Group #31806808 — subgroups

Subgroups — #31806808

Order 17792.

344 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 97 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	#22581431	C₂	1	1	yes	maximal, characteristic, complemented
2	#22580958	C₂	1	2	yes	maximal
3	C₂×#17984761	C₂	1	1	yes	maximal, characteristic, complemented
4	#22580713	C₂	1	2	yes	maximal, complemented
5	C₂×#124×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555769	C₂²	1	2	yes	—
7	Q₈×Dic₁₃₉	—	2	4	no	—
8	#17984801	—	2	2	no	—
9	#17984801	C₂²	1	2	yes	—
10	#17984763	—	2	4	no	—
11	#17984761	—	2	2	no	—
12	C₂²×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
14	#124×C₁₃₉	—	2	2	no	—
15	#124×C₁₃₉	C₄	1	4	yes	complemented
16	C₂×C₂²⋊C₄×C₁₃₉	—	2	2	no	—
17	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
18	#14555769	—	2	2	no	—
19	#14555769	—	2	2	no	—
20	C₂×C₁₃₉⋊C₈	—	2	2	no	—
21	C₄⋊Dic₁₃₉	—	2	4	no	—
22	C₄⋊Dic₁₃₉	D₄	1	4	yes	—
23	C₂²⋊Dic₁₃₉	—	2	2	no	—
24	C₄×Dic₁₃₉	—	2	4	no	—
25	C₄×Dic₁₃₉	—	2	2	no	—
26	C₂⁴×C₁₃₉	—	2	1	no	—
27	C₂×Q₈×C₁₃₉	—	2	2	no	—
28	C₂×Q₈×C₁₃₉	—	2	2	no	—
29	C₂×Q₈×C₁₃₉	C₂×C₄	1	2	yes	—
30	C₂×Q₈×C₁₃₉	D₄	1	2	yes	—
31	C₂×D₄×C₁₃₉	—	2	2	no	—
32	C₂×D₄×C₁₃₉	—	2	1	no	—
33	C₂×D₄×C₁₃₉	—	2	2	no	—
34	C₂×D₄×C₁₃₉	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₁₃₉	—	2	1	no	—
36	C₂²×C₄×C₁₃₉	D₄	1	2	yes	—
37	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
38	C₂²⋊C₄×C₁₃₉	—	2	8	no	—
39	C₄²×C₁₃₉	—	2	1	no	—
40	C₄²×C₁₃₉	C₂×C₄	1	2	yes	—
41	C₂×Dic₁₃₉	—	2	4	no	—
42	C₁₃₉⋊C₈	—	4	2	no	—
43	C₂³×C₁₃₉	—	2	2	no	—
44	C₂³×C₁₃₉	—	4	4	no	—
45	C₂³×C₁₃₉	—	4	1	no	—
46	C₂³×C₁₃₉	—	4	1	no	—
47	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
48	Q₈×C₁₃₉	—	2	4	no	—
49	Q₈×C₁₃₉	—	2	4	no	—
50	D₄×C₁₃₉	—	2	4	no	—
51	D₄×C₁₃₉	—	4	2	no	—
52	C₂×C₄×C₁₃₉	—	2	2	no	—
53	C₂×C₄×C₁₃₉	—	2	2	no	—
54	C₂×C₄×C₁₃₉	—	2	2	no	—
55	C₂×C₄×C₁₃₉	—	4	2	no	—
56	C₂×C₄×C₁₃₉	C₂²⋊C₄	1	4	yes	—
57	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₁₃₉	#41	1	2	yes	—
61	C₂²×C₁₃₉	—	2	2	no	—
62	C₂²×C₁₃₉	—	4	4	no	—
63	C₂²×C₁₃₉	—	4	2	no	—
64	C₂²×C₁₃₉	—	8	2	no	—
65	C₂²×C₁₃₉	—	8	2	no	—
66	C₂²×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₁₃₉	#120	1	1	yes	characteristic
68	C₂²×C₁₃₉	#123	1	1	yes	characteristic
69	C₄×C₁₃₉	—	2	4	no	—
70	C₄×C₁₃₉	—	4	2	no	—
71	C₄×C₁₃₉	C₄×D₄	1	2	yes	—
72	C₄×C₁₃₉	#121	1	2	yes	—
73	Dic₁₃₉	—	4	4	no	—
74	C₂×C₁₃₉	—	2	1	no	—
75	C₂×C₁₃₉	—	2	1	no	—
