Group #31805565 — subgroups

Subgroups — #31805565

Order 17792.

188 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 57 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	#22581682	C₂	1	1	yes	maximal, characteristic
2	#22581304	C₂	1	1	yes	maximal, characteristic
3	#22581249	C₂	1	1	yes	maximal, characteristic
4	#22581248	C₂	1	1	yes	maximal, characteristic
5	#22580700	C₂	1	1	yes	maximal, characteristic
6	#22580685	C₂	1	1	yes	maximal, characteristic
7	Q₈⋊C₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic
8	#17984842	—	2	1	no	—
9	#17984842	C₂²	1	1	yes	characteristic
10	#17984842	C₂²	1	1	yes	characteristic
11	#17984833	—	2	2	no	—
12	#17984833	—	2	1	no	—
13	Q₈×Dic₁₃₉	—	2	1	no	—
14	C₄×Dic₂₇₈	—	2	1	no	—
15	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
16	#17984776	—	2	1	no	—
17	#17984775	—	2	1	no	—
18	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
19	#17984767	—	2	1	no	—
20	#17984767	—	2	2	no	—
21	#17984766	—	2	1	no	—
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	—	2	1	no	—
27	C₂×Dic₂₇₈	D₄	1	1	yes	characteristic
28	C₂×C₁₃₉⋊C₈	—	2	1	no	—
29	C₄⋊Dic₁₃₉	—	2	2	no	—
30	C₄⋊Dic₁₃₉	—	2	2	no	—
31	C₄⋊Dic₁₃₉	—	2	1	no	—
32	C₄⋊Dic₁₃₉	—	2	1	no	—
33	C₄⋊Dic₁₃₉	—	2	1	no	—
34	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
35	C₄⋊Dic₁₃₉	D₄	1	2	yes	—
36	Dic₁₃₉⋊C₄	—	2	2	no	—
37	Dic₁₃₉⋊C₄	—	2	1	no	—
38	C₄×Dic₁₃₉	—	2	2	no	—
39	C₄×Dic₁₃₉	—	2	1	no	—
40	C₂×Q₈×C₁₃₉	D₄	1	1	yes	characteristic
41	C₂×C₈×C₁₃₉	—	2	1	no	—
42	C₂×C₈×C₁₃₉	—	2	1	no	—
43	C₄⋊C₄×C₁₃₉	—	2	1	no	—
44	C₄⋊C₄×C₁₃₉	D₄	1	1	yes	characteristic
45	C₄²×C₁₃₉	—	2	1	no	—
46	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
47	Dic₂₇₈	—	2	1	no	—
48	Dic₂₇₈	—	2	1	no	—
49	Dic₂₇₈	—	4	1	no	—
50	Dic₂₇₈	—	4	1	no	—
51	Dic₂₇₈	—	4	2	no	—
52	C₂×Dic₁₃₉	—	2	1	no	—
53	C₂×Dic₁₃₉	—	2	1	no	—
54	C₂×Dic₁₃₉	—	2	2	no	—
55	C₂×Dic₁₃₉	—	4	1	no	—
56	C₁₃₉⋊C₈	—	4	1	no	—
57	Q₈×C₁₃₉	—	2	1	no	—
58	Q₈×C₁₃₉	Q₁₆	1	2	yes	—
59	C₂×C₄×C₁₃₉	—	2	1	no	—
60	C₂×C₄×C₁₃₉	—	2	1	no	—
61	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
64	C₈×C₁₃₉	—	4	1	no	—
65	C₈×C₁₃₉	—	4	1	no	—
66	C₂²×C₁₃₉	#120	1	1	yes	characteristic
67	C₄×C₁₃₉	—	2	2	no	—
68	C₄×C₁₃₉	—	2	1	no	—
69	C₄×C₁₃₉	—	4	1	no	—
70	C₄×C₁₃₉	C₂×Q₁₆	1	1	yes	characteristic
71	C₄×C₁₃₉	#135	1	1	yes	characteristic
72	C₄×C₁₃₉	#137	1	1	yes	characteristic
73	C₄×C₁₃₉	#137	1	1	yes	characteristic
74	Dic₁₃₉	—	4	1	no	—
75	Dic₁₃₉	—	4	2	no	—
76	Dic₁₃₉	—	4	1	no	—
77	Dic₁₃₉	—	8	1	no	—
