Group #31806096 — subgroups

Subgroups — #31806096

Order 17792.

480 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 165 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	C₂²×#14555768	C₂	1	2	yes	maximal
2	C₂×C₄²×D₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#22580708	C₂	1	1	yes	maximal, characteristic, complemented
4	#22580707	C₂	1	2	yes	maximal, complemented
5	#336×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₁₃₉	C₄	1	2	yes	—
7	C₂²×C₄×D₁₃₉	C₂²	1	1	yes	characteristic
8	C₂×#14555768	—	2	4	no	—
9	C₂×#14555768	—	2	4	no	—
10	C₂×#14555768	C₄	1	8	yes	complemented
11	C₂²×C₁₃₉⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄×Dic₁₃₉	C₄	1	2	yes	—
13	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
14	C₄²×D₁₃₉	—	2	4	no	—
15	C₂²×C₈×C₁₃₉	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
17	C₂³×D₁₃₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×Dic₁₃₉	C₂×C₄	1	2	yes	—
19	C₂²×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×D₁₃₉	—	2	4	no	—
21	C₂×C₄×D₁₃₉	—	2	4	no	—
22	C₂×C₄×D₁₃₉	—	2	4	no	—
23	C₂×C₄×D₁₃₉	C₂×C₄	1	4	yes	—
24	C₂×C₄×D₁₃₉	D₄	1	2	yes	—
25	C₂×C₄×D₁₃₉	D₄	1	4	yes	—
26	C₂×C₄×D₁₃₉	Q₈	1	2	yes	—
27	#14555768	—	2	16	no	—
28	C₂×C₁₃₉⋊C₈	—	2	2	no	—
29	C₂×C₁₃₉⋊C₈	—	2	2	no	—
30	C₂×C₁₃₉⋊C₈	C₂×C₄	1	4	yes	complemented
31	C₄×Dic₁₃₉	—	2	4	no	—
32	C₄×Dic₁₃₉	—	2	2	no	—
33	C₂²×C₄×C₁₃₉	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
35	C₂×C₈×C₁₃₉	—	2	2	no	—
36	C₂×C₈×C₁₃₉	—	2	2	no	—
37	C₂×C₈×C₁₃₉	C₂×C₄	1	4	yes	complemented
38	C₄²×C₁₃₉	—	2	2	no	—
39	C₂²×D₁₃₉	—	2	2	no	—
40	C₂²×D₁₃₉	—	2	2	no	—
41	C₂²×D₁₃₉	C₄²	1	2	yes	—
42	C₂²×D₁₃₉	C₂²⋊C₄	1	2	yes	—
43	C₂²×D₁₃₉	C₄⋊C₄	1	2	yes	—
44	C₂×Dic₁₃₉	—	2	2	no	—
45	C₂×Dic₁₃₉	—	2	2	no	—
46	C₂×Dic₁₃₉	C₄²	1	2	yes	—
47	C₂×Dic₁₃₉	C₂²⋊C₄	1	2	yes	—
48	C₂×Dic₁₃₉	C₄⋊C₄	1	2	yes	—
49	C₂×Dic₁₃₉	M₁₆	1	4	yes	complemented
50	C₄×D₁₃₉	—	2	16	no	—
51	C₄×D₁₃₉	C₂²⋊C₄	1	8	yes	—
52	C₄×D₁₃₉	C₄⋊C₄	1	8	yes	—
53	C₁₃₉⋊C₈	—	2	4	no	—
54	C₂³×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₃₉	—	2	2	no	—
56	C₂×C₄×C₁₃₉	—	2	2	no	—
57	C₂×C₄×C₁₃₉	M₁₆	1	4	yes	complemented
58	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
62	C₈×C₁₃₉	—	2	4	no	—
63	C₂²×C₁₃₉	C₂×C₄²	1	1	yes	characteristic
64	C₂²×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₁₃₉	C₂×M₁₆	1	2	yes	—
67	C₂²×C₁₃₉	C₂×M₁₆	1	2	yes	—
68	C₂×D₁₃₉	—	2	2	no	—
69	C₂×D₁₃₉	—	2	2	no	—
70	C₂×D₁₃₉	—	2	4	no	—
71	C₂×D₁₃₉	—	2	4	no	—
72	C₂×D₁₃₉	#95	1	4	yes	—
73	C₄×C₁₃₉	—	2	4	no	—
74	C₄×C₁₃₉	C₂×C₂²⋊C₄	1	2	yes	—
75	C₄×C₁₃₉	C₂×C₄⋊C₄	1	2	yes	—
76	Dic₁₃₉	—	2	4	no	—
77	Dic₁₃₉	#95	1	4	yes	—
78	C₂×C₁₃₉	C₂×#95	1	1	yes	characteristic
