Finite Group Databasebeta

← #17064379

Subgroups — #17064379

Order 4352.

752 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#10587583C211yesmaximal, characteristic
2#14531180C211yesmaximal, characteristic
3#14530357C211yesmaximal, characteristic, complemented
4#14530333C211yesmaximal, characteristic, complemented
5#14530269C211yesmaximal, characteristic, complemented
6#14530218C211yesmaximal, characteristic
7#14530211C211yesmaximal, characteristic
8#14530206C211yesmaximal, characteristic, complemented
9#14529990C211yesmaximal, characteristic
10#14529982C211yesmaximal, characteristic
11C4×#10587478C211yesmaximal, characteristic, complemented
12C4×#10587481C211yesmaximal, characteristic, complemented
13C4×#10587487C211yesmaximal, characteristic, complemented
14C4×#10587491C211yesmaximal, characteristic
15C4×C17×#106C211yesmaximal, characteristic, complemented
16C2×#10587583C2211yescharacteristic
17C2×C4×Dic34C2211yescharacteristic
18C2×C42×D17C2211yescharacteristic
19C2×#10587478C2211yescharacteristic
20C2×#10587478C2211yescharacteristic
21C2×#10587487C2211yescharacteristic
22C2×#10587487C2211yescharacteristic
23C2×#10587481C2211yescharacteristic
24C2×#10587481C2211yescharacteristic
25C2×#10587491C2211yescharacteristic
26C2×#10587491C2211yescharacteristic
27C2×C4×#3622C2211yescharacteristic
28#11762714C2211yescharacteristic
29#11762708C2211yescharacteristic
30#11762694—21no—
31#11762683—21no—
32#11762676—21no—
33#11762674C2211yescharacteristic
34#11762674C2211yescharacteristic
35#11762671—21no—
36#11762666C2211yescharacteristic
37#11762666C2211yescharacteristic
38#11762666C2211yescharacteristic
39C4×C4⋊Dic17—21no—
40C4×C4⋊Dic17C2211yescharacteristic
41C4×#63227—21no—
42C4×#63227—21no—
43#11762655—21no—
44#11762572C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C2×C17×#106C2211yescharacteristic
53C2×C17×#106C2211yescharacteristic
54C2×C4×C8×C17C2211yescharacteristic
55C4×C4⋊C4×C17C2211yescharacteristic
56C4×C4⋊C4×C17C2211yescharacteristic
57C17×#340C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C22×C4×D17D411yescharacteristic
60C22×C4×D17D411yescharacteristic
61C22×C4×D17C2311yescharacteristic
62C22×Dic34C2311yescharacteristic
63C22×C8×C17C2311yescharacteristic
64C22×C8×C17C2311yescharacteristic
65C2×C4⋊C4×C17C2311yescharacteristic
66C2×C4⋊C4×C17C2311yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C42×C17—21no—
70C2×C42×C17—21no—
71C2×C42×C17C2311yescharacteristic
72C2×#63227—21no—
73C2×#63227—21no—
74C2×#63227—21no—
75C2×#63227—21no—
76C2×C4⋊Dic17—21no—
77C2×C4⋊Dic17—21no—
78C2×C4⋊Dic17C2311yescharacteristic
79C2×C4⋊Dic17C2311yescharacteristic
80C2×C4⋊Dic17C2311yescharacteristic
81C2×#63225—21no—
82C2×C4×Dic17—21no—
83C2×C4×Dic17—21no—
84C2×C4×Dic17D411yescharacteristic
85C2×C4×Dic17D411yescharacteristic
86C2×C4×Dic17C2311yescharacteristic
87C22×#3622C2311yescharacteristic
88C22×#3622C2311yescharacteristic
89#10587583—22no—
90#10587583—22no—
91C42×D17—22no—
92C42×D17—22no—
93C4×Dic34—22no—
94C4×Dic34—22no—
95C17×#106—22no—
96C17×#106—22no—
97C4×C8×C17—22no—
98C17×#95—21no—
99C17×#95—21no—
100#10587502—21no—
101#10587502—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#10587502—21no—
111#10587491—22no—
112#10587491—22no—
113#10587487—22no—
114#10587487—22no—
115#10587481—22no—
