Group #31806486 — subgroups

Subgroups — #31806486

Order 17792.

272 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 77 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×#17984800	C₂	1	1	yes	maximal, characteristic
2	#22580916	C₂	1	2	yes	maximal
3	#22580722	C₂	1	1	yes	maximal, characteristic, complemented
4	#22580721	C₂	1	2	yes	maximal, complemented
5	#343×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₁₃₉	C₂²	1	1	yes	characteristic
7	C₂×#14555769	C₂²	1	2	yes	—
8	#17984800	—	2	4	no	—
9	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
10	C₂×(C₂×D₁₃₉)⋊C₄	C₄	1	2	yes	—
11	C₂×C₄×Dic₁₃₉	C₄	1	2	yes	—
12	(C₂×D₁₃₉)⋊C₈	—	2	4	no	—
13	C₂×M₁₆×C₁₃₉	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₂³×D₁₃₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×Dic₁₃₉	C₂×C₄	1	2	yes	—
18	C₂×C₄×D₁₃₉	—	2	1	no	—
19	C₂×C₄×D₁₃₉	—	2	2	no	—
20	C₂×C₄×D₁₃₉	—	2	1	no	—
21	C₂×C₄×D₁₃₉	D₄	1	4	yes	—
22	#14555769	—	2	2	no	—
23	#14555769	—	2	2	no	—
24	C₂×C₁₃₉⋊C₈	—	2	2	no	—
25	C₄⋊Dic₁₃₉	—	2	2	no	—
26	(C₂×D₁₃₉)⋊C₄	—	2	4	no	—
27	C₄×Dic₁₃₉	—	2	4	no	—
28	C₂²×C₄×C₁₃₉	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
30	M₁₆×C₁₃₉	—	2	2	no	—
31	M₁₆×C₁₃₉	—	2	2	no	—
32	C₂×C₈×C₁₃₉	—	2	2	no	—
33	C₄⋊C₄×C₁₃₉	—	2	2	no	—
34	C₂²×D₁₃₉	—	2	1	no	—
35	C₂²×D₁₃₉	—	2	1	no	—
36	C₂²×D₁₃₉	—	4	1	no	—
37	C₂²×D₁₃₉	—	4	1	no	—
38	C₂²×D₁₃₉	C₂²⋊C₄	1	2	yes	—
39	C₂×Dic₁₃₉	—	2	1	no	—
40	C₂×Dic₁₃₉	—	2	1	no	—
41	C₂×Dic₁₃₉	—	2	2	no	—
42	C₂×Dic₁₃₉	—	4	2	no	—
43	C₂×Dic₁₃₉	C₂²⋊C₄	1	2	yes	—
44	C₄×D₁₃₉	—	2	4	no	—
45	C₄×D₁₃₉	—	2	4	no	—
46	C₁₃₉⋊C₈	—	4	2	no	—
47	C₂³×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₃₉	—	2	2	no	—
49	C₂×C₄×C₁₃₉	—	4	2	no	—
50	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
52	C₂×C₄×C₁₃₉	#41	1	2	yes	—
53	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
54	C₈×C₁₃₉	—	4	2	no	—
55	C₂²×C₁₃₉	—	2	1	no	—
56	C₂²×C₁₃₉	—	2	1	no	—
57	C₂²×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₁₃₉	#117	1	1	yes	characteristic
59	C₂²×C₁₃₉	#117	1	1	yes	characteristic
60	C₂×D₁₃₉	—	2	2	no	—
61	C₂×D₁₃₉	—	2	2	no	—
62	C₂×D₁₃₉	—	2	2	no	—
63	C₂×D₁₃₉	—	4	2	no	—
64	C₂×D₁₃₉	—	4	2	no	—
65	C₄×C₁₃₉	—	4	2	no	—
66	C₄×C₁₃₉	#123	1	2	yes	—
67	C₄×C₁₃₉	#123	1	2	yes	—
68	Dic₁₃₉	—	2	2	no	—
69	Dic₁₃₉	—	4	2	no	—
70	C₂×C₁₃₉	—	2	1	no	—
71	C₂×C₁₃₉	—	2	1	no	—
