Group #31807486 — subgroups

Subgroups — #31807486

Order 17792.

406 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 97 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×#14555773	C₂	1	2	yes	maximal
2	C₂×#17984842	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17984765	C₂	1	2	yes	maximal, complemented
4	#22580708	C₂	1	1	yes	maximal, characteristic, complemented
5	#390×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₇₈	C₂²	1	2	yes	—
7	C₂×#14555773	—	2	4	no	—
8	C₂×#14555773	—	2	4	no	—
9	C₂×#14555773	—	2	4	no	—
10	C₂²×C₁₃₉⋊C₈	C₂²	1	2	yes	—
11	#17984842	—	2	4	no	—
12	C₂×C₄⋊Dic₁₃₉	—	2	2	no	—
13	#17984765	—	2	4	no	—
14	C₂²×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊C₄×C₁₃₉	—	2	2	no	—
17	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
18	C₂×Dic₂₇₈	—	2	2	no	—
19	C₂×Dic₂₇₈	—	2	2	no	—
20	C₂×Dic₂₇₈	—	2	2	no	—
21	C₂×Dic₂₇₈	—	2	2	no	—
22	C₂×Dic₂₇₈	—	2	2	no	—
23	C₂×Dic₂₇₈	D₄	1	4	yes	—
24	C₂²×Dic₁₃₉	—	2	2	no	—
25	#14555773	—	2	16	no	—
26	C₂×C₁₃₉⋊C₈	—	2	2	no	—
27	C₂×C₁₃₉⋊C₈	—	2	2	no	—
28	C₂×C₁₃₉⋊C₈	D₄	1	4	yes	—
29	C₄⋊Dic₁₃₉	—	2	8	no	—
30	C₂⁴×C₁₃₉	—	2	1	no	—
31	C₂×D₄×C₁₃₉	—	2	2	no	—
32	C₂×D₄×C₁₃₉	—	2	2	no	—
33	C₂×D₄×C₁₃₉	—	2	1	no	—
34	C₂×D₄×C₁₃₉	D₄	1	2	yes	—
35	C₂²×C₄×C₁₃₉	—	2	1	no	—
36	C₂²×C₄×C₁₃₉	D₄	1	2	yes	—
37	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₁₃₉	—	2	2	no	—
39	C₂²⋊C₄×C₁₃₉	—	4	4	no	—
40	C₄²×C₁₃₉	—	2	2	no	—
41	Dic₂₇₈	—	2	4	no	—
42	Dic₂₇₈	—	2	4	no	—
43	Dic₂₇₈	—	4	2	no	—
44	Dic₂₇₈	—	4	2	no	—
45	C₂×Dic₁₃₉	—	2	4	no	—
46	C₂×Dic₁₃₉	—	4	2	no	—
47	C₂×Dic₁₃₉	—	4	2	no	—
48	C₁₃₉⋊C₈	—	2	4	no	—
49	C₂³×C₁₃₉	—	2	2	no	—
50	C₂³×C₁₃₉	—	4	1	no	—
51	C₂³×C₁₃₉	—	4	1	no	—
52	C₂³×C₁₃₉	—	4	2	no	—
53	C₂³×C₁₃₉	—	4	2	no	—
54	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
55	D₄×C₁₃₉	—	2	4	no	—
56	D₄×C₁₃₉	—	4	2	no	—
57	C₂×C₄×C₁₃₉	—	2	2	no	—
58	C₂×C₄×C₁₃₉	—	2	2	no	—
59	C₂×C₄×C₁₃₉	—	4	1	no	—
60	C₂×C₄×C₁₃₉	—	4	2	no	—
61	C₂×C₄×C₁₃₉	SD₁₆	1	4	yes	complemented
62	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
66	C₂²×C₁₃₉	—	4	2	no	—
67	C₂²×C₁₃₉	—	4	4	no	—
68	C₂²×C₁₃₉	—	8	2	no	—
69	C₂²×C₁₃₉	—	8	1	no	—
70	C₂²×C₁₃₉	—	8	1	no	—
71	C₂²×C₁₃₉	#120	1	1	yes	characteristic
72	C₂²×C₁₃₉	#121	1	1	yes	characteristic
73	C₂²×C₁₃₉	#127	1	1	yes	characteristic
74	C₂²×C₁₃₉	C₂×SD₁₆	1	2	yes	—
75	C₂²×C₁₃₉	#137	1	2	yes	—
76	C₄×C₁₃₉	—	2	4	no	—
77	C₄×C₁₃₉	—	8	1	no	—
78	C₄×C₁₃₉	#120	1	2	yes	—
79	C₄×C₁₃₉	#121	1	2	yes	—
80	Dic₁₃₉	—	4	4	no	—
81	C₂×C₁₃₉	—	8	2	no	—
82	C₂×C₁₃₉	#392	1	1	yes	characteristic
83	C₂×C₁₃₉	#448	1	2	yes	—
84	C₂×C₁₃₉	#461	1	2	yes	—
