Finite Group Databasebeta

← #17092391

Subgroups — #17092391

Order 4352.

696 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14532510C211yesmaximal, characteristic
2#14531665C211yesmaximal, characteristic
3#14530354C211yesmaximal, characteristic, complemented
4#14530286C211yesmaximal, characteristic, complemented
5#14530283C211yesmaximal, characteristic, complemented
6#14530123C211yesmaximal, characteristic, complemented
7#14529741C211yesmaximal, characteristic, complemented
8#14529555C211yesmaximal, characteristic, complemented
9#14529361C211yesmaximal, characteristic, complemented
10#14528978C211yesmaximal, characteristic
11#14528968C211yesmaximal, characteristic
12#14528878C211yesmaximal, characteristic
13#14528814C211yesmaximal, characteristic
14#14528502C211yesmaximal, characteristic
15C17×#1919C211yesmaximal, characteristic, complemented
16C22×#63218C2211yescharacteristic
17C2×#10587556C2211yescharacteristic
18C2×#10587578C2211yescharacteristic
19C2×C4⋊C4×D17C2211yescharacteristic
20C2×#10587489C2211yescharacteristic
21C2×#10587488C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×C4⋊#3622C2211yescharacteristic
24C2×#10587485C2211yescharacteristic
25C2×#10587491C2211yescharacteristic
26#11762750—21no—
27#11762740—21no—
28#11762707C2211yescharacteristic
29#11762705C2211yescharacteristic
30#11762705C2211yescharacteristic
31#11762700—21no—
32#11762698—21no—
33#11762680—21no—
34#11762674C2211yescharacteristic
35#11762673—21no—
36#11762668—21no—
37#11762666C2211yescharacteristic
38#11762666C2211yescharacteristic
39C4⋊C4×Dic17C2211yescharacteristic
40C4×#63227C2211yescharacteristic
41#11762655—21no—
42#11762572C2211yescharacteristic
43#11762572C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762563C2211yescharacteristic
52C2×C17×#106C2211yescharacteristic
53C17×#389C2211yescharacteristic
54C4×C4⋊C4×C17C2211yescharacteristic
55C17×#340C2211yescharacteristic
56C17×#340C2211yescharacteristic
57C17×#340C2211yescharacteristic
58C17×#336C2211yescharacteristic
59C22×C4×D17—21no—
60C22×C4×D17C2311yescharacteristic
61C22×C8×C17C2311yescharacteristic
62C22×C8×C17C2311yescharacteristic
63C2×C4⋊C4×C17—21no—
64C2×C4⋊C4×C17C2311yescharacteristic
65C2×C4⋊C4×C17C2311yescharacteristic
66C2×C4⋊C4×C17C2311yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C42×C17—21no—
69C2×C42×C17C2311yescharacteristic
70C2×#63227—21no—
71C2×#63227—21no—
72C2×#63227—21no—
73C2×#63227D411yescharacteristic
74C2×#63227D411yescharacteristic
75C2×C4⋊Dic17C2311yescharacteristic
76C2×C4⋊Dic17C2311yescharacteristic
77C2×C4⋊Dic17C2311yescharacteristic
78C2×C4⋊Dic17C2311yescharacteristic
79C2×#63225—21no—
80C2×#63225—21no—
81C2×#63225—21no—
82C2×#63225—21no—
83C2×#63225—21no—
84C2×#63225—21no—
85C2×C4×Dic17—21no—
86C2×C4×Dic17—21no—
87C2×C4×Dic17C2311yescharacteristic
88C22×#3622C2311yescharacteristic
89C22×#3622C2311yescharacteristic
90C2×#63218—22no—
91C2×#63218—22no—
92C2×#63218—41no—
93#10587578—22no—
94#10587578Q814yes—
95C4⋊C4×D17—22no—
96C4⋊C4×D17—41no—
97#10587556—22no—
98#10587556—41no—
99C17×#106—22no—
100C17×#95—21no—
101C17×#95—21no—
102#10587502—21no—
103#10587502—21no—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502D411yescharacteristic
111#10587502D411yescharacteristic
112#10587491—22no—
