Group #30869534 — subgroups

Subgroups — #30869534

Order 17152.

578 conjugacy classes of subgroups; only the 243 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 243 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	#22569150	C₂	1	2	yes	maximal
2	#22561191	C₂	1	2	yes	maximal
3	#22560810	C₂	1	8	yes	maximal
4	#22560347	C₂	1	1	yes	maximal, characteristic
5	#22559960	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#3160	C₂	1	1	yes	maximal, characteristic
7	#16863527	—	2	8	no	—
8	#16863423	C₂²	1	4	yes	—
9	#16863420	C₂²	1	2	yes	—
10	#16863352	C₂²	1	4	yes	—
11	C₂×#14523954	C₂²	1	1	yes	characteristic
12	#16863242	C₂²	1	1	yes	characteristic
13	#16863161	C₂²	1	4	yes	—
14	#16863155	C₂²	1	8	yes	—
15	#16863150	C₂²	1	4	yes	—
16	#533×C₆₇	C₂²	1	2	yes	—
17	#500×C₆₇	C₄	1	2	yes	complemented
18	#499×C₆₇	C₄	1	4	yes	complemented
19	#498×C₆₇	C₄	1	2	yes	complemented
20	#427×C₆₇	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#11761612	C₂³	1	4	yes	—
23	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
24	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
25	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
26	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
27	#14523954	—	2	4	no	—
28	#14523954	D₄	1	4	yes	—
29	C₄×C₆₇⋊C₈	—	2	4	no	—
30	C₆₇⋊#95	—	2	2	no	—
31	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
32	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
33	#125×C₆₇	C₂×C₄	1	4	yes	—
34	C₂²⋊Q₈×C₆₇	—	2	4	no	—
35	C₄×Q₈×C₆₇	—	2	4	no	—
36	#117×C₆₇	C₂³	1	4	yes	—
37	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
38	#107×C₆₇	C₂×C₄	1	8	yes	—
39	#106×C₆₇	C₂×C₄	1	8	yes	—
40	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×Dic₆₇	—	2	2	no	—
44	#11761612	—	2	8	no	—
45	C₂×C₆₇⋊C₈	—	2	4	no	—
46	C₂×C₆₇⋊C₈	—	2	4	no	—
47	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
48	C₄×Dic₆₇	—	2	4	no	—
49	C₂×Q₈×C₆₇	—	2	4	no	—
50	C₂²×C₄×C₆₇	#41	1	2	yes	—
51	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
52	C₂×C₈×C₆₇	—	2	4	no	—
53	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
54	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
55	C₄⋊C₄×C₆₇	—	2	8	no	—
56	C₄⋊C₄×C₆₇	—	2	4	no	—
57	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
58	C₂²⋊C₄×C₆₇	—	2	4	no	—
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	Q₈×C₆₇	—	8	4	no	—
65	C₂×C₄×C₆₇	—	2	8	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	—	4	4	no	—
73	C₂×Dic₆₇	—	4	2	no	—
74	C₈×C₆₇	—	2	2	no	—
75	C₈×C₆₇	—	2	2	no	—
76	C₆₇⋊C₈	—	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	—	4	8	no	—
84	C₄×C₆₇	#519	1	2	yes	—
85	C₄×C₆₇	#536	1	2	yes	—
86	Dic₆₇	—	8	2	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	D₆₇	1	1	yes	characteristic
92	#2961	—	67	2	no	—
