Group #31805647 — subgroups

Subgroups — #31805647

Order 17792.

194 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 55 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	#22581678	C₂	1	1	yes	maximal, characteristic
2	#22581356	C₂	1	1	yes	maximal, characteristic
3	#22581239	C₂	1	1	yes	maximal, characteristic
4	#22581164	C₂	1	1	yes	maximal, characteristic
5	#22580683	C₂	1	1	yes	maximal, characteristic
6	#22580683	C₂	1	1	yes	maximal, characteristic
7	#332×C₁₃₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₅₅₆	—	2	1	no	—
9	#17984842	C₂²	1	1	yes	characteristic
10	#17984840	—	2	1	no	—
11	#17984833	—	2	1	no	—
12	#17984832	—	2	2	no	—
13	Q₈×Dic₁₃₉	—	2	1	no	—
14	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
15	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
16	#17984807	—	2	1	no	—
17	#17984775	—	2	2	no	—
18	#17984774	—	2	1	no	—
19	#17984767	—	2	1	no	—
20	#17984767	—	2	1	no	—
21	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
22	C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
25	C₂×Dic₂₇₈	—	2	1	no	—
26	C₂×Dic₂₇₈	D₄	1	1	yes	characteristic
27	C₂×Dic₂₇₈	D₄	1	1	yes	characteristic
28	Dic₅₅₆	—	4	2	no	—
29	C₂×C₁₃₉⋊C₈	—	2	1	no	—
30	C₄⋊Dic₁₃₉	—	2	2	no	—
31	C₄⋊Dic₁₃₉	—	2	1	no	—
32	C₄⋊Dic₁₃₉	—	2	1	no	—
33	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
34	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
35	Dic₁₃₉⋊C₄	—	2	2	no	—
36	Dic₁₃₉⋊C₄	—	2	2	no	—
37	Dic₁₃₉⋊C₄	—	2	1	no	—
38	Dic₁₃₉⋊C₄	—	2	1	no	—
39	C₄×Dic₁₃₉	—	2	2	no	—
40	C₄×Dic₁₃₉	—	2	1	no	—
41	C₄×Dic₁₃₉	—	2	1	no	—
42	C₂×Q₈×C₁₃₉	—	2	1	no	—
43	C₂×C₈×C₁₃₉	—	2	1	no	—
44	C₂×C₈×C₁₃₉	—	2	1	no	—
45	C₄⋊C₄×C₁₃₉	—	2	1	no	—
46	C₄⋊C₄×C₁₃₉	D₄	1	2	yes	—
47	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
48	Dic₂₇₈	—	2	1	no	—
49	Dic₂₇₈	—	2	1	no	—
50	Dic₂₇₈	—	2	1	no	—
51	Dic₂₇₈	—	4	1	no	—
52	Dic₂₇₈	—	4	1	no	—
53	Dic₂₇₈	Q₁₆	1	2	yes	complemented
54	C₂×Dic₁₃₉	—	2	1	no	—
55	C₂×Dic₁₃₉	—	2	1	no	—
56	C₂×Dic₁₃₉	—	2	1	no	—
57	C₂×Dic₁₃₉	—	2	1	no	—
58	C₂×Dic₁₃₉	—	4	1	no	—
59	C₁₃₉⋊C₈	—	2	2	no	—
60	Q₈×C₁₃₉	—	4	1	no	—
61	Q₈×C₁₃₉	—	4	1	no	—
62	C₂×C₄×C₁₃₉	—	2	2	no	—
63	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
66	C₈×C₁₃₉	—	4	1	no	—
67	C₈×C₁₃₉	—	4	1	no	—
68	C₂²×C₁₃₉	#120	1	1	yes	characteristic
69	C₄×C₁₃₉	—	2	1	no	—
70	C₄×C₁₃₉	—	4	2	no	—
71	C₄×C₁₃₉	C₂×Q₁₆	1	1	yes	characteristic
72	C₄×C₁₃₉	#135	1	1	yes	characteristic
73	C₄×C₁₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₄×C₁₃₉	#137	1	1	yes	characteristic
75	Dic₁₃₉	—	2	2	no	—
76	Dic₁₃₉	—	4	1	no	—
77	Dic₁₃₉	—	4	1	no	—
