Finite Group Databasebeta

← #17194778

Subgroups — #17194778

Order 4352.

2110 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763365C211yesmaximal, characteristic
2#14531868C214yesmaximal
3#14531861C214yesmaximal
4#14531738C214yesmaximal, complemented
5#14531737C214yesmaximal, complemented
6#14531531C212yesmaximal, complemented
7#14531528C212yesmaximal, complemented
8#14531406C212yesmaximal
9#125×Dic17C211yesmaximal, characteristic, complemented
10C4×#10587556C211yesmaximal, characteristic, complemented
11#14531339C211yesmaximal, characteristic, complemented
12D17×#382C211yesmaximal, characteristic
13#14531138C212yesmaximal
14C4×#10587558C211yesmaximal, characteristic
15C4×C17×#125C211yesmaximal, characteristic, complemented
16#11763365C4116yescomplemented
17C2×#10587574C2212yes—
18C2×#10587556C2211yescharacteristic
19C2×#10587582C2214yes—
20C2×#10587573C2214yes—
21C2×#10587578C2212yes—
22C2×#10587558C2211yescharacteristic
23C2×#10587558C2211yescharacteristic
24C2×C4⋊C4×D17C2212yes—
25C2×C42×D17C2211yescharacteristic
26#11762715C2214yes—
27#11762714C2214yes—
28#11762713C2214yes—
29#11762713C2214yes—
30#11762708C2212yes—
31#11762707C2212yes—
32#11762705C2212yes—
33#11762696C2214yes—
34#11762695C2214yes—
35#11762693C2214yes—
36#11762693C2214yes—
37#11762691C2214yes—
38#11762691C2214yes—
39#11762690C2214yes—
40#11762683C2214yes—
41#11762680C2214yes—
42#11762678C2214yes—
43#11762675C2211yescharacteristic
44#11762674C2212yes—
45#11762673C2214yes—
46#11762673C2214yes—
47C4⋊C4×Dic17C2212yes—
48C4⋊C4×Dic17C2214yes—
49C4⋊C4×Dic17C2212yes—
50C4×#63225C2214yes—
51C4×#63225C2212yes—
52C4×#63225C2212yes—
53C4×#63225C2214yes—
54C4×C4⋊Dic17C2212yes—
55#11762660C2214yes—
56#11762660C2214yes—
57C4×#63227C2212yes—
58C4×#63227C2212yes—
59C4×#63227C2214yes—
60#11762656C2214yes—
61#11762655C2211yescharacteristic
62#11762655C2211yescharacteristic
63#95×D17C2214yes—
64C42×Dic17C2211yescharacteristic
65C2×C17×#125C2211yescharacteristic
66C17×#389C2212yes—
67C17×#387C2214yes—
68C17×#382C2211yescharacteristic
69C4×C4⋊C4×C17C2214yes—
70C4×C4⋊C4×C17C2212yes—
71C43×C17C2211yescharacteristic
72C22×C4×D17C2314yes—
73C22×C4×D17C2312yes—
74C22×C4×D17C2311yescharacteristic
75C22×Dic34C2314yes—
76C17×#125C2×C418yescomplemented
77C2×C4⋊C4×C17C2312yes—
78C2×C4⋊C4×C17C2314yes—
79C2×C4⋊C4×C17C2314yes—
80C2×C4⋊C4×C17C2312yes—
81C2×C42×C17C2311yescharacteristic
82C2×C42×C17C2312yes—
83C2×C42×C17C2314yes—
84C2×C42×C17C2312yes—
85C2×C42×C17C2311yescharacteristic
86C2×C42×C17C2311yescharacteristic
87C2×#63227C2312yes—
88C2×#63227C2314yes—