72	C₂×C₁₃₉	#381	1	1	yes	characteristic
73	C₂×C₁₃₉	#413	1	1	yes	characteristic
74	C₂×C₁₃₉	#413	1	1	yes	characteristic
75	D₁₃₉	—	4	2	no	—
76	C₁₃₉	#1822	1	1	yes	characteristic, complemented, Sylow
77	#1822	—	139	1	no	maximal, Sylow
78	C₂×#117	—	139	1	no	—
79	#407	—	139	2	no	—
80	#343	—	139	2	no	—
81	#343	—	139	1	no	—
82	#343	D₁₃₉	1	1	yes	characteristic, complemented
83	C₂³×C₄	—	139	1	no	—
84	C₂×M₁₆	—	139	2	no	—
85	C₂×M₁₆	C₂×D₁₃₉	1	2	yes	—
86	#117	—	278	4	no	—
87	C₂×C₄⋊C₄	—	139	1	no	—
88	C₂×C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
89	C₂×C₂²⋊C₄	—	139	2	no	—
90	C₂×C₄²	—	139	2	no	—
91	C₂²⋊C₈	—	278	4	no	—
92	C₂⁴	—	139	1	no	—
93	C₂²×C₄	—	278	1	no	—
94	C₂²×C₄	—	278	2	no	—
95	C₂²×C₄	—	278	1	no	—
96	C₂²×C₄	—	139	1	no	—
97	C₂²×C₄	—	139	4	no	—
98	C₂²×C₄	—	139	2	no	—
99	C₂²×C₄	C₄×D₁₃₉	1	2	yes	—
100	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
101	M₁₆	—	2	2	no	—
102	M₁₆	—	2	2	no	—
103	M₁₆	—	278	2	no	—
104	M₁₆	—	278	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	278	2	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	278	2	no	—
109	C₂²⋊C₄	—	278	4	no	—
110	C₄²	—	278	4	no	—
111	C₂³	—	278	1	no	—
112	C₂³	—	278	1	no	—
113	C₂³	—	556	1	no	—
114	C₂³	—	556	1	no	—
115	C₂³	—	139	2	no	—
116	C₂³	C₂×C₄×D₁₃₉	1	1	yes	characteristic
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	278	1	no	—
120	C₂×C₄	—	278	1	no	—
121	C₂×C₄	—	278	4	no	—
122	C₂×C₄	—	278	4	no	—
123	C₂×C₄	—	278	2	no	—
124	C₂×C₄	—	556	2	no	—
125	C₂×C₄	—	139	2	no	—
126	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
127	C₂×C₄	#14555784	1	1	yes	characteristic
128	C₂×C₄	#14555785	1	2	yes	—
129	C₂×C₄	#14555786	1	1	yes	characteristic
130	C₈	—	4	2	no	—
131	C₈	—	556	2	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	278	2	no	—
135	C₂²	—	278	2	no	—
136	C₂²	—	278	2	no	—
137	C₂²	—	556	2	no	—
138	C₂²	—	556	2	no	—
139	C₂²	C₂²⋊C₄×D₁₃₉	1	1	yes	characteristic
140	C₂²	#17984799	1	1	yes	characteristic
141	C₂²	#17984800	1	1	yes	characteristic
142	C₄	—	4	2	no	—
143	C₄	—	278	2	no	—
144	C₄	—	556	2	no	—
145	C₄	#17984824	1	2	yes	—
146	C₄	#17984827	1	2	yes	—
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	—	556	2	no	—
150	C₂	#22580815	1	1	yes	characteristic
151	C₂	#22580934	1	1	yes	characteristic
152	C₂	#22580936	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984800

