Group #31806399 — subgroups

Subgroups — #31806399

Order 17792.

272 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 87 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×#17984807	C₂	1	1	yes	maximal, characteristic
2	#22580976	C₂	1	2	yes	maximal
3	C₂×#17984760	C₂	1	2	yes	maximal
4	#22580722	C₂	1	1	yes	maximal, characteristic
5	#343×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₇₈	C₂²	1	1	yes	characteristic
7	C₂×#14555769	C₂²	1	2	yes	—
8	#17984807	—	2	2	no	—
9	#17984807	—	2	2	no	—
10	C₂×Dic₁₃₉⋊C₄	C₄	1	2	yes	—
11	C₂×C₄×Dic₁₃₉	C₄	1	2	yes	—
12	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
13	Dic₁₃₉⋊C₈	—	2	2	no	—
14	Dic₁₃₉⋊C₈	—	2	2	no	—
15	#17984760	—	2	4	no	—
16	C₂×M₁₆×C₁₃₉	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
18	C₂×Dic₂₇₈	—	2	2	no	—
19	C₂×Dic₂₇₈	—	2	2	no	—
20	C₂×Dic₂₇₈	D₄	1	2	yes	—
21	C₂×Dic₂₇₈	Q₈	1	2	yes	—
22	C₂²×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₁₃₉	C₂×C₄	1	2	yes	—
24	C₂²×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
25	#14555769	—	2	2	no	—
26	#14555769	—	2	2	no	—
27	C₂×C₁₃₉⋊C₈	—	2	2	no	—
28	Dic₁₃₉⋊C₄	—	2	4	no	—
29	C₄×Dic₁₃₉	—	2	2	no	—
30	C₄×Dic₁₃₉	—	2	2	no	—
31	C₄×Dic₁₃₉	D₄	1	2	yes	—
32	C₄×Dic₁₃₉	D₄	1	2	yes	—
33	C₂²×C₄×C₁₃₉	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
35	M₁₆×C₁₃₉	—	2	2	no	—
36	M₁₆×C₁₃₉	—	2	2	no	—
37	C₂×C₈×C₁₃₉	—	2	2	no	—
38	C₄⋊C₄×C₁₃₉	—	2	2	no	—
39	Dic₂₇₈	—	2	4	no	—
40	Dic₂₇₈	—	2	4	no	—
41	C₂×Dic₁₃₉	—	2	2	no	—
42	C₂×Dic₁₃₉	—	2	2	no	—
43	C₂×Dic₁₃₉	—	2	2	no	—
44	C₂×Dic₁₃₉	—	4	1	no	—
45	C₂×Dic₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×Dic₁₃₉	C₂²⋊C₄	1	2	yes	—
47	C₂×Dic₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×Dic₁₃₉	C₄⋊C₄	1	4	yes	—
49	C₁₃₉⋊C₈	—	4	2	no	—
50	C₂³×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₃₉	—	2	2	no	—
52	C₂×C₄×C₁₃₉	—	4	2	no	—
53	C₂×C₄×C₁₃₉	C₂×D₄	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	—	4	2	no	—
58	C₂²×C₁₃₉	—	2	1	no	—
59	C₂²×C₁₃₉	—	2	1	no	—
60	C₂²×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₁₃₉	#117	1	1	yes	characteristic
63	C₄×C₁₃₉	—	4	2	no	—
64	C₄×C₁₃₉	#121	1	2	yes	—
65	C₄×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
66	Dic₁₃₉	—	2	4	no	—
67	Dic₁₃₉	—	2	1	no	—
68	Dic₁₃₉	—	2	2	no	—
69	Dic₁₃₉	#101	1	2	yes	complemented
70	C₂×C₁₃₉	—	2	1	no	—
