Finite Group Databasebeta

← AΣL(1,𝔽16)×D101 #27172762

Subgroups — AΣL(1,𝔽16)×D101 #27172762

Order 12928.

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×D101C211yesmaximal, characteristic
2#100×D101C211yesmaximal, characteristic, complemented
3#99×D101C211yesmaximal, characteristic, complemented
4#20264965C211yesmaximal, characteristic, complemented
5#20264964C211yesmaximal, characteristic, complemented
6#20264963C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C101C211yesmaximal, characteristic, complemented
8C24×D101C411yescharacteristic, complemented
9C2×D4×D101—21no—
10C2×D4×D101C2211yescharacteristic
11M16×D101—21no—
12#15692225—21no—
13#15692225C411yescharacteristic, complemented
14#15692224C411yescharacteristic, complemented
15#15692223—21no—
16#15692223C2211yescharacteristic
17C22⋊C4×D101—21no—
18C22⋊C4×D101—21no—
19C22⋊C4×D101C411yescharacteristic, complemented
20#15692172C2211yescharacteristic, complemented
21#15692171—21no—
22#15692170C2211yescharacteristic, complemented
23#15692169—21no—
24#120×C101C2211yescharacteristic
25#100×C101C2211yescharacteristic, complemented
26#99×C101C2211yescharacteristic, complemented
27C23×D101—21no—
28C23×D101—21no—
29C23×D101—44no—
30C23×D101—41no—
31C23×D101—41no—
32C23×D101C2×C411yescharacteristic
33C23×D101C2×C411yescharacteristic, complemented
34C23×D101D411yescharacteristic
35C2×D404—21no—
36C2×D404D411yescharacteristic
37C2×C22⋊D101—21no—
38C2×C22⋊D101—21no—
39C2×C22⋊D101C2×C411yescharacteristic
40C2×C22⋊D101D411yescharacteristic
41D4×D101—41no—
42D4×D101—41no—
43D4×D101—44no—
44C2×C4×D101—21no—
45C2×C4×D101—41no—
46C2×C4×D101D411yescharacteristic
47#13108058—21no—
48#13108057—41no—
49C8×D101—41no—
50C22⋊Dic101—21no—
51C22⋊Dic101—21no—
52C22⋊Dic101C2×C411yescharacteristic, complemented
53#13108051—21no—
54#13108051—21no—
55#13108051—21no—
56#13108051—41no—
57C24×C101C2×C411yescharacteristic, complemented
58C2×D4×C101—21no—
59C2×D4×C101C2311yescharacteristic
60M16×C101—21no—
61C22⋊C4×C101—21no—
62C22⋊C4×C101—21no—
63C22⋊C4×C101C2×C411yescharacteristic, complemented
64C23×C101—21no—
65C23×C101—42no—
66C23×C101—41no—
67C23×C101C22×C411yescharacteristic
68C23×C101C2×D411yescharacteristic
69C22×D101—24no—
70C22×D101—21no—
71C22×D101—21no—
72C22×D101—21no—
73C22×D101—21no—
74C22×D101—21no—
75C22×D101—42no—
76C22×D101—42no—
77C22×D101—42no—
78C22×D101—42no—
79C22×D101—42no—
80C22×D101—42no—
81C22×D101—42no—
82C22×D101—44no—
83C22×D101—44no—
84C22×D101—44no—
85C22×D101—41no—
86C22×D101—41no—
87C22×D101—41no—
88C22×D101—44no—
89C22×D101C22⋊C411yescharacteristic
90C22×D101C22⋊C411yescharacteristic
91C22×D101C22⋊C411yescharacteristic
92D4×C101—41no—
93D4×C101—42no—
94C2×C4×C101—21no—
95C2×C4×C101—41no—
96C2×C4×C101C2×D411yescharacteristic
97C22⋊D101—41no—
98C22⋊D101—41no—
99C22⋊D101—42no—
100C22⋊D101—42no—
101C2×Dic101—21no—
102C2×Dic101—41no—
103C2×Dic101C22⋊C411yescharacteristic
104D404—41no—
105D404—42no—
106C4×D101—21no—
107C4×D101—21no—
108C4×D101—42no—
109C4×D101—82no—
110C8×C101—41no—
111#11717579—41no—
112C22×C101—22no—
113C22×C101—21no—
114C22×C101—21no—
115C22×C101—41no—
116C22×C101—41no—
117C22×C101—41no—
118C22×C101—42no—
119C22×C101—42no—
120C22×C101—41no—
121C22×C101C2×C22⋊C411yescharacteristic
122C2×D101—22no—
123C2×D101—22no—
124C2×D101—21no—
125C2×D101—21no—
126C2×D101—44no—
127C2×D101—42no—
128C2×D101—41no—
129C2×D101—48no—
130C2×D101—42no—
131C2×D101—41no—
132C2×D101—42no—
133C2×D101—44no—
134C2×D101—42no—
135C2×D101—41no—
136C2×D101—41no—
137C2×D101—42no—
138C2×D101—81no—
139C2×D101#9911yescharacteristic
140C2×D101#9911yescharacteristic
141C4×C101—21no—
142C4×C101—41no—
143C4×C101—81no—
144Dic101—21no—
145Dic101—41no—
146Dic101—81no—
147C2×C101—21no—
148C2×C101—41no—
149C2×C101—42no—
150C2×C101—41no—
151C2×C101C2×#9911yescharacteristic
152D101—21no—
153D101—42no—
154D101—41no—
155D101—41no—
156D101AΣL(1,𝔽16)12yesdirect product factor
157C2×AΣL(1,𝔽16)—1011nomaximal, Sylow
158C101C2×AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
159C2×#120—1011no—
160C2×#100—1011no—
161C2×#99—1011no—
162AΣL(1,𝔽16)—1011no—
163AΣL(1,𝔽16)—1011no—
164AΣL(1,𝔽16)—1011no—
165AΣL(1,𝔽16)D10111yescharacteristic, direct product factor
166C25—1011no—
167C22×D4—1011no—
168C22×D4—2021no—
169C2×M16—2021no—
170#120—1011no—
171#120—1011no—
172#120—1011no—
173#120—2021no—
174#120—2021no—
175#120C2×D10111yescharacteristic
176C2×C22⋊C4—1011no—
177C2×C22⋊C4—2021no—
178C2×C22⋊C4—2021no—
179#100—1011no—
180#100—2021no—
181#100C2×D10111yescharacteristic, complemented
182#99—1011no—
183#99—2021no—
184#99C2×D10111yescharacteristic, complemented
185C24—1011no—
186C24—1011no—
187C24—1011no—
188C24—4041no—
189C24—4041no—
190C24—4044no—
191C24—2021no—
192C24—2021no—
193C24C4×D10111yescharacteristic, complemented
194C2×D4—21no—
195C2×D4—1011no—
196C2×D4—1011no—
197C2×D4—1011no—
198C2×D4—4044no—
199C2×D4—4041no—
200C2×D4—4041no—
201C2×D4—2021no—
202C2×D4—2021no—
203C2×D4—2021no—
204C2×D4C22×D10111yescharacteristic
205C22×C4—1011no—
206C22×C4—4041no—
207C22×C4—2021no—
208M16—21no—
209M16—4041no—
210M16—2021no—
211C2×C8—4041no—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—1011no—
215C22⋊C4—4041no—
216C22⋊C4—2021no—
217C22⋊C4—2021no—
218C22⋊C4—2021no—
219C22⋊C4—2021no—
220C22⋊C4—2021no—
221C22⋊C4C4×D10111yescharacteristic, complemented
222C23—21no—
223C23—42no—
224C23—41no—
225C23—1011no—
226C23—1011no—
227C23—1011no—
228C23—4041no—
229C23—4042no—
230C23—4042no—
231C23—4041no—
232C23—4041no—
233C23—4044no—
234C23—4044no—
235C23—4044no—
236C23—4044no—
237C23—4042no—
238C23—4042no—
239C23—4042no—
240C23—4042no—
241C23—4042no—
242C23—2021no—
243C23—2021no—
244C23—2021no—
245C23—2021no—
246C23—2024no—
247C23—2021no—
248C23C2×C4×D10111yescharacteristic
249C23D4×D10111yescharacteristic
250D4—41no—
251D4—42no—
252D4—4041no—
253D4—4042no—
254D4—4042no—
255D4—4042no—
256D4—4041no—
257D4—4041no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—8082no—
261C2×C4—1011no—
262C2×C4—4042no—
263C2×C4—4041no—
264C2×C4—2021no—
265C2×C4—2021no—
266C2×C4—2021no—
267C2×C4D4×D10111yescharacteristic
268C8—41no—
269C8—4041no—
270C22—22no—
271C22—21no—
272C22—21no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—42no—
277C22—42no—
278C22—41no—
279C22—8081no—
280C22—1011no—
281C22—1011no—
282C22—4041no—
283C22—4041no—
284C22—4041no—
285C22—4042no—
286C22—4044no—
287C22—4042no—
288C22—4042no—
289C22—4042no—
290C22—4048no—
291C22—4044no—
292C22—4042no—
293C22—4041no—
294C22—2022no—
295C22—2022no—
296C22—2021no—
297C22—2021no—
298C22C22⋊C4×D10111yescharacteristic
299C4—21no—
300C4—41no—
301C4—81no—
302C4—8081no—
303C4—4041no—
304C4—2021no—
305C2—21no—
306C2—41no—
307C2—42no—
308C2—41no—
309C2—1012no—
310C2—4041no—
311C2—4041no—
312C2—4042no—
313C2—2021no—
314C2#99×D10111yescharacteristic
subgroupG/H
centerC2#99×D101
commutatorC23×C101C22×C4
FrattiniC2×D4C22×D101
FittingAΣL(1,𝔽16)×C101C2
socleC2×C101C2×#99
prime radical#120×D101C2
cosocleC2×D4×C101C23
hypercenterAΣL(1,𝔽16)D101
hypocenterC101C2×AΣL(1,𝔽16)
LCSG ▷ C23×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ AΣL(1,𝔽16)