113#10587489—41no—
114#10587488—22no—
115#10587485—22no—
116#10587479—41no—
117C4⋊#3622—41no—
118C23×D17C2×D411yescharacteristic
119C22×C4×C17—21no—
120C22×C4×C17—21no—
121C22×C4×C17—21no—
122C22×C4×C17—21no—
123C22×C4×C17C2×D411yescharacteristic
124C22×C4×C17C2×D411yescharacteristic
125C22×C4×C17C2411yescharacteristic
126C22×Dic17—21no—
127C22×Dic17—21no—
128C22×Dic17—21no—
129C22×Dic17—21no—
130C22×Dic17C2×D411yescharacteristic
131C22×Dic17C2×D411yescharacteristic
132C22×Dic17C2×D411yescharacteristic
133C2×C4×D17—21no—
134C2×C4×D17—21no—
135C2×C4×D17—21no—
136C2×C4×D17—21no—
137C2×C4×D17—42no—
138C2×C4×D17—42no—
139C2×C4×D17—81no—
140C2×C4×D17C2×Q812yes—
141C2×C4×D17#4112yes—
142C2×C8×C17—21no—
143C2×C8×C17—21no—
144C2×C8×C17—21no—
145C2×C8×C17—21no—
146C2×C8×C17—21no—
147C2×C8×C17#4112yes—
148C4⋊C4×C17—21no—
149C4⋊C4×C17—21no—
150C4⋊C4×C17—21no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17—21no—
154C4⋊C4×C17—21no—
155C4⋊C4×C17—81no—
156C4⋊C4×C17C2×Q812yes—
157C42×C17—21no—
158C42×C17—21no—
159C42×C17—42no—
160#63227—24no—
161#63227—41no—
162#63227—41no—
163#63227—42no—
164#63227—81no—
165C4⋊Dic17—21no—
166C4⋊Dic17—21no—
167C4⋊Dic17—21no—
168C4⋊Dic17—21no—
169C4⋊Dic17—21no—
170C4⋊Dic17—21no—
171C4⋊Dic17C2×Q812yes—
172C4⋊Dic17#4112yes—
173#63225—42no—
174#63225—42no—
175#63225—42no—
176#63225—81no—
177#63225—81no—
178#63225—81no—
179C4×Dic17—21no—
180C4×Dic17—21no—
181C4×Dic17—24no—
182C4×Dic17—42no—
183C2×#3622—21no—
184C2×#3622—21no—
185C2×#3622—21no—
186C2×#3622—21no—
187C2×#3622—21no—
188C2×#3622—21no—
189#63218—44no—
190#26019—171nomaximal, Sylow
191C23×C17C22×D411yescharacteristic
192C22×D17—21no—
193C22×D17—21no—
194C22×D17—21no—
195C22×D17—21no—
196C22×D17—21no—
197C22×D17—21no—
198C22×D17C22⋊Q812yes—
199C2×C4×C17—22no—
200C2×C4×C17—21no—
201C2×C4×C17—21no—
202C2×C4×C17—21no—
203C2×C4×C17—21no—
204C2×C4×C17—21no—
205C2×C4×C17—21no—
206C2×C4×C17—41no—
207C2×C4×C17—41no—
208C2×C4×C17—41no—
209C2×C4×C17—41no—
210C2×C4×C17—41no—
211C2×C4×C17—41no—
212C2×C4×C17—41no—
213C2×C4×C17—41no—
214C2×C4×C17—41no—
215C2×C4×C17—41no—
216C2×C4×C17—41no—
217C2×C4×C17C22×Q811yescharacteristic
218C2×C4×C17C2×#4111yescharacteristic
219C2×C4×C17C2×#4111yescharacteristic
220C2×C4×C17C2×#4111yescharacteristic
221C2×C4×C17H(5,𝔽2)11yescharacteristic
222C2×C4×C172−1+411yescharacteristic
223C2×Dic17—21no—
224C2×Dic17—21no—
225C2×Dic17—22no—
226C2×Dic17—22no—
227C2×Dic17—21no—
228C2×Dic17—21no—
229C2×Dic17—21no—
230C2×Dic17—21no—
231C2×Dic17—21no—
232C2×Dic17—21no—
233C2×Dic17—41no—
234C2×Dic17—41no—
235C2×Dic17—41no—
236C2×Dic17—41no—
237C2×Dic17—41no—
238C2×Dic17—41no—
239C2×Dic17—41no—
240C2×Dic17—41no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17C22⋊Q812yes—
244C4×D17—22no—
245C4×D17—22no—
246C4×D17—22no—
247C4×D17—22no—
248C4×D17—84no—
249C8×C17—22no—
250C8×C17—41no—
251#3622—41no—
252#3622—41no—
253#2442—171no—
254#2346—171no—
255#2101—171no—
256#2088—171no—
257#2088—171no—
258#2059—171no—
259#1989—171no—
260#1949—171no—
261#1919—171no—
262#1919D1711yescharacteristic, complemented
263#1853—171no—
264#1851—171no—
265#1825—171no—
266#1809—171no—
267#1736—171no—
268C22×C17C2×C22⋊Q811yescharacteristic
269C22×C17#54611yescharacteristic
270C22×C17#54711yescharacteristic
