Group #30749620 — subgroups

Subgroups — #30749620

Order 17152.

438 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 227 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	#22560369	C₂	1	1	yes	maximal, characteristic
2	#22560369	C₂	1	2	yes	maximal
3	#22560366	C₂	1	1	yes	maximal, characteristic
4	#22560337	C₂	1	2	yes	maximal
5	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523977	—	2	4	no	—
7	C₂×#14523977	—	2	2	no	—
8	C₂×#14523975	—	2	2	no	—
9	#16863266	C₂²	1	1	yes	characteristic
10	#16863266	C₂²	1	2	yes	—
11	#16863266	C₂²	1	1	yes	characteristic
12	#16863159	—	2	4	no	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#322×C₆₇	C₄	1	4	yes	complemented
16	C₂×C₄⋊Dic₆₇	—	2	2	no	—
17	C₂×C₄⋊Dic₆₇	—	2	2	no	—
18	C₂×C₄⋊Dic₆₇	—	2	1	no	—
19	C₂×C₄⋊Dic₆₇	—	2	1	no	—
20	C₂×C₄⋊Dic₆₇	—	2	2	no	—
21	C₂×C₄⋊Dic₆₇	—	2	2	no	—
22	C₂×C₄⋊Dic₆₇	—	2	2	no	—
23	#14523977	—	4	8	no	—
24	#14523977	—	4	4	no	—
25	#14523975	—	4	4	no	—
26	C₂²×C₈×C₆₇	D₄	1	4	yes	—
27	C₂²×C₈×C₆₇	D₄	1	2	yes	—
28	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
29	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₆₇	Q₈	1	4	yes	—
32	C₄×C₈×C₆₇	Q₈	1	2	yes	—
33	C₂²×Dic₆₇	—	4	2	no	—
34	C₂²×Dic₆₇	—	4	1	no	—
35	C₂²×Dic₆₇	—	4	1	no	—
36	C₄⋊Dic₆₇	—	4	2	no	—
37	C₄⋊Dic₆₇	—	4	2	no	—
38	C₄⋊Dic₆₇	—	4	4	no	—
39	C₄⋊Dic₆₇	—	4	4	no	—
40	C₄⋊Dic₆₇	—	4	4	no	—
41	C₄⋊Dic₆₇	—	4	4	no	—
42	C₄⋊Dic₆₇	—	4	4	no	—
43	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
45	C₂×C₈×C₆₇	—	2	4	no	—
46	C₂×C₈×C₆₇	—	2	4	no	—
47	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	—
48	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
49	C₂×C₈×C₆₇	D₈	1	4	yes	—
50	C₂×C₈×C₆₇	Q₁₆	1	4	yes	—
51	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
54	C₂³×C₆₇	#127	1	1	yes	characteristic
55	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
59	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
60	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
61	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
62	C₂×C₄×C₆₇	#137	1	4	yes	—
63	C₂×Dic₆₇	—	8	1	no	—
64	C₂×Dic₆₇	—	8	2	no	—
65	C₂×Dic₆₇	—	8	1	no	—
66	C₂×Dic₆₇	—	8	2	no	—
67	C₂×Dic₆₇	—	8	2	no	—
68	C₂×Dic₆₇	—	8	2	no	—
69	C₂×Dic₆₇	—	8	2	no	—
70	C₈×C₆₇	—	2	8	no	—
71	C₈×C₆₇	#107	1	8	yes	complemented
72	C₂²×C₆₇	#384	1	1	yes	characteristic
73	C₂²×C₆₇	#493	1	1	yes	characteristic
74	C₂²×C₆₇	#494	1	1	yes	characteristic
75	C₂²×C₆₇	#496	1	2	yes	—
76	C₂²×C₆₇	#497	1	2	yes	—
77	C₄×C₆₇	C₂×#107	1	2	yes	—
78	C₄×C₆₇	#428	1	4	yes	—
79	C₄×C₆₇	#500	1	2	yes	—
80	C₄×C₆₇	#501	1	4	yes	—
81	Dic₆₇	—	16	1	no	—
82	Dic₆₇	—	16	2	no	—
83	Dic₆₇	—	16	1	no	—
84	#8820	—	67	1	no	maximal, Sylow
85	C₂×C₆₇	#1721	1	1	yes	characteristic
86	C₂×C₆₇	#1722	1	1	yes	characteristic
87	C₂×C₆₇	#1723	1	2	yes	—
88	C₂×C₆₇	#1845	1	1	yes	characteristic
89	C₂×C₆₇	#1847	1	2	yes	—
