Finite Group Databasebeta

← AΣL(1,𝔽16)×D127 #30640360

Subgroups — AΣL(1,𝔽16)×D127 #30640360

Order 16256.

438 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 57 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
1#120×D127C211yesmaximal, characteristic
2#100×D127C211yesmaximal, characteristic, complemented
3#99×D127C211yesmaximal, characteristic, complemented
4#22549664C211yesmaximal, characteristic, complemented
5#22549663C211yesmaximal, characteristic, complemented
6#22549662C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C127C211yesmaximal, characteristic, complemented
8C24×D127C411yescharacteristic, complemented
9C2×D4×D127—21no—
10C2×D4×D127C2211yescharacteristic
11M16×D127—21no—
12#16861179—21no—
13#16861179C411yescharacteristic, complemented
14#16861178C411yescharacteristic, complemented
15#16861177—21no—
16#16861177C2211yescharacteristic
17C22⋊C4×D127—21no—
18C22⋊C4×D127—21no—
19C22⋊C4×D127C411yescharacteristic, complemented
20#16861126C2211yescharacteristic, complemented
21#16861125—21no—
22#16861124C2211yescharacteristic, complemented
23#16861123—21no—
24#120×C127C2211yescharacteristic
25#100×C127C2211yescharacteristic, complemented
26#99×C127C2211yescharacteristic, complemented
27C23×D127—21no—
28C23×D127—21no—
29C23×D127—44no—
30C23×D127—41no—
31C23×D127—41no—
32C23×D127C2×C411yescharacteristic
33C23×D127C2×C411yescharacteristic, complemented
34C23×D127D411yescharacteristic
35C2×D508—21no—
36C2×D508D411yescharacteristic
37C2×C22⋊D127—21no—
38C2×C22⋊D127—21no—
39C2×C22⋊D127C2×C411yescharacteristic
40C2×C22⋊D127D411yescharacteristic
41D4×D127—41no—
42D4×D127—41no—
43D4×D127—44no—
44C2×C4×D127—21no—
45C2×C4×D127—41no—
46C2×C4×D127D411yescharacteristic
47#14523232—21no—
48#14523231—41no—
49C8×D127—41no—
50C22⋊Dic127—21no—
51C22⋊Dic127—21no—
52C22⋊Dic127C2×C411yescharacteristic, complemented
53(C2×D127)⋊C4—21no—
54(C2×D127)⋊C4—21no—
55(C2×D127)⋊C4—21no—
56(C2×D127)⋊C4—41no—
57C24×C127C2×C411yescharacteristic, complemented
58C2×D4×C127—21no—
59C2×D4×C127C2311yescharacteristic
60M16×C127—21no—
61C22⋊C4×C127—21no—
62C22⋊C4×C127—21no—
63C22⋊C4×C127C2×C411yescharacteristic, complemented
64C22×D127—24no—
65C22×D127—21no—
66C22×D127—21no—
67C22×D127—21no—
68C22×D127—21no—
69C22×D127—21no—
70C22×D127—42no—
71C22×D127—42no—
72C22×D127—42no—
73C22×D127—42no—
74C22×D127—42no—
75C22×D127—42no—
76C22×D127—42no—
77C22×D127—44no—
78C22×D127—44no—
79C22×D127—44no—
80C22×D127—41no—
81C22×D127—41no—
82C22×D127—41no—
83C22×D127—44no—
84C22×D127C22⋊C411yescharacteristic
85C22×D127C22⋊C411yescharacteristic
86C22×D127C22⋊C411yescharacteristic
87D508—41no—
88D508—42no—
89C22⋊D127—41no—
90C22⋊D127—41no—
91C22⋊D127—42no—
92C22⋊D127—42no—
93C2×Dic127—21no—
94C2×Dic127—41no—
95C2×Dic127C22⋊C411yescharacteristic
96C4×D127—21no—
97C4×D127—21no—
98C4×D127—42no—
99C4×D127—82no—
100C127⋊C8—41no—
101C23×C127—21no—
102C23×C127—42no—
103C23×C127—41no—
104C23×C127C22×C411yescharacteristic
105C23×C127C2×D411yescharacteristic
106D4×C127—41no—
107D4×C127—42no—
108C2×C4×C127—21no—
109C2×C4×C127—41no—
110C2×C4×C127C2×D411yescharacteristic
111C8×C127—41no—
112C22×C127—22no—
113C22×C127—21no—
114C22×C127—21no—
115C22×C127—41no—
116C22×C127—41no—
117C22×C127—41no—
118C22×C127—42no—
119C22×C127—42no—
120C22×C127—41no—
121C22×C127C2×C22⋊C411yescharacteristic
122C2×D127—22no—
123C2×D127—22no—
124C2×D127—21no—
125C2×D127—21no—
126C2×D127—44no—
127C2×D127—42no—
128C2×D127—41no—
129C2×D127—48no—
130C2×D127—42no—
131C2×D127—41no—
132C2×D127—42no—
133C2×D127—44no—
134C2×D127—42no—
135C2×D127—41no—
136C2×D127—41no—
