Group #30770583 — subgroups

Subgroups — #30770583

Order 17152.

340 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 91 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	#22562459	C₂	1	2	yes	maximal, complemented
2	#22562433	C₂	1	2	yes	maximal, complemented
3	#22559824	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559801	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₀₇₂	—	2	4	no	—
7	#16863801	C₂²	1	1	yes	characteristic
8	#16863773	—	2	1	no	—
9	#16863773	—	2	1	no	—
10	#16863773	C₂²	1	1	yes	characteristic
11	#16863773	C₂²	1	2	yes	—
12	#16863769	—	2	1	no	—
13	#16863743	—	2	1	no	—
14	#16863198	—	2	4	no	—
15	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
16	C₈²×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#11761617	—	4	2	no	—
18	C₂×D₅₃₆	—	2	2	no	—
19	C₂×D₅₃₆	—	2	2	no	—
20	C₂×D₅₃₆	—	2	2	no	—
21	C₂×D₅₃₆	—	4	2	no	—
22	C₂×D₅₃₆	—	4	2	no	—
23	#14524043	—	2	1	no	—
24	#14524040	—	2	1	no	—
25	#14524040	—	2	2	no	—
26	D₁₀₇₂	—	4	8	no	—
27	#14523977	—	2	2	no	—
28	#14523965	—	4	2	no	—
29	C₂×C₁₆×C₆₇	D₄	1	4	yes	—
30	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
31	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
32	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₃₄	—	4	1	no	—
34	C₂×D₂₆₈	—	4	1	no	—
35	C₂×D₂₆₈	—	4	2	no	—
36	C₂×D₂₆₈	—	4	1	no	—
37	C₂×D₂₆₈	—	4	2	no	—
38	C₂×D₂₆₈	—	4	2	no	—
39	C₂×D₂₆₈	—	4	1	no	—
40	#11761617	—	8	4	no	—
41	D₅₃₆	—	4	4	no	—
42	D₅₃₆	—	4	4	no	—
43	D₅₃₆	—	4	4	no	—
44	D₅₃₆	—	8	4	no	—
45	D₅₃₆	—	8	4	no	—
46	C₄⋊Dic₆₇	—	4	1	no	—
47	C₄⋊Dic₆₇	—	4	1	no	—
48	C₂×C₈×C₆₇	D₈	1	2	yes	—
49	C₂×C₈×C₆₇	D₈	1	2	yes	—
50	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
51	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₁₆×C₆₇	—	2	4	no	—
53	C₂²×D₆₇	—	8	1	no	—
54	C₂²×D₆₇	—	8	2	no	—
55	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
56	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
58	C₂×Dic₆₇	—	8	1	no	—
59	D₂₆₈	—	8	2	no	—
60	D₂₆₈	—	8	2	no	—
61	D₂₆₈	—	8	2	no	—
62	D₂₆₈	—	8	4	no	—
63	D₂₆₈	—	8	2	no	—
64	D₂₆₈	—	8	4	no	—
65	D₂₆₈	—	8	2	no	—
66	Dic₁₃₄	—	8	2	no	—
67	C₈×C₆₇	—	2	2	no	—
68	C₈×C₆₇	D₁₆	1	4	yes	complemented
69	C₈×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
70	C₂²×C₆₇	#493	1	1	yes	characteristic
71	C₂×D₆₇	—	16	1	no	—
72	C₂×D₆₇	—	16	2	no	—
73	C₂×D₆₇	—	16	2	no	—
74	C₂×D₆₇	—	16	1	no	—
75	C₂×D₆₇	—	16	2	no	—
76	C₂×D₆₇	—	16	1	no	—
77	C₄×C₆₇	#496	1	2	yes	—
78	C₄×C₆₇	C₂×D₁₆	1	2	yes	—
79	C₄×C₆₇	#509	1	2	yes	—
80	Dic₆₇	—	16	1	no	—
81	#12525	—	67	1	no	maximal, Sylow
