Group #31807120 — subgroups

Subgroups — #31807120

Order 17792.

260 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 129 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×Dic₁₃₉⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
4	#22580707	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₃₉	C₄	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₃₉	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₃₉	C₂²	1	2	yes	—
11	Dic₁₃₉⋊C₈	—	2	4	no	—
12	C₄×C₁₃₉⋊C₈	—	2	2	no	—
13	C₁₃₉⋊#95	C₄	1	2	yes	—
14	C₂²×C₈×C₁₃₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₃₉	—	2	2	no	—
17	C₂²×Dic₁₃₉	C₂×C₄	1	2	yes	—
18	C₂²×Dic₁₃₉	C₂×C₄	1	2	yes	—
19	C₂×C₁₃₉⋊C₈	—	2	1	no	—
20	C₂×C₁₃₉⋊C₈	—	2	1	no	—
21	C₂×C₁₃₉⋊C₈	—	2	1	no	—
22	C₂×C₁₃₉⋊C₈	—	2	1	no	—
23	C₂×C₁₃₉⋊C₈	D₄	1	2	yes	—
24	C₂×C₁₃₉⋊C₈	Q₈	1	2	yes	—
25	C₄⋊Dic₁₃₉	—	2	2	no	—
26	C₄⋊Dic₁₃₉	C₈	1	4	yes	complemented
27	C₄×Dic₁₃₉	—	2	2	no	—
28	C₄×Dic₁₃₉	—	2	2	no	—
29	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₁₃₉	—	2	2	no	—
33	C₂×C₈×C₁₃₉	—	2	2	no	—
34	C₂×C₈×C₁₃₉	—	2	2	no	—
35	C₄²×C₁₃₉	D₄	1	2	yes	—
36	C₄²×C₁₃₉	Q₈	1	2	yes	—
37	C₂×Dic₁₃₉	—	2	2	no	—
38	C₂×Dic₁₃₉	—	2	2	no	—
39	C₂×Dic₁₃₉	—	2	2	no	—
40	C₂×Dic₁₃₉	—	2	2	no	—
41	C₂×Dic₁₃₉	—	2	2	no	—
42	C₂×Dic₁₃₉	C₂×C₈	1	4	yes	—
43	C₁₃₉⋊C₈	—	2	2	no	—
44	C₁₃₉⋊C₈	—	2	2	no	—
45	C₂³×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃₉	—	2	1	no	—
47	C₂×C₄×C₁₃₉	—	2	1	no	—
48	C₂×C₄×C₁₃₉	C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₁₃₉	C₄⋊C₄	1	2	yes	—
50	C₂×C₄×C₁₃₉	C₂×C₈	1	2	yes	—
51	C₂×C₄×C₁₃₉	M₁₆	1	2	yes	—
52	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
57	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
58	C₈×C₁₃₉	—	4	2	no	—
59	C₂²×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₁₃₉	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₁₃₉	C₄×Q₈	1	1	yes	characteristic
62	C₂²×C₁₃₉	C₂²×C₈	1	1	yes	characteristic
63	C₂²×C₁₃₉	C₂×M₁₆	1	1	yes	characteristic
64	C₂²×C₁₃₉	#131	1	1	yes	characteristic
65	C₂²×C₁₃₉	#131	1	1	yes	characteristic
66	C₄×C₁₃₉	—	2	2	no	—
67	C₄×C₁₃₉	C₄⋊C₈	1	4	yes	complemented
68	C₄×C₁₃₉	#121	1	1	yes	characteristic
69	C₄×C₁₃₉	C₂²⋊Q₈	1	1	yes	characteristic
70	C₄×C₁₃₉	#125	1	1	yes	characteristic
71	C₄×C₁₃₉	C₄⋊Q₈	1	1	yes	characteristic
72	Dic₁₃₉	—	2	4	no	—
73	Dic₁₃₉	—	4	2	no	—
74	C₂×C₁₃₉	#389	1	1	yes	characteristic
75	C₂×C₁₃₉	C₂×C₄⋊C₈	1	1	yes	characteristic
76	C₂×C₁₃₉	#424	1	1	yes	characteristic
77	C₂×C₁₃₉	C₈×D₄	1	1	yes	characteristic
78	C₂×C₁₃₉	#436	1	1	yes	characteristic
79	C₂×C₁₃₉	C₈×Q₈	1	1	yes	characteristic
80	C₂×C₁₃₉	#446	1	1	yes	characteristic
