Group #22573130 — subgroups

Subgroups — #22573130

Order 8576.

360 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 133 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	#16864124	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863802	C₂	1	2	yes	maximal, complemented
3	#16863799	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863696	C₂	1	2	yes	maximal
5	#16863688	C₂	1	2	yes	maximal, complemented
6	#16863678	C₂	1	1	yes	maximal, characteristic, complemented
7	#16863676	C₂	1	1	yes	maximal, characteristic
8	#16863527	C₂	1	1	yes	maximal, characteristic, complemented
9	#16863496	C₂	1	2	yes	maximal
10	C₄×#11761616	C₂	1	1	yes	maximal, characteristic
11	D₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂×#11761616	C₂²	1	1	yes	characteristic
13	#14524043	C₂²	1	1	yes	characteristic
14	#14524041	—	2	1	no	—
15	#14524035	C₂²	1	2	yes	—
16	#14524033	—	2	2	no	—
17	C₄×Dic₁₃₄	C₂²	1	2	yes	—
18	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
19	#14524008	—	2	1	no	—
20	#14523976	C₂²	1	2	yes	—
21	#14523976	C₂²	1	2	yes	—
22	#14523976	C₂²	1	2	yes	—
23	#14523974	C₂²	1	1	yes	characteristic
24	#14523974	C₂²	1	2	yes	—
25	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
26	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
27	#14523966	C₂²	1	2	yes	—
28	#14523966	C₂²	1	1	yes	characteristic
29	#14523964	C₂²	1	2	yes	—
30	#14523964	C₂²	1	1	yes	characteristic
31	#14523954	C₂²	1	2	yes	—
32	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
33	C₂²×Q₈×C₆₇	—	2	1	no	—
34	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
35	C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
36	C₂²⋊Q₈×C₆₇	—	2	1	no	—
37	C₂²⋊Q₈×C₆₇	—	2	2	no	—
38	C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
39	C₄×D₄×C₆₇	C₂²	1	2	yes	—
40	C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×Dic₁₃₄	—	2	1	no	—
42	C₂×Dic₁₃₄	C₂³	1	1	yes	characteristic
43	#11761616	Q₈	1	4	yes	complemented
44	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
45	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
46	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
47	C₄⋊Dic₆₇	—	2	1	no	—
48	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
49	C₄⋊Dic₆₇	C₂³	1	2	yes	—
50	Dic₆₇⋊C₄	—	2	2	no	—
51	Dic₆₇⋊C₄	—	2	1	no	—
52	Dic₆₇⋊C₄	—	2	2	no	—
53	Dic₆₇⋊C₄	—	2	2	no	—
54	C₄×Dic₆₇	—	2	2	no	—
55	C₄×Dic₆₇	—	2	1	no	—
56	C₂×Q₈×C₆₇	—	2	2	no	—
57	C₂×Q₈×C₆₇	—	2	1	no	—
58	C₂×Q₈×C₆₇	—	4	4	no	—
59	C₂×Q₈×C₆₇	D₄	1	1	yes	characteristic
60	C₂×D₄×C₆₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	—	2	2	no	—
62	C₂²×C₄×C₆₇	—	2	1	no	—
63	C₄⋊C₄×C₆₇	—	2	2	no	—
64	C₄⋊C₄×C₆₇	—	2	1	no	—
65	C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₆₇	D₄	1	2	yes	—
67	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
68	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
69	C₂²⋊C₄×C₆₇	—	2	2	no	—
70	C₂²⋊C₄×C₆₇	—	2	1	no	—
71	C₄²×C₆₇	C₂³	1	1	yes	characteristic
72	C₄²×C₆₇	C₂³	1	2	yes	—
73	C₂³×C₆₇	—	2	1	no	—
74	Q₈×C₆₇	—	2	2	no	—
75	Q₈×C₆₇	—	4	4	no	—
76	Q₈×C₆₇	—	4	2	no	—
77	D₄×C₆₇	—	2	1	no	—
78	D₄×C₆₇	C₂×Q₈	1	2	yes	—
79	C₂×C₄×C₆₇	—	2	2	no	—
80	C₂×C₄×C₆₇	—	2	1	no	—
81	C₂×C₄×C₆₇	—	4	4	no	—
82	C₂×C₄×C₆₇	—	4	2	no	—
83	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
88	C₂×Dic₆₇	—	2	1	no	—
89	C₂×Dic₆₇	—	2	1	no	—
90	C₂×Dic₆₇	—	2	2	no	—
91	Dic₁₃₄	—	2	1	no	—
92	Dic₁₃₄	—	4	2	no	—
93	Dic₁₃₄	C₂×Q₈	1	2	yes	complemented
94	C₆₇⋊C₈	—	2	1	no	—
95	C₆₇⋊C₈	—	2	2	no	—
96	C₆₇⋊C₈	C₂×Q₈	1	2	yes	complemented
97	C₂²×C₆₇	—	2	2	no	—
98	C₂²×C₆₇	—	4	1	no	—
99	C₂²×C₆₇	—	4	1	no	—
100	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
101	C₄×C₆₇	—	2	2	no	—
102	C₄×C₆₇	—	2	2	no	—
103	C₄×C₆₇	—	2	1	no	—
