Group #31807151 — subgroups

Subgroups — #31807151

Order 17792.

208 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 83 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×#17984777	C₂	1	1	yes	maximal, characteristic
2	#22580989	C₂	1	1	yes	maximal, characteristic
3	#22580981	C₂	1	1	yes	maximal, characteristic
4	#22580980	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#22580713	C₂	1	1	yes	maximal, characteristic
7	#339×C₁₃₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#14555769	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
11	#17984801	C₂²	1	1	yes	characteristic
12	#17984777	—	2	2	no	—
13	#17984775	C₄	1	2	yes	complemented
14	#17984773	C₄	1	2	yes	complemented
15	C₄⋊(C₁₃₉⋊C₈)	—	2	2	no	—
16	Dic₁₃₉⋊C₈	—	2	2	no	—
17	C₄×C₁₃₉⋊C₈	—	2	2	no	—
18	C₂×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
19	#117×C₁₃₉	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
21	#14555769	—	2	1	no	—
22	#14555769	—	2	1	no	—
23	C₂×C₁₃₉⋊C₈	—	2	1	no	—
24	C₂×C₁₃₉⋊C₈	—	2	1	no	—
25	C₂×C₁₃₉⋊C₈	—	2	1	no	—
26	C₂×C₁₃₉⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₃₉⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₃₉⋊C₈	D₄	1	2	yes	—
29	C₂×C₁₃₉⋊C₈	D₄	1	2	yes	—
30	C₂×C₁₃₉⋊C₈	Q₈	1	2	yes	—
31	C₄⋊Dic₁₃₉	C₂×C₄	1	2	yes	—
32	C₂²⋊Dic₁₃₉	—	2	1	no	—
33	C₄×Dic₁₃₉	—	2	1	no	—
34	C₂²×C₄×C₁₃₉	—	2	1	no	—
35	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
36	M₁₆×C₁₃₉	—	2	1	no	—
37	M₁₆×C₁₃₉	—	2	1	no	—
38	C₂×C₈×C₁₃₉	—	2	1	no	—
39	C₄⋊C₄×C₁₃₉	C₂×C₄	1	2	yes	—
40	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
41	C₄²×C₁₃₉	—	2	1	no	—
42	C₄²×C₁₃₉	—	2	1	no	—
43	C₄²×C₁₃₉	Q₈	1	2	yes	—
44	C₂×Dic₁₃₉	—	2	2	no	—
45	C₁₃₉⋊C₈	—	2	2	no	—
46	C₁₃₉⋊C₈	—	4	1	no	—
47	C₁₃₉⋊C₈	C₄⋊C₄	1	2	yes	—
48	C₁₃₉⋊C₈	C₄⋊C₄	1	2	yes	—
49	C₂³×C₁₃₉	#41	1	1	yes	characteristic
50	C₂×C₄×C₁₃₉	—	2	4	no	—
51	C₂×C₄×C₁₃₉	—	2	2	no	—
52	C₂×C₄×C₁₃₉	—	2	2	no	—
53	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
59	C₈×C₁₃₉	—	4	1	no	—
60	C₂²×C₁₃₉	—	2	2	no	—
61	C₂²×C₁₃₉	C₄×D₄	1	1	yes	characteristic
62	C₂²×C₁₃₉	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₁₃₉	#125	1	1	yes	characteristic
64	C₄×C₁₃₉	—	2	4	no	—
65	C₄×C₁₃₉	—	4	2	no	—
66	C₄×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₄×C₁₃₉	C₄×Q₈	1	1	yes	characteristic
68	C₄×C₁₃₉	#121	1	1	yes	characteristic
69	C₄×C₁₃₉	C₄⋊Q₈	1	1	yes	characteristic
