Finite Group Databasebeta

← AΣL(1,𝔽16)×D251 #44551995

Subgroups — AΣL(1,𝔽16)×D251 #44551995

Order 32128.

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×D251C211yesmaximal, characteristic
2#100×D251C211yesmaximal, characteristic, complemented
3#99×D251C211yesmaximal, characteristic, complemented
4#30631475C211yesmaximal, characteristic, complemented
5#30631474C211yesmaximal, characteristic, complemented
6#30631473C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C251C211yesmaximal, characteristic, complemented
8C24×D251C411yescharacteristic, complemented
9C2×D4×D251—21no—
10C2×D4×D251C2211yescharacteristic
11M16×D251—21no—
12#22548685—21no—
13#22548685C411yescharacteristic, complemented
14#22548684C411yescharacteristic, complemented
15#22548683—21no—
16#22548683C2211yescharacteristic
17C22⋊C4×D251—21no—
18C22⋊C4×D251—21no—
19C22⋊C4×D251C411yescharacteristic, complemented
20#22548632C2211yescharacteristic, complemented
21#22548631—21no—
22#22548630C2211yescharacteristic, complemented
23#22548629—21no—
24#120×C251C2211yescharacteristic
25#100×C251C2211yescharacteristic, complemented
26#99×C251C2211yescharacteristic, complemented
27C23×D251—21no—
28C23×D251—21no—
29C23×D251—44no—
30C23×D251—41no—
31C23×D251—41no—
32C23×D251C2×C411yescharacteristic
33C23×D251C2×C411yescharacteristic, complemented
34C23×D251D411yescharacteristic
35C2×D1004—21no—
36C2×D1004D411yescharacteristic
37C2×C22⋊D251—21no—
38C2×C22⋊D251—21no—
39C2×C22⋊D251C2×C411yescharacteristic
40C2×C22⋊D251D411yescharacteristic
41D4×D251—41no—
42D4×D251—41no—
43D4×D251—44no—
44C2×C4×D251—21no—
45C2×C4×D251—41no—
46C2×C4×D251D411yescharacteristic
47#16860794—21no—
48#16860793—41no—
49C8×D251—41no—
50C22⋊Dic251—21no—
51C22⋊Dic251—21no—
52C22⋊Dic251C2×C411yescharacteristic, complemented
53(C2×D251)⋊C4—21no—
54(C2×D251)⋊C4—21no—
55(C2×D251)⋊C4—21no—
56(C2×D251)⋊C4—41no—
57C24×C251C2×C411yescharacteristic, complemented
58C2×D4×C251—21no—
59C2×D4×C251C2311yescharacteristic
60M16×C251—21no—
61C22⋊C4×C251—21no—
62C22⋊C4×C251—21no—
63C22⋊C4×C251C2×C411yescharacteristic, complemented
64C22×D251—24no—
65C22×D251—21no—
66C22×D251—21no—
67C22×D251—21no—
68C22×D251—21no—
69C22×D251—21no—
70C22×D251—42no—
71C22×D251—42no—
72C22×D251—42no—
73C22×D251—42no—
74C22×D251—42no—
75C22×D251—42no—
76C22×D251—42no—
77C22×D251—44no—
78C22×D251—44no—
79C22×D251—44no—
80C22×D251—41no—
81C22×D251—41no—
82C22×D251—41no—
83C22×D251—44no—
84C22×D251C22⋊C411yescharacteristic
85C22×D251C22⋊C411yescharacteristic
86C22×D251C22⋊C411yescharacteristic
87D1004—41no—
88D1004—42no—
89C22⋊D251—41no—
90C22⋊D251—41no—
91C22⋊D251—42no—
92C22⋊D251—42no—
93C2×Dic251—21no—
94C2×Dic251—41no—
95C2×Dic251C22⋊C411yescharacteristic
96C4×D251—21no—
97C4×D251—21no—
98C4×D251—42no—
99C4×D251—82no—
100C251⋊C8—41no—
101C23×C251—21no—
102C23×C251—42no—
103C23×C251—41no—
104C23×C251C22×C411yescharacteristic
105C23×C251C2×D411yescharacteristic
106D4×C251—41no—
107D4×C251—42no—
108C2×C4×C251—21no—
109C2×C4×C251—41no—
110C2×C4×C251C2×D411yescharacteristic
111C8×C251—41no—
112C22×C251—22no—
113C22×C251—21no—
114C22×C251—21no—
115C22×C251—41no—
116C22×C251—41no—
117C22×C251—41no—
118C22×C251—42no—
119C22×C251—42no—
120C22×C251—41no—
121C22×C251C2×C22⋊C411yescharacteristic
122C2×D251—22no—
123C2×D251—22no—
124C2×D251—21no—
125C2×D251—21no—
126C2×D251—44no—
127C2×D251—42no—
128C2×D251—41no—
129C2×D251—48no—
130C2×D251—42no—
131C2×D251—41no—
132C2×D251—42no—
133C2×D251—44no—
134C2×D251—42no—
135C2×D251—41no—
136C2×D251—41no—
137C2×D251—42no—
138C2×D251—81no—
139C2×D251#9911yescharacteristic
