Group #30755294 — subgroups

Subgroups — #30755294

Order 17152.

478 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 155 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	C₂×#16863776	C₂	1	1	yes	maximal, characteristic
2	#22560381	C₂	1	1	yes	maximal, characteristic
3	#22558880	C₂	1	4	yes	maximal
4	C₆₇×#1450	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₆₈	—	2	2	no	—
6	C₂×#14524043	C₂²	1	2	yes	—
7	#16863776	—	2	1	no	—
8	#16863776	—	2	1	no	—
9	#16863776	C₄	1	4	yes	—
10	C₂×#14523979	—	2	2	no	—
11	#16863420	C₂²	1	2	yes	—
12	#16863141	—	2	8	no	—
13	#431×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#322×C₆₇	—	2	1	no	—
16	C₈²×C₆₇	—	2	1	no	—
17	C₂²×Dic₁₃₄	—	2	2	no	—
18	C₂×Dic₂₆₈	—	2	4	no	—
19	C₂×Dic₂₆₈	—	2	4	no	—
20	C₂×Dic₂₆₈	—	4	4	no	—
21	C₂×Dic₂₆₈	—	4	4	no	—
22	C₂×#11761612	—	2	2	no	—
23	#14524043	—	2	4	no	—
24	#14524043	—	2	2	no	—
25	#14524043	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊Dic₆₇	—	2	4	no	—
27	#14523979	—	4	4	no	—
28	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
29	C₂²×C₈×C₆₇	D₄	1	2	yes	—
30	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₆₇	D₄	1	4	yes	—
32	C₂²⋊C₈×C₆₇	—	2	2	no	—
33	C₄×C₈×C₆₇	—	2	2	no	—
34	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
36	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
37	C₂×Dic₁₃₄	—	2	4	no	—
38	C₂×Dic₁₃₄	—	4	4	no	—
39	C₂×Dic₁₃₄	—	4	4	no	—
40	C₂×Dic₁₃₄	—	4	2	no	—
41	C₂²×Dic₆₇	—	4	2	no	—
42	Dic₂₆₈	—	4	8	no	—
43	Dic₂₆₈	—	4	8	no	—
44	#11761612	—	4	4	no	—
45	C₂×C₆₇⋊C₈	—	4	2	no	—
46	C₄⋊Dic₆₇	—	2	8	no	—
47	C₄⋊Dic₆₇	—	4	4	no	—
48	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	#41	1	2	yes	—
50	C₂×C₈×C₆₇	—	2	4	no	—
51	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
52	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
53	C₂×C₈×C₆₇	D₈	1	4	yes	—
54	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
55	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₆₇	C₂×D₄	1	2	yes	—
58	C₂³×C₆₇	#124	1	1	yes	characteristic
59	C₂×C₄×C₆₇	—	2	4	no	—
60	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
62	C₂×C₄×C₆₇	#121	1	2	yes	—
63	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
65	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
66	C₂×Dic₆₇	—	4	4	no	—
67	C₂×Dic₆₇	—	8	4	no	—
68	Dic₁₃₄	—	4	8	no	—
69	Dic₁₃₄	—	8	2	no	—
70	Dic₁₃₄	—	8	2	no	—
71	C₈×C₆₇	—	4	4	no	—
72	C₈×C₆₇	#102	1	4	yes	—
73	C₈×C₆₇	#102	1	4	yes	—
74	C₆₇⋊C₈	—	8	2	no	—
75	C₂²×C₆₇	—	2	2	no	—
76	C₂²×C₆₇	#390	1	1	yes	characteristic
77	C₂²×C₆₇	#486	1	2	yes	—
78	C₄×C₆₇	—	2	2	no	—
79	C₄×C₆₇	C₂×#102	1	2	yes	—
80	C₄×C₆₇	#473	1	4	yes	—
81	C₄×C₆₇	#492	1	1	yes	characteristic
82	C₄×C₆₇	#493	1	1	yes	characteristic
83	Dic₆₇	—	8	4	no	—
84	#9772	—	67	1	no	maximal, Sylow
85	C₂×C₆₇	—	2	2	no	—
86	C₂×C₆₇	#1957	1	1	yes	characteristic