70	Dic₁₃₉	—	4	2	no	—
71	C₂×C₁₃₉	—	2	1	no	—
72	C₂×C₁₃₉	—	2	1	no	—
73	C₂×C₁₃₉	#389	1	1	yes	characteristic
74	C₂×C₁₃₉	#443	1	2	yes	—
75	C₁₃₉	#1945	1	1	yes	characteristic, complemented, Sylow
76	#1945	—	139	1	no	maximal, Sylow
77	C₂×#108	—	139	1	no	—
78	#427	—	139	1	no	—
79	#424	—	139	1	no	—
80	#424	—	139	1	no	—
81	C₂×C₄×C₈	—	139	1	no	—
82	#339	—	139	1	no	—
83	#339	D₁₃₉	1	1	yes	characteristic
84	C₂×M₁₆	—	139	1	no	—
85	C₂×M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
86	C₂²×C₈	—	139	1	no	—
87	C₂²×C₈	—	139	1	no	—
88	#117	—	139	1	no	—
89	#117	C₂×D₁₃₉	1	1	yes	characteristic
90	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
91	#108	—	278	2	no	—
92	#107	—	139	2	no	—
93	#106	—	139	2	no	—
94	C₄⋊C₈	—	278	2	no	—
95	C₄⋊C₈	—	278	2	no	—
96	C₄×C₈	—	278	2	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
99	M₁₆	—	2	1	no	—
100	M₁₆	—	2	1	no	—
101	M₁₆	—	278	1	no	—
102	M₁₆	—	278	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	278	1	no	—
105	C₂×C₈	—	278	1	no	—
106	C₂×C₈	—	278	1	no	—
107	C₂×C₈	—	139	2	no	—
108	C₂×C₈	—	139	1	no	—
109	C₂×C₈	—	139	1	no	—
110	C₂×C₈	—	139	2	no	—
111	C₂×C₈	—	139	2	no	—
112	C₄⋊C₄	—	139	2	no	—
113	C₄⋊C₄	C₄×D₁₃₉	1	2	yes	—
114	C₂²⋊C₄	—	2	1	no	—
115	C₂²⋊C₄	—	278	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	278	1	no	—
119	C₄²	Dic₂₇₈	1	2	yes	—
120	C₂³	#14555784	1	1	yes	characteristic
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	278	2	no	—
125	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
126	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
127	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₁₃₉	1	1	yes	characteristic
129	C₂×C₄	C₂×Dic₂₇₈	1	1	yes	characteristic
130	C₂×C₄	#14555785	1	1	yes	characteristic
131	C₈	—	4	1	no	—
132	C₈	—	278	2	no	—
133	C₈	—	556	1	no	—
134	C₈	—	139	2	no	—
135	C₈	—	139	2	no	—
136	C₂²	—	2	2	no	—
137	C₂²	(C₂²⋊D₁₃₉)⋊C₄	1	1	yes	characteristic
138	C₂²	#17984819	1	1	yes	characteristic
139	C₂²	#17984832	1	1	yes	characteristic
140	C₄	—	2	4	no	—
141	C₄	—	4	2	no	—
142	C₄	—	556	2	no	—
143	C₄	C₄⋊C₄×D₁₃₉	1	1	yes	characteristic
144	C₄	C₄×Dic₂₇₈	1	1	yes	characteristic
145	C₄	#17984815	1	1	yes	characteristic
146	C₄	#17984841	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#22580850	1	1	yes	characteristic
150	C₂	#22581067	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984777

