Finite Group Databasebeta

← #17044902

Subgroups — #17044902

Order 4352.

786 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762865C211yesmaximal, characteristic
2#14531190C211yesmaximal, characteristic
3#14531180C211yesmaximal, characteristic
4#14529554C211yesmaximal, characteristic, complemented
5#14529552C211yesmaximal, characteristic, complemented
6#14529550C211yesmaximal, characteristic, complemented
7#14528986C211yesmaximal, characteristic
8#14528984C211yesmaximal, characteristic
9#14528982C211yesmaximal, characteristic
10#14528980C211yesmaximal, characteristic
11#14528951C211yesmaximal, characteristic, complemented
12#14528949C211yesmaximal, characteristic, complemented
13#14528916C211yesmaximal, characteristic, complemented
14#14528916C211yesmaximal, characteristic, complemented
15C17×#1949C211yesmaximal, characteristic, complemented
16C2×#10587578C2211yescharacteristic
17C2×#10587578C2211yescharacteristic
18C2×C42×D17C2211yescharacteristic
19#11762865C418yescomplemented
20C2×#10587489C2211yescharacteristic
21C2×#10587478C2211yescharacteristic
22C2×#10587478C2211yescharacteristic
23C2×#10587488C2211yescharacteristic
24C2×#10587479C2211yescharacteristic
25C2×#10587479C2211yescharacteristic
26C2×C4⋊#3622C2211yescharacteristic
27C2×#10587491C2211yescharacteristic
28C2×#10587491C2211yescharacteristic
29C2×#10587486C2211yescharacteristic
30C2×#10587486C2211yescharacteristic
31#11762678C2211yescharacteristic
32#11762678C2211yescharacteristic
33#11762676—21no—
34#11762675C2211yescharacteristic
35#11762674C2211yescharacteristic
36#11762674C2211yescharacteristic
37#11762671—21no—
38#11762668—21no—
39#11762668—21no—
40#11762666C2211yescharacteristic
41#11762666C2211yescharacteristic
42C4×C4⋊Dic17C2211yescharacteristic
43#11762660—21no—
44#11762660—21no—
45C4×#63227—21no—
46#11762655—21no—
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762571C2211yescharacteristic
53C2×C17×#107C2211yescharacteristic
54C2×C17×#106C2211yescharacteristic
55C2×C17×C4⋊C8C2211yescharacteristic
56C17×#382C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#340C2211yescharacteristic
60C22×C4×D17D411yescharacteristic
61C22×C4×D17D411yescharacteristic
62C22×C4×D17C2311yescharacteristic
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×C17C2311yescharacteristic
71C2×#63227—21no—
72C2×#63227—21no—
73C2×#63227—21no—
74C2×#63227—21no—
75C2×C4⋊Dic17C2311yescharacteristic
76C2×C4⋊Dic17C2311yescharacteristic
77C2×C4⋊Dic17C2311yescharacteristic
78C2×C4⋊Dic17C2311yescharacteristic
79C2×#63225—21no—
80C2×#63225—21no—
81C2×#63225—21no—
82C2×#63225—21no—
83C2×C4×Dic17—21no—
84C2×C4×Dic17—21no—
85C2×C4×Dic17—21no—
86C2×C4×Dic17D411yescharacteristic
87C2×C4×Dic17D411yescharacteristic
88C2×C4×Dic17C2311yescharacteristic
89C22×#3622C2311yescharacteristic
90C22×#3622C2311yescharacteristic
91#10587578—41no—
92#10587578—41no—
93#10587578—41no—
94#10587578—41no—
95C42×D17C2×C414yes—
96C42×D17Q814yes—
97C17×#107—41no—
98C17×#106—41no—
99C17×C4⋊C8C2×C414yes—
100C17×#95—21no—
101C17×#95—21no—
102C17×#95—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#10587502—21no—
112#10587491C2×C414yes—
113#10587491C2×C414yes—