78	Dic₁₃₉	—	4	1	no	—
79	Dic₁₃₉	—	8	1	no	—
80	C₂×C₁₃₉	#449	1	1	yes	characteristic
81	C₂×C₁₃₉	#451	1	1	yes	characteristic
82	C₂×C₁₃₉	#456	1	1	yes	characteristic
83	C₁₃₉	#1648	1	1	yes	characteristic, complemented, Sylow
84	#1648	—	139	1	no	maximal, Sylow
85	#557	—	139	1	no	—
86	#500	—	139	1	no	—
87	#475	—	139	1	no	—
88	#462	—	139	1	no	—
89	#332	D₁₃₉	1	1	yes	characteristic
90	Q₈⋊C₈	—	139	1	no	—
91	Q₈⋊C₈	—	139	1	no	—
92	C₂×Q₁₆	—	278	1	no	—
93	C₄⋊Q₈	—	278	1	no	—
94	C₄⋊Q₈	—	139	1	no	—
95	C₄⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
96	#125	—	278	1	no	—
97	#125	—	278	2	no	—
98	C₄×Q₈	—	278	1	no	—
99	C₄×Q₈	—	278	1	no	—
100	C₄×Q₈	—	139	1	no	—
101	C₄×Q₈	—	139	1	no	—
102	#107	—	278	2	no	—
103	#106	—	278	1	no	—
104	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
105	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
106	#103	—	278	1	no	—
107	#103	—	278	1	no	—
108	C₄×C₈	—	139	1	no	—
109	C₂×Q₈	—	2	1	no	—
110	C₂×Q₈	—	278	1	no	—
111	C₂×Q₈	—	139	1	no	—
112	C₂×Q₈	—	139	1	no	—
113	Q₁₆	—	556	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	278	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	278	2	no	—
119	C₄⋊C₄	—	278	2	no	—
120	C₄⋊C₄	—	278	2	no	—
121	C₄⋊C₄	—	278	1	no	—
122	C₄⋊C₄	—	278	1	no	—
123	C₄⋊C₄	—	278	1	no	—
124	C₄⋊C₄	—	278	1	no	—
125	C₄⋊C₄	—	139	1	no	—
126	C₄⋊C₄	—	139	1	no	—
127	C₄⋊C₄	D₅₅₆	1	2	yes	—
128	C₄²	—	278	2	no	—
129	C₄²	—	278	1	no	—
130	C₄²	—	278	1	no	—
131	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
132	Q₈	—	4	1	no	—
133	Q₈	—	4	1	no	—
134	Q₈	—	278	1	no	—
135	Q₈	—	278	1	no	—
136	Q₈	—	278	1	no	—
137	Q₈	—	556	1	no	—
138	Q₈	—	556	1	no	—
139	Q₈	—	139	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	278	1	no	—
142	C₂×C₄	—	278	1	no	—
143	C₂×C₄	—	278	1	no	—
144	C₂×C₄	—	278	1	no	—
145	C₂×C₄	—	556	1	no	—
146	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
147	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
148	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	4	1	no	—
151	C₈	—	278	2	no	—
152	C₂²	#17984812	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	4	2	no	—
155	C₄	—	278	2	no	—
156	C₄	—	556	1	no	—
157	C₄	—	556	1	no	—
158	C₄	—	556	1	no	—
159	C₄	—	1112	1	no	—
160	C₄	Q₁₆×D₁₃₉	1	1	yes	characteristic
161	C₄	#17984863	1	1	yes	characteristic
162	C₄	#17984866	1	1	yes	characteristic
163	C₄	#17984871	1	1	yes	characteristic
164	C₂	#22581099	1	1	yes	characteristic
165	C₂	#22581110	1	1	yes	characteristic
166	C₂	#22581138	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581678

