Finite Group Databasebeta

← AΣL(1,𝔽16)×D173 #36355193

Subgroups — AΣL(1,𝔽16)×D173 #36355193

Order 22144.

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×D173C211yesmaximal, characteristic
2#100×D173C211yesmaximal, characteristic, complemented
3#99×D173C211yesmaximal, characteristic, complemented
4#26011279C211yesmaximal, characteristic, complemented
5#26011278C211yesmaximal, characteristic, complemented
6#26011277C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C173C211yesmaximal, characteristic, complemented
8C24×D173C411yescharacteristic, complemented
9C2×D4×D173—21no—
10C2×D4×D173C2211yescharacteristic
11M16×D173—21no—
12#19149649—21no—
13#19149649C411yescharacteristic, complemented
14#19149648C411yescharacteristic, complemented
15#19149647—21no—
16#19149647C2211yescharacteristic
17C22⋊C4×D173—21no—
18C22⋊C4×D173—21no—
19C22⋊C4×D173C411yescharacteristic, complemented
20#19149596C2211yescharacteristic, complemented
21#19149595—21no—
22#19149594C2211yescharacteristic, complemented
23#19149593—21no—
24#120×C173C2211yescharacteristic
25#100×C173C2211yescharacteristic, complemented
26#99×C173C2211yescharacteristic, complemented
27C23×D173—21no—
28C23×D173—21no—
29C23×D173—44no—
30C23×D173—41no—
31C23×D173—41no—
32C23×D173C2×C411yescharacteristic
33C23×D173C2×C411yescharacteristic, complemented
34C23×D173D411yescharacteristic
35C2×D692—21no—
36C2×D692D411yescharacteristic
37C2×C22⋊D173—21no—
38C2×C22⋊D173—21no—
39C2×C22⋊D173C2×C411yescharacteristic
40C2×C22⋊D173D411yescharacteristic
41D4×D173—41no—
42D4×D173—41no—
43D4×D173—44no—
44C2×C4×D173—21no—
45C2×C4×D173—41no—
46C2×C4×D173D411yescharacteristic
47#14583810—21no—
48#14583809—41no—
49C8×D173—41no—
50C22⋊Dic173—21no—
51C22⋊Dic173—21no—
52C22⋊Dic173C2×C411yescharacteristic, complemented
53#14583803—21no—
54#14583803—21no—
55#14583803—21no—
56#14583803—41no—
57C24×C173C2×C411yescharacteristic, complemented
58C2×D4×C173—21no—
59C2×D4×C173C2311yescharacteristic
60M16×C173—21no—
61C22⋊C4×C173—21no—
62C22⋊C4×C173—21no—
63C22⋊C4×C173C2×C411yescharacteristic, complemented
64C22×D173—24no—
65C22×D173—21no—
66C22×D173—21no—
67C22×D173—21no—
68C22×D173—21no—
69C22×D173—21no—
70C22×D173—42no—
71C22×D173—42no—
72C22×D173—42no—
73C22×D173—42no—
74C22×D173—42no—
75C22×D173—42no—
76C22×D173—42no—
77C22×D173—44no—
78C22×D173—44no—
79C22×D173—44no—
80C22×D173—41no—
81C22×D173—41no—
82C22×D173—41no—
83C22×D173—44no—
84C22×D173C22⋊C411yescharacteristic
85C22×D173C22⋊C411yescharacteristic
86C22×D173C22⋊C411yescharacteristic
87D692—41no—
88D692—42no—
89C22⋊D173—41no—
90C22⋊D173—41no—
91C22⋊D173—42no—
92C22⋊D173—42no—
93C2×Dic173—21no—
94C2×Dic173—41no—
95C2×Dic173C22⋊C411yescharacteristic
96C4×D173—21no—
97C4×D173—21no—
98C4×D173—42no—
99C4×D173—82no—
100#13063436—41no—
101C23×C173—21no—
102C23×C173—42no—
103C23×C173—41no—
104C23×C173C22×C411yescharacteristic
105C23×C173C2×D411yescharacteristic
106D4×C173—41no—
107D4×C173—42no—
108C2×C4×C173—21no—
109C2×C4×C173—41no—
110C2×C4×C173C2×D411yescharacteristic
111C8×C173—41no—
112C22×C173—22no—
113C22×C173—21no—
114C22×C173—21no—
115C22×C173—41no—
116C22×C173—41no—
117C22×C173—41no—
118C22×C173—42no—
119C22×C173—41no—
120C22×C173—42no—
121C22×C173C2×C22⋊C411yescharacteristic
122C2×D173—22no—
123C2×D173—22no—
124C2×D173—21no—
125C2×D173—21no—
126C2×D173—44no—
127C2×D173—42no—
128C2×D173—41no—
129C2×D173—48no—
130C2×D173—42no—
131C2×D173—41no—
132C2×D173—42no—
133C2×D173—44no—
134C2×D173—42no—
135C2×D173—41no—
136C2×D173—41no—
137C2×D173—42no—
138C2×D173—81no—
139C2×D173#9911yescharacteristic
140C2×D173#9911yescharacteristic
141C4×C173—21no—
142C4×C173—41no—
143C4×C173—81no—
144Dic173—21no—
145Dic173—41no—
146Dic173—81no—
147C2×C173—21no—
148C2×C173—41no—
149C2×C173—42no—
150C2×C173—41no—
151C2×C173C2×#9911yescharacteristic
152D173—21no—
153D173—42no—
154D173—41no—
155D173—41no—
156D173AΣL(1,𝔽16)12yesdirect product factor
157C173C2×AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
158C2×AΣL(1,𝔽16)—1731nomaximal, Sylow
159C2×#120—1731no—
160C2×#100—1731no—
161C2×#99—1731no—
162AΣL(1,𝔽16)—1731no—
163AΣL(1,𝔽16)—1731no—
164AΣL(1,𝔽16)—1731no—
165AΣL(1,𝔽16)D17311yescharacteristic, direct product factor
166C25—1731no—
167C22×D4—3461no—
168C22×D4—1731no—
169C2×M16—3461no—
170#120—3461no—
171#120—3461no—
172#120—1731no—
173#120—1731no—
174#120—1731no—
175#120C2×D17311yescharacteristic
176C2×C22⋊C4—3461no—
177C2×C22⋊C4—3461no—
178C2×C22⋊C4—1731no—
179#100—3461no—
180#100—1731no—
181#100C2×D17311yescharacteristic, complemented
182#99—3461no—
183#99—1731no—
184#99C2×D17311yescharacteristic, complemented
185C24—3461no—
186C24—3461no—
187C24—1731no—
188C24—1731no—
189C24—1731no—
190C24—6924no—
191C24—6921no—
192C24—6921no—
193C24C4×D17311yescharacteristic, complemented
194C2×D4—21no—
195C2×D4—3461no—
196C2×D4—3461no—
197C2×D4—3461no—
198C2×D4—1731no—
199C2×D4—1731no—
200C2×D4—1731no—
201C2×D4—6921no—
202C2×D4—6921no—
203C2×D4—6924no—
204C2×D4C22×D17311yescharacteristic
205C22×C4—3461no—
206C22×C4—1731no—
207C22×C4—6921no—
208M16—21no—
209M16—3461no—
210M16—6921no—
211C2×C8—6921no—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—3461no—
215C22⋊C4—3461no—
216C22⋊C4—3461no—
217C22⋊C4—3461no—
218C22⋊C4—3461no—
219C22⋊C4—1731no—
220C22⋊C4—6921no—
221C22⋊C4C4×D17311yescharacteristic, complemented
222C23—21no—
223C23—42no—
224C23—41no—
225C23—3464no—
226C23—3461no—
227C23—3461no—
228C23—3461no—
229C23—3461no—
230C23—3461no—
231C23—1731no—
232C23—1731no—
233C23—1731no—
234C23—6921no—
235C23—6922no—
236C23—6922no—
237C23—6922no—
238C23—6922no—
239C23—6922no—
240C23—6922no—
241C23—6922no—
242C23—6921no—
243C23—6921no—
244C23—6924no—
245C23—6924no—
246C23—6924no—
247C23—6924no—
248C23C2×C4×D17311yescharacteristic
249C23D4×D17311yescharacteristic
250D4—41no—
251D4—42no—
252D4—6921no—
253D4—6922no—
254D4—6921no—
255D4—6921no—
256D4—6922no—
257D4—6922no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—3461no—
261C2×C4—3461no—
262C2×C4—3461no—
263C2×C4—13842no—
264C2×C4—1731no—
265C2×C4—6922no—
266C2×C4—6921no—
267C2×C4D4×D17311yescharacteristic
268C8—41no—
269C8—6921no—
270C22—22no—
271C22—21no—
272C22—21no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—42no—
277C22—42no—
278C22—41no—
279C22—3462no—
280C22—3462no—
281C22—3461no—
282C22—3461no—
283C22—13841no—
284C22—1731no—
285C22—1731no—
286C22—6921no—
287C22—6921no—
288C22—6921no—
289C22—6924no—
290C22—6922no—
291C22—6922no—
292C22—6924no—
293C22—6922no—
294C22—6922no—
295C22—6922no—
296C22—6928no—
297C22—6921no—
298C22C22⋊C4×D17311yescharacteristic
299C4—21no—
300C4—41no—
301C4—81no—
302C4—3461no—
303C4—13841no—
304C4—6921no—
305C2—21no—
306C2—41no—
307C2—42no—
308C2—41no—
309C2—3461no—
310C2—1732no—
311C2—6921no—
312C2—6922no—
313C2—6921no—
314C2#99×D17311yescharacteristic
subgroupG/H
centerC2#99×D173
commutatorC23×C173C22×C4
FrattiniC2×D4C22×D173
FittingAΣL(1,𝔽16)×C173C2
socleC2×C173C2×#99
prime radical#120×D173C2
cosocleC2×D4×C173C23
hypercenterAΣL(1,𝔽16)D173
hypocenterC173C2×AΣL(1,𝔽16)
LCSG ▷ C23×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ AΣL(1,𝔽16)

