Group #22572925 — subgroups

Subgroups — #22572925

Order 8576.

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

normal subgroups only

This group has 372 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	#16864116	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863754	C₂	1	2	yes	maximal, complemented
4	#16863747	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863690	C₂	1	2	yes	maximal, complemented
6	#16863676	C₂	1	1	yes	maximal, characteristic, complemented
7	#16863609	C₂	1	2	yes	maximal
8	#16863601	C₂	1	1	yes	maximal, characteristic, complemented
9	#16863526	C₂	1	1	yes	maximal, characteristic, complemented
10	#16863493	C₂	1	2	yes	maximal, complemented
11	C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₁₃₄	—	2	1	no	—
13	C₂×#11761616	C₂²	1	2	yes	—
14	#14524043	C₂²	1	1	yes	characteristic
15	#14524043	C₂²	1	1	yes	characteristic
16	#14524042	C₂²	1	2	yes	—
17	#14524036	—	2	2	no	—
18	#14524030	C₂²	1	2	yes	—
19	#14524022	—	2	1	no	—
20	#14524022	—	2	1	no	—
21	#14524020	—	2	2	no	—
22	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
23	#14524008	C₂²	1	2	yes	—
24	D₄×Dic₆₇	C₂²	1	2	yes	—
25	#14524002	—	2	1	no	—
26	#14523976	C₂²	1	2	yes	—
27	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
28	Dic₆₇⋊C₈	C₂²	1	2	yes	—
29	#14523968	C₂²	1	2	yes	—
30	#14523968	C₂²	1	2	yes	—
31	#14523966	C₂²	1	2	yes	—
32	#14523964	C₂²	1	2	yes	—
33	#14523953	C₂²	1	2	yes	—
34	C₂×D₈×C₆₇	C₂²	1	1	yes	characteristic
35	C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
37	#107×C₆₇	C₂²	1	1	yes	characteristic
38	#102×C₆₇	C₂²	1	1	yes	characteristic
39	#102×C₆₇	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×Dic₁₃₄	—	2	1	no	—
42	C₂×Dic₁₃₄	—	2	2	no	—
43	C₂×Dic₁₃₄	—	4	4	no	—
44	C₂×Dic₁₃₄	D₄	1	1	yes	characteristic
45	C₂×Dic₁₃₄	C₂³	1	2	yes	—
46	C₂²×Dic₆₇	—	2	2	no	—
47	#11761616	—	2	4	no	—
48	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
49	C₄⋊Dic₆₇	—	2	1	no	—
50	C₄⋊Dic₆₇	—	2	1	no	—
51	C₄⋊Dic₆₇	—	2	1	no	—
52	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
53	C₄⋊Dic₆₇	C₂³	1	2	yes	—
54	Dic₆₇⋊C₄	—	2	2	no	—
55	Dic₆₇⋊C₄	—	2	2	no	—
56	Dic₆₇⋊C₄	D₄	1	2	yes	—
57	C₂²⋊Dic₆₇	—	2	2	no	—
58	C₂²⋊Dic₆₇	—	2	2	no	—
59	C₂²⋊Dic₆₇	—	2	2	no	—
60	C₄×Dic₆₇	—	2	2	no	—
61	C₄×Dic₆₇	C₂³	1	2	yes	—
62	C₂×D₄×C₆₇	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₆₇	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	—	2	1	no	—
65	C₂²×C₄×C₆₇	—	2	1	no	—
