Group #31806202 — subgroups

Subgroups — #101×Dic₁₃₉ #31806202

Order 17792.

296 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 119 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	C₂×Q₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984761	C₂	1	1	yes	maximal, characteristic
3	M₁₆×Dic₁₃₉	C₂	1	2	yes	maximal
4	#22580721	C₂	1	2	yes	maximal
5	C₂×#101×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555769	C₂²	1	2	yes	—
7	Q₈×Dic₁₃₉	—	2	2	no	—
8	Q₈×Dic₁₃₉	—	2	2	no	—
9	C₂×C₄⋊Dic₁₃₉	C₄	1	2	yes	—
10	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₃₉	C₄	1	2	yes	—
12	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
13	#17984761	C₄	1	4	yes	complemented
14	#17984752	—	2	2	no	—
15	C₈×Dic₁₃₉	—	2	2	no	—
16	C₂²×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₁₃₉	C₂²	1	2	yes	—
18	#101×C₁₃₉	C₄	1	4	yes	complemented
19	C₂²×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×Dic₁₃₉	C₂×C₄	1	2	yes	—
21	C₂²×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
22	#14555769	—	2	2	no	—
23	#14555769	C₂×C₄	1	4	yes	—
24	C₂×C₁₃₉⋊C₈	—	2	2	no	—
25	C₄⋊Dic₁₃₉	—	2	4	no	—
26	C₄⋊Dic₁₃₉	—	2	2	no	—
27	C₄×Dic₁₃₉	—	2	2	no	—
28	C₄×Dic₁₃₉	—	2	2	no	—
29	C₄×Dic₁₃₉	D₄	1	2	yes	—
30	C₄×Dic₁₃₉	D₄	1	2	yes	—
31	C₂×Q₈×C₁₃₉	—	2	2	no	—
32	C₂×Q₈×C₁₃₉	—	2	2	no	—
33	C₂×Q₈×C₁₃₉	C₂×C₄	1	2	yes	—
34	C₂×Q₈×C₁₃₉	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₁₃₉	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
37	M₁₆×C₁₃₉	—	2	2	no	—
38	M₁₆×C₁₃₉	C₂×C₄	1	4	yes	—
39	C₂×C₈×C₁₃₉	—	2	2	no	—
40	C₂×Dic₁₃₉	—	2	2	no	—
41	C₂×Dic₁₃₉	—	2	2	no	—
42	C₂×Dic₁₃₉	—	4	1	no	—
43	C₂×Dic₁₃₉	—	4	2	no	—
44	C₂×Dic₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×Dic₁₃₉	C₂²⋊C₄	1	2	yes	—
46	C₂×Dic₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₁₃₉⋊C₈	—	2	4	no	—
48	C₂³×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
49	Q₈×C₁₃₉	—	2	4	no	—
50	Q₈×C₁₃₉	—	2	4	no	—
51	C₂×C₄×C₁₃₉	—	2	2	no	—
52	C₂×C₄×C₁₃₉	—	2	2	no	—
53	C₂×C₄×C₁₃₉	C₄²	1	4	yes	—
54	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₁₃₉	#41	1	2	yes	—
58	C₈×C₁₃₉	—	2	4	no	—
59	C₂²×C₁₃₉	—	2	1	no	—
60	C₂²×C₁₃₉	—	2	1	no	—
61	C₂²×C₁₃₉	C₂×C₄²	1	1	yes	characteristic
62	C₂²×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₁₃₉	#117	1	1	yes	characteristic
64	C₄×C₁₃₉	—	2	4	no	—
65	C₄×C₁₃₉	C₄×D₄	1	2	yes	—
66	C₄×C₁₃₉	C₄×D₄	1	2	yes	—
67	Dic₁₃₉	—	2	1	no	—
68	Dic₁₃₉	—	2	2	no	—
69	Dic₁₃₉	—	4	2	no	—
70	Dic₁₃₉	#101	1	2	yes	direct product factor
