Group #30760823 — subgroups

Subgroups — #30760823

Order 17152.

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

normal subgroups only

This group has 320 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	#22561149	C₂	1	1	yes	maximal, characteristic
2	#22560384	C₂	1	1	yes	maximal, characteristic
3	#22558980	C₂	1	2	yes	maximal
4	#22558639	C₂	1	2	yes	maximal
5	C₆₇×#1275	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
7	C₂×#14523979	—	2	2	no	—
8	#16863420	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
10	#16863150	—	2	2	no	—
11	#16863147	—	2	4	no	—
12	#16863128	—	2	4	no	—
13	#431×C₆₇	C₂²	1	2	yes	—
14	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₃₄	Q₈	1	1	yes	characteristic
16	C₂×#11761612	—	2	2	no	—
17	C₂²×C₆₇⋊C₈	—	2	1	no	—
18	C₄×Dic₁₃₄	—	2	2	no	—
19	C₄×Dic₁₃₄	—	2	2	no	—
20	C₂×C₄⋊Dic₆₇	Q₈	1	1	yes	characteristic
21	C₂×Dic₆₇⋊C₄	—	2	1	no	—
22	C₂×C₄×Dic₆₇	—	2	1	no	—
23	#14523979	—	4	4	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
26	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
27	C₂²×C₈×C₆₇	D₄	1	2	yes	—
28	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₆₇	—	2	2	no	—
30	C₂²⋊C₈×C₆₇	—	2	2	no	—
31	#97×C₆₇	—	2	2	no	—
32	C₄×C₈×C₆₇	D₄	1	4	yes	—
33	C₂×Dic₁₃₄	—	2	2	no	—
34	C₂×Dic₁₃₄	—	4	2	no	—
35	C₂²×Dic₆₇	—	2	1	no	—
36	C₂²×Dic₆₇	—	2	1	no	—
37	#11761612	—	4	4	no	—
38	C₂×C₆₇⋊C₈	—	4	1	no	—
39	C₂×C₆₇⋊C₈	—	4	2	no	—
40	C₂×C₆₇⋊C₈	—	4	2	no	—
41	C₄⋊Dic₆₇	—	2	2	no	—
42	Dic₆₇⋊C₄	—	4	2	no	—
43	C₄×Dic₆₇	—	4	1	no	—
44	C₄×Dic₆₇	—	4	1	no	—
45	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
48	C₂×C₈×C₆₇	—	2	2	no	—
49	C₂×C₈×C₆₇	—	2	2	no	—
50	C₂×C₈×C₆₇	—	2	4	no	—
51	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
52	C₄²×C₆₇	C₂×D₄	1	2	yes	—
53	C₄²×C₆₇	#41	1	2	yes	—
54	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
55	C₂×C₄×C₆₇	—	4	1	no	—
56	C₂×C₄×C₆₇	—	4	1	no	—
57	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
58	C₂×C₄×C₆₇	#123	1	2	yes	—
59	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
60	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
62	C₂×C₄×C₆₇	#137	1	2	yes	—
63	C₂×Dic₆₇	—	2	2	no	—
64	C₂×Dic₆₇	—	4	1	no	—
65	C₂×Dic₆₇	—	4	1	no	—
66	C₂×Dic₆₇	—	4	2	no	—
67	C₂×Dic₆₇	—	4	1	no	—
68	Dic₁₃₄	—	2	4	no	—
69	Dic₁₃₄	—	4	2	no	—
70	C₈×C₆₇	—	2	4	no	—
71	C₈×C₆₇	—	4	4	no	—
72	C₆₇⋊C₈	—	8	2	no	—
73	C₆₇⋊C₈	—	8	1	no	—
74	C₂²×C₆₇	—	2	1	no	—
75	C₂²×C₆₇	—	2	1	no	—
76	C₂²×C₆₇	#399	1	1	yes	characteristic
77	C₂²×C₆₇	#475	1	2	yes	—
78	C₄×C₆₇	—	2	2	no	—
79	C₄×C₆₇	—	2	2	no	—
80	C₄×C₆₇	#483	1	2	yes	—
81	C₄×C₆₇	#492	1	1	yes	characteristic
82	C₄×C₆₇	#497	1	1	yes	characteristic
83	Dic₆₇	—	4	2	no	—
84	Dic₆₇	—	8	1	no	—
