Group #31806428 — subgroups

Subgroups — #31806428

Order 17792.

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

normal subgroups only

This group has 336 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×C₈×D₁₃₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984773	C₂	1	2	yes	maximal, complemented
3	#22580810	C₂	1	2	yes	maximal
4	#22580716	C₂	1	1	yes	maximal, characteristic, complemented
5	#340×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₁₃₉	C₂²	1	1	yes	characteristic
7	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₈×D₁₃₉	—	2	4	no	—
9	C₂×C₈×D₁₃₉	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
11	C₂×(C₂×D₁₃₉)⋊C₄	—	2	2	no	—
12	#17984773	—	2	4	no	—
13	C₁₃₉⋊#95	—	2	2	no	—
14	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
16	C₂³×D₁₃₉	D₄	1	1	yes	characteristic
17	C₂²×Dic₁₃₉	—	2	2	no	—
18	C₂²×Dic₁₃₉	D₄	1	1	yes	characteristic
19	C₂×C₄×D₁₃₉	—	2	2	no	—
20	C₂×C₄×D₁₃₉	—	2	2	no	—
21	C₂×C₄×D₁₃₉	C₂×C₄	1	2	yes	—
22	C₂×C₄×D₁₃₉	Q₈	1	2	yes	—
23	C₂×C₁₃₉⋊C₈	C₂×C₄	1	2	yes	—
24	C₂×C₁₃₉⋊C₈	D₄	1	4	yes	—
25	C₈×D₁₃₉	—	2	8	no	—
26	C₄⋊Dic₁₃₉	—	2	2	no	—
27	C₄⋊Dic₁₃₉	—	2	2	no	—
28	(C₂×D₁₃₉)⋊C₄	—	4	4	no	—
29	C₂²×C₄×C₁₃₉	—	2	2	no	—
30	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₁₃₉	—	2	2	no	—
32	C₂×C₈×C₁₃₉	C₂×C₄	1	2	yes	—
33	C₄⋊C₄×C₁₃₉	—	2	2	no	—
34	C₄⋊C₄×C₁₃₉	—	2	2	no	—
35	C₂²×D₁₃₉	—	2	2	no	—
36	C₂²×D₁₃₉	—	2	1	no	—
37	C₂²×D₁₃₉	—	2	1	no	—
38	C₂²×D₁₃₉	C₄⋊C₄	1	2	yes	—
39	C₂²×D₁₃₉	SD₁₆	1	4	yes	—
40	C₂×Dic₁₃₉	—	2	2	no	—
41	C₂×Dic₁₃₉	—	4	2	no	—
42	C₂×Dic₁₃₉	—	4	2	no	—
43	C₂×Dic₁₃₉	—	4	2	no	—
44	C₂×Dic₁₃₉	C₄⋊C₄	1	2	yes	—
45	C₄×D₁₃₉	—	2	2	no	—
46	C₄×D₁₃₉	—	2	2	no	—
47	C₄×D₁₃₉	—	2	4	no	—
48	C₁₃₉⋊C₈	C₂²⋊C₄	1	4	yes	complemented
49	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₃₉	—	4	2	no	—
51	C₂×C₄×C₁₃₉	—	4	2	no	—
52	C₂×C₄×C₁₃₉	—	4	2	no	—
53	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₁₃₉	#41	1	2	yes	—
57	C₈×C₁₃₉	—	2	2	no	—
58	C₂²×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₃₉	#121	1	2	yes	—
60	C₂²×C₁₃₉	C₂×SD₁₆	1	2	yes	—
61	C₂²×C₁₃₉	#135	1	1	yes	characteristic
62	C₂²×C₁₃₉	#135	1	1	yes	characteristic
63	C₂×D₁₃₉	—	2	2	no	—
64	C₂×D₁₃₉	—	2	2	no	—
65	C₂×D₁₃₉	—	2	4	no	—
66	C₂×D₁₃₉	—	2	4	no	—
67	C₂×D₁₃₉	#106	1	4	yes	complemented
68	C₄×C₁₃₉	—	8	2	no	—
69	C₄×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₄×C₁₃₉	#117	1	1	yes	characteristic
71	C₄×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
72	Dic₁₃₉	—	2	2	no	—