71	C₂×C₁₃₉	—	2	1	no	—
72	C₂×C₁₃₉	—	2	1	no	—
73	C₂×C₁₃₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₁₃₉	C₂×#101	1	1	yes	characteristic
75	C₂×C₁₃₉	#413	1	1	yes	characteristic
76	C₁₃₉	C₄×#101	1	1	yes	characteristic, complemented, Sylow
77	C₄×#101	—	139	1	no	maximal, Sylow
78	C₂×C₄×Q₈	—	139	1	no	—
79	C₂×#101	—	139	1	no	—
80	C₂×#101	D₁₃₉	1	1	yes	characteristic
81	C₄×M₁₆	—	139	2	no	—
82	#343	—	139	2	no	—
83	C₂²×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
84	C₂×M₁₆	—	139	2	no	—
85	C₂×M₁₆	C₂×D₁₃₉	1	2	yes	—
86	C₄×Q₈	—	278	2	no	—
87	C₄×Q₈	—	278	2	no	—
88	C₂×C₄⋊C₄	—	139	2	no	—
89	C₂×C₄⋊C₄	—	139	1	no	—
90	C₂×C₄²	—	139	2	no	—
91	C₂×C₄²	—	139	1	no	—
92	#101	—	139	4	no	—
93	#101	Dic₁₃₉	1	4	yes	direct product factor
94	#97	—	278	2	no	—
95	C₄×C₈	—	278	2	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	C₄×D₁₃₉	1	2	yes	—
99	C₂×Q₈	C₂×Dic₁₃₉	1	2	yes	—
100	C₂²×C₄	—	139	1	no	—
101	C₂²×C₄	—	139	2	no	—
102	C₂²×C₄	—	139	1	no	—
103	C₂²×C₄	C₄×D₁₃₉	1	2	yes	—
104	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
105	M₁₆	—	2	2	no	—
106	M₁₆	—	278	2	no	—
107	M₁₆	—	139	4	no	—
108	M₁₆	C₂×Dic₁₃₉	1	4	yes	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	278	2	no	—
111	C₄⋊C₄	—	278	4	no	—
112	C₄⋊C₄	—	278	2	no	—
113	C₄²	—	278	2	no	—
114	C₄²	—	278	2	no	—
115	C₄²	—	139	2	no	—
116	C₄²	—	139	2	no	—
117	C₂³	C₂×C₄×D₁₃₉	1	1	yes	characteristic
118	Q₈	—	2	4	no	—
119	Q₈	—	2	4	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	278	2	no	—
123	C₂×C₄	—	278	2	no	—
124	C₂×C₄	—	556	1	no	—
125	C₂×C₄	—	556	2	no	—
126	C₂×C₄	—	139	1	no	—
127	C₂×C₄	—	139	2	no	—
128	C₂×C₄	—	139	1	no	—
129	C₂×C₄	C₄×Dic₁₃₉	1	4	yes	—
130	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
131	C₂×C₄	C₂²×Dic₁₃₉	1	1	yes	characteristic
132	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
133	C₂×C₄	#14555784	1	2	yes	—
134	C₈	—	2	4	no	—
135	C₈	—	278	4	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	C₂×C₄×Dic₁₃₉	1	1	yes	characteristic
139	C₂²	C₂²⋊C₄×D₁₃₉	1	1	yes	characteristic
140	C₂²	#17984798	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	278	1	no	—
143	C₄	—	278	2	no	—
144	C₄	—	556	2	no	—
145	C₄	—	139	2	no	—
146	C₄	(C₂²⋊D₁₃₉)⋊C₄	1	2	yes	—
147	C₄	D₄×Dic₁₃₉	1	2	yes	—
148	C₂	—	2	1	no	—
149	C₂	—	2	1	no	—
150	C₂	C₂²⋊C₄×Dic₁₃₉	1	1	yes	characteristic
151	C₂	#101×D₁₃₉	1	1	yes	characteristic
152	C₂	#22580936	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₁₃₉