C₂

yes

maximal, characteristic

#22580989

C₂

yes

maximal, characteristic

#22580981

C₂

yes

maximal, characteristic

#22580980

C₂

yes

maximal, characteristic

C₂×C₄×C₁₃₉⋊C₈

C₂

yes

maximal, characteristic

#22580713

C₂

yes

maximal, characteristic

#339×C₁₃₉

C₂

yes

maximal, characteristic

C₂×#14555769

C₂²

yes

characteristic

C₂²×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₂²×C₁₃₉⋊C₈

C₂²

yes

characteristic

#17984801

C₂²

yes

characteristic

#17984777

—

#17984775

C₄

yes

complemented

#17984773

C₄

yes

complemented

C₄⋊(C₁₃₉⋊C₈)

—

Dic₁₃₉⋊C₈

—

C₄×C₁₃₉⋊C₈

—

C₂×M₁₆×C₁₃₉

C₂²

yes

characteristic

#117×C₁₃₉

C₂²

yes

characteristic

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

#14555769

—

#14555769

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₃₉⋊C₈

D₄

yes

—

C₂×C₁₃₉⋊C₈

D₄

yes

—

C₂×C₁₃₉⋊C₈

Q₈

yes

—

C₄⋊Dic₁₃₉

C₂×C₄

yes

—

C₂²⋊Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

M₁₆×C₁₃₉

—

M₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₄⋊C₄×C₁₃₉

C₂×C₄

yes

—

C₂²⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

—

C₄²×C₁₃₉

—

C₄²×C₁₃₉

Q₈

yes

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

C₄⋊C₄

yes

—

C₁₃₉⋊C₈

C₄⋊C₄

yes

—

C₂³×C₁₃₉

#41

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

characteristic

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₄×D₄

yes

characteristic

C₂²×C₁₃₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₃₉

#125

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₃₉

C₄×Q₈

yes

characteristic

C₄×C₁₃₉

#121

yes

characteristic

C₄×C₁₃₉

C₄⋊Q₈

yes

characteristic

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#389

yes

characteristic

C₂×C₁₃₉

#443

yes

—

C₁₃₉

#1945

yes

characteristic, complemented, Sylow

#1945

—

139

maximal, Sylow

C₂×#108

—

139

—

#427

—

139

—

#424

—

139

—

#424

—

139

—

C₂×C₄×C₈

—

139

—

#339

—

139

—

#339

D₁₃₉

yes

characteristic

C₂×M₁₆

—

139

—

C₂×M₁₆

C₂×D₁₃₉

yes

characteristic

C₂²×C₈

—

139

—

C₂²×C₈

—

139

—

#117

—

139

—

#117

C₂×D₁₃₉

yes

characteristic

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

#108

—

278

—

#107

—

139

—

#106

—

139

—

C₄⋊C₈

—

278

—

C₄⋊C₈

—

278

—

C₄×C₈

—

278

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

M₁₆

—

100

M₁₆

—

101

M₁₆

—

278

—

102

M₁₆

—

278

—

103

C₂×C₈

—

104

C₂×C₈

—

278

—

105

C₂×C₈

—

278

—

106

C₂×C₈

—

278

—

107

C₂×C₈

—

139

—

108

C₂×C₈

—

139

—

109

C₂×C₈

—

139

—

110

C₂×C₈

—

139

—

111

C₂×C₈

—

139

—

112

C₄⋊C₄

—

139

—

113

C₄⋊C₄

C₄×D₁₃₉

yes

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

278

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

278

—

119

C₄²

Dic₂₇₈

yes

—

120

C₂³

#14555784

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

278

—

125

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

126

C₂×C₄

D₄×D₁₃₉

yes

characteristic

127

C₂×C₄

D₄×D₁₃₉

yes

characteristic

128

C₂×C₄

Q₈×D₁₃₉

yes

characteristic

129

C₂×C₄

C₂×Dic₂₇₈

yes

characteristic

130

C₂×C₄

#14555785

yes

characteristic

131

C₈

—

132

C₈

—

278

—

133

C₈

—

556

—

134

C₈

—

139

—

135

C₈

—

139

—

136

C₂²

—

137

C₂²

(C₂²⋊D₁₃₉)⋊C₄

yes

characteristic

138

C₂²

#17984819

yes

characteristic

139

C₂²

#17984832

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

556

—

143

C₄

C₄⋊C₄×D₁₃₉

yes

characteristic

144

C₄

C₄×Dic₂₇₈

yes

characteristic

145

C₄

#17984815

yes

characteristic

146

C₄

#17984841

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

#22580850

yes

characteristic

150

C₂

#22581067

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₃₉
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#339×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₄×D₄
prime radical	C₂³×C₁₃₉	#41
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	#339	D₁₃₉
hypocenter	C₁₃₉	#1945
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

subgroup

G/H

center

C₂×C₄

D₄×D₁₃₉

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

#339×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₄×D₄

prime radical

C₂³×C₁₃₉

#41

cosocle

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

N	H ≅ G/N	type
13. #17984775	C₄	semidirect
14. #17984773	C₄	semidirect
75. C₁₃₉	#1945	semidirect

H ≅ G/N

type

13. #17984775

C₄

semidirect

14. #17984773

C₄

semidirect

75. C₁₃₉

#1945

semidirect

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

subgroup	index
C₂×#17984777	2
#22580989	2
#22580981	2
#22580980	2
C₂×C₄×C₁₃₉⋊C₈	2
#22580713	2
#339×C₁₃₉	2
#1945	139

subgroup

index

C₂×C₄×C₁₃₉⋊C₈

#339×C₁₃₉

139

Subgroups — #31807151

Conjugacy classes of subgroups

Complements

Maximal subgroups