85	C₂×C₁₃₉	#492	1	1	yes	characteristic
86	C₂×C₁₃₉	#497	1	1	yes	characteristic
87	C₁₃₉	#2013	1	1	yes	characteristic, complemented, Sylow
88	#2013	—	139	1	no	maximal, Sylow
89	C₂²×SD₁₆	—	139	2	no	—
90	C₂×C₄⋊Q₈	—	139	1	no	—
91	C₂×#103	—	139	2	no	—
92	#390	D₁₃₉	1	1	yes	characteristic
93	#336	—	139	1	no	—
94	C₂²×Q₈	—	139	2	no	—
95	C₂²×D₄	C₂×D₁₃₉	1	1	yes	characteristic
96	C₂×SD₁₆	—	278	4	no	—
97	C₂×SD₁₆	—	278	4	no	—
98	C₂×SD₁₆	—	278	4	no	—
99	C₂²×C₈	—	139	2	no	—
100	C₄⋊Q₈	—	278	4	no	—
101	C₂×C₄⋊C₄	—	278	2	no	—
102	C₂×C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
103	C₂×C₂²⋊C₄	—	2	2	no	—
104	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
105	#103	—	278	4	no	—
106	C₂⁴	—	2	1	no	—
107	C₂×Q₈	—	278	2	no	—
108	C₂×Q₈	—	278	2	no	—
109	C₂×Q₈	—	278	2	no	—
110	C₂×Q₈	—	278	2	no	—
111	C₂×Q₈	—	278	2	no	—
112	C₂×Q₈	—	139	4	no	—
113	C₂×D₄	—	2	2	no	—
114	C₂×D₄	—	2	2	no	—
115	C₂×D₄	—	2	1	no	—
116	C₂×D₄	D₅₅₆	1	2	yes	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	278	2	no	—
119	C₂²×C₄	C₂²⋊D₁₃₉	1	2	yes	—
120	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
121	SD₁₆	—	278	16	no	—
122	C₂×C₈	—	278	2	no	—
123	C₂×C₈	—	278	2	no	—
124	C₂×C₈	—	139	4	no	—
125	C₄⋊C₄	—	2	2	no	—
126	C₄⋊C₄	—	278	8	no	—
127	C₂²⋊C₄	—	4	4	no	—
128	C₄²	—	2	2	no	—
129	C₂³	—	2	2	no	—
130	C₂³	—	4	1	no	—
131	C₂³	—	4	1	no	—
132	C₂³	—	4	2	no	—
133	C₂³	—	4	2	no	—
134	C₂³	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
135	Q₈	—	278	4	no	—
136	Q₈	—	278	4	no	—
137	Q₈	—	556	2	no	—
138	Q₈	—	556	2	no	—
139	D₄	—	2	4	no	—
140	D₄	—	4	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	278	4	no	—
146	C₂×C₄	—	556	2	no	—
147	C₂×C₄	—	556	2	no	—
148	C₂×C₄	#14555773	1	4	yes	complemented
149	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
150	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
151	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
152	C₂×C₄	#14555785	1	1	yes	characteristic
153	C₈	—	278	4	no	—
154	C₂²	—	4	2	no	—
155	C₂²	—	4	4	no	—
156	C₂²	—	8	1	no	—
157	C₂²	—	8	2	no	—
158	C₂²	—	8	1	no	—
159	C₂²	#17984810	1	1	yes	characteristic
160	C₂²	#17984816	1	1	yes	characteristic
161	C₂²	#17984838	1	1	yes	characteristic
162	C₂²	C₂×#14555773	1	2	yes	—
163	C₂²	#17984872	1	2	yes	—
164	C₄	—	2	4	no	—
165	C₄	—	8	1	no	—
166	C₄	—	556	4	no	—
167	C₄	#17984812	1	2	yes	—
168	C₄	#17984815	1	2	yes	—
169	C₂	—	8	2	no	—
170	C₂	#22580862	1	1	yes	characteristic
171	C₂	#22581091	1	2	yes	—
172	C₂	#22581158	1	2	yes	—
173	C₂	#22581325	1	1	yes	characteristic
174	C₂	#22581348	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#14555773