271C22×C17#57511yescharacteristic
272C22×C17#57511yescharacteristic
273C22×C17#57711yescharacteristic
274C22×C17#57711yescharacteristic
275C2×D17—22no—
276C2×D17—22no—
277C2×D17—22no—
278C2×D17—41no—
279C2×D17—41no—
280C2×D17—41no—
281C2×D17—41no—
282C4×C17—42no—
283C4×C17—41no—
284C4×C17—41no—
285C4×C17—81no—
286C4×C17—81no—
287C4×C17—81no—
288C4×C17#53311yescharacteristic
289C4×C17#54411yescharacteristic
290C4×C17#55111yescharacteristic
291C4×C17#55211yescharacteristic
292Dic17—22no—
293Dic17—42no—
294Dic17—41no—
295Dic17—41no—
296Dic17—42no—
297Dic17—81no—
298Dic17—81no—
299C22×M16—171no—
300C2×#125—171no—
301C2×#117—171no—
302C22×C4⋊C4—171no—
303C2×#107—171no—
304C2×#106—171no—
305C2×#106—171no—
306C2×#106C2×D1711yescharacteristic
307C2×C4⋊C8—171no—
308C2×C4⋊C8—171no—
309C2×C22⋊C8—171no—
310#400—341no—
311#398—341no—
312#389—171no—
313#389—171no—
314#389—171no—
315#389C2×D1711yescharacteristic
316#388—341no—
317#388—341no—
318#385—341no—
319#382—171no—
320#382—341no—
321#380—171no—
322#380—171no—
323#380—341no—
324C4×C4⋊C4—171no—
325C4×C4⋊C4C2×D1711yescharacteristic
326C4×C22⋊C4—171no—
327#376—341no—
328#340—171no—
329#340—171no—
330#340—171no—
331#340—171no—
332#340—171no—
333#340—171no—
334#340—171no—
335#340—171no—
336#340—171no—
337#340C2×D1711yescharacteristic
338#340C2×D1711yescharacteristic
339#340C2×D1711yescharacteristic
340#336—171no—
341#336C2×D1711yescharacteristic
342C2×C17#243811yescharacteristic
343C2×C17#308011yescharacteristic
344C2×C17#308411yescharacteristic
345C2×C17#332111yescharacteristic
346C2×C17#332311yescharacteristic
347C2×C17#334811yescharacteristic
348C2×C17#335111yescharacteristic
349D17—42no—
350C23×C4—171no—
351C23×C4—341no—
352C2×M16—342no—
353C2×M16—342no—
354C2×M16—681no—
355C22×C8—171no—
356C22×C8—171no—
357C22×C8C22×D1711yescharacteristic
358C22×C8C22×D1711yescharacteristic
359#125—342no—
360#125—681no—
361#117—174no—
362#117—342no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—171no—
365C2×C4⋊C4—171no—
366C2×C4⋊C4—171no—
367C2×C4⋊C4—171no—
368C2×C4⋊C4—342no—
369C2×C4⋊C4—341no—
370C2×C4⋊C4—341no—
371C2×C4⋊C4—341no—
372C2×C4⋊C4—341no—
373C2×C4⋊C4—341no—
374C2×C4⋊C4—341no—
375C2×C4⋊C4—681no—
376C2×C4⋊C4C22×D1711yescharacteristic
377C2×C4⋊C4C22×D1711yescharacteristic
378C2×C4⋊C4C22×D1711yescharacteristic
379C2×C4⋊C4C22×D1711yescharacteristic
380C2×C22⋊C4—171no—
381C2×C22⋊C4—171no—
382C2×C22⋊C4—341no—
383C2×C22⋊C4—341no—
384C2×C22⋊C4—341no—
385C2×C42—21no—
386C2×C42—171no—
387C2×C42—341no—
388C2×C42—341no—
389C2×C42C22×D1711yescharacteristic
390#107—681no—
391#106—22no—
392#106—342no—
393#106—681no—
394C4⋊C8—342no—
395C4⋊C8—681no—
396C22⋊C8—342no—
397#95—21no—
398#95—21no—
399#95—171no—
400#95—171no—
401#95—341no—
402#95—341no—
403#95—341no—
404#95—341no—
405#95—341no—
406#95—341no—
407#95—341no—
408#95—341no—
409C17#2601911yescharacteristic, complemented, Sylow
410C24—171no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—171no—
416C22×C4—172no—
417C22×C4—172no—
418C22×C4—171no—
419C22×C4—171no—
420C22×C4—341no—
421C22×C4—341no—
422C22×C4—341no—
423C22×C4—341no—
424C22×C4—341no—
425C22×C4—341no—
426C22×C4—341no—
427C22×C4—341no—
428C22×C4—682no—
429C22×C4—682no—
430C22×C4—1361no—
431C22×C4D4×D1711yescharacteristic
432C22×C4D4×D1711yescharacteristic