Complements

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

NH ≅ G/Ntype
2. #100×D101C2semidirect
3. #99×D101C2semidirect
4. #20264965C2semidirect
5. #20264964C2semidirect
6. #20264963C2semidirect
7. AΣL(1,𝔽16)×C101C2semidirect
8. C24×D101C4semidirect
13. #15692225C4semidirect
14. #15692224C4semidirect
19. C22⋊C4×D101C4semidirect
20. #15692172C22semidirect
22. #15692170C22semidirect
25. #100×C101C22semidirect
26. #99×C101C22semidirect
33. C23×D101C2×C4semidirect
52. C22⋊Dic101C2×C4semidirect
57. C24×C101C2×C4semidirect
63. C22⋊C4×C101C2×C4semidirect
156. D101AΣL(1,𝔽16)direct (N × H)
158. C101C2×AΣL(1,𝔽16)semidirect
165. AΣL(1,𝔽16)D101direct (N × H)
181. #100C2×D101semidirect
184. #99C2×D101semidirect
193. C24C4×D101semidirect
221. C22⋊C4C4×D101semidirect

Maximal subgroups

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

subgroupindex
#120×D1012
#100×D1012
#99×D1012
#202649652
#202649642
#202649632
AΣL(1,𝔽16)×C1012
C2×AΣL(1,𝔽16)101