Group #30869533 — subgroups

Subgroups — #30869533

Order 17152.

578 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 211 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	#22570723	C₂	1	1	yes	maximal, characteristic
2	#22569149	C₂	1	2	yes	maximal
3	#22561190	C₂	1	2	yes	maximal
4	#22560809	C₂	1	8	yes	maximal
5	#22560347	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1746	C₂	1	1	yes	maximal, characteristic
7	#16863953	C₂²	1	2	yes	—
8	#16863801	C₄	1	2	yes	complemented
9	#16863798	C₄	1	4	yes	complemented
10	#16863794	C₄	1	2	yes	complemented
11	#16863524	—	2	8	no	—
12	#16863434	C₂²	1	4	yes	—
13	#16863422	C₂²	1	4	yes	—
14	#16863420	C₂²	1	2	yes	—
15	#16863351	C₂²	1	4	yes	—
16	#16863161	C₂²	1	4	yes	—
17	#16863156	C₂²	1	8	yes	—
18	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	#376×C₆₇	C₂²	1	1	yes	characteristic
21	#336×C₆₇	C₂²	1	4	yes	—
22	C₂×#11761612	C₂³	1	4	yes	—
23	#14524043	C₂×C₄	1	4	yes	—
24	#14524035	C₂×C₄	1	4	yes	—
25	#14524022	—	2	4	no	—
26	C₄×Dic₁₃₄	—	2	4	no	—
27	#14524002	C₂³	1	4	yes	—
28	#14523977	C₂×C₄	1	8	yes	—
29	#14523975	C₂×C₄	1	8	yes	—
30	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
31	Dic₆₇⋊C₈	—	2	8	no	—
32	#14523953	—	2	4	no	—
33	C₈×Dic₆₇	—	2	4	no	—
34	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
35	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
36	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
37	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
38	#97×C₆₇	D₄	1	4	yes	—
39	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
42	#95×C₆₇	—	2	2	no	—
43	C₂×Dic₁₃₄	—	2	4	no	—
44	#11761612	—	2	8	no	—
45	C₂×C₆₇⋊C₈	—	2	4	no	—
46	C₄⋊Dic₆₇	—	2	4	no	—
47	C₄⋊Dic₆₇	C₂²×C₄	1	8	yes	—
48	Dic₆₇⋊C₄	—	2	8	no	—
49	C₂²⋊Dic₆₇	—	2	4	no	—
50	C₄×Dic₆₇	—	2	4	no	—
51	C₂²×C₄×C₆₇	—	2	2	no	—
52	C₂²×C₄×C₆₇	#41	1	2	yes	—
53	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
54	C₂×C₈×C₆₇	—	2	4	no	—
55	C₂×C₈×C₆₇	—	2	4	no	—
56	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
57	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
58	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
59	C₄²×C₆₇	—	2	4	no	—
60	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₆₇	C₂×D₄	1	2	yes	—
63	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
64	C₂×C₄×C₆₇	—	4	2	no	—
65	C₂×C₄×C₆₇	—	4	4	no	—
66	C₂×C₄×C₆₇	—	4	2	no	—
67	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
68	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
69	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
70	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
71	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
72	C₂×Dic₆₇	—	2	8	no	—
73	Dic₁₃₄	—	8	4	no	—
74	C₈×C₆₇	—	2	2	no	—
75	C₈×C₆₇	—	2	2	no	—
76	C₈×C₆₇	—	2	4	no	—
77	C₆₇⋊C₈	—	4	4	no	—
78	C₂²×C₆₇	—	2	2	no	—
79	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
80	C₂²×C₆₇	#541	1	2	yes	—
81	C₄×C₆₇	—	2	2	no	—
82	C₄×C₆₇	—	2	2	no	—
83	C₄×C₆₇	—	8	2	no	—
84	C₄×C₆₇	#519	1	2	yes	—
85	C₄×C₆₇	#536	1	2	yes	—
86	Dic₆₇	—	4	8	no	—
87	#22258	—	67	1	no	maximal, Sylow
88	C₂×C₆₇	—	2	2	no	—
89	C₂×C₆₇	#2327	1	1	yes	characteristic
90	C₂×C₆₇	#2957	1	2	yes	—
91	#3160	—	67	1	no	—
92	#2961	—	67	2	no	—
93	#1995	—	67	2	no	—
94	#1922	—	67	8	no	—
95	#1837	—	67	1	no	—
96	#1746	D₆₇	1	1	yes	characteristic
97	C₆₇	#22258	1	1	yes	characteristic, complemented, Sylow
98	#533	—	67	2	no	—
99	#500	—	67	2	no	—
100	#499	—	67	4	no	—
101	#498	—	67	2	no	—
102	#446	—	134	8	no	—
103	#427	—	67	4	no	—
104	#424	—	67	4	no	—
105	#423	—	67	2	no	—
106	#405	—	67	4	no	—
107	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
109	#376	C₂×D₆₇	1	1	yes	characteristic
110	#341	—	67	4	no	—
111	#339	—	67	8	no	—
112	#336	C₂×D₆₇	1	4	yes	—
113	C₂×M₁₆	—	67	4	no	—
114	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
115	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
116	C₄⋊Q₈	—	67	4	no	—
117	#125	—	67	4	no	—
118	C₂²⋊Q₈	—	134	4	no	—
119	C₄×Q₈	—	134	4	no	—
120	#117	—	67	4	no	—
121	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
122	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
123	#107	—	67	8	no	—
124	#106	—	67	8	no	—
125	C₄⋊C₈	—	134	8	no	—
126	C₄⋊C₈	—	134	4	no	—
127	#97	—	134	4	no	—
128	#97	D₂₆₈	1	4	yes	—
129	C₄×C₈	—	134	4	no	—
130	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
131	C₄×C₈	C₄×D₆₇	1	2	yes	—
132	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
133	#95	—	2	2	no	—
134	C₂×Q₈	—	134	4	no	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	#11761628	1	2	yes	—
137	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
138	M₁₆	—	134	8	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	134	4	no	—
142	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
143	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
144	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
145	C₄⋊C₄	—	67	8	no	—
146	C₄⋊C₄	—	134	8	no	—
147	C₄⋊C₄	—	134	4	no	—
148	C₂²⋊C₄	—	134	4	no	—
149	C₄²	—	2	4	no	—
150	C₄²	—	134	4	no	—
151	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
152	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
153	C₄²	C₂×D₂₆₈	1	2	yes	—
154	C₂³	C₂×#11761628	1	1	yes	characteristic
155	Q₈	—	536	4	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	134	8	no	—
160	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
161	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
162	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
163	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
164	C₂×C₄	#14524090	1	4	yes	—
165	C₈	—	2	2	no	—
166	C₈	—	2	2	no	—
167	C₈	—	2	4	no	—
168	C₈	—	268	4	no	—
169	C₂²	—	2	2	no	—
170	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
171	C₂²	#16864008	1	2	yes	—
172	C₄	—	2	2	no	—
173	C₄	—	2	2	no	—
174	C₄	—	8	2	no	—
175	C₄	—	268	8	no	—
176	C₄	#16863873	1	2	yes	—
177	C₄	#16863972	1	2	yes	—
178	C₂	—	2	2	no	—
179	C₂	#22562928	1	1	yes	characteristic
180	C₂	#22569127	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22570723

