Group #31806509 — subgroups

Subgroups — #31806509

Order 17792.

262 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 81 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	#41×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#22580927	C₂	1	1	yes	maximal, characteristic
3	#22580714	C₂	1	2	yes	maximal, complemented
4	#22580710	C₂	1	2	yes	maximal, complemented
5	#413×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₁₃₉	—	2	1	no	—
7	D₄×Dic₁₃₉	—	2	1	no	—
8	D₄×Dic₁₃₉	—	2	2	no	—
9	#17984801	C₄	1	2	yes	complemented
10	#17984801	C₂²	1	2	yes	—
11	#17984801	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₁₃₉	C₄	1	2	yes	complemented
13	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
14	#17984757	—	2	2	no	—
15	C₂×#41×C₁₃₉	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₁₃₉	C₂²	1	2	yes	—
17	#101×C₁₃₉	—	2	1	no	—
18	#100×C₁₃₉	—	2	1	no	—
19	C₂²×Dic₁₃₉	—	2	2	no	—
20	C₂²×Dic₁₃₉	—	2	1	no	—
21	C₄⋊Dic₁₃₉	—	2	2	no	—
22	C₄⋊Dic₁₃₉	—	2	1	no	—
23	C₄⋊Dic₁₃₉	—	2	2	no	—
24	C₂²⋊Dic₁₃₉	—	2	2	no	—
25	C₂²⋊Dic₁₃₉	—	2	1	no	—
26	C₂²⋊Dic₁₃₉	—	2	2	no	—
27	C₄×Dic₁₃₉	—	2	2	no	—
28	C₄×Dic₁₃₉	—	2	1	no	—
29	C₄×Dic₁₃₉	—	2	2	no	—
30	C₄×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₁₃₉	C₂×C₄	1	2	yes	—
32	C₄×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
33	#41×C₁₃₉	—	2	2	no	—
34	#41×C₁₃₉	—	2	2	no	—
35	C₂×Q₈×C₁₃₉	D₄	1	1	yes	characteristic
36	C₂×D₄×C₁₃₉	D₄	1	1	yes	characteristic
37	C₂×D₄×C₁₃₉	D₄	1	2	yes	—
38	C₂²×C₄×C₁₃₉	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
40	M₁₆×C₁₃₉	—	2	2	no	—
41	M₁₆×C₁₃₉	—	2	2	no	—
42	C₂×C₈×C₁₃₉	—	2	2	no	—
43	C₂×Dic₁₃₉	—	2	1	no	—
44	C₂×Dic₁₃₉	—	2	2	no	—
45	C₂×Dic₁₃₉	—	2	1	no	—
46	C₂×Dic₁₃₉	—	4	2	no	—
47	C₂×Dic₁₃₉	—	4	4	no	—
48	C₂×Dic₁₃₉	—	4	2	no	—
49	C₂³×C₁₃₉	C₂²⋊C₄	1	2	yes	—
50	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₁₃₉	—	2	2	no	—
52	D₄×C₁₃₉	—	2	4	no	—
53	D₄×C₁₃₉	—	2	2	no	—
54	C₂×C₄×C₁₃₉	—	2	2	no	—
55	C₂×C₄×C₁₃₉	—	2	2	no	—
56	C₂×C₄×C₁₃₉	C₂²⋊C₄	1	2	yes	—
57	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₁₃₉	#41	1	2	yes	—
60	C₂×C₄×C₁₃₉	#41	1	2	yes	—
61	C₈×C₁₃₉	—	4	2	no	—
62	C₂²×C₁₃₉	—	2	2	no	—
63	C₂²×C₁₃₉	—	4	1	no	—
64	C₂²×C₁₃₉	—	4	2	no	—
65	C₂²×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₁₃₉	#123	1	2	yes	—
67	C₄×C₁₃₉	—	2	2	no	—
68	C₄×C₁₃₉	#117	1	1	yes	characteristic
69	C₄×C₁₃₉	#117	1	1	yes	characteristic