85	#10663	—	67	1	no	maximal, Sylow
86	C₂×C₆₇	—	2	2	no	—
87	C₂×C₆₇	#2091	1	1	yes	characteristic
88	C₂×C₆₇	#2096	1	1	yes	characteristic
89	C₂×C₆₇	#2098	1	1	yes	characteristic
90	#1985	—	67	1	no	—
91	#1849	—	67	1	no	—
92	#1570	—	67	2	no	—
93	#1487	—	67	2	no	—
94	#1275	D₆₇	1	1	yes	characteristic
95	C₆₇	#10663	1	1	yes	characteristic, complemented, Sylow
96	C₂×C₄×Q₈	—	67	1	no	—
97	#431	C₂×D₆₇	1	2	yes	—
98	C₂×#108	—	134	2	no	—
99	#423	—	67	2	no	—
100	C₂×C₄⋊C₈	—	67	1	no	—
101	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
102	#336	—	134	2	no	—
103	#334	—	134	4	no	—
104	Q₈⋊C₈	—	134	4	no	—
105	C₂²×Q₈	—	67	1	no	—
106	C₂×M₁₆	—	134	2	no	—
107	C₂²×C₈	—	134	1	no	—
108	C₂²×C₈	D₂₆₈	1	2	yes	—
109	C₄×Q₈	—	134	2	no	—
110	C₄×Q₈	—	134	2	no	—
111	C₂×C₄⋊C₄	—	67	1	no	—
112	C₂×C₄⋊C₄	—	134	1	no	—
113	C₂×C₄²	—	134	1	no	—
114	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
115	#108	—	268	4	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	134	2	no	—
118	C₄⋊C₈	—	134	2	no	—
119	C₄⋊C₈	—	134	2	no	—
120	C₂²⋊C₈	—	2	2	no	—
121	#97	—	2	2	no	—
122	C₄×C₈	D₂₆₈	1	4	yes	—
123	C₂×Q₈	—	268	2	no	—
124	C₂×Q₈	—	134	2	no	—
125	C₂²×C₄	—	134	1	no	—
126	C₂²×C₄	—	134	1	no	—
127	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
128	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
129	C₂²×C₄	#11761627	1	1	yes	characteristic
130	M₁₆	—	268	4	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	—	268	1	no	—
135	C₂×C₈	—	268	2	no	—
136	C₂×C₈	—	268	2	no	—
137	C₂×C₈	#11761617	1	4	yes	—
138	C₄⋊C₄	—	268	2	no	—
139	C₄⋊C₄	—	134	2	no	—
140	C₄²	—	268	1	no	—
141	C₄²	—	268	1	no	—
142	C₄²	C₂×D₂₆₈	1	2	yes	—
143	C₄²	#11761627	1	2	yes	—
144	C₂³	#14524023	1	1	yes	characteristic
145	Q₈	—	268	2	no	—
146	Q₈	—	134	4	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	268	1	no	—
150	C₂×C₄	—	268	1	no	—
151	C₂×C₄	—	268	2	no	—
152	C₂×C₄	—	268	1	no	—
153	C₂×C₄	—	134	2	no	—
154	C₂×C₄	#14524023	1	2	yes	—
155	C₂×C₄	#14524027	1	2	yes	—
156	C₂×C₄	#14524027	1	1	yes	characteristic
157	C₂×C₄	#14524040	1	1	yes	characteristic
158	C₂×C₄	C₂×#11761617	1	2	yes	—
159	C₂×C₄	#14524076	1	2	yes	—
160	C₈	—	2	4	no	—
161	C₈	—	4	4	no	—
162	C₈	—	536	2	no	—
163	C₈	—	536	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	#16863333	1	1	yes	characteristic
167	C₂²	#16863682	1	2	yes	—
168	C₄	—	2	2	no	—
169	C₄	—	2	2	no	—
170	C₄	—	268	2	no	—
171	C₄	—	536	1	no	—
172	C₄	#16863727	1	2	yes	—
173	C₄	#16863769	1	1	yes	characteristic
174	C₄	#16863790	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	#22561726	1	1	yes	characteristic
177	C₂	#22561751	1	1	yes	characteristic
178	C₂	#22561763	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22561149

