Group #31780894 — subgroups

Subgroups — #31780894

Order 17536.

262 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 81 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	#41×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#22577377	C₂	1	1	yes	maximal, characteristic
3	#22577164	C₂	1	2	yes	maximal, complemented
4	#22577160	C₂	1	2	yes	maximal, complemented
5	#413×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₁₃₇	—	2	1	no	—
7	D₄×Dic₁₃₇	—	2	1	no	—
8	D₄×Dic₁₃₇	—	2	2	no	—
9	#17984047	C₄	1	2	yes	complemented
10	#17984047	C₂²	1	2	yes	—
11	#17984047	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₁₃₇	C₄	1	2	yes	complemented
13	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
14	#17984003	—	2	2	no	—
15	C₂×#41×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₁₃₇	C₂²	1	2	yes	—
17	#101×C₁₃₇	—	2	1	no	—
18	#100×C₁₃₇	—	2	1	no	—
19	C₂²×Dic₁₃₇	—	2	2	no	—
20	C₂²×Dic₁₃₇	—	2	1	no	—
21	C₄⋊Dic₁₃₇	—	2	2	no	—
22	C₄⋊Dic₁₃₇	—	2	1	no	—
23	C₄⋊Dic₁₃₇	—	2	2	no	—
24	C₂²⋊Dic₁₃₇	—	2	2	no	—
25	C₂²⋊Dic₁₃₇	—	2	1	no	—
26	C₂²⋊Dic₁₃₇	—	2	2	no	—
27	C₄×Dic₁₃₇	—	2	2	no	—
28	C₄×Dic₁₃₇	—	2	1	no	—
29	C₄×Dic₁₃₇	—	2	2	no	—
30	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
32	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
33	#41×C₁₃₇	—	2	2	no	—
34	#41×C₁₃₇	—	2	2	no	—
35	C₂×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
36	C₂×D₄×C₁₃₇	D₄	1	1	yes	characteristic
37	C₂×D₄×C₁₃₇	D₄	1	2	yes	—
38	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
40	M₁₆×C₁₃₇	—	2	2	no	—
41	M₁₆×C₁₃₇	—	2	2	no	—
42	C₂×C₈×C₁₃₇	—	2	2	no	—
43	C₂×Dic₁₃₇	—	2	1	no	—
44	C₂×Dic₁₃₇	—	2	2	no	—
45	C₂×Dic₁₃₇	—	2	1	no	—
46	C₂×Dic₁₃₇	—	4	2	no	—
47	C₂×Dic₁₃₇	—	4	4	no	—
48	C₂×Dic₁₃₇	—	4	2	no	—
49	C₂³×C₁₃₇	C₂²⋊C₄	1	2	yes	—
50	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₁₃₇	—	2	2	no	—
52	D₄×C₁₃₇	—	2	4	no	—
53	D₄×C₁₃₇	—	2	2	no	—
54	C₂×C₄×C₁₃₇	—	2	2	no	—
55	C₂×C₄×C₁₃₇	—	2	2	no	—
56	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
57	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₁₃₇	#41	1	2	yes	—
60	C₂×C₄×C₁₃₇	#41	1	2	yes	—
61	C₈×C₁₃₇	—	4	2	no	—
62	C₂²×C₁₃₇	—	2	2	no	—
63	C₂²×C₁₃₇	—	4	1	no	—
64	C₂²×C₁₃₇	—	4	2	no	—
65	C₂²×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₁₃₇	#123	1	2	yes	—
67	C₄×C₁₃₇	—	2	2	no	—
68	C₄×C₁₃₇	#117	1	1	yes	characteristic
69	C₄×C₁₃₇	#117	1	1	yes	characteristic
70	C₄×C₁₃₇	#123	1	2	yes	—
71	Dic₁₃₇	—	2	2	no	—
72	Dic₁₃₇	—	4	2	no	—
73	Dic₁₃₇	—	4	1	no	—
74	Dic₁₃₇	—	8	2	no	—
75	C₂×C₁₃₇	—	2	1	no	—
76	C₂×C₁₃₇	—	2	2	no	—
77	C₂×C₁₃₇	—	4	2	no	—
78	C₂×C₁₃₇	#381	1	1	yes	characteristic
79	C₁₃₇	#1827	1	1	yes	characteristic, complemented, Sylow
80	#1827	—	137	1	no	maximal, Sylow
81	C₄×#41	—	137	1	no	—
82	#413	D₁₃₇	1	1	yes	characteristic
83	#410	—	137	1	no	—
84	#339	—	137	2	no	—
85	H(3,ℤ/4)	—	137	2	no	—
86	C₂×#41	C₂×D₁₃₇	1	1	yes	characteristic
87	C₂×M₁₆	C₂×D₁₃₇	1	2	yes	—
88	C₄×Q₈	—	274	1	no	—
89	C₄×D₄	—	274	1	no	—
90	C₄×D₄	—	274	2	no	—
91	#117	—	137	2	no	—
92	#117	—	137	2	no	—
93	#117	—	137	1	no	—
94	C₂×C₄²	—	137	2	no	—
95	C₂×C₄²	—	137	1	no	—
96	#101	—	2	1	no	—
97	#100	—	2	1	no	—
98	#99	—	274	2	no	—
99	#41	—	2	2	no	—
100	#41	—	2	2	no	—
101	C₂×Q₈	C₂²⋊D₁₃₇	1	1	yes	characteristic
102	C₂×D₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
103	C₂×D₄	D₅₄₈	1	2	yes	—
104	C₂²×C₄	—	274	2	no	—
105	C₂²×C₄	—	274	1	no	—
106	C₂²×C₄	C₄×D₁₃₇	1	2	yes	—
107	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₄⋊C₄	—	274	2	no	—
112	C₄⋊C₄	—	274	1	no	—
113	C₄⋊C₄	—	274	2	no	—
114	C₂²⋊C₄	—	274	2	no	—
115	C₂²⋊C₄	—	274	1	no	—
116	C₂²⋊C₄	—	274	2	no	—
117	C₄²	—	274	2	no	—
118	C₄²	—	274	1	no	—
119	C₄²	—	274	2	no	—
120	C₄²	—	137	1	no	—
121	C₄²	—	137	2	no	—
122	C₄²	—	137	1	no	—
123	C₂³	#14555487	1	2	yes	—
124	C₂³	C₂×D₅₄₈	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	D₄	—	2	4	no	—
127	D₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	274	1	no	—
131	C₂×C₄	—	274	2	no	—
132	C₂×C₄	—	274	1	no	—
133	C₂×C₄	—	548	2	no	—
134	C₂×C₄	—	548	4	no	—
135	C₂×C₄	—	548	2	no	—
136	C₂×C₄	#14555487	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
138	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
139	C₂×C₄	#14555509	1	2	yes	—
140	C₂×C₄	#14555509	1	2	yes	—
141	C₈	—	4	2	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	4	1	no	—
144	C₂²	—	4	2	no	—
145	C₂²	C₂×#14555487	1	1	yes	characteristic
146	C₂²	#17984072	1	2	yes	—
147	C₄	—	2	2	no	—
148	C₄	—	274	2	no	—
149	C₄	—	548	2	no	—
150	C₄	—	548	1	no	—
151	C₄	—	1096	2	no	—
152	C₄	#17984044	1	1	yes	characteristic
153	C₄	#17984044	1	1	yes	characteristic
154	C₄	#17984069	1	2	yes	—
155	C₂	—	2	1	no	—
156	C₂	—	2	2	no	—
157	C₂	—	4	2	no	—
158	C₂	#22577264	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#41×Dic₁₃₇

