Group #31806586 — subgroups

Subgroups — #31806586

Order 17792.

324 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 109 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×#14555769	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₃₉⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₄×C₂²⋊Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
4	#22580707	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊C₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555769	—	2	4	no	—
7	C₂×#14555769	—	2	2	no	—
8	C₂²×C₁₃₉⋊C₈	C₂²	1	2	yes	—
9	C₂×C₂²⋊Dic₁₃₉	C₄	1	2	yes	—
10	C₂×C₄×Dic₁₃₉	C₂²	1	2	yes	—
11	Dic₁₃₉⋊C₈	—	2	4	no	—
12	C₁₃₉⋊#95	C₄	1	1	yes	characteristic
13	C₁₃₉⋊#95	C₄	1	1	yes	characteristic
14	C₂³×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
17	C₂²⋊C₈×C₁₃₉	—	2	2	no	—
18	C₂²×Dic₁₃₉	C₂×C₄	1	2	yes	—
19	C₂²×Dic₁₃₉	C₂×C₄	1	2	yes	—
20	#14555769	—	2	8	no	—
21	C₂×C₁₃₉⋊C₈	—	2	2	no	—
22	C₂×C₁₃₉⋊C₈	—	2	2	no	—
23	C₂×C₁₃₉⋊C₈	D₄	1	4	yes	—
24	C₂²⋊Dic₁₃₉	—	2	4	no	—
25	C₄×Dic₁₃₉	—	2	2	no	—
26	C₄×Dic₁₃₉	—	2	2	no	—
27	C₂⁴×C₁₃₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₃₉	—	2	2	no	—
29	C₂²×C₄×C₁₃₉	—	2	2	no	—
30	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₃₉	D₄	1	2	yes	—
32	C₂²×C₄×C₁₃₉	Q₈	1	2	yes	—
33	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₁₃₉	—	2	2	no	—
35	C₂×C₈×C₁₃₉	—	2	1	no	—
36	C₂×C₈×C₁₃₉	—	2	1	no	—
37	C₂×C₈×C₁₃₉	—	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	—	2	2	no	—
43	C₂×Dic₁₃₉	M₁₆	1	4	yes	complemented
44	C₁₃₉⋊C₈	—	2	4	no	—
45	C₂³×C₁₃₉	—	2	2	no	—
46	C₂³×C₁₃₉	—	2	2	no	—
47	C₂³×C₁₃₉	—	2	1	no	—
48	C₂³×C₁₃₉	—	2	1	no	—
49	C₂³×C₁₃₉	C₄⋊C₄	1	2	yes	—
50	C₂³×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₃₉	—	2	2	no	—
52	C₂×C₄×C₁₃₉	—	2	2	no	—
53	C₂×C₄×C₁₃₉	—	2	4	no	—
54	C₂×C₄×C₁₃₉	—	2	2	no	—
55	C₂×C₄×C₁₃₉	C₄⋊C₄	1	2	yes	—
56	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₁₃₉	#41	1	2	yes	—
60	C₈×C₁₃₉	—	4	1	no	—
61	C₈×C₁₃₉	—	4	1	no	—
62	C₂²×C₁₃₉	—	2	2	no	—
63	C₂²×C₁₃₉	—	2	2	no	—
64	C₂²×C₁₃₉	—	2	2	no	—
65	C₂²×C₁₃₉	—	4	2	no	—
66	C₂²×C₁₃₉	—	4	2	no	—
67	C₂²×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₁₃₉	C₄×D₄	1	2	yes	—
69	C₂²×C₁₃₉	C₂×M₁₆	1	2	yes	—
70	C₂²×C₁₃₉	#131	1	1	yes	characteristic
71	C₂²×C₁₃₉	#131	1	1	yes	characteristic
72	C₄×C₁₃₉	—	2	2	no	—
73	C₄×C₁₃₉	#120	1	1	yes	characteristic
74	C₄×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
75	C₄×C₁₃₉	#123	1	1	yes	characteristic
76	Dic₁₃₉	—	2	4	no	—
77	Dic₁₃₉	—	4	2	no	—
78	C₂×C₁₃₉	—	4	2	no	—
79	C₂×C₁₃₉	#385	1	1	yes	characteristic
