Group #30775790 — subgroups

Subgroups — #30775790

Order 17152.

268 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 101 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	#22561997	C₂	1	2	yes	maximal
2	#22560957	C₂	1	1	yes	maximal, characteristic
3	#22560373	C₂	1	1	yes	maximal, characteristic
4	#22558318	C₂	1	2	yes	maximal
5	C₆₇×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523979	C₂²	1	1	yes	characteristic
7	#16863434	C₂²	1	1	yes	characteristic
8	C₂×#14523977	C₂²	1	2	yes	—
9	C₂×#14523975	—	2	1	no	—
10	#16863422	—	2	1	no	—
11	#16863262	—	2	1	no	—
12	#16863224	—	2	2	no	—
13	#16863222	—	2	2	no	—
14	#16863156	—	2	1	no	—
15	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#11761612	—	2	1	no	—
18	#14524002	—	2	1	no	—
19	C₂×C₄⋊Dic₆₇	—	2	2	no	—
20	#14523979	—	2	2	no	—
21	#14523977	—	2	2	no	—
22	#14523977	—	2	2	no	—
23	#14523977	D₄	1	2	yes	—
24	#14523975	—	2	1	no	—
25	#14523975	—	4	2	no	—
26	Dic₆₇⋊C₈	—	4	2	no	—
27	C₆₇⋊#95	—	4	2	no	—
28	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
31	M₃₂×C₆₇	—	2	2	no	—
32	M₃₂×C₆₇	Q₈	1	4	yes	—
33	C₂×C₁₆×C₆₇	—	2	2	no	—
34	C₄×C₈×C₆₇	—	2	2	no	—
35	C₂²×Dic₆₇	—	4	2	no	—
36	#11761612	—	4	1	no	—
37	#11761612	—	4	1	no	—
38	C₂×C₆₇⋊C₈	—	4	1	no	—
39	C₄⋊Dic₆₇	—	2	2	no	—
40	C₄⋊Dic₆₇	—	4	2	no	—
41	C₄⋊Dic₆₇	—	4	2	no	—
42	C₂²⋊Dic₆₇	—	4	1	no	—
43	C₄×Dic₆₇	—	4	1	no	—
44	C₂²×C₄×C₆₇	—	2	1	no	—
45	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₆₇	—	2	1	no	—
47	C₂×C₈×C₆₇	—	2	1	no	—
48	C₂×C₈×C₆₇	D₈	1	2	yes	—
49	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
51	C₂×C₈×C₆₇	#41	1	2	yes	—
52	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
53	C₄²×C₆₇	—	2	1	no	—
54	C₄²×C₆₇	Q₁₆	1	2	yes	—
55	C₁₆×C₆₇	—	2	4	no	—
56	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
57	C₂×C₄×C₆₇	—	2	2	no	—
58	C₂×C₄×C₆₇	—	4	2	no	—
59	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
60	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
64	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
65	C₂×Dic₆₇	—	4	2	no	—
66	C₂×Dic₆₇	—	8	4	no	—
67	C₂×Dic₆₇	—	8	2	no	—
68	C₈×C₆₇	—	2	2	no	—
69	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
70	C₈×C₆₇	#125	1	2	yes	—
71	C₆₇⋊C₈	—	8	1	no	—
72	C₂²×C₆₇	—	2	2	no	—
73	C₂²×C₆₇	#459	1	1	yes	characteristic
74	C₂²×C₆₇	#484	1	1	yes	characteristic
75	C₂²×C₆₇	#501	1	1	yes	characteristic
76	C₄×C₆₇	—	4	2	no	—
77	C₄×C₆₇	#398	1	1	yes	characteristic
78	C₄×C₆₇	#462	1	1	yes	characteristic
79	C₄×C₆₇	#480	1	1	yes	characteristic
80	C₄×C₆₇	#500	1	1	yes	characteristic
81	Dic₆₇	—	8	2	no	—
82	Dic₆₇	—	16	2	no	—
83	#13509	—	67	1	no	maximal, Sylow
84	C₂×C₆₇	—	2	1	no	—
85	C₂×C₆₇	—	2	1	no	—
86	C₂×C₆₇	#2083	1	1	yes	characteristic
87	C₂×C₆₇	#2212	1	2	yes	—
