Finite Group Databasebeta

← #17092382

Subgroups — #17092382

Order 4352.

704 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14532495C211yesmaximal, characteristic
2#14531657C211yesmaximal, characteristic
3#14530348C211yesmaximal, characteristic, complemented
4#14530285C211yesmaximal, characteristic, complemented
5#14530276C211yesmaximal, characteristic, complemented
6#14530124C211yesmaximal, characteristic, complemented
7#14529738C211yesmaximal, characteristic, complemented
8#14529542C211yesmaximal, characteristic, complemented
9#14529372C211yesmaximal, characteristic, complemented
10#14528982C211yesmaximal, characteristic
11#14528968C211yesmaximal, characteristic
12#14528869C211yesmaximal, characteristic
13#14528816C211yesmaximal, characteristic
14#14528502C211yesmaximal, characteristic
15C17×#1919C211yesmaximal, characteristic, complemented
16C22×#63218C2211yescharacteristic
17C2×#10587576C2211yescharacteristic
18C2×#10587578C2211yescharacteristic
19C2×C4⋊C4×D17C2211yescharacteristic
20C2×#10587478C2211yescharacteristic
21C2×#10587478C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×C4⋊#3622C2211yescharacteristic
24C2×#10587485C2211yescharacteristic
25C2×#10587491C2211yescharacteristic
26#11762746—21no—
27#11762731—21no—
28#11762708—21no—
29#11762708C2211yescharacteristic
30#11762707C2211yescharacteristic
31#11762705C2211yescharacteristic
32#11762694—21no—
33#11762693—21no—
34#11762691—21no—
35#11762679C2211yescharacteristic
36#11762668—21no—
37#11762666C2211yescharacteristic
38#11762666C2211yescharacteristic
39C4⋊C4×Dic17C2211yescharacteristic
40C4×#63227—21no—
41C4×#63227C2211yescharacteristic
42#11762572C2211yescharacteristic
43#11762572C2211yescharacteristic
44#11762572C2211yescharacteristic
45#11762572C2211yescharacteristic
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⋊Dic17—21no—
76C2×C4⋊Dic17—21no—
77C2×C4⋊Dic17—21no—
78C2×C4⋊Dic17—21no—
79C2×C4⋊Dic17C2311yescharacteristic
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×#63225—21no—
84C2×#63225—21no—
85C2×#63225—21no—
86C2×#63225—21no—
87C2×C4×Dic17—21no—
88C2×C4×Dic17—21no—
89C2×C4×Dic17C2311yescharacteristic
90C22×#3622C2311yescharacteristic
91C22×#3622C2311yescharacteristic
92C2×#63218—22no—
93C2×#63218—22no—
94C2×#63218—41no—
95#10587578—22no—
96#10587578—41no—
97C4⋊C4×D17—22no—
98C4⋊C4×D17Q814yes—
99#10587576—22no—
100#10587576—41no—
101C17×#106—22no—
102C17×#95—21no—
103C17×#95—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#10587485—41no—
114#10587479—22no—
115#10587478—41no—
116#10587478—41no—
117C4⋊#3622—22no—
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—24no—
138C2×C4×D17—81no—
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—21no—
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—42no—
155C4⋊C4×C17C2×Q812yes—
156C4⋊C4×C17#4112yes—
157C42×C17—21no—
158C42×C17—21no—
159C42×C17—81no—
160#63227—22no—
161#63227—41no—
162#63227—42no—
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⋊Dic17—21no—
172C4⋊Dic17—42no—
173C4⋊Dic17—42no—
174C4⋊Dic17—81no—
175C4⋊Dic17—81no—
176C4⋊Dic17C2×Q812yes—
177#63225—24no—
178#63225—42no—
179#63225—81no—
180#63225—81no—
181C4×Dic17—21no—
182C4×Dic17—21no—
183C4×Dic17—42no—
184C4×Dic17—81no—
185C2×#3622—21no—
186C2×#3622—21no—
187C2×#3622—21no—
188C2×#3622—21no—
189C2×#3622—21no—
190C2×#3622#4112yes—
191#63218—44no—
192#26018—171nomaximal, Sylow
193C23×C17C22×D411yescharacteristic
194C22×D17—21no—
195C22×D17—21no—
196C22×D17—21no—
197C22×D17—21no—
198C22×D17—21no—
199C22×D17—21no—
200C22×D17C22⋊Q812yes—
201C2×C4×C17—21no—
202C2×C4×C17—21no—
203C2×C4×C17—22no—
204C2×C4×C17—22no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—21no—
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#13712yes—