90	#1845	—	67	1	no	—
91	#1845	—	67	2	no	—
92	#1844	—	67	1	no	—
93	#1834	—	67	2	no	—
94	C₂×#322	D₆₇	1	1	yes	characteristic
95	C₆₇	#8820	1	1	yes	characteristic, complemented, Sylow
96	C₂×#107	—	134	4	no	—
97	C₂×#107	—	134	2	no	—
98	C₂×#106	—	134	2	no	—
99	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
100	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
101	#384	—	67	1	no	—
102	#384	—	67	2	no	—
103	#384	—	67	1	no	—
104	#340	—	134	4	no	—
105	#322	Dic₆₇	1	4	yes	complemented
106	C₂²×C₈	D₂₆₈	1	4	yes	—
107	C₂²×C₈	D₂₆₈	1	2	yes	—
108	C₂×C₄⋊C₄	—	134	2	no	—
109	C₂×C₄⋊C₄	—	134	2	no	—
110	C₂×C₄⋊C₄	—	134	1	no	—
111	C₂×C₄⋊C₄	—	134	1	no	—
112	C₂×C₄⋊C₄	—	134	2	no	—
113	C₂×C₄⋊C₄	—	134	2	no	—
114	C₂×C₄⋊C₄	—	134	2	no	—
115	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
116	#107	—	268	8	no	—
117	#107	—	268	4	no	—
118	#106	—	268	4	no	—
119	C₄×C₈	Dic₁₃₄	1	4	yes	—
120	C₄×C₈	Dic₁₃₄	1	2	yes	—
121	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
122	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
123	C₂²×C₄	—	268	2	no	—
124	C₂²×C₄	—	268	1	no	—
125	C₂²×C₄	—	268	1	no	—
126	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
127	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	C₄⋊Dic₆₇	1	8	yes	—
131	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
132	C₂×C₈	D₅₃₆	1	4	yes	—
133	C₂×C₈	Dic₂₆₈	1	4	yes	—
134	C₄⋊C₄	—	268	2	no	—
135	C₄⋊C₄	—	268	2	no	—
136	C₄⋊C₄	—	268	4	no	—
137	C₄⋊C₄	—	268	4	no	—
138	C₄⋊C₄	—	268	4	no	—
139	C₄⋊C₄	—	268	4	no	—
140	C₄⋊C₄	—	268	4	no	—
141	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
142	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
143	C₄²	C₂×Dic₁₃₄	1	2	yes	—
144	C₂³	#14524040	1	1	yes	characteristic
145	C₂×C₄	—	536	1	no	—
146	C₂×C₄	—	536	2	no	—
147	C₂×C₄	—	536	1	no	—
148	C₂×C₄	—	536	2	no	—
149	C₂×C₄	—	536	2	no	—
150	C₂×C₄	—	536	2	no	—
151	C₂×C₄	—	536	2	no	—
152	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
154	C₂×C₄	#14524043	1	1	yes	characteristic
155	C₂×C₄	#14524043	1	2	yes	—
156	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
157	C₂×C₄	C₂×Dic₂₆₈	1	2	yes	—
158	C₂×C₄	#14524071	1	4	yes	—
159	C₂×C₄	#14524076	1	4	yes	—
160	C₈	—	2	8	no	—
161	C₈	#14523977	1	8	yes	complemented
162	C₂²	#16863266	1	1	yes	characteristic
163	C₂²	#16863773	1	1	yes	characteristic
164	C₂²	#16863776	1	1	yes	characteristic
165	C₂²	#16863785	1	2	yes	—
166	C₂²	#16863790	1	2	yes	—
167	C₄	—	1072	1	no	—
168	C₄	—	1072	2	no	—
169	C₄	—	1072	1	no	—
170	C₄	C₂×#14523977	1	2	yes	—
171	C₄	#16863438	1	4	yes	—
172	C₄	#16863801	1	2	yes	—
173	C₄	#16863805	1	4	yes	—
174	C₂	#22559875	1	1	yes	characteristic
175	C₂	#22559880	1	1	yes	characteristic
176	C₂	#22559886	1	2	yes	—
177	C₂	#22560369	1	1	yes	characteristic
178	C₂	#22560377	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22560369

