Group #31806263 — subgroups

Subgroups — #31806263

Order 17792.

260 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 133 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×C₄⋊(C₁₃₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984752	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
4	#22580799	C₂	1	1	yes	maximal, characteristic
5	#22580707	C₂	1	1	yes	maximal, characteristic
6	#22580707	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊C₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
12	C₄⋊(C₁₃₉⋊C₈)	C₄	1	4	yes	complemented
13	#17984752	—	2	2	no	—
14	C₈×Dic₁₃₉	—	2	2	no	—
15	C₁₃₉⋊#95	C₄	1	2	yes	—
16	C₁₃₉⋊#95	C₄	1	2	yes	—
17	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₁₃₉	C₄	1	4	yes	complemented
21	C₂²×Dic₁₃₉	C₂×C₄	1	2	yes	—
22	C₂²×Dic₁₃₉	C₂×C₄	1	2	yes	—
23	C₂×C₁₃₉⋊C₈	—	2	1	no	—
24	C₂×C₁₃₉⋊C₈	—	2	1	no	—
25	C₂×C₁₃₉⋊C₈	—	2	1	no	—
26	C₂×C₁₃₉⋊C₈	—	2	1	no	—
27	C₂×C₁₃₉⋊C₈	C₂×C₄	1	2	yes	—
28	C₂×C₁₃₉⋊C₈	C₂×C₄	1	2	yes	—
29	C₄×Dic₁₃₉	—	2	1	no	—
30	C₄×Dic₁₃₉	—	2	1	no	—
31	C₄×Dic₁₃₉	D₄	1	2	yes	—
32	C₄×Dic₁₃₉	Q₈	1	2	yes	—
33	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₁₃₉	—	2	1	no	—
37	C₂×C₈×C₁₃₉	—	2	1	no	—
38	C₂×C₈×C₁₃₉	—	2	1	no	—
39	C₂×C₈×C₁₃₉	—	2	1	no	—
40	C₂×C₈×C₁₃₉	C₂×C₄	1	2	yes	—
41	C₂×C₈×C₁₃₉	C₂×C₄	1	2	yes	—
42	C₄²×C₁₃₉	C₂×C₄	1	2	yes	—
43	C₄²×C₁₃₉	C₂×C₄	1	2	yes	—
44	C₂×Dic₁₃₉	—	2	2	no	—
45	C₂×Dic₁₃₉	—	2	2	no	—
46	C₂×Dic₁₃₉	—	2	2	no	—
47	C₂×Dic₁₃₉	—	2	2	no	—
48	C₂×Dic₁₃₉	—	2	2	no	—
49	C₂×Dic₁₃₉	C₄⋊C₄	1	4	yes	—
50	C₁₃₉⋊C₈	—	2	2	no	—
51	C₁₃₉⋊C₈	—	2	2	no	—
52	C₂³×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₃₉	—	2	2	no	—
54	C₂×C₄×C₁₃₉	—	2	2	no	—
55	C₂×C₄×C₁₃₉	C₄²	1	2	yes	—
56	C₂×C₄×C₁₃₉	C₄²	1	2	yes	—
57	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
62	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
63	C₈×C₁₃₉	—	2	2	no	—
64	C₈×C₁₃₉	—	2	2	no	—
65	C₂²×C₁₃₉	C₂×C₄²	1	1	yes	characteristic
66	C₂²×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₁₃₉	#117	1	1	yes	characteristic
68	C₂²×C₁₃₉	#131	1	2	yes	—
69	C₂²×C₁₃₉	#131	1	2	yes	—
70	C₄×C₁₃₉	—	2	2	no	—
71	C₄×C₁₃₉	—	2	2	no	—
72	C₄×C₁₃₉	C₄×D₄	1	1	yes	characteristic
73	C₄×C₁₃₉	C₄×D₄	1	1	yes	characteristic
74	C₄×C₁₃₉	C₄×Q₈	1	1	yes	characteristic
75	C₄×C₁₃₉	C₄×Q₈	1	1	yes	characteristic
76	Dic₁₃₉	—	2	4	no	—
77	Dic₁₃₉	—	4	2	no	—
78	C₂×C₁₃₉	C₄×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₁₃₉	#405	1	1	yes	characteristic
80	C₂×C₁₃₉	#405	1	1	yes	characteristic
81	C₂×C₁₃₉	#424	1	2	yes	—
82	C₂×C₁₃₉	#433	1	2	yes	—