89C2×#63227C2311yescharacteristic
90C2×#63227C2311yescharacteristic
91C2×C4⋊Dic17C2312yes—
92C2×C4⋊Dic17C2314yes—
93C2×C4⋊Dic17C2312yes—
94C2×#63225C2314yes—
95C2×#63225C2312yes—
96C2×#63225C2314yes—
97C2×#63225C2314yes—
98C2×#63225C2314yes—
99C2×#63225C2314yes—
100C2×#63225C2311yescharacteristic
101C2×#63225C2311yescharacteristic
102C2×C4×Dic17C2311yescharacteristic
103C2×C4×Dic17C2314yes—
104C2×C4×Dic17C2312yes—
105C2×C4×Dic17C2314yes—
106C2×C4×Dic17C2312yes—
107C2×C4×Dic17C2314yes—
108C2×C4×Dic17C2312yes—
109C2×C4×Dic17C2311yescharacteristic
110C2×C4×Dic17C2314yes—
111C2×C4×Dic17C2311yescharacteristic
112#10587582C2×C4132yes—
113#10587578C2×C4116yes—
114C4⋊C4×D17—28no—
115#10587574C2×C4116yescomplemented
116#10587573C2×C4132yescomplemented
117#10587558—24no—
118#10587558Q818yescomplemented
119C42×D17C2×C418yes—
120#10587556C2×C418yescomplemented
121C17×#95C2314yes—
122C17×#95C2314yes—
123C17×#95C2314yes—
124#10587502C2312yes—
125#10587502C2314yes—
126#10587502C2314yes—
127#10587502C2314yes—
128#10587502C2314yes—
129#10587502C2312yes—
130#10587502C2314yes—
131#10587502C2314yes—
132#10587502C2312yes—
133#10587502C2314yes—
134#10587502C2314yes—
135#10587502C2314yes—
136#10587502C2312yes—
137C23×D17C2411yescharacteristic
138C22×C4×C17C2414yes—
139C22×C4×C17C2412yes—
140C22×C4×C17C2414yes—
141C22×C4×C17C2412yes—
142C22×C4×C17C2411yescharacteristic
143C22×C4×C17C2411yescharacteristic
144C22×C4×C17C2411yescharacteristic
145C22×Dic17C2414yes—
146C22×Dic17C2412yes—
147C22×Dic17C2412yes—
148C22×Dic17C2414yes—
149C22×Dic17C2411yescharacteristic
150C22×Dic17C2411yescharacteristic
151C22×Dic17C2411yescharacteristic
152C2×C4×D17—28no—
153C2×C4×D17—42no—
154C2×C4×D17—42no—
155C2×C4×D17—44no—
156C2×C4×D17—44no—
157C2×C4×D17C22×C418yes—
158C2×C4×D17C22×C414yes—
159C2×C4×D17C2×Q814yes—
160C2×C4×D17#4114yes—
161C2×Dic34—44no—
162C2×Dic34—44no—
163C2×Dic34C22×C4116yes—
164C4⋊C4×C17—24no—
165C4⋊C4×C17—44no—
166C4⋊C4×C17C22×C4116yes—
167C4⋊C4×C17C22×C418yes—
168C42×C17—22no—
169C42×C17—44no—
170C42×C17—42no—
171C42×C17—42no—
172C42×C17C22×C414yes—
173C42×C17C2×Q814yes—
174#63227—24no—
175#63227C22×C4116yes—
176#63227C2×Q814yes—
177#63227C2×Q814yes—
178C4⋊Dic17—24no—
179C4⋊Dic17—44no—
180C4⋊Dic17C22×C418yes—
181#63225—24no—
182#63225—28no—
183#63225—44no—
184#63225—44no—
185#63225C22×C4116yes—
186#63225C22×C4116yes—
187#63225C2×Q814yes—
188#63225C2×Q814yes—
189C4×Dic17—22no—
190C4×Dic17—44no—
191C4×Dic17—44no—
192C4×Dic17—44no—
193C4×Dic17—42no—
194C4×Dic17—42no—
195C4×Dic17C22×C418yes—
196C4×Dic17C22×C414yes—
197C4×Dic17C22×C4116yes—