71	C₂×C₁₃₉	—	2	1	no	—
72	C₂×C₁₃₉	#380	1	1	yes	characteristic
73	C₂×C₁₃₉	C₂×#101	1	1	yes	characteristic
74	C₂×C₁₃₉	#413	1	1	yes	characteristic
75	C₁₃₉	#1801	1	1	yes	characteristic, complemented, Sylow
76	#1801	—	139	1	no	maximal, Sylow
77	C₂×C₄×Q₈	—	139	1	no	—
78	#423	—	139	2	no	—
79	C₂×#101	—	139	2	no	—
80	#343	—	139	1	no	—
81	#343	D₁₃₉	1	1	yes	characteristic
82	C₂²×Q₈	—	139	1	no	—
83	C₂×M₁₆	—	139	2	no	—
84	C₂×M₁₆	C₂×D₁₃₉	1	2	yes	—
85	C₄×Q₈	—	278	2	no	—
86	C₄×Q₈	—	278	2	no	—
87	C₂×C₄⋊C₄	—	139	2	no	—
88	C₂×C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
89	C₂×C₄²	—	139	2	no	—
90	C₂×C₄²	—	139	1	no	—
91	C₄⋊C₈	—	278	2	no	—
92	C₄⋊C₈	—	278	2	no	—
93	#101	—	278	4	no	—
94	C₂×Q₈	—	278	2	no	—
95	C₂×Q₈	—	278	2	no	—
96	C₂×Q₈	—	139	2	no	—
97	C₂×Q₈	—	139	2	no	—
98	C₂²×C₄	—	139	1	no	—
99	C₂²×C₄	—	139	2	no	—
100	C₂²×C₄	—	139	1	no	—
101	C₂²×C₄	C₄×D₁₃₉	1	2	yes	—
102	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
103	M₁₆	—	2	2	no	—
104	M₁₆	—	2	2	no	—
105	M₁₆	—	278	2	no	—
106	M₁₆	—	278	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	278	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	278	4	no	—
111	C₄²	—	278	2	no	—
112	C₄²	—	278	2	no	—
113	C₄²	—	139	2	no	—
114	C₄²	—	139	2	no	—
115	C₂³	C₂×C₄×D₁₃₉	1	1	yes	characteristic
116	Q₈	—	278	4	no	—
117	Q₈	—	278	4	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	278	2	no	—
121	C₂×C₄	—	278	2	no	—
122	C₂×C₄	—	278	2	no	—
123	C₂×C₄	—	556	1	no	—
124	C₂×C₄	—	139	1	no	—
125	C₂×C₄	—	139	2	no	—
126	C₂×C₄	—	139	4	no	—
127	C₂×C₄	—	139	1	no	—
128	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
129	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
130	C₂×C₄	Q₈×D₁₃₉	1	1	yes	characteristic
131	C₂×C₄	#14555785	1	2	yes	—
132	C₈	—	4	2	no	—
133	C₈	—	556	2	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	C₂²⋊C₄×D₁₃₉	1	1	yes	characteristic
137	C₂²	C₄⋊C₄×D₁₃₉	1	1	yes	characteristic
138	C₂²	#17984799	1	1	yes	characteristic
139	C₄	—	4	2	no	—
140	C₄	—	278	4	no	—
141	C₄	—	278	1	no	—
142	C₄	—	278	2	no	—
143	C₄	—	139	2	no	—
144	C₄	#17984815	1	2	yes	—
145	C₄	#17984820	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	1	no	—
148	C₂	#22580812	1	1	yes	characteristic
149	C₂	#101×D₁₃₉	1	1	yes	characteristic
150	C₂	#22580934	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984807