83	C₁₃₉	#1768	1	1	yes	characteristic, complemented, Sylow
84	#1768	—	139	1	no	maximal, Sylow
85	C₂×C₄⋊C₈	—	139	1	no	—
86	C₂×C₄⋊C₈	D₁₃₉	1	1	yes	characteristic
87	C₂×#97	—	139	1	no	—
88	C₂×C₄×C₈	—	139	1	no	—
89	#376	—	139	1	no	—
90	#336	—	139	1	no	—
91	#336	—	139	1	no	—
92	C₂²×C₈	—	139	1	no	—
93	C₂²×C₈	—	139	1	no	—
94	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
95	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
96	C₂×C₄²	—	139	1	no	—
97	C₂×C₄²	—	139	1	no	—
98	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
99	C₄⋊C₈	—	139	4	no	—
100	C₄⋊C₈	Dic₁₃₉	1	4	yes	complemented
101	#97	—	278	2	no	—
102	C₄×C₈	—	278	2	no	—
103	#95	—	139	2	no	—
104	#95	—	139	2	no	—
105	C₂²×C₄	—	139	2	no	—
106	C₂²×C₄	—	139	2	no	—
107	C₂²×C₄	C₄×D₁₃₉	1	1	yes	characteristic
108	C₂²×C₄	C₄×D₁₃₉	1	1	yes	characteristic
109	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	278	1	no	—
115	C₂×C₈	—	278	1	no	—
116	C₂×C₈	—	278	1	no	—
117	C₂×C₈	—	278	1	no	—
118	C₂×C₈	—	139	2	no	—
119	C₂×C₈	—	139	2	no	—
120	C₂×C₈	C₂×Dic₁₃₉	1	2	yes	—
121	C₂×C₈	C₂×Dic₁₃₉	1	2	yes	—
122	C₄²	—	278	1	no	—
123	C₄²	—	278	1	no	—
124	C₄²	—	139	2	no	—
125	C₄²	—	139	2	no	—
126	C₄²	C₄×D₁₃₉	1	2	yes	—
127	C₄²	C₂×Dic₁₃₉	1	2	yes	—
128	C₂³	C₂×C₄×D₁₃₉	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	278	2	no	—
132	C₂×C₄	—	278	2	no	—
133	C₂×C₄	—	278	2	no	—
134	C₂×C₄	—	278	2	no	—
135	C₂×C₄	—	278	2	no	—
136	C₂×C₄	—	139	4	no	—
137	C₂×C₄	C₄×Dic₁₃₉	1	2	yes	—
138	C₂×C₄	C₄×Dic₁₃₉	1	2	yes	—
139	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
140	C₂×C₄	C₂²×Dic₁₃₉	1	1	yes	characteristic
141	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
142	C₂×C₄	Q₈×D₁₃₉	1	1	yes	characteristic
143	C₂×C₄	#14555784	1	1	yes	characteristic
144	C₂×C₄	#14555786	1	1	yes	characteristic
145	C₈	—	2	2	no	—
146	C₈	—	2	2	no	—
147	C₈	—	278	2	no	—
148	C₈	—	278	2	no	—
149	C₂²	C₂×C₄×Dic₁₃₉	1	1	yes	characteristic
150	C₂²	C₄⋊C₄×D₁₃₉	1	1	yes	characteristic
151	C₂²	#17984800	1	1	yes	characteristic
152	C₂²	#17984848	1	2	yes	—
153	C₂²	#17984850	1	2	yes	—
154	C₄	—	2	2	no	—
155	C₄	—	2	2	no	—
156	C₄	—	278	4	no	—
157	C₄	—	556	2	no	—
158	C₄	#17984804	1	1	yes	characteristic
159	C₄	D₄×Dic₁₃₉	1	1	yes	characteristic
160	C₄	#17984807	1	1	yes	characteristic
161	C₄	Q₈×Dic₁₃₉	1	1	yes	characteristic
162	C₂	C₄⋊C₄×Dic₁₃₉	1	1	yes	characteristic
163	C₂	#22580908	1	1	yes	characteristic
164	C₂	#22580911	1	1	yes	characteristic
165	C₂	#22580978	1	2	yes	—
166	C₂	#22581013	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₃₉⋊C₈)

