Finite Group Databasebeta

← C2×AΣL(1,𝔽16)×C67 #22556549

Subgroups — C2×AΣL(1,𝔽16)×C67 #22556549

Order 8576.

438 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#120×C67C211yesmaximal, characteristic
2C2×#100×C67C211yesmaximal, characteristic, complemented
3C2×#99×C67C211yesmaximal, characteristic, complemented
4AΣL(1,𝔽16)×C67C214yesmaximal, direct product factor
5C25×C67C411yescharacteristic, complemented
6C22×D4×C67—21no—
7C22×D4×C67C2211yescharacteristic
8C2×M16×C67—21no—
9#120×C67—21no—
10#120×C67—21no—
11#120×C67C412yescomplemented
12#120×C67C2212yes—
13C2×C22⋊C4×C67—21no—
14C2×C22⋊C4×C67—21no—
15C2×C22⋊C4×C67C411yescharacteristic, complemented
16#100×C67—21no—
17#100×C67C2212yescomplemented
18#99×C67—21no—
19#99×C67C2212yescomplemented
20C24×C67—21no—
21C24×C67—21no—
22C24×C67—44no—
23C24×C67—41no—
24C24×C67—41no—
25C24×C67C2×C411yescharacteristic
26C24×C67C2×C412yescomplemented
27C24×C67D411yescharacteristic
28C2×D4×C67—22no—
29C2×D4×C67—22no—
30C2×D4×C67—41no—
31C2×D4×C67—41no—
32C2×D4×C67—44no—
33C2×D4×C67C2×C411yescharacteristic
34C2×D4×C67D411yescharacteristic
35C2×D4×C67D411yescharacteristic
36C2×D4×C67C2311yescharacteristic
37C22×C4×C67—21no—
38C22×C4×C67—41no—
39C22×C4×C67D411yescharacteristic
40M16×C67—22no—
41M16×C67—41no—
42C2×C8×C67—41no—
43C22⋊C4×C67—21no—
44C22⋊C4×C67—22no—
45C22⋊C4×C67—22no—
46C22⋊C4×C67—22no—
47C22⋊C4×C67—41no—
48C22⋊C4×C67C2×C412yescomplemented
49C23×C67—24no—
50C23×C67—21no—
51C23×C67—21no—
52C23×C67—21no—
53C23×C67—22no—
54C23×C67—21no—
55C23×C67—42no—
56C23×C67—44no—
57C23×C67—42no—
58C23×C67—42no—
59C23×C67—44no—
60C23×C67—42no—
61C23×C67—42no—
62C23×C67—44no—
63C23×C67—48no—
64C23×C67—44no—
65C23×C67—41no—
66C23×C67—41no—
67C23×C67C22⋊C411yescharacteristic
68C23×C67C22⋊C411yescharacteristic
69C23×C67C22⋊C411yescharacteristic
70C23×C67C22×C411yescharacteristic
71C23×C67C2×D411yescharacteristic
72D4×C67—41no—
73D4×C67—41no—
74D4×C67—41no—
75D4×C67—41no—
76D4×C67—44no—
77D4×C67—44no—
78C2×C4×C67—21no—
79C2×C4×C67—21no—
80C2×C4×C67—22no—
81C2×C4×C67—42no—
82C2×C4×C67—42no—
83C2×C4×C67—82no—
84C2×C4×C67C22⋊C411yescharacteristic
85C2×C4×C67C2×D411yescharacteristic
86C8×C67—42no—
87C22×C67—22no—
88C22×C67—24no—
89C22×C67—21no—
90C22×C67—21no—
91C22×C67—21no—
92C22×C67—21no—
93C22×C67—41no—
94C22×C67—41no—
95C22×C67—41no—
96C22×C67—41no—
97C22×C67—42no—
98C22×C67—44no—
99C22×C67—42no—
100C22×C67—44no—
101C22×C67—44no—
102C22×C67—48no—
103C22×C67—42no—
104C22×C67—44no—
105C22×C67—42no—
106C22×C67—42no—
107C22×C67—81no—
108C22×C67#9911yescharacteristic
109C22×C67#9911yescharacteristic
110C22×C67C2×C22⋊C411yescharacteristic
111C4×C67—21no—
112C4×C67—21no—
113C4×C67—42no—
114C4×C67—82no—
115C2×C67—21no—
116C2×C67—21no—
117C2×C67—41no—
118C2×C67—44no—
119C2×C67—41no—
120C2×C67—41no—
121C2×C67—41no—
122C2×C67AΣL(1,𝔽16)12yesdirect product factor
123C2×C67C2×#9911yescharacteristic
124C2×AΣL(1,𝔽16)C6711yesmaximal, characteristic, direct product factor, Sylow
125C67C2×AΣL(1,𝔽16)11yescharacteristic, direct product factor, Sylow
126C2×#120C2×C6711yescharacteristic
127C2×#100C2×C6711yescharacteristic, complemented
128C2×#99C2×C6711yescharacteristic, complemented
129AΣL(1,𝔽16)C2×C6714yesdirect product factor