76	C₂×C₁₃₉	—	8	2	no	—
77	C₂×C₁₃₉	#386	1	1	yes	characteristic
78	C₂×C₁₃₉	#454	1	2	yes	—
79	C₁₃₉	#1888	1	1	yes	characteristic, complemented, Sylow
80	#1888	—	139	1	no	maximal, Sylow
81	C₂×#124	D₁₃₉	1	1	yes	characteristic
82	#519	—	139	1	no	—
83	#418	—	139	2	no	—
84	C₂×#101	—	139	1	no	—
85	#339	—	139	2	no	—
86	C₂²×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
87	C₂²×D₄	C₂×D₁₃₉	1	1	yes	characteristic
88	C₂×M₁₆	—	139	2	no	—
89	#124	—	2	2	no	—
90	#124	Dic₁₃₉	1	4	yes	complemented
91	C₄×Q₈	—	278	4	no	—
92	#117	—	278	2	no	—
93	#117	—	139	2	no	—
94	C₂×C₂²⋊C₄	—	2	2	no	—
95	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
96	#102	—	278	4	no	—
97	#101	—	278	2	no	—
98	C₂⁴	—	2	1	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	C₂×Dic₁₃₉	1	2	yes	—
102	C₂×Q₈	C₂²⋊D₁₃₉	1	2	yes	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	—	2	2	no	—
106	C₂×D₄	C₂×Dic₁₃₉	1	2	yes	—
107	C₂²×C₄	—	2	1	no	—
108	C₂²×C₄	C₂²⋊D₁₃₉	1	2	yes	—
109	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
110	M₁₆	—	278	2	no	—
111	M₁₆	—	278	2	no	—
112	C₂×C₈	—	278	2	no	—
113	C₄⋊C₄	—	278	4	no	—
114	C₄⋊C₄	—	139	4	no	—
115	C₂²⋊C₄	—	2	8	no	—
116	C₂²⋊C₄	—	278	2	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	278	4	no	—
119	C₄²	—	278	2	no	—
120	C₄²	C₂×Dic₁₃₉	1	2	yes	—
121	C₂³	—	2	2	no	—
122	C₂³	—	4	4	no	—
123	C₂³	—	4	1	no	—
124	C₂³	—	4	1	no	—
125	C₂³	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
126	Q₈	—	2	4	no	—
127	Q₈	—	2	4	no	—
128	D₄	—	2	4	no	—
129	D₄	—	4	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	278	4	no	—
135	C₂×C₄	C₂²⋊Dic₁₃₉	1	4	yes	—
136	C₂×C₄	C₂²×Dic₁₃₉	1	1	yes	characteristic
137	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
138	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
139	C₂×C₄	#14555784	1	2	yes	—
140	C₈	—	556	2	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	4	4	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	8	2	no	—
145	C₂²	—	8	2	no	—
146	C₂²	C₂×C₂²⋊Dic₁₃₉	1	1	yes	characteristic
147	C₂²	#17984810	1	1	yes	characteristic
148	C₂²	#17984823	1	1	yes	characteristic
149	C₄	—	2	4	no	—
150	C₄	—	4	2	no	—
151	C₄	—	556	4	no	—
152	C₄	D₄×Dic₁₃₉	1	2	yes	—
153	C₄	#17984814	1	2	yes	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	—	8	2	no	—
157	C₂	#22580833	1	1	yes	characteristic
158	C₂	#22581125	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22581431

