Finite Group Databasebeta

← #17093322

Subgroups — #17093322

Order 4352.

750 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533817C211yesmaximal, characteristic, complemented
2#14531732C211yesmaximal, characteristic
3#14530315C211yesmaximal, characteristic
4#14530158C211yesmaximal, characteristic
5#14530154C211yesmaximal, characteristic
6#14529545C211yesmaximal, characteristic, complemented
7#14529544C211yesmaximal, characteristic, complemented
8#14529544C211yesmaximal, characteristic, complemented
9#14529501C211yesmaximal, characteristic, complemented
10#14529371C211yesmaximal, characteristic, complemented
11#14529370C211yesmaximal, characteristic, complemented
12#14528939C211yesmaximal, characteristic, complemented
13#14528782C211yesmaximal, characteristic
14#102×Dic17C211yesmaximal, characteristic
15C17×#2094C211yesmaximal, characteristic
16C2×#10587575—21no—
17C2×D4×Dic17C2211yescharacteristic
18C2×#10587563—21no—
19C2×#10587488C2211yescharacteristic
20C2×#10587479C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×C4⋊#3622C2211yescharacteristic
24C2×#10587485C2211yescharacteristic
25C2×#10587485C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×#10587503C2211yescharacteristic
28C2×C8×Dic17C2211yescharacteristic
29#11762742—21no—
30#11762740—21no—
31#11762712C2211yescharacteristic
32#11762708C2211yescharacteristic
33#11762707C2211yescharacteristic
34#11762704—21no—
35#11762703—21no—
36#11762697—21no—
37#11762678C2211yescharacteristic
38#11762678C2211yescharacteristic
39#11762678C2211yescharacteristic
40#11762667—21no—
41C4⋊C4×Dic17C2211yescharacteristic
42C4⋊C4×Dic17C2211yescharacteristic
43C4×#63225C2211yescharacteristic
44C4×C4⋊Dic17C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762563C2211yescharacteristic
52C2×C17×#106C2211yescharacteristic
53C2×C17×#106C2211yescharacteristic
54C2×C17×#102C2211yescharacteristic
55C2×C17×#102C2211yescharacteristic
56C17×#390C2211yescharacteristic
57C17×#384C2211yescharacteristic
58C17×#336C2211yescharacteristic
59C22×D4×C17C2311yescharacteristic
60C23×Dic17—21no—
61C22×C8×C17C2311yescharacteristic
62C22×C8×C17C2311yescharacteristic
63C2×C4⋊C4×C17—21no—
64C2×C4⋊C4×C17—21no—
65C2×C4⋊C4×C17C2311yescharacteristic
66C2×C4⋊C4×C17C2311yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C22⋊C4×C17—21no—
69C2×C22⋊C4×C17—21no—
70C2×C42×C17C2311yescharacteristic
71C2×C22⋊Dic17—21no—
72C2×C22⋊Dic17—21no—
73C2×C22⋊Dic17—21no—
74D4×Dic17—22no—
75D4×Dic17Q814yes—
76C2×C4⋊Dic17—21no—
77C2×C4⋊Dic17C2311yescharacteristic
78C2×C4⋊Dic17C2311yescharacteristic
79C2×C4⋊Dic17C2311yescharacteristic
80C2×C4⋊Dic17C2311yescharacteristic
81C2×#63225—21no—
82C2×#63225—21no—
83C2×#63225—21no—
84C2×#63225—21no—
85C2×#63225—21no—
86C2×#63225—21no—
87C2×#63225—21no—
88C2×#63225D411yescharacteristic
89C2×#63225D411yescharacteristic
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C2×C4×Dic17—21no—
93C2×C4×Dic17C2311yescharacteristic
94C2×C4×Dic17C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97#10587575—44no—
98#10587563—44no—
99C17×#106—22no—
100C17×#106—41no—
101C17×#102—22no—
102C17×#102—41no—
103#10587503—22no—
104#10587503—41no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502D411yescharacteristic
111#10587502D411yescharacteristic