104	C₄×C₆₇	—	4	2	no	—
105	C₄×C₆₇	—	4	1	no	—
106	C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
107	C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
108	C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
109	Dic₆₇	—	2	2	no	—
110	Dic₆₇	—	4	2	no	—
111	Dic₆₇	—	4	1	no	—
112	C₂×C₆₇	—	4	2	no	—
113	C₂×C₆₇	D₄×Q₈	1	1	yes	characteristic
114	C₂×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₂×C₆₇	#578	1	1	yes	characteristic
116	#3386	—	67	1	no	maximal, Sylow
117	C₆₇	#3386	1	1	yes	characteristic, complemented, Sylow
118	#557	—	67	1	no	—
119	D₄×Q₈	D₆₇	1	1	yes	characteristic
120	#501	—	67	2	no	—
121	#500	—	67	1	no	—
122	#478	—	67	2	no	—
123	#477	—	67	2	no	—
124	#475	—	67	1	no	—
125	#474	—	67	1	no	—
126	#446	—	67	1	no	—
127	#440	—	67	2	no	—
128	C₄×SD₁₆	—	67	1	no	—
129	C₂²×Q₈	—	2	1	no	—
130	C₂×SD₁₆	—	67	1	no	—
131	C₄⋊Q₈	—	67	1	no	—
132	C₄⋊Q₈	—	134	1	no	—
133	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
134	C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
135	#125	—	67	2	no	—
136	#125	—	134	2	no	—
137	C₂²⋊Q₈	—	2	1	no	—
138	C₂²⋊Q₈	—	2	2	no	—
139	C₄×Q₈	—	67	1	no	—
140	C₄×Q₈	—	67	2	no	—
141	C₄×Q₈	—	134	1	no	—
142	C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
143	C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
144	C₄×D₄	C₂×D₆₇	1	2	yes	—
145	#107	—	67	2	no	—
146	#107	—	67	2	no	—
147	#107	—	67	2	no	—
148	#106	—	67	1	no	—
149	#106	—	67	2	no	—
150	C₄⋊C₈	—	67	1	no	—
151	C₄⋊C₈	—	67	2	no	—
152	#103	—	67	1	no	—
153	#103	—	67	2	no	—
154	#102	—	67	1	no	—
155	#102	—	67	2	no	—
156	#97	—	67	2	no	—
157	C₄×C₈	—	67	1	no	—
158	C₂×Q₈	—	2	2	no	—
159	C₂×Q₈	—	2	1	no	—
160	C₂×Q₈	—	4	4	no	—
161	C₂×Q₈	—	67	1	no	—
162	C₂×Q₈	—	134	1	no	—
163	C₂×Q₈	C₂²⋊D₆₇	1	1	yes	characteristic
164	C₂×D₄	C₂²×D₆₇	1	1	yes	characteristic
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	1	no	—
167	SD₁₆	—	67	4	no	—
168	C₂×C₈	—	67	1	no	—
169	C₂×C₈	—	67	1	no	—
170	C₂×C₈	—	67	2	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	1	no	—
173	C₄⋊C₄	—	67	1	no	—
174	C₄⋊C₄	—	67	2	no	—
175	C₄⋊C₄	—	134	1	no	—
176	C₄⋊C₄	—	134	2	no	—
177	C₄⋊C₄	—	134	2	no	—
178	C₄⋊C₄	—	134	2	no	—
179	C₄⋊C₄	—	134	1	no	—
180	C₄⋊C₄	C₂²⋊D₆₇	1	1	yes	characteristic
181	C₄⋊C₄	C₂²⋊D₆₇	1	2	yes	—
182	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
183	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	1	no	—
186	C₄²	—	134	2	no	—
187	C₄²	—	134	1	no	—
188	C₄²	C₂²×D₆₇	1	1	yes	characteristic
189	C₄²	C₂²×D₆₇	1	2	yes	—
190	C₂³	—	2	1	no	—
191	Q₈	—	2	2	no	—
192	Q₈	—	4	4	no	—
193	Q₈	—	4	2	no	—
194	Q₈	—	268	2	no	—
195	Q₈	—	67	2	no	—
196	Q₈	—	134	1	no	—
197	D₄	—	2	1	no	—
198	D₄	Q₈×D₆₇	1	2	yes	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	134	1	no	—
204	C₂×C₄	—	134	2	no	—
205	C₂×C₄	—	134	1	no	—
206	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
207	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
208	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
209	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
210	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
211	C₈	—	67	2	no	—
212	C₈	—	134	1	no	—
213	C₈	—	134	2	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	4	1	no	—
216	C₂²	—	4	1	no	—
217	C₂²	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	1	no	—
221	C₄	—	4	2	no	—
222	C₄	—	4	1	no	—
223	C₄	—	268	2	no	—
224	C₄	—	268	1	no	—
225	C₄	—	134	2	no	—
226	C₄	#14524069	1	2	yes	—
227	C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
228	C₄	#14524091	1	1	yes	characteristic
229	C₂	—	4	2	no	—
230	C₂	Q₈×C₂²⋊D₆₇	1	1	yes	characteristic
231	C₂	C₂×#14524069	1	1	yes	characteristic
232	C₂	#16864211	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16864124