130C25C4×C6711yescharacteristic, complemented
131C22×D4—21no—
132C22×D4C22×C6711yescharacteristic
133C2×M16—21no—
134#120—21no—
135#120—21no—
136#120C4×C6712yescomplemented
137#120C22×C6712yes—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4C4×C6711yescharacteristic, complemented
141#100—21no—
142#100C22×C6712yescomplemented
143#99—21no—
144#99C22×C6712yescomplemented
145C24—21no—
146C24—21no—
147C24—44no—
148C24—41no—
149C24—41no—
150C24C2×C4×C6711yescharacteristic
151C24C2×C4×C6712yescomplemented
152C24D4×C6711yescharacteristic
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—44no—
158C2×D4C2×C4×C6711yescharacteristic
159C2×D4D4×C6711yescharacteristic
160C2×D4D4×C6711yescharacteristic
161C2×D4C23×C6711yescharacteristic
162C22×C4—21no—
163C22×C4—41no—
164C22×C4D4×C6711yescharacteristic
165M16—22no—
166M16—41no—
167C2×C8—41no—
168C22⋊C4—21no—
169C22⋊C4—22no—
170C22⋊C4—22no—
171C22⋊C4—22no—
172C22⋊C4—41no—
173C22⋊C4C2×C4×C6712yescomplemented
174C23—24no—
175C23—22no—
176C23—21no—
177C23—21no—
178C23—21no—
179C23—21no—
180C23—42no—
181C23—44no—
182C23—42no—
183C23—42no—
184C23—44no—
185C23—42no—
186C23—42no—
187C23—44no—
188C23—48no—
189C23—44no—
190C23—41no—
191C23—41no—
192C23C22⋊C4×C6711yescharacteristic
193C23C22⋊C4×C6711yescharacteristic
194C23C22⋊C4×C6711yescharacteristic
195C23C22×C4×C6711yescharacteristic
196C23C2×D4×C6711yescharacteristic
197D4—41no—
198D4—41no—
199D4—41no—
200D4—41no—
201D4—44no—
202D4—44no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—82no—
209C2×C4C22⋊C4×C6711yescharacteristic
210C2×C4C2×D4×C6711yescharacteristic
211C8—42no—
212C22—22no—
213C22—24no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—42no—
223C22—44no—
224C22—42no—
225C22—44no—
226C22—44no—
227C22—48no—
228C22—42no—
229C22—44no—
230C22—42no—
231C22—42no—
232C22—81no—
233C22#99×C6711yescharacteristic
234C22#99×C6711yescharacteristic
235C22C2×C22⋊C4×C6711yescharacteristic
236C4—21no—
237C4—21no—
238C4—42no—
239C4—82no—
240C2—21no—
241C2—21no—
242C2—41no—
243C2—44no—
244C2—41no—
245C2—41no—
246C2—41no—
247C2AΣL(1,𝔽16)×C6712yesdirect product factor
248C2C2×#99×C6711yescharacteristic
subgroupG/H
center, socleC22×C67#99
commutatorC23C22×C4×C67
Frattini, cosocleC2×D4C23×C67
prime radicalC2×#120×C67C2
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C23×C67 ◁ C22×D4×C67 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#100×C67C2semidirect
3. C2×#99×C67C2semidirect
4. AΣL(1,𝔽16)×C67C2direct (N × H)
5. C25×C67C4semidirect
11. #120×C67C4semidirect
15. C2×C22⋊C4×C67C4semidirect
17. #100×C67C22semidirect
19. #99×C67C22semidirect
26. C24×C67C2×C4semidirect
48. C22⋊C4×C67C2×C4semidirect
122. C2×C67AΣL(1,𝔽16)direct (N × H)
124. C2×AΣL(1,𝔽16)C67direct (N × H)
125. C67C2×AΣL(1,𝔽16)direct (N × H)
127. C2×#100C2×C67semidirect
128. C2×#99C2×C67semidirect
129. AΣL(1,𝔽16)C2×C67direct (N × H)
130. C25C4×C67semidirect
136. #120C4×C67semidirect
140. C2×C22⋊C4C4×C67semidirect
142. #100C22×C67semidirect
144. #99C22×C67semidirect
151. C24C2×C4×C67semidirect
173. C22⋊C4C2×C4×C67semidirect
247. C2AΣL(1,𝔽16)×C67direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#120×C672
C2×#100×C672
C2×#99×C672
AΣL(1,𝔽16)×C672
C2×AΣL(1,𝔽16)67