112#10587488—22no—
113#10587485—22no—
114#10587485—22no—
115C8×Dic17—22no—
116#10587479—41no—
117#10587479—41no—
118#10587479—41no—
119C4⋊#3622—22no—
120C24×C17—21no—
121C2×D4×C17—21no—
122C2×D4×C17—21no—
123C2×D4×C17—21no—
124C2×D4×C17—21no—
125C2×D4×C17—21no—
126C2×D4×C17C2×Q812yes—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17C2×D411yescharacteristic
131C22×C4×C17C2×D411yescharacteristic
132C22×C4×C17C2411yescharacteristic
133C22×Dic17—21no—
134C22×Dic17—24no—
135C22×Dic17—21no—
136C22×Dic17—21no—
137C22×Dic17—21no—
138C22×Dic17—42no—
139C22×Dic17—42no—
140C22×Dic17C2×D411yescharacteristic
141C22×Dic17C2×D411yescharacteristic
142C22×Dic17C2×D411yescharacteristic
143C22×Dic17C2×D411yescharacteristic
144C2×C8×C17—21no—
145C2×C8×C17—21no—
146C2×C8×C17—21no—
147C2×C8×C17—21no—
148C2×C8×C17—21no—
149C2×C8×C17#4112yes—
150C4⋊C4×C17—21no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17—21no—
154C4⋊C4×C17—21no—
155C4⋊C4×C17—21no—
156C4⋊C4×C17—42no—
157C4⋊C4×C17—81no—
158C22⋊C4×C17—42no—
159C22⋊C4×C17—81no—
160C42×C17—21no—
161C42×C17—21no—
162C22⋊Dic17—24no—
163C22⋊Dic17—42no—
164C22⋊Dic17—81no—
165C4⋊Dic17—21no—
166C4⋊Dic17—21no—
167C4⋊Dic17—21no—
168C4⋊Dic17—21no—
169C4⋊Dic17—21no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—42no—
173C4⋊Dic17C2×Q812yes—
174#63225—22no—
175#63225—42no—
176#63225—42no—
177#63225—42no—
178#63225—42no—
179#63225—81no—
180#63225—81no—
181#63225—81no—
182#63225SD1614yes—
183C4×Dic17—21no—
184C4×Dic17—21no—
185C4×Dic17—42no—
186C4×Dic17—42no—
187C4×Dic17—42no—
188C4×Dic17C2×Q812yes—
189C4×Dic17#4112yes—
190C2×#3622—21no—
191C2×#3622—21no—
192C2×#3622—21no—
193C2×#3622—21no—
194C2×#3622—21no—
195C2×#3622#4112yes—
196#26090—171nomaximal, Sylow
197C23×C17—22no—
198C23×C17—41no—
199C23×C17—41no—
200C23×C17—41no—
201C23×C17—41no—
202C23×C17—41no—
203C23×C17—41no—
204C23×C17C22×D411yescharacteristic
205D4×C17—22no—
206D4×C17—22no—
207D4×C17—41no—
208D4×C17—41no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—41no—
215C2×C4×C17—41no—
216C2×C4×C17—41no—
217C2×C4×C17—41no—
218C2×C4×C17—41no—
219C2×C4×C17—41no—
220C2×C4×C17—41no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17C2×SD1612yes—
224C2×C4×C17C22×Q811yescharacteristic
225C2×C4×C17C2×#4111yescharacteristic
226C2×C4×C17C2×#4111yescharacteristic
227C2×C4×C17C2×#4111yescharacteristic
228C2×C4×C17H(5,𝔽2)11yescharacteristic
229C2×C4×C172−1+411yescharacteristic
230C2×Dic17—21no—
231C2×Dic17—21no—
232C2×Dic17—21no—
233C2×Dic17—21no—
234C2×Dic17—22no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—41no—
239C2×Dic17—41no—
240C2×Dic17—48no—
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—
250C2×Dic17C22⋊Q812yes—
251C2×Dic17C22⋊Q812yes—
252C2×Dic17C2×SD1612yes—
253C2×Dic17C2×SD1612yes—
254C2×Dic17AGL(1,ℤ/8)12yes—
255C8×C17—22no—
256C8×C17—41no—
257#3622—22no—
258#3622—41no—
259#2552—171no—
260#2354—171no—
261#2094—171no—
262#2094D1711yescharacteristic
263#2065—171no—
264#2065—171no—
265#1948—171no—
266#1948—171no—
267#1948—171no—
268#1939—171no—
269#1920—171no—
270#1920—171no—
271#1844—171no—
272#1802—171no—
273C4×#102—171no—
274C22×C17—42no—
275C22×C17—42no—
276C22×C17—42no—
277C22×C17—81no—
278C22×C17—81no—
279C22×C17—81no—
280C22×C17—81no—
281C22×C17C2×C22⋊Q811yescharacteristic