73	Dic₁₃₉	—	8	2	no	—
74	C₂×C₁₃₉	#380	1	1	yes	characteristic
75	C₂×C₁₃₉	C₂×#106	1	1	yes	characteristic
76	C₂×C₁₃₉	#427	1	1	yes	characteristic
77	C₂×C₁₃₉	#465	1	2	yes	—
78	C₂×C₁₃₉	#465	1	2	yes	—
79	D₁₃₉	—	2	4	no	—
80	C₁₃₉	#1807	1	1	yes	characteristic, complemented, Sylow
81	#1807	—	139	1	no	maximal, Sylow
82	C₂³×C₈	—	139	1	no	—
83	C₂×#106	—	139	2	no	—
84	#380	—	139	2	no	—
85	#340	—	139	1	no	—
86	#340	D₁₃₉	1	1	yes	characteristic, complemented
87	C₂³×C₄	—	139	1	no	—
88	C₂²×C₈	—	278	4	no	—
89	C₂²×C₈	—	139	4	no	—
90	C₂²×C₈	—	139	1	no	—
91	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
92	C₂×C₄⋊C₄	—	139	2	no	—
93	C₂×C₄⋊C₄	C₂×D₁₃₉	1	2	yes	—
94	C₂×C₂²⋊C₄	—	278	2	no	—
95	#106	—	278	4	no	—
96	#95	—	278	2	no	—
97	C₂⁴	—	139	1	no	—
98	C₂²×C₄	—	2	2	no	—
99	C₂²×C₄	—	278	2	no	—
100	C₂²×C₄	—	278	2	no	—
101	C₂²×C₄	—	278	2	no	—
102	C₂²×C₄	—	139	1	no	—
103	C₂²×C₄	—	139	2	no	—
104	C₂²×C₄	—	139	2	no	—
105	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	278	8	no	—
108	C₂×C₈	—	139	2	no	—
109	C₂×C₈	—	139	4	no	—
110	C₂×C₈	C₄×D₁₃₉	1	2	yes	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	278	2	no	—
114	C₄⋊C₄	—	278	2	no	—
115	C₂²⋊C₄	—	556	4	no	—
116	C₂³	—	278	2	no	—
117	C₂³	—	278	1	no	—
118	C₂³	—	278	1	no	—
119	C₂³	—	139	2	no	—
120	C₂³	—	139	4	no	—
121	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	278	2	no	—
126	C₂×C₄	—	278	2	no	—
127	C₂×C₄	—	278	2	no	—
128	C₂×C₄	—	278	4	no	—
129	C₂×C₄	—	556	2	no	—
130	C₂×C₄	—	556	2	no	—
131	C₂×C₄	—	556	2	no	—
132	C₂×C₄	—	139	2	no	—
133	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
134	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
135	C₂×C₄	Q₈×D₁₃₉	1	1	yes	characteristic
136	C₂×C₄	#14555785	1	2	yes	—
137	C₈	—	2	2	no	—
138	C₈	—	139	4	no	—
139	C₂²	—	278	2	no	—
140	C₂²	—	278	4	no	—
141	C₂²	—	278	4	no	—
142	C₂²	—	278	2	no	—
143	C₂²	—	139	4	no	—
144	C₂²	C₄⋊C₄×D₁₃₉	1	1	yes	characteristic
145	C₂²	#17984815	1	2	yes	—
146	C₂²	SD₁₆×D₁₃₉	1	2	yes	—
147	C₂²	#17984861	1	1	yes	characteristic
148	C₂²	#17984865	1	1	yes	characteristic
149	C₄	—	8	2	no	—
150	C₄	—	278	2	no	—
151	C₄	—	1112	2	no	—
152	C₄	C₂²⋊C₄×D₁₃₉	1	1	yes	characteristic
153	C₄	#17984799	1	1	yes	characteristic
154	C₄	#17984820	1	2	yes	—
155	C₂	—	278	4	no	—
156	C₂	#22580812	1	1	yes	characteristic
157	C₂	#106×D₁₃₉	1	1	yes	characteristic
158	C₂	#22580990	1	1	yes	characteristic
159	C₂	#22581181	1	2	yes	—
160	C₂	#22581185	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×D₁₃₉