C₂

yes

maximal, characteristic

#22581356

C₂

yes

maximal, characteristic

#22581239

C₂

yes

maximal, characteristic

#22581164

C₂

yes

maximal, characteristic

#22580683

C₂

yes

maximal, characteristic

#22580683

C₂

yes

maximal, characteristic

#332×C₁₃₉

C₂

yes

maximal, characteristic

C₂×Dic₅₅₆

—

#17984842

C₂²

yes

—

—

—

Q₈×Dic₁₃₉

—

C₄×Dic₂₇₈

C₂²

yes

characteristic

C₄×Dic₂₇₈

C₂²

yes

—

—

—

—

—

C₄×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

D₄

yes

characteristic

C₂×Dic₂₇₈

D₄

yes

characteristic

Dic₅₅₆

—

C₂×C₁₃₉⋊C₈

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

D₄

yes

characteristic

C₄⋊Dic₁₃₉

D₄

yes

characteristic

Dic₁₃₉⋊C₄

—

Dic₁₃₉⋊C₄

—

Dic₁₃₉⋊C₄

—

Dic₁₃₉⋊C₄

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₂×Q₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

D₄

yes

—

C₄²×C₁₃₉

C₂³

yes

characteristic

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

Q₁₆

yes

complemented

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

Q₈×C₁₃₉

—

Q₈×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₈×C₁₃₉

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

#120

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×Q₁₆

yes

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#1648

—

139

maximal, Sylow

#557

—

139

—

#500

—

139

—

#475

—

139

—

#462

—

139

—

#332

D₁₃₉

yes

characteristic

Q₈⋊C₈

—

139

—

Q₈⋊C₈

—

139

—

C₂×Q₁₆

—

278

—

C₄⋊Q₈

—

278

—

C₄⋊Q₈

—

139

—

C₄⋊Q₈

C₂×D₁₃₉

yes

characteristic

#125

—

278

—

#125

—

278

—

C₄×Q₈

—

278

—

C₄×Q₈

—

278

—

100

C₄×Q₈

—

139

—

101

C₄×Q₈

—

139

—

102

#107

—

278

—

103

#106

—

278

—

104

C₄⋊C₈

C₂×D₁₃₉

yes

characteristic

105

C₄⋊C₈

C₂×D₁₃₉

yes

characteristic

106

#103

—

278

—

107

#103

—

278

—

108

C₄×C₈

—

139

—

109

C₂×Q₈

—

110

C₂×Q₈

—

278

—

111

C₂×Q₈

—

139

—

112

C₂×Q₈

—

139

—

113

Q₁₆

—

556

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

278

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

278

—

119

C₄⋊C₄

—

278

—

120

C₄⋊C₄

—

278

—

121

C₄⋊C₄

—

278

—

122

C₄⋊C₄

—

278

—

123

C₄⋊C₄

—

278

—

124

C₄⋊C₄

—

278

—

125

C₄⋊C₄

—

139

—

126

C₄⋊C₄

—

139

—

127

C₄⋊C₄

D₅₅₆

yes

—

128

C₄²

—

278

—

129

C₄²

—

278

—

130

C₄²

—

278

—

131

C₄²

C₂²×D₁₃₉

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

278

—

135

Q₈

—

278

—

136

Q₈

—

278

—

137

Q₈

—

556

—

138

Q₈

—

556

—

139

Q₈

—

139

—

140

C₂×C₄

—

141

C₂×C₄

—

278

—

142

C₂×C₄

—

278

—

143

C₂×C₄

—

278

—

144

C₂×C₄

—

278

—

145

C₂×C₄

—

556

—

146

C₂×C₄

D₄×D₁₃₉

yes

characteristic

147

C₂×C₄

D₄×D₁₃₉

yes

characteristic

148

C₂×C₄

C₂×D₅₅₆

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

278

—

152

C₂²

#17984812

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

278

—

156

C₄

—

556

—

157

C₄

—

556

—

158

C₄

—

556

—

159

C₄

—

1112

—

160

C₄

Q₁₆×D₁₃₉

yes

characteristic

161

C₄

#17984863

yes

characteristic

162

C₄

#17984866

yes

characteristic

163

C₄

#17984871

yes

characteristic

164

C₂

#22581099

yes

characteristic

165

C₂

#22581110

yes

characteristic

166

C₂

#22581138

yes

characteristic

	subgroup	G/H
center	C₂²	#17984812
commutator, cosocle	C₄²×C₁₃₉	C₂³
Frattini	C₄²	C₂²×D₁₃₉
Fitting	#332×C₁₃₉	C₂
socle, prime radical	C₂²×C₁₃₉	#120
hypercenter	#332	D₁₃₉
hypocenter	C₁₃₉	#1648
LCS	G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#17984812

commutator, cosocle

C₄²×C₁₃₉

C₂³

Frattini

C₄²

C₂²×D₁₃₉

Fitting

#332×C₁₃₉

C₂

C₂²×C₁₃₉

D₁₃₉

C₁₃₉

G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
53. Dic₂₇₈	Q₁₆	semidirect
83. C₁₃₉	#1648	semidirect

H ≅ G/N

type

53. Dic₂₇₈

Q₁₆

semidirect

83. C₁₃₉

#1648

semidirect

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

subgroup	index
#22581678	2
#22581356	2
#22581239	2
#22581164	2
#22580683	2
#332×C₁₃₉	2
#1648	139

subgroup

index

#332×C₁₃₉

139

Subgroups — #31805647

Conjugacy classes of subgroups

Complements

Maximal subgroups