C₂

yes

maximal, characteristic

C₂×#17984761

C₂

yes

maximal, characteristic

M₁₆×Dic₁₃₉

C₂

yes

maximal

#22580721

C₂

yes

maximal

C₂×#101×C₁₃₉

C₂

yes

maximal, characteristic

C₂×#14555769

C₂²

yes

—

Q₈×Dic₁₃₉

—

Q₈×Dic₁₃₉

—

C₂×C₄⋊Dic₁₃₉

C₄

yes

—

C₂×C₄⋊Dic₁₃₉

C₂²

yes

characteristic

C₂×C₄×Dic₁₃₉

C₄

yes

—

C₂×C₄×Dic₁₃₉

C₂²

yes

characteristic

#17984761

C₄

yes

complemented

#17984752

—

C₈×Dic₁₃₉

—

C₂²×Q₈×C₁₃₉

C₂²

yes

characteristic

C₂×M₁₆×C₁₃₉

C₂²

yes

—

#101×C₁₃₉

C₄

yes

complemented

C₂²×Dic₁₃₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₃₉

C₂×C₄

yes

—

C₂²×Dic₁₃₉

C₂×C₄

yes

characteristic

#14555769

—

#14555769

C₂×C₄

yes

—

C₂×C₁₃₉⋊C₈

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

D₄

yes

—

C₄×Dic₁₃₉

D₄

yes

—

C₂×Q₈×C₁₃₉

—

C₂×Q₈×C₁₃₉

—

C₂×Q₈×C₁₃₉

C₂×C₄

yes

—

C₂×Q₈×C₁₃₉

C₂×C₄

yes

—

C₂²×C₄×C₁₃₉

C₂×C₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

M₁₆×C₁₃₉

—

M₁₆×C₁₃₉

C₂×C₄

yes

—

C₂×C₈×C₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₃₉

C₂²⋊C₄

yes

—

C₂×Dic₁₃₉

C₂²⋊C₄

yes

characteristic

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

C₂²×C₄

yes

characteristic

Q₈×C₁₃₉

—

Q₈×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₄²

yes

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

#41

yes

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×C₄²

yes

characteristic

C₂²×C₁₃₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃₉

#117

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

C₄×D₄

yes

—

C₄×C₁₃₉

C₄×D₄

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

#101

yes

direct product factor

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

C₄×C₂²⋊C₄

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

C₄×#101

—

139

maximal, Sylow

C₂×C₄×Q₈

—

139

—

C₂×#101

—

139

—

C₂×#101

D₁₃₉

yes

characteristic

C₄×M₁₆

—

139

—

#343

—

139

—

C₂²×Q₈

C₂×D₁₃₉

yes

characteristic

C₂×M₁₆

—

139

—

C₂×M₁₆

C₂×D₁₃₉

yes

—

C₄×Q₈

—

278

—

C₄×Q₈

—

278

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄²

—

139

—

C₂×C₄²

—

139

—

#101

—

139

—

#101

Dic₁₃₉

yes

direct product factor

#97

—

278

—

C₄×C₈

—

278

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₄×D₁₃₉

yes

—

C₂×Q₈

C₂×Dic₁₃₉

yes

—

100

C₂²×C₄

—

139

—

101

C₂²×C₄

—

139

—

102

C₂²×C₄

—

139

—

103

C₂²×C₄

C₄×D₁₃₉

yes

—

104

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

105

M₁₆

—

106

M₁₆

—

278

—

107

M₁₆

—

139

—

108

M₁₆

C₂×Dic₁₃₉

yes

—

109

C₂×C₈

—

110

C₂×C₈

—

278

—

111

C₄⋊C₄

—

278

—

112

C₄⋊C₄

—

278

—

113

C₄²

—

278

—

114

C₄²

—

278

—

115

C₄²

—

139

—

116

C₄²

—

139

—

117

C₂³

C₂×C₄×D₁₃₉

yes

characteristic

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

278

—

123

C₂×C₄

—

278

—

124

C₂×C₄

—

556

—

125

C₂×C₄

—

556

—

126

C₂×C₄

—

139

—

127

C₂×C₄

—

139

—

128

C₂×C₄

—

139

—

129

C₂×C₄

C₄×Dic₁₃₉

yes

—

130

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

131

C₂×C₄

C₂²×Dic₁₃₉

yes

characteristic

132

C₂×C₄

D₄×D₁₃₉

yes

—

133

C₂×C₄

#14555784

yes

—

134

C₈

—

135

C₈

—

278

—

136

C₂²

—

137

C₂²

—

138

C₂²

C₂×C₄×Dic₁₃₉

yes

characteristic

139

C₂²

C₂²⋊C₄×D₁₃₉

yes

characteristic

140

C₂²

#17984798

yes

characteristic

141

C₄

—

142

C₄

—

278

—

143

C₄

—

278

—

144

C₄

—

556

—

145

C₄

—

139

—

146

C₄

(C₂²⋊D₁₃₉)⋊C₄

yes

—

147

C₄

D₄×Dic₁₃₉

yes

—

148

C₂

—

149

C₂

—

150

C₂

C₂²⋊C₄×Dic₁₃₉

yes

characteristic

151

C₂

#101×D₁₃₉

yes

characteristic

152

C₂

#22580936

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₃₉
commutator	C₂²×C₁₃₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	C₂×#101×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₃₉	C₂²×C₄
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	C₂×#101	D₁₃₉
hypocenter	C₁₃₉	C₄×#101
LCS	G ▷ C₂²×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₃₉

commutator

C₂²×C₁₃₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

C₂×#101×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₃₉

C₂²×C₄

cosocle

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂²×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
13. #17984761	C₄	semidirect
18. #101×C₁₃₉	C₄	semidirect
70. Dic₁₃₉	#101	direct (N × H)
76. C₁₃₉	C₄×#101	semidirect
93. #101	Dic₁₃₉	direct (N × H)

H ≅ G/N

type

13. #17984761

C₄

semidirect

18. #101×C₁₃₉

C₄

semidirect

70. Dic₁₃₉

#101

direct (N × H)

76. C₁₃₉

C₄×#101

semidirect

93. #101

Dic₁₃₉

direct (N × H)

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

subgroup	index
C₂×Q₈×Dic₁₃₉	2
C₂×#17984761	2
M₁₆×Dic₁₃₉	2
#22580721	2
C₂×#101×C₁₃₉	2
C₄×#101	139

subgroup

index

C₂×Q₈×Dic₁₃₉

C₂×#17984761

M₁₆×Dic₁₃₉

#22580721

C₂×#101×C₁₃₉

C₄×#101

139

Subgroups — #101×Dic₁₃₉ #31806202

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #101×Dic139 #31806202

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #101×Dic₁₃₉ #31806202