433C22×C4C23×D1711yescharacteristic
434M16—684no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—341no—
441C2×C8—341no—
442C2×C8—341no—
443C2×C8—341no—
444C2×C8—341no—
445C2×C8—341no—
446C2×C8#6325212yes—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—81no—
455C4⋊C4—172no—
456C4⋊C4—172no—
457C4⋊C4—341no—
458C4⋊C4—341no—
459C4⋊C4—341no—
460C4⋊C4—341no—
461C4⋊C4—341no—
462C4⋊C4—341no—
463C4⋊C4—682no—
464C4⋊C4—682no—
465C4⋊C4—682no—
466C4⋊C4—1361no—
467C4⋊C4—1361no—
468C4⋊C4—1361no—
469C4⋊C4Q8×D1712yes—
470C22⋊C4—344no—
471C22⋊C4—681no—
472C22⋊C4—681no—
473C22⋊C4—682no—
474C22⋊C4—1361no—
475C42—21no—
476C42—21no—
477C42—42no—
478C42—344no—
479C42—341no—
480C42—341no—
481C42—682no—
482C23—172no—
483C23—341no—
484C23—341no—
485C23—341no—
486C23—341no—
487C23—341no—
488C23—341no—
489C23C2×D4×D1711yescharacteristic
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—172no—
509C2×C4—341no—
510C2×C4—341no—
511C2×C4—342no—
512C2×C4—342no—
513C2×C4—342no—
514C2×C4—342no—
515C2×C4—341no—
516C2×C4—341no—
517C2×C4—342no—
518C2×C4—341no—
519C2×C4—341no—
520C2×C4—341no—
521C2×C4—341no—
522C2×C4—342no—
523C2×C4—681no—
524C2×C4—681no—
525C2×C4—681no—
526C2×C4—681no—
527C2×C4—681no—
528C2×C4—681no—
529C2×C4—681no—
530C2×C4—681no—
531C2×C4—681no—
532C2×C4—681no—
533C2×C4—1364no—
534C2×C4C2×#6325211yescharacteristic
535C2×C4C2×Q8×D1711yescharacteristic
536C2×C4#41×D1711yescharacteristic
537C2×C4#41×D1711yescharacteristic
538C2×C4#1058769611yescharacteristic
539C2×C4#1058769711yescharacteristic
540C8—22no—
541C8—41no—
542C8—681no—
543C8—681no—
544C22—342no—
545C22—342no—
546C22—342no—
547C22—681no—
548C22—681no—
549C22—681no—
550C22—681no—
551C22C22⋊Q8×D1711yescharacteristic
552C22#1176345111yescharacteristic
553C22#1176346411yescharacteristic
554C22#1176360511yescharacteristic
555C22#1176360611yescharacteristic
556C22#1176361511yescharacteristic
557C22#1176361811yescharacteristic
558C4—42no—
559C4—41no—
560C4—41no—
561C4—81no—
562C4—81no—
563C4—81no—
564C4—342no—
565C4—681no—
566C4—682no—
567C4—682no—
568C4—681no—
569C4—1361no—
570C4—1361no—
571C4#1176336311yescharacteristic
572C4#1176343911yescharacteristic
573C4#1176348311yescharacteristic
574C4#1176349511yescharacteristic
575C2—682no—
576C2#1453245911yescharacteristic
577C2#1453860111yescharacteristic
578C2#1453864311yescharacteristic
579C2#1454096811yescharacteristic
580C2#1454100311yescharacteristic
581C2#1454130411yescharacteristic
582C2#1454134911yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1919C2
socleC23×C17C22×D4
prime radicalC23×D17C2×D4
hypercenter#1919D17
hypocenterC17#26019
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1919

Complements

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

NH ≅ G/Ntype
3. #14530354C2semidirect
4. #14530286C2semidirect
5. #14530283C2semidirect
6. #14530123C2semidirect
7. #14529741C2semidirect
8. #14529555C2semidirect
9. #14529361C2semidirect
15. C17×#1919C2semidirect
262. #1919D17semidirect
409. C17#26019semidirect

Maximal subgroups

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

subgroupindex
#145325102
#145316652
#145303542
#145302862
#145302832
#145301232
#145297412
#145295552
#145293612
#145289782
#145289682
#145288782
#145288142
#145285022
C17×#19192
#2601917