Group #31816944 — subgroups

Subgroups — #31816944

Order 17792.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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	#22581581	C₂	1	2	yes	maximal
2	#22581360	C₂	1	4	yes	maximal, complemented
3	#22581357	C₂	1	2	yes	maximal, complemented
4	#22581354	C₂	1	2	yes	maximal, complemented
5	#22580996	C₂	1	2	yes	maximal
6	C₂×#17984776	C₂	1	1	yes	maximal, characteristic
7	#22580908	C₂	1	1	yes	maximal, characteristic
8	#427×C₁₃₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#14555769	C₂²	1	1	yes	characteristic
10	#17984842	—	2	2	no	—
11	#17984841	C₂²	1	4	yes	—
12	#17984834	—	2	2	no	—
13	#17984833	C₂²	1	4	yes	—
14	#17984819	—	2	4	no	—
15	#17984801	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
17	#17984776	C₂²	1	2	yes	—
18	#17984776	C₂²	1	2	yes	—
19	#17984775	C₂²	1	4	yes	—
20	#17984773	C₂²	1	4	yes	—
21	#17984752	C₂²	1	2	yes	—
22	C₈×Dic₁₃₉	C₂²	1	2	yes	—
23	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
24	#117×C₁₃₉	C₂²	1	2	yes	—
25	#107×C₁₃₉	C₂²	1	1	yes	characteristic
26	#107×C₁₃₉	C₂²	1	1	yes	characteristic
27	#106×C₁₃₉	C₂²	1	2	yes	—
28	C₂×Dic₂₇₈	—	2	4	no	—
29	C₂²×Dic₁₃₉	—	2	2	no	—
30	#14555769	Q₈	1	4	yes	complemented
31	C₂×C₁₃₉⋊C₈	C₂³	1	2	yes	—
32	C₄⋊Dic₁₃₉	—	2	4	no	—
33	C₄⋊Dic₁₃₉	—	2	2	no	—
34	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
35	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
36	C₄⋊Dic₁₃₉	C₂³	1	4	yes	—
37	Dic₁₃₉⋊C₄	—	2	4	no	—
38	Dic₁₃₉⋊C₄	—	2	4	no	—
39	C₂²⋊Dic₁₃₉	D₄	1	2	yes	—
40	C₄×Dic₁₃₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
42	C₂×C₈×C₁₃₉	Q₈	1	4	yes	complemented
43	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
46	C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
47	C₂²⋊C₄×C₁₃₉	—	2	2	no	—
48	C₄²×C₁₃₉	—	2	2	no	—
49	Dic₂₇₈	—	8	4	no	—
50	C₂×Dic₁₃₉	—	2	4	no	—
51	C₂×Dic₁₃₉	—	8	2	no	—
52	C₂×Dic₁₃₉	C₂×D₄	1	2	yes	—
53	C₂×Dic₁₃₉	C₂×D₄	1	2	yes	—
54	C₁₃₉⋊C₈	C₂×Q₈	1	4	yes	—
55	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₃₉	—	2	2	no	—
57	C₂×C₄×C₁₃₉	—	2	2	no	—
58	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
60	C₂×C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
62	C₈×C₁₃₉	C₂×Q₈	1	2	yes	—
63	C₈×C₁₃₉	C₂×Q₈	1	2	yes	—
64	C₂²×C₁₃₉	—	2	1	no	—
65	C₂²×C₁₃₉	—	2	1	no	—
66	C₂²×C₁₃₉	C₄⋊Q₈	1	2	yes	—
67	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₁₃₉	—	4	2	no	—
69	C₄×C₁₃₉	—	4	2	no	—
70	C₄×C₁₃₉	C₄⋊Q₈	1	2	yes	—
71	C₄×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
72	C₄×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
73	Dic₁₃₉	—	2	2	no	—
74	Dic₁₃₉	—	2	2	no	—