198C4×Dic17C2×Q814yes—
199#34215—171nomaximal, Sylow
200C23×C17C2511yescharacteristic
201C22×D17—22no—
202C22×D17—22no—
203C22×D17C23×C412yes—
204C22×D17C22×Q812yes—
205C22×D17C2×#4112yes—
206C2×C4×C17—22no—
207C2×C4×C17—24no—
208C2×C4×C17—22no—
209C2×C4×C17—22no—
210C2×C4×C17—22no—
211C2×C4×C17—24no—
212C2×C4×C17—22no—
213C2×C4×C17—24no—
214C2×C4×C17—22no—
215C2×C4×C17—24no—
216C2×C4×C17—22no—
217C2×C4×C17—24no—
218C2×C4×C17C23×C414yes—
219C2×C4×C17C23×C418yes—
220C2×C4×C17C23×C412yes—
221C2×C4×C17C22×Q812yes—
222C2×C4×C17C22×Q812yes—
223C2×C4×C17C22×Q812yes—
224C2×C4×C17C2×#4112yes—
225C2×Dic17—22no—
226C2×Dic17—22no—
227C2×Dic17—24no—
228C2×Dic17—24no—
229C2×Dic17—22no—
230C2×Dic17—22no—
231C2×Dic17—24no—
232C2×Dic17—22no—
233C2×Dic17—24no—
234C2×Dic17—24no—
235C2×Dic17—22no—
236C2×Dic17—22no—
237C2×Dic17C23×C414yes—
238C2×Dic17C23×C418yes—
239C2×Dic17C23×C412yes—
240C2×Dic17C22×Q812yes—
241C2×Dic17C22×Q812yes—
242C2×Dic17C22×Q812yes—
243C2×Dic17C2×#4112yes—
244C4×D17—48no—
245C4×D17—88no—
246C4×D17C4×Q8116yes—
247Dic34—416no—
248C2×#533—171no—
249#2368—174no—
250#2368—174no—
251#2355—174no—
252#2355—174no—
253#2331—172no—
254#2331—172no—
255#2317—172no—
256C4×#125—171no—
257C4×#125—171no—
258C4×#125—171no—
259C4×#125D1711yescharacteristic, complemented
260C2×#382—171no—
261#2274—172no—
262C4×#117—171no—
263C22×C17C24×C411yescharacteristic
264C22×C17C23×Q811yescharacteristic
265C22×C17C22×#4111yescharacteristic
266C22×C17#58512yes—
267C22×C17#58512yes—
268C2×D17—24no—
269C2×D17—22no—
270C2×D17—22no—
271C2×D17—24no—
272C2×D17C2×C4×Q814yes—
273C4×C17—28no—
274C4×C17—24no—
275C4×C17—22no—
276C4×C17—22no—
277C4×C17—44no—
278C4×C17—42no—
279C4×C17C2×C4×Q814yes—
280Dic17—24no—
281Dic17—28no—
282Dic17—22no—
283Dic17—22no—
284Dic17—44no—
285Dic17—42no—
286Dic17C2×C4×Q814yes—
287#533—1716no—
288C2×C4⋊Q8—172no—
289C2×#125—171no—
290C2×#125C2×D1711yescharacteristic
291C2×C22⋊Q8—174no—
292C2×C4×Q8—174no—
293C2×#117—172no—
294C2×#117—171no—
295C2×#117—171no—
296C22×C4⋊C4—172no—
297C22×C42—171no—
298#391—174no—
299#391—174no—
300#391—174no—
301#391—174no—
302#389—172no—
303#389—172no—
304#389—172no—
305#389C2×D1712yes—
306#387—174no—
307#387—174no—
308#387—174no—
309#387—174no—
310#387—174no—
311#387—174no—
312#387—174no—
313#387C2×D1714yes—
314#385—174no—
315#385—174no—
316#384—174no—
317#382—172no—
318#382—171no—
319#382—174no—
320#382—174no—
321#382C2×D1711yescharacteristic
322C4×C4⋊C4—172no—
323C4×C4⋊C4—172no—
324C4×C4⋊C4—174no—
325C4×C4⋊C4—172no—
326C4×C4⋊C4—174no—
327C4×C4⋊C4—172no—
328C4×C4⋊C4—172no—