66	D₈×C₆₇	—	2	2	no	—
67	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
71	C₂²⋊C₄×C₆₇	—	2	1	no	—
72	C₂²⋊C₄×C₆₇	—	2	1	no	—
73	C₄²×C₆₇	C₂³	1	1	yes	characteristic
74	C₂³×C₆₇	—	2	1	no	—
75	C₂³×C₆₇	—	2	1	no	—
76	D₄×C₆₇	—	2	1	no	—
77	D₄×C₆₇	—	2	1	no	—
78	D₄×C₆₇	—	2	1	no	—
79	D₄×C₆₇	#41	1	2	yes	—
80	C₂×C₄×C₆₇	—	2	1	no	—
81	C₂×C₄×C₆₇	—	2	1	no	—
82	C₂×C₄×C₆₇	—	4	2	no	—
83	C₂×C₄×C₆₇	—	4	2	no	—
84	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
87	C₂×Dic₆₇	—	2	2	no	—
88	C₂×Dic₆₇	—	2	2	no	—
89	C₂×Dic₆₇	—	4	4	no	—
90	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
91	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
92	Dic₁₃₄	—	2	2	no	—
93	Dic₁₃₄	—	2	2	no	—
94	Dic₁₃₄	—	2	2	no	—
95	Dic₁₃₄	—	4	4	no	—
96	Dic₁₃₄	—	4	2	no	—
97	C₈×C₆₇	—	2	1	no	—
98	C₈×C₆₇	—	2	1	no	—
99	C₆₇⋊C₈	—	2	2	no	—
100	C₂²×C₆₇	—	2	2	no	—
101	C₂²×C₆₇	—	4	1	no	—
102	C₂²×C₆₇	—	4	1	no	—
103	C₂²×C₆₇	—	4	1	no	—
104	C₂²×C₆₇	—	4	1	no	—
105	C₂²×C₆₇	—	4	1	no	—
106	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
107	C₄×C₆₇	—	2	1	no	—
108	C₄×C₆₇	—	4	1	no	—
109	C₄×C₆₇	—	4	1	no	—
110	C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
111	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
112	C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
113	Dic₆₇	—	2	2	no	—
114	Dic₆₇	—	2	2	no	—
115	Dic₆₇	—	4	2	no	—
116	Dic₆₇	—	4	2	no	—
117	C₂×C₆₇	—	4	2	no	—
118	C₂×C₆₇	—	8	1	no	—
119	C₂×C₆₇	#548	1	1	yes	characteristic
120	C₂×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₆₇	#578	1	1	yes	characteristic
122	#3367	—	67	1	no	maximal, Sylow
123	C₆₇	#3367	1	1	yes	characteristic, complemented, Sylow
124	#556	—	67	1	no	—
125	D₄×Q₈	—	67	1	no	—
126	#489	—	67	2	no	—
127	#487	—	67	1	no	—
128	#477	—	67	2	no	—
129	#474	—	67	1	no	—
130	#463	—	67	2	no	—
131	#461	—	67	1	no	—
132	#446	—	67	1	no	—
133	#440	—	67	2	no	—
134	C₄×D₈	D₆₇	1	1	yes	characteristic
135	C₂²×Q₈	—	134	1	no	—
136	C₂×SD₁₆	—	67	2	no	—
137	C₂×D₈	C₂×D₆₇	1	1	yes	characteristic
138	C₄⋊Q₈	—	67	2	no	—
139	C₄⋊Q₈	—	67	1	no	—
140	C₄⋊Q₈	—	67	1	no	—
141	#126	—	134	2	no	—
142	#124	—	67	2	no	—
143	C₂²⋊Q₈	—	134	2	no	—
144	C₂²⋊Q₈	—	134	1	no	—
145	C₂²⋊Q₈	—	134	1	no	—
146	C₄×Q₈	—	67	1	no	—
147	C₄×Q₈	—	67	2	no	—
148	C₄×D₄	—	67	2	no	—
149	C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
150	C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
151	#117	—	134	1	no	—
152	#107	—	67	2	no	—
153	#107	C₂×D₆₇	1	1	yes	characteristic