88	#2159	—	67	2	no	—
89	#1945	—	67	1	no	—
90	#1846	—	67	1	no	—
91	#1387	—	67	2	no	—
92	#1376	D₆₇	1	1	yes	characteristic
93	C₆₇	#13509	1	1	yes	characteristic, complemented, Sylow
94	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
95	C₂×#108	—	67	1	no	—
96	#427	—	67	1	no	—
97	C₂×#107	—	67	2	no	—
98	C₂×#106	—	134	1	no	—
99	#424	—	134	1	no	—
100	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
101	#382	—	134	1	no	—
102	#367	—	134	2	no	—
103	#366	—	134	2	no	—
104	#339	—	134	1	no	—
105	C₂×M₁₆	—	134	1	no	—
106	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
107	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
108	#117	—	134	1	no	—
109	C₂×C₄⋊C₄	—	134	2	no	—
110	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
111	M₃₂	—	2	2	no	—
112	M₃₂	Dic₁₃₄	1	4	yes	—
113	C₂×C₁₆	—	2	2	no	—
114	#108	—	134	2	no	—
115	#107	—	67	2	no	—
116	#107	—	134	2	no	—
117	#107	—	134	2	no	—
118	#106	—	268	2	no	—
119	#106	—	134	1	no	—
120	C₄⋊C₈	—	268	2	no	—
121	C₄×C₈	—	2	2	no	—
122	#95	—	268	2	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	—	268	2	no	—
125	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
126	M₁₆	—	268	1	no	—
127	M₁₆	—	268	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	268	1	no	—
131	C₂×C₈	D₅₃₆	1	2	yes	—
132	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
133	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
134	C₂×C₈	#11761627	1	2	yes	—
135	C₂×C₈	#11761629	1	1	yes	characteristic
136	C₄⋊C₄	—	268	2	no	—
137	C₄⋊C₄	—	268	2	no	—
138	C₄⋊C₄	—	134	2	no	—
139	C₂²⋊C₄	—	268	1	no	—
140	C₄²	—	2	1	no	—
141	C₄²	—	268	1	no	—
142	C₄²	Dic₂₆₈	1	2	yes	—
143	C₁₆	—	2	4	no	—
144	C₂³	#14524071	1	1	yes	characteristic
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	268	2	no	—
148	C₂×C₄	—	536	4	no	—
149	C₂×C₄	—	536	2	no	—
150	C₂×C₄	#14524018	1	1	yes	characteristic
151	C₂×C₄	#14524027	1	1	yes	characteristic
152	C₂×C₄	#14524043	1	1	yes	characteristic
153	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
154	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
155	C₂×C₄	#14524076	1	1	yes	characteristic
156	C₈	—	2	2	no	—
157	C₈	—	536	1	no	—
158	C₈	#14524020	1	2	yes	—
159	C₈	#14524034	1	2	yes	—
160	C₂²	—	2	2	no	—
161	C₂²	#16863592	1	1	yes	characteristic
162	C₂²	#16863732	1	1	yes	characteristic
163	C₂²	#16863805	1	1	yes	characteristic
164	C₄	—	4	2	no	—
165	C₄	—	536	2	no	—
166	C₄	—	1072	2	no	—
167	C₄	#16863328	1	1	yes	characteristic
168	C₄	#16863607	1	1	yes	characteristic
169	C₄	#16863710	1	1	yes	characteristic
170	C₄	#16863801	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	#22561682	1	1	yes	characteristic
174	C₂	#22562279	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22561997