329C4×C4⋊C4—174no—
330C4×C4⋊C4—174no—
331C4×C4⋊C4—174no—
332C4×C4⋊C4C2×D1712yes—
333C4×C4⋊C4C2×D1714yes—
334C4×C22⋊C4—172no—
335C4×C22⋊C4—172no—
336C4×C22⋊C4—174no—
337C4×C22⋊C4—174no—
338#376—171no—
339#376—171no—
340C2×#95—174no—
341C43—171no—
342C43C2×D1711yescharacteristic
343C2×C17C22×C4×Q811yescharacteristic
344C2×C17#342812yes—
345C2×C17#350411yescharacteristic
346C2×C17#350411yescharacteristic
347C2×C17#351012yes—
348D17—24no—
349C22×Q8—174no—
350C23×C4—171no—
351C23×C4—172no—
352C23×C4—174no—
353C4⋊Q8—1716no—
354#125—178no—
355#125C4×D1718yescomplemented
356C22⋊Q8—1732no—
357C4×Q8—1732no—
358#117—1716no—
359#117—178no—
360#117—344no—
361C2×C4⋊C4—172no—
362C2×C4⋊C4—171no—
363C2×C4⋊C4—171no—
364C2×C4⋊C4—174no—
365C2×C4⋊C4—174no—
366C2×C4⋊C4—174no—
367C2×C4⋊C4—172no—
368C2×C4⋊C4—172no—
369C2×C4⋊C4—174no—
370C2×C4⋊C4—174no—
371C2×C4⋊C4—174no—
372C2×C4⋊C4—348no—
373C2×C4⋊C4C22×D1712yes—
374C2×C4⋊C4C22×D1714yes—
375C2×C4⋊C4C22×D1712yes—
376C2×C4⋊C4C22×D1714yes—
377C2×C22⋊C4—174no—
378C2×C22⋊C4—171no—
379C2×C22⋊C4—171no—
380C2×C22⋊C4—172no—
381C2×C42—174no—
382C2×C42—171no—
383C2×C42—172no—
384C2×C42—171no—
385C2×C42—178no—
386C2×C42—171no—
387C2×C42—172no—
388C2×C42—172no—
389C2×C42—174no—
390C2×C42—174no—
391C2×C42—174no—
392C2×C42C22×D1711yescharacteristic
393C2×C42C22×D1712yes—
394C2×C42C22×D1712yes—
395C2×C42C22×D1714yes—
396C2×C42C22×D1711yescharacteristic
397C2×C42C22×D1711yescharacteristic
398#95—174no—
399#95—174no—
400#95—174no—
401#95—172no—
402#95—172no—
403#95—172no—
404#95—172no—
405#95—174no—
406#95—174no—
407#95—174no—
408#95—174no—
409#95—174no—
410#95—174no—
411#95C22×D1714yes—
412#95C22×D1714yes—
413#95C22×D1714yes—
414C17#3421511yescharacteristic, complemented, Sylow
415C24—171no—
416C2×Q8—1716no—
417C2×Q8—684no—
418C2×Q8—684no—
419C22×C4—171no—
420C22×C4—174no—
421C22×C4—174no—
422C22×C4—174no—
423C22×C4—172no—
424C22×C4—171no—
425C22×C4—171no—
426C22×C4—174no—
427C22×C4—178no—
428C22×C4—174no—
429C22×C4—172no—
430C22×C4—348no—
431C22×C4—682no—
432C22×C4—682no—
433C22×C4—684no—
434C22×C4—684no—
435C22×C4C23×D1714yes—
436C22×C4C23×D1712yes—
437C22×C4C23×D1712yes—
438C22×C4C23×D1714yes—
439C22×C4C23×D1711yescharacteristic
440C22×C4C23×D1711yescharacteristic
441C22×C4C23×D1711yescharacteristic
442C4⋊C4—24no—
443C4⋊C4—44no—
444C4⋊C4—178no—
445C4⋊C4—1716no—
446C4⋊C4—1716no—
447C4⋊C4—174no—
448C4⋊C4—174no—
449C4⋊C4—348no—
450C4⋊C4—344no—
451C4⋊C4—344no—
452C4⋊C4—684no—
453C4⋊C4—684no—
454C4⋊C4—684no—
455C4⋊C4C2×C4×D1718yes—
456C4⋊C4C2×C4×D17116yes—
457C22⋊C4—1716no—
458C22⋊C4—174no—
459C22⋊C4—174no—