140C2×D251#9911yescharacteristic
141C4×C251—21no—
142C4×C251—41no—
143C4×C251—81no—
144Dic251—21no—
145Dic251—41no—
146Dic251—81no—
147C2×C251—21no—
148C2×C251—41no—
149C2×C251—42no—
150C2×C251—41no—
151C2×C251C2×#9911yescharacteristic
152D251—21no—
153D251—42no—
154D251—41no—
155D251—41no—
156D251AΣL(1,𝔽16)12yesdirect product factor
157C251C2×AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
158C2×AΣL(1,𝔽16)—2511nomaximal, Sylow
159C2×#120—2511no—
160C2×#100—2511no—
161C2×#99—2511no—
162AΣL(1,𝔽16)—2511no—
163AΣL(1,𝔽16)—2511no—
164AΣL(1,𝔽16)—2511no—
165AΣL(1,𝔽16)D25111yescharacteristic, direct product factor
166C25—2511no—
167C22×D4—5021no—
168C22×D4—2511no—
169C2×M16—5021no—
170#120—5021no—
171#120—5021no—
172#120—2511no—
173#120—2511no—
174#120—2511no—
175#120C2×D25111yescharacteristic
176C2×C22⋊C4—5021no—
177C2×C22⋊C4—5021no—
178C2×C22⋊C4—2511no—
179#100—5021no—
180#100—2511no—
181#100C2×D25111yescharacteristic, complemented
182#99—5021no—
183#99—2511no—
184#99C2×D25111yescharacteristic, complemented
185C24—10041no—
186C24—10041no—
187C24—10044no—
188C24—5021no—
189C24—5021no—
190C24—2511no—
191C24—2511no—
192C24—2511no—
193C24C4×D25111yescharacteristic, complemented
194C2×D4—21no—
195C2×D4—10044no—
196C2×D4—10041no—
197C2×D4—10041no—
198C2×D4—5021no—
199C2×D4—5021no—
200C2×D4—5021no—
201C2×D4—2511no—
202C2×D4—2511no—
203C2×D4—2511no—
204C2×D4C22×D25111yescharacteristic
205C22×C4—10041no—
206C22×C4—5021no—
207C22×C4—2511no—
208M16—21no—
209M16—10041no—
210M16—5021no—
211C2×C8—10041no—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—10041no—
215C22⋊C4—5021no—
216C22⋊C4—5021no—
217C22⋊C4—5021no—
218C22⋊C4—5021no—
219C22⋊C4—5021no—
220C22⋊C4—2511no—
221C22⋊C4C4×D25111yescharacteristic, complemented
222C23—21no—
223C23—42no—
224C23—41no—
225C23—10041no—
226C23—10042no—
227C23—10042no—
228C23—10044no—
229C23—10044no—
230C23—10044no—
231C23—10044no—
232C23—10042no—
233C23—10042no—
234C23—10042no—
235C23—10041no—
236C23—10041no—
237C23—10042no—
238C23—10042no—
239C23—5021no—
240C23—5024no—
241C23—5021no—
242C23—5021no—
243C23—5021no—
244C23—5021no—
245C23—2511no—
246C23—2511no—
247C23—2511no—
248C23C2×C4×D25111yescharacteristic
249C23D4×D25111yescharacteristic
250D4—41no—
251D4—42no—
252D4—10041no—
253D4—10042no—
254D4—10042no—
255D4—10042no—
256D4—10041no—
257D4—10041no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—20082no—
261C2×C4—10042no—
262C2×C4—10041no—
263C2×C4—5021no—
264C2×C4—5021no—
265C2×C4—5021no—
266C2×C4—2511no—
267C2×C4D4×D25111yescharacteristic
268C8—41no—
269C8—10041no—
270C22—22no—
271C22—21no—
272C22—21no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—42no—
277C22—42no—
278C22—41no—
279C22—20081no—
280C22—10041no—
281C22—10041no—
282C22—10041no—
283C22—10042no—
284C22—10042no—
285C22—10048no—
286C22—10044no—
287C22—10042no—
288C22—10042no—
289C22—10042no—
290C22—10044no—
291C22—10041no—
292C22—5022no—
293C22—5022no—
294C22—5021no—
295C22—5021no—
296C22—2511no—
297C22—2511no—
298C22C22⋊C4×D25111yescharacteristic
299C4—21no—
300C4—41no—
301C4—81no—
302C4—20081no—
303C4—10041no—
304C4—5021no—
305C2—21no—
306C2—41no—
307C2—42no—
308C2—41no—
309C2—10041no—
310C2—10041no—
311C2—10042no—
312C2—5021no—
313C2—2512no—
314C2#99×D25111yescharacteristic
subgroupG/H
centerC2#99×D251
commutatorC23×C251C22×C4
FrattiniC2×D4C22×D251
FittingAΣL(1,𝔽16)×C251C2
socleC2×C251C2×#99
prime radical#120×D251C2
cosocleC2×D4×C251C23
hypercenterAΣL(1,𝔽16)D251
hypocenterC251C2×AΣL(1,𝔽16)
LCSG ▷ C23×C251 ▷ C22×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ AΣL(1,𝔽16)