C₂

yes

maximal, characteristic

C₂×#17984752

C₂

yes

maximal, characteristic

C₂×C₈×Dic₁₃₉

C₂

yes

maximal, characteristic

#22580799

C₂

yes

maximal, characteristic

#22580707

C₂

yes

maximal, characteristic

#22580707

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₂²×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₁₃₉

C₂²

yes

characteristic

C₂×C₄×Dic₁₃₉

C₂²

yes

characteristic

C₄⋊(C₁₃₉⋊C₈)

C₄

yes

complemented

#17984752

—

C₈×Dic₁₃₉

—

C₁₃₉⋊#95

C₄

yes

—

C₁₃₉⋊#95

C₄

yes

—

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₉

C₄

yes

complemented

C₂²×Dic₁₃₉

C₂×C₄

yes

—

C₂²×Dic₁₃₉

C₂×C₄

yes

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

—

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

D₄

yes

—

C₄×Dic₁₃₉

Q₈

yes

—

C₂²×C₄×C₁₃₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

—

C₄²×C₁₃₉

C₂×C₄

yes

—

C₄²×C₁₃₉

C₂×C₄

yes

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

C₄⋊C₄

yes

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₄²

yes

—

C₂×C₄×C₁₃₉

C₄²

yes

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

C₂×C₄×C₁₃₉

yes

C₂×C₄×C₁₃₉

yes

C₈×C₁₃₉

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

C₂×C₄²

yes

characteristic

C₂²×C₁₃₉

C₂×C₄⋊C₄

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

—

C₂²×C₁₃₉

#131

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₄×D₄

yes

characteristic

C₄×C₁₃₉

C₄×D₄

yes

characteristic

C₄×C₁₃₉

C₄×Q₈

yes

characteristic

C₄×C₁₃₉

C₄×Q₈

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

C₄×C₄⋊C₄

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

—

C₂×C₁₃₉

#433

yes

—

C₁₃₉

#1768

yes

characteristic, complemented, Sylow

#1768

—

139

maximal, Sylow

C₂×C₄⋊C₈

—

139

—

C₂×C₄⋊C₈

D₁₃₉

yes

characteristic

C₂×#97

—

139

—

C₂×C₄×C₈

—

139

—

#376

—

139

—

#336

—

139

—

#336

—

139

—

C₂²×C₈

—

139

—

C₂²×C₈

—

139

—

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

C₂×C₄²

—

139

—

C₂×C₄²

—

139

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

C₄⋊C₈

—

139

—

100

C₄⋊C₈

Dic₁₃₉

yes

complemented

101

#97

—

278

—

102

C₄×C₈

—

278

—

103

#95

—

139

—

104

#95

—

139

—

105

C₂²×C₄

—

139

—

106

C₂²×C₄

—

139

—

107

C₂²×C₄

C₄×D₁₃₉

yes

characteristic

108

C₂²×C₄

C₄×D₁₃₉

yes

characteristic

109

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

278

—

115

C₂×C₈

—

278

—

116

C₂×C₈

—

278

—

117

C₂×C₈

—

278

—

118

C₂×C₈

—

139

—

119

