Group #30869682 — subgroups

Subgroups — #30869682

Order 17152.

826 conjugacy classes of subgroups; only the 259 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 259 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	#22562929	C₂	1	2	yes	maximal
2	#22561030	C₂	1	2	yes	maximal
3	#22560552	C₂	1	8	yes	maximal
4	#22560340	C₂	1	1	yes	maximal, characteristic
5	#22559960	C₂	1	1	yes	maximal, characteristic
6	C₂×#494×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₈×Dic₆₇	C₂²	1	4	yes	—
8	C₂×#14523967	C₂²	1	8	yes	—
9	C₂×#14523954	C₂²	1	1	yes	characteristic
10	#16863303	C₂²	1	4	yes	—
11	#16863282	—	2	8	no	—
12	#16863262	C₂²	1	2	yes	—
13	#16863242	C₂²	1	1	yes	characteristic
14	#16863159	C₂²	1	4	yes	—
15	#16863150	C₂²	1	4	yes	—
16	C₂²×Q₁₆×C₆₇	C₂²	1	4	yes	—
17	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
18	#494×C₆₇	C₄	1	8	yes	complemented
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
21	Q₈×Dic₆₇	—	4	8	no	—
22	C₂×C₄⋊Dic₆₇	—	2	4	no	—
23	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
24	C₂×C₄×Dic₆₇	—	2	4	no	—
25	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
26	#14523967	—	4	8	no	—
27	#14523954	—	2	2	no	—
28	C₆₇⋊#95	—	2	8	no	—
29	C₆₇⋊#95	—	2	4	no	—
30	C₆₇⋊#95	D₄	1	4	yes	—
31	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
32	C₂×Q₁₆×C₆₇	—	4	8	no	—
33	C₂×Q₁₆×C₆₇	C₂×C₄	1	16	yes	—
34	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
35	C₄⋊Q₈×C₆₇	—	2	4	no	—
36	C₄⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
37	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
38	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
39	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
40	C₂²×Dic₆₇	—	2	4	no	—
41	C₂²×Dic₆₇	C₂×D₄	1	4	yes	—
42	C₂×C₆₇⋊C₈	—	2	2	no	—
43	C₂×C₆₇⋊C₈	—	2	4	no	—
44	C₄⋊Dic₆₇	—	2	8	no	—
45	C₄⋊Dic₆₇	—	4	8	no	—
46	C₄×Dic₆₇	—	2	4	no	—
47	C₄×Dic₆₇	—	4	8	no	—
48	C₂×Q₈×C₆₇	—	2	4	no	—
49	C₂×Q₈×C₆₇	—	2	8	no	—
50	C₂×Q₈×C₆₇	—	2	8	no	—
51	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
52	C₂²×C₄×C₆₇	—	2	4	no	—
53	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
55	Q₁₆×C₆₇	—	4	16	no	—
56	C₂×C₈×C₆₇	—	2	4	no	—
57	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
58	C₄⋊C₄×C₆₇	—	2	16	no	—
59	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
60	C₄²×C₆₇	#41	1	2	yes	—
61	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
62	Q₈×C₆₇	—	2	16	no	—
63	Q₈×C₆₇	—	4	8	no	—
64	C₂×C₄×C₆₇	—	2	4	no	—
65	C₂×C₄×C₆₇	—	2	8	no	—
66	C₂×C₄×C₆₇	—	4	4	no	—
67	C₂×C₄×C₆₇	—	4	4	no	—
68	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
69	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
70	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
71	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
72	C₂×Dic₆₇	—	2	4	no	—
73	C₂×Dic₆₇	—	2	8	no	—
74	C₂×Dic₆₇	—	4	4	no	—
75	C₂×Dic₆₇	—	4	8	no	—
76	C₈×C₆₇	—	2	4	no	—
77	C₆₇⋊C₈	—	4	2	no	—
78	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
79	C₂²×C₆₇	#535	1	2	yes	—
80	C₂²×C₆₇	#578	1	4	yes	—
81	C₄×C₆₇	—	2	4	no	—
82	C₄×C₆₇	—	4	8	no	—
83	C₄×C₆₇	#518	1	2	yes	—
84	C₄×C₆₇	#540	1	2	yes	—
85	Dic₆₇	—	4	4	no	—
86	Dic₆₇	—	8	4	no	—
87	#22277	—	67	1	no	maximal, Sylow
88	C₂×C₆₇	#2328	1	1	yes	characteristic
89	C₂×C₆₇	#2950	1	2	yes	—
90	C₂×C₆₇	#3223	1	4	yes	—
91	C₂×#494	D₆₇	1	1	yes	characteristic
92	#2327	—	67	2	no	—
93	#1960	—	67	2	no	—
94	#1878	—	67	8	no	—
95	#1835	—	67	1	no	—
96	#1746	—	67	1	no	—
97	C₆₇	#22277	1	1	yes	characteristic, complemented, Sylow
98	C₂²×Q₁₆	C₂×D₆₇	1	4	yes	—
99	C₂×C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
100	C₂×C₄×Q₈	—	67	4	no	—
101	#494	Dic₆₇	1	8	yes	complemented
102	C₂×#103	—	67	8	no	—
103	C₂×#97	—	67	1	no	—
104	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
105	#391	—	67	4	no	—
106	#387	—	134	8	no	—
107	#382	—	67	2	no	—
108	#376	—	67	1	no	—
109	#340	—	67	4	no	—
110	#336	—	67	4	no	—
111	C₂²×Q₈	C₂²×D₆₇	1	4	yes	—
112	C₂×Q₁₆	—	4	8	no	—
113	C₂×Q₁₆	C₂×Dic₆₇	1	16	yes	—
114	C₂²×C₈	—	67	2	no	—
115	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
116	C₄⋊Q₈	—	2	4	no	—
117	C₄⋊Q₈	C₂×Dic₆₇	1	8	yes	—
118	C₄×Q₈	—	268	8	no	—
119	C₂×C₄⋊C₄	—	2	4	no	—
120	C₂×C₄⋊C₄	—	67	4	no	—
121	C₂×C₄⋊C₄	—	134	4	no	—
122	C₂×C₄²	—	67	2	no	—
123	C₂×C₄²	—	134	4	no	—
124	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
125	#103	—	268	8	no	—
126	#97	—	134	2	no	—
127	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
128	#95	—	67	4	no	—
129	#95	—	134	8	no	—
130	#95	—	134	4	no	—
131	C₂×Q₈	—	2	4	no	—
132	C₂×Q₈	—	2	8	no	—
133	C₂×Q₈	—	2	8	no	—
134	C₂×Q₈	C₂²×Dic₆₇	1	8	yes	—
135	C₂²×C₄	—	2	4	no	—
136	C₂²×C₄	—	67	4	no	—
137	C₂²×C₄	—	134	4	no	—
138	C₂²×C₄	D₄×D₆₇	1	2	yes	—
139	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
140	Q₁₆	—	4	16	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	134	2	no	—
143	C₂×C₈	—	134	4	no	—
144	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
145	C₄⋊C₄	—	2	16	no	—
146	C₄⋊C₄	—	268	8	no	—
147	C₄⋊C₄	—	134	8	no	—
148	C₄²	—	268	8	no	—
149	C₄²	—	134	4	no	—
150	C₄²	C₂²×Dic₆₇	1	2	yes	—
151	C₄²	#11761627	1	2	yes	—
152	C₂³	C₂×D₄×D₆₇	1	1	yes	characteristic
153	Q₈	—	2	16	no	—
154	Q₈	—	4	8	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	8	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	268	8	no	—
160	C₂×C₄	—	268	4	no	—
161	C₂×C₄	—	134	8	no	—
162	C₂×C₄	—	134	4	no	—
163	C₂×C₄	D₄×Dic₆₇	1	4	yes	—
164	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
165	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
166	C₂×C₄	#14524088	1	4	yes	—
167	C₈	—	2	4	no	—
168	C₈	—	268	2	no	—
169	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
170	C₂²	#16863959	1	2	yes	—
171	C₂²	#16864209	1	4	yes	—
172	C₄	—	2	4	no	—
173	C₄	—	4	8	no	—
174	C₄	—	268	4	no	—
175	C₄	—	536	4	no	—
176	C₄	#16863871	1	2	yes	—
177	C₄	#16864000	1	2	yes	—
178	C₂	#22562935	1	1	yes	characteristic
179	C₂	#22569077	1	2	yes	—
180	C₂	#22571285	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22562929