70	C₄×C₁₃₉	#123	1	2	yes	—
71	Dic₁₃₉	—	2	2	no	—
72	Dic₁₃₉	—	4	2	no	—
73	Dic₁₃₉	—	4	1	no	—
74	Dic₁₃₉	—	8	2	no	—
75	C₂×C₁₃₉	—	2	1	no	—
76	C₂×C₁₃₉	—	2	2	no	—
77	C₂×C₁₃₉	—	4	2	no	—
78	C₂×C₁₃₉	#381	1	1	yes	characteristic
79	C₁₃₉	#1827	1	1	yes	characteristic, complemented, Sylow
80	#1827	—	139	1	no	maximal, Sylow
81	C₄×#41	—	139	1	no	—
82	#413	D₁₃₉	1	1	yes	characteristic
83	#410	—	139	1	no	—
84	#339	—	139	2	no	—
85	H(3,ℤ/4)	—	139	2	no	—
86	C₂×#41	C₂×D₁₃₉	1	1	yes	characteristic
87	C₂×M₁₆	C₂×D₁₃₉	1	2	yes	—
88	C₄×Q₈	—	278	1	no	—
89	C₄×D₄	—	278	1	no	—
90	C₄×D₄	—	278	2	no	—
91	#117	—	139	2	no	—
92	#117	—	139	2	no	—
93	#117	—	139	1	no	—
94	C₂×C₄²	—	139	2	no	—
95	C₂×C₄²	—	139	1	no	—
96	#101	—	2	1	no	—
97	#100	—	2	1	no	—
98	#99	—	278	2	no	—
99	#41	—	2	2	no	—
100	#41	—	2	2	no	—
101	C₂×Q₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
102	C₂×D₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
103	C₂×D₄	D₅₅₆	1	2	yes	—
104	C₂²×C₄	—	278	2	no	—
105	C₂²×C₄	—	278	1	no	—
106	C₂²×C₄	C₄×D₁₃₉	1	2	yes	—
107	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₄⋊C₄	—	278	2	no	—
112	C₄⋊C₄	—	278	1	no	—
113	C₄⋊C₄	—	278	2	no	—
114	C₂²⋊C₄	—	278	2	no	—
115	C₂²⋊C₄	—	278	1	no	—
116	C₂²⋊C₄	—	278	2	no	—
117	C₄²	—	278	2	no	—
118	C₄²	—	278	1	no	—
119	C₄²	—	278	2	no	—
120	C₄²	—	139	1	no	—
121	C₄²	—	139	2	no	—
122	C₄²	—	139	1	no	—
123	C₂³	(C₂×D₁₃₉)⋊C₄	1	2	yes	—
124	C₂³	C₂×D₅₅₆	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	D₄	—	2	4	no	—
127	D₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	278	1	no	—
131	C₂×C₄	—	278	2	no	—
132	C₂×C₄	—	278	1	no	—
133	C₂×C₄	—	556	2	no	—
134	C₂×C₄	—	556	4	no	—
135	C₂×C₄	—	556	2	no	—
136	C₂×C₄	(C₂×D₁₃₉)⋊C₄	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
138	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
139	C₂×C₄	#14555784	1	2	yes	—
140	C₂×C₄	#14555784	1	2	yes	—
141	C₈	—	4	2	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	4	1	no	—
144	C₂²	—	4	2	no	—
145	C₂²	C₂×(C₂×D₁₃₉)⋊C₄	1	1	yes	characteristic
146	C₂²	#17984826	1	2	yes	—
147	C₄	—	2	2	no	—
148	C₄	—	278	2	no	—
149	C₄	—	556	2	no	—
150	C₄	—	556	1	no	—
151	C₄	—	1112	2	no	—
152	C₄	#17984798	1	1	yes	characteristic
153	C₄	#17984798	1	1	yes	characteristic
154	C₄	#17984823	1	2	yes	—
155	C₂	—	2	1	no	—
156	C₂	—	2	2	no	—
157	C₂	—	4	2	no	—
158	C₂	#22580814	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#41×Dic₁₃₉