81	C₁₃₉	#1939	1	1	yes	characteristic, complemented, Sylow
82	#1939	—	139	1	no	maximal, Sylow
83	C₂×C₄⋊C₈	—	139	2	no	—
84	C₂×C₄⋊C₈	D₁₃₉	1	1	yes	characteristic
85	C₂×C₄×C₈	—	139	1	no	—
86	C₄×C₄⋊C₄	—	139	1	no	—
87	#336	—	139	2	no	—
88	C₂²×C₈	—	139	1	no	—
89	C₂²×C₈	—	139	1	no	—
90	C₂²×C₈	C₂×D₁₃₉	1	2	yes	—
91	C₂×C₄⋊C₄	—	139	1	no	—
92	C₂×C₄⋊C₄	—	139	1	no	—
93	C₂×C₄²	—	139	2	no	—
94	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
95	C₄⋊C₈	—	2	2	no	—
96	C₄⋊C₈	—	278	4	no	—
97	C₄×C₈	—	278	2	no	—
98	#95	—	139	2	no	—
99	C₂²×C₄	—	139	2	no	—
100	C₂²×C₄	—	139	2	no	—
101	C₂²×C₄	C₄×D₁₃₉	1	1	yes	characteristic
102	C₂²×C₄	C₄×D₁₃₉	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	278	1	no	—
108	C₂×C₈	—	278	1	no	—
109	C₂×C₈	—	278	1	no	—
110	C₂×C₈	—	278	1	no	—
111	C₂×C₈	—	139	2	no	—
112	C₂×C₈	—	139	2	no	—
113	C₄⋊C₄	—	278	2	no	—
114	C₄⋊C₄	—	139	4	no	—
115	C₄²	—	278	2	no	—
116	C₄²	—	278	2	no	—
117	C₄²	C₂²⋊D₁₃₉	1	2	yes	—
118	C₄²	Dic₂₇₈	1	2	yes	—
119	C₂³	C₂×C₄×D₁₃₉	1	1	yes	characteristic
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	278	2	no	—
123	C₂×C₄	—	278	2	no	—
124	C₂×C₄	—	278	2	no	—
125	C₂×C₄	—	278	2	no	—
126	C₂×C₄	—	278	2	no	—
127	C₂×C₄	—	139	4	no	—
128	C₂×C₄	Dic₁₃₉⋊C₄	1	2	yes	—
129	C₂×C₄	Dic₁₃₉⋊C₄	1	2	yes	—
130	C₂×C₄	C₈×D₁₃₉	1	2	yes	—
131	C₂×C₄	#14555768	1	2	yes	—
132	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
133	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
134	C₂×C₄	Q₈×D₁₃₉	1	1	yes	characteristic
135	C₂×C₄	C₂×Dic₂₇₈	1	1	yes	characteristic
136	C₂×C₄	#14555784	1	1	yes	characteristic
137	C₂×C₄	#14555786	1	1	yes	characteristic
138	C₈	—	4	2	no	—
139	C₈	—	278	2	no	—
140	C₈	—	278	2	no	—
141	C₂²	C₂×Dic₁₃₉⋊C₄	1	1	yes	characteristic
142	C₂²	#17984804	1	1	yes	characteristic
143	C₂²	#17984807	1	1	yes	characteristic
144	C₂²	C₂×C₈×D₁₃₉	1	1	yes	characteristic
145	C₂²	C₂×#14555768	1	1	yes	characteristic
146	C₂²	#17984850	1	1	yes	characteristic
147	C₂²	#17984850	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	278	4	no	—
150	C₄	—	556	2	no	—
151	C₄	Dic₁₃₉⋊C₈	1	4	yes	complemented
152	C₄	#17984813	1	1	yes	characteristic
153	C₄	#17984818	1	1	yes	characteristic
154	C₄	#17984833	1	1	yes	characteristic
155	C₄	#17984841	1	1	yes	characteristic
156	C₂	#22580849	1	1	yes	characteristic
157	C₂	C₂×Dic₁₃₉⋊C₈	1	1	yes	characteristic
158	C₂	#22580981	1	1	yes	characteristic
159	C₂	#22581018	1	1	yes	characteristic
160	C₂	#22581030	1	1	yes	characteristic
161	C₂	#22581077	1	1	yes	characteristic
162	C₂	#22581080	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₃₉⋊C₈

