Finite Group Databasebeta

← #17241489

Subgroups — #17241489

Order 4352.

1854 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14537071C212yesmaximal
2#14535979C214yesmaximal, complemented
3#14535689C214yesmaximal, complemented
4#14535442C214yesmaximal, complemented
5#14535250C214yesmaximal, complemented
6#14534898C211yesmaximal, characteristic, complemented
7#14534754C211yesmaximal, characteristic, complemented
8#14533024C214yesmaximal
9#14532919C214yesmaximal, complemented
10#14531517C212yesmaximal, complemented
11C2×C17×#380C211yesmaximal, characteristic
12C2×#10587565C2214yes—
13C2×#10587575C2214yescomplemented
14C2×#10587610C2212yes—
15C2×#10587610C2212yes—
16C2×#10587610C2212yes—
17#11762751C2214yescomplemented
18#11762747C2214yescomplemented
19#11762743C2214yescomplemented
20#11762742C2214yescomplemented
21#11762741C2214yescomplemented
22#11762740C2214yescomplemented
23#11762739C2214yescomplemented
24#11762737C2214yes—
25#11762737C2214yes—
26#11762734C2214yes—
27#11762732C2214yes—
28#11762730C2214yescomplemented
29#11762713C2214yescomplemented
30#11762706C2214yescomplemented
31#11762704C2214yescomplemented
32#11762703C2214yes—
33#11762702C2214yes—
34#11762697C2214yes—
35#11762696C2214yescomplemented
36#11762695C2214yescomplemented
37#11762692C2214yescomplemented
38#11762690C2214yescomplemented
39#11762682C2214yes—
40#11762682C2214yes—
41#11762682C2214yes—
42#11762682C2214yes—
43#11762682C2214yes—
44#11762669C2212yes—
45#11762669C2212yes—
46#11762669C2214yes—
47#11762669C2212yes—
48#11762665C2214yes—
49C4×C22⋊Dic17C2214yes—
50C24×C4×C17C2211yescharacteristic
51C22×C4⋊C4×C17C2212yes—
52C22×C22⋊C4×C17C2212yes—
53C17×#380C2214yes—
54C17×#380C2214yes—
55C2×C17×#95C2211yescharacteristic
56C2×C17×#95C2211yescharacteristic
57C25×C17C2311yescharacteristic
58C23×C4×C17—24no—
59C23×C4×C17—44no—
60C23×C4×C17C2311yescharacteristic
61C23×C4×C17C2312yes—
62C23×C4×C17C2312yes—
63C23×C4×C17C2312yes—
64C23×C4×C17C2312yes—
65C23×C4×C17C2311yescharacteristic
66C22×Dic34C2312yes—
67C2×C4⋊C4×C17—44no—
68C2×C4⋊C4×C17C2314yes—
69C2×C4⋊C4×C17C2314yes—
70C2×C22⋊C4×C17—44no—
71C2×C22⋊C4×C17C2314yes—
72C2×C22⋊C4×C17C2314yes—
73C2×C22⋊Dic17C2314yes—
74C2×C22⋊Dic17C2314yes—
75C2×C22⋊Dic17C2314yes—
76C2×C22⋊Dic17C2314yes—
77C2×C22⋊Dic17C2314yes—
78C2×C22⋊Dic17C2314yes—
79C2×C4⋊Dic17C2312yes—
80C2×C4⋊Dic17C2314yes—
81C2×C4⋊Dic17C2312yes—
82C2×C4⋊Dic17C2314yes—
83C2×C4⋊Dic17C2312yes—
84#10587610—28no—
85#10587610—44no—
86#10587610—44no—
87C2×#63225C2314yes—
88C2×#63225C2314yes—
89C2×#63225C2314yes—
90C2×#63225C2314yes—
91C2×#63225C2314yes—
92C2×#63225C2314yes—
93C2×#63225C2314yes—
94C2×C4×Dic17C2314yes—
95C2×C4×Dic17C2314yes—
96#10587575—48no—
97#10587565—48no—
98C17×#95C2314yes—
99C17×#95C2314yes—
100#10587502C2314yes—
101#10587502C2314yes—
102#10587502C2314yes—
103#10587502C2314yes—
