Finite Group Databasebeta

← #17090548

Subgroups — #17090548

Order 4352.

1334 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763359C211yesmaximal, characteristic
2C2×#11763179C211yesmaximal, characteristic, complemented
3C2×#11763175C211yesmaximal, characteristic, complemented
4#14530025C214yesmaximal, complemented
5#14529607C211yesmaximal, characteristic, complemented
6#14529603C211yesmaximal, characteristic, complemented
7#14528722C214yesmaximal, complemented
8#14528546C211yesmaximal, characteristic, complemented
9C17×#2851C211yesmaximal, characteristic, complemented
10C22×#63252C2212yes—
11#11763359—24no—
12#11763359—24no—
13C2×#10587555C2211yescharacteristic
14C2×#10587560C2211yescharacteristic
15C2×#10587561C2212yes—
16C2×#10587616—22no—
17C2×C4×D68—22no—
18#11763179—24no—
19#11763175—24no—
20C2×#10587481C2214yes—
21C2×#10587481C2214yes—
22C2×#10587480C2214yes—
23C2×#10587480C2214yes—
24C2×C22⋊#3622C2212yes—
25C2×#10587474C2211yescharacteristic
26C2×#10587474C2211yescharacteristic
27#11762564C2212yes—
28C2×C17×#125C2211yescharacteristic
29C22×C42×C17C2211yescharacteristic
30C17×#389C2212yes—
31C17×#387—22no—
32C17×#385—22no—
33C17×#382C2211yescharacteristic
34C17×#380C2212yes—
35C23×C4×C17D412yes—
36C23×C4×C17C2311yescharacteristic
37C22×C22⋊D17—22no—
38C2×#63252—24no—
39C2×#63252—28no—
40C2×#63252—24no—
41C2×#63252—44no—
42C22×D68—22no—
43C22×D68C2312yes—
44C22×C4×D17—22no—
45C22×Dic34C2312yes—
46C17×#125—22no—
47C17×#125—22no—
48C2×C4⋊C4×C17—22no—
49C2×C4⋊C4×C17—22no—
50C2×C4⋊C4×C17C2312yes—
51C2×C4⋊C4×C17C2312yes—
52C2×C22⋊C4×C17—22no—
53C2×C42×C17—22no—
54C2×C42×C17—22no—
55C2×C42×C17D412yes—
56C2×C42×C17C2311yescharacteristic
57C2×C42×C17C2311yescharacteristic
58#10587616—216no—
59C2×#63227—24no—
60C2×C4⋊Dic17—22no—
61C22×#3622C2312yes—
62C22×#3622C2312yes—
63#10587561—24no—
64#10587561—24no—
65#10587560—22no—
66#10587560D414yes—
67C4×D68—216no—
68#10587555—22no—
69#10587555D414yes—
70C17×#95—22no—
71C17×#95—22no—
72C17×#95—22no—
73C22⋊#3622—24no—
74#10587481—28no—
75#10587481—44no—
76#10587480—28no—
77#10587480—44no—
78#10587474—22no—
79#10587474—22no—
80C24×C17C2×D411yescharacteristic
81C23×D17—22no—
82C22×C4×C17—21no—
83C22×C4×C17—21no—
84C22×C4×C17—22no—
85C22×C4×C17—24no—
86C22×C4×C17—22no—
87C22×C4×C17—22no—
88C22×C4×C17—42no—
89C22×C4×C17—42no—
90C22×C4×C17C2×D412yes—
91C22×C4×C17C2×D412yes—
92C22×C4×C17C2×D411yescharacteristic
93C22×C4×C17#4114yes—
94C22×C4×C17C2411yescharacteristic
95C2×C22⋊D17—28no—
96C2×C22⋊D17—44no—
97C2×C22⋊D17—44no—
98C22×Dic17—22no—
99#63252—416no—
100#63252—416no—
101C2×D68—24no—
102C2×D68—24no—
103C2×D68—28no—
104C2×D68—22no—
105C2×D68—44no—
106C2×D68—44no—
107C2×D68C2×D414yes—
108C2×C4×D17—28no—
109C2×C4×D17—44no—
110C2×C4×D17—44no—
111C2×Dic34—24no—
112C2×Dic34—24no—
113C2×Dic34—22no—
114C2×Dic34C2×D414yes—
115C4⋊C4×C17—22no—
116C4⋊C4×C17—22no—
117C4⋊C4×C17—24no—
118C4⋊C4×C17—44no—
119C4⋊C4×C17—44no—
120C22⋊C4×C17—44no—
121C42×C17—24no—
122C42×C17—21no—
123C42×C17—21no—
124C42×C17C2×D412yes—