87	C₂×C₆₇	#1979	1	2	yes	—
88	C₂×#494	—	67	1	no	—
89	#1848	—	67	1	no	—
90	#1546	—	67	4	no	—
91	#1450	D₆₇	1	1	yes	characteristic
92	C₆₇	#9772	1	1	yes	characteristic, complemented, Sylow
93	C₂²×Q₁₆	—	134	2	no	—
94	C₂×C₄⋊Q₈	—	67	2	no	—
95	#494	—	67	4	no	—
96	#494	—	134	1	no	—
97	#494	—	134	1	no	—
98	#431	C₂×D₆₇	1	2	yes	—
99	C₂×#108	—	134	2	no	—
100	#423	—	67	2	no	—
101	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
102	#332	—	134	8	no	—
103	#322	—	2	1	no	—
104	C₈²	—	2	1	no	—
105	C₂²×Q₈	—	134	2	no	—
106	C₂×Q₁₆	—	268	4	no	—
107	C₂×Q₁₆	—	268	4	no	—
108	C₂×Q₁₆	—	134	4	no	—
109	C₂×Q₁₆	—	134	4	no	—
110	C₂×M₁₆	—	134	2	no	—
111	C₂²×C₈	D₂₆₈	1	2	yes	—
112	C₄⋊Q₈	—	67	4	no	—
113	C₄⋊Q₈	—	134	2	no	—
114	C₄⋊Q₈	—	134	4	no	—
115	C₂×C₄⋊C₄	—	134	4	no	—
116	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
117	#108	—	268	4	no	—
118	C₄⋊C₈	—	134	4	no	—
119	C₄⋊C₈	D₂₆₈	1	4	yes	—
120	C₂²⋊C₈	—	2	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	C₄×D₆₇	1	2	yes	—
123	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
124	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
125	C₂×Q₈	—	268	2	no	—
126	C₂×Q₈	—	268	4	no	—
127	C₂×Q₈	—	268	4	no	—
128	C₂×Q₈	—	134	4	no	—
129	C₂²×C₄	—	268	2	no	—
130	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
131	C₂²×C₄	#11761628	1	2	yes	—
132	Q₁₆	—	268	8	no	—
133	Q₁₆	—	268	8	no	—
134	M₁₆	—	268	4	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	268	2	no	—
137	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
138	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
139	C₂×C₈	D₅₃₆	1	4	yes	—
140	C₂×C₈	#11761617	1	4	yes	—
141	C₄⋊C₄	—	268	4	no	—
142	C₄⋊C₄	—	134	8	no	—
143	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
144	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
145	C₄²	C₂×D₂₆₈	1	2	yes	—
146	C₂³	#14524031	1	1	yes	characteristic
147	Q₈	—	268	8	no	—
148	Q₈	—	536	2	no	—
149	Q₈	—	536	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	268	4	no	—
152	C₂×C₄	—	536	4	no	—
153	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
154	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
155	C₂×C₄	#14524017	1	2	yes	—
156	C₂×C₄	#14524040	1	1	yes	characteristic
157	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
158	C₂×C₄	C₂×#11761617	1	2	yes	—
159	C₈	—	4	4	no	—
160	C₈	—	536	2	no	—
161	C₈	#14523965	1	4	yes	—
162	C₈	#14523965	1	4	yes	—
163	C₂²	—	2	2	no	—
164	C₂²	#16863298	1	1	yes	characteristic
165	C₂²	#16863743	1	2	yes	—
166	C₄	—	2	2	no	—
167	C₄	—	536	4	no	—
168	C₄	C₂×#14523965	1	2	yes	—
169	C₄	#16863671	1	4	yes	—
170	C₄	#16863769	1	1	yes	characteristic
171	C₄	#16863773	1	1	yes	characteristic
172	C₂	—	2	2	no	—
173	C₂	#22561015	1	1	yes	characteristic
174	C₂	#22561124	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863776