154	C₄⋊C₈	—	67	2	no	—
155	C₄⋊C₈	—	67	1	no	—
156	#103	—	67	2	no	—
157	#103	—	67	2	no	—
158	#103	—	67	2	no	—
159	#102	—	67	2	no	—
160	#102	C₂×D₆₇	1	1	yes	characteristic
161	#102	C₂×D₆₇	1	1	yes	characteristic
162	#97	—	67	2	no	—
163	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
164	C₂×Q₈	—	268	4	no	—
165	C₂×Q₈	—	67	1	no	—
166	C₂×Q₈	—	67	2	no	—
167	C₂×Q₈	—	134	2	no	—
168	C₂×Q₈	—	134	1	no	—
169	C₂×D₄	C₂²×D₆₇	1	1	yes	characteristic
170	C₂×D₄	C₂²×D₆₇	1	1	yes	characteristic
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	134	2	no	—
174	SD₁₆	—	134	4	no	—
175	D₈	—	2	2	no	—
176	C₂×C₈	—	67	2	no	—
177	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
179	C₄⋊C₄	—	67	1	no	—
180	C₄⋊C₄	—	67	2	no	—
181	C₄⋊C₄	—	67	2	no	—
182	C₄⋊C₄	—	134	2	no	—
183	C₄⋊C₄	—	134	2	no	—
184	C₄⋊C₄	—	134	1	no	—
185	C₄⋊C₄	—	134	1	no	—
186	C₄⋊C₄	—	134	1	no	—
187	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
188	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
189	C₂²⋊C₄	—	2	1	no	—
190	C₂²⋊C₄	—	2	1	no	—
191	C₂²⋊C₄	—	134	2	no	—
192	C₂²⋊C₄	—	134	2	no	—
193	C₂²⋊C₄	—	134	2	no	—
194	C₄²	—	67	2	no	—
195	C₄²	—	134	2	no	—
196	C₄²	C₂²×D₆₇	1	1	yes	characteristic
197	C₂³	—	2	1	no	—
198	C₂³	—	2	1	no	—
199	Q₈	—	268	4	no	—
200	Q₈	—	268	2	no	—
201	Q₈	—	134	2	no	—
202	Q₈	—	134	2	no	—
203	Q₈	—	134	2	no	—
204	D₄	—	2	1	no	—
205	D₄	—	2	1	no	—
206	D₄	—	2	1	no	—
207	D₄	#11761628	1	2	yes	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	268	4	no	—
213	C₂×C₄	—	67	2	no	—
214	C₂×C₄	—	67	2	no	—
215	C₂×C₄	—	134	2	no	—
216	C₂×C₄	—	134	2	no	—
217	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
218	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
219	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
220	C₈	—	2	1	no	—
221	C₈	—	2	1	no	—
222	C₈	—	134	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	1	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	4	1	no	—
227	C₂²	—	4	1	no	—
228	C₂²	—	4	1	no	—
229	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	4	1	no	—
232	C₄	—	4	1	no	—
233	C₄	—	268	2	no	—
234	C₄	—	268	2	no	—
235	C₄	—	134	2	no	—
236	C₄	—	134	2	no	—
237	C₄	#14524067	1	2	yes	—
238	C₄	C₂×#11761628	1	1	yes	characteristic
239	C₄	#14524090	1	1	yes	characteristic
240	C₂	—	4	2	no	—
241	C₂	—	8	1	no	—
242	C₂	#16864055	1	1	yes	characteristic
243	C₂	C₂×#14524067	1	1	yes	characteristic
244	C₂	#16864210	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16864116

