄×C₁₃₉	—	4	4	no	—
66	C₂×C₄×C₁₃₉	—	8	4	no	—
67	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₃₉	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₃₉	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₁₃₉	#121	1	1	yes	characteristic
71	C₂×C₄×C₁₃₉	#121	1	1	yes	characteristic
72	C₂×C₄×C₁₃₉	#127	1	1	yes	characteristic
73	C₂×C₄×C₁₃₉	#137	1	4	yes	—
74	C₈×C₁₃₉	—	2	4	no	—
75	C₈×C₁₃₉	—	4	2	no	—
76	C₈×C₁₃₉	—	4	2	no	—
77	C₈×C₁₃₉	—	8	2	no	—
78	C₂²×C₁₃₉	—	2	1	no	—
79	C₂²×C₁₃₉	—	2	1	no	—
80	C₂²×C₁₃₉	#390	1	1	yes	characteristic
81	C₂²×C₁₃₉	#491	1	2	yes	—
82	C₄×C₁₃₉	—	2	1	no	—
83	C₄×C₁₃₉	—	2	1	no	—
84	C₄×C₁₃₉	—	8	4	no	—
85	C₄×C₁₃₉	#419	1	2	yes	—
86	C₄×C₁₃₉	#472	1	2	yes	—
87	C₄×C₁₃₉	#473	1	2	yes	—
88	C₄×C₁₃₉	#497	1	2	yes	—
89	C₂×C₁₃₉	—	2	2	no	—
90	C₂×C₁₃₉	#1960	1	1	yes	characteristic
91	C₂×C₁₃₉	#1978	1	1	yes	characteristic
92	C₂×C₁₃₉	#1979	1	1	yes	characteristic
93	#9778	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₁₃₉	#9778	1	1	yes	characteristic, direct product factor, Sylow
95	C₂×#500	C₂×C₁₃₉	1	1	yes	characteristic
96	#1837	C₂×C₁₃₉	1	1	yes	characteristic
97	#1549	C₂×C₁₃₉	1	2	yes	—
98	#1548	C₂×C₁₃₉	1	2	yes	—
99	#1452	C₂×C₁₃₉	1	1	yes	characteristic
100	C₂×C₄⋊Q₈	C₂²×C₁₃₉	1	2	yes	—
101	#500	—	2	2	no	—
102	#500	C₄×C₁₃₉	1	4	yes	—
103	#431	C₂²×C₁₃₉	1	1	yes	characteristic
104	#431	C₂²×C₁₃₉	1	1	yes	characteristic
105	C₂×#107	—	2	2	no	—
106	#423	C₂²×C₁₃₉	1	2	yes	—
107	C₂×C₄×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
108	#341	—	2	2	no	—
109	#333	—	2	4	no	—
110	#332	—	2	4	no	—
111	#322	—	2	2	no	—
112	C₂²×Q₈	—	2	2	no	—
113	C₂×M₁₆	—	2	2	no	—
114	C₂²×C₈	D₄×C₁₃₉	1	2	yes	—
115	C₄⋊Q₈	—	2	2	no	—
116	C₄⋊Q₈	—	2	2	no	—
117	C₄⋊Q₈	—	2	2	no	—
118	C₄⋊Q₈	C₂×C₄×C₁₃₉	1	4	yes	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
122	#107	—	2	8	no	—
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	D₄×C₁₃₉	1	2	yes	—
126	C₄⋊C₈	D₄×C₁₃₉	1	2	yes	—
127	C₂²⋊C₈	—	2	1	no	—
128	C₂²⋊C₈	—	2	1	no	—
129	C₄×C₈	—	2	1	no	—
130	C₄×C₈	—	2	1	no	—
131	C₄×C₈	C₂×C₄×C₁₃₉	1	2	yes	—
132	C₄×C₈	D₄×C₁₃₉	1	2	yes	—
133	C₂×Q₈	—	2	4	no	—
134	C₂×Q₈	—	4	2	no	—
135	C₂×Q₈	—	8	4	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
138	C₂²×C₄	#41×C₁₃₉	1	1	yes	characteristic
139	C₂²×C₄	#41×C₁₃₉	1	1	yes	characteristic
140	M₁₆	—	4	4	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	C₂²⋊C₄×C₁₃₉	1	4	yes	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄²	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
151	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
152	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
153	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
154	C₂³	#124×C₁₃₉	1	1	yes	characteristic
155	Q₈	—	8	4	no	—
156	Q₈	—	8	4	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	—	8	4	no	—
161	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
162	C₂×C₄	C₄×D₄×C₁₃₉	1	1	yes	characteristic
163	C₂×C₄	C₄×D₄×C₁₃₉	1	1	yes	characteristic
164	C₂×C₄	#121×C₁₃₉	1	1	yes	characteristic
165	C₂×C₄	#121×C₁₃₉	1	1	yes	characteristic
166	C₂×C₄	#127×C₁₃₉	1	1	yes	characteristic
167	C₂×C₄	#137×C₁₃₉	1	4	yes	—
168	C₈	—	2	4	no	—
169	C₈	—	4	2	no	—
170	C₈	—	4	2	no	—
171	C₈	—	8	2	no	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	#390×C₁₃₉	1	1	yes	characteristic
175	C₂²	#491×C₁₃₉	1	2	yes	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	8	4	no	—
179	C₄	#419×C₁₃₉	1	2	yes	—
180	C₄	#472×C₁₃₉	1	2	yes	—
181	C₄	#473×C₁₃₉	1	2	yes	—
182	C₄	#497×C₁₃₉	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	#1960×C₁₃₉	1	1	yes	characteristic
185	C₂	#1978×C₁₃₉	1	1	yes	characteristic
186	C₂	#1979×C₁₃₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#500×C₁₃₉