79	C₂×C₁₃₉	C₄×M₁₆	1	1	yes	characteristic
80	C₂×C₁₃₉	C₄×M₁₆	1	1	yes	characteristic
81	C₂×C₁₃₉	#407	1	2	yes	—
82	C₂×C₁₃₉	#423	1	2	yes	—
83	D₁₃₉	—	2	4	no	—
84	C₁₃₉	#1728	1	1	yes	characteristic, complemented, Sylow
85	#1728	—	139	1	no	maximal, Sylow
86	C₂²×M₁₆	—	139	2	no	—
87	C₂²×C₄²	—	139	1	no	—
88	#336	—	139	2	no	—
89	#336	—	139	1	no	—
90	#336	D₁₃₉	1	1	yes	characteristic, complemented
91	C₂³×C₄	—	139	2	no	—
92	C₂³×C₄	—	139	1	no	—
93	C₂×M₁₆	—	278	4	no	—
94	C₂×M₁₆	—	278	4	no	—
95	C₂×M₁₆	—	139	8	no	—
96	C₂²×C₈	—	139	2	no	—
97	C₂²×C₈	C₂×D₁₃₉	1	2	yes	—
98	C₂×C₄²	—	278	4	no	—
99	C₂×C₄²	—	139	2	no	—
100	C₂×C₄²	—	139	1	no	—
101	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
102	C₂⁴	—	139	1	no	—
103	C₂²×C₄	—	278	4	no	—
104	C₂²×C₄	—	278	4	no	—
105	C₂²×C₄	—	278	4	no	—
106	C₂²×C₄	—	139	2	no	—
107	C₂²×C₄	—	139	1	no	—
108	C₂²×C₄	—	139	4	no	—
109	C₂²×C₄	—	139	4	no	—
110	C₂²×C₄	—	139	2	no	—
111	C₂²×C₄	—	139	2	no	—
112	C₂²×C₄	C₄×D₁₃₉	1	2	yes	—
113	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
114	M₁₆	—	278	16	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	278	2	no	—
118	C₂×C₈	—	278	2	no	—
119	C₂×C₈	—	139	4	no	—
120	C₂×C₈	C₄×D₁₃₉	1	4	yes	complemented
121	C₄²	—	2	2	no	—
122	C₄²	—	278	4	no	—
123	C₄²	—	278	2	no	—
124	C₂³	—	278	2	no	—
125	C₂³	—	278	2	no	—
126	C₂³	—	139	2	no	—
127	C₂³	—	139	2	no	—
128	C₂³	—	139	2	no	—
129	C₂³	C₂×C₄×D₁₃₉	1	1	yes	characteristic
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	278	16	no	—
133	C₂×C₄	—	278	2	no	—
134	C₂×C₄	—	278	2	no	—
135	C₂×C₄	—	139	4	no	—
136	C₂×C₄	—	139	2	no	—
137	C₂×C₄	—	139	2	no	—
138	C₂×C₄	—	139	2	no	—
139	C₂×C₄	—	139	8	no	—
140	C₂×C₄	—	139	8	no	—
141	C₂×C₄	#14555768	1	4	yes	complemented
142	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
143	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
144	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
145	C₂×C₄	Q₈×D₁₃₉	1	1	yes	characteristic
146	C₈	—	2	4	no	—
147	C₈	—	278	4	no	—
148	C₂²	—	278	2	no	—
149	C₂²	—	278	2	no	—
150	C₂²	—	278	4	no	—
151	C₂²	—	278	4	no	—
152	C₂²	—	139	4	no	—
153	C₂²	C₄²×D₁₃₉	1	1	yes	characteristic
154	C₂²	C₂²⋊C₄×D₁₃₉	1	1	yes	characteristic
155	C₂²	C₄⋊C₄×D₁₃₉	1	1	yes	characteristic
156	C₂²	M₁₆×D₁₃₉	1	2	yes	—
157	C₂²	C₂×#14555768	1	2	yes	—
158	C₄	—	2	4	no	—
159	C₄	—	278	4	no	—
160	C₄	—	139	4	no	—
161	C₄	C₂²⋊C₄×D₁₃₉	1	2	yes	—
162	C₄	C₄⋊C₄×D₁₃₉	1	2	yes	—
163	C₂	—	278	4	no	—
164	C₂	#95×D₁₃₉	1	1	yes	characteristic
165	C₂	C₄×#14555768	1	1	yes	characteristic
166	C₂	#22580905	1	1	yes	characteristic
167	C₂	#22580915	1	2	yes	—
168	C₂	#22580975	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#14555768