C₂

yes

maximal, characteristic

#22560384

C₂

yes

maximal, characteristic

#22558980

C₂

yes

maximal

#22558639

C₂

yes

maximal

C₆₇×#1275

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₃₄

C₂²

yes

characteristic

C₂×#14523979

—

#16863420

C₂²

yes

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

—

—

#431×C₆₇

C₂²

yes

—

C₂×#97×C₆₇

C₂²

yes

characteristic

C₂²×Dic₁₃₄

Q₈

yes

characteristic

C₂×#11761612

—

C₂²×C₆₇⋊C₈

—

C₄×Dic₁₃₄

—

C₄×Dic₁₃₄

—

C₂×C₄⋊Dic₆₇

Q₈

yes

characteristic

C₂×Dic₆₇⋊C₄

—

C₂×C₄×Dic₆₇

—

#14523979

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

#97×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₇

#123

yes

—

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

C₂×SD₁₆

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#399

yes

characteristic

C₂²×C₆₇

#475

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#483

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

#10663

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

—

—

—

D₆₇

yes

characteristic

C₆₇

#10663

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

#431

C₂×D₆₇

yes

—

C₂×#108

—

134

—

#423

—

100

C₂×C₄⋊C₈

—

101

C₂×#97

C₂×D₆₇

yes

characteristic

102

#336

—

134

—

103

#334

—

134

—

104

Q₈⋊C₈

—

134

—

105

C₂²×Q₈

—

106

C₂×M₁₆

—

134

—

107

C₂²×C₈

—

134

—

108

C₂²×C₈

D₂₆₈

yes

—

109

C₄×Q₈

—

134

—

110

C₄×Q₈

—

134

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

134

—

113

C₂×C₄²

—

134

—

114

C₂×C₄²

C₂²×D₆₇

yes

characteristic

115

#108

—

268

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

134

—

118

C₄⋊C₈

—

134

—

119

C₄⋊C₈

—

134

—

120

C₂²⋊C₈

—

121

#97

—

122

C₄×C₈

D₂₆₈

yes

—

123

C₂×Q₈

—

268

—

124

C₂×Q₈

—

134

—

125

C₂²×C₄

—

134

—

126

C₂²×C₄

—

134

—

127

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

128

C₂²×C₄

Q₈×D₆₇

yes

characteristic

129

C₂²×C₄

#11761627

yes

characteristic

130

M₁₆

—

268

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

268

—

135

C₂×C₈

—

268

—

136

C₂×C₈

—

268

—

137

C₂×C₈

#11761617

yes

—

138

C₄⋊C₄

—

268

—

139

C₄⋊C₄

—

134

—

140

C₄²

—

268

—

141

C₄²

—

268

—

142

C₄²

C₂×D₂₆₈

yes

—

143

C₄²

#11761627

yes

—

144

C₂³

#14524023

yes

characteristic

145

Q₈

—

268

—

146

Q₈

—

134

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

268

—

150

C₂×C₄

—

268

—

151

C₂×C₄

—

268

—

152

C₂×C₄

—

268

—

153

C₂×C₄

—

134

—

154

C₂×C₄

#14524023

yes

—

155

C₂×C₄

#14524027

yes

—

156

C₂×C₄

#14524027

yes

characteristic

157

C₂×C₄

#14524040

yes

characteristic

158

C₂×C₄

C₂×#11761617

yes

—

159

C₂×C₄

#14524076

yes

—

160

C₈

—

161

C₈

—

162

C₈

—

536

—

163

C₈

—

536

—

164

C₂²

—

165

C₂²

—

166

C₂²

#16863333

yes

characteristic

167

C₂²

#16863682

yes

—

168

C₄

—

169

C₄

—

170

C₄

—

268

—

171

C₄

—

536

—

172

C₄

#16863727

yes

—

173

C₄

#16863769

yes

characteristic

174

C₄

#16863790

yes

characteristic

175

C₂

—

176

C₂

#22561726

yes

characteristic

177

C₂

#22561751

yes

characteristic

178

C₂

#22561763

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14524040
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1275	C₂
socle	C₂²×C₆₇	#399
prime radical	C₂³×C₆₇	C₂²⋊Q₈
hypercenter	#1275	D₆₇
hypocenter	C₆₇	#10663
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1275

subgroup

G/H

center

C₂×C₄

#14524040

commutator, cosocle

C₂×C₄²×C₆₇

C₂³

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1275

C₂

socle

C₂²×C₆₇

#399

prime radical

C₂³×C₆₇

C₂²⋊Q₈

D₆₇

C₆₇

G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1275

N	H ≅ G/N	type
95. C₆₇	#10663	semidirect

H ≅ G/N

type

95. C₆₇

#10663

semidirect

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

subgroup	index
#22561149	2
#22560384	2
#22558980	2
#22558639	2
C₆₇×#1275	2
#10663	67

subgroup

index

C₆₇×#1275

Subgroups — #30760823

Conjugacy classes of subgroups

Complements

Maximal subgroups