Complements

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

NH ≅ G/Ntype
2. #100×D173C2semidirect
3. #99×D173C2semidirect
4. #26011279C2semidirect
5. #26011278C2semidirect
6. #26011277C2semidirect
7. AΣL(1,𝔽16)×C173C2semidirect
8. C24×D173C4semidirect
13. #19149649C4semidirect
14. #19149648C4semidirect
19. C22⋊C4×D173C4semidirect
20. #19149596C22semidirect
22. #19149594C22semidirect
25. #100×C173C22semidirect
26. #99×C173C22semidirect
33. C23×D173C2×C4semidirect
52. C22⋊Dic173C2×C4semidirect
57. C24×C173C2×C4semidirect
63. C22⋊C4×C173C2×C4semidirect
156. D173AΣL(1,𝔽16)direct (N × H)
157. C173C2×AΣL(1,𝔽16)semidirect
165. AΣL(1,𝔽16)D173direct (N × H)
181. #100C2×D173semidirect
184. #99C2×D173semidirect
193. C24C4×D173semidirect
221. C22⋊C4C4×D173semidirect

Maximal subgroups

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

subgroupindex
#120×D1732
#100×D1732
#99×D1732
#260112792
#260112782
#260112772
AΣL(1,𝔽16)×C1732
C2×AΣL(1,𝔽16)173