Group #22570556 — subgroups

Subgroups — #22570556

Order 8576.

392 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 117 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	#16863952	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863947	C₂	1	1	yes	maximal, characteristic
3	#16863764	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863762	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863761	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863757	C₂	1	4	yes	maximal, complemented
7	#16863391	C₂	1	4	yes	maximal, complemented
8	#97×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#491×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11761629	C₂²	1	2	yes	—
11	C₂×C₈×D₆₇	C₂²	1	2	yes	—
12	#14524042	C₂²	1	2	yes	—
13	#14524040	C₂²	1	1	yes	characteristic
14	#14524035	C₂²	1	1	yes	characteristic
15	#14524032	C₂²	1	2	yes	—
16	#14524021	—	2	2	no	—
17	#14524018	—	2	2	no	—
18	#14524008	—	2	2	no	—
19	#14524005	—	2	2	no	—
20	#14524001	C₂²	1	2	yes	—
21	C₄²×D₆₇	C₂²	1	1	yes	characteristic
22	#14523967	C₂²	1	4	yes	—
23	#14523965	C₂²	1	4	yes	—
24	#14523963	C₂²	1	4	yes	—
25	#14523954	C₂²	1	1	yes	characteristic
26	#14523953	C₂²	1	2	yes	—
27	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
28	#125×C₆₇	C₂²	1	1	yes	characteristic
29	#103×C₆₇	C₂²	1	4	yes	—
30	#97×C₆₇	C₂²	1	1	yes	characteristic
31	#11761629	—	2	8	no	—
32	C₂×Dic₁₃₄	—	2	2	no	—
33	C₂×D₂₆₈	—	2	2	no	—
34	C₂×D₂₆₈	—	2	2	no	—
35	C₂×D₂₆₈	C₂³	1	2	yes	—
36	C₂×C₄×D₆₇	—	2	2	no	—
37	C₂×C₄×D₆₇	D₄	1	2	yes	—
38	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
39	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
40	C₈×D₆₇	—	2	4	no	—
41	C₄⋊Dic₆₇	C₂³	1	2	yes	—
42	Dic₆₇⋊C₄	—	2	2	no	—
43	Dic₆₇⋊C₄	—	2	2	no	—
44	(C₂×D₆₇)⋊C₄	—	2	2	no	—
45	(C₂×D₆₇)⋊C₄	—	2	4	no	—
46	C₄×Dic₆₇	—	2	2	no	—
47	C₄×Dic₆₇	D₄	1	2	yes	—
48	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
49	C₂×Q₈×C₆₇	C₂³	1	2	yes	—
50	C₂×C₈×C₆₇	C₂³	1	2	yes	—
51	C₄⋊C₄×C₆₇	—	2	2	no	—
52	C₄⋊C₄×C₆₇	—	2	2	no	—
53	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
54	C₄²×C₆₇	C₂³	1	1	yes	characteristic
55	C₂²×D₆₇	—	2	2	no	—
56	C₂²×D₆₇	C₂×D₄	1	1	yes	characteristic
57	Q₈×C₆₇	—	2	4	no	—
58	C₂×C₄×C₆₇	—	2	2	no	—
59	C₂×C₄×C₆₇	—	2	2	no	—
60	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
62	C₂×Dic₆₇	—	2	2	no	—
63	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
64	C₂×Dic₆₇	C₂×D₄	1	1	yes	characteristic
65	D₂₆₈	—	2	4	no	—
66	D₂₆₈	—	4	4	no	—
67	D₂₆₈	—	4	4	no	—
68	C₄×D₆₇	—	4	2	no	—
69	C₄×D₆₇	—	4	4	no	—
70	C₄×D₆₇	#41	1	4	yes	—
71	Dic₁₃₄	—	4	4	no	—
72	C₈×C₆₇	—	2	2	no	—
73	C₆₇⋊C₈	—	2	2	no	—
74	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
75	C₂×D₆₇	—	2	2	no	—