C₂

yes

maximal, characteristic, complemented

D₄×Dic₁₃₄

C₂

yes

maximal, characteristic, complemented

#16863754

C₂

yes

maximal, complemented

#16863747

C₂

yes

maximal, characteristic, complemented

#16863690

C₂

yes

maximal, complemented

#16863676

C₂

yes

maximal, characteristic, complemented

#16863609

C₂

yes

maximal

#16863601

C₂

yes

maximal, characteristic, complemented

#16863526

C₂

yes

maximal, characteristic, complemented

#16863493

C₂

yes

maximal, complemented

C₄×D₈×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

—

C₂×#11761616

C₂²

yes

—

#14524043

C₂²

yes

characteristic

#14524043

C₂²

yes

characteristic

#14524042

C₂²

yes

—

#14524036

—

#14524030

C₂²

yes

—

#14524022

—

#14524022

—

#14524020

—

C₄×Dic₁₃₄

C₂²

yes

characteristic

#14524008

C₂²

yes

—

D₄×Dic₆₇

C₂²

yes

—

#14524002

—

#14523976

C₂²

yes

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

Dic₆₇⋊C₈

C₂²

yes

—

#14523968

C₂²

yes

—

#14523968

C₂²

yes

—

#14523966

C₂²

yes

—

#14523964

C₂²

yes

—

#14523953

C₂²

yes

—

C₂×D₈×C₆₇

C₂²

yes

characteristic

C₄×D₄×C₆₇

C₂²

yes

characteristic

C₄×D₄×C₆₇

C₂²

yes

characteristic

#107×C₆₇

C₂²

yes

characteristic

#102×C₆₇

C₂²

yes

characteristic

#102×C₆₇

C₂²

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

D₄

yes

characteristic

C₂×Dic₁₃₄

C₂³

yes

—

C₂²×Dic₆₇

—

#11761616

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

characteristic

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

D₄

yes

—

C₂²⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

C₂³

yes

—

C₂×D₄×C₆₇

C₂³

yes

characteristic

C₂×D₄×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

D₈×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

C₂³

yes

characteristic

C₂²⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

—

C₂³×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×D₄

yes

—

C₂×Dic₆₇

C₂×D₄

yes

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

100

C₂²×C₆₇

—

101

C₂²×C₆₇

—

102

C₂²×C₆₇

—

103

C₂²×C₆₇

—

104

C₂²×C₆₇

—

105

C₂²×C₆₇

—

106

C₂²×C₆₇

C₂²×D₄

yes

characteristic

107

C₄×C₆₇

—

108

C₄×C₆₇

—

109

C₄×C₆₇

—

110

C₄×C₆₇

AGL(1,ℤ/8)

yes

—

111

C₄×C₆₇

C₂×#41

yes

characteristic

112

C₄×C₆₇

2₋¹⁺⁴

yes

characteristic

113

Dic₆₇

—

114

Dic₆₇

—

115

Dic₆₇

—

116

Dic₆₇

—

117

C₂×C₆₇

—

118

C₂×C₆₇

—

119

C₂×C₆₇

#548

yes

characteristic

120

C₂×C₆₇

C₂×AGL(1,ℤ/8)