C₂

yes

maximal

C₂×C₄×C₁₃₉⋊C₈

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₁₃₉

C₂

yes

maximal, characteristic

#22580707

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₂²×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃₉

C₄

yes

characteristic

C₂×C₄⋊Dic₁₃₉

C₄

yes

characteristic

C₂×C₄×Dic₁₃₉

C₂²

yes

—

Dic₁₃₉⋊C₈

—

C₄×C₁₃₉⋊C₈

—

C₁₃₉⋊#95

C₄

yes

—

C₂²×C₈×C₁₃₉

C₂²

yes

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₉

—

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

D₄

yes

—

C₂×C₁₃₉⋊C₈

Q₈

yes

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

C₈

yes

complemented

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

—

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

—

C₄²×C₁₃₉

D₄

yes

—

C₄²×C₁₃₉

Q₈

yes

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

C₂×C₈

yes

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₃₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₃₉

C₂×C₈

yes

—

C₂×C₄×C₁₃₉

M₁₆

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

C₂×C₄×C₁₃₉

yes

C₂×C₄×C₁₃₉

yes

C₈×C₁₃₉

—

C₂²×C₁₃₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₃₉

C₄×D₄

yes

characteristic

C₂²×C₁₃₉

C₄×Q₈

yes

characteristic

C₂²×C₁₃₉

C₂²×C₈

yes

characteristic

C₂²×C₁₃₉

C₂×M₁₆

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₄×C₁₃₉

—

C₄×C₁₃₉

C₄⋊C₈

yes

complemented

C₄×C₁₃₉

#121

yes

characteristic

C₄×C₁₃₉

C₂²⋊Q₈

yes

characteristic

C₄×C₁₃₉

#125

yes

characteristic

C₄×C₁₃₉

C₄⋊Q₈

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

#389

yes

characteristic

C₂×C₁₃₉

C₂×C₄⋊C₈

yes

characteristic

C₂×C₁₃₉

#424

yes

characteristic

C₂×C₁₃₉

C₈×D₄

yes

characteristic

C₂×C₁₃₉

#436

yes

characteristic

C₂×C₁₃₉

C₈×Q₈

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#1939

—

139

maximal, Sylow

C₂×C₄⋊C₈

—

139

—

C₂×C₄⋊C₈

D₁₃₉

yes

characteristic

C₂×C₄×C₈

—

139

—

C₄×C₄⋊C₄

—

139

—

#336

—

139

—

C₂²×C₈

—

139

—

C₂²×C₈

—

139

—

C₂²×C₈

C₂×D₁₃₉

yes

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄²

—

139

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

278

—

C₄×C₈

—

278

—

#95

—

139

—

C₂²×C₄

—

139

—

100

C₂²×C₄

—

139

—

101

C₂²×C₄

C₄×D₁₃₉

yes

characteristic

102

C₂²×C₄

C₄×D₁₃₉

yes

characteristic

103

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

278

—

108

C₂×C₈

—

278

—

109

C₂×C₈

—

278

—

110

C₂×C₈

—

278

—

111

C₂×C₈

—

139

—

112

C₂×C₈

—

139

—

113