80	C₂×C₁₃₉	#423	1	1	yes	characteristic
81	C₂×C₁₃₉	#424	1	1	yes	characteristic
82	C₂×C₁₃₉	#435	1	2	yes	—
83	C₂×C₁₃₉	#436	1	2	yes	—
84	C₁₃₉	#1851	1	1	yes	characteristic, complemented, Sylow
85	#1851	—	139	1	no	maximal, Sylow
86	C₂²×M₁₆	—	139	1	no	—
87	C₂×C₄⋊C₈	—	139	2	no	—
88	C₂×C₂²⋊C₈	D₁₃₉	1	1	yes	characteristic
89	C₄×C₂²⋊C₄	—	139	1	no	—
90	#336	—	139	2	no	—
91	C₂³×C₄	C₂×D₁₃₉	1	1	yes	characteristic
92	C₂×M₁₆	—	278	2	no	—
93	C₂×M₁₆	—	278	4	no	—
94	C₂²×C₈	—	139	2	no	—
95	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
96	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
97	C₂×C₂²⋊C₄	—	139	2	no	—
98	C₂×C₄²	—	139	2	no	—
99	C₄⋊C₈	—	278	4	no	—
100	C₂²⋊C₈	—	2	2	no	—
101	#95	—	139	1	no	—
102	#95	—	139	1	no	—
103	C₂⁴	C₄×D₁₃₉	1	1	yes	characteristic
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	139	2	no	—
107	C₂²×C₄	—	139	2	no	—
108	C₂²×C₄	C₄×D₁₃₉	1	1	yes	characteristic
109	C₂²×C₄	C₂²⋊D₁₃₉	1	2	yes	—
110	C₂²×C₄	Dic₂₇₈	1	2	yes	—
111	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
112	M₁₆	—	278	8	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	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	C₂²⋊C₄	—	278	4	no	—
121	C₄²	—	278	2	no	—
122	C₄²	—	278	2	no	—
123	C₂³	—	2	2	no	—
124	C₂³	—	2	2	no	—
125	C₂³	—	2	1	no	—
126	C₂³	—	2	1	no	—
127	C₂³	Dic₁₃₉⋊C₄	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	—	2	4	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	278	2	no	—
134	C₂×C₄	—	278	2	no	—
135	C₂×C₄	—	278	2	no	—
136	C₂×C₄	—	278	2	no	—
137	C₂×C₄	—	278	2	no	—
138	C₂×C₄	—	139	4	no	—
139	C₂×C₄	Dic₁₃₉⋊C₄	1	2	yes	—
140	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
141	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
142	C₂×C₄	C₂×Dic₂₇₈	1	1	yes	characteristic
143	C₂×C₄	#14555784	1	2	yes	—
144	C₈	—	4	1	no	—
145	C₈	—	4	1	no	—
146	C₈	—	278	4	no	—
147	C₂²	—	2	2	no	—
148	C₂²	—	2	2	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	4	2	no	—
151	C₂²	—	4	2	no	—
152	C₂²	C₂×Dic₁₃₉⋊C₄	1	1	yes	characteristic
153	C₂²	(C₂²⋊D₁₃₉)⋊C₄	1	2	yes	—
154	C₂²	M₁₆×D₁₃₉	1	2	yes	—
155	C₂²	#17984848	1	1	yes	characteristic
156	C₂²	#17984848	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	278	4	no	—
159	C₄	—	556	2	no	—
160	C₄	#17984810	1	1	yes	characteristic
161	C₄	#17984819	1	2	yes	—
162	C₄	#17984823	1	1	yes	characteristic
163	C₂	—	4	2	no	—
164	C₂	#22580825	1	1	yes	characteristic
165	C₂	#22580976	1	1	yes	characteristic
166	C₂	#22580979	1	1	yes	characteristic
167	C₂	#22581027	1	2	yes	—
168	C₂	#22581028	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#14555769

