Group #30749637 — subgroups

Subgroups — #30749637

Order 17152.

322 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 143 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	#22560346	C₂	1	2	yes	maximal
2	#22558967	C₂	1	2	yes	maximal
3	#22558965	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1566	C₂	1	1	yes	maximal, characteristic
6	#16863420	C₂²	1	2	yes	—
7	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	#16863160	—	2	4	no	—
10	#16863148	—	2	4	no	—
11	#16863147	—	2	2	no	—
12	C₈×C₆₇⋊C₈	C₄	1	1	yes	characteristic
13	C₈×C₆₇⋊C₈	C₄	1	1	yes	characteristic
14	C₈×C₆₇⋊C₈	C₄	1	1	yes	characteristic
15	C₈×C₆₇⋊C₈	C₄	1	1	yes	characteristic
16	#423×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#334×C₆₇	—	2	2	no	—
19	C₂×#11761612	—	2	2	no	—
20	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
21	C₂²×C₆₇⋊C₈	Q₈	1	2	yes	—
22	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
29	C₄×C₆₇⋊C₈	Q₈	1	2	yes	—
30	C₂×M₁₆×C₆₇	—	2	2	no	—
31	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
32	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₆₇	—	2	2	no	—
34	C₄⋊C₈×C₆₇	—	2	2	no	—
35	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₆₇	D₄	1	2	yes	—
38	#11761612	—	4	4	no	—
39	C₂×C₆₇⋊C₈	—	2	4	no	—
40	C₂×C₆₇⋊C₈	—	2	4	no	—
41	C₂×C₆₇⋊C₈	—	4	2	no	—
42	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
43	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
44	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
45	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
49	M₁₆×C₆₇	—	4	4	no	—
50	C₂×C₈×C₆₇	—	2	4	no	—
51	C₂×C₈×C₆₇	—	4	2	no	—
52	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
54	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
58	C₂³×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₆₇	—	2	2	no	—
60	C₂×C₄×C₆₇	—	2	2	no	—
61	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₆₇	#135	1	4	yes	—
68	C₈×C₆₇	—	2	2	no	—
69	C₈×C₆₇	—	8	2	no	—
70	C₈×C₆₇	#108	1	4	yes	—
71	C₆₇⋊C₈	—	2	2	no	—
72	C₆₇⋊C₈	—	2	2	no	—
73	C₆₇⋊C₈	—	2	2	no	—
74	C₆₇⋊C₈	—	8	2	no	—
75	C₆₇⋊C₈	#108	1	4	yes	—
76	C₂²×C₆₇	—	2	1	no	—
77	C₂²×C₆₇	—	2	1	no	—
78	C₂²×C₆₇	#384	1	1	yes	characteristic
79	C₂²×C₆₇	#499	1	2	yes	—
80	C₄×C₆₇	—	2	1	no	—
81	C₄×C₆₇	—	2	1	no	—
82	C₄×C₆₇	#427	1	2	yes	—
83	C₄×C₆₇	C₂×#108	1	2	yes	—
84	C₄×C₆₇	C₂×#108	1	2	yes	—
85	C₄×C₆₇	#495	1	2	yes	—
86	#8824	—	67	1	no	maximal, Sylow
87	C₂×C₆₇	—	2	2	no	—
88	C₂×C₆₇	#1846	1	1	yes	characteristic
89	C₂×C₆₇	#1848	1	1	yes	characteristic
90	C₂×C₆₇	#1848	1	1	yes	characteristic
91	#1837	—	67	2	no	—
92	#1567	—	67	2	no	—
93	#1566	—	67	1	no	—
94	#1566	D₆₇	1	1	yes	characteristic