C₂

yes

maximal

#22561030

C₂

yes

maximal

#22560552

C₂

yes

maximal

#22560340

C₂

yes

maximal, characteristic

#22559960

C₂

yes

maximal, characteristic

C₂×#494×C₆₇

C₂

yes

maximal, characteristic

C₂×Q₈×Dic₆₇

C₂²

yes

—

C₂×#14523967

C₂²

yes

—

C₂×#14523954

C₂²

yes

characteristic

#16863303

C₂²

yes

—

#16863282

—

#16863262

C₂²

yes

—

#16863242

C₂²

yes

characteristic

#16863159

C₂²

yes

—

#16863150

C₂²

yes

—

C₂²×Q₁₆×C₆₇

C₂²

yes

—

C₂×C₄⋊Q₈×C₆₇

C₂²

yes

—

#494×C₆₇

C₄

yes

complemented

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂³

yes

—

Q₈×Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

C₂³

yes

—

C₂×C₄×Dic₆₇

—

C₂×C₄×Dic₆₇

C₂³

yes

—

#14523967

—

#14523954

—

C₆₇⋊#95

—

C₆₇⋊#95

—

C₆₇⋊#95

D₄

yes

—

C₂²×Q₈×C₆₇

C₂³

yes

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

C₂×C₄

yes

—

C₂²×C₈×C₆₇

C₂³

yes

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

C₂×D₄

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

C₂²×C₄

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂⁴

yes

characteristic

Q₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²×C₄

yes

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂²×C₄

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

C₂²×D₄

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

C₂³×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×#41

yes

characteristic

C₂×C₄×C₆₇

H(5,𝔽₂)