104#10587502C2314yes—
105#10587502C2314yes—
106#10587502C2314yes—
107#10587502C2314yes—
108#10587502C2314yes—
109#10587502C2314yes—
110#10587502C2314yes—
111C24×C17—22no—
112C24×C17—22no—
113C24×C17—22no—
114C24×C17—24no—
115C24×C17—42no—
116C24×C17C2411yescharacteristic
117C24×C17C2411yescharacteristic
118C24×C17C2411yescharacteristic
119C22×C4×C17—22no—
120C22×C4×C17—24no—
121C22×C4×C17—24no—
122C22×C4×C17—24no—
123C22×C4×C17—24no—
124C22×C4×C17—24no—
125C22×C4×C17—24no—
126C22×C4×C17—24no—
127C22×C4×C17—42no—
128C22×C4×C17—42no—
129C22×C4×C17—42no—
130C22×C4×C17—42no—
131C22×C4×C17—42no—
132C22×C4×C17—42no—
133C22×C4×C17—42no—
134C22×C4×C17—416no—
135C22×C4×C17#4114yes—
136C22×C4×C17C2414yes—
137C22×C4×C17C2414yes—
138C22×C4×C17C2412yes—
139C22×C4×C17C2412yes—
140C22×Dic17C2414yes—
141C22×Dic17C2414yes—
142C22×Dic17C2414yes—
143C22×Dic17C2414yes—
144C2×Dic34—24no—
145C2×Dic34—42no—
146C2×Dic34—42no—
147C4⋊C4×C17—28no—
148C4⋊C4×C17—44no—
149C22⋊C4×C17—28no—
150C22⋊C4×C17—44no—
151C22⋊Dic17—28no—
152C22⋊Dic17—28no—
153C22⋊Dic17—28no—
154C22⋊Dic17—28no—
155C22⋊Dic17—44no—
156C22⋊Dic17—44no—
157C4⋊Dic17—24no—
158C4⋊Dic17—44no—
159C4⋊Dic17—44no—
160C4⋊Dic17—42no—
161C4⋊Dic17—42no—
162#63225—28no—
163#63225—28no—
164#63225—28no—
165#63225—44no—
166#63225—44no—
167#63225—44no—
168#63225—44no—
169C4×Dic17—44no—
170C4×Dic17—44no—
171#36728—171nomaximal, Sylow
172C23×C17—22no—
173C23×C17—21no—
174C23×C17—21no—
175C23×C17—28no—
176C23×C17—21no—
177C23×C17—21no—
178C23×C17—22no—
179C23×C17—22no—
180C23×C17—21no—
181C23×C17—22no—
182C23×C17—22no—
183C23×C17—21no—
184C23×C17—48no—
185C23×C17—48no—
186C23×C17—44no—
187C23×C17—44no—
188C23×C17C2×#4112yes—
189C23×C17H(5,𝔽2)12yes—
190C23×C172−1+412yes—
191C23×C172−1+414yes—
192C23×C17C2511yescharacteristic
193C2×C4×C17—28no—
194C2×C4×C17—22no—
195C2×C4×C17—22no—
196C2×C4×C17—22no—
197C2×C4×C17—22no—
198C2×C4×C17—24no—
199C2×C4×C17—24no—
200C2×C4×C17—24no—
201C2×C4×C17—24no—
202C2×C4×C17—24no—
203C2×C4×C17—24no—
204C2×C4×C17—44no—
205C2×C4×C17—48no—
206C2×C4×C17—48no—
207C2×C4×C17—84no—
208C2×C4×C17—84no—
209C2×C4×C17C2×#4114yes—
210C2×C4×C17H(5,𝔽2)14yes—
211C2×Dic17—24no—
212C2×Dic17—24no—
213C2×Dic17—24no—
214C2×Dic17—24no—
215C2×Dic17—24no—
216C2×Dic17—24no—
217C2×Dic17—24no—
218C2×Dic17—24no—
219C2×Dic17—24no—
220C2×Dic17—24no—
221C2×Dic17—24no—
222C2×Dic17—24no—
223Dic34—84no—
224#2848—172no—
225#2747—174no—
226#2722—174no—
227#2704—174no—
228#2690—174no—
229#2659—171no—
230#2646—171no—
231#2480—174no—
232#2473—174no—
233#2329—172no—
234C2×#380D1711yescharacteristic
235C22×C17—22no—
236C22×C17—22no—
237C22×C17—24no—
238C22×C17—24no—
239C22×C17—22no—
240C22×C17—22no—
241C22×C17—24no—
242C22×C17—24no—
243C22×C17—416no—
244C22×C17—41no—
245C22×C17—41no—
246C22×C17—41no—
247C22×C17—41no—
248C22×C17—42no—
249C22×C17—42no—
250C22×C17#54114yes—