C₄⋊C₄

—

278

—

114

C₄⋊C₄

—

139

—

115

C₄²

—

278

—

116

C₄²

—

278

—

117

C₄²

C₂²⋊D₁₃₉

yes

—

118

C₄²

Dic₂₇₈

yes

—

119

C₂³

C₂×C₄×D₁₃₉

yes

characteristic

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

278

—

123

C₂×C₄

—

278

—

124

C₂×C₄

—

278

—

125

C₂×C₄

—

278

—

126

C₂×C₄

—

278

—

127

C₂×C₄

—

139

—

128

C₂×C₄

Dic₁₃₉⋊C₄

yes

—

129

C₂×C₄

Dic₁₃₉⋊C₄

yes

—

130

C₂×C₄

C₈×D₁₃₉

yes

—

131

C₂×C₄

#14555768

yes

—

132

C₂×C₄

D₄×D₁₃₉

yes

characteristic

133

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

134

C₂×C₄

Q₈×D₁₃₉

yes

characteristic

135

C₂×C₄

C₂×Dic₂₇₈

yes

characteristic

136

C₂×C₄

#14555784

yes

characteristic

137

C₂×C₄

#14555786

yes

characteristic

138

C₈

—

139

C₈

—

278

—

140

C₈

—

278

—

141

C₂²

C₂×Dic₁₃₉⋊C₄

yes

characteristic

142

C₂²

#17984804

yes

characteristic

143

C₂²

#17984807

yes

characteristic

144

C₂²

C₂×C₈×D₁₃₉

yes

characteristic

145

C₂²

C₂×#14555768

yes

characteristic

146

C₂²

#17984850

yes

characteristic

147

C₂²

#17984850

yes

characteristic

148

C₄

—

149

C₄

—

278

—

150

C₄

—

556

—

151

C₄

Dic₁₃₉⋊C₈

yes

complemented

152

C₄

#17984813

yes

characteristic

153

C₄

#17984818

yes

characteristic

154

C₄

#17984833

yes

characteristic

155

C₄

#17984841

yes

characteristic

156

C₂

#22580849

yes

characteristic

157

C₂

C₂×Dic₁₃₉⋊C₈

yes

characteristic

158

C₂

#22580981

yes

characteristic

159

C₂

#22581018

yes

characteristic

160

C₂

#22581030

yes

characteristic

161

C₂

#22581077

yes

characteristic

162

C₂

#22581080

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₃₉
commutator	C₂²×C₁₃₉	C₂²×C₈
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	#1939
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₃₉

commutator

C₂²×C₁₃₉

C₂²×C₈

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
26. C₄⋊Dic₁₃₉	C₈	semidirect
67. C₄×C₁₃₉	C₄⋊C₈	semidirect
81. C₁₃₉	#1939	semidirect
151. C₄	Dic₁₃₉⋊C₈	semidirect

H ≅ G/N

type

26. C₄⋊Dic₁₃₉

C₈

semidirect

67. C₄×C₁₃₉

C₄⋊C₈

semidirect

81. C₁₃₉

#1939

semidirect

151. C₄

Dic₁₃₉⋊C₈

semidirect

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

subgroup	index
C₂×Dic₁₃₉⋊C₈	2
C₂×C₄×C₁₃₉⋊C₈	2
C₄×C₄⋊Dic₁₃₉	2
#22580707	2
C₂×C₄⋊C₈×C₁₃₉	2
#1939	139

subgroup

index

C₂×Dic₁₃₉⋊C₈

C₂×C₄×C₁₃₉⋊C₈

C₄×C₄⋊Dic₁₃₉

#22580707

C₂×C₄⋊C₈×C₁₃₉

#1939

139

Subgroups — #31807120

Conjugacy classes of subgroups

Complements

Maximal subgroups