C₂

yes

maximal, characteristic, complemented

#22580958

C₂

yes

maximal

C₂×#17984761

C₂

yes

maximal, characteristic, complemented

#22580713

C₂

yes

maximal, complemented

C₂×#124×C₁₃₉

C₂

yes

maximal, characteristic

C₂×#14555769

C₂²

yes

—

Q₈×Dic₁₃₉

—

#17984801

—

#17984801

C₂²

yes

—

#17984763

—

#17984761

—

C₂²×Q₈×C₁₃₉

C₂²

yes

characteristic

C₂²×D₄×C₁₃₉

C₂²

yes

characteristic

#124×C₁₃₉

—

#124×C₁₃₉

C₄

yes

complemented

C₂×C₂²⋊C₄×C₁₃₉

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

#14555769

—

#14555769

—

C₂×C₁₃₉⋊C₈

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

D₄

yes

—

C₂²⋊Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₂⁴×C₁₃₉

—

C₂×Q₈×C₁₃₉

—

C₂×Q₈×C₁₃₉

—

C₂×Q₈×C₁₃₉

C₂×C₄

yes

—

C₂×Q₈×C₁₃₉

D₄

yes

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

C₂×C₄

yes

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₂²⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

—

C₄²×C₁₃₉

C₂×C₄

yes

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

Q₈×C₁₃₉

—

Q₈×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×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

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×C₂²⋊C₄

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₄×D₄

yes

—

C₄×C₁₃₉

#121

yes

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#386

yes

characteristic

C₂×C₁₃₉

#454

yes

—

C₁₃₉

#1888

yes

characteristic, complemented, Sylow

#1888

—

139

maximal, Sylow

C₂×#124

D₁₃₉

yes

characteristic

#519

—

139

—

#418

—

139

—

C₂×#101

—

139

—

#339

—

139

—

C₂²×Q₈

C₂×D₁₃₉

yes

characteristic

C₂²×D₄

C₂×D₁₃₉

yes

characteristic

C₂×M₁₆

—

139

—

#124

—

#124

Dic₁₃₉

yes

complemented

C₄×Q₈

—

278

—

#117

—

278

—

#117

—

139

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

#102

—

278

—

#101

—

278

—

C₂⁴

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

C₂×Dic₁₃₉

yes

—

102

C₂×Q₈

C₂²⋊D₁₃₉

yes

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

C₂×Dic₁₃₉

yes

—

107

C₂²×C₄

—

108

C₂²×C₄

C₂²⋊D₁₃₉

yes

—

109

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

110

M₁₆

—

278

—

111

M₁₆

—

278

—

112

C₂×C₈

—

278

—

113

C₄⋊C₄

—

278

—

114

C₄⋊C₄

—

139

—

115

C₂²⋊C₄

—

116

C₂²⋊C₄

—

278

—

117

C₄²

—

118

C₄²

—

278

—

119

C₄²

—

278

—

120

C₄²

C₂×Dic₁₃₉

yes

—

121

C₂³

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

C₂×C₂²⋊D₁₃₉

yes

characteristic

126

Q₈

—

127

Q₈

—

128

D₄

—

129

D₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

278

—

135

C₂×C₄

C₂²⋊Dic₁₃₉

yes

—

136

C₂×C₄

C₂²×Dic₁₃₉

yes

characteristic

137

C₂×C₄

D₄×D₁₃₉

yes

—

138

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

139

C₂×C₄

#14555784

yes

—

140

C₈

—

556

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×C₂²⋊Dic₁₃₉

yes

characteristic

147

C₂²

#17984810

yes

characteristic

148

C₂²

#17984823

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

556

—

152

C₄

D₄×Dic₁₃₉

yes

—

153

C₄

#17984814

yes

—

154

C₂

—

155

C₂

—

156

C₂

—

157

C₂

#22580833

yes

characteristic

158

C₂

#22581125

yes

—

	subgroup	G/H
center	C₂²	#17984810
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	C₂×#124×C₁₃₉	C₂
socle	C₂²×C₁₃₉	#120
prime radical	C₂²×D₄×C₁₃₉	C₂²
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	C₂×#124	D₁₃₉
hypocenter	C₁₃₉	#1888
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#124

subgroup

G/H

center

C₂²

#17984810

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

C₂×#124×C₁₃₉

C₂

socle

C₂²×C₁₃₉

#120

prime radical

C₂²×D₄×C₁₃₉

C₂²

cosocle

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#124

N	H ≅ G/N	type
1. #22581431	C₂	semidirect
3. C₂×#17984761	C₂	semidirect
4. #22580713	C₂	semidirect
15. #124×C₁₃₉	C₄	semidirect
79. C₁₃₉	#1888	semidirect
90. #124	Dic₁₃₉	semidirect

H ≅ G/N

type

1. #22581431

C₂

semidirect

3. C₂×#17984761

C₂

semidirect

4. #22580713

C₂

semidirect

15. #124×C₁₃₉

C₄

semidirect

79. C₁₃₉

#1888

semidirect

90. #124

Dic₁₃₉

semidirect

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

subgroup	index
#22581431	2
#22580958	2
C₂×#17984761	2
#22580713	2
C₂×#124×C₁₃₉	2
#1888	139

subgroup

index

C₂×#124×C₁₃₉

139

Subgroups — #31806808

Conjugacy classes of subgroups

Complements

Maximal subgroups