137C2×D127—42no—
138C2×D127—81no—
139C2×D127#9911yescharacteristic
140C2×D127#9911yescharacteristic
141C4×C127—21no—
142C4×C127—41no—
143C4×C127—81no—
144Dic127—21no—
145Dic127—41no—
146Dic127—81no—
147C2×C127—21no—
148C2×C127—41no—
149C2×C127—42no—
150C2×C127—41no—
151C2×C127C2×#9911yescharacteristic
152D127—21no—
153D127—42no—
154D127—41no—
155D127—41no—
156D127AΣL(1,𝔽16)12yesdirect product factor
157C2×AΣL(1,𝔽16)—1271nomaximal, Sylow
158C127C2×AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
159C2×#120—1271no—
160C2×#100—1271no—
161C2×#99—1271no—
162AΣL(1,𝔽16)—1271no—
163AΣL(1,𝔽16)—1271no—
164AΣL(1,𝔽16)—1271no—
165AΣL(1,𝔽16)D12711yescharacteristic, direct product factor
166C25—1271no—
167C22×D4—1271no—
168C22×D4—2541no—
169C2×M16—2541no—
170#120—1271no—
171#120—1271no—
172#120—1271no—
173#120—2541no—
174#120—2541no—
175#120C2×D12711yescharacteristic
176C2×C22⋊C4—1271no—
177C2×C22⋊C4—2541no—
178C2×C22⋊C4—2541no—
179#100—1271no—
180#100—2541no—
181#100C2×D12711yescharacteristic, complemented
182#99—1271no—
183#99—2541no—
184#99C2×D12711yescharacteristic, complemented
185C24—1271no—
186C24—1271no—
187C24—1271no—
188C24—5081no—
189C24—5081no—
190C24—5084no—
191C24—2541no—
192C24—2541no—
193C24C4×D12711yescharacteristic, complemented
194C2×D4—21no—
195C2×D4—1271no—
196C2×D4—1271no—
197C2×D4—1271no—
198C2×D4—5084no—
199C2×D4—5081no—
200C2×D4—5081no—
201C2×D4—2541no—
202C2×D4—2541no—
203C2×D4—2541no—
204C2×D4C22×D12711yescharacteristic
205C22×C4—1271no—
206C22×C4—5081no—
207C22×C4—2541no—
208M16—21no—
209M16—5081no—
210M16—2541no—
211C2×C8—5081no—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—1271no—
215C22⋊C4—5081no—
216C22⋊C4—2541no—
217C22⋊C4—2541no—
218C22⋊C4—2541no—
219C22⋊C4—2541no—
220C22⋊C4—2541no—
221C22⋊C4C4×D12711yescharacteristic, complemented
222C23—21no—
223C23—42no—
224C23—41no—
225C23—1271no—
226C23—1271no—
227C23—1271no—
228C23—5081no—
229C23—5082no—
230C23—5081no—
231C23—5081no—
232C23—5082no—
233C23—5084no—
234C23—5084no—
235C23—5084no—
236C23—5084no—
237C23—5082no—
238C23—5082no—
239C23—5082no—
240C23—5082no—
241C23—5082no—
242C23—2541no—
243C23—2541no—
244C23—2541no—
245C23—2541no—
246C23—2544no—
247C23—2541no—
248C23C2×C4×D12711yescharacteristic
249C23D4×D12711yescharacteristic
250D4—41no—
251D4—42no—
252D4—5081no—
253D4—5082no—
254D4—5082no—
255D4—5082no—
256D4—5081no—
257D4—5081no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—1271no—
261C2×C4—10162no—
262C2×C4—5082no—
263C2×C4—5081no—
264C2×C4—2541no—
265C2×C4—2541no—
266C2×C4—2541no—
267C2×C4D4×D12711yescharacteristic
268C8—41no—
269C8—5081no—
270C22—22no—
271C22—21no—
272C22—21no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—42no—
277C22—42no—
278C22—41no—
279C22—1271no—
280C22—1271no—
281C22—10161no—
282C22—5081no—
283C22—5081no—
284C22—5081no—
285C22—5082no—
286C22—5084no—
287C22—5082no—
288C22—5082no—
289C22—5082no—
290C22—5088no—
291C22—5084no—
292C22—5082no—
293C22—5081no—
294C22—2542no—
295C22—2542no—
296C22—2541no—
297C22—2541no—
298C22C22⋊C4×D12711yescharacteristic
299C4—21no—
300C4—41no—
301C4—81no—
302C4—10161no—
303C4—5081no—
304C4—2541no—
305C2—21no—
306C2—41no—
307C2—42no—
308C2—41no—
309C2—1272no—
310C2—5081no—
311C2—5081no—
312C2—5082no—
313C2—2541no—
314C2#99×D12711yescharacteristic
subgroupG/H
centerC2#99×D127
commutatorC23×C127C22×C4
FrattiniC2×D4C22×D127
FittingAΣL(1,𝔽16)×C127C2
socleC2×C127C2×#99
prime radical#120×D127C2
cosocleC2×D4×C127C23
hypercenterAΣL(1,𝔽16)D127
hypocenterC127C2×AΣL(1,𝔽16)
LCSG ▷ C23×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ AΣL(1,𝔽16)