C₂

yes

maximal, characteristic

#22560369

C₂

yes

maximal

#22560366

C₂

yes

maximal, characteristic

#22560337

C₂

yes

maximal

C₂×#322×C₆₇

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

characteristic

#16863266

C₂²

yes

—

#16863266

C₂²

yes

characteristic

#16863159

—

C₂×C₄×C₈×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#322×C₆₇

C₄

yes

complemented

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

#14523977

—

#14523977

—

#14523975

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

Q₈

yes

—

C₄×C₈×C₆₇

Q₈

yes

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

Q₁₆

yes

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₄²×C₆₇

C₂×Q₈

yes

—

C₂³×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

—

C₂×C₄×C₆₇

C₂×Q₁₆

yes

—

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₂²×C₆₇

#497

yes

—

C₄×C₆₇

C₂×#107

yes

—

C₄×C₆₇

#428

yes

—

C₄×C₆₇

#500

yes

—

C₄×C₆₇

#501

yes

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×C₆₇

#1845

yes

characteristic

C₂×C₆₇

#1847

yes

—

—

—

—

—

D₆₇

yes

characteristic

C₆₇

#8820

yes

characteristic, complemented, Sylow

C₂×#107

—

134

—

C₂×#107

—

134

—

C₂×#106

—

134

—

C₂×C₄×C₈

C₂×D₆₇

yes

—

100

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

101

#384

—

102

#384

—

103

#384

—

104

#340

—

134

—

105

#322

Dic₆₇

yes

complemented

106

C₂²×C₈

D₂₆₈

yes

—

107

C₂²×C₈

D₂₆₈

yes

—

108

C₂×C₄⋊C₄

—

134

—

109

C₂×C₄⋊C₄

—

134

—

110

C₂×C₄⋊C₄

—

134

—

111

C₂×C₄⋊C₄

—

134

—

112

C₂×C₄⋊C₄

—

134

—

113

C₂×C₄⋊C₄

—

134

—

114

C₂×C₄⋊C₄

—

134

—

115

C₂×C₄²

C₂²×D₆₇

yes

characteristic

116

#107

—

268

—

117

#107

—

268

—

118

#106

—

268

—

119

C₄×C₈

Dic₁₃₄

yes

—

120

C₄×C₈

Dic₁₃₄

yes

—

121

C₄×C₈

C₂×Dic₆₇

yes

—

122

C₄×C₈

C₂×Dic₆₇

yes

—

123

C₂²×C₄

—

268

—

124

C₂²×C₄

—

268

—

125

C₂²×C₄

—

268

—

126

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

127

C₂²×C₄

C₂×D₂₆₈

yes

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

C₄⋊Dic₆₇

yes

—

131

C₂×C₈

C₄⋊Dic₆₇

yes

—

132

C₂×C₈

D₅₃₆

yes

—

133

C₂×C₈

Dic₂₆₈

yes

—

134

C₄⋊C₄

—

268

—

135

C₄⋊C₄

—

268

—

136

C₄⋊C₄

—

268

—

137

C₄⋊C₄

—

268

—

138

C₄⋊C₄

—

268

—

139

C₄⋊C₄

—

268

—

140

C₄⋊C₄

—

268

—

141

C₄²

C₂²×Dic₆₇

yes

characteristic

142

C₄²

C₂×Dic₁₃₄

yes

characteristic

143

C₄²

C₂×Dic₁₃₄

yes

—

144

C₂³

#14524040

yes

characteristic

145

C₂×C₄

—

536

—

146

C₂×C₄

—

536

—

147

C₂×C₄

—

536

—

148

C₂×C₄

—

536

—

149

C₂×C₄

—

536

—

150

C₂×C₄

—

536

—

151

C₂×C₄

—

536

—

152

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

153

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

—

154

C₂×C₄

#14524043

yes

characteristic

155

C₂×C₄

#14524043

yes

—

156

C₂×C₄

C₂×D₅₃₆

yes

—

157

C₂×C₄

C₂×Dic₂₆₈

yes

—

158

C₂×C₄

#14524071

yes

—

159

C₂×C₄

#14524076

yes

—

160

C₈

—

161

C₈

#14523977

yes

complemented

162

C₂²

#16863266

yes

characteristic

163

C₂²

#16863773

yes

characteristic

164

C₂²

#16863776

yes

characteristic

165

C₂²

#16863785

yes

—

166

C₂²

#16863790

yes

—

167

C₄

—

1072

—

168

C₄

—

1072

—

169

C₄

—

1072

—

170

C₄

C₂×#14523977

yes

—

171

C₄

#16863438

yes

—

172

C₄

#16863801

yes

—

173

C₄

#16863805

yes

—

174

C₂

#22559875

yes

characteristic

175

C₂

#22559880

yes

characteristic

176

C₂

#22559886

yes

—

177

C₂

#22560369

yes

characteristic

178

C₂

#22560377

yes

—

	subgroup	G/H
center	C₂³	#14524040
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#322×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#127
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#322	D₆₇
hypocenter	C₆₇	#8820
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂³

#14524040

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₂×#322×C₆₇

C₂

socle, prime radical

C₂³×C₆₇

#127

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
15. #322×C₆₇	C₄	semidirect
71. C₈×C₆₇	#107	semidirect
95. C₆₇	#8820	semidirect
105. #322	Dic₆₇	semidirect
161. C₈	#14523977	semidirect

H ≅ G/N

type

15. #322×C₆₇

C₄

semidirect

71. C₈×C₆₇

#107

semidirect

95. C₆₇

#8820

semidirect

105. #322

Dic₆₇

semidirect

161. C₈

#14523977

semidirect

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

subgroup	index
#22560369	2
#22560366	2
#22560337	2
C₂×#322×C₆₇	2
#8820	67

subgroup

index

C₂×#322×C₆₇

Subgroups — #30749620

Conjugacy classes of subgroups

Complements

Maximal subgroups