116#10587481—22no—
117#10587478—22no—
118#10587478—22no—
119C4×#3622—22no—
120C23×D17C2×D411yescharacteristic
121C22×C4×C17—21no—
122C22×C4×C17—21no—
123C22×C4×C17—21no—
124C22×C4×C17—21no—
125C22×C4×C17C2×D411yescharacteristic
126C22×C4×C17C2×D411yescharacteristic
127C22×C4×C17C2411yescharacteristic
128C22×Dic17—21no—
129C22×Dic17—21no—
130C22×Dic17—21no—
131C22×Dic17—21no—
132C22×Dic17C2×D411yescharacteristic
133C22×Dic17C2×D411yescharacteristic
134C22×Dic17C2×D411yescharacteristic
135C2×C4×D17—22no—
136C2×C4×D17—22no—
137C2×C4×D17—22no—
138C2×C4×D17—22no—
139C2×C4×D17—22no—
140C2×C4×D17—21no—
141C2×C4×D17—21no—
142C2×C4×D17—21no—
143C2×C4×D17—21no—
144C2×C4×D17—21no—
145C2×C4×D17—21no—
146C2×C4×D17—41no—
147C2×Dic34—21no—
148C2×Dic34—21no—
149C2×Dic34—21no—
150C2×Dic34—21no—
151C2×Dic34—21no—
152C2×Dic34#4112yes—
153C2×C8×C17—21no—
154C2×C8×C17—21no—
155C2×C8×C17—21no—
156C2×C8×C17—21no—
157C2×C8×C17—21no—
158C2×C8×C17#4112yes—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17#4112yes—
167C42×C17—42no—
168C42×C17—42no—
169C42×C17#4112yes—
170C42×C17#4112yes—
171#63227—24no—
172#63227—42no—
173#63227—42no—
174#63227—42no—
175C4⋊Dic17—24no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—42no—
183#63225—42no—
184C4×Dic17—22no—
185C4×Dic17—22no—
186C4×Dic17—21no—
187C4×Dic17—21no—
188C4×Dic17—42no—
189C4×Dic17—42no—
190C2×#3622—21no—
191C2×#3622—21no—
192C2×#3622—21no—
193C2×#3622—21no—
194C2×#3622—21no—
195C2×#3622#4112yes—
196#23759—171nomaximal, Sylow
197C23×C17C22×D411yescharacteristic
198C22×D17—21no—
199C22×D17—21no—
200C22×D17—21no—
201C22×D17—21no—
202C22×D17—21no—
203C22×D17—21no—
204C22×D17#13512yes—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—22no—
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×C17—41no—
218C2×C4×C17—41no—
219C2×C4×C17#12312yes—
220C2×C4×C17#12312yes—
221C2×C4×C17#13512yes—
222C2×C4×C17#13712yes—
223C2×C4×C17C2×#4111yescharacteristic
224C2×C4×C17C2×#4111yescharacteristic
225C2×C4×C17C2×#4111yescharacteristic
226C2×C4×C17C2×#4111yescharacteristic
227C2×C4×C17C2×#4111yescharacteristic
228C2×C4×C17C2×#4111yescharacteristic
229C2×Dic17—21no—
230C2×Dic17—21no—
231C2×Dic17—21no—
232C2×Dic17—22no—
233C2×Dic17—21no—
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—41no—
240C2×Dic17—41no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C4×D17—44no—
251C4×D17—41no—
252C4×D17—41no—
253C4×D17—41no—
254C4×D17—41no—
255C4×D17—44no—
256Dic34—22no—
257Dic34—22no—
258Dic34—41no—
259Dic34—41no—
260C8×C17—22no—
261C8×C17—41no—
262#3622—22no—
263#3622—41no—
264C4×C22⋊Q8—171no—
265#2283—171no—
266#2101—171no—
267#2097—171no—
268#2085—171no—
269#2077—171no—
270#2075—171no—
271#2074—171no—
272#2036—171no—
273#2035—171no—
274C4×#107—171no—
275C4×#106D1711yescharacteristic, complemented
276C4×#103—171no—
277C4×#103—171no—
278C4×C22⋊C8—171no—
279C22×C17C2×#12311yescharacteristic
280C22×C17#52511yescharacteristic
281C22×C17#53211yescharacteristic
282C22×C17C2×#13511yescharacteristic
283C22×C17C2×#13511yescharacteristic
284C22×C17C2×#13711yescharacteristic
285C22×C17#57511yescharacteristic
286C2×D17—22no—
287C2×D17—22no—
288C2×D17—22no—
289C2×D17—41no—
290C2×D17—41no—
291C2×D17—41no—
292C2×D17—41no—
293C4×C17—22no—
294C4×C17—42no—
295C4×C17—81no—
296C4×C17—81no—
297C4×C17—81no—