C₂

yes

maximal

#22560957

C₂

yes

maximal, characteristic

#22560373

C₂

yes

maximal, characteristic

#22558318

C₂

yes

maximal

C₆₇×#1376

C₂

yes

maximal, characteristic

C₂×#14523979

C₂²

yes

characteristic

#16863434

C₂²

yes

characteristic

C₂×#14523977

C₂²

yes

—

—

—

—

—

—

—

C₂×M₃₂×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×#11761612

—

#14524002

—

C₂×C₄⋊Dic₆₇

—

—

—

—

D₄

yes

—

#14523975

—

#14523975

—

Dic₆₇⋊C₈

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

M₃₂×C₆₇

—

M₃₂×C₆₇

Q₈

yes

—

C₂×C₁₆×C₆₇

—

C₄×C₈×C₆₇

—

C₂²×Dic₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

—

C₂×C₈×C₆₇

#41

yes

—

C₂×C₈×C₆₇

#41

yes

characteristic

C₄²×C₆₇

—

C₄²×C₆₇

Q₁₆

yes

—

C₁₆×C₆₇

—

C₂³×C₆₇

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₇

#137

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₂²⋊Q₈

yes

—

C₈×C₆₇

#125

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

#13509

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#2083

yes

characteristic

C₂×C₆₇

#2212

yes

—

—

—

—

—

D₆₇

yes

characteristic

C₆₇

#13509

yes

characteristic, complemented, Sylow

C₂×M₃₂

C₂×D₆₇

yes

—

—

—

—

—

134

—

#424

—

134

—

100

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

101

#382

—

134

—

102

#367

—

134

—

103

#366

—

134

—

104

#339

—

134

—

105

C₂×M₁₆

—

134

—

106

C₂²×C₈

D₂₆₈

yes

characteristic

107

C₂²×C₈

C₂²×D₆₇

yes

characteristic

108

#117

—

134

—

109

C₂×C₄⋊C₄

—

134

—

110

C₂×C₄²

D₂₆₈

yes

characteristic

111

M₃₂

—

112

M₃₂

Dic₁₃₄

yes

—

113

C₂×C₁₆

—

114

#108

—

134

—

115

#107

—

116

#107

—

134

—

117

#107

—

134

—

118

#106

—

268

—

119

#106

—

134

—

120

C₄⋊C₈

—

268

—

121

C₄×C₈

—

122

#95

—

268

—

123

C₂²×C₄

—

124

C₂²×C₄

—

268

—

125

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

126

M₁₆

—

268

—

127

M₁₆

—

268

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

268

—

131

C₂×C₈

D₅₃₆

yes

—

132

C₂×C₈

D₄×D₆₇

yes

characteristic

133

C₂×C₈

C₂×Dic₁₃₄

yes

—

134

C₂×C₈

#11761627

yes

—

135

C₂×C₈

#11761629

yes

characteristic

136

C₄⋊C₄

—

268

—

137

C₄⋊C₄

—

268

—

138

C₄⋊C₄

—

134

—

139

C₂²⋊C₄

—

268

—

140

C₄²

—

141

C₄²

—

268

—

142

C₄²

Dic₂₆₈

yes

—

143

C₁₆

—

144

C₂³

#14524071

yes

characteristic

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

268

—

148

C₂×C₄

—

536

—

149

C₂×C₄

—

536

—

150

C₂×C₄

#14524018

yes

characteristic

151

C₂×C₄

#14524027

yes

characteristic

152

C₂×C₄

#14524043

yes

characteristic

153

C₂×C₄

C₂×D₅₃₆

yes

characteristic

154

C₂×C₄

C₂×Dic₂₆₈

yes

characteristic

155

C₂×C₄

#14524076

yes

characteristic

156

C₈

—

157

C₈

—

536

—

158

C₈

#14524020

yes

—

159

C₈

#14524034

yes

—

160

C₂²

—

161

C₂²

#16863592

yes

characteristic

162

C₂²

#16863732

yes

characteristic

163

C₂²

#16863805

yes

characteristic

164

C₄

—

165

C₄

—

536

—

166

C₄

—

1072

—

167

C₄

#16863328

yes

characteristic

168

C₄

#16863607

yes

characteristic

169

C₄

#16863710

yes

characteristic

170

C₄

#16863801

yes

characteristic

171

C₂

—

172

C₂

—

173

C₂

#22561682

yes

characteristic

174

C₂

#22562279

yes

—

	subgroup	G/H
center	C₂²	#16863592
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1376	C₂
socle	C₂²×C₆₇	#459
prime radical	C₂³×C₆₇	AGL(1,ℤ/8)
hypercenter	#1376	D₆₇
hypocenter	C₆₇	#13509
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

subgroup

G/H

center

C₂²

#16863592

commutator, cosocle

C₂²×C₈×C₆₇

C₂³

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1376

C₂

C₂²×C₆₇

C₂³×C₆₇

D₆₇

C₆₇

G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

N	H ≅ G/N	type
93. C₆₇	#13509	semidirect

H ≅ G/N

type

93. C₆₇

#13509

semidirect

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

subgroup	index
#22561997	2
#22560957	2
#22560373	2
#22558318	2
C₆₇×#1376	2
#13509	67

subgroup

index

C₆₇×#1376

Subgroups — #30775790

Conjugacy classes of subgroups

Complements

Maximal subgroups