78	C₂×C₁₃₉	#451	1	1	yes	characteristic
79	C₂×C₁₃₉	#452	1	1	yes	characteristic
80	C₂×C₁₃₉	#454	1	1	yes	characteristic
81	C₁₃₉	#1636	1	1	yes	characteristic, complemented, Sylow
82	#1636	—	139	1	no	maximal, Sylow
83	#557	—	139	1	no	—
84	#487	—	139	1	no	—
85	#477	—	139	1	no	—
86	#477	—	139	1	no	—
87	#332	—	139	1	no	—
88	Q₈⋊C₈	—	139	1	no	—
89	Q₈⋊C₈	D₁₃₉	1	1	yes	characteristic
90	C₄⋊Q₈	—	278	1	no	—
91	C₄⋊Q₈	—	139	1	no	—
92	C₄⋊Q₈	—	139	1	no	—
93	#125	—	278	2	no	—
94	#125	—	278	1	no	—
95	C₄×Q₈	—	278	1	no	—
96	C₄×Q₈	—	278	1	no	—
97	C₄×Q₈	—	139	1	no	—
98	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
99	#107	—	278	1	no	—
100	#107	—	278	1	no	—
101	C₄⋊C₈	—	139	1	no	—
102	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
103	#103	—	278	1	no	—
104	#103	—	278	1	no	—
105	#103	—	278	2	no	—
106	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
107	C₂×Q₈	—	278	1	no	—
108	C₂×Q₈	—	278	1	no	—
109	C₂×Q₈	—	139	1	no	—
110	C₂×Q₈	D₅₅₆	1	1	yes	characteristic
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	278	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	278	2	no	—
116	C₄⋊C₄	—	278	2	no	—
117	C₄⋊C₄	—	278	2	no	—
118	C₄⋊C₄	—	278	1	no	—
119	C₄⋊C₄	—	278	1	no	—
120	C₄⋊C₄	—	278	1	no	—
121	C₄⋊C₄	—	278	1	no	—
122	C₄⋊C₄	—	139	2	no	—
123	C₄⋊C₄	—	139	1	no	—
124	C₄⋊C₄	D₅₅₆	1	1	yes	characteristic
125	C₄²	—	2	1	no	—
126	C₄²	—	278	2	no	—
127	C₄²	—	278	1	no	—
128	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
129	Q₈	—	2	1	no	—
130	Q₈	—	278	1	no	—
131	Q₈	—	278	1	no	—
132	Q₈	—	556	2	no	—
133	Q₈	—	556	1	no	—
134	Q₈	—	556	1	no	—
135	Q₈	Dic₅₅₆	1	2	yes	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	278	1	no	—
139	C₂×C₄	—	278	1	no	—
140	C₂×C₄	—	278	2	no	—
141	C₂×C₄	—	556	1	no	—
142	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
143	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
144	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
145	C₈	—	4	1	no	—
146	C₈	—	4	1	no	—
147	C₈	—	556	1	no	—
148	C₂²	#17984812	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	1	no	—
151	C₄	—	4	1	no	—
152	C₄	—	556	1	no	—
153	C₄	—	556	2	no	—
154	C₄	—	556	1	no	—
155	C₄	—	1112	1	no	—
156	C₄	C₂×Dic₅₅₆	1	1	yes	characteristic
157	C₄	#17984863	1	1	yes	characteristic
158	C₄	#17984874	1	1	yes	characteristic
159	C₄	#17984875	1	1	yes	characteristic
160	C₂	#22581111	1	1	yes	characteristic
161	C₂	#22581118	1	1	yes	characteristic
162	C₂	#22581127	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581682

