Finite Group Databasebeta

← #17093192

Subgroups — #17093192

Order 4352.

782 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762850C211yesmaximal, characteristic
2#14532504C211yesmaximal, characteristic
3#14532471C211yesmaximal, characteristic
4#14530269C211yesmaximal, characteristic, complemented
5#14530268C211yesmaximal, characteristic, complemented
6#14530119C211yesmaximal, characteristic, complemented
7#14530109C211yesmaximal, characteristic, complemented
8#14529522C211yesmaximal, characteristic, complemented
9#14529518C211yesmaximal, characteristic, complemented
10#14528982C211yesmaximal, characteristic
11#14528970C211yesmaximal, characteristic
12#14528945C211yesmaximal, characteristic, complemented
13#14528814C211yesmaximal, characteristic
14#14528569C211yesmaximal, characteristic
15C2×C17×#501C211yesmaximal, characteristic, complemented
16C22×#63218C2211yescharacteristic
17C2×#10587578C2211yescharacteristic
18C2×C4⋊C4×D17C2211yescharacteristic
19#11762850—24no—
20C2×#10587489C2211yescharacteristic
21C2×#10587478C2211yescharacteristic
22C2×#10587478C2211yescharacteristic
23C2×#10587478C2211yescharacteristic
24C2×#10587488C2211yescharacteristic
25C2×#10587479C2211yescharacteristic
26C2×#10587491C2211yescharacteristic
27C2×C8×Dic17C2211yescharacteristic
28#11762747—21no—
29#11762742—21no—
30#11762731—21no—
31#11762720—21no—
32#11762716—21no—
33#11762708C2211yescharacteristic
34#11762708C2211yescharacteristic
35#11762707C2211yescharacteristic
36#11762707C2211yescharacteristic
37#11762690—21no—
38#11762680—21no—
39#11762674C2211yescharacteristic
40#11762674C2211yescharacteristic
41#11762666—21no—
42#11762666C2211yescharacteristic
43#11762666C2211yescharacteristic
44C4×#63227C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762564C2211yescharacteristic
52#11762563C2211yescharacteristic
53C2×C17×C4⋊Q8C2211yescharacteristic
54C2×C17×#125C2211yescharacteristic
55C17×#501—24no—
56C2×C17×#107C2211yescharacteristic
57C2×C17×#107C2211yescharacteristic
58C2×C17×#106C2211yescharacteristic
59C2×C17×#106C2211yescharacteristic
60C2×C17×#97C2211yescharacteristic
61C22×Q8×C17—21no—
62C22×C4×D17—21no—
63C22×C4×D17C2311yescharacteristic
64C22×C8×C17C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C17×C4⋊Q8—22no—
67C17×C4⋊Q8—41no—
68C17×#125—22no—
69C17×#125—41no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C42×C17C2311yescharacteristic
78C2×#63227—21no—
79C2×#63227—21no—
80C2×#63227—21no—
81C2×#63227D411yescharacteristic
82C2×#63227D411yescharacteristic
83C2×C4⋊Dic17—21no—
84C2×C4⋊Dic17—21no—
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×#63225—21no—
90C2×#63225—21no—
91C2×#63225—21no—
92C2×C4×Dic17—21no—
93C2×C4×Dic17C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C2×#63218—22no—
97C2×#63218—22no—
98C2×#63218Q814yescomplemented
99#10587578—22no—
100#10587578—41no—
101C4⋊C4×D17—22no—
102C4⋊C4×D17—41no—
103C17×#107—22no—
104C17×#107D414yes—
105C17×#106—22no—
106C17×#106—22no—
107C17×#97—22no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502D411yescharacteristic
119#10587502D411yescharacteristic
120#10587491Q814yescomplemented
121#10587489—22no—