Complements

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

NH ≅ G/Ntype
2. #100×D127C2semidirect
3. #99×D127C2semidirect
4. #22549664C2semidirect
5. #22549663C2semidirect
6. #22549662C2semidirect
7. AΣL(1,𝔽16)×C127C2semidirect
8. C24×D127C4semidirect
13. #16861179C4semidirect
14. #16861178C4semidirect
19. C22⋊C4×D127C4semidirect
20. #16861126C22semidirect
22. #16861124C22semidirect
25. #100×C127C22semidirect
26. #99×C127C22semidirect
33. C23×D127C2×C4semidirect
52. C22⋊Dic127C2×C4semidirect
57. C24×C127C2×C4semidirect
63. C22⋊C4×C127C2×C4semidirect
156. D127AΣL(1,𝔽16)direct (N × H)
158. C127C2×AΣL(1,𝔽16)semidirect
165. AΣL(1,𝔽16)D127direct (N × H)
181. #100C2×D127semidirect
184. #99C2×D127semidirect
193. C24C4×D127semidirect
221. C22⋊C4C4×D127semidirect

Maximal subgroups

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

subgroupindex
#120×D1272
#100×D1272
#99×D1272
#225496642
#225496632
#225496622
AΣL(1,𝔽16)×C1272
C2×AΣL(1,𝔽16)127