C₂

yes

maximal, characteristic, complemented

#16863802

C₂

yes

maximal, complemented

#16863799

C₂

yes

maximal, characteristic, complemented

#16863696

C₂

yes

maximal

#16863688

C₂

yes

maximal, complemented

#16863678

C₂

yes

maximal, characteristic, complemented

#16863676

C₂

yes

maximal, characteristic

#16863527

C₂

yes

maximal, characteristic, complemented

#16863496

C₂

yes

maximal

C₄×#11761616

C₂

yes

maximal, characteristic

D₄×Q₈×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761616

C₂²

yes

characteristic

#14524043

C₂²

yes

characteristic

#14524041

—

#14524035

C₂²

yes

—

#14524033

—

C₄×Dic₁₃₄

C₂²

yes

—

C₄×Dic₁₃₄

C₂²

yes

characteristic

#14524008

—

#14523976

C₂²

yes

—

#14523976

C₂²

yes

—

#14523976

C₂²

yes

—

#14523974

C₂²

yes

characteristic

#14523974

C₂²

yes

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

#14523966

C₂²

yes

—

#14523966

C₂²

yes

characteristic

#14523964

C₂²

yes

—

#14523964

C₂²

yes

characteristic

#14523954

C₂²

yes

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₂²×Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

C₄⋊Q₈×C₆₇

C₂²

yes

—

C₂²⋊Q₈×C₆₇

—

C₂²⋊Q₈×C₆₇

—

C₄×Q₈×C₆₇

C₂²

yes

characteristic

C₄×D₄×C₆₇

C₂²

yes

—

C₄×D₄×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

C₂³

yes

characteristic

#11761616

Q₈

yes

complemented

C₂×C₆₇⋊C₈

C₂³

yes

characteristic

C₂×C₆₇⋊C₈

C₂³

yes

—

C₂×C₆₇⋊C₈

C₂³

yes

characteristic

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂³

yes

characteristic

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

D₄

yes

characteristic

C₂×D₄×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

D₄

yes

characteristic

C₄⋊C₄×C₆₇

D₄

yes

—

C₄⋊C₄×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

C₂³

yes

characteristic

C₂²⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₄²×C₆₇

C₂³

yes

—

C₂³×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

C₂×Q₈

yes

complemented

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

C₂×Q₈

yes

complemented

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

100

C₂²×C₆₇

C₂²×D₄

yes

characteristic

101

C₄×C₆₇

—

102

C₄×C₆₇

—

103

C₄×C₆₇

—

104

C₄×C₆₇

—

105

C₄×C₆₇

—

106

C₄×C₆₇

AGL(1,ℤ/8)

yes

—

107

C₄×C₆₇

C₂²×Q₈

yes

characteristic

108

C₄×C₆₇

2₋¹⁺⁴

yes

characteristic

109

Dic₆₇

—

110

Dic₆₇

—

111

Dic₆₇

—

112

C₂×C₆₇

—

113

C₂×C₆₇

D₄×Q₈

yes

characteristic

114

C₂×C₆₇

C₂×AGL(1,ℤ/8)