C₂

yes

maximal, characteristic

#22580976

C₂

yes

maximal

C₂×#17984760

C₂

yes

maximal

#22580722

C₂

yes

maximal, characteristic

#343×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₇₈

C₂²

yes

characteristic

C₂×#14555769

C₂²

yes

—

#17984807

—

#17984807

—

C₂×Dic₁₃₉⋊C₄

C₄

yes

—

C₂×C₄×Dic₁₃₉

C₄

yes

—

C₂×C₄×Dic₁₃₉

C₂²

yes

characteristic

Dic₁₃₉⋊C₈

—

Dic₁₃₉⋊C₈

—

#17984760

—

C₂×M₁₆×C₁₃₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

D₄

yes

—

C₂×Dic₂₇₈

Q₈

yes

—

C₂²×Dic₁₃₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₃₉

C₂×C₄

yes

—

C₂²×Dic₁₃₉

C₂×C₄

yes

characteristic

#14555769

—

#14555769

—

C₂×C₁₃₉⋊C₈

—

Dic₁₃₉⋊C₄

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

D₄

yes

—

C₄×Dic₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

C₂×C₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

M₁₆×C₁₃₉

—

M₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

Dic₂₇₈

—

Dic₂₇₈

—

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

C₂×Dic₁₃₉

C₄⋊C₄

yes

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₃₉

#117

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

#121

yes

—

C₄×C₁₃₉

C₂²⋊Q₈

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

#101

yes

complemented

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#1801

—

139

maximal, Sylow

C₂×C₄×Q₈

—

139

—

#423

—

139

—

C₂×#101

—

139

—

#343

—

139

—

#343

D₁₃₉

yes

characteristic

C₂²×Q₈

—

139

—

C₂×M₁₆

—

139

—

C₂×M₁₆

C₂×D₁₃₉

yes

—

C₄×Q₈

—

278

—

C₄×Q₈

—

278

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄⋊C₄

C₂×D₁₃₉

yes

characteristic

C₂×C₄²

—

139

—

C₂×C₄²

—

139

—

C₄⋊C₈

—

278

—

C₄⋊C₈

—

278

—

#101

—

278

—

C₂×Q₈

—

278

—

C₂×Q₈

—

278

—

C₂×Q₈

—

139

—

C₂×Q₈

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

100

C₂²×C₄

—

139

—

101

C₂²×C₄

C₄×D₁₃₉

yes

—

102

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

278

—

106

M₁₆

—

278

—

107

C₂×C₈

—

108

C₂×C₈

—

278

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

278

—

111

C₄²

—

278

—

112

C₄²

—

278

—

113

C₄²

—

139

—

114

C₄²

—

139

—

115

C₂³

C₂×C₄×D₁₃₉

yes

characteristic

116

Q₈

—

278

—

117

Q₈

—

278

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

278

—

121

C₂×C₄

—

278

—

122

C₂×C₄

—

278

—

123

C₂×C₄

—

556

—

124

C₂×C₄

—

139

—

125

C₂×C₄

—

139

—

126

C₂×C₄

—

139

—

127

C₂×C₄

—

139

—

128

C₂×C₄

D₄×D₁₃₉

yes

characteristic

129

C₂×C₄

D₄×D₁₃₉

yes

—

130

C₂×C₄

Q₈×D₁₃₉

yes

characteristic

131

C₂×C₄

#14555785

yes

—

132

C₈

—

133

C₈

—

556

—

134

C₂²

—

135

C₂²

—

136

C₂²

C₂²⋊C₄×D₁₃₉

yes

characteristic

137

C₂²

C₄⋊C₄×D₁₃₉

yes

characteristic

138

C₂²

#17984799

yes

characteristic

139

C₄

—

140

C₄

—

278

—

141

C₄

—

278

—

142

C₄

—

278

—

143

C₄

—

139

—

144

C₄

#17984815

yes

—

145

C₄

#17984820

yes

—

146

C₂

—

147

C₂

—

148

C₂

#22580812

yes

characteristic

149

C₂

#101×D₁₃₉

yes

characteristic

150

C₂

#22580934

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₃₉
commutator, prime radical	C₂³×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#343×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	#343	D₁₃₉
hypocenter	C₁₃₉	#1801
LCS	G ▷ C₂³×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₃₉

commutator, prime radical

C₂³×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

#343×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂×C₂²⋊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
69. Dic₁₃₉	#101	semidirect
75. C₁₃₉	#1801	semidirect

H ≅ G/N

type

69. Dic₁₃₉

#101

semidirect

75. C₁₃₉

#1801

semidirect

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

subgroup	index
C₂×#17984807	2
#22580976	2
C₂×#17984760	2
#22580722	2
#343×C₁₃₉	2
#1801	139

subgroup

index

#343×C₁₃₉

139

Subgroups — #31806399

Conjugacy classes of subgroups

Complements

Maximal subgroups