C₂

yes

maximal, characteristic

#22560381

C₂

yes

maximal, characteristic

#22558880

C₂

yes

maximal

C₆₇×#1450

C₂

yes

maximal, characteristic

C₂²×Dic₂₆₈

—

C₂×#14524043

C₂²

yes

—

#16863776

—

#16863776

—

#16863776

C₄

yes

—

C₂×#14523979

—

#16863420

C₂²

yes

—

#16863141

—

#431×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#322×C₆₇

—

C₈²×C₆₇

—

C₂²×Dic₁₃₄

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊Dic₆₇

—

#14523979

—

C₄⋊(C₆₇⋊C₈)

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

D₄

yes

—

C₂²⋊C₈×C₆₇

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

Dic₂₆₈

—

Dic₂₆₈

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

—

C₂³×C₆₇

#124

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

—

C₂×C₄×C₆₇

C₂×SD₁₆

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

#102

yes

—

C₈×C₆₇

#102

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#390

yes

characteristic

C₂²×C₆₇

#486

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₂×#102

yes

—

C₄×C₆₇

#473

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

#9772

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1957

yes

characteristic

C₂×C₆₇

#1979

yes

—

—

—

—

D₆₇

yes

characteristic

C₆₇

#9772

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

134

—

C₂×C₄⋊Q₈

—

#494

—

#494

—

134

—

#494

—

134

—

#431

C₂×D₆₇

yes

—

C₂×#108

—

134

—

100

#423

—

101

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

102

#332

—

134

—

103

#322

—

104

C₈²

—

105

C₂²×Q₈

—

134

—

106

C₂×Q₁₆

—

268

—

107

C₂×Q₁₆

—

268

—

108

C₂×Q₁₆

—

134

—

109

C₂×Q₁₆

—

134

—

110

C₂×M₁₆

—

134

—

111

C₂²×C₈

D₂₆₈

yes

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

134

—

114

C₄⋊Q₈

—

134

—

115

C₂×C₄⋊C₄

—

134

—

116

C₂×C₄²

C₂²×D₆₇

yes

characteristic

117

#108

—

268

—

118

C₄⋊C₈

—

134

—

119

C₄⋊C₈

D₂₆₈

yes

—

120

C₂²⋊C₈

—

121

C₄×C₈

—

122

C₄×C₈

C₄×D₆₇

yes

—

123

C₄×C₈

C₂²⋊D₆₇

yes

characteristic

124

C₄×C₈

C₂²⋊D₆₇

yes

characteristic

125

C₂×Q₈

—

268

—

126

C₂×Q₈

—

268

—

127

C₂×Q₈

—

268

—

128

C₂×Q₈

—

134

—

129

C₂²×C₄

—

268

—

130

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

131

C₂²×C₄

#11761628

yes

—

132

Q₁₆

—

268

—

133

Q₁₆

—

268

—

134

M₁₆

—

268

—

135

C₂×C₈

—

136

C₂×C₈

—

268

—

137

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

138

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

139

C₂×C₈

D₅₃₆

yes

—

140

C₂×C₈

#11761617

yes

—

141

C₄⋊C₄

—

268

—

142

C₄⋊C₄

—

134

—

143

C₄²

C₂×C₄×D₆₇

yes

characteristic

144

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

145

C₄²

C₂×D₂₆₈

yes

—

146

C₂³

#14524031

yes

characteristic

147

Q₈

—

268

—

148

Q₈

—

536

—

149

Q₈

—

536

—

150

C₂×C₄

—

151

C₂×C₄

—

268

—

152

C₂×C₄

—

536

—

153

C₂×C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

154

C₂×C₄

C₄×D₂₆₈

yes

—

155

C₂×C₄

#14524017

yes

—

156

C₂×C₄

#14524040

yes

characteristic

157

C₂×C₄

C₂×D₅₃₆

yes

—

158

C₂×C₄

C₂×#11761617

yes

—

159

C₈

—

160

C₈

—

536

—

161

C₈

#14523965

yes

—

162

C₈

#14523965

yes

—

163

C₂²

—

164

C₂²

#16863298

yes

characteristic

165

C₂²

#16863743

yes

—

166

C₄

—

167

C₄

—

536

—

168

C₄

C₂×#14523965

yes

—

169

C₄

#16863671

yes

—

170

C₄

#16863769

yes

characteristic

171

C₄

#16863773

yes

characteristic

172

C₂

—

173

C₂

#22561015

yes

characteristic

174

C₂

#22561124

yes

—

	subgroup	G/H
center	C₂²	#16863298
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1450	C₂
socle	C₂²×C₆₇	#390
prime radical	C₂³×C₆₇	#124
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1450	D₆₇
hypocenter	C₆₇	#9772
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

subgroup

G/H

center

C₂²

#16863298

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1450

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

N	H ≅ G/N	type
92. C₆₇	#9772	semidirect

H ≅ G/N

type

92. C₆₇

#9772

semidirect

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

subgroup	index
C₂×#16863776	2
#22560381	2
#22558880	2
C₆₇×#1450	2
#9772	67

subgroup

index

C₆₇×#1450

Subgroups — #30755294

Conjugacy classes of subgroups

Complements

Maximal subgroups