C₂

yes

maximal, characteristic

#22577377

C₂

yes

maximal, characteristic

#22577164

C₂

yes

maximal, complemented

#22577160

C₂

yes

maximal, complemented

#413×C₁₃₇

C₂

yes

maximal, characteristic

Q₈×Dic₁₃₇

—

D₄×Dic₁₃₇

—

D₄×Dic₁₃₇

—

#17984047

C₄

yes

complemented

#17984047

C₂²

yes

—

#17984047

C₂²

yes

characteristic

C₂×C₄×Dic₁₃₇

C₄

yes

complemented

C₂×C₄×Dic₁₃₇

C₂²

yes

characteristic

#17984003

—

C₂×#41×C₁₃₇

C₂²

yes

characteristic

C₂×M₁₆×C₁₃₇

C₂²

yes

—

#101×C₁₃₇

—

#100×C₁₃₇

—

C₂²×Dic₁₃₇

—

C₂²×Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₂²⋊Dic₁₃₇

—

C₂²⋊Dic₁₃₇

—

C₂²⋊Dic₁₃₇

—

C₄×Dic₁₃₇

—

C₄×Dic₁₃₇

—

C₄×Dic₁₃₇

—

C₄×Dic₁₃₇

C₂×C₄

yes

characteristic

C₄×Dic₁₃₇

C₂×C₄

yes

—

C₄×Dic₁₃₇

C₂×C₄

yes

characteristic

#41×C₁₃₇

—

#41×C₁₃₇

—

C₂×Q₈×C₁₃₇

D₄

yes

characteristic

C₂×D₄×C₁₃₇

D₄

yes

characteristic

C₂×D₄×C₁₃₇

D₄

yes

—

C₂²×C₄×C₁₃₇

C₂×C₄

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

M₁₆×C₁₃₇

—

M₁₆×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂³×C₁₃₇

C₂²⋊C₄

yes

—

C₂³×C₁₃₇

C₂×D₄

yes

characteristic

Q₈×C₁₃₇

—

D₄×C₁₃₇

—

D₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

#41

yes

—

C₂×C₄×C₁₃₇

#41

yes

—

C₈×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃₇

#123

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

#381

yes

characteristic

C₁₃₇

#1827

yes

characteristic, complemented, Sylow

#1827

—

137

maximal, Sylow

C₄×#41

—

137

—

#413

D₁₃₇

yes

characteristic

#410

—

137

—

#339

—

137

—

H(3,ℤ/4)