C₂

yes

maximal

C₂×C₄²×D₁₃₉

C₂

yes

maximal, characteristic

#22580708

C₂

yes

maximal, characteristic, complemented

#22580707

C₂

yes

maximal, complemented

#336×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₁₃₉

C₄

yes

—

C₂²×C₄×D₁₃₉

C₂²

yes

—

—

C₄

yes

complemented

C₂²×C₁₃₉⋊C₈

C₂²

yes

—

C₂×C₄×Dic₁₃₉

C₄

yes

—

C₂×C₄×Dic₁₃₉

C₂²

yes

characteristic

C₄²×D₁₃₉

—

C₂²×C₈×C₁₃₉

C₂²

yes

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₂³×D₁₃₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₃₉

C₂×C₄

yes

—

C₂²×Dic₁₃₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₃₉

—

C₂×C₄×D₁₃₉

—

C₂×C₄×D₁₃₉

—

C₂×C₄×D₁₃₉

C₂×C₄

yes

—

C₂×C₄×D₁₃₉

D₄

yes

—

C₂×C₄×D₁₃₉

D₄

yes

—

C₂×C₄×D₁₃₉

Q₈

yes

—

#14555768

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

complemented

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₂²×C₄×C₁₃₉

C₂×C₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

complemented

C₄²×C₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

C₄²

yes

—

C₂²×D₁₃₉

C₂²⋊C₄

yes

—

C₂²×D₁₃₉

C₄⋊C₄

yes

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

C₄²

yes

—

C₂×Dic₁₃₉

C₂²⋊C₄

yes

—

C₂×Dic₁₃₉

C₄⋊C₄

yes

—

C₂×Dic₁₃₉

M₁₆

yes

complemented

C₄×D₁₃₉

—

C₄×D₁₃₉

C₂²⋊C₄

yes

—

C₄×D₁₃₉

C₄⋊C₄

yes

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

M₁₆

yes

complemented

C₂×C₄×C₁₃₉

C₂²×C₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×Q₈

yes

characteristic

C₈×C₁₃₉

—

C₂²×C₁₃₉

C₂×C₄²

yes

characteristic

C₂²×C₁₃₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₃₉

C₂×M₁₆

yes

—

C₂²×C₁₃₉

C₂×M₁₆

yes

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

#95

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×C₂²⋊C₄

yes

—

C₄×C₁₃₉

C₂×C₄⋊C₄

yes

—

Dic₁₃₉

—

Dic₁₃₉

#95

yes

—

C₂×C₁₃₉

C₂×#95

yes

characteristic

C₂×C₁₃₉

C₄×M₁₆

yes

characteristic

C₂×C₁₃₉

C₄×M₁₆

yes

characteristic

C₂×C₁₃₉

#407

yes

—

C₂×C₁₃₉

#423

yes

—

D₁₃₉

—

C₁₃₉

#1728

yes

characteristic, complemented, Sylow

#1728

—

139

maximal, Sylow

C₂²×M₁₆

—

139

—

C₂²×C₄²

—

139

—

#336

—

139

—

#336

—

139

—

#336

D₁₃₉

yes

characteristic, complemented

C₂³×C₄

—

139

—

C₂³×C₄

—

139

—

C₂×M₁₆

—

278

—

C₂×M₁₆

—

278

—

C₂×M₁₆

—

139

—

C₂²×C₈

—

139

—

C₂²×C₈

C₂×D₁₃₉

yes

—

C₂×C₄²

—

278

—

C₂×C₄²

—

139

—

100

C₂×C₄²

—

139

—

101

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

102

C₂⁴

—

139

—

103

C₂²×C₄

—

278

—

104

C₂²×C₄

—

278

—

105

C₂²×C₄

—

278

—

106

C₂²×C₄

—

139

—

107

C₂²×C₄

—

139

—

108

C₂²×C₄

—

139

—

109

C₂²×C₄

—

139

—

110

C₂²×C₄

—

139

—

111

C₂²×C₄

—

139

—

112

C₂²×C₄

C₄×D₁₃₉

yes

—

113

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

114

M₁₆

—

278

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

278

—

118