125C42×C17C2×D412yes—
126#63227—216no—
127C4⋊Dic17—28no—
128C2×#3622—22no—
129C2×#3622—22no—
130C2×#3622—22no—
131C2×#3622—22no—
132C2×#3622—22no—
133C2×#3622—22no—
134#25862—171nomaximal, Sylow
135C23×C17—21no—
136C23×C17—21no—
137C23×C17—22no—
138C23×C17—22no—
139C23×C17#12412yes—
140C23×C17C22×D411yescharacteristic
141C22×D17—24no—
142C22×D17—42no—
143C22×D17—42no—
144C22×D17—44no—
145C22×D17—44no—
146C2×C4×C17—21no—
147C2×C4×C17—21no—
148C2×C4×C17—22no—
149C2×C4×C17—22no—
150C2×C4×C17—24no—
151C2×C4×C17—24no—
152C2×C4×C17—22no—
153C2×C4×C17—22no—
154C2×C4×C17—22no—
155C2×C4×C17—48no—
156C2×C4×C17—42no—
157C2×C4×C17—42no—
158C2×C4×C17—42no—
159C2×C4×C17—42no—
160C2×C4×C17—42no—
161C2×C4×C17—42no—
162C2×C4×C17#12014yes—
163C2×C4×C17#12412yes—
164C2×C4×C17C22×D411yescharacteristic
165C2×C4×C17C22×D411yescharacteristic
166C2×C4×C17C2×#4112yes—
167C2×C4×C17H(5,𝔽2)12yes—
168C22⋊D17—416no—
169C2×Dic17—24no—
170C2×Dic17—42no—
171C2×Dic17—42no—
172D68—28no—
173D68—416no—
174D68—44no—
175C4×D17—416no—
176Dic34—28no—
177Dic34—44no—
178#3622—42no—
179#3622—42no—
180C2×#532—171no—
181C2×#491—171no—
182C2×#490—171no—
183#2851D1711yescharacteristic, complemented
184#2042—174no—
185#1961—171no—
186#1960—171no—
187#1788—174no—
188#1749—171no—
189C22×C17—22no—
190C22×C17—22no—
191C22×C17—22no—
192C22×C17—42no—
193C22×C17—42no—
194C22×C17C2×#12011yescharacteristic
195C22×C17C2×#12411yescharacteristic
196C22×C17#53511yescharacteristic
197C22×C17#57512yes—
198C22×C17#57512yes—
199C2×D17—44no—
200C2×D17—48no—
201C2×D17—82no—
202C2×D17—82no—
203C2×D17—84no—
204C4×C17—22no—
205C4×C17—24no—
206C4×C17—42no—
207C4×C17—82no—
208C4×C17—82no—
209C4×C17#54612yes—
210C4×C17#54612yes—
211Dic17—44no—
212C22×#41—172no—
213#532—344no—
214#532—344no—
215C2×C4⋊Q8—171no—
216C2×#127—171no—
217C2×#125C2×D1711yescharacteristic
218C2×#124—172no—
219C2×#121—342no—
220C2×C4×D4—342no—
221C22×C42C2×D1711yescharacteristic
222#491—344no—
223#490—344no—
224C2×#103—174no—
225C2×#103—174no—
226C2×#102—174no—
227C2×#102—174no—
228C2×C22⋊C8—172no—
229C2×#97—171no—
230C2×#97—171no—
231#389C2×D1712yes—
232#387—22no—
233#385—22no—
234#382C2×D1711yescharacteristic
235#380C2×D1712yes—
236#336—172no—
237C2×C17—42no—
238C2×C17#243511yescharacteristic
239C2×C17#300512yes—
240C2×C17#313511yescharacteristic
241C2×C17#314211yescharacteristic
242C2×C17#321912yes—
243D17—84no—
244C2×#41—344no—
245C2×#41—344no—
246C2×#41—348no—
247C2×#41—684no—
248C22×Q8—172no—
249C22×D4—172no—
250C22×D4—342no—
251C22×D4—342no—
252C23×C4—342no—
253C23×C4C22⋊D1712yes—
254C23×C4C22×D1711yescharacteristic
255C22×C8—172no—
256C22×C8—172no—
257C4⋊Q8—174no—
258C4⋊Q8—342no—
259#127—174no—
260#127—342no—
261#125—22no—
262#125—22no—
263#124—344no—
264#124—344no—
265#121—3416no—
266C4×D4—3416no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—342no—
270C2×C4⋊C4C22×D1712yes—
271C2×C4⋊C4C22×D1712yes—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—344no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42C22⋊D1712yes—