122#10587488—22no—
123C8×Dic17—22no—
124#10587479—22no—
125#10587478—22no—
126#10587478—41no—
127#10587478—41no—
128C23×D17C2×D411yescharacteristic
129C2×Q8×C17—42no—
130C2×Q8×C17—42no—
131C2×Q8×C17—81no—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2411yescharacteristic
139C22×Dic17—21no—
140C22×Dic17—21no—
141C22×Dic17—21no—
142C22×Dic17—21no—
143C22×Dic17C2×D411yescharacteristic
144C22×Dic17C2×D411yescharacteristic
145C22×Dic17C2×D411yescharacteristic
146C2×C4×D17—21no—
147C2×C4×D17—21no—
148C2×C4×D17—21no—
149C2×C4×D17—21no—
150C2×C4×D17—42no—
151C2×C4×D17—42no—
152C2×C4×D17—81no—
153C2×C4×D17C2×Q812yes—
154C2×C4×D17C2×Q812yes—
155C2×C8×C17—21no—
156C2×C8×C17—21no—
157C2×C8×C17C2×D412yes—
158C2×C8×C17C2×Q812yes—
159C2×C8×C17C2×Q812yes—
160C2×C8×C17#4112yes—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—42no—
168C4⋊C4×C17—42no—
169C4⋊C4×C17—42no—
170C4⋊C4×C17C2×D412yes—
171C4⋊C4×C17C2×D412yes—
172C42×C17—21no—
173C42×C17—21no—
174#63227—41no—
175#63227—41no—
176#63227—42no—
177#63227—42no—
178#63227—81no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17—21no—
186C4⋊Dic17—21no—
187C4⋊Dic17—81no—
188C4⋊Dic17—81no—
189#63225—42no—
190#63225—42no—
191#63225—81no—
192C4×Dic17—21no—
193C4×Dic17—21no—
194C4×Dic17—42no—
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622C2×Q812yes—
200C2×#3622C2×Q812yes—
201#63218—28no—
202#26079—171nomaximal, Sylow
203C23×C17C22×D411yescharacteristic
204C22×D17—21no—
205C22×D17—21no—
206C22×D17—21no—
207C22×D17—21no—
208C22×D17—21no—
209C22×D17—21no—
210C22×D17C4⋊Q812yes—
211Q8×C17—84no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—22no—
215C2×C4×C17—22no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—41no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17C22×D411yescharacteristic
231C2×C4×C17C22×Q811yescharacteristic
232C2×C4×C17C22×Q811yescharacteristic
233C2×C4×C17C2×#4111yescharacteristic
234C2×C4×C17H(5,𝔽2)11yescharacteristic
235C2×C4×C172−1+411yescharacteristic
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—21no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256C2×Dic17C4⋊Q812yes—
257C4×D17—22no—
258C4×D17—22no—
259C4×D17—22no—
260C4×D17—22no—
261C4×D17—84no—
262C8×C17—22no—
263C8×C17C22⋊Q814yes—
264#3622—22no—
265#3622—22no—
266C2×#501D1711yescharacteristic, complemented
267C2×#428—171no—
268#2442—171no—
269#2439—171no—
270#2085—171no—
271#2085—171no—
272#2059—171no—
273#2057—171no—
274#1943—171no—
275#1942—171no—
276#1854—171no—
277#1852—171no—
278#1846—171no—
279#1809—171no—
280#1753—171no—
281C22×C17C2×C4⋊Q811yescharacteristic
282C22×C17D4211yescharacteristic
283C22×C17#54711yescharacteristic
284C22×C17#57511yescharacteristic
285C22×C17#57511yescharacteristic
286C22×C17#57611yescharacteristic
287C22×C17#57811yescharacteristic
288C2×D17—22no—
289C2×D17—22no—
290C2×D17—22no—
291C2×D17—41no—
292C2×D17—41no—
293C2×D17—41no—
294C2×D17—41no—
295C4×C17—42no—
296C4×C17—41no—
297C4×C17—42no—
298C4×C17—41no—
299C4×C17—81no—
300C4×C17—81no—
301C4×C17C2×C22⋊Q811yescharacteristic
302C4×C17#54411yescharacteristic
303C4×C17D4×Q811yescharacteristic
304C4×C17#55411yescharacteristic