C₂

yes

maximal, characteristic

#22580927

C₂

yes

maximal, characteristic

#22580714

C₂

yes

maximal, complemented

#22580710

C₂

yes

maximal, complemented

#413×C₁₃₉

C₂

yes

maximal, characteristic

Q₈×Dic₁₃₉

—

D₄×Dic₁₃₉

—

D₄×Dic₁₃₉

—

#17984801

C₄

yes

complemented

#17984801

C₂²

yes

—

#17984801

C₂²

yes

characteristic

C₂×C₄×Dic₁₃₉

C₄

yes

complemented

C₂×C₄×Dic₁₃₉

C₂²

yes

characteristic

#17984757

—

C₂×#41×C₁₃₉

C₂²

yes

characteristic

C₂×M₁₆×C₁₃₉

C₂²

yes

—

#101×C₁₃₉

—

#100×C₁₃₉

—

C₂²×Dic₁₃₉

—

C₂²×Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₂²⋊Dic₁₃₉

—

C₂²⋊Dic₁₃₉

—

C₂²⋊Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

C₂×C₄

yes

characteristic

C₄×Dic₁₃₉

C₂×C₄

yes

—

C₄×Dic₁₃₉

C₂×C₄

yes

characteristic

#41×C₁₃₉

—

#41×C₁₃₉

—

C₂×Q₈×C₁₃₉

D₄

yes

characteristic

C₂×D₄×C₁₃₉

D₄

yes

characteristic

C₂×D₄×C₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

C₂×C₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

M₁₆×C₁₃₉

—

M₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂³×C₁₃₉

C₂²⋊C₄

yes

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

Q₈×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

—

C₂×C₄×C₁₃₉

#41

yes

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃₉

#123

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#381

yes

characteristic

C₁₃₉

#1827

yes

characteristic, complemented, Sylow

#1827

—

139

maximal, Sylow

C₄×#41

—

139

—

#413

D₁₃₉

yes

characteristic

#410

—

139

—

#339

—

139

—

H(3,ℤ/4)

—

139

—

C₂×#41

C₂×D₁₃₉

yes

characteristic

C₂×M₁₆

C₂×D₁₃₉

yes

—

C₄×Q₈

—

278

—

C₄×D₄

—

278

—

C₄×D₄

—

278

—

#117

—

139

—

#117

—

139

—

#117

—

139

—

C₂×C₄²

—

139

—

C₂×C₄²

—

139

—

#101

—

#100

—

#99

—

278

—

#41

—

100

#41

—

101

C₂×Q₈

C₂²⋊D₁₃₉

yes

characteristic

102

C₂×D₄

C₂²⋊D₁₃₉

yes

characteristic

103

C₂×D₄

D₅₅₆

yes

—

104

C₂²×C₄

—

278

—

105

C₂²×C₄

—

278

—

106

C₂²×C₄

C₄×D₁₃₉

yes

—

107

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

C₂×C₈

—

111

C₄⋊C₄

—

278

—

112

C₄⋊C₄

—

278

—

113

C₄⋊C₄

—

278

—

114

C₂²⋊C₄

—

278

—

115

C₂²⋊C₄

—

278

—

116

C₂²⋊C₄

—

278

—

117

C₄²

—

278

—

118

C₄²

—

278

—

119

C₄²

—

278

—

120

C₄²

—

139

—

121

C₄²

—

139

—

122

C₄²

—

139

—

123

C₂³

(C₂×D₁₃₉)⋊C₄

yes

—

124

C₂³

C₂×D₅₅₆

yes

characteristic

125

Q₈

—

126

D₄

—

127

D₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

278

—

131

C₂×C₄

—

278

—

132

C₂×C₄

—

278

—

133

C₂×C₄

—

556

—

134

C₂×C₄

—

556

—

135

C₂×C₄

—

556

—

136

C₂×C₄

(C₂×D₁₃₉)⋊C₄

yes

—

137

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

138

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

139

C₂×C₄

#14555784

yes

—

140

C₂×C₄

#14555784

yes

—

141

C₈

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

C₂×(C₂×D₁₃₉)⋊C₄

yes

characteristic

146

C₂²

#17984826

yes

—

147

C₄

—

148

C₄

—

278

—

149

C₄

—

556

—

150

C₄

—

556

—

151

C₄

—

1112

—

152

C₄

#17984798

yes

characteristic

153

C₄

#17984798

yes

characteristic

154

C₄

#17984823

yes

—

155

C₂

—

156

C₂

—

157

C₂

—

158

C₂

#22580814

yes

characteristic

	subgroup	G/H
center	C₄	#17984798
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#413×C₁₃₉	C₂
socle	C₂×C₁₃₉	#381
prime radical	C₂×#41×C₁₃₉	C₂²
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	#413	D₁₃₉
hypocenter	C₁₃₉	#1827
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₄

#17984798

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

#413×C₁₃₉

C₂

socle

C₂×C₁₃₉

#381

prime radical

C₂×#41×C₁₃₉

C₂²

cosocle

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
3. #22580714	C₂	semidirect
4. #22580710	C₂	semidirect
9. #17984801	C₄	semidirect
12. C₂×C₄×Dic₁₃₉	C₄	semidirect
79. C₁₃₉	#1827	semidirect

H ≅ G/N

type

3. #22580714

C₂

semidirect

4. #22580710

C₂

semidirect

9. #17984801

C₄

semidirect

12. C₂×C₄×Dic₁₃₉

C₄

semidirect

79. C₁₃₉

#1827

semidirect

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

subgroup	index
#41×Dic₁₃₉	2
#22580927	2
#22580714	2
#22580710	2
#413×C₁₃₉	2
#1827	139

subgroup

index

#41×Dic₁₃₉

#413×C₁₃₉

139

Subgroups — #31806509

Conjugacy classes of subgroups

Complements

Maximal subgroups