Finite Group Databasebeta

← #17090526

Subgroups — #17090526

Order 4352.

1422 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587561C211yesmaximal, characteristic
2C2×#11763166C212yesmaximal, complemented
3#14530025C214yesmaximal, complemented
4#14529593C212yesmaximal, complemented
5#14528718C212yesmaximal, complemented
6#14528714C212yesmaximal
7C4×C22⋊#3622C211yesmaximal, characteristic, complemented
8C17×#2851C211yesmaximal, characteristic, complemented
9C23×Dic34C2211yescharacteristic
10C23×D68C2211yescharacteristic
11C2×#10587561—22no—
12C2×#10587561—28no—
13C2×#10587561C2212yes—
14C2×#10587561C2212yes—
15C22×#63227—22no—
16#11763166—28no—
17C2×#10587481C2214yescomplemented
18C2×#10587481C2214yescomplemented
19C2×#10587480C2214yes—
20C2×#10587480C2214yes—
21C2×C22⋊#3622C2212yes—
22C2×C4×#3622C2212yescomplemented
23#11762564C2212yescomplemented
24C2×C17×#125C2211yescharacteristic, complemented
25C22×C42×C17C2211yescharacteristic
26C17×#389C2212yescomplemented
27C17×#387—22no—
28C17×#385—22no—
29C17×#382C2211yescharacteristic, complemented
30C17×#380C2212yes—
31C24×D17—21no—
32C23×C4×C17D412yes—
33C23×C4×C17C2311yescharacteristic
34C23×Dic17—21no—
35C22×D68—21no—
36C22×D68—24no—
37C22×D68—24no—
38C22×D68—42no—
39C22×D68C2312yes—
40C22×Dic34—21no—
41C22×Dic34—24no—
42C22×Dic34—24no—
43C22×Dic34—42no—
44C22×Dic34C2312yes—
45C17×#125—24no—
46C2×C4⋊C4×C17—22no—
47C2×C4⋊C4×C17—22no—
48C2×C4⋊C4×C17C2312yes—
49C2×C4⋊C4×C17C2312yes—
50C2×C22⋊C4×C17—22no—
51C2×C42×C17—24no—
52C2×C42×C17D412yes—
53C2×C42×C17C2311yescharacteristic
54C2×C42×C17C2311yescharacteristic
55C2×#63227—216no—
56C2×#63227—24no—
57C2×#63227—24no—
58C22×#3622C2314yes—
59#10587561—216no—
60#10587561—24no—
61#10587561—24no—
62#10587561—24no—
63#10587561D418yes—
64C17×#95—22no—
65C17×#95—22no—
66C17×#95—22no—
67C22⋊#3622—24no—
68#10587481—28no—
69#10587481—44no—
70#10587480—28no—
71#10587480—44no—
72C4×#3622—24no—
73C24×C17C2×D411yescharacteristic
74C23×D17—24no—
75C23×D17—22no—
76C23×D17—42no—
77C23×D17—44no—
78C23×D17—42no—
79C23×D17—44no—
80C22×C4×C17—22no—
81C22×C4×C17—22no—
82C22×C4×C17—24no—
83C22×C4×C17—24no—
84C22×C4×C17—22no—
85C22×C4×C17—22no—
86C22×C4×C17—42no—
87C22×C4×C17C2×D412yes—
88C22×C4×C17C2×D412yes—
89C22×C4×C17C2×D411yescharacteristic
90C22×C4×C17#4114yes—
91C22×C4×C17C2411yescharacteristic
92C22×Dic17—24no—
93C22×Dic17—22no—
94C22×Dic17—42no—
95C22×Dic17—42no—
96C2×D68—24no—
97C2×D68—24no—
98C2×D68—24no—
99C2×D68—22no—
100C2×D68—416no—
101C2×D68—42no—
102C2×D68—42no—
103C2×D68C2×D414yes—
104C2×Dic34—24no—
105C2×Dic34—24no—
106C2×Dic34—24no—
107C2×Dic34—22no—
108C2×Dic34—416no—
109C2×Dic34—42no—
110C2×Dic34—42no—
111C2×Dic34C2×D414yes—
112C4⋊C4×C17—24no—
113C4⋊C4×C17—24no—
114C4⋊C4×C17—44no—
115C4⋊C4×C17—44no—
116C22⋊C4×C17—44no—
117C42×C17—24no—
118C42×C17—22no—
119C42×C17C2×D414yes—
120#63227—216no—
121#63227—216no—
122C2×#3622—24no—
123C2×#3622—24no—
124C2×#3622—24no—
125#25859—171nomaximal, Sylow
126C23×C17—21no—
127C23×C17—21no—
128C23×C17—22no—
129C23×C17#12412yes—
130C23×C17#13514yes—