C₂

yes

maximal, characteristic

C₂×Dic₁₃₉⋊C₈

C₂

yes

maximal, complemented

C₄×C₂²⋊Dic₁₃₉

C₂

yes

maximal, characteristic

#22580707

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₁₃₉

C₂

yes

maximal, characteristic

C₂×#14555769

—

C₂×#14555769

—

C₂²×C₁₃₉⋊C₈

C₂²

yes

—

C₂×C₂²⋊Dic₁₃₉

C₄

yes

—

C₂×C₄×Dic₁₃₉

C₂²

yes

—

Dic₁₃₉⋊C₈

—

C₁₃₉⋊#95

C₄

yes

characteristic

C₁₃₉⋊#95

C₄

yes

characteristic

C₂³×C₄×C₁₃₉

C₂²

yes

characteristic

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

C₂²⋊C₈×C₁₃₉

—

C₂²×Dic₁₃₉

C₂×C₄

yes

—

C₂²×Dic₁₃₉

C₂×C₄

yes

—

#14555769

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

D₄

yes

—

C₂²⋊Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₂⁴×C₁₃₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

Q₈

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

M₁₆

yes

complemented

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

C₄⋊C₄

yes

—

C₂³×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

—

C₈×C₁₃₉

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₃₉

C₄×D₄

yes

—

C₂²×C₁₃₉

C₂×M₁₆

yes

—

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₄×C₁₃₉

—

C₄×C₁₃₉

#120

yes

characteristic

C₄×C₁₃₉

C₂²⋊Q₈

yes

—

C₄×C₁₃₉

#123

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

—

C₂×C₁₃₉

#436

yes

—

C₁₃₉

#1851

yes

characteristic, complemented, Sylow

#1851

—

139

maximal, Sylow

C₂²×M₁₆

—

139

—

C₂×C₄⋊C₈

—

139

—

C₂×C₂²⋊C₈

D₁₃₉

yes

characteristic

C₄×C₂²⋊C₄

—

139

—

#336

—

139

—

C₂³×C₄

C₂×D₁₃₉

yes

characteristic

C₂×M₁₆

—

278

—

C₂×M₁₆

—

278

—

C₂²×C₈

—

139

—

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

C₂×C₂²⋊C₄

—

139

—

C₂×C₄²

—

139

—

C₄⋊C₈

—

278

—

100

C₂²⋊C₈

—

101

#95

—

139

—

102

#95

—

139

—

103

C₂⁴

C₄×D₁₃₉

yes

characteristic

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

139

—

107

C₂²×C₄

—

139

—

108

C₂²×C₄

C₄×D₁₃₉

yes

characteristic

109

C₂²×C₄

C₂²⋊D₁₃₉

yes

—

110

C₂²×C₄

Dic₂₇₈

yes

—

111

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

112

M₁₆

—

278

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

278

—

118

C₂×C₈

—

278

—

119

C₂×C₈

—

139

—

120

C₂²⋊C₄

—

278

—

121

C₄²

—

278

—

122

C₄²

—

278

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

Dic₁₃₉⋊C₄

yes

—

128

C₂³

C₂×C₄×D₁₃₉

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

278

—

134

C₂×C₄

—

278

—

135

C₂×C₄

—

278

—

136

C₂×C₄

—

278

—

137

C₂×C₄

—

278

—

138

C₂×C₄

—

139

—

139

C₂×C₄

Dic₁₃₉⋊C₄

yes

—

140

C₂×C₄

D₄×D₁₃₉

yes

—

141

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

142

C₂×C₄

C₂×Dic₂₇₈

yes

characteristic

143

C₂×C₄

#14555784

yes

—

144

C₈

—

145

C₈

—

146

C₈

—

278

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₂×Dic₁₃₉⋊C₄

yes

characteristic

153

C₂²

(C₂²⋊D₁₃₉)⋊C₄

yes

—

154

C₂²

M₁₆×D₁₃₉

yes

—

155

C₂²

#17984848

yes

characteristic

156

C₂²

#17984848

yes

characteristic

157

C₄

—

158

C₄

—

278

—

159

C₄

—

556

—

160

C₄

#17984810

yes

characteristic

161

C₄

#17984819

yes

—

162

C₄

#17984823

yes

characteristic

163

C₂

—

164

C₂

#22580825

yes

characteristic

165

C₂

#22580976

yes

characteristic

166

C₂

#22580979

yes

characteristic

167

C₂

#22581027

yes

—

168

C₂

#22581028

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₃₉
commutator, socle	C₂³×C₁₃₉	C₂²×C₄
Fitting	C₂×C₂²⋊C₈×C₁₃₉	C₂
prime radical	C₂⁴×C₁₃₉	C₂×C₄
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	C₂×C₂²⋊C₈	D₁₃₉
hypocenter	C₁₃₉	#1851
LCS	G ▷ C₂³×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₃₉

commutator, socle

C₂³×C₁₃₉

C₂²×C₄

Fitting

C₂×C₂²⋊C₈×C₁₃₉

C₂

prime radical

C₂⁴×C₁₃₉

C₂×C₄

cosocle

C₂²×C₄×C₁₃₉

C₂³

hypercenter

C₂×C₂²⋊C₈

D₁₃₉

C₁₃₉

G ▷ C₂³×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
2. C₂×Dic₁₃₉⋊C₈	C₂	semidirect
4. #22580707	C₂	semidirect
43. C₂×Dic₁₃₉	M₁₆	semidirect
84. C₁₃₉	#1851	semidirect

H ≅ G/N

type

2. C₂×Dic₁₃₉⋊C₈

C₂

semidirect

4. #22580707

C₂

semidirect

43. C₂×Dic₁₃₉

M₁₆

semidirect

84. C₁₃₉

#1851

semidirect

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

subgroup	index
C₂²×#14555769	2
C₂×Dic₁₃₉⋊C₈	2
C₄×C₂²⋊Dic₁₃₉	2
#22580707	2
C₂×C₂²⋊C₈×C₁₃₉	2
#1851	139

subgroup

index

C₂²×#14555769

C₂×Dic₁₃₉⋊C₈

C₄×C₂²⋊Dic₁₃₉

#22580707

C₂×C₂²⋊C₈×C₁₃₉

#1851

139

Subgroups — #31806586

Conjugacy classes of subgroups

Complements

Maximal subgroups