251C22×C17C22×#4111yescharacteristic
252C22×C17C2×H(5,𝔽2)12yes—
253C22×C17C2×H(5,𝔽2)11yescharacteristic
254C22×C17C2×2−1+412yes—
255C22×C17C2×2−1+411yescharacteristic
256C4×C17—24no—
257C4×C17—24no—
258C4×C17—44no—
259C4×C17—44no—
260Dic17—44no—
261Dic17—44no—
262Dic17—44no—
263Dic17—44no—
264C24×C4C2×D1711yescharacteristic
265C2×#126—174no—
266C2×#125—174no—
267C2×C22⋊Q8—172no—
268C2×C22⋊Q8—172no—
269C2×C22⋊Q8—172no—
270C22×C4⋊C4C2×D1712yes—
271C22×C22⋊C4C2×D1712yes—
272#400—174no—
273#400—174no—
274#398—174no—
275#398—174no—
276#398—174no—
277#398—174no—
278#398—174no—
279#397—174no—
280#397—174no—
281#397—174no—
282#396—174no—
283#395—174no—
284#391—174no—
285#389—174no—
286#389—174no—
287#388—174no—
288#388—174no—
289#388—174no—
290#387—174no—
291#387—174no—
292#387—174no—
293#387—174no—
294#385—174no—
295#385—174no—
296#385—174no—
297#385—174no—
298#385—174no—
299#380—172no—
300#380—172no—
301#380—172no—
302#380—174no—
303#380C2×D1714yes—
304#380C2×D1714yes—
305#379—174no—
306C4×C22⋊C4—174no—
307C2×#95C2×D1711yescharacteristic
308C2×#95C2×D1711yescharacteristic
309C2×C17—24no—
310C2×C17—42no—
311C2×C17—42no—
312C2×C17C2×#54111yescharacteristic
313C2×C17#349011yescharacteristic
314C2×C17#349211yescharacteristic
315C2×C17#353612yes—
316C2×C17#353812yes—
317C25C22×D1711yescharacteristic
318C22×Q8—172no—
319C23×C4—24no—
320C23×C4—44no—
321C23×C4C22×D1711yescharacteristic
322C23×C4C22×D1712yes—
323C23×C4C22×D1712yes—
324C23×C4C22×D1712yes—
325C23×C4C22×D1712yes—
326C23×C4C22×D1711yescharacteristic
327#126—688no—
328#125—688no—
329C22⋊Q8—348no—
330C22⋊Q8—684no—
331C22⋊Q8—684no—
332C2×C4⋊C4—44no—
333C2×C4⋊C4—174no—
334C2×C4⋊C4—174no—
335C2×C4⋊C4—172no—
336C2×C4⋊C4—174no—
337C2×C4⋊C4—172no—
338C2×C4⋊C4—172no—
339C2×C4⋊C4—174no—
340C2×C4⋊C4—174no—
341C2×C4⋊C4—174no—
342C2×C4⋊C4—174no—
343C2×C4⋊C4—174no—
344C2×C4⋊C4—174no—
345C2×C4⋊C4C22×D1714yes—
346C2×C4⋊C4C22×D1714yes—
347C2×C22⋊C4—44no—
348C2×C22⋊C4—174no—
349C2×C22⋊C4—174no—
350C2×C22⋊C4—174no—
351C2×C22⋊C4—174no—
352C2×C22⋊C4—174no—
353C2×C22⋊C4—174no—
354C2×C22⋊C4C22×D1714yes—
355C2×C22⋊C4C22×D1714yes—
356C2×C42—174no—
357C2×C42—174no—
358#95—174no—
359#95—174no—
360#95—174no—
361#95—174no—
362#95—174no—
363#95—174no—
364#95—174no—
365#95—174no—
366#95—174no—
367#95—174no—
368#95—174no—
369#95C22×D1714yes—
370#95C22×D1714yes—
371C17#3672811yescharacteristic, complemented, Sylow
372C24—22no—
373C24—22no—
374C24—22no—
375C24—24no—
376C24—42no—
377C24C23×D1711yescharacteristic
378C24C23×D1711yescharacteristic
379C24C23×D1711yescharacteristic
380C2×Q8—344no—
381C2×Q8—682no—
382C2×Q8—682no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—416no—
399C22×C4—174no—
400C22×C4—174no—
401C22×C4—174no—
402C22×C4—174no—
403C22×C4#6325214yes—
404C22×C4C23×D1714yes—
405C22×C4C23×D1714yes—
406C22×C4C23×D1712yes—
407C22×C4C23×D1712yes—