131C23×C17C22×D411yescharacteristic
132C22×D17—24no—
133C22×D17—48no—
134C22×D17—416no—
135C22×D17—42no—
136C22×D17—42no—
137C22×D17—44no—
138C22×D17—44no—
139C22×D17—84no—
140C22×D17—88no—
141C22×D17—84no—
142C2×C4×C17—21no—
143C2×C4×C17—21no—
144C2×C4×C17—22no—
145C2×C4×C17—22no—
146C2×C4×C17—28no—
147C2×C4×C17—22no—
148C2×C4×C17—22no—
149C2×C4×C17—22no—
150C2×C4×C17—48no—
151C2×C4×C17—42no—
152C2×C4×C17—42no—
153C2×C4×C17—42no—
154C2×C4×C17—42no—
155C2×C4×C17—42no—
156C2×C4×C17—42no—
157C2×C4×C17#12014yes—
158C2×C4×C17#12412yes—
159C2×C4×C17C22×D412yes—
160C2×C4×C17C2×#4112yes—
161C2×C4×C17H(5,𝔽2)12yes—
162C2×Dic17—24no—
163C2×Dic17—48no—
164C2×Dic17—42no—
165C2×Dic17—42no—
166D68—28no—
167D68—48no—
168D68—44no—
169Dic34—28no—
170Dic34—48no—
171Dic34—44no—
172#3622—44no—
173C22×#124—171no—
174C2×#488—172no—
175#2851D1711yescharacteristic, complemented
176#2042—174no—
177#1958—172no—
178#1787—172no—
179#1786—172no—
180C4×C22⋊C8—171no—
181C22×C17—22no—
182C22×C17—22no—
183C22×C17—24no—
184C22×C17—24no—
185C22×C17#48814yescomplemented
186C22×C17C2×#12011yescharacteristic
187C22×C17C2×#12411yescharacteristic
188C22×C17#53511yescharacteristic
189C22×C17C2×#13512yes—
190C22×C17#57512yes—
191C2×D17—44no—
192C2×D17—48no—
193C2×D17—816no—
194C2×D17—82no—
195C2×D17—82no—
196C2×D17—84no—
197C4×C17—22no—
198C4×C17—24no—
199C4×C17—42no—
200C4×C17—82no—
201C4×C17—82no—
202C4×C17#54614yes—
203Dic17—44no—
204C23×Q8—171no—
205C23×D4—171no—
206C2×#125C2×D1711yescharacteristic, complemented
207C2×#124—172no—
208C2×#124—172no—
209C2×#124—348no—
210C2×#124—342no—
211C22×C22⋊C4—342no—
212C22×C42C2×D1711yescharacteristic
213#488—348no—
214C2×#103—174no—
215C2×#103—174no—
216C2×#102—174no—
217C2×#102—174no—
218C2×C22⋊C8—172no—
219C2×C4×C8—172no—
220#389C2×D1712yescomplemented
221#387—22no—
222#385—22no—
223#382C2×D1711yescharacteristic, complemented
224#380C2×D1712yes—
225#336—172no—
226C2×C17—24no—
227C2×C17#243511yescharacteristic
228C2×C17#300212yes—
229C2×C17C2×#48811yescharacteristic
230C2×C17#314011yescharacteristic
231C2×C17#321112yes—
232D17—84no—
233C25—341no—
234C22×Q8—172no—
235C22×Q8—344no—
236C22×Q8—341no—
237C22×Q8—344no—
238C22×Q8—682no—
239C22×D4—172no—
240C22×D4—344no—
241C22×D4—344no—
242C22×D4—341no—
243C22×D4—682no—
244C23×C4—341no—
245C23×C4C22⋊D1712yes—
246C23×C4C22×D1711yescharacteristic
247C22×C8—174no—
248#125—24no—
249#124—178no—
250#124—344no—
251#124—344no—
252#124—3416no—
253#124—344no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4C22×D1712yes—
257C2×C4⋊C4C22×D1712yes—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—3416no—
260C2×C22⋊C4—344no—
261C2×C22⋊C4—344no—
262C2×C42—24no—
263C2×C42C22⋊D1712yes—
264C2×C42C22×D1711yescharacteristic
265C2×C42C22×D1711yescharacteristic
266#103—348no—
267#103—684no—
268#102—348no—
269#102—684no—
270C22⋊C8—344no—
271C4×C8—344no—
272#95—22no—
273#95—22no—
274#95—22no—
275C17#2585911yescharacteristic, complemented, Sylow
276C24—344no—
277C24—342no—
278C24—682no—
279C24—684no—
280C24—682no—
281C24—684no—
282C24C2×C22⋊D1711yescharacteristic