95	C₂×C₈²	—	67	1	no	—
96	C₆₇	#8824	1	1	yes	characteristic, complemented, Sylow
97	#423	—	67	2	no	—
98	#423	C₂×D₆₇	1	2	yes	—
99	C₂×C₄×C₈	—	67	1	no	—
100	C₂×C₄×C₈	—	67	1	no	—
101	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
102	#341	—	134	4	no	—
103	#335	—	134	4	no	—
104	#334	—	2	2	no	—
105	#334	—	134	2	no	—
106	C₈²	—	67	1	no	—
107	C₈²	—	67	1	no	—
108	C₈²	—	67	1	no	—
109	C₈²	—	67	1	no	—
110	C₂×M₁₆	—	2	2	no	—
111	C₂×M₁₆	—	134	2	no	—
112	C₂²×C₈	—	67	2	no	—
113	C₂²×C₈	—	67	2	no	—
114	C₂²×C₈	Dic₁₃₄	1	2	yes	—
115	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	134	2	no	—
119	C₄⋊C₈	—	134	2	no	—
120	C₄×C₈	—	67	1	no	—
121	C₄×C₈	—	67	1	no	—
122	C₄×C₈	—	67	1	no	—
123	C₄×C₈	—	67	1	no	—
124	C₄×C₈	—	67	2	no	—
125	C₄×C₈	—	67	2	no	—
126	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
127	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
128	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
129	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
130	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
131	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
132	M₁₆	—	4	4	no	—
133	M₁₆	—	268	4	no	—
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	268	2	no	—
137	C₂×C₈	—	67	2	no	—
138	C₂×C₈	—	67	2	no	—
139	C₂×C₈	—	67	2	no	—
140	C₂×C₈	—	67	2	no	—
141	C₂×C₈	—	134	4	no	—
142	C₂×C₈	—	134	4	no	—
143	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
144	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
145	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
146	C₄²	D₄×D₆₇	1	1	yes	characteristic
147	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
148	C₄²	Q₈×D₆₇	1	1	yes	characteristic
149	C₂³	#14524042	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
153	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
154	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
155	C₂×C₄	#14524039	1	1	yes	characteristic
156	C₂×C₄	#14524041	1	1	yes	characteristic
157	C₂×C₄	#14524042	1	1	yes	characteristic
158	C₂×C₄	#14524064	1	4	yes	—
159	C₈	—	2	2	no	—
160	C₈	—	8	2	no	—
161	C₈	—	536	2	no	—
162	C₈	—	67	4	no	—
163	C₈	—	134	2	no	—
164	C₈	—	134	2	no	—
165	C₈	—	134	2	no	—
166	C₈	#14523978	1	4	yes	—
167	C₂²	—	2	1	no	—
168	C₂²	—	2	1	no	—
169	C₂²	#16863265	1	1	yes	characteristic
170	C₂²	#16863796	1	2	yes	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	#16863431	1	2	yes	—
174	C₄	#108×D₆₇	1	2	yes	—
175	C₄	C₂×#14523978	1	2	yes	—
176	C₄	#16863777	1	2	yes	—
177	C₂	—	2	2	no	—
178	C₂	#22560370	1	1	yes	characteristic
179	C₂	#22560379	1	1	yes	characteristic
180	C₂	#22560380	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560346