—

137

—

C₂×#41

C₂×D₁₃₇

yes

characteristic

C₂×M₁₆

C₂×D₁₃₇

yes

—

C₄×Q₈

—

274

—

C₄×D₄

—

274

—

C₄×D₄

—

274

—

#117

—

137

—

#117

—

137

—

#117

—

137

—

C₂×C₄²

—

137

—

C₂×C₄²

—

137

—

#101

—

#100

—

#99

—

274

—

#41

—

100

#41

—

101

C₂×Q₈

C₂²⋊D₁₃₇

yes

characteristic

102

C₂×D₄

C₂²⋊D₁₃₇

yes

characteristic

103

C₂×D₄

D₅₄₈

yes

—

104

C₂²×C₄

—

274

—

105

C₂²×C₄

—

274

—

106

C₂²×C₄

C₄×D₁₃₇

yes

—

107

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

C₂×C₈

—

111

C₄⋊C₄

—

274

—

112

C₄⋊C₄

—

274

—

113

C₄⋊C₄

—

274

—

114

C₂²⋊C₄

—

274

—

115

C₂²⋊C₄

—

274

—

116

C₂²⋊C₄

—

274

—

117

C₄²

—

274

—

118

C₄²

—

274

—

119

C₄²

—

274

—

120

C₄²

—

137

—

121

C₄²

—

137

—

122

C₄²

—

137

—

123

C₂³

#14555487

yes

—

124

C₂³

C₂×D₅₄₈

yes

characteristic

125

Q₈

—

126

D₄

—

127

D₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

274

—

131

C₂×C₄

—

274

—

132

C₂×C₄

—

274

—

133

C₂×C₄

—

548

—

134

C₂×C₄

—

548

—

135

C₂×C₄

—

548

—

136

C₂×C₄

#14555487

yes

—

137

C₂×C₄

C₂×C₄×D₁₃₇

yes

characteristic

138

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

characteristic

139

C₂×C₄

#14555509

yes

—

140

C₂×C₄

#14555509

yes

—

141

C₈

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

C₂×#14555487

yes

characteristic

146

C₂²

#17984072

yes

—

147

C₄

—

148

C₄

—

274

—

149

C₄

—

548

—

150

C₄

—

548

—

151

C₄

—

1096

—

152

C₄

#17984044

yes

characteristic

153

C₄

#17984044

yes

characteristic

154

C₄

#17984069

yes

—

155

C₂

—

156

C₂

—

157

C₂

—

158

C₂

#22577264

yes

characteristic

	subgroup	G/H
center	C₄	#17984044
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₇
Fitting	#413×C₁₃₇	C₂
socle	C₂×C₁₃₇	#381
prime radical	C₂×#41×C₁₃₇	C₂²
cosocle	C₂²×C₄×C₁₃₇	C₂³
hypercenter	#413	D₁₃₇
hypocenter	C₁₃₇	#1827
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₄

#17984044

commutator

C₂×C₄×C₁₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₇

Fitting

#413×C₁₃₇

C₂

socle

C₂×C₁₃₇

#381

prime radical

C₂×#41×C₁₃₇

C₂²

cosocle

C₂²×C₄×C₁₃₇

C₂³

D₁₃₇

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
3. #22577164	C₂	semidirect
4. #22577160	C₂	semidirect
9. #17984047	C₄	semidirect
12. C₂×C₄×Dic₁₃₇	C₄	semidirect
79. C₁₃₇	#1827	semidirect

H ≅ G/N

type

3. #22577164

C₂

semidirect

4. #22577160

C₂

semidirect

9. #17984047

C₄

semidirect

12. C₂×C₄×Dic₁₃₇

C₄

semidirect

79. C₁₃₇

#1827

semidirect

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

subgroup	index
#41×Dic₁₃₇	2
#22577377	2
#22577164	2
#22577160	2
#413×C₁₃₇	2
#1827	137

subgroup

index

#41×Dic₁₃₇

#413×C₁₃₇

137

Subgroups — #31780894

Conjugacy classes of subgroups

Complements

Maximal subgroups