76	C₂×D₆₇	—	4	2	no	—
77	C₂×D₆₇	—	4	4	no	—
78	C₂×D₆₇	#124	1	2	yes	—
79	C₄×C₆₇	—	2	2	no	—
80	C₄×C₆₇	—	4	2	no	—
81	C₄×C₆₇	—	4	2	no	—
82	C₄×C₆₇	C₂×#41	1	2	yes	—
83	Dic₆₇	—	2	2	no	—
84	Dic₆₇	—	4	2	no	—
85	Dic₆₇	#124	1	2	yes	—
86	C₂×C₆₇	C₂×#124	1	1	yes	characteristic
87	C₂×C₆₇	#575	1	2	yes	—
88	D₆₇	—	2	2	no	—
89	D₆₇	—	8	2	no	—
90	#3135	—	67	1	no	maximal, Sylow
91	C₆₇	#3135	1	1	yes	characteristic, complemented, Sylow
92	#533	—	67	1	no	—
93	#532	—	67	1	no	—
94	#491	—	67	1	no	—
95	#491	D₆₇	1	1	yes	characteristic, complemented
96	#490	—	67	1	no	—
97	#490	—	67	1	no	—
98	#489	—	67	4	no	—
99	#416	—	67	4	no	—
100	C₂×#97	—	67	1	no	—
101	C₂×#41	—	67	2	no	—
102	C₂²×C₈	—	67	2	no	—
103	C₄⋊Q₈	—	67	2	no	—
104	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
105	#127	—	67	1	no	—
106	#125	—	67	1	no	—
107	#125	C₂×D₆₇	1	1	yes	characteristic
108	#124	—	67	2	no	—
109	C₂²⋊Q₈	—	134	2	no	—
110	#121	—	134	2	no	—
111	C₄×Q₈	—	134	2	no	—
112	C₄×D₄	—	134	2	no	—
113	#117	—	67	2	no	—
114	C₂×C₄²	—	67	1	no	—
115	#103	—	67	4	no	—
116	#103	C₂×D₆₇	1	4	yes	—
117	#102	—	67	4	no	—
118	#102	—	67	4	no	—
119	#97	—	67	2	no	—
120	#97	—	67	1	no	—
121	#97	C₂×D₆₇	1	1	yes	characteristic
122	#41	—	134	8	no	—
123	C₂×Q₈	—	134	2	no	—
124	C₂×Q₈	C₂²×D₆₇	1	2	yes	—
125	C₂×D₄	—	67	2	no	—
126	C₂×D₄	—	134	2	no	—
127	C₂×D₄	—	134	2	no	—
128	C₂²×C₄	—	67	2	no	—
129	C₂²×C₄	—	67	1	no	—
130	C₂²×C₄	—	134	2	no	—
131	C₂×C₈	—	67	2	no	—
132	C₂×C₈	—	134	4	no	—
133	C₂×C₈	C₂²×D₆₇	1	2	yes	—
134	C₄⋊C₄	—	2	2	no	—
135	C₄⋊C₄	—	2	2	no	—
136	C₄⋊C₄	—	67	2	no	—
137	C₄⋊C₄	—	134	2	no	—
138	C₄⋊C₄	—	134	2	no	—
139	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
140	C₂²⋊C₄	—	134	4	no	—
141	C₂²⋊C₄	—	134	2	no	—
142	C₄²	—	67	2	no	—
143	C₄²	—	67	1	no	—
144	C₄²	—	134	2	no	—
145	C₄²	C₂²×D₆₇	1	1	yes	characteristic
146	C₂³	—	67	1	no	—
147	C₂³	—	134	2	no	—
148	Q₈	—	2	4	no	—
149	Q₈	—	268	4	no	—
150	D₄	—	268	4	no	—
151	D₄	—	268	4	no	—
152	D₄	—	134	4	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	268	2	no	—
156	C₂×C₄	—	268	4	no	—
157	C₂×C₄	—	67	1	no	—
158	C₂×C₄	—	67	4	no	—
159	C₂×C₄	—	67	2	no	—
160	C₂×C₄	—	134	2	no	—
161	C₂×C₄	D₄×D₆₇	1	2	yes	—
162	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
163	C₈	—	2	2	no	—
164	C₈	—	134	2	no	—
165	C₂²	—	268	2	no	—
166	C₂²	—	268	4	no	—
167	C₂²	—	67	2	no	—
168	C₂²	—	134	2	no	—
169	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
170	C₄	—	2	2	no	—
171	C₄	—	4	2	no	—
172	C₄	—	4	2	no	—
173	C₄	—	268	2	no	—
174	C₄	—	67	2	no	—
175	C₄	—	134	2	no	—
176	C₄	#41×D₆₇	1	2	yes	—
177	C₂	—	536	2	no	—
178	C₂	—	134	2	no	—
179	C₂	#124×D₆₇	1	1	yes	characteristic
180	C₂	#16864195	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16863952