298C4×C17#48514yes—
299C4×C17#52911yescharacteristic
300C4×C17#52911yescharacteristic
301C4×C17#52911yescharacteristic
302C4×C17#52911yescharacteristic
303Dic17—41no—
304Dic17—41no—
305Dic17—42no—
306Dic17—41no—
307Dic17—81no—
308Dic17—81no—
309Dic17—81no—
310C2×C22⋊Q8—171no—
311C2×C4×Q8—171no—
312C22×C42—171no—
313C2×#107—171no—
314C2×#107—171no—
315C2×#106C2×D1711yescharacteristic
316C2×#106C2×D1711yescharacteristic
317C2×#103—171no—
318C2×#103—171no—
319C2×#103—171no—
320C2×#103—171no—
321C2×C22⋊C8—171no—
322C2×C22⋊C8—171no—
323C2×C4×C8—171no—
324C2×C4×C8C2×D1711yescharacteristic
325#391—171no—
326#389—171no—
327#387—341no—
328#385—341no—
329#383—341no—
330#382—171no—
331#382—171no—
332#381—341no—
333#380—171no—
334#380—171no—
335#380—171no—
336C4×C4⋊C4—171no—
337C4×C4⋊C4—341no—
338C4×C4⋊C4C2×D1711yescharacteristic
339C4×C4⋊C4C2×D1711yescharacteristic
340C4×C22⋊C4—341no—
341C4×C22⋊C4—341no—
342#376—341no—
343#340—171no—
344#340—171no—
345#340—171no—
346#340—171no—
347#340—171no—
348#340—171no—
349#340C2×D1711yescharacteristic
350#340C2×D1711yescharacteristic
351#336—171no—
352#336—171no—
353C2×C17#239511yescharacteristic
354C2×C17C2×#48511yescharacteristic
355C2×C17#309111yescharacteristic
356C2×C17#310211yescharacteristic
357C2×C17#310811yescharacteristic
358C2×C17#316811yescharacteristic
359C2×C17#317211yescharacteristic
360D17—42no—
361C22×Q8—171no—
362C23×C4—171no—
363C23×C4—171no—
364C23×C4—171no—
365C22×C8—171no—
366C22×C8—171no—
367C22×C8C22×D1711yescharacteristic
368C22×C8C22×D1711yescharacteristic
369C22⋊Q8—342no—
370C22⋊Q8—342no—
371C4×Q8—342no—
372C4×Q8—342no—
373C2×C4⋊C4—171no—
374C2×C4⋊C4—171no—
375C2×C4⋊C4—171no—
376C2×C4⋊C4—341no—
377C2×C4⋊C4—341no—
378C2×C4⋊C4—341no—
379C2×C4⋊C4C22×D1711yescharacteristic
380C2×C4⋊C4C22×D1711yescharacteristic
381C2×C4⋊C4C22×D1711yescharacteristic
382C2×C4⋊C4C22×D1711yescharacteristic
383C2×C22⋊C4—341no—
384C2×C22⋊C4—341no—
385C2×C22⋊C4—341no—
386C2×C22⋊C4—341no—
387C2×C42—21no—
388C2×C42—21no—
389C2×C42—171no—
390C2×C42—171no—
391C2×C42—171no—
392C2×C42—342no—
393C2×C42—342no—
394C2×C42—341no—
395C2×C42—341no—
396C2×C42C22×D1711yescharacteristic
397#107—342no—
398#107—342no—
399#106—22no—
400#106—22no—
401#103—342no—
402#103—342no—
403#103—342no—
404#103—342no—
405C22⋊C8—342no—
406C22⋊C8—342no—
407C4×C8—22no—
408C4×C8—342no—
409#95—21no—
410#95—21no—
411#95—341no—
412#95—341no—
413#95—341no—
414#95—341no—
415#95—341no—
416#95—341no—
417#95—341no—
418#95—341no—
419#95—341no—
420#95—341no—
421#95—341no—
422C17#2375911yescharacteristic, complemented, Sylow
423C24—171no—
424C2×Q8—172no—
425C2×Q8—341no—
426C2×Q8—341no—
427C2×Q8—341no—
428C2×Q8—341no—
429C2×Q8—341no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—171no—
435C22×C4—171no—
436C22×C4—171no—
437C22×C4—342no—
438C22×C4—341no—
439C22×C4—341no—
440C22×C4—341no—
441C22×C4—341no—
442C22×C4—341no—
443C22×C4—341no—
444C22×C4—341no—
445C22×C4—342no—
446C22×C4—342no—
447C22×C4—342no—
448C22×C4—342no—
449C22×C4—341no—
450C22×C4—341no—
451C22×C4—341no—
452C22×C4—681no—
453C22×C4D4×D1711yescharacteristic
454C22×C4D4×D1711yescharacteristic
455C22×C4C23×D1711yescharacteristic
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—172no—
462C2×C8—341no—
463C2×C8—341no—
464C2×C8—341no—
465C2×C8—341no—
