Finite Group Databasebeta

← #17092375

Subgroups — #17092375

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#14532497C211yesmaximal, characteristic
2#14531659C211yesmaximal, characteristic
3#14530348C211yesmaximal, characteristic, complemented
4#14530282C211yesmaximal, characteristic, complemented
5#14530272C211yesmaximal, characteristic, complemented
6#14530123C211yesmaximal, characteristic, complemented
7#14529739C211yesmaximal, characteristic, complemented
8#14529543C211yesmaximal, characteristic, complemented
9#14529370C211yesmaximal, characteristic, complemented
10#14529363C211yesmaximal, characteristic, complemented
11#14528985C211yesmaximal, characteristic
12#14528871C211yesmaximal, characteristic
13#14528817C211yesmaximal, characteristic
14#14528504C211yesmaximal, characteristic
15C17×#1851C211yesmaximal, characteristic
16C2×#10587556C2211yescharacteristic
17C2×#10587612C2211yescharacteristic
18C22×C4⋊Dic17C2211yescharacteristic
19C2×#10587489C2211yescharacteristic
20C2×#10587489C2211yescharacteristic
21C2×#10587488C2211yescharacteristic
22C2×#10587488C2211yescharacteristic
23#11762745—21no—
24#11762732—21no—
25#11762707C2211yescharacteristic
26#11762707C2211yescharacteristic
27#11762707C2211yescharacteristic
28#11762707C2211yescharacteristic
29#11762706—21no—
30#11762692—21no—
31#11762692—21no—
32#11762692—21no—
33#11762679C2211yescharacteristic
34#11762669C2211yescharacteristic
35#11762669C2211yescharacteristic
36#11762667—21no—
37C4×C4⋊Dic17C2211yescharacteristic
38C4×C4⋊Dic17C2211yescharacteristic
39C22⋊C4×Dic17—21no—
40#11762572C2211yescharacteristic
41#11762572C2211yescharacteristic
42#11762572C2211yescharacteristic
43#11762572C2211yescharacteristic
44#11762572C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762572C2211yescharacteristic
51#11762572C2211yescharacteristic
52C22×M16×C17C2211yescharacteristic
53C2×C17×C4⋊C8C2211yescharacteristic
54C2×C17×C4⋊C8C2211yescharacteristic
55C2×C17×C22⋊C8C2211yescharacteristic
56C4×C22⋊C4×C17C2211yescharacteristic
57C17×#336C2211yescharacteristic
58C17×#336C2211yescharacteristic
59C23×C4×C17C2311yescharacteristic
60C23×Dic17—21no—
61C2×M16×C17—22no—
62C2×M16×C17—22no—
63C2×M16×C17—41no—
64C22×C8×C17C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C2×C22⋊C4×C17D411yescharacteristic
69C2×C22⋊C4×C17D411yescharacteristic
70C2×C42×C17C2311yescharacteristic
71C2×C42×C17C2311yescharacteristic
72C2×C22⋊Dic17—21no—
73C2×C22⋊Dic17—21no—
74C2×C22⋊Dic17—21no—
75#10587612—22no—
76#10587612—41no—
77C2×C4⋊Dic17—21no—
78C2×C4⋊Dic17—21no—
79C2×C4⋊Dic17—21no—
80C2×C4⋊Dic17—22no—
81C2×C4⋊Dic17Q814yes—
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90C2×#63225—21no—
91C2×#63225—21no—
92C2×#63225—21no—
93C2×#63225—21no—
94C2×#63225—21no—
95C2×#63225—21no—
96C2×C4×Dic17—21no—
97C2×C4×Dic17—21no—
98C2×C4×Dic17—21no—
99#10587556—22no—
100#10587556—41no—
101C17×C4⋊C8—22no—
102C17×C4⋊C8—41no—
103C17×C22⋊C8—22no—
104C17×#95D411yescharacteristic