114#10587489—41no—
115#10587488—41no—
116#10587486C2×C414yes—
117#10587486C2×C414yes—
118#10587479—41no—
119#10587479—41no—
120#10587478—41no—
121#10587478—41no—
122C4⋊#3622C2×C414yes—
123C23×D17C2×D411yescharacteristic
124C22×C4×C17—21no—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17C2×D411yescharacteristic
129C22×C4×C17C2×D411yescharacteristic
130C22×C4×C17C2411yescharacteristic
131C22×Dic17—21no—
132C22×Dic17—21no—
133C22×Dic17—21no—
134C22×Dic17—21no—
135C22×Dic17C2×D411yescharacteristic
136C22×Dic17C2×D411yescharacteristic
137C22×Dic17C2×D411yescharacteristic
138C2×C4×D17—21no—
139C2×C4×D17—21no—
140C2×C4×D17—21no—
141C2×C4×D17—21no—
142C2×C4×D17—41no—
143C2×C4×D17—41no—
144C2×C4×D17—41no—
145C2×C4×D17—41no—
146C2×C4×D17C4⋊C414yes—
147C2×C4×D17C4⋊C414yes—
148C2×C4×D17C22×C412yes—
149C2×C4×D17C2×Q812yes—
150C2×C8×C17—21no—
151C2×C8×C17—21no—
152C2×C8×C17—21no—
153C2×C8×C17—21no—
154C2×C8×C17C22×C412yes—
155C2×C8×C17C22×C412yes—
156C4⋊C4×C17—21no—
157C4⋊C4×C17—21no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C42×C17—42no—
165C42×C17C22×C412yes—
166C42×C17C2×Q812yes—
167#63227—42no—
168#63227—42no—
169#63227—42no—
170#63227—42no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179#63225—42no—
180#63225—42no—
181#63225—42no—
182#63225—42no—
183C4×Dic17—42no—
184C4×Dic17—42no—
185C4×Dic17—42no—
186C4×Dic17C4⋊C414yes—
187C4×Dic17C4⋊C414yes—
188C4×Dic17C22×C412yes—
189C4×Dic17C2×Q812yes—
190C2×#3622—21no—
191C2×#3622—21no—
192C2×#3622—21no—
193C2×#3622—21no—
194C2×#3622C22×C412yes—
195C2×#3622C22×C412yes—
196#21989—171nomaximal, Sylow
197C23×C17C22×D411yescharacteristic
198C22×D17—21no—
199C22×D17—21no—
200C22×D17—21no—
201C22×D17—21no—
202C22×D17—21no—
203C22×D17—21no—
204C22×D17C2×C4⋊C412yes—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—21no—
208C2×C4×C17—21no—
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—41no—
220C2×C4×C17—41no—
221C2×C4×C17C2×C4⋊C412yes—
222C2×C4×C17C2×C4⋊C412yes—
223C2×C4×C17C23×C411yescharacteristic
224C2×C4×C17C22×Q811yescharacteristic
225C2×C4×C17H(5,𝔽2)11yescharacteristic
226C2×C4×C17H(5,𝔽2)11yescharacteristic
227C2×C4×C172−1+411yescharacteristic
228C2×C4×C172−1+411yescharacteristic
229C2×Dic17—21no—
230C2×Dic17—21no—
231C2×Dic17—21no—
232C2×Dic17—21no—
233C2×Dic17—21no—
234C2×Dic17—21no—
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×Dic17C2×C4⋊C412yes—
248C2×Dic17C2×C4⋊C412yes—
249C2×Dic17C2×C4⋊C412yes—
250C4×D17—22no—
251C4×D17—22no—
252C4×D17—22no—
253C4×D17—22no—
254C4×D17—44no—
255C4×D17—44no—
256C8×C17—22no—
257C8×C17—22no—
258#3622—22no—
259#3622—22no—
260C2×#432—171no—
261#2285—171no—
262#2283—171no—
263#1949—171no—
264#1949—171no—
265#1949—171no—
266#1949D1711yescharacteristic, complemented
267#1854—171no—
268#1854—171no—
269#1854—171no—
270#1854—171no—
271#1847—171no—
272#1847—171no—
273#1836—171no—
274#1836—171no—
275C22×C17C22×C4⋊C411yescharacteristic
276C22×C17#53511yescharacteristic
277C22×C17#53611yescharacteristic
278C22×C17#57511yescharacteristic
279C22×C17#57511yescharacteristic
280C22×C17#57511yescharacteristic
281C22×C17#57511yescharacteristic
282C2×D17—22no—
283C2×D17—22no—
284C2×D17—22no—