219C2×C4×C17C22×Q811yescharacteristic
220C2×C4×C17C2×#4111yescharacteristic
221C2×C4×C17C2×#4111yescharacteristic
222C2×C4×C17C2×#4111yescharacteristic
223C2×C4×C17H(5,𝔽2)11yescharacteristic
224C2×C4×C172−1+411yescharacteristic
225C2×Dic17—21no—
226C2×Dic17—21no—
227C2×Dic17—22no—
228C2×Dic17—21no—
229C2×Dic17—21no—
230C2×Dic17—21no—
231C2×Dic17—21no—
232C2×Dic17—21no—
233C2×Dic17—21no—
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×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17C22⋊Q812yes—
246C4×D17—22no—
247C4×D17—22no—
248C4×D17—22no—
249C4×D17—22no—
250C4×D17—84no—
251C8×C17—41no—
252C8×C17—41no—
253#3622—22no—
254#3622—41no—
255#2441—171no—
256#2345—171no—
257#2100—171no—
258#2088—171no—
259#2086—171no—
260#2059—171no—
261#1989—171no—
262#1947—171no—
263#1920—171no—
264#1919D1711yescharacteristic, complemented
265#1854—171no—
266#1851—171no—
267#1823—171no—
268#1809—171no—
269#1736—171no—
270C22×C17C2×C22⋊Q811yescharacteristic
271C22×C17#54611yescharacteristic
272C22×C17#54711yescharacteristic
273C22×C17C2×#13711yescharacteristic
274C22×C17#57511yescharacteristic
275C22×C17#57611yescharacteristic
276C22×C17#57711yescharacteristic
277C2×D17—22no—
278C2×D17—22no—
279C2×D17—22no—
280C2×D17—41no—
281C2×D17—41no—
282C2×D17—41no—
283C2×D17—41no—
284C4×C17—22no—
285C4×C17—42no—
286C4×C17—42no—
287C4×C17—41no—
288C4×C17—81no—
289C4×C17—81no—
290C4×C17#53311yescharacteristic
291C4×C17#54411yescharacteristic
292C4×C17#55111yescharacteristic
293C4×C17#55211yescharacteristic
294Dic17—22no—
295Dic17—41no—
296Dic17—42no—
297Dic17—41no—
298Dic17—81no—
299Dic17—81no—
300Dic17—81no—
301C22×M16—171no—
302C2×#125—171no—
303C2×#117—171no—
304C22×C4⋊C4—171no—
305C2×#107—171no—
306C2×#107—171no—
307C2×#106—171no—
308C2×#106C2×D1711yescharacteristic
309C2×C4⋊C8—171no—
310C2×C4⋊C8—171no—
311C2×C22⋊C8—171no—
312#399—341no—
313#396—341no—
314#389—171no—
315#389—171no—
316#389—171no—
317#389—341no—
318#389C2×D1711yescharacteristic
319#387—341no—
320#387—341no—
321#387—341no—
322#384—171no—
323#380—171no—
324#380—171no—
325#380—341no—
326C4×C4⋊C4—171no—
327C4×C4⋊C4C2×D1711yescharacteristic
328C4×C22⋊C4—171no—
329C4×C22⋊C4—341no—
330#340—171no—
331#340—171no—
332#340—171no—
333#340—171no—
334#340—171no—
335#340—171no—
336#340—171no—
337#340—171no—
338#340—171no—
339#340C2×D1711yescharacteristic
340#340C2×D1711yescharacteristic
341#340C2×D1711yescharacteristic
342#336—171no—
343#336C2×D1711yescharacteristic
344C2×C17#243811yescharacteristic
345C2×C17#307811yescharacteristic
346C2×C17#308111yescharacteristic
347C2×C17#330911yescharacteristic
348C2×C17#333111yescharacteristic
349C2×C17#334611yescharacteristic
350C2×C17#335111yescharacteristic
351D17—42no—
352C23×C4—171no—
353C23×C4—341no—
354C2×M16—342no—
355C2×M16—342no—
356C2×M16—681no—
357C22×C8—171no—
358C22×C8—171no—
359C22×C8C22×D1711yescharacteristic
360C22×C8C22×D1711yescharacteristic
361#125—342no—
362#125—681no—
363#117—342no—
364#117—681no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—171no—
367C2×C4⋊C4—174no—
368C2×C4⋊C4—171no—
369C2×C4⋊C4—171no—
370C2×C4⋊C4—171no—
371C2×C4⋊C4—342no—
372C2×C4⋊C4—341no—
373C2×C4⋊C4—341no—
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⋊C4C22×D1711yescharacteristic
381C2×C4⋊C4C22×D1711yescharacteristic
382C2×C4⋊C4C22×D1711yescharacteristic
383C2×C4⋊C4C22×D1711yescharacteristic
384C2×C22⋊C4—171no—
385C2×C22⋊C4—171no—
386C2×C22⋊C4—341no—
387C2×C22⋊C4—341no—
388C2×C22⋊C4—341no—
389C2×C42—21no—
390C2×C42—171no—
391C2×C42—341no—
392C2×C42—341no—
393C2×C42C22×D1711yescharacteristic
394#107—681no—
395#107—681no—
396#106—22no—
397#106—342no—
398C4⋊C8—342no—
399C4⋊C8—681no—
400C22⋊C8—342no—
401#95—21no—