yes

characteristic

121

C₂×C₆₇

yes

122

—

123

C₆₇

#3367

yes

characteristic, complemented, Sylow

124

#556

—

125

D₄×Q₈

—

126

#489

—

127

#487

—

128

#477

—

129

#474

—

130

#463

—

131

#461

—

132

#446

—

133

#440

—

134

C₄×D₈

D₆₇

yes

characteristic

135

C₂²×Q₈

—

134

—

136

C₂×SD₁₆

—

137

C₂×D₈

C₂×D₆₇

yes

characteristic

138

C₄⋊Q₈

—

139

C₄⋊Q₈

—

140

C₄⋊Q₈

—

141

#126

—

134

—

142

#124

—

143

C₂²⋊Q₈

—

134

—

144

C₂²⋊Q₈

—

134

—

145

C₂²⋊Q₈

—

134

—

146

C₄×Q₈

—

147

C₄×Q₈

—

148

C₄×D₄

—

149

C₄×D₄

C₂×D₆₇

yes

characteristic

150

C₄×D₄

C₂×D₆₇

yes

characteristic

151

#117

—

134

—

152

#107

—

153

#107

C₂×D₆₇

yes

characteristic

154

C₄⋊C₈

—

155

C₄⋊C₈

—

156

#103

—

157

#103

—

158

#103

—

159

#102

—

160

#102

C₂×D₆₇

yes

characteristic

161

#102

C₂×D₆₇

yes

characteristic

162

#97

—

163

C₄×C₈

C₂×D₆₇

yes

characteristic

164

C₂×Q₈

—

268

—

165

C₂×Q₈

—

166

C₂×Q₈

—

167

C₂×Q₈

—

134

—

168

C₂×Q₈

—

134

—

169

C₂×D₄

C₂²×D₆₇

yes

characteristic

170

C₂×D₄

C₂²×D₆₇

yes

characteristic

171

C₂²×C₄

—

172

C₂²×C₄

—

173

C₂²×C₄

—

134

—

174

SD₁₆

—

134

—

175

D₈

—

176

C₂×C₈

—

177

C₂×C₈

C₂²×D₆₇

yes

characteristic

178

C₂×C₈

C₂²×D₆₇

yes

characteristic

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

134

—

183

C₄⋊C₄

—

134

—

184

C₄⋊C₄

—

134

—

185

C₄⋊C₄

—

134

—

186

C₄⋊C₄

—

134

—

187

C₄⋊C₄

C₂²×D₆₇

yes

characteristic

188

C₄⋊C₄

C₂²×D₆₇

yes

characteristic

189

C₂²⋊C₄

—

190

C₂²⋊C₄

—

191

C₂²⋊C₄

—

134

—

192

C₂²⋊C₄

—

134

—

193

C₂²⋊C₄

—

134

—

194

C₄²

—

195

C₄²

—

134

—

196

C₄²

C₂²×D₆₇

yes

characteristic

197

C₂³

—

198

C₂³

—

199

Q₈

—

268

—

200

Q₈

—

268

—

201

Q₈

—

134

—

202

Q₈

—

134

—

203

Q₈

—

134

—

204

D₄

—

205

D₄

—

206

D₄

—

207

D₄

#11761628

yes

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

—

268

—

213

C₂×C₄

—

214

C₂×C₄

—

215

C₂×C₄

—

134

—

216

C₂×C₄

—

134

—

217

C₂×C₄

D₄×D₆₇

yes

characteristic

218

C₂×C₄

D₄×D₆₇

yes

characteristic

219

C₂×C₄

C₂³×D₆₇

yes

characteristic

220

C₈

—

221

C₈

—

222

C₈

—

134

—

223

C₂²

—

224

C₂²

—

225

C₂²

—

226

C₂²

—

227

C₂²

—

228

C₂²

—

229

C₂²

C₂×D₄×D₆₇

yes

characteristic

230

C₄

—

231

C₄

—

232

C₄

—

233

C₄

—

268

—

234

C₄

—

268

—

235

C₄

—

134

—

236

C₄

—

134

—

237

C₄

#14524067

yes

—

238

C₄

C₂×#11761628

yes

characteristic

239

C₄

#14524090

yes

characteristic

240

C₂

—

241

C₂

—

242

C₂

#16864055

yes

characteristic

243

C₂

C₂×#14524067

yes

characteristic

244

C₂

#16864210

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₄×D₈×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	C₄×D₈	D₆₇
hypocenter	C₆₇	#3367
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₈

subgroup

G/H

center

C₂²

C₂×D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₄×D₈×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

hypercenter

C₄×D₈

D₆₇

hypocenter

C₆₇

#3367

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₈

N	H ≅ G/N	type
1. #16864116	C₂	semidirect
2. D₄×Dic₁₃₄	C₂	semidirect
3. #16863754	C₂	semidirect
4. #16863747	C₂	semidirect
5. #16863690	C₂	semidirect
6. #16863676	C₂	semidirect
8. #16863601	C₂	semidirect
9. #16863526	C₂	semidirect
10. #16863493	C₂	semidirect
123. C₆₇	#3367	semidirect

H ≅ G/N

type

1. #16864116

C₂

semidirect

2. D₄×Dic₁₃₄

C₂

semidirect

3. #16863754

C₂

semidirect

4. #16863747

C₂

semidirect

5. #16863690

C₂

semidirect

6. #16863676

C₂

semidirect

8. #16863601

C₂

semidirect

9. #16863526

C₂

semidirect

10. #16863493

C₂

semidirect

123. C₆₇

#3367

semidirect

Subgroups — #22572925

Conjugacy classes of subgroups

Complements