460C22⋊C4—344no—
461C42—22no—
462C42—42no—
463C42—44no—
464C42—42no—
465C42—178no—
466C42—174no—
467C42—1716no—
468C42—174no—
469C42—342no—
470C42—684no—
471C42—682no—
472C42—682no—
473C42—684no—
474C42—684no—
475C42C2×C4×D1714yes—
476C42Q8×D1714yes—
477C23—172no—
478C23—172no—
479C23—172no—
480C23—342no—
481C23—342no—
482C23C24×D1711yescharacteristic
483Q8—6816no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—172no—
497C2×C4—172no—
498C2×C4—172no—
499C2×C4—1716no—
500C2×C4—174no—
501C2×C4—178no—
502C2×C4—172no—
503C2×C4—172no—
504C2×C4—342no—
505C2×C4—342no—
506C2×C4—344no—
507C2×C4—344no—
508C2×C4—342no—
509C2×C4—342no—
510C2×C4—342no—
511C2×C4—344no—
512C2×C4—342no—
513C2×C4—342no—
514C2×C4—344no—
515C2×C4—344no—
516C2×C4—688no—
517C2×C4—1368no—
518C2×C4C22×C4×D1714yes—
519C2×C4C22×C4×D1718yes—
520C2×C4C22×C4×D1712yes—
521C2×C4C2×Q8×D1712yes—
522C2×C4C2×Q8×D1712yes—
523C2×C4C2×Q8×D1712yes—
524C2×C4#41×D1712yes—
525C22—174no—
526C22—342no—
527C22—342no—
528C22—344no—
529C22—344no—
530C22C23×C4×D1711yescharacteristic
531C22C22×Q8×D1711yescharacteristic
532C22C2×#41×D1711yescharacteristic
533C22#1176364512yes—
534C22#1176364512yes—
535C4—22no—
536C4—22no—
537C4—24no—
538C4—28no—
539C4—44no—
540C4—42no—
541C4—174no—
542C4—342no—
543C4—342no—
544C4—344no—
545C4—348no—
546C4—684no—
547C4—682no—
548C4C4×Q8×D1714yes—
549C2—344no—
550C2C2×C4×Q8×D1711yescharacteristic
551C2#1454198312yes—
552C2#1454260511yescharacteristic
553C2#1454260811yescharacteristic
554C2#1454267312yes—
subgroupG/H
center, FrattiniC23C24×D17
commutatorC22×C17C24×C4
FittingC4×C17×#125C2
socle, cosocleC23×C17C25
prime radicalC23×D17C24
hypercenterC4×#125D17
hypocenterC17#34215
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×#125

Complements

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

NH ≅ G/Ntype
4. #14531738C2semidirect
5. #14531737C2semidirect
6. #14531531C2semidirect
7. #14531528C2semidirect
9. #125×Dic17C2semidirect
10. C4×#10587556C2semidirect
11. #14531339C2semidirect
15. C4×C17×#125C2semidirect
16. #11763365C4semidirect
76. C17×#125C2×C4semidirect
115. #10587574C2×C4semidirect
116. #10587573C2×C4semidirect
118. #10587558Q8semidirect
120. #10587556C2×C4semidirect
259. C4×#125D17semidirect
355. #125C4×D17semidirect
414. C17#34215semidirect

Maximal subgroups

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

subgroupindex
C2×#117633652
#145318682
#145318612
#145317382
#145317372
#145315312
#145315282
#145314062
#125×Dic172
C4×#105875562
#145313392
D17×#3822
#145311382
C4×#105875582
C4×C17×#1252
#3421517