466C2×C8—341no—
467C2×C8#6325212yes—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—344no—
476C4⋊C4—341no—
477C4⋊C4—341no—
478C4⋊C4—341no—
479C4⋊C4—341no—
480C4⋊C4—341no—
481C4⋊C4—341no—
482C4⋊C4—682no—
483C4⋊C4—682no—
484C4⋊C4#6325212yes—
485C22⋊C4—344no—
486C22⋊C4—682no—
487C22⋊C4—682no—
488C22⋊C4—682no—
489C42—42no—
490C42—42no—
491C42—341no—
492C42—341no—
493C42—342no—
494C42—342no—
495C42—682no—
496C42—682no—
497C42#6325212yes—
498C42#6325612yes—
499C23—172no—
500C23—341no—
501C23—341no—
502C23—341no—
503C23—341no—
504C23—341no—
505C23—341no—
506C23C2×D4×D1711yescharacteristic
507Q8—342no—
508Q8—342no—
509Q8—681no—
510Q8—681no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—341no—
526C2×C4—341no—
527C2×C4—341no—
528C2×C4—342no—
529C2×C4—341no—
530C2×C4—341no—
531C2×C4—341no—
532C2×C4—341no—
533C2×C4—341no—
534C2×C4—341no—
535C2×C4—684no—
536C2×C4—681no—
537C2×C4—681no—
538C2×C4—681no—
539C2×C4—681no—
540C2×C4—681no—
541C2×C4—681no—
542C2×C4—684no—
543C2×C4—681no—
544C2×C4—681no—
545C2×C4—681no—
546C2×C4—681no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—681no—
551C2×C4—681no—
552C2×C4#1058758012yes—
553C2×C4#1058758012yes—
554C2×C4#1058760112yes—
555C2×C4#1058760212yes—
556C2×C4C2×#6325211yescharacteristic
557C2×C4C2×#6325211yescharacteristic
558C2×C4C2×#6325211yescharacteristic
559C2×C4C2×#6325611yescharacteristic
560C2×C4#41×D1711yescharacteristic
561C2×C4#41×D1711yescharacteristic
562C8—22no—
563C8—41no—
564C8—342no—
565C8—681no—
566C22—342no—
567C22—342no—
568C22—342no—
569C22—681no—
570C22—681no—
571C22—681no—
572C22—681no—
573C22C2×#1058758011yescharacteristic
574C22#1176331811yescharacteristic
575C22#1176335611yescharacteristic
576C22#135×D1711yescharacteristic
577C22C2×#1058760211yescharacteristic
578C22C2×#1058760111yescharacteristic
579C22#1176360611yescharacteristic
580C4—22no—
581C4—42no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—681no—
586C4—681no—
587C4—681no—
588C4—682no—
589C4—1361no—
590C4—1361no—
591C4—1361no—
592C4#1176315314yes—
593C4#1176333711yescharacteristic
594C4#1176334311yescharacteristic
595C4#1176334611yescharacteristic
596C4#1176334611yescharacteristic
597C2—682no—
598C2#1453204711yescharacteristic
599C2C2×#1176315311yescharacteristic
600C2#1453870111yescharacteristic
601C2#1453880611yescharacteristic
602C2#1453886211yescharacteristic
603C2#1453935011yescharacteristic
604C2#1453938611yescharacteristic
subgroupG/H
centerC22×C4D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC4×C17×#106C2
socleC23×C17C22×D4
prime radicalC23×D17C2×D4
hypercenterC4×#106D17
hypocenterC17#23759
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#106

Complements

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

NH ≅ G/Ntype
3. #14530357C2semidirect
4. #14530333C2semidirect
5. #14530269C2semidirect
8. #14530206C2semidirect
11. C4×#10587478C2semidirect
12. C4×#10587481C2semidirect
13. C4×#10587487C2semidirect
15. C4×C17×#106C2semidirect
275. C4×#106D17semidirect
422. C17#23759semidirect

Maximal subgroups

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

subgroupindex
C4×#105875832
#145311802
#145303572
#145303332
#145302692
#145302182
#145302112
#145302062
#145299902
#145299822
C4×#105874782
C4×#105874812
C4×#105874872
C4×#105874912
C4×C17×#1062
#2375917