82	C₂×C₆₇	#1721	1	1	yes	characteristic
83	C₂×C₆₇	#2246	1	1	yes	characteristic
84	C₂×C₆₇	#2250	1	1	yes	characteristic
85	D₆₇	—	32	2	no	—
86	D₆₇	—	32	1	no	—
87	#2246	—	67	2	no	—
88	#2240	—	67	2	no	—
89	#1712	—	67	1	no	—
90	#1706	—	67	1	no	—
91	#1312	D₆₇	1	1	yes	characteristic, complemented
92	C₆₇	#12525	1	1	yes	characteristic, complemented, Sylow
93	C₂×D₁₆	—	134	4	no	—
94	#500	—	67	1	no	—
95	#493	—	67	1	no	—
96	#493	—	67	2	no	—
97	#493	—	134	1	no	—
98	#493	—	134	1	no	—
99	#492	—	134	1	no	—
100	#486	—	134	1	no	—
101	#357	—	134	4	no	—
102	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
103	C₈²	C₂×D₆₇	1	1	yes	characteristic
104	C₂×SD₁₆	—	268	2	no	—
105	C₂×D₈	—	268	2	no	—
106	C₂×D₈	—	268	2	no	—
107	C₂×D₈	—	134	2	no	—
108	C₂×D₈	—	134	2	no	—
109	C₂×D₈	—	134	2	no	—
110	C₄⋊Q₈	—	134	1	no	—
111	#127	—	134	1	no	—
112	#127	—	134	2	no	—
113	D₁₆	—	268	8	no	—
114	C₂×C₁₆	D₂₆₈	1	4	yes	—
115	#107	—	134	2	no	—
116	#102	—	268	2	no	—
117	C₄×C₈	D₂₆₈	1	1	yes	characteristic
118	C₄×C₈	D₂₆₈	1	1	yes	characteristic
119	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
120	C₂×Q₈	—	268	1	no	—
121	C₂×D₄	—	268	1	no	—
122	C₂×D₄	—	268	2	no	—
123	C₂×D₄	—	268	1	no	—
124	C₂×D₄	—	268	2	no	—
125	C₂×D₄	—	268	2	no	—
126	C₂×D₄	—	268	1	no	—
127	SD₁₆	—	536	4	no	—
128	D₈	—	268	4	no	—
129	D₈	—	268	4	no	—
130	D₈	—	268	4	no	—
131	D₈	—	536	4	no	—
132	D₈	—	536	4	no	—
133	C₂×C₈	D₅₃₆	1	2	yes	—
134	C₂×C₈	D₅₃₆	1	2	yes	—
135	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
136	C₄⋊C₄	—	268	1	no	—
137	C₄⋊C₄	—	268	1	no	—
138	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
139	C₁₆	—	2	4	no	—
140	C₂³	—	536	1	no	—
141	C₂³	—	536	2	no	—
142	Q₈	—	536	2	no	—
143	D₄	—	536	2	no	—
144	D₄	—	536	2	no	—
145	D₄	—	536	2	no	—
146	D₄	—	536	4	no	—
147	D₄	—	536	2	no	—
148	D₄	—	536	4	no	—
149	D₄	—	536	2	no	—
150	C₂×C₄	—	536	1	no	—
151	C₂×C₄	#14524040	1	1	yes	characteristic
152	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
153	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	D₁₀₇₂	1	4	yes	complemented
156	C₈	#14524071	1	4	yes	—
157	C₂²	—	1072	1	no	—
158	C₂²	—	1072	2	no	—
159	C₂²	—	1072	2	no	—
160	C₂²	—	1072	1	no	—
161	C₂²	—	1072	2	no	—
162	C₂²	—	1072	1	no	—
163	C₂²	#16863773	1	1	yes	characteristic
164	C₄	—	1072	1	no	—
165	C₄	#16863785	1	2	yes	—
166	C₄	C₂×D₁₀₇₂	1	2	yes	—
167	C₄	#16863834	1	2	yes	—
168	C₂	—	2144	2	no	—
169	C₂	—	2144	1	no	—
170	C₂	#22559875	1	1	yes	characteristic
171	C₂	#22562459	1	1	yes	characteristic
172	C₂	#22562475	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562459