C₂

yes

maximal, characteristic

#22581304

C₂

yes

maximal, characteristic

#22581249

C₂

yes

maximal, characteristic

#22581248

C₂

yes

maximal, characteristic

#22580700

C₂

yes

maximal, characteristic

#22580685

C₂

yes

maximal, characteristic

Q₈⋊C₈×C₁₃₉

C₂

yes

maximal, characteristic

#17984842

—

#17984842

C₂²

yes

characteristic

#17984842

C₂²

yes

characteristic

#17984833

—

#17984833

—

Q₈×Dic₁₃₉

—

C₄×Dic₂₇₈

—

C₄×Dic₂₇₈

C₂²

yes

characteristic

#17984776

—

#17984775

—

C₄⋊(C₁₃₉⋊C₈)

C₂²

yes

—

—

—

C₄×Q₈×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₉

C₂²

yes

characteristic

C₄×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

D₄

yes

characteristic

C₂×C₁₃₉⋊C₈

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

D₄

yes

characteristic

C₄⋊Dic₁₃₉

D₄

yes

—

Dic₁₃₉⋊C₄

—

Dic₁₃₉⋊C₄

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₂×Q₈×C₁₃₉

D₄

yes

characteristic

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

D₄

yes

characteristic

C₄²×C₁₃₉

—

C₄²×C₁₃₉

C₂³

yes

characteristic

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

Q₈×C₁₃₉

—

Q₈×C₁₃₉

Q₁₆

yes

—

C₂×C₄×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

—

C₄×C₁₃₉

C₂×Q₁₆

yes

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#1636

—

139

maximal, Sylow

#557

—

139

—

#487

—

139

—

#477

—

139

—

#477

—

139

—

#332

—

139

—

Q₈⋊C₈

—

139

—

Q₈⋊C₈

D₁₃₉

yes

characteristic

C₄⋊Q₈

—

278

—

C₄⋊Q₈

—

139

—

C₄⋊Q₈

—

139

—

#125

—

278

—

#125

—

278

—

C₄×Q₈

—

278

—

C₄×Q₈

—

278

—

C₄×Q₈

—

139

—

C₄×Q₈

C₂×D₁₃₉

yes

characteristic

#107

—

278

—

100

#107

—

278

—

101

C₄⋊C₈

—

139

—

102

C₄⋊C₈

C₂×D₁₃₉

yes

characteristic

103

#103

—

278

—

104

#103

—

278

—

105

#103

—

278

—

106

C₄×C₈

C₂×D₁₃₉

yes

characteristic

107

C₂×Q₈

—

278

—

108

C₂×Q₈

—

278

—

109

C₂×Q₈

—

139

—

110

C₂×Q₈

D₅₅₆

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

278

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

278

—

116

C₄⋊C₄

—

278

—

117

C₄⋊C₄

—

278

—

118

C₄⋊C₄

—

278

—

119

C₄⋊C₄

—

278

—

120

C₄⋊C₄

—

278

—

121

C₄⋊C₄

—

278

—

122

C₄⋊C₄

—

139

—

123

C₄⋊C₄

—

139

—

124

C₄⋊C₄

D₅₅₆

yes

characteristic

125

C₄²

—

126

C₄²

—

278

—

127

C₄²

—

278

—

128

C₄²

C₂²×D₁₃₉

yes

characteristic

129

Q₈

—

130

Q₈

—

278

—

131

Q₈

—

278

—

132

Q₈

—

556

—

133

Q₈

—

556

—

134

Q₈

—

556

—

135

Q₈

Dic₅₅₆

yes

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

278

—

139

C₂×C₄

—

278

—

140

C₂×C₄

—

278

—

141

C₂×C₄

—

556

—

142

C₂×C₄

D₄×D₁₃₉

yes

characteristic

143

C₂×C₄

D₄×D₁₃₉

yes

characteristic

144

C₂×C₄

C₂×D₅₅₆

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

556

—

148

C₂²

#17984812

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

556

—

153

C₄

—

556

—

154

C₄

—

556

—

155

C₄

—

1112

—

156

C₄

C₂×Dic₅₅₆

yes

characteristic

157

C₄

#17984863

yes

characteristic

158

C₄

#17984874

yes

characteristic

159

C₄

#17984875

yes

characteristic

160

C₂

#22581111

yes

characteristic

161

C₂

#22581118

yes

characteristic

162

C₂

#22581127

yes

characteristic

	subgroup	G/H
center	C₂²	#17984812
commutator, cosocle	C₄²×C₁₃₉	C₂³
Frattini	C₄²	C₂²×D₁₃₉
Fitting	Q₈⋊C₈×C₁₃₉	C₂
socle, prime radical	C₂²×C₁₃₉	#120
hypercenter	Q₈⋊C₈	D₁₃₉
hypocenter	C₁₃₉	#1636
LCS	G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

subgroup

G/H

center

C₂²

#17984812

commutator, cosocle

C₄²×C₁₃₉

C₂³

Frattini

C₄²

C₂²×D₁₃₉

Fitting

Q₈⋊C₈×C₁₃₉

C₂

socle, prime radical

C₂²×C₁₃₉

#120

hypercenter

Q₈⋊C₈

D₁₃₉

hypocenter

C₁₃₉

#1636

LCS

G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

N	H ≅ G/N	type
81. C₁₃₉	#1636	semidirect

H ≅ G/N

type

81. C₁₃₉

#1636

semidirect

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

subgroup	index
#22581682	2
#22581304	2
#22581249	2
#22581248	2
#22580700	2
#22580685	2
Q₈⋊C₈×C₁₃₉	2
#1636	139

subgroup

index

Q₈⋊C₈×C₁₃₉

139

Subgroups — #31805565

Conjugacy classes of subgroups

Complements

Maximal subgroups