283C2×Q8—174no—
284C2×Q8—344no—
285C2×Q8—342no—
286C2×Q8—344no—
287C2×Q8—344no—
288C2×Q8—6816no—
289C2×Q8—682no—
290C2×Q8—682no—
291C2×D4—174no—
292C2×D4—344no—
293C2×D4—342no—
294C2×D4—344no—
295C2×D4—344no—
296C2×D4—6816no—
297C2×D4—682no—
298C2×D4—682no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—42no—
306C22×C4—344no—
307C22×C4—342no—
308C22×C4—682no—
309C22×C4—682no—
310C22×C4#6325614yes—
311C22×C4C2×C22⋊D1712yes—
312C22×C4C2×C22⋊D1712yes—
313C22×C4C2×C22⋊D1711yescharacteristic
314C22×C4C23×D1711yescharacteristic
315C2×C8—344no—
316C2×C8—344no—
317C2×C8—344no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C22⋊C4—44no—
323C22⋊C4—3416no—
324C22⋊C4—3416no—
325C42—22no—
326C42—24no—
327C42D4×D1714yes—
328C23—21no—
329C23—22no—
330C23—21no—
331C23—344no—
332C23—688no—
333C23—6816no—
334C23—682no—
335C23—684no—
336C23—684no—
337C23—682no—
338C23—1364no—
339C23—1368no—
340C23—1364no—
341C23#1058763312yes—
342C23#1058763614yes—
343C23C22×C22⋊D1711yescharacteristic
344Q8—348no—
345Q8—688no—
346Q8—684no—
347D4—348no—
348D4—688no—
349D4—684no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—28no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—48no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—344no—
366C2×C4—688no—
367C2×C4—682no—
368C2×C4—682no—
369C2×C4#1058762314yes—
370C2×C4#1058763312yes—
371C2×C4C2×D4×D1712yes—
372C2×C4C2×#6325612yes—
373C2×C4#1058769612yes—
374C8—684no—
375C22—24no—
376C22—22no—
377C22—22no—
378C22—24no—
379C22—688no—
380C22—684no—
381C22—13616no—
382C22—1362no—
383C22—1362no—
384C22—1364no—
385C22#1176316614yescomplemented
386C22C2×#1058762311yescharacteristic
387C22C2×#1058763311yescharacteristic
388C22#1176337311yescharacteristic
389C22C2×#1058763612yes—
390C22#1176360712yes—
391C4—24no—
392C4—22no—
393C4—42no—
394C4—82no—
395C4—82no—
396C4—684no—
397C4#1176345014yes—
398C2—24no—
399C2—1364no—
400C2#1453243611yescharacteristic
401C2#1453800012yes—
402C2C2×#1176316611yescharacteristic
403C2#1453917311yescharacteristic
404C2#1453974512yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2851C2
socleC23×C17C22×D4
prime radicalC23×D68C22
hypercenter#2851D17
hypocenterC17#25859
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2851

Complements

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

NH ≅ G/Ntype
2. C2×#11763166C2semidirect
3. #14530025C2semidirect
4. #14529593C2semidirect
5. #14528718C2semidirect
7. C4×C22⋊#3622C2semidirect
8. C17×#2851C2semidirect
17. C2×#10587481C22semidirect
18. C2×#10587481C22semidirect
22. C2×C4×#3622C22semidirect
23. #11762564C22semidirect
24. C2×C17×#125C22semidirect
26. C17×#389C22semidirect
29. C17×#382C22semidirect
175. #2851D17semidirect
185. C22×C17#488semidirect
206. C2×#125C2×D17semidirect
220. #389C2×D17semidirect
223. #382C2×D17semidirect
275. C17#25859semidirect
385. C22#11763166semidirect

Maximal subgroups

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

subgroupindex
C22×#105875612
C2×#117631662
#145300252
#145295932
#145287182
#145287142
C4×C22⋊#36222
C17×#28512
#2585917