C₂

yes

maximal

#22558967

C₂

yes

maximal

#22558965

C₂

yes

maximal, characteristic

C₂×C₈×C₆₇⋊C₈

C₂

yes

maximal, characteristic

C₆₇×#1566

C₂

yes

maximal, characteristic

#16863420

C₂²

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

—

—

C₈×C₆₇⋊C₈

C₄

yes

characteristic

C₈×C₆₇⋊C₈

C₄

yes

characteristic

C₈×C₆₇⋊C₈

C₄

yes

characteristic

C₈×C₆₇⋊C₈

C₄

yes

characteristic

#423×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#334×C₆₇

—

C₂×#11761612

—

C₂²×C₆₇⋊C₈

D₄

yes

—

C₂²×C₆₇⋊C₈

Q₈

yes

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₄×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₄×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₄×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₄×C₆₇⋊C₈

D₄

yes

—

C₄×C₆₇⋊C₈

Q₈

yes

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

Q₈

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₆₇

C₂×Q₈

yes

characteristic

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₂³×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

#135

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#108

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

#108

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#384

yes

characteristic

C₂²×C₆₇

#499

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#427

yes

—

C₄×C₆₇

C₂×#108

yes

—

C₄×C₆₇

C₂×#108

yes

—

C₄×C₆₇

#495

yes

—

#8824

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

—

—

D₆₇

yes

characteristic

C₂×C₈²

—

C₆₇

#8824

yes

characteristic, complemented, Sylow

#423

—

#423

C₂×D₆₇

yes

—

C₂×C₄×C₈

—

100

C₂×C₄×C₈

—

101

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

102

#341

—

134

—

103

#335

—

134

—

104

#334

—

105

#334

—

134

—

106

C₈²

—

107

C₈²

—

108

C₈²

—

109

C₈²

—

110

C₂×M₁₆

—

111

C₂×M₁₆

—

134

—

112

C₂²×C₈

—

113

C₂²×C₈

—

114

C₂²×C₈

Dic₁₃₄

yes

—

115

C₂×C₄²

C₂²×D₆₇

yes

characteristic

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

134

—

119

C₄⋊C₈

—

134

—

120

C₄×C₈

—

121

C₄×C₈

—

122

C₄×C₈

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

C₄×D₆₇

yes

characteristic

127

C₄×C₈

C₄×D₆₇

yes

characteristic

128

C₄×C₈

C₂²⋊D₆₇

yes

—

129

C₂²×C₄

D₄×D₆₇

yes

characteristic

130

C₂²×C₄

Q₈×D₆₇

yes

characteristic

131

C₂²×C₄

C₂×Dic₁₃₄

yes

characteristic

132

M₁₆

—

133

M₁₆

—

268

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

268

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

134

—

142

C₂×C₈

—

134

—

143

C₂×C₈

Dic₆₇⋊C₄

yes

—

144

C₂×C₈

Dic₆₇⋊C₄

yes

—

145

C₄²

C₂×C₄×D₆₇

yes

characteristic

146

C₄²

D₄×D₆₇

yes

characteristic

147

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

148

C₄²

Q₈×D₆₇

yes

characteristic

149

C₂³

#14524042

yes

characteristic

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

C₄⋊C₄×D₆₇

yes

characteristic

153

C₂×C₄

C₄⋊C₄×D₆₇

yes

characteristic

154

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

155

C₂×C₄

#14524039

yes

characteristic

156

C₂×C₄

#14524041

yes

characteristic

157

C₂×C₄

#14524042

yes

characteristic

158

C₂×C₄

#14524064

yes

—

159

C₈

—

160

C₈

—

161

C₈

—

536

—

162

C₈

—

163

C₈

—

134

—

164

C₈

—

134

—

165

C₈

—

134

—

166

C₈

#14523978

yes

—

167

C₂²

—

168

C₂²

—

169

C₂²

#16863265

yes

characteristic

170

C₂²

#16863796

yes

—

171

C₄

—

172

C₄

—

173

C₄

#16863431

yes

—

174

C₄

#108×D₆₇

yes

—

175

C₄

C₂×#14523978

yes

—

176

C₄

#16863777

yes

—

177

C₂

—

178

C₂

#22560370

yes

characteristic

179

C₂

#22560379

yes

characteristic

180

C₂

#22560380

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14524039
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1566	C₂
socle	C₂²×C₆₇	#384
prime radical	C₂³×C₆₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1566	D₆₇
hypocenter	C₆₇	#8824
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂×C₄

#14524039

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1566

C₂

socle

C₂²×C₆₇

#384

prime radical

C₂³×C₆₇

C₄⋊Q₈

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

N	H ≅ G/N	type
96. C₆₇	#8824	semidirect

H ≅ G/N

type

96. C₆₇

#8824

semidirect

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

subgroup	index
#22560346	2
#22558967	2
#22558965	2
C₂×C₈×C₆₇⋊C₈	2
C₆₇×#1566	2
#8824	67

subgroup

index

#22560346

#22558967

#22558965

C₂×C₈×C₆₇⋊C₈

C₆₇×#1566

#8824

Subgroups — #30749637

Conjugacy classes of subgroups

Complements

Maximal subgroups