305Dic17—22no—
306Dic17—41no—
307Dic17—41no—
308Dic17—81no—
309Dic17—81no—
310Dic17—81no—
311Dic17—81no—
312C22×M16—171no—
313C2×C4⋊Q8C2×D1711yescharacteristic
314C2×#125C2×D1711yescharacteristic
315C2×#117—171no—
316C22×C4⋊C4—171no—
317#501—24no—
318#428—344no—
319C2×#107—171no—
320C2×#107—171no—
321C2×#107—171no—
322C2×#107—171no—
323C2×#107C2×D1711yescharacteristic
324C2×#107C2×D1711yescharacteristic
325C2×#106—171no—
326C2×#106—171no—
327C2×#106C2×D1711yescharacteristic
328C2×#106C2×D1711yescharacteristic
329C2×C22⋊C8—171no—
330C2×#97C2×D1711yescharacteristic
331C2×C4×C8—171no—
332#400—341no—
333#398—341no—
334#396—341no—
335#393—341no—
336#391—341no—
337#389—171no—
338#389—171no—
339#389—171no—
340#389—171no—
341#387—341no—
342#385—341no—
343#382—171no—
344#382—171no—
345#380—171no—
346#380—171no—
347#380—341no—
348C4×C22⋊C4—171no—
349#340—171no—
350#340—171no—
351#340—171no—
352#340—171no—
353#340—171no—
354#340—171no—
355#336—171no—
356#336—171no—
357C2×C17#244011yescharacteristic
358C2×C17#316411yescharacteristic
359C2×C17#316611yescharacteristic
360C2×C17#327311yescharacteristic
361C2×C17#327711yescharacteristic
362C2×C17#334711yescharacteristic
363C2×C17#334911yescharacteristic
364D17—42no—
365C22×Q8—21no—
366C23×C4—171no—
367C23×C4—341no—
368C2×M16—174no—
369C2×M16—342no—
370C2×M16—342no—
371C22×C8—171no—
372C22×C8—171no—
373C22×C8C22×D1711yescharacteristic
374C22×C8C22×D1711yescharacteristic
375C4⋊Q8—22no—
376C4⋊Q8—41no—
377#125—22no—
378#125—41no—
379#117—342no—
380#117—681no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—171no—
385C2×C4⋊C4—171no—
386C2×C4⋊C4—171no—
387C2×C4⋊C4—171no—
388C2×C4⋊C4—342no—
389C2×C4⋊C4—341no—
390C2×C4⋊C4—341no—
391C2×C4⋊C4—341no—
392C2×C4⋊C4—341no—
393C2×C4⋊C4—341no—
394C2×C4⋊C4—681no—
395C2×C4⋊C4C22×D1711yescharacteristic
396C2×C4⋊C4C22×D1711yescharacteristic
397C2×C4⋊C4C22×D1711yescharacteristic
398C2×C4⋊C4C22×D1711yescharacteristic
399C2×C22⋊C4—171no—
400C2×C22⋊C4—171no—
401C2×C22⋊C4—341no—
402C2×C22⋊C4—341no—
403C2×C22⋊C4—341no—
404C2×C42—171no—
405C2×C42—341no—
406C2×C42C22×D1711yescharacteristic
407#107—22no—
408#107—342no—
409#107—342no—
410#107—681no—
411#107—681no—
412#107C22⋊D1714yes—
413#106—22no—
414#106—22no—
415#106—342no—
416#106—342no—
417C22⋊C8—174no—
418#97—22no—
419C4×C8—342no—
420#95—171no—
421#95—171no—
422#95—341no—
423#95—341no—
424#95—341no—
425#95—341no—
426#95—341no—
427#95—341no—
428#95—341no—
429#95—341no—
430#95—341no—
431#95—341no—
432C17#2607911yescharacteristic, complemented, Sylow
433C24—171no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—81no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—171no—
442C22×C4—172no—
443C22×C4—172no—
444C22×C4—171no—
445C22×C4—171no—
446C22×C4—341no—
447C22×C4—341no—
448C22×C4—341no—
449C22×C4—341no—
450C22×C4—341no—
451C22×C4—341no—
452C22×C4—341no—
453C22×C4—341no—
454C22×C4—682no—
455C22×C4—682no—
456C22×C4—1361no—
457C22×C4D4×D1711yescharacteristic
458C22×C4D4×D1711yescharacteristic
459C22×C4C23×D1711yescharacteristic
460M16—348no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—172no—
464C2×C8—172no—
465C2×C8—341no—
466C2×C8—341no—
467C2×C8—341no—