282C22×C17#54711yescharacteristic
283C22×C17#55011yescharacteristic
284C22×C17C22×SD1611yescharacteristic
285C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C17#57711yescharacteristic
287C22×C17#57711yescharacteristic
288C4×C17—22no—
289C4×C17—41no—
290C4×C17—81no—
291C4×C17—81no—
292C4×C17—81no—
293C4×C17#53311yescharacteristic
294C4×C17#54411yescharacteristic
295C4×C17#55211yescharacteristic
296C4×C17#55311yescharacteristic
297Dic17—22no—
298Dic17—22no—
299Dic17—41no—
300Dic17—42no—
301Dic17—81no—
302Dic17—81no—
303Dic17—81no—
304Dic17#47614yes—
305C2×#125—341no—
306C2×C4×D4—171no—
307C2×#117—341no—
308C2×#106—171no—
309C2×#106—171no—
310C2×#106—171no—
311C2×#106—171no—
312C2×#106C2×D1711yescharacteristic
313C2×#106C2×D1711yescharacteristic
314C2×C4⋊C8—171no—
315C2×C4⋊C8—171no—
316C2×C4⋊C8—171no—
317C2×#102—171no—
318C2×#102—171no—
319C2×#102C2×D1711yescharacteristic
320C2×#102C2×D1711yescharacteristic
321C2×C4×C8—171no—
322#398—341no—
323#398—341no—
324#390—171no—
325#390C2×D1711yescharacteristic
326#389—171no—
327#389—171no—
328#389—341no—
329#388—341no—
330#388—341no—
331#384—171no—
332#384—171no—
333#384—171no—
334#384C2×D1711yescharacteristic
335#380—341no—
336C4×C4⋊C4—171no—
337C4×C4⋊C4—171no—
338C4×C4⋊C4—171no—
339C4×C4⋊C4—171no—
340#340—171no—
341#340—171no—
342#340—171no—
343#340—171no—
344#340—171no—
345#340—171no—
346#336—171no—
347#336C2×D1711yescharacteristic
348C2×C17—82no—
349C2×C17#244111yescharacteristic
350C2×C17C2×#47611yescharacteristic
351C2×C17#308411yescharacteristic
352C2×C17#333511yescharacteristic
353C2×C17#335411yescharacteristic
354C2×C17#339211yescharacteristic
355C2×C17#339711yescharacteristic
356C22×D4C22×D1711yescharacteristic
357C23×C4—341no—
358C22×C8—171no—
359C22×C8—171no—
360C22×C8C22×D1711yescharacteristic
361C22×C8C22×D1711yescharacteristic
362#125—684no—
363C4×D4—174no—
364C4×D4—342no—
365#117—684no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—171no—
369C2×C4⋊C4—171no—
370C2×C4⋊C4—171no—
371C2×C4⋊C4—171no—
372C2×C4⋊C4—171no—
373C2×C4⋊C4—171no—
374C2×C4⋊C4—341no—
375C2×C4⋊C4—341no—
376C2×C4⋊C4—341no—
377C2×C4⋊C4—341no—
378C2×C4⋊C4—341no—
379C2×C4⋊C4—341no—
380C2×C4⋊C4—341no—
381C2×C4⋊C4—341no—
382C2×C4⋊C4C22×D1711yescharacteristic
383C2×C4⋊C4C22×D1711yescharacteristic
384C2×C4⋊C4C22×D1711yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—341no—
388C2×C22⋊C4—341no—
389C2×C22⋊C4—341no—
390C2×C42—171no—
391C2×C42—171no—
392C2×C42—341no—
393C2×C42—341no—
394C2×C42—341no—
395C2×C42C22×D1711yescharacteristic
396#106—22no—
397#106—41no—
398#106—342no—
399#106—681no—
400#106—681no—
401#106—681no—
402C4⋊C8—342no—
403C4⋊C8—342no—
404C4⋊C8—342no—
405#102—22no—
406#102—41no—
407#102—342no—
408#102—681no—
409C4×C8—342no—
410#95—171no—
411#95—171no—
412#95—341no—
413#95—341no—
414#95—341no—
415#95—341no—
416#95—341no—
417C17#2609011yescharacteristic, complemented, Sylow
418C24—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4Q8×D1712yes—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—171no—
429C22×C4—171no—
430C22×C4—171no—
431C22×C4—171no—
432C22×C4—344no—
433C22×C4—341no—
434C22×C4—341no—
435C22×C4—341no—
436C22×C4—341no—
437C22×C4—682no—
438C22×C4—682no—
439C22×C4D4×D1711yescharacteristic
440C22×C4D4×D1711yescharacteristic