C₂

yes

maximal, characteristic

#22569149

C₂

yes

maximal

#22561190

C₂

yes

maximal

#22560809

C₂

yes

maximal

#22560347

C₂

yes

maximal, characteristic

C₆₇×#1746

C₂

yes

maximal, characteristic

#16863953

C₂²

yes

—

#16863801

C₄

yes

complemented

#16863798

C₄

yes

complemented

#16863794

C₄

yes

complemented

#16863524

—

#16863434

C₂²

yes

—

#16863422

C₂²

yes

—

#16863420

C₂²

yes

—

#16863351

C₂²

yes

—

#16863161

C₂²

yes

—

#16863156

C₂²

yes

—

C₂×#97×C₆₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#376×C₆₇

C₂²

yes

characteristic

#336×C₆₇

C₂²

yes

—

C₂×#11761612

C₂³

yes

—

#14524043

C₂×C₄

yes

—

#14524035

C₂×C₄

yes

—

#14524022

—

C₄×Dic₁₃₄

—

#14524002

C₂³

yes

—

#14523977

C₂×C₄

yes

—

#14523975

C₂×C₄

yes

—

C₄⋊(C₆₇⋊C₈)

—

Dic₆₇⋊C₈

—

#14523953

—

C₈×Dic₆₇

—

C₂²×C₈×C₆₇

C₂³

yes

—

C₂²×C₈×C₆₇

C₂³

yes

—

C₂×C₄²×C₆₇

C₂³

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#97×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