408C4⋊C4—28no—
409C4⋊C4—44no—
410C4⋊C4—348no—
411C4⋊C4—344no—
412C4⋊C4—348no—
413C4⋊C4—348no—
414C4⋊C4—682no—
415C4⋊C4—682no—
416C4⋊C4—684no—
417C4⋊C4—684no—
418C4⋊C4—684no—
419C4⋊C4—684no—
420C4⋊C4—684no—
421C4⋊C4—684no—
422C22⋊C4—28no—
423C22⋊C4—44no—
424C22⋊C4—348no—
425C22⋊C4—348no—
426C22⋊C4—348no—
427C22⋊C4—348no—
428C22⋊C4—684no—
429C22⋊C4—684no—
430C42—684no—
431C42—684no—
432C23—22no—
433C23—21no—
434C23—21no—
435C23—28no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—21no—
444C23—48no—
445C23—48no—
446C23—44no—
447C23—44no—
448C23C2×#6325212yes—
449C23#1058769112yes—
450C23#1058769412yes—
451C23#1058769714yes—
452C23C24×D1711yescharacteristic
453Q8—1364no—
454C2×C4—28no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—44no—
466C2×C4—48no—
467C2×C4—48no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4—344no—
471C2×C4—344no—
472C2×C4—344no—
473C2×C4—344no—
474C2×C4—344no—
475C2×C4—344no—
476C2×C4—344no—
477C2×C4—344no—
478C2×C4—344no—
479C2×C4—344no—
480C2×C4—344no—
481C2×C4—344no—
482C2×C4C2×#6325214yes—
483C2×C4#1058769114yes—
484C22—22no—
485C22—22no—
486C22—24no—
487C22—24no—
488C22—22no—
489C22—22no—
490C22—24no—
491C22—24no—
492C22—416no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—42no—
498C22—42no—
499C22#1176342114yes—
500C22C22×#6325211yescharacteristic
501C22C2×#1058769112yes—
502C22C2×#1058769111yescharacteristic
503C22C2×#1058769712yes—
504C22C2×#1058769411yescharacteristic
505C4—24no—
506C4—24no—
507C4—44no—
508C4—44no—
509C4—684no—
510C4—684no—
511C4—684no—
512C4—684no—
513C2—24no—
514C2—42no—
515C2—42no—
516C2C2×#1176342111yescharacteristic
517C2#1454247311yescharacteristic
518C2#1454249311yescharacteristic
519C2#1454300612yes—
520C2#1454302912yes—
subgroupG/H
center, FrattiniC23C24×D17
commutator, socle, cosocleC23×C17C25
FittingC2×C17×#380C2
prime radicalC25×C17C23
hypercenterC2×#380D17
hypocenterC17#36728
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. #14535979C2semidirect
3. #14535689C2semidirect
4. #14535442C2semidirect
5. #14535250C2semidirect
6. #14534898C2semidirect
7. #14534754C2semidirect
9. #14532919C2semidirect
10. #14531517C2semidirect
13. C2×#10587575C22semidirect
17. #11762751C22semidirect
18. #11762747C22semidirect
19. #11762743C22semidirect
20. #11762742C22semidirect
21. #11762741C22semidirect
22. #11762740C22semidirect
23. #11762739C22semidirect
28. #11762730C22semidirect
29. #11762713C22semidirect
30. #11762706C22semidirect
31. #11762704C22semidirect
35. #11762696C22semidirect
36. #11762695C22semidirect
37. #11762692C22semidirect
38. #11762690C22semidirect
371. C17#36728semidirect

Maximal subgroups

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

subgroupindex
#145370712
#145359792
#145356892
#145354422
#145352502
#145348982
#145347542
#145330242
#145329192
#145315172
C2×C17×#3802
#3672817