105C17×#95D411yescharacteristic
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587489—41no—
115#10587489—41no—
116#10587488—22no—
117#10587488—22no—
118C24×C17C2×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×C17C2×D411yescharacteristic
126C22×C4×C17C2×D411yescharacteristic
127C22×C4×C17C2×D411yescharacteristic
128C22×C4×C17C2×Q812yes—
129C22×C4×C17#4112yes—
130C22×C4×C17C2411yescharacteristic
131C22×Dic17—21no—
132C22×Dic17—21no—
133C22×Dic17—21no—
134C22×Dic17—24no—
135C22×Dic17—21no—
136C22×Dic17—21no—
137C22×Dic17—21no—
138C22×Dic17—21no—
139C22×Dic17—21no—
140C22×Dic17—81no—
141C22×Dic17—81no—
142M16×C17—44no—
143C2×C8×C17—21no—
144C2×C8×C17—21no—
145C2×C8×C17—21no—
146C2×C8×C17—21no—
147C2×C8×C17—21no—
148C2×C8×C17—21no—
149C2×C8×C17—21no—
150C2×C8×C17—21no—
151C2×C8×C17—21no—
152C2×C8×C17—21no—
153C2×C8×C17—21no—
154C2×C8×C17#4112yes—
155C22⋊C4×C17—22no—
156C22⋊C4×C17—41no—
157C42×C17—21no—
158C42×C17—21no—
159C42×C17—21no—
160C42×C17—21no—
161C22⋊Dic17—42no—
162C22⋊Dic17—42no—
163C22⋊Dic17—81no—
164C4⋊Dic17—24no—
165C4⋊Dic17—21no—
166C4⋊Dic17—21no—
167C4⋊Dic17—21no—
168C4⋊Dic17—21no—
169C4⋊Dic17—21no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—42no—
179C4⋊Dic17—81no—
180C4⋊Dic17C2×Q812yes—
181C4⋊Dic17C2×Q812yes—
182C4⋊Dic17#4112yes—
183#63225—42no—
184#63225—42no—
185#63225—42no—
186#63225—81no—
187#63225—81no—
188#63225—81no—
189C4×Dic17—42no—
190C4×Dic17—81no—
191C4×Dic17—81no—
192#26018—171nomaximal, Sylow
193C23×C17—21no—
194C23×C17—21no—
195C23×C17—21no—
196C23×C17—21no—
197C23×C17—21no—
198C23×C17—21no—
199C23×C17C22⋊Q812yes—
200C23×C17C22×D411yescharacteristic
201C2×C4×C17—21no—
202C2×C4×C17—21no—
203C2×C4×C17—21no—
204C2×C4×C17—21no—
205C2×C4×C17—22no—
206C2×C4×C17—22no—
207C2×C4×C17—22no—
208C2×C4×C17—22no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17C22⋊Q812yes—
219C2×C4×C17#13712yes—
220C2×C4×C17C22×Q811yescharacteristic
221C2×C4×C17C2×#4111yescharacteristic
222C2×C4×C17C2×#4111yescharacteristic
223C2×C4×C17C2×#4111yescharacteristic
224C2×C4×C17H(5,𝔽2)11yescharacteristic
225C2×C4×C172−1+411yescharacteristic
226C2×Dic17—22no—
227C2×Dic17—22no—
228C2×Dic17—22no—
229C2×Dic17—41no—
230C2×Dic17—41no—
231C2×Dic17—41no—
232C2×Dic17—41no—
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×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—84no—
251C8×C17—22no—
252C8×C17—41no—
253C8×C17—41no—
254C8×C17—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#1919—171no—
265#1854—171no—
266#1851D1711yescharacteristic
267#1823—171no—
268#1809—171no—
269#1736—171no—
270C22×C17—22no—
271C22×C17—22no—
272C22×C17—22no—
273C22×C17—41no—
274C22×C17—41no—
275C22×C17—41no—
276C22×C17—41no—
277C22×C17C2×C22⋊Q811yescharacteristic
278C22×C17#54611yescharacteristic
279C22×C17#54711yescharacteristic
280C22×C17C2×#13711yescharacteristic
281C22×C17#57511yescharacteristic
282C22×C17#57611yescharacteristic
283C22×C17#57711yescharacteristic
284C4×C17—22no—