C₂

yes

maximal, complemented

#22562433

C₂

yes

maximal, complemented

#22559824

C₂

yes

maximal, characteristic, complemented

#22559801

C₂

yes

maximal, characteristic, complemented

C₆₇×#1312

C₂

yes

maximal, characteristic, complemented

C₂×D₁₀₇₂

—

#16863801

C₂²

yes

—

—

C₂²

yes

characteristic

#16863773

C₂²

yes

—

#16863769

—

#16863743

—

#16863198

—

C₄×C₁₆×C₆₇

C₂²

yes

—

C₈²×C₆₇

C₂²

yes

characteristic

C₂×#11761617

—

C₂×D₅₃₆

—

C₂×D₅₃₆

—

C₂×D₅₃₆

—

C₂×D₅₃₆

—

C₂×D₅₃₆

—

#14524043

—

#14524040

—

#14524040

—

D₁₀₇₂

—

#14523977

—

#14523965

—

C₂×C₁₆×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×Dic₁₃₄

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

#11761617

—

D₅₃₆

—

D₅₃₆

—

D₅₃₆

—

D₅₃₆

—

D₅₃₆

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₁₆×C₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

D₁₆

yes

complemented

C₈×C₆₇

AGL(1,ℤ/8)

yes

—

C₂²×C₆₇

#493

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

#496

yes

—

C₄×C₆₇

C₂×D₁₆

yes

—

C₄×C₆₇

#509

yes

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

—

D₆₇

—

—

—

—

—

D₆₇

yes

characteristic, complemented

C₆₇

#12525

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

134

—

—

—

—

—

134

—

#493

—

134

—

#492

—

134

—

100

#486

—

134

—

101

#357

—

134

—

102

C₄×C₁₆

C₂×D₆₇

yes

—

103

C₈²

C₂×D₆₇

yes

characteristic

104

C₂×SD₁₆

—

268

—

105

C₂×D₈

—

268

—

106

C₂×D₈

—

268

—

107

C₂×D₈

—

134

—

108

C₂×D₈

—

134

—

109

C₂×D₈

—

134

—

110

C₄⋊Q₈

—

134

—

111

#127

—

134

—

112

#127

—

134

—

113

D₁₆

—

268

—

114

C₂×C₁₆

D₂₆₈

yes

—

115

#107

—

134

—

116

#102

—

268

—

117

C₄×C₈

D₂₆₈

yes

characteristic

118

C₄×C₈

D₂₆₈

yes

characteristic

119

C₄×C₈

C₂²×D₆₇

yes

characteristic

120

C₂×Q₈

—

268

—

121

C₂×D₄

—

268

—

122

C₂×D₄

—

268

—

123

C₂×D₄

—

268

—

124

C₂×D₄

—

268

—

125

C₂×D₄

—

268

—

126

C₂×D₄

—

268

—

127

SD₁₆

—

536

—

128

D₈

—

268

—

129

D₈

—

268

—

130

D₈

—

268

—

131

D₈

—

536

—

132

D₈

—

536

—

133

C₂×C₈

D₅₃₆

yes

—

134

C₂×C₈

D₅₃₆

yes

—

135

C₂×C₈

C₂×D₂₆₈

yes

—

136

C₄⋊C₄

—

268

—

137

C₄⋊C₄

—

268

—

138

C₄²

C₂×D₂₆₈

yes

characteristic

139

C₁₆

—

140

C₂³

—

536

—

141

C₂³

—

536

—

142

Q₈

—

536

—

143

D₄

—

536

—

144

D₄

—

536

—

145

D₄

—

536

—

146

D₄

—

536

—

147

D₄

—

536

—

148

D₄

—

536

—

149

D₄

—

536

—

150

C₂×C₄

—

536

—

151

C₂×C₄

#14524040

yes

characteristic

152

C₂×C₄

C₂×D₅₃₆

yes

characteristic

153

C₂×C₄

C₂×D₅₃₆

yes

characteristic

154

C₈

—

155

C₈

D₁₀₇₂

yes

complemented

156

C₈

#14524071

yes

—

157

C₂²

—

1072

—

158

C₂²

—

1072

—

159

C₂²

—

1072

—

160

C₂²

—

1072

—

161

C₂²

—

1072

—

162

C₂²

—

1072

—

163

C₂²

#16863773

yes

characteristic

164

C₄

—

1072

—

165

C₄

#16863785

yes

—

166

C₄

C₂×D₁₀₇₂

yes

—

167

C₄

#16863834

yes

—

168

C₂

—

2144

—

169

C₂

—

2144

—

170

C₂

#22559875

yes

characteristic

171

C₂

#22562459

yes

characteristic

172

C₂

#22562475

yes

characteristic

	subgroup	G/H
center	C₂²	#16863773
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1312	C₂
socle	C₂²×C₆₇	#493
hypercenter	#1312	D₆₇
hypocenter	C₆₇	#12525
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

subgroup

G/H

center

C₂²

#16863773

commutator, cosocle

C₄×C₈×C₆₇

C₂³

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#1312

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

N	H ≅ G/N	type
1. #22562459	C₂	semidirect
2. #22562433	C₂	semidirect
3. #22559824	C₂	semidirect
4. #22559801	C₂	semidirect
5. C₆₇×#1312	C₂	semidirect
68. C₈×C₆₇	D₁₆	semidirect
91. #1312	D₆₇	semidirect
92. C₆₇	#12525	semidirect
155. C₈	D₁₀₇₂	semidirect

H ≅ G/N

type

1. #22562459

C₂

semidirect

2. #22562433

C₂

semidirect

3. #22559824

C₂

semidirect

4. #22559801

C₂

semidirect

5. C₆₇×#1312

C₂

semidirect

68. C₈×C₆₇

D₁₆

semidirect

91. #1312

D₆₇

semidirect

92. C₆₇

#12525

semidirect

155. C₈

D₁₀₇₂

semidirect

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

subgroup	index
#22562459	2
#22562433	2
#22559824	2
#22559801	2
C₆₇×#1312	2
#12525	67

subgroup

index

C₆₇×#1312

Subgroups — #30770583

Conjugacy classes of subgroups

Complements

Maximal subgroups