Complements

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

NH ≅ G/Ntype
2. #100×D251C2semidirect
3. #99×D251C2semidirect
4. #30631475C2semidirect
5. #30631474C2semidirect
6. #30631473C2semidirect
7. AΣL(1,𝔽16)×C251C2semidirect
8. C24×D251C4semidirect
13. #22548685C4semidirect
14. #22548684C4semidirect
19. C22⋊C4×D251C4semidirect
20. #22548632C22semidirect
22. #22548630C22semidirect
25. #100×C251C22semidirect
26. #99×C251C22semidirect
33. C23×D251C2×C4semidirect
52. C22⋊Dic251C2×C4semidirect
57. C24×C251C2×C4semidirect
63. C22⋊C4×C251C2×C4semidirect
156. D251AΣL(1,𝔽16)direct (N × H)
157. C251C2×AΣL(1,𝔽16)semidirect
165. AΣL(1,𝔽16)D251direct (N × H)
181. #100C2×D251semidirect
184. #99C2×D251semidirect
193. C24C4×D251semidirect
221. C22⋊C4C4×D251semidirect

Maximal subgroups

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

subgroupindex
#120×D2512
#100×D2512
#99×D2512
#306314752
#306314742
#306314732
AΣL(1,𝔽16)×C2512
C2×AΣL(1,𝔽16)251