285C4×C17—22no—
286C4×C17—41no—
287C4×C17—41no—
288C4×C17—41no—
289C4×C17#53311yescharacteristic
290C4×C17#54411yescharacteristic
291C4×C17#55111yescharacteristic
292C4×C17#55211yescharacteristic
293Dic17—42no—
294Dic17—42no—
295Dic17—42no—
296Dic17—81no—
297Dic17—81no—
298Dic17—81no—
299Dic17—81no—
300Dic17—81no—
301C22×M16C2×D1711yescharacteristic
302C2×#125—171no—
303C2×#117—171no—
304C22×C4⋊C4—171no—
305C2×#107—171no—
306C2×#107—171no—
307C2×#106—171no—
308C2×#106—171no—
309C2×C4⋊C8C2×D1711yescharacteristic
310C2×C4⋊C8C2×D1711yescharacteristic
311C2×C22⋊C8C2×D1711yescharacteristic
312#399—341no—
313#396—341no—
314#389—171no—
315#389—171no—
316#389—171no—
317#389—171no—
318#389—341no—
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⋊C4—171no—
328C4×C22⋊C4—341no—
329C4×C22⋊C4C2×D1711yescharacteristic
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#340—171no—
340#340—171no—
341#340—171no—
342#336C2×D1711yescharacteristic
343#336C2×D1711yescharacteristic
344C2×C17—42no—
345C2×C17#243811yescharacteristic
346C2×C17#307811yescharacteristic
347C2×C17#308111yescharacteristic
348C2×C17#330911yescharacteristic
349C2×C17#333111yescharacteristic
350C2×C17#334611yescharacteristic
351C2×C17#335111yescharacteristic
352C23×C4—341no—
353C23×C4C22×D1711yescharacteristic
354C2×M16—22no—
355C2×M16—22no—
356C2×M16—41no—
357C22×C8C22×D1711yescharacteristic
358C22×C8C22×D1711yescharacteristic
359C22×C8C22×D1711yescharacteristic
360C22×C8C22×D1711yescharacteristic
361#125—342no—
362#125—681no—
363#117—342no—
364#117—681no—
365C2×C4⋊C4—174no—
366C2×C4⋊C4—171no—
367C2×C4⋊C4—171no—
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—342no—
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⋊C4—341no—
383C2×C4⋊C4—341no—
384C2×C22⋊C4—341no—
385C2×C22⋊C4—341no—
386C2×C22⋊C4—341no—
387C2×C22⋊C4D6811yescharacteristic
388C2×C22⋊C4D6811yescharacteristic
389C2×C42—341no—
390C2×C42—341no—
391C2×C42—341no—
392C2×C42C22×D1711yescharacteristic
393C2×C42C22×D1711yescharacteristic
394#107—681no—
395#107—681no—
396#106—342no—
397#106—342no—
398C4⋊C8—22no—
399C4⋊C8—41no—
400C22⋊C8—22no—
401#95—341no—
402#95—341no—
403#95—341no—
404#95—341no—
405#95—341no—
406#95—341no—
407#95—341no—
408#95—341no—
409#95D6811yescharacteristic
410#95D6811yescharacteristic
411C17#2601811yescharacteristic, complemented, Sylow
412C24C2×D6811yescharacteristic
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—341no—
418C22×C4—341no—
419C22×C4—341no—
420C22×C4—344no—
421C22×C4—341no—
422C22×C4—341no—
423C22×C4—341no—
424C22×C4—341no—
425C22×C4—341no—
426C22×C4—1361no—
427C22×C4—1361no—
428C22×C4C2×D6811yescharacteristic
429C22×C4C2×D6811yescharacteristic
430C22×C4C2×D6811yescharacteristic
431C22×C4C2×D6811yescharacteristic
432C22×C4C2×D6811yescharacteristic
433C22×C4#6325412yes—
434C22×C4Q8×D1712yes—
435C22×C4C23×D1711yescharacteristic
436M16—44no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8#6325412yes—
449C4⋊C4—172no—