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₂×C₄×D₄

yes

characteristic

C₂²×C₆₇

#535

yes

—

C₂²×C₆₇

#578

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#518

yes

—

C₄×C₆₇

#540

yes

—

Dic₆₇

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×C₆₇

#3223

yes

—

C₂×#494

D₆₇

yes

—

—

—

—

—

C₆₇

#22277

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

C₂×D₆₇

yes

—

C₂×C₄⋊Q₈

C₂×D₆₇

yes

—

100

C₂×C₄×Q₈

—

101

#494

Dic₆₇

yes

complemented

102

C₂×#103

—

103

C₂×#97

—

104

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

105

#391

—

106

#387

—

134

—

107

#382

—

108

#376

—

109

#340

—

110

#336

—

111

C₂²×Q₈

C₂²×D₆₇

yes

—

112

C₂×Q₁₆

—

113

C₂×Q₁₆

C₂×Dic₆₇

yes

—

114

C₂²×C₈

—

115

C₂²×C₈

C₂²×D₆₇

yes

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

C₂×Dic₆₇

yes

—

118

C₄×Q₈

—

268

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

134

—

122

C₂×C₄²

—

123

C₂×C₄²

—

134

—

124

C₂×C₄²

C₂²×D₆₇

yes

characteristic

125

#103

—

268

—

126

#97

—

134

—

127

C₄×C₈

C₂×Dic₆₇

yes

—

128

#95

—

129

#95

—

134

—

130

#95

—

134

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

C₂²×Dic₆₇

yes

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

134

—

138

C₂²×C₄

D₄×D₆₇

yes

—

139

C₂²×C₄

C₂³×D₆₇

yes

characteristic

140

Q₁₆

—

141

C₂×C₈

—

142

C₂×C₈

—

134

—

143

C₂×C₈

—

134

—

144

C₂×C₈

C₂²×Dic₆₇

yes

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

268

—

147

C₄⋊C₄

—

134

—

148

C₄²

—

268

—

149

C₄²

—

134

—

150

C₄²

C₂²×Dic₆₇

yes

—

151

C₄²

#11761627

yes

—

152

C₂³

C₂×D₄×D₆₇

yes

characteristic

153

Q₈

—

154

Q₈

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

268

—

160

C₂×C₄

—

268

—

161

C₂×C₄

—

134

—

162

C₂×C₄

—

134

—

163

C₂×C₄

D₄×Dic₆₇

yes

—

164

C₂×C₄

C₂³×Dic₆₇

yes

characteristic

165

C₂×C₄

C₂×#11761627

yes

characteristic

166

C₂×C₄

#14524088

yes

—

167

C₈

—

168

C₈

—

268

—

169

C₂²

C₂×D₄×Dic₆₇

yes

characteristic

170

C₂²

#16863959

yes

—

171

C₂²

#16864209

yes

—

172

C₄

—

173

C₄

—

174

C₄

—

268

—

175

C₄

—

536

—

176

C₄

#16863871

yes

—

177

C₄

#16864000

yes

—

178

C₂

#22562935

yes

characteristic

179

C₂

#22569077

yes

—

180

C₂

#22571285

yes

—

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#494×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#494	D₆₇
hypocenter	C₆₇	#22277
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

subgroup

G/H

center

C₂³

C₂×D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂³×C₄

Frattini

C₂²×C₄

C₂³×D₆₇

Fitting

C₂×#494×C₆₇

C₂

socle, prime radical

C₂³×C₆₇

C₂²×D₄

cosocle

C₂²×C₄×C₆₇

C₂⁴

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

N	H ≅ G/N	type
18. #494×C₆₇	C₄	semidirect
97. C₆₇	#22277	semidirect
101. #494	Dic₆₇	semidirect

H ≅ G/N

type

18. #494×C₆₇

C₄

semidirect

97. C₆₇

#22277

semidirect

101. #494

Dic₆₇

semidirect

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

subgroup	index
#22562929	2
#22561030	2
#22560552	2
#22560340	2
#22559960	2
C₂×#494×C₆₇	2
#22277	67

subgroup

index

C₂×#494×C₆₇

Subgroups — #30869682

Conjugacy classes of subgroups

Complements

Maximal subgroups