277C2×C42C22×D1711yescharacteristic
278C2×C42C22×D1711yescharacteristic
279#103—348no—
280#103—684no—
281#102—348no—
282#102—684no—
283C22⋊C8—344no—
284#97—342no—
285#97—342no—
286#95—22no—
287#95—22no—
288#95—22no—
289C17#2586211yescharacteristic, complemented, Sylow
290C24—342no—
291C24C2×C22⋊D1711yescharacteristic
292#41—6816no—
293#41—6816no—
294C2×Q8—174no—
295C2×Q8—342no—
296C2×Q8—344no—
297C2×Q8—344no—
298C2×D4—174no—
299C2×D4—348no—
300C2×D4—342no—
301C2×D4—344no—
302C2×D4—344no—
303C2×D4—348no—
304C2×D4—684no—
305C2×D4—684no—
306C2×D4—684no—
307C2×D4—684no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—348no—
317C22×C4—342no—
318C22×C4—684no—
319C22×C4—684no—
320C22×C4#6325614yes—
321C22×C4C2×C22⋊D1712yes—
322C22×C4C2×C22⋊D1712yes—
323C22×C4C2×C22⋊D1711yescharacteristic
324C22×C4C23×D1711yescharacteristic
325C2×C8—342no—
326C2×C8—342no—
327C2×C8—342no—
328C2×C8—342no—
329C2×C8—342no—
330C2×C8—342no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—24no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—348no—
337C22⋊C4—44no—
338C22⋊C4—3416no—
339C42—21no—
340C42—21no—
341C42—24no—
342C42D4×D1712yes—
343C42D4×D1712yes—
344C23—21no—
345C23—22no—
346C23—22no—
347C23—21no—
348C23—344no—
349C23—682no—
350C23—684no—
351C23—684no—
352C23—682no—
353C23#1058763312yes—
354C23C22×C22⋊D1711yescharacteristic
355Q8—348no—
356Q8—684no—
357D4—348no—
358D4—6816no—
359D4—6816no—
360D4—684no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—48no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—344no—
378C2×C4—6816no—
379C2×C4—682no—
380C2×C4—682no—
381C2×C4#1058762314yes—
382C2×C4#1058763312yes—
383C2×C4C2×D4×D1711yescharacteristic
384C2×C4C2×D4×D1711yescharacteristic
385C2×C4C2×#6325612yes—
386C2×C4#1058769612yes—
387C8—682no—
388C8—682no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—42no—
393C22—42no—
394C22—688no—
395C22—684no—
396C22—1364no—
397C22—1362no—
398C22—1362no—
399C22C2×#1058762311yescharacteristic
400C22C2×#1058763311yescharacteristic
401C22#1176337311yescharacteristic
402C22#1176360712yes—
403C22#1176360712yes—
404C4—24no—
405C4—22no—
406C4—42no—
407C4—82no—
408C4—82no—
409C4—684no—
410C4#1176345012yes—
411C4#1176345012yes—
412C2—42no—
413C2—1364no—
414C2#1453243611yescharacteristic
415C2#1453802412yes—
416C2#1453913211yescharacteristic
417C2#1453918711yescharacteristic
418C2#1453981712yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2851C2
socleC23×C17C22×D4
prime radicalC2×#11763359C2
hypercenter#2851D17
hypocenterC17#25862
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×#11763179C2semidirect
3. C2×#11763175C2semidirect
4. #14530025C2semidirect
5. #14529607C2semidirect
6. #14529603C2semidirect
7. #14528722C2semidirect
8. #14528546C2semidirect
9. C17×#2851C2semidirect
183. #2851D17semidirect
289. C17#25862semidirect

Maximal subgroups

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

subgroupindex
C2×#117633592
C2×#117631792
C2×#117631752
#145300252
#145296072
#145296032
#145287222
#145285462
C17×#28512
#2586217