#95×C₆₇

—

C₂×Dic₁₃₄

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂²×C₄

yes

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

#41

yes

—

C₂²×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²×C₄

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

—

C₂³×C₆₇

C₂×#41

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

C₂³×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×D₄

yes

characteristic

C₂×C₄×C₆₇

2₋¹⁺⁴

yes

—

C₂×Dic₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₄×D₄

yes

characteristic

C₂²×C₆₇

#541

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#519

yes

—

C₄×C₆₇

#536

yes

—

Dic₆₇

—

#22258

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#2327

yes

characteristic

C₂×C₆₇

#2957

yes

—

—

—

—

—

—

D₆₇

yes

characteristic

C₆₇

#22258

yes

characteristic, complemented, Sylow

#533

—

#500

—

100

#499

—

101

#498

—

102

#446

—

134

—

103

#427

—

104

#424

—

105

#423

—

106

#405

—

107

C₂×#97

C₂×D₆₇

yes

characteristic

108

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

109

#376

C₂×D₆₇

yes

characteristic

110

#341

—

111

#339

—

112

#336

C₂×D₆₇

yes

—

113

C₂×M₁₆

—

114

C₂²×C₈

C₂²×D₆₇

yes

—

115

C₂²×C₈

C₂²×D₆₇

yes

—

116

C₄⋊Q₈

—

117

#125

—

118

C₂²⋊Q₈

—

134

—

119

C₄×Q₈

—

134

—

120

#117

—

121

C₂×C₄²

C₂²×D₆₇

yes

—

122

C₂×C₄²

C₂²×D₆₇

yes

characteristic

123

#107

—

124

#106

—

125

C₄⋊C₈

—

134

—

126

C₄⋊C₈

—

134

—

127

#97

—

134

—

128

#97

D₂₆₈

yes

—

129

C₄×C₈

—

134

—

130

C₄×C₈

C₄×D₆₇

yes

characteristic

131

C₄×C₈

C₄×D₆₇

yes

—

132

C₄×C₈

C₄×D₆₇

yes

characteristic

133

#95

—

134

C₂×Q₈

—

134

—

135

C₂²×C₄

—

136

C₂²×C₄

#11761628

yes

—

137

C₂²×C₄

C₂³×D₆₇

yes

characteristic

138

M₁₆

—

134

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

134

—

142

C₂×C₈

C₂×C₄×D₆₇

yes

—

143

C₂×C₈

C₂×C₄×D₆₇

yes

—

144

C₂×C₈

C₂×D₂₆₈

yes

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

134

—

147

C₄⋊C₄

—

134

—

148

C₂²⋊C₄

—

134

—

149

C₄²

—

150

C₄²

—

134

—

151

C₄²

C₂×C₄×D₆₇

yes

characteristic

152

C₄²

C₂×C₄×D₆₇

yes

characteristic

153

C₄²

C₂×D₂₆₈

yes

—

154

C₂³

C₂×#11761628

yes

characteristic

155

Q₈

—

536

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

134

—

160

C₂×C₄

C₄×D₂₆₈

yes

—

161

C₂×C₄

C₄×D₂₆₈

yes

—

162

C₂×C₄

C₂²×C₄×D₆₇

yes

characteristic

163

C₂×C₄

C₂²×D₂₆₈

yes

characteristic

164

C₂×C₄

#14524090

yes

—

165

C₈

—

166

C₈

—

167

C₈

—

168

C₈

—

268

—

169

C₂²

—

170

C₂²

C₂×C₄×D₂₆₈

yes

characteristic

171

C₂²

#16864008

yes

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

268

—

176

C₄

#16863873

yes

—

177

C₄

#16863972

yes

—

178

C₂

—

179

C₂

#22562928

yes

characteristic

180

C₂

#22569127

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1746	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1746	D₆₇
hypocenter	C₆₇	#22258
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1746

subgroup

G/H

center

C₂×C₄

C₂²×D₂₆₈

commutator

C₂×C₄×C₆₇

C₂³×C₄

Frattini

C₂²×C₄

C₂³×D₆₇

Fitting

C₆₇×#1746

C₂

socle

C₂²×C₆₇

C₂×C₄×D₄

prime radical

C₂³×C₆₇

C₂×#41

cosocle

C₂²×C₄×C₆₇

C₂⁴

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1746

N	H ≅ G/N	type
8. #16863801	C₄	semidirect
9. #16863798	C₄	semidirect
10. #16863794	C₄	semidirect
97. C₆₇	#22258	semidirect

H ≅ G/N

type

8. #16863801

C₄

semidirect

9. #16863798

C₄

semidirect

10. #16863794

C₄

semidirect

97. C₆₇

#22258

semidirect

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

subgroup	index
#22570723	2
#22569149	2
#22561190	2
#22560809	2
#22560347	2
C₆₇×#1746	2
#22258	67

subgroup

index

C₆₇×#1746

Subgroups — #30869533

Conjugacy classes of subgroups

Complements

Maximal subgroups