C₂×C₈

—

139

—

120

C₂×C₈

C₂×Dic₁₃₉

yes

—

121

C₂×C₈

C₂×Dic₁₃₉

yes

—

122

C₄²

—

278

—

123

C₄²

—

278

—

124

C₄²

—

139

—

125

C₄²

—

139

—

126

C₄²

C₄×D₁₃₉

yes

—

127

C₄²

C₂×Dic₁₃₉

yes

—

128

C₂³

C₂×C₄×D₁₃₉

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

278

—

132

C₂×C₄

—

278

—

133

C₂×C₄

—

278

—

134

C₂×C₄

—

278

—

135

C₂×C₄

—

278

—

136

C₂×C₄

—

139

—

137

C₂×C₄

C₄×Dic₁₃₉

yes

—

138

C₂×C₄

C₄×Dic₁₃₉

yes

—

139

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

140

C₂×C₄

C₂²×Dic₁₃₉

yes

characteristic

141

C₂×C₄

D₄×D₁₃₉

yes

characteristic

142

C₂×C₄

Q₈×D₁₃₉

yes

characteristic

143

C₂×C₄

#14555784

yes

characteristic

144

C₂×C₄

#14555786

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

278

—

148

C₈

—

278

—

149

C₂²

C₂×C₄×Dic₁₃₉

yes

characteristic

150

C₂²

C₄⋊C₄×D₁₃₉

yes

characteristic

151

C₂²

#17984800

yes

characteristic

152

C₂²

#17984848

yes

—

153

C₂²

#17984850

yes

—

154

C₄

—

155

C₄

—

156

C₄

—

278

—

157

C₄

—

556

—

158

C₄

#17984804

yes

characteristic

159

C₄

D₄×Dic₁₃₉

yes

characteristic

160

C₄

#17984807

yes

characteristic

161

C₄

Q₈×Dic₁₃₉

yes

characteristic

162

C₂

C₄⋊C₄×Dic₁₃₉

yes

characteristic

163

C₂

#22580908

yes

characteristic

164

C₂

#22580911

yes

characteristic

165

C₂

#22580978

yes

—

166

C₂

#22581013

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₃₉
commutator	C₂²×C₁₃₉	C₂×C₄²
Fitting	C₂×C₄⋊C₈×C₁₃₉	C₂
socle, prime radical	C₂³×C₁₃₉	C₂²×C₄
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₃₉
hypocenter	C₁₃₉	#1768
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₃₉

commutator

C₂²×C₁₃₉

C₂×C₄²

Fitting

C₂×C₄⋊C₈×C₁₃₉

C₂

socle, prime radical

C₂³×C₁₃₉

C₂²×C₄

cosocle

C₂²×C₄×C₁₃₉

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₃₉

C₁₃₉

G ▷ C₂²×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
12. C₄⋊(C₁₃₉⋊C₈)	C₄	semidirect
20. C₄⋊C₈×C₁₃₉	C₄	semidirect
83. C₁₃₉	#1768	semidirect
100. C₄⋊C₈	Dic₁₃₉	semidirect

H ≅ G/N

type

12. C₄⋊(C₁₃₉⋊C₈)

C₄

semidirect

20. C₄⋊C₈×C₁₃₉

C₄

semidirect

83. C₁₃₉

#1768

semidirect

100. C₄⋊C₈

Dic₁₃₉

semidirect

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

subgroup	index
C₂×C₄⋊(C₁₃₉⋊C₈)	2
C₂×#17984752	2
C₂×C₈×Dic₁₃₉	2
#22580799	2
#22580707	2
C₂×C₄⋊C₈×C₁₃₉	2
#1768	139

subgroup

index

C₂×C₄⋊(C₁₃₉⋊C₈)

C₂×C₈×Dic₁₃₉

C₂×C₄⋊C₈×C₁₃₉

139

Subgroups — #31806263

Conjugacy classes of subgroups

Complements

Maximal subgroups