75	Dic₁₃₉	—	4	4	no	—
76	C₂×C₁₃₉	—	2	2	no	—
77	C₂×C₁₃₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₁₃₉	#576	1	1	yes	characteristic
79	C₂×C₁₃₉	#578	1	1	yes	characteristic
80	C₁₃₉	#3164	1	1	yes	characteristic, complemented, Sylow
81	#3164	—	139	1	no	maximal, Sylow
82	#544	—	139	2	no	—
83	#501	—	139	4	no	—
84	#500	—	139	2	no	—
85	#499	—	139	2	no	—
86	#428	—	139	2	no	—
87	#427	D₁₃₉	1	1	yes	characteristic
88	C₂×#107	—	139	1	no	—
89	#405	—	139	1	no	—
90	C₂×M₁₆	—	139	1	no	—
91	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
92	C₄⋊Q₈	—	278	2	no	—
93	C₄⋊Q₈	—	139	4	no	—
94	#125	—	278	2	no	—
95	#125	—	139	4	no	—
96	C₂²⋊Q₈	—	278	4	no	—
97	#117	—	139	1	no	—
98	#117	C₂×D₁₃₉	1	2	yes	—
99	C₂×C₄⋊C₄	—	139	2	no	—
100	#107	—	139	2	no	—
101	#107	—	139	2	no	—
102	#107	—	139	4	no	—
103	#107	C₂×D₁₃₉	1	1	yes	characteristic
104	#107	C₂×D₁₃₉	1	1	yes	characteristic
105	#106	—	139	4	no	—
106	#106	C₂×D₁₃₉	1	2	yes	—
107	#97	—	139	2	no	—
108	C₄×C₈	—	139	2	no	—
109	C₂×Q₈	—	278	4	no	—
110	C₂²×C₄	—	278	2	no	—
111	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
112	M₁₆	—	139	4	no	—
113	C₂×C₈	—	139	2	no	—
114	C₂×C₈	Dic₂₇₈	1	4	yes	complemented
115	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
116	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
117	C₄⋊C₄	—	278	4	no	—
118	C₄⋊C₄	—	278	4	no	—
119	C₄⋊C₄	—	278	4	no	—
120	C₄⋊C₄	—	278	2	no	—
121	C₄⋊C₄	—	139	4	no	—
122	C₄⋊C₄	—	139	1	no	—
123	C₄⋊C₄	—	139	1	no	—
124	C₄⋊C₄	C₂²×D₁₃₉	1	2	yes	—
125	C₄⋊C₄	C₂²×D₁₃₉	1	2	yes	—
126	C₂²⋊C₄	—	2	2	no	—
127	C₂²⋊C₄	—	139	2	no	—
128	C₄²	—	2	2	no	—
129	C₄²	—	139	2	no	—
130	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
131	Q₈	—	1112	4	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	278	4	no	—
135	C₂×C₄	—	1112	2	no	—
136	C₂×C₄	—	139	2	no	—
137	C₂×C₄	—	139	2	no	—
138	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
139	C₂×C₄	Q₈×D₁₃₉	1	2	yes	—
140	C₂×C₄	C₂×Dic₂₇₈	1	2	yes	—
141	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
142	C₈	—	139	4	no	—
143	C₈	C₂×Dic₂₇₈	1	2	yes	—
144	C₈	C₂×Dic₂₇₈	1	2	yes	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#17984841	1	2	yes	—
148	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
149	C₄	—	4	2	no	—
150	C₄	—	4	2	no	—
151	C₄	—	278	2	no	—
152	C₄	—	278	2	no	—
153	C₄	—	556	4	no	—
154	C₄	#17984841	1	2	yes	—
155	C₄	C₂×Q₈×D₁₃₉	1	1	yes	characteristic
156	C₄	C₂²×Dic₂₇₈	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	C₂×#17984841	1	1	yes	characteristic
159	C₂	#22581755	1	1	yes	characteristic
160	C₂	#22581767	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581581