C₂

yes

maximal, characteristic

C₂×#17984773

C₂

yes

maximal, complemented

#22580810

C₂

yes

maximal

#22580716

C₂

yes

maximal, characteristic, complemented

#340×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₁₃₉

C₂²

yes

characteristic

C₂²×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₁₃₉

—

C₂×C₈×D₁₃₉

C₄

yes

complemented

C₂×C₄⋊Dic₁₃₉

C₂²

yes

—

C₂×(C₂×D₁₃₉)⋊C₄

—

#17984773

—

C₁₃₉⋊#95

—

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₃₉

C₂²

yes

—

C₂³×D₁₃₉

D₄

yes

characteristic

C₂²×Dic₁₃₉

—

C₂²×Dic₁₃₉

D₄

yes

characteristic

C₂×C₄×D₁₃₉

—

C₂×C₄×D₁₃₉

—

C₂×C₄×D₁₃₉

C₂×C₄

yes

—

C₂×C₄×D₁₃₉

Q₈

yes

—

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

—

C₂×C₁₃₉⋊C₈

D₄

yes

—

C₈×D₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

(C₂×D₁₃₉)⋊C₄

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

C₄⋊C₄

yes

—

C₂²×D₁₃₉

SD₁₆

yes

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

C₄⋊C₄

yes

—

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₁₃₉⋊C₈

C₂²⋊C₄

yes

complemented

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₃₉

#121

yes

—

C₂²×C₁₃₉

C₂×SD₁₆

yes

—

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

#106

yes

complemented

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₃₉

#117

yes

characteristic

C₄×C₁₃₉

C₂²⋊Q₈

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

—

C₂×C₁₃₉

#465

yes

—

D₁₃₉

—

C₁₃₉

#1807

yes

characteristic, complemented, Sylow

#1807

—

139

maximal, Sylow

C₂³×C₈

—

139

—

C₂×#106

—

139

—

#380

—

139

—

#340

—

139

—

#340

D₁₃₉

yes

characteristic, complemented

C₂³×C₄

—

139

—

C₂²×C₈

—

278

—

C₂²×C₈

—

139

—

C₂²×C₈

—

139

—

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

C₂×C₄⋊C₄

—

139

—

C₂×C₄⋊C₄

C₂×D₁₃₉

yes

—

C₂×C₂²⋊C₄

—

278

—

#106

—

278

—

#95

—

278

—

C₂⁴

—

139

—

C₂²×C₄

—

C₂²×C₄

—

278

—

100

C₂²×C₄

—

278

—

101

C₂²×C₄

—

278

—

102

C₂²×C₄

—

139

—

103

C₂²×C₄

—

139

—

104

C₂²×C₄

—

139

—

105

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

278

—

108

C₂×C₈

—

139

—

109

C₂×C₈

—

139

—

110

C₂×C₈

C₄×D₁₃₉

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

278

—

114

C₄⋊C₄

—

278

—

115

C₂²⋊C₄

—

556

—

116

C₂³

—

278

—

117

C₂³

—

278

—

118

C₂³

—

278

—

119

C₂³

—

139

—

120

C₂³

—

139

—

121

C₂³

D₄×D₁₃₉

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

278

—

126

C₂×C₄

—

278

—

127

C₂×C₄

—

278

—

128

C₂×C₄

—

278

—

129

C₂×C₄

—

556

—

130

C₂×C₄

—

556

—

131

C₂×C₄

—

556

—

132

C₂×C₄

—

139

—

133

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

134

C₂×C₄

D₄×D₁₃₉

yes

—

135

C₂×C₄

Q₈×D₁₃₉

yes

characteristic

136

C₂×C₄

#14555785

yes

—

137

C₈

—

138

C₈

—

139

—

139

C₂²

—

278

—

140

C₂²

—

278

—

141

C₂²

—

278

—

142

C₂²

—

278

—

143

C₂²

—

139

—

144

C₂²

C₄⋊C₄×D₁₃₉

yes

characteristic

145

C₂²

#17984815

yes

—

146

C₂²

SD₁₆×D₁₃₉

yes

—

147

C₂²

#17984861

yes

characteristic

148

C₂²

#17984865

yes

characteristic

149

C₄

—

150

C₄

—

278

—

151

C₄

—

1112

—

152

C₄

C₂²⋊C₄×D₁₃₉

yes

characteristic

153

C₄

#17984799

yes

characteristic

154

C₄

#17984820

yes

—

155

C₂

—

278

—

156

C₂

#22580812

yes

characteristic

157

C₂

#106×D₁₃₉

yes

characteristic

158

C₂

#22580990

yes

characteristic

159

C₂

#22581181

yes

—

160

C₂

#22581185

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₃₉
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#340×C₁₃₉	C₂
socle	C₂³×C₁₃₉	C₂×D₄
prime radical	C₂³×D₁₃₉	D₄
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	#340	D₁₃₉
hypocenter	C₁₃₉	#1807
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₁₃₉

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

#340×C₁₃₉

C₂

socle

C₂³×C₁₃₉

C₂×D₄

prime radical

C₂³×D₁₃₉

D₄

cosocle

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
2. C₂×#17984773	C₂	semidirect
4. #22580716	C₂	semidirect
5. #340×C₁₃₉	C₂	semidirect
9. C₂×C₈×D₁₃₉	C₄	semidirect
48. C₁₃₉⋊C₈	C₂²⋊C₄	semidirect
67. C₂×D₁₃₉	#106	semidirect
80. C₁₃₉	#1807	semidirect
86. #340	D₁₃₉	semidirect

H ≅ G/N

type

2. C₂×#17984773

C₂

semidirect

4. #22580716

C₂

semidirect

5. #340×C₁₃₉

C₂

semidirect

9. C₂×C₈×D₁₃₉

C₄

semidirect

48. C₁₃₉⋊C₈

C₂²⋊C₄

semidirect

67. C₂×D₁₃₉

#106

semidirect

80. C₁₃₉

#1807

semidirect

86. #340

D₁₃₉

semidirect

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

subgroup	index
C₂²×C₈×D₁₃₉	2
C₂×#17984773	2
#22580810	2
#22580716	2
#340×C₁₃₉	2
#1807	139

subgroup

index

C₂²×C₈×D₁₃₉

#340×C₁₃₉

139

Subgroups — #31806428

Conjugacy classes of subgroups

Complements

Maximal subgroups