402#95—21no—
403#95—171no—
404#95—171no—
405#95—341no—
406#95—341no—
407#95—341no—
408#95—341no—
409#95—341no—
410#95—341no—
411C17#2601811yescharacteristic, complemented, Sylow
412C24—171no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—171no—
418C22×C4—172no—
419C22×C4—172no—
420C22×C4—171no—
421C22×C4—171no—
422C22×C4—344no—
423C22×C4—341no—
424C22×C4—341no—
425C22×C4—341no—
426C22×C4—341no—
427C22×C4—341no—
428C22×C4—341no—
429C22×C4—341no—
430C22×C4—341no—
431C22×C4—1361no—
432C22×C4—1361no—
433C22×C4D4×D1711yescharacteristic
434C22×C4D4×D1711yescharacteristic
435C22×C4C23×D1711yescharacteristic
436M16—684no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—172no—
444C2×C8—341no—
445C2×C8—341no—
446C2×C8—341no—
447C2×C8—341no—
448C2×C8—341no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—42no—
456C4⋊C4—172no—
457C4⋊C4—344no—
458C4⋊C4—341no—
459C4⋊C4—341no—
460C4⋊C4—341no—
461C4⋊C4—341no—
462C4⋊C4—341no—
463C4⋊C4—341no—
464C4⋊C4—341no—
465C4⋊C4—682no—
466C4⋊C4—682no—
467C4⋊C4—682no—
468C4⋊C4—1361no—
469C4⋊C4—1361no—
470C4⋊C4—1361no—
471C4⋊C4—1361no—
472C4⋊C4#6325212yes—
473C4⋊C4Q8×D1712yes—
474C22⋊C4—342no—
475C22⋊C4—681no—
476C22⋊C4—682no—
477C22⋊C4—682no—
478C22⋊C4—1361no—
479C42—21no—
480C42—21no—
481C42—81no—
482C42—341no—
483C42—341no—
484C42—682no—
485C42—1361no—
486C23—172no—
487C23—341no—
488C23—341no—
489C23—341no—
490C23—341no—
491C23—341no—
492C23—341no—
493C23C2×D4×D1711yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—172no—
512C2×C4—341no—
513C2×C4—341no—
514C2×C4—342no—
515C2×C4—342no—
516C2×C4—342no—
517C2×C4—342no—
518C2×C4—342no—
519C2×C4—341no—
520C2×C4—341no—
521C2×C4—341no—
522C2×C4—341no—
523C2×C4—341no—
524C2×C4—341no—
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—681no—
534C2×C4—681no—
535C2×C4—681no—
536C2×C4—1364no—
537C2×C4#1058760112yes—
538C2×C4C2×#6325211yescharacteristic
539C2×C4C2×Q8×D1711yescharacteristic
540C2×C4#41×D1711yescharacteristic
541C2×C4#41×D1711yescharacteristic
542C2×C4#1058769611yescharacteristic
543C2×C4#1058769711yescharacteristic
544C8—41no—
545C8—41no—
546C8—342no—
547C8—681no—
548C22—342no—
549C22—342no—
550C22—342no—
551C22—681no—
552C22—681no—
553C22—681no—
554C22—681no—
555C22C22⋊Q8×D1711yescharacteristic
556C22#1176345511yescharacteristic
557C22#1176345711yescharacteristic
558C22C2×#1058760111yescharacteristic
559C22#1176360811yescharacteristic
560C22#1176361311yescharacteristic
561C22#1176361611yescharacteristic
562C4—22no—
563C4—41no—
564C4—42no—
565C4—42no—
566C4—81no—
567C4—81no—
568C4—342no—
569C4—681no—
570C4—682no—
571C4—681no—
572C4—1361no—
573C4—1361no—
574C4—1361no—
575C4#1176336311yescharacteristic
576C4#1176343911yescharacteristic
577C4#1176348411yescharacteristic
578C4#1176349811yescharacteristic
579C2—682no—
580C2#1453246511yescharacteristic
581C2#1453858411yescharacteristic
582C2#1453861411yescharacteristic
583C2#1454079511yescharacteristic
584C2#1454110211yescharacteristic
585C2#1454126711yescharacteristic
586C2#1454134211yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1919C2
socleC23×C17C22×D4
prime radicalC23×D17C2×D4
hypercenter#1919D17
hypocenterC17#26018
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. #14530348C2semidirect
4. #14530285C2semidirect
5. #14530276C2semidirect
6. #14530124C2semidirect
7. #14529738C2semidirect
8. #14529542C2semidirect
9. #14529372C2semidirect
15. C17×#1919C2semidirect
264. #1919D17semidirect
411. C17#26018semidirect

Maximal subgroups

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

subgroupindex
#145324952
#145316572
#145303482
#145302852
#145302762
#145301242
#145297382
#145295422
#145293722
#145289822
#145289682
#145288692
#145288162
#145285022
C17×#19192
#2601817