C₂

yes

maximal

C₂×#17984842

C₂

yes

maximal, characteristic, complemented

C₂×#17984765

C₂

yes

maximal, complemented

#22580708

C₂

yes

maximal, characteristic, complemented

#390×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₇₈

C₂²

yes

—

C₂×#14555773

—

C₂×#14555773

—

C₂×#14555773

—

C₂²×C₁₃₉⋊C₈

C₂²

yes

—

#17984842

—

C₂×C₄⋊Dic₁₃₉

—

#17984765

—

C₂²×D₄×C₁₃₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₃₉

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₁₃₉

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

D₄

yes

—

C₂²×Dic₁₃₉

—

#14555773

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

D₄

yes

—

C₄⋊Dic₁₃₉

—

C₂⁴×C₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₉

—

C₂²⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

D₄×C₁₃₉

—

D₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

SD₁₆

yes

complemented

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

characteristic

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

C₂×SD₁₆

yes

—

C₂²×C₁₃₉

#137

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#120

yes

—

C₄×C₁₃₉

#121

yes

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#392

yes

characteristic

C₂×C₁₃₉

#448

yes

—

C₂×C₁₃₉

#461

yes

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#2013

—

139

maximal, Sylow

C₂²×SD₁₆

—

139

—

C₂×C₄⋊Q₈

—

139

—

C₂×#103

—

139

—

#390

D₁₃₉

yes

characteristic

#336

—

139

—

C₂²×Q₈

—

139

—

C₂²×D₄

C₂×D₁₃₉

yes

characteristic

C₂×SD₁₆

—

278

—

C₂×SD₁₆

—

278

—

C₂×SD₁₆

—

278

—

C₂²×C₈

—

139

—

100

C₄⋊Q₈

—

278

—

101

C₂×C₄⋊C₄

—

278

—

102

C₂×C₄⋊C₄

C₂×D₁₃₉

yes

characteristic

103

C₂×C₂²⋊C₄

—

104

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

105

#103

—

278

—

106

C₂⁴

—

107

C₂×Q₈

—

278

—

108

C₂×Q₈

—

278

—

109

C₂×Q₈

—

278

—

110

C₂×Q₈

—

278

—

111

C₂×Q₈

—

278

—

112

C₂×Q₈

—

139

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

D₅₅₆

yes

—

117

C₂²×C₄

—

118

C₂²×C₄

—

278

—

119

C₂²×C₄

C₂²⋊D₁₃₉

yes

—

120

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

121

SD₁₆

—

278

—

122

C₂×C₈

—

278

—

123

C₂×C₈

—

278

—

124

C₂×C₈

—

139

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

278

—

127

C₂²⋊C₄

—

128

C₄²

—

129

C₂³

—

130

C₂³

—

131

C₂³

—

132

C₂³

—

133

C₂³

—

134

C₂³

C₂×C₂²⋊D₁₃₉

yes

characteristic

135

Q₈

—

278

—

136

Q₈

—

278

—

137

Q₈

—

556

—

138

Q₈

—

556

—

139

D₄

—

140

D₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

278

—

146

C₂×C₄

—

556

—

147

C₂×C₄

—

556

—

148

C₂×C₄

#14555773

yes

complemented

149

C₂×C₄

D₄×D₁₃₉

yes

—

150

C₂×C₄

D₄×D₁₃₉

yes

—

151

C₂×C₄

C₂×D₅₅₆

yes

characteristic

152

C₂×C₄

#14555785

yes

characteristic

153

C₈

—

278

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

#17984810

yes

characteristic

160

C₂²

#17984816

yes

characteristic

161

C₂²

#17984838

yes

characteristic

162

C₂²

C₂×#14555773

yes

—

163

C₂²

#17984872

yes

—

164

C₄

—

165

C₄

—

166

C₄

—

556

—

167

C₄

#17984812

yes

—

168

C₄

#17984815

yes

—

169

C₂

—

170

C₂

#22580862

yes

characteristic

171

C₂

#22581091

yes

—

172

C₂

#22581158

yes

—

173

C₂

#22581325

yes

characteristic

174

C₂

#22581348

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃₉
commutator, cosocle	C₂²×C₄×C₁₃₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#390×C₁₃₉	C₂
socle	C₂³×C₁₃₉	C₂×D₄
prime radical	C₂²×D₄×C₁₃₉	C₂²
hypercenter	#390	D₁₃₉
hypocenter	C₁₃₉	#2013
LCS	G ▷ C₂²×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃₉

commutator, cosocle

C₂²×C₄×C₁₃₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

#390×C₁₃₉

C₂

socle

C₂³×C₁₃₉

C₂×D₄

prime radical

C₂²×D₄×C₁₃₉

C₂²

D₁₃₉

C₁₃₉

G ▷ C₂²×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

N	H ≅ G/N	type
2. C₂×#17984842	C₂	semidirect
3. C₂×#17984765	C₂	semidirect
4. #22580708	C₂	semidirect
61. C₂×C₄×C₁₃₉	SD₁₆	semidirect
87. C₁₃₉	#2013	semidirect
148. C₂×C₄	#14555773	semidirect

H ≅ G/N

type

2. C₂×#17984842

C₂

semidirect

3. C₂×#17984765

C₂

semidirect

4. #22580708

C₂

semidirect

61. C₂×C₄×C₁₃₉

SD₁₆

semidirect

87. C₁₃₉

#2013

semidirect

148. C₂×C₄

#14555773

semidirect

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

subgroup	index
C₂²×#14555773	2
C₂×#17984842	2
C₂×#17984765	2
#22580708	2
#390×C₁₃₉	2
#2013	139

subgroup

index

#390×C₁₃₉

139

Subgroups — #31807486

Conjugacy classes of subgroups

Complements

Maximal subgroups