93	#1995	—	67	2	no	—
94	#1922	—	67	8	no	—
95	#1837	—	67	1	no	—
96	#1746	—	67	1	no	—
97	C₆₇	#22258	1	1	yes	characteristic, complemented, Sylow
98	#533	C₂×D₆₇	1	2	yes	—
99	#500	Dic₆₇	1	2	yes	complemented
100	#499	Dic₆₇	1	4	yes	complemented
101	#498	Dic₆₇	1	2	yes	complemented
102	#446	—	134	8	no	—
103	#427	C₂×D₆₇	1	4	yes	—
104	#424	—	67	4	no	—
105	#423	—	67	2	no	—
106	#405	—	67	4	no	—
107	C₂×#97	—	67	1	no	—
108	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
109	#376	—	67	1	no	—
110	#341	—	67	4	no	—
111	#339	—	67	8	no	—
112	#336	—	67	4	no	—
113	C₂×M₁₆	—	67	4	no	—
114	C₂²×C₈	—	67	2	no	—
115	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
116	C₄⋊Q₈	C₂×Dic₆₇	1	4	yes	—
117	#125	C₂×Dic₆₇	1	4	yes	—
118	C₂²⋊Q₈	—	2	4	no	—
119	C₄×Q₈	—	2	4	no	—
120	#117	C₂²×D₆₇	1	4	yes	—
121	C₂×C₄²	—	67	2	no	—
122	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
123	#107	C₂×Dic₆₇	1	8	yes	—
124	#106	C₂×Dic₆₇	1	8	yes	—
125	C₄⋊C₈	—	134	8	no	—
126	C₄⋊C₈	—	134	4	no	—
127	#97	—	67	4	no	—
128	#97	—	134	4	no	—
129	C₄×C₈	—	134	4	no	—
130	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
131	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
132	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
133	#95	—	134	2	no	—
134	C₂×Q₈	—	2	4	no	—
135	C₂²×C₄	—	134	2	no	—
136	C₂²×C₄	#11761627	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	—	67	4	no	—
141	C₂×C₈	—	134	4	no	—
142	C₂×C₈	—	134	4	no	—
143	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
144	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
145	C₄⋊C₄	—	2	8	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	C₂²×Dic₆₇	1	8	yes	—
148	C₂²⋊C₄	—	2	4	no	—
149	C₄²	—	2	4	no	—
150	C₄²	—	134	4	no	—
151	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
152	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
153	C₄²	D₄×D₆₇	1	2	yes	—
154	C₂³	C₂×#11761627	1	1	yes	characteristic
155	Q₈	—	8	4	no	—
156	C₂×C₄	—	2	8	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	268	4	no	—
159	C₂×C₄	—	268	2	no	—
160	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
161	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
162	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
163	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
164	C₂×C₄	#14524091	1	4	yes	—
165	C₈	—	2	2	no	—
166	C₈	—	2	2	no	—
167	C₈	—	268	4	no	—
168	C₈	—	134	4	no	—
169	C₂²	—	2	2	no	—
170	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
171	C₂²	#16864007	1	2	yes	—
172	C₄	—	2	2	no	—
173	C₄	—	2	2	no	—
174	C₄	—	4	8	no	—
175	C₄	—	536	2	no	—
176	C₄	#16863874	1	2	yes	—
177	C₄	#16863971	1	2	yes	—
178	C₂	—	2	2	no	—
179	C₂	#22562929	1	1	yes	characteristic
180	C₂	#22569126	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22569150