C₂

yes

maximal

#22581360

C₂

yes

maximal, complemented

#22581357

C₂

yes

maximal, complemented

#22581354

C₂

yes

maximal, complemented

#22580996

C₂

yes

maximal

C₂×#17984776

C₂

yes

maximal, characteristic

#22580908

C₂

yes

maximal, characteristic

#427×C₁₃₉

C₂

yes

maximal, characteristic

C₂×#14555769

C₂²

yes

characteristic

#17984842

—

#17984841

C₂²

yes

—

#17984834

—

#17984833

C₂²

yes

—

#17984819

—

#17984801

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃₉

C₂²

yes

—

#17984776

C₂²

yes

—

#17984776

C₂²

yes

—

#17984775

C₂²

yes

—

#17984773

C₂²

yes

—

#17984752

C₂²

yes

—

C₈×Dic₁₃₉

C₂²

yes

—

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

#117×C₁₃₉

C₂²

yes

—

#107×C₁₃₉

C₂²

yes

characteristic

#107×C₁₃₉

C₂²

yes

characteristic

#106×C₁₃₉

C₂²

yes

—

C₂×Dic₂₇₈

—

C₂²×Dic₁₃₉

—

#14555769

Q₈

yes

complemented

C₂×C₁₃₉⋊C₈

C₂³

yes

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

D₄

yes

characteristic

C₄⋊Dic₁₃₉

D₄

yes

characteristic

C₄⋊Dic₁₃₉

C₂³

yes

—

Dic₁₃₉⋊C₄

—

Dic₁₃₉⋊C₄

—

C₂²⋊Dic₁₃₉

D₄

yes

—

C₄×Dic₁₃₉

C₂³

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₈×C₁₃₉

Q₈

yes

complemented

C₂×C₈×C₁₃₉

C₂³

yes

characteristic

C₂×C₈×C₁₃₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₉

C₂³

yes

—

C₄⋊C₄×C₁₃₉

C₂³

yes

—

C₂²⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

C₂×D₄

yes

—

C₂×Dic₁₃₉

C₂×D₄

yes

—

C₁₃₉⋊C₈

C₂×Q₈

yes

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

—

C₂×C₄×C₁₃₉

C₂×Q₈

yes

—

C₂×C₄×C₁₃₉

C₂⁴

yes

characteristic

C₈×C₁₃₉

C₂×Q₈

yes

—

C₈×C₁₃₉

C₂×Q₈

yes

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₄⋊Q₈

yes

—

C₂²×C₁₃₉

C₂²×D₄

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₄⋊Q₈

yes

—

C₄×C₁₃₉

C₂²×Q₈

yes

characteristic

C₄×C₁₃₉

C₂²×Q₈

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

C₂×C₄⋊Q₈

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#3164

—

139

maximal, Sylow

#544

—

139

—

#501

—

139

—

#500

—

139

—

#499

—

139

—

#428

—

139

—

#427

D₁₃₉

yes

characteristic

C₂×#107

—

139

—

#405

—

139

—

C₂×M₁₆

—

139

—

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

C₄⋊Q₈

—

278

—

C₄⋊Q₈

—

139

—

#125

—

278

—

#125

—

139

—

C₂²⋊Q₈

—

278

—

#117

—

139

—

#117

C₂×D₁₃₉

yes

—

C₂×C₄⋊C₄

—

139

—

100

#107

—

139

—

101

#107

—

139

—

102

#107

—

139

—

103

#107

C₂×D₁₃₉

yes

characteristic

104

#107

C₂×D₁₃₉

yes

characteristic

105

#106

—

139

—

106

#106

C₂×D₁₃₉

yes

—

107

#97

—

139

—

108

C₄×C₈

—

139

—

109

C₂×Q₈

—

278

—

110

C₂²×C₄

—

278

—

111

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

112

M₁₆

—

139

—

113

C₂×C₈

—

139

—

114

C₂×C₈

Dic₂₇₈

yes

complemented

115

C₂×C₈

C₂²×D₁₃₉

yes

characteristic

116

C₂×C₈

C₂²×D₁₃₉

yes

characteristic

117

C₄⋊C₄

—

278

—

118

C₄⋊C₄

—

278

—

119

C₄⋊C₄

—

278

—

120

C₄⋊C₄

—

278

—

121

C₄⋊C₄

—

139

—

122

C₄⋊C₄

—

139

—

123

C₄⋊C₄

—

139

—

124

C₄⋊C₄

C₂²×D₁₃₉

yes

—

125

C₄⋊C₄

C₂²×D₁₃₉

yes

—

126

C₂²⋊C₄

—

127

C₂²⋊C₄

—

139

—

128

C₄²

—

129

C₄²

—

139

—

130

C₂³

D₄×D₁₃₉

yes

characteristic

131

Q₈

—

1112

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

278

—

135

C₂×C₄

—

1112

—

136

C₂×C₄

—

139

—

137

C₂×C₄

—

139

—

138

C₂×C₄

D₄×D₁₃₉

yes

characteristic

139

C₂×C₄

Q₈×D₁₃₉

yes

—

140

C₂×C₄

C₂×Dic₂₇₈

yes

—

141

C₂×C₄

C₂³×D₁₃₉

yes

characteristic

142

C₈

—

139

—

143

C₈

C₂×Dic₂₇₈

yes

—

144

C₈

C₂×Dic₂₇₈

yes

—

145

C₂²

—

146

C₂²

—

147

C₂²

#17984841

yes

—

148

C₂²

C₂×D₄×D₁₃₉

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

278

—

152

C₄

—

278

—

153

C₄

—

556

—

154

C₄

#17984841

yes

—

155

C₄

C₂×Q₈×D₁₃₉

yes

characteristic

156

C₄

C₂²×Dic₂₇₈

yes

characteristic

157

C₂

—

158

C₂

C₂×#17984841

yes

characteristic

159

C₂

#22581755

yes

characteristic

160

C₂

#22581767

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#427×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂³×C₁₃₉	C₂×D₄
hypercenter	#427	D₁₃₉
hypocenter	C₁₃₉	#3164
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂²

C₂×D₄×D₁₃₉

commutator, cosocle

C₂×C₄×C₁₃₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₉

Fitting

#427×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂²×D₄

prime radical

C₂³×C₁₃₉

C₂×D₄

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #22581360	C₂	semidirect
3. #22581357	C₂	semidirect
4. #22581354	C₂	semidirect
30. #14555769	Q₈	semidirect
42. C₂×C₈×C₁₃₉	Q₈	semidirect
80. C₁₃₉	#3164	semidirect
114. C₂×C₈	Dic₂₇₈	semidirect

H ≅ G/N

type

2. #22581360

C₂

semidirect

3. #22581357

C₂

semidirect

4. #22581354

C₂

semidirect

30. #14555769

Q₈

semidirect

42. C₂×C₈×C₁₃₉

Q₈

semidirect

80. C₁₃₉

#3164

semidirect

114. C₂×C₈

Dic₂₇₈

semidirect

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

subgroup	index
#22581581	2
#22581360	2
#22581357	2
#22581354	2
#22580996	2
C₂×#17984776	2
#22580908	2
#427×C₁₃₉	2
#3164	139

subgroup

index

#427×C₁₃₉

139

Subgroups — #31816944

Conjugacy classes of subgroups

Complements

Maximal subgroups