C₂

yes

maximal, characteristic

#1837×C₁₃₉

C₂

yes

maximal, characteristic

#1549×C₁₃₉

C₂

yes

maximal

#1548×C₁₃₉

C₂

yes

maximal

#1452×C₁₃₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₃₉

C₂²

yes

—

#500×C₁₃₉

—

#500×C₁₃₉

C₄

yes

—

#431×C₁₃₉

C₂²

yes

characteristic

#431×C₁₃₉

C₂²

yes

characteristic

C₂×#107×C₁₃₉

—

#423×C₁₃₉

C₂²

yes

—

C₂×C₄×C₈×C₁₃₉

C₂²

yes

characteristic

#341×C₁₃₉

—

#333×C₁₃₉

—

#332×C₁₃₉

—

#322×C₁₃₉

—

C₂²×Q₈×C₁₃₉

—

C₂×M₁₆×C₁₃₉

—

C₂²×C₈×C₁₃₉

D₄

yes

—

C₄⋊Q₈×C₁₃₉

—

C₄⋊Q₈×C₁₃₉

—

C₄⋊Q₈×C₁₃₉

—

C₄⋊Q₈×C₁₃₉

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₁₃₉

—

C₂×C₄⋊C₄×C₁₃₉

—

C₂×C₄²×C₁₃₉

C₂³

yes

characteristic

#107×C₁₃₉

—

C₄⋊C₈×C₁₃₉

—

C₄⋊C₈×C₁₃₉

—

C₄⋊C₈×C₁₃₉

D₄

yes

—

C₄⋊C₈×C₁₃₉

D₄

yes

—

C₂²⋊C₈×C₁₃₉

—

C₂²⋊C₈×C₁₃₉

—

C₄×C₈×C₁₃₉

—

C₄×C₈×C₁₃₉

—

C₄×C₈×C₁₃₉

C₂×C₄

yes

—

C₄×C₈×C₁₃₉

D₄

yes

—

C₂×Q₈×C₁₃₉

—

C₂×Q₈×C₁₃₉

—

C₂×Q₈×C₁₃₉

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

C₂×D₄

yes

C₂²×C₄×C₁₃₉

yes

C₂²×C₄×C₁₃₉

yes

M₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂²⋊C₄

yes

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

C₂²×C₄

yes

characteristic

C₄²×C₁₃₉

C₂×D₄

yes

characteristic

C₄²×C₁₃₉

C₂×D₄

yes

characteristic

C₄²×C₁₃₉

C₂×D₄

yes

characteristic

C₂³×C₁₃₉

#124

yes

characteristic

Q₈×C₁₃₉

—

Q₈×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₄×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₄×D₄