C₂

yes

maximal

#22561191

C₂

yes

maximal

#22560810

C₂

yes

maximal

#22560347

C₂

yes

maximal, characteristic

#22559960

C₂

yes

maximal, characteristic

C₆₇×#3160

C₂

yes

maximal, characteristic

#16863527

—

#16863423

C₂²

yes

—

#16863420

C₂²

yes

—

#16863352

C₂²

yes

—

C₂×#14523954

C₂²

yes

characteristic

#16863242

C₂²

yes

characteristic

#16863161

C₂²

yes

—

#16863155

C₂²

yes

—

#16863150

C₂²

yes

—

#533×C₆₇

C₂²

yes

—

#500×C₆₇

C₄

yes

complemented

#499×C₆₇

C₄

yes

complemented

#498×C₆₇

C₄

yes

complemented

#427×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×#11761612

C₂³

yes

—

C₂²×C₆₇⋊C₈

C₂³

yes

—

C₂×C₄×Dic₆₇

C₂³

yes

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

#14523954

—

#14523954

D₄

yes

—

C₄×C₆₇⋊C₈

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

C₂³

yes

—

C₄⋊Q₈×C₆₇

C₂×C₄

yes

—

#125×C₆₇

C₂×C₄

yes

—

C₂²⋊Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

#117×C₆₇

C₂³

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#107×C₆₇

C₂×C₄

yes

—

#106×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×Dic₆₇

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×D₄

yes

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

#41

yes

—

C₂²×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²×C₄

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂²×C₄

yes

—

C₂²⋊C₄×C₆₇

—

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

Q₈×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

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

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

—

#3160

D₆₇

yes

—

—

—

—

—

C₆₇

#22258

yes

characteristic, complemented, Sylow

#533

C₂×D₆₇

yes

—

#500

Dic₆₇

yes

complemented

100

#499

Dic₆₇

yes

complemented

101

#498

Dic₆₇

yes

complemented

102

#446

—

134

—

103

#427

C₂×D₆₇

yes

—

104

#424

—

105

#423

—

106

#405

—

107

C₂×#97

—

108

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

109

#376

—

110

#341

—

111

#339

—

112

#336

—

113

C₂×M₁₆

—

114

C₂²×C₈

—

115

C₂²×C₈

C₂²×D₆₇

yes

—

116

C₄⋊Q₈

C₂×Dic₆₇

yes

—

117

#125

C₂×Dic₆₇

yes

—

118

C₂²⋊Q₈

—

119

C₄×Q₈

—

120

#117

C₂²×D₆₇

yes

—

121

C₂×C₄²

—

122

C₂×C₄²

C₂²×D₆₇

yes

characteristic

123

#107

C₂×Dic₆₇

yes

—

124

#106

C₂×Dic₆₇

yes

—

125

C₄⋊C₈

—

134

—

126

C₄⋊C₈

—

134

—

127

#97

—

128

#97

—

134

—

129

C₄×C₈

—

134

—

130

C₄×C₈

C₂×Dic₆₇

yes

characteristic

131

C₄×C₈

C₂×Dic₆₇

yes

—

132

C₄×C₈

C₂×Dic₆₇

yes

characteristic

133

#95

—

134

—

134

C₂×Q₈

—

135

C₂²×C₄

—

134

—

136

C₂²×C₄

#11761627

yes

—

137

C₂²×C₄

C₂³×D₆₇

yes

characteristic

138

M₁₆

—

134

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

134

—

142

C₂×C₈

—

134

—

143

C₂×C₈

C₂²×Dic₆₇

yes

—

144

C₂×C₈

C₂²×Dic₆₇

yes

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

C₂²×Dic₆₇

yes

—

148

C₂²⋊C₄

—

149

C₄²

—

150

C₄²

—

134

—

151

C₄²

C₂²×Dic₆₇

yes

characteristic

152

C₄²

C₂²×Dic₆₇

yes

characteristic

153

C₄²

D₄×D₆₇

yes

—

154

C₂³

C₂×#11761627

yes

characteristic

155

Q₈

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

268

—

159

C₂×C₄

—

268

—

160

C₂×C₄

D₄×Dic₆₇

yes

—

161

C₂×C₄

D₄×Dic₆₇

yes

—

162

C₂×C₄

C₂³×Dic₆₇

yes

characteristic

163

C₂×C₄

C₂×D₄×D₆₇

yes

characteristic

164

C₂×C₄

#14524091

yes

—

165

C₈

—

166

C₈

—

167

C₈

—

268

—

168

C₈

—

134

—

169

C₂²

—

170

C₂²

C₂×D₄×Dic₆₇

yes

characteristic

171

C₂²

#16864007

yes

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

536

—

176

C₄

#16863874

yes

—

177

C₄

#16863971

yes

—

178

C₂

—

179

C₂

#22562929

yes

characteristic

180

C₂

#22569126

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#3160	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#3160	D₆₇
hypocenter	C₆₇	#22258
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

subgroup

G/H

center

C₂×C₄

C₂×D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂³×C₄

Frattini

C₂²×C₄

C₂³×D₆₇

Fitting

C₆₇×#3160

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×C₄² ◁ #3160

N	H ≅ G/N	type
17. #500×C₆₇	C₄	semidirect
18. #499×C₆₇	C₄	semidirect
19. #498×C₆₇	C₄	semidirect
97. C₆₇	#22258	semidirect
99. #500	Dic₆₇	semidirect
100. #499	Dic₆₇	semidirect
101. #498	Dic₆₇	semidirect

H ≅ G/N

type

17. #500×C₆₇

C₄

semidirect

18. #499×C₆₇

C₄

semidirect

19. #498×C₆₇

C₄

semidirect

97. C₆₇

#22258

semidirect

99. #500

Dic₆₇

semidirect

100. #499

Dic₆₇

semidirect

101. #498

Dic₆₇

semidirect

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

subgroup	index
#22569150	2
#22561191	2
#22560810	2
#22560347	2
#22559960	2
C₆₇×#3160	2
#22258	67

subgroup

index

C₆₇×#3160

Subgroups — #30869534

Conjugacy classes of subgroups

Complements

Maximal subgroups