450C4⋊C4—172no—
451C4⋊C4—172no—
452C4⋊C4—341no—
453C4⋊C4—341no—
454C4⋊C4—341no—
455C4⋊C4—344no—
456C4⋊C4—341no—
457C4⋊C4—341no—
458C4⋊C4—341no—
459C4⋊C4—341no—
460C4⋊C4—341no—
461C4⋊C4—341no—
462C4⋊C4—341no—
463C4⋊C4—341no—
464C4⋊C4—341no—
465C4⋊C4—341no—
466C4⋊C4—682no—
467C4⋊C4—682no—
468C4⋊C4—682no—
469C4⋊C4—682no—
470C4⋊C4—1361no—
471C4⋊C4—1361no—
472C4⋊C4—1361no—
473C4⋊C4—1361no—
474C22⋊C4—22no—
475C22⋊C4—41no—
476C22⋊C4—682no—
477C22⋊C4—682no—
478C22⋊C4—1361no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—682no—
484C42—1361no—
485C42—1361no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23#1058758312yes—
493C23C22×D6811yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—342no—
512C2×C4—342no—
513C2×C4—342no—
514C2×C4—681no—
515C2×C4—681no—
516C2×C4—681no—
517C2×C4—681no—
518C2×C4—681no—
519C2×C4—681no—
520C2×C4—681no—
521C2×C4—681no—
522C2×C4—681no—
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—681no—
534C2×C4—681no—
535C2×C4—1364no—
536C2×C4#1058758312yes—
537C2×C4#1058759512yes—
538C2×C4C2×#6325411yescharacteristic
539C2×C4C2×#6325411yescharacteristic
540C2×C4#1058769111yescharacteristic
541C2×C4C2×Q8×D1711yescharacteristic
542C2×C4#41×D1711yescharacteristic
543C2×C4#1058769711yescharacteristic
544C8—22no—
545C8—41no—
546C8—41no—
547C8—41no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×#1058758311yescharacteristic
556C22#1176345411yescharacteristic
557C22#1176346311yescharacteristic
558C22C2×#1058759511yescharacteristic
559C22#1176360911yescharacteristic
560C22#1176361411yescharacteristic
561C22#1176362111yescharacteristic
562C4—22no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—682no—
568C4—682no—
569C4—682no—
570C4—1361no—
571C4—1361no—
572C4—1361no—
573C4—1361no—
574C4—1361no—
575C4#1176336111yescharacteristic
576C4#1176343811yescharacteristic
577C4#1176349011yescharacteristic
578C4#1176349611yescharacteristic
579C2—42no—
580C2#1453245811yescharacteristic
581C2#1453859111yescharacteristic
582C2#1453861611yescharacteristic
583C2#1454079211yescharacteristic
584C2#1454110111yescharacteristic
585C2#1454127111yescharacteristic
586C2#1454134611yescharacteristic
subgroupG/H
centerC23C22×D68
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1851C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#1851D17
hypocenterC17#26018
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1851

Complements

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

NH ≅ G/Ntype
3. #14530348C2semidirect
4. #14530282C2semidirect
5. #14530272C2semidirect
6. #14530123C2semidirect
7. #14529739C2semidirect
8. #14529543C2semidirect
9. #14529370C2semidirect
10. #14529363C2semidirect
411. C17#26018semidirect

Maximal subgroups

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

subgroupindex
#145324972
#145316592
#145303482
#145302822
#145302722
#145301232
#145297392
#145295432
#145293702
#145293632
#145289852
#145288712
#145288172
#145285042
C17×#18512
#2601817