285C2×D17—41no—
286C2×D17—41no—
287C2×D17—41no—
288C2×D17—41no—
289C4×C17—22no—
290C4×C17—22no—
291C4×C17—81no—
292C4×C17—81no—
293C4×C17—81no—
294C4×C17—81no—
295C4×C17#52011yescharacteristic
296C4×C17#52011yescharacteristic
297C4×C17#54411yescharacteristic
298C4×C17#54411yescharacteristic
299Dic17—22no—
300Dic17—22no—
301Dic17—22no—
302Dic17—81no—
303Dic17—81no—
304Dic17—81no—
305Dic17—81no—
306C2×#117—171no—
307C2×#117—171no—
308C22×C42—171no—
309#432—178no—
310C2×#107—171no—
311C2×#107—171no—
312C2×#107—171no—
313C2×#107C2×D1711yescharacteristic
314C2×#106—171no—
315C2×#106—171no—
316C2×#106—171no—
317C2×#106C2×D1711yescharacteristic
318C2×C4⋊C8—171no—
319C2×C4⋊C8C2×D1711yescharacteristic
320C2×C22⋊C8—171no—
321C2×C22⋊C8—171no—
322C2×#97—171no—
323C2×#97—171no—
324#384—171no—
325#384—171no—
326#383—341no—
327#382—171no—
328#382—171no—
329#382—171no—
330#382C2×D1711yescharacteristic
331#381—341no—
332#380—171no—
333#380—171no—
334#380—341no—
335#380—341no—
336C4×C4⋊C4—171no—
337C4×C4⋊C4—341no—
338C4×C4⋊C4—341no—
339C4×C4⋊C4C2×D1711yescharacteristic
340C4×C22⋊C4—341no—
341#376—341no—
342#340—171no—
343#340—171no—
344#340—171no—
345#340—171no—
346#340—171no—
347#340—171no—
348#340C2×D1711yescharacteristic
349#340C2×D1711yescharacteristic
350C2×C17#231711yescharacteristic
351C2×C17#291111yescharacteristic
352C2×C17#291111yescharacteristic
353C2×C17#321911yescharacteristic
354C2×C17#321911yescharacteristic
355C2×C17#324011yescharacteristic
356C2×C17#324011yescharacteristic
357D17—42no—
358C23×C4—171no—
359C23×C4—171no—
360C23×C4—171no—
361C22×C8—171no—
362C22×C8—171no—
363C22×C8C22×D1711yescharacteristic
364C22×C8C22×D1711yescharacteristic
365#117—681no—
366#117—681no—
367#117—681no—
368#117—681no—
369C2×C4⋊C4—171no—
370C2×C4⋊C4—171no—
371C2×C4⋊C4—171no—
372C2×C4⋊C4—171no—
373C2×C4⋊C4—341no—
374C2×C4⋊C4—341no—
375C2×C4⋊C4—341no—
376C2×C4⋊C4—341no—
377C2×C4⋊C4C22×D1711yescharacteristic
378C2×C4⋊C4C22×D1711yescharacteristic
379C2×C4⋊C4C22×D1711yescharacteristic
380C2×C4⋊C4C22×D1711yescharacteristic
381C2×C22⋊C4—341no—
382C2×C22⋊C4—341no—
383C2×C22⋊C4—341no—
384C2×C22⋊C4—341no—
385C2×C42—21no—
386C2×C42—171no—
387C2×C42—171no—
388C2×C42—171no—
389C2×C42—174no—
390C2×C42—174no—
391C2×C42—341no—
392C2×C42—341no—
393C2×C42—341no—
394C2×C42C22×D1711yescharacteristic
395#107—41no—
396#107—681no—
397#107—681no—
398#107—681no—
399#106—41no—
400#106—681no—
401#106—681no—
402#106—681no—
403C4⋊C8—174no—
404C4⋊C8C4×D1714yes—
405C22⋊C8—174no—
406C22⋊C8—174no—
407#97—174no—
408#97—174no—
409#95—21no—
410#95—21no—
411#95—21no—
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—
421C17#2198911yescharacteristic, complemented, Sylow
422C24—171no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—171no—
428C22×C4—171no—
429C22×C4—171no—
430C22×C4—172no—
431C22×C4—172no—
432C22×C4—174no—
433C22×C4—174no—
434C22×C4—341no—
435C22×C4—341no—
436C22×C4—341no—
437C22×C4—341no—
438C22×C4—341no—
439C22×C4—341no—
440C22×C4—341no—
441C22×C4—341no—
442C22×C4—681no—
443C22×C4—681no—
444C22×C4—681no—
445C22×C4—681no—
446C22×C4D4×D1711yescharacteristic
447C22×C4D4×D1711yescharacteristic
448C22×C4C23×D1711yescharacteristic