yes

C₂×C₄×C₁₃₉

yes

C₂×C₄×C₁₃₉

yes

C₂×C₄×C₁₃₉

yes

C₂×C₄×C₁₃₉

yes

—

C₈×C₁₃₉

—

C₈×C₁₃₉

—

C₈×C₁₃₉

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

#390

yes

characteristic

C₂²×C₁₃₉

#491

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#419

yes

—

C₄×C₁₃₉

#472

yes

—

C₄×C₁₃₉

#473

yes

—

C₄×C₁₃₉

#497

yes

—

C₂×C₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

maximal, characteristic, direct product factor, Sylow

C₁₃₉

#9778

yes

characteristic, direct product factor, Sylow

C₂×#500

C₂×C₁₃₉

yes

characteristic

#1837

C₂×C₁₃₉

yes

characteristic

#1549

C₂×C₁₃₉

yes

—

#1548

C₂×C₁₃₉

yes

—

#1452

C₂×C₁₃₉

yes

characteristic

100

C₂×C₄⋊Q₈

C₂²×C₁₃₉

yes

—

101

#500

—

102

#500

C₄×C₁₃₉

yes

—

103

#431

C₂²×C₁₃₉

yes

characteristic

104

#431

C₂²×C₁₃₉

yes

characteristic

105

C₂×#107

—

106

#423

C₂²×C₁₃₉

yes

—

107

C₂×C₄×C₈

C₂²×C₁₃₉

yes

characteristic

108

#341

—

109

#333

—

110

#332

—

111

#322

—

112

C₂²×Q₈

—

113

C₂×M₁₆

—

114

C₂²×C₈

D₄×C₁₃₉

yes

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

C₂×C₄×C₁₃₉

yes

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄²

C₂³×C₁₃₉

yes

characteristic

122

#107

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄⋊C₈

D₄×C₁₃₉

yes

—

126

C₄⋊C₈

D₄×C₁₃₉

yes

—

127

C₂²⋊C₈

—

128

C₂²⋊C₈

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

C₂×C₄×C₁₃₉

yes

—

132

C₄×C₈

D₄×C₁₃₉

yes

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂²×C₄

—

137

C₂²×C₄

C₂×D₄×C₁₃₉

yes

characteristic

138

C₂²×C₄

#41×C₁₃₉

yes

characteristic

139

C₂²×C₄

#41×C₁₃₉

yes

characteristic

140

M₁₆

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

C₂²⋊C₄×C₁₃₉

yes

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄²

C₂²×C₄×C₁₃₉

yes

characteristic

151

C₄²

C₂×D₄×C₁₃₉

yes

characteristic

152

C₄²

C₂×D₄×C₁₃₉

yes

characteristic

153

C₄²

C₂×D₄×C₁₃₉

yes

characteristic

154

C₂³

#124×C₁₃₉

yes

characteristic

155

Q₈

—

156

Q₈

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

C₂×C₂²⋊C₄×C₁₃₉

yes

characteristic

162

C₂×C₄

C₄×D₄×C₁₃₉

yes

characteristic

163

C₂×C₄

C₄×D₄×C₁₃₉

yes

characteristic

164

C₂×C₄

#121×C₁₃₉

yes

characteristic

165

C₂×C₄

#121×C₁₃₉

yes

characteristic

166

C₂×C₄

#127×C₁₃₉

yes

characteristic

167

C₂×C₄

#137×C₁₃₉

yes

—

168

C₈

—

169

C₈

—

170

C₈

—

171

C₈

—

172

C₂²

—

173

C₂²

—

174

C₂²

#390×C₁₃₉

yes

characteristic

175

C₂²

#491×C₁₃₉

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

#419×C₁₃₉

yes

—

180

C₄

#472×C₁₃₉

yes

—

181

C₄

#473×C₁₃₉

yes

—

182

C₄

#497×C₁₃₉

yes

—

183

C₂

—

184

C₂

#1960×C₁₃₉

yes

characteristic

185

C₂

#1978×C₁₃₉

yes

characteristic

186

C₂

#1979×C₁₃₉

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₉	#390
commutator	C₄²	C₂²×C₄×C₁₃₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₉
prime radical	C₂³×C₁₃₉	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₉ ◁ C₂×C₄²×C₁₃₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₃₉

#390

commutator

C₄²

C₂²×C₄×C₁₃₉

Frattini, cosocle

C₂×C₄²

C₂³×C₁₃₉

C₂³×C₁₃₉

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₁₃₉ ◁ C₂×C₄²×C₁₃₉ ◁ G

N	H ≅ G/N	type
93. #9778	C₁₃₉	direct (N × H)
94. C₁₃₉	#9778	direct (N × H)

H ≅ G/N

type

93. #9778

C₁₃₉

direct (N × H)

94. C₁₃₉

#9778

direct (N × H)

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

subgroup	index
C₂×#500×C₁₃₉	2
#1837×C₁₃₉	2
#1549×C₁₃₉	2
#1548×C₁₃₉	2
#1452×C₁₃₉	2
#9778	139

subgroup

index

C₂×#500×C₁₃₉

#1837×C₁₃₉

#1549×C₁₃₉

#1548×C₁₃₉

#1452×C₁₃₉

#9778

139

Subgroups — C₁₃₉×#9778 #48004374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C139×#9778 #48004374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#9778 #48004374