yes

characteristic

115

C₂×C₆₇

yes

116

—

117

C₆₇

#3386

yes

characteristic, complemented, Sylow

118

#557

—

119

D₄×Q₈

D₆₇

yes

characteristic

120

#501

—

121

#500

—

122

#478

—

123

#477

—

124

#475

—

125

#474

—

126

#446

—

127

#440

—

128

C₄×SD₁₆

—

129

C₂²×Q₈

—

130

C₂×SD₁₆

—

131

C₄⋊Q₈

—

132

C₄⋊Q₈

—

134

—

133

C₄⋊Q₈

C₂×D₆₇

yes

characteristic

134

C₄⋊Q₈

C₂×D₆₇

yes

—

135

#125

—

136

#125

—

134

—

137

C₂²⋊Q₈

—

138

C₂²⋊Q₈

—

139

C₄×Q₈

—

140

C₄×Q₈

—

141

C₄×Q₈

—

134

—

142

C₄×Q₈

C₂×D₆₇

yes

characteristic

143

C₄×D₄

C₂×D₆₇

yes

characteristic

144

C₄×D₄

C₂×D₆₇

yes

—

145

#107

—

146

#107

—

147

#107

—

148

#106

—

149

#106

—

150

C₄⋊C₈

—

151

C₄⋊C₈

—

152

#103

—

153

#103

—

154

#102

—

155

#102

—

156

#97

—

157

C₄×C₈

—

158

C₂×Q₈

—

159

C₂×Q₈

—

160

C₂×Q₈

—

161

C₂×Q₈

—

162

C₂×Q₈

—

134

—

163

C₂×Q₈

C₂²⋊D₆₇

yes

characteristic

164

C₂×D₄

C₂²×D₆₇

yes

characteristic

165

C₂²×C₄

—

166

C₂²×C₄

—

167

SD₁₆

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

134

—

176

C₄⋊C₄

—

134

—

177

C₄⋊C₄

—

134

—

178

C₄⋊C₄

—

134

—

179

C₄⋊C₄

—

134

—

180

C₄⋊C₄

C₂²⋊D₆₇

yes

characteristic

181

C₄⋊C₄

C₂²⋊D₆₇

yes

—

182

C₄⋊C₄

C₂²×D₆₇

yes

characteristic

183

C₄⋊C₄

C₂²×D₆₇

yes

—

184

C₂²⋊C₄

—

185

C₂²⋊C₄

—

186

C₄²

—

134

—

187

C₄²

—

134

—

188

C₄²

C₂²×D₆₇

yes

characteristic

189

C₄²

C₂²×D₆₇

yes

—

190

C₂³

—

191

Q₈

—

192

Q₈

—

193

Q₈

—

194

Q₈

—

268

—

195

Q₈

—

196

Q₈

—

134

—

197

D₄

—

198

D₄

Q₈×D₆₇

yes

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

134

—

204

C₂×C₄

—

134

—

205

C₂×C₄

—

134

—

206

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

207

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

208

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

209

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

210

C₂×C₄

C₂³×D₆₇

yes

characteristic

211

C₈

—

212

C₈

—

134

—

213

C₈

—

134

—

214

C₂²

—

215

C₂²

—

216

C₂²

—

217

C₂²

C₂²×C₂²⋊D₆₇

yes

characteristic

218

C₄

—

219

C₄

—

220

C₄

—

221

C₄

—

222

C₄

—

223

C₄

—

268

—

224

C₄

—

268

—

225

C₄

—

134

—

226

C₄

#14524069

yes

—

227

C₄

C₂×Q₈×D₆₇

yes

characteristic

228

C₄

#14524091

yes

characteristic

229

C₂

—

230

C₂

Q₈×C₂²⋊D₆₇

yes

characteristic

231

C₂

C₂×#14524069

yes

characteristic

232

C₂

#16864211

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	D₄×Q₈×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	D₄×Q₈	D₆₇
hypocenter	C₆₇	#3386
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

D₄×Q₈×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

hypercenter

D₄×Q₈

D₆₇

hypocenter

C₆₇

#3386

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

N	H ≅ G/N	type
1. #16864124	C₂	semidirect
2. #16863802	C₂	semidirect
3. #16863799	C₂	semidirect
5. #16863688	C₂	semidirect
6. #16863678	C₂	semidirect
8. #16863527	C₂	semidirect
43. #11761616	Q₈	semidirect
93. Dic₁₃₄	C₂×Q₈	semidirect
96. C₆₇⋊C₈	C₂×Q₈	semidirect
117. C₆₇	#3386	semidirect

H ≅ G/N

type

1. #16864124

C₂

semidirect

2. #16863802

C₂

semidirect

3. #16863799

C₂

semidirect

5. #16863688

C₂

semidirect

6. #16863678

C₂

semidirect

8. #16863527

C₂

semidirect

43. #11761616

Q₈

semidirect

93. Dic₁₃₄

C₂×Q₈

semidirect

96. C₆₇⋊C₈

C₂×Q₈

semidirect

117. C₆₇

#3386

semidirect

Subgroups — #22573130

Conjugacy classes of subgroups

Complements