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—172no—
454C2×C8—172no—
455C2×C8—341no—
456C2×C8—341no—
457C2×C8—341no—
458C2×C8—341no—
459C2×C8C2×C4×D1712yes—
460C2×C8C2×C4×D1712yes—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—341no—
470C4⋊C4—341no—
471C4⋊C4—341no—
472C4⋊C4—341no—
473C4⋊C4—341no—
474C4⋊C4—341no—
475C4⋊C4—341no—
476C4⋊C4—341no—
477C4⋊C4—682no—
478C4⋊C4—682no—
479C4⋊C4—682no—
480C4⋊C4—682no—
481C22⋊C4—682no—
482C22⋊C4—682no—
483C22⋊C4—682no—
484C22⋊C4—682no—
485C42—42no—
486C42—172no—
487C42—172no—
488C42—174no—
489C42—174no—
490C42—682no—
491C42—682no—
492C42—682no—
493C42C2×C4×D1712yes—
494C42Q8×D1712yes—
495C23—172no—
496C23—341no—
497C23—341no—
498C23—341no—
499C23—341no—
500C23—341no—
501C23—341no—
502C23C2×D4×D1711yescharacteristic
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
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—172no—
520C2×C4—172no—
521C2×C4—172no—
522C2×C4—341no—
523C2×C4—341no—
524C2×C4—341no—
525C2×C4—341no—
526C2×C4—341no—
527C2×C4—341no—
528C2×C4—342no—
529C2×C4—342no—
530C2×C4—342no—
531C2×C4—342no—
532C2×C4—684no—
533C2×C4—684no—
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×C4—681no—
546C2×C4C4⋊C4×D1712yes—
547C2×C4C4⋊C4×D1712yes—
548C2×C4C22×C4×D1711yescharacteristic
549C2×C4#1058769111yescharacteristic
550C2×C4C2×Q8×D1711yescharacteristic
551C2×C4#1058769411yescharacteristic
552C2×C4#1058769611yescharacteristic
553C2×C4#1058769711yescharacteristic
554C8—22no—
555C8—22no—
556C8—342no—
557C8—342no—
558C22—342no—
559C22—342no—
560C22—342no—
561C22—681no—
562C22—681no—
563C22—681no—
564C22—681no—
565C22C2×C4⋊C4×D1711yescharacteristic
566C22#1176337511yescharacteristic
567C22#1176338211yescharacteristic
568C22#1176360511yescharacteristic
569C22#1176360611yescharacteristic
570C22#1176360611yescharacteristic
571C22#1176360811yescharacteristic
572C4—22no—
573C4—22no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4—342no—
579C4—342no—
580C4—342no—
581C4—1361no—
582C4—1361no—
583C4—1361no—
584C4—1361no—
585C4#1176328811yescharacteristic
586C4#1176329111yescharacteristic
587C4#1176343611yescharacteristic
588C4#1176343911yescharacteristic
589C2—682no—
590C2#1453140611yescharacteristic
591C2#1453737111yescharacteristic
592C2#1453737211yescharacteristic
593C2#1453982011yescharacteristic
594C2#1453982611yescharacteristic
595C2#1454002311yescharacteristic
596C2#1454002911yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1949C2
socleC23×C17C22×D4
prime radicalC23×D17C2×D4
cosocleC22×C4×C17C24
hypercenter#1949D17
hypocenterC17#21989
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
4. #14529554C2semidirect
5. #14529552C2semidirect
6. #14529550C2semidirect
11. #14528951C2semidirect
12. #14528949C2semidirect
13. #14528916C2semidirect
14. #14528916C2semidirect
15. C17×#1949C2semidirect
19. #11762865C4semidirect
266. #1949D17semidirect
421. C17#21989semidirect

Maximal subgroups

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

subgroupindex
C2×#117628652
#145311902
#145311802
#145295542
#145295522
#145295502
#145289862
#145289842
#145289822
#145289802
#145289512
#145289492
#145289162
C17×#19492
#2198917