C₂

yes

maximal, characteristic

#22580916

C₂

yes

maximal

#22580722

C₂

yes

maximal, characteristic, complemented

#22580721

C₂

yes

maximal, complemented

#343×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₁₃₉

C₂²

yes

characteristic

C₂×#14555769

C₂²

yes

—

#17984800

—

C₂×C₄⋊Dic₁₃₉

C₂²

yes

characteristic

C₂×(C₂×D₁₃₉)⋊C₄

C₄

yes

—

C₂×C₄×Dic₁₃₉

C₄

yes

—

(C₂×D₁₃₉)⋊C₈

—

C₂×M₁₆×C₁₃₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₃₉

C₂²

yes

characteristic

C₂³×D₁₃₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₃₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₃₉

C₂×C₄

yes

—

C₂×C₄×D₁₃₉

—

C₂×C₄×D₁₃₉

—

C₂×C₄×D₁₃₉

—

C₂×C₄×D₁₃₉

D₄

yes

—

#14555769

—

#14555769

—

C₂×C₁₃₉⋊C₈

—

C₄⋊Dic₁₃₉

—

(C₂×D₁₃₉)⋊C₄

—

C₄×Dic₁₃₉

—

C₂²×C₄×C₁₃₉

C₂×C₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

M₁₆×C₁₃₉

—

M₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

C₂²⋊C₄

yes

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

C₂²⋊C₄

yes

—

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

#41

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

—

C₂×C₄×C₁₃₉

#41

yes

characteristic

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×C₂²⋊C₄

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#123

yes

—

C₄×C₁₃₉

#123

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

D₁₃₉

—

C₁₃₉

#1822

yes

characteristic, complemented, Sylow

#1822

—

139

maximal, Sylow

C₂×#117

—

139

—

#407

—

139

—

#343

—

139

—

#343

—

139

—

#343

D₁₃₉

yes

characteristic, complemented

C₂³×C₄

—

139

—

C₂×M₁₆

—

139

—

C₂×M₁₆

C₂×D₁₃₉

yes

—

#117

—

278

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄⋊C₄

C₂×D₁₃₉

yes

characteristic

C₂×C₂²⋊C₄

—

139

—

C₂×C₄²

—

139

—

C₂²⋊C₈

—

278

—

C₂⁴

—

139

—

C₂²×C₄

—

278

—

C₂²×C₄

—

278

—

C₂²×C₄

—

278

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

C₄×D₁₃₉

yes

—

100

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

278

—

104

M₁₆

—

278

—

105

C₂×C₈

—

106

C₂×C₈

—

278

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

278

—

109

C₂²⋊C₄

—

278

—

110

C₄²

—

278

—

111

C₂³

—

278

—

112

C₂³

—

278

—

113

C₂³

—

556

—

114

C₂³

—

556

—

115

C₂³

—

139

—

116

C₂³

C₂×C₄×D₁₃₉

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

278

—

120

C₂×C₄

—

278

—

121

C₂×C₄

—

278

—

122

C₂×C₄

—

278

—

123

C₂×C₄

—

278

—

124

C₂×C₄

—

556

—

125

C₂×C₄

—

139

—

126

C₂×C₄

D₄×D₁₃₉

yes

—

127

C₂×C₄

#14555784

yes

characteristic

128

C₂×C₄

#14555785

yes

—

129

C₂×C₄

#14555786

yes

characteristic

130

C₈

—

131

C₈

—

556

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

278

—

135

C₂²

—

278

—

136

C₂²

—

278

—

137

C₂²

—

556

—

138

C₂²

—

556

—

139

C₂²

C₂²⋊C₄×D₁₃₉

yes

characteristic

140

C₂²

#17984799

yes

characteristic

141

C₂²

#17984800

yes

characteristic

142

C₄

—

143

C₄

—

278

—

144

C₄

—

556

—

145

C₄

#17984824

yes

—

146

C₄

#17984827

yes

—

147

C₂

—

148

C₂

—

149

C₂

—

556

—

150

C₂

#22580815

yes

characteristic

151

C₂

#22580934

yes

characteristic

152

C₂

#22580936

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₃₉
commutator	C₂³×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#343×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂×C₂²⋊C₄
prime radical	C₂³×D₁₃₉	C₂×C₄
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	#343	D₁₃₉
hypocenter	C₁₃₉	#1822
LCS	G ▷ C₂³×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₃₉

commutator

C₂³×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

#343×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂×C₂²⋊C₄

prime radical

C₂³×D₁₃₉

C₂×C₄

cosocle

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂³×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
3. #22580722	C₂	semidirect
4. #22580721	C₂	semidirect
5. #343×C₁₃₉	C₂	semidirect
76. C₁₃₉	#1822	semidirect
82. #343	D₁₃₉	semidirect

H ≅ G/N

type

3. #22580722

C₂

semidirect

4. #22580721

C₂

semidirect

5. #343×C₁₃₉

C₂

semidirect

76. C₁₃₉

#1822

semidirect

82. #343

D₁₃₉

semidirect

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

subgroup	index
C₂×#17984800	2
#22580916	2
#22580722	2
#22580721	2
#343×C₁₃₉	2
#1822	139

subgroup

index

#343×C₁₃₉

139

Subgroups — #31806486

Conjugacy classes of subgroups

Complements

Maximal subgroups