C₂×C₈

—

278

—

119

C₂×C₈

—

139

—

120

C₂×C₈

C₄×D₁₃₉

yes

complemented

121

C₄²

—

122

C₄²

—

278

—

123

C₄²

—

278

—

124

C₂³

—

278

—

125

C₂³

—

278

—

126

C₂³

—

139

—

127

C₂³

—

139

—

128

C₂³

—

139

—

129

C₂³

C₂×C₄×D₁₃₉

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

278

—

133

C₂×C₄

—

278

—

134

C₂×C₄

—

278

—

135

C₂×C₄

—

139

—

136

C₂×C₄

—

139

—

137

C₂×C₄

—

139

—

138

C₂×C₄

—

139

—

139

C₂×C₄

—

139

—

140

C₂×C₄

—

139

—

141

C₂×C₄

#14555768

yes

complemented

142

C₂×C₄

C₂×C₄×D₁₃₉

yes

—

143

C₂×C₄

D₄×D₁₃₉

yes

characteristic

144

C₂×C₄

D₄×D₁₃₉

yes

—

145

C₂×C₄

Q₈×D₁₃₉

yes

characteristic

146

C₈

—

147

C₈

—

278

—

148

C₂²

—

278

—

149

C₂²

—

278

—

150

C₂²

—

278

—

151

C₂²

—

278

—

152

C₂²

—

139

—

153

C₂²

C₄²×D₁₃₉

yes

characteristic

154

C₂²

C₂²⋊C₄×D₁₃₉

yes

characteristic

155

C₂²

C₄⋊C₄×D₁₃₉

yes

characteristic

156

C₂²

M₁₆×D₁₃₉

yes

—

157

C₂²

C₂×#14555768

yes

—

158

C₄

—

159

C₄

—

278

—

160

C₄

—

139

—

161

C₄

C₂²⋊C₄×D₁₃₉

yes

—

162

C₄

C₄⋊C₄×D₁₃₉

yes

—

163

C₂

—

278

—

164

C₂

#95×D₁₃₉

yes

characteristic

165

C₂

C₄×#14555768

yes

characteristic

166

C₂

#22580905

yes

characteristic

167

C₂

#22580915

yes

—

168

C₂

#22580975

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₃₉
commutator	C₂²×C₁₃₉	C₂×C₄²
Fitting	#336×C₁₃₉	C₂
socle	C₂³×C₁₃₉	C₂²×C₄
prime radical	C₂³×D₁₃₉	C₂×C₄
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	#336	D₁₃₉
hypocenter	C₁₃₉	#1728
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₃₉

commutator

C₂²×C₁₃₉

C₂×C₄²

Fitting

#336×C₁₃₉

C₂

socle

C₂³×C₁₃₉

C₂²×C₄

prime radical

C₂³×D₁₃₉

C₂×C₄

cosocle

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂²×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
3. #22580708	C₂	semidirect
4. #22580707	C₂	semidirect
5. #336×C₁₃₉	C₂	semidirect
10. C₂×#14555768	C₄	semidirect
30. C₂×C₁₃₉⋊C₈	C₂×C₄	semidirect
37. C₂×C₈×C₁₃₉	C₂×C₄	semidirect
49. C₂×Dic₁₃₉	M₁₆	semidirect
57. C₂×C₄×C₁₃₉	M₁₆	semidirect
84. C₁₃₉	#1728	semidirect
90. #336	D₁₃₉	semidirect
120. C₂×C₈	C₄×D₁₃₉	semidirect
141. C₂×C₄	#14555768	semidirect

H ≅ G/N

type

3. #22580708

C₂

semidirect

4. #22580707

C₂

semidirect

5. #336×C₁₃₉

C₂

semidirect

10. C₂×#14555768

C₄

semidirect

30. C₂×C₁₃₉⋊C₈

C₂×C₄

semidirect

37. C₂×C₈×C₁₃₉

C₂×C₄

semidirect

49. C₂×Dic₁₃₉

M₁₆

semidirect

57. C₂×C₄×C₁₃₉

M₁₆

semidirect

84. C₁₃₉

#1728

semidirect

90. #336

D₁₃₉

semidirect

120. C₂×C₈

C₄×D₁₃₉

semidirect

141. C₂×C₄

#14555768

semidirect

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

subgroup	index
C₂²×#14555768	2
C₂×C₄²×D₁₃₉	2
#22580708	2
#22580707	2
#336×C₁₃₉	2
#1728	139

subgroup

index

C₂×C₄²×D₁₃₉

#336×C₁₃₉

139

Subgroups — #31806096

Conjugacy classes of subgroups

Complements

Maximal subgroups