468C2×C8—341no—
469C2×C8Q8×D1712yes—
470C2×C8Q8×D1712yes—
471C2×C8#6325612yes—
472C2×C8C2×C22⋊D1712yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—341no—
483C4⋊C4—341no—
484C4⋊C4—341no—
485C4⋊C4—341no—
486C4⋊C4—341no—
487C4⋊C4—341no—
488C4⋊C4—341no—
489C4⋊C4—341no—
490C4⋊C4—682no—
491C4⋊C4—682no—
492C4⋊C4—1361no—
493C4⋊C4—1361no—
494C4⋊C4—1361no—
495C4⋊C4C2×C22⋊D1712yes—
496C4⋊C4C2×C22⋊D1712yes—
497C22⋊C4—681no—
498C22⋊C4—681no—
499C22⋊C4—682no—
500C22⋊C4—682no—
501C22⋊C4—1361no—
502C42—21no—
503C42—21no—
504C42—341no—
505C42—341no—
506C42—682no—
507C23—172no—
508C23—341no—
509C23—341no—
510C23—341no—
511C23—341no—
512C23—341no—
513C23—341no—
514C23C2×D4×D1711yescharacteristic
515Q8—84no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—172no—
535C2×C4—341no—
536C2×C4—341no—
537C2×C4—342no—
538C2×C4—342no—
539C2×C4—342no—
540C2×C4—342no—
541C2×C4—341no—
542C2×C4—341no—
543C2×C4—341no—
544C2×C4—341no—
545C2×C4—341no—
546C2×C4—341no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—681no—
551C2×C4—681no—
552C2×C4—681no—
553C2×C4—681no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4—681no—
557C2×C4—681no—
558C2×C4—681no—
559C2×C4—1364no—
560C2×C4#1058769111yescharacteristic
561C2×C4C2×Q8×D1711yescharacteristic
562C2×C4C2×Q8×D1711yescharacteristic
563C2×C4C2×#6325611yescharacteristic
564C2×C4#1058769411yescharacteristic
565C2×C4C22×C22⋊D1711yescharacteristic
566C8—22no—
567C8—342no—
568C8—342no—
569C8#1058763214yes—
570C22—342no—
571C22—342no—
572C22—342no—
573C22—681no—
574C22—681no—
575C22—681no—
576C22—681no—
577C22C4⋊Q8×D1711yescharacteristic
578C22D4×C22⋊D1711yescharacteristic
579C22#1176346211yescharacteristic
580C22#1176360511yescharacteristic
581C22#1176360811yescharacteristic
582C22#1176361311yescharacteristic
583C22#1176362511yescharacteristic
584C4—41no—
585C4—42no—
586C4—42no—
587C4—41no—
588C4—81no—
589C4—81no—
590C4—342no—
591C4—681no—
592C4—681no—
593C4—1361no—
594C4—1361no—
595C4—1361no—
596C4—1361no—
597C4C2×#1058763211yescharacteristic
598C4#1176343611yescharacteristic
599C4Q8×C22⋊D1711yescharacteristic
600C4#1176352011yescharacteristic
601C2—682no—
602C2#1453248211yescharacteristic
603C2#1453931911yescharacteristic
604C2#1453933511yescharacteristic
605C2#1454038911yescharacteristic
606C2#1454043511yescharacteristic
607C2#1454128311yescharacteristic
608C2#1454131311yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#501C2
socleC23×C17C22×D4
prime radicalC23×D17C2×D4
hypercenterC2×#501D17
hypocenterC17#26079
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
4. #14530269C2semidirect
5. #14530268C2semidirect
6. #14530119C2semidirect
7. #14530109C2semidirect
8. #14529522C2semidirect
9. #14529518C2semidirect
12. #14528945C2semidirect
15. C2×C17×#501C2semidirect
98. C2×#63218Q8semidirect
120. #10587491Q8semidirect
266. C2×#501D17semidirect
432. C17#26079semidirect

Maximal subgroups

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

subgroupindex
C2×#117628502
#145325042
#145324712
#145302692
#145302682
#145301192
#145301092
#145295222
#145295182
#145289822
#145289702
#145289452
#145288142
#145285692
C2×C17×#5012
#2607917