C₂

yes

maximal, characteristic, complemented

#16863947

C₂

yes

maximal, characteristic

#16863764

C₂

yes

maximal, characteristic, complemented

#16863762

C₂

yes

maximal, characteristic, complemented

#16863761

C₂

yes

maximal, characteristic, complemented

#16863757

C₂

yes

maximal, complemented

#16863391

C₂

yes

maximal, complemented

#97×D₆₇

C₂

yes

maximal, characteristic, complemented

#491×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#11761629

C₂²

yes

—

C₂×C₈×D₆₇

C₂²

yes

—

#14524042

C₂²

yes

—

#14524040

C₂²

yes

characteristic

#14524035

C₂²

yes

characteristic

#14524032

C₂²

yes

—

—

—

—

—

C₂²

yes

—

C₄²×D₆₇

C₂²

yes

characteristic

#14523967

C₂²

yes

—

#14523965

C₂²

yes

—

#14523963

C₂²

yes

—

#14523954

C₂²

yes

characteristic

#14523953

C₂²

yes

—

C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

#125×C₆₇

C₂²

yes

characteristic

#103×C₆₇

C₂²

yes

—

#97×C₆₇

C₂²

yes

characteristic

#11761629

—

C₂×Dic₁₃₄

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

C₂³

yes

—

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

D₄

yes

—

C₂×C₄×D₆₇

C₂³

yes

characteristic

C₂×C₆₇⋊C₈

C₂³

yes

—

C₈×D₆₇

—

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

(C₂×D₆₇)⋊C₄

—

(C₂×D₆₇)⋊C₄

—

C₄×Dic₆₇

—

C₄×Dic₆₇

D₄

yes

—

C₄×Dic₆₇

C₂³

yes

characteristic

C₂×Q₈×C₆₇

C₂³

yes

—

C₂×C₈×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂³

yes

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×D₄

yes

—

C₂×Dic₆₇

C₂×D₄

yes

characteristic

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₄×D₆₇

—

C₄×D₆₇

—

C₄×D₆₇

#41

yes

—

Dic₁₃₄

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

#124

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×#41

yes

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

#124

yes

—

C₂×C₆₇

C₂×#124

yes

characteristic

C₂×C₆₇

#575

yes

—

D₆₇

—

D₆₇

—

#3135

—

maximal, Sylow

C₆₇

#3135

yes

characteristic, complemented, Sylow

—

—

—

D₆₇

yes

characteristic, complemented

—

—

—

—

100

C₂×#97

—

101

C₂×#41

—

102

C₂²×C₈

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

C₂×D₆₇

yes

characteristic

105

#127

—

106

#125

—

107

#125

C₂×D₆₇

yes

characteristic

108

#124

—

109

C₂²⋊Q₈

—

134

—

110

#121

—

134

—

111

C₄×Q₈

—

134

—

112

C₄×D₄

—

134

—

113

#117

—

114

C₂×C₄²

—

115

#103

—

116

#103

C₂×D₆₇

yes

—

117

#102

—

118

#102

—

119

#97

—

120

#97

—

121

#97

C₂×D₆₇

yes

characteristic

122

#41

—

134

—

123

C₂×Q₈

—

134

—

124

C₂×Q₈

C₂²×D₆₇

yes

—

125

C₂×D₄

—

126

C₂×D₄

—

134

—

127

C₂×D₄

—

134

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

134

—

131

C₂×C₈

—

132

C₂×C₈

—

134

—

133

C₂×C₈

C₂²×D₆₇

yes

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

134

—

138

C₄⋊C₄

—

134

—

139

C₄⋊C₄

C₂²×D₆₇

yes

—

140

C₂²⋊C₄

—

134

—

141

C₂²⋊C₄

—

134

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

134

—

145

C₄²

C₂²×D₆₇

yes

characteristic

146

C₂³

—

147

C₂³

—

134

—

148

Q₈

—

149

Q₈

—

268

—

150

D₄

—

268

—

151

D₄

—

268

—

152

D₄

—

134

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

268

—

156

C₂×C₄

—

268

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

134

—

161

C₂×C₄

D₄×D₆₇

yes

—

162

C₂×C₄

C₂³×D₆₇

yes

characteristic

163

C₈

—

164

C₈

—

134

—

165

C₂²

—

268

—

166

C₂²

—

268

—

167

C₂²

—

168

C₂²

—

134

—

169

C₂²

C₂×D₄×D₆₇

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

268

—

174

C₄

—

175

C₄

—

134

—

176

C₄

#41×D₆₇

yes

—

177

C₂

—

536

—

178

C₂

—

134

—

179

C₂

#124×D₆₇

yes

characteristic

180

C₂

#16864195

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	#491×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	#491	D₆₇
hypocenter	C₆₇	#3135
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #491

subgroup

G/H

center

C₂²

C₂×D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

#491×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #491

N	H ≅ G/N	type
1. #16863952	C₂	semidirect
3. #16863764	C₂	semidirect
4. #16863762	C₂	semidirect
5. #16863761	C₂	semidirect
6. #16863757	C₂	semidirect
7. #16863391	C₂	semidirect
8. #97×D₆₇	C₂	semidirect
9. #491×C₆₇	C₂	semidirect
91. C₆₇	#3135	semidirect
95. #491	D₆₇	semidirect

H ≅ G/N

type

1. #16863952

C₂

semidirect

3. #16863764

C₂

semidirect

4. #16863762

C₂

semidirect

5. #16863761

C₂

semidirect

6. #16863757

C₂

semidirect

7. #16863391

C₂

semidirect

8. #97×D₆₇

C₂

semidirect

9. #491×C₆₇

C₂

semidirect

91. C₆₇

#3135

semidirect

95. #491

D₆₇

semidirect

Subgroups — #22570556

Conjugacy classes of subgroups

Complements