441C22×C4C23×D1711yescharacteristic
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—172no—
448C2×C8—341no—
449C2×C8—341no—
450C2×C8—341no—
451C2×C8—341no—
452C2×C8—341no—
453C2×C8#6325412yes—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—42no—
461C4⋊C4—81no—
462C4⋊C4—174no—
463C4⋊C4—172no—
464C4⋊C4—342no—
465C4⋊C4—341no—
466C4⋊C4—341no—
467C4⋊C4—341no—
468C4⋊C4—341no—
469C4⋊C4—341no—
470C4⋊C4—341no—
471C4⋊C4—341no—
472C4⋊C4—682no—
473C4⋊C4—682no—
474C4⋊C4—682no—
475C4⋊C4—682no—
476C4⋊C4—682no—
477C4⋊C4—1361no—
478C4⋊C4—1361no—
479C4⋊C4—1361no—
480C22⋊C4—42no—
481C22⋊C4—81no—
482C22⋊C4—344no—
483C22⋊C4—682no—
484C22⋊C4—1361no—
485C42—21no—
486C42—21no—
487C42—172no—
488C42—172no—
489C42—341no—
490C42—341no—
491C42—682no—
492C42—682no—
493C42—682no—
494C23—22no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23C2×D4×D1711yescharacteristic
502D4—22no—
503D4—22no—
504D4—41no—
505D4—41no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—172no—
521C2×C4—172no—
522C2×C4—172no—
523C2×C4—172no—
524C2×C4—172no—
525C2×C4—341no—
526C2×C4—341no—
527C2×C4—341no—
528C2×C4—341no—
529C2×C4—341no—
530C2×C4—342no—
531C2×C4—341no—
532C2×C4—341no—
533C2×C4—688no—
534C2×C4—681no—
535C2×C4—681no—
536C2×C4—681no—
537C2×C4—681no—
538C2×C4—681no—
539C2×C4—681no—
540C2×C4—681no—
541C2×C4—681no—
542C2×C4—681no—
543C2×C4—681no—
544C2×C4—681no—
545C2×C4SD16×D1712yes—
546C2×C4C2×#6325411yescharacteristic
547C2×C4#1058769111yescharacteristic
548C2×C4C2×Q8×D1711yescharacteristic
549C2×C4#41×D1711yescharacteristic
550C2×C4#41×D1711yescharacteristic
551C2×C4#1058769711yescharacteristic
552C8—22no—
553C8—41no—
554C8—342no—
555C8—681no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22C22⋊Q8×D1711yescharacteristic
564C22#1176345711yescharacteristic
565C22#1176347911yescharacteristic
566C22C2×SD16×D1711yescharacteristic
567C22AGL(1,ℤ/8)×D1711yescharacteristic
568C22#1176361511yescharacteristic
569C22#1176361911yescharacteristic
570C4—22no—
571C4—41no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—174no—
576C4—342no—
577C4—342no—
578C4—682no—
579C4—681no—
580C4—1361no—
581C4—1361no—
582C4—1361no—
583C4#1176336111yescharacteristic
584C4#1176343811yescharacteristic
585C4#1176349611yescharacteristic
586C4#1176350411yescharacteristic
587C2—82no—
588C2#1453249011yescharacteristic
589C2D17×#47611yescharacteristic
590C2#1453864111yescharacteristic
591C2#1454113911yescharacteristic
592C2#1454137211yescharacteristic
593C2#1454176211yescharacteristic
594C2#1454180111yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2094C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
hypercenter#2094D17
hypocenterC17#26090
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2094

Complements

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

NH ≅ G/Ntype
1. #14533817C2semidirect
6. #14529545C2semidirect
7. #14529544C2semidirect
8. #14529544C2semidirect
9. #14529501C2semidirect
10. #14529371C2semidirect
11. #14529370C2semidirect
12. #14528939C2semidirect
417. C17#26090semidirect

Maximal subgroups

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

subgroupindex
#145338172
#145317322
#145303152
#145301582
#145301542
#145295452
#145295442
#145295012
#145293712
#145293702
#145289392
#145287822
#102×Dic172
C17×#20942
#2609017