Finite Group Databasebeta

← #17054030

Subgroups — #17054030

Order 4352.

710 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#10587636C211yesmaximal, characteristic
2#14537449C211yesmaximal, characteristic, complemented
3#14529676C212yesmaximal
4#14529432C212yesmaximal, complemented
5#14529429C212yesmaximal, complemented
6#14528955C211yesmaximal, characteristic, complemented
7#14528804C211yesmaximal, characteristic, complemented
8#14528801C211yesmaximal, characteristic, complemented
9#14528669C212yesmaximal, complemented
10#14528654C211yesmaximal, characteristic, complemented
11C17×#2925C211yesmaximal, characteristic, complemented
12C2×#10587636C2211yescharacteristic
13C4×#63252C2211yescharacteristic
14C4×#63231C411yescharacteristic, complemented
15C4×#63231C411yescharacteristic, complemented
16C4×#63230C411yescharacteristic, complemented
17C4×#63230C411yescharacteristic, complemented
18C4×#63229C411yescharacteristic, complemented
19C4×#63229C411yescharacteristic, complemented
20C4×#63228C411yescharacteristic, complemented
21C4×#63228C411yescharacteristic, complemented
22#11762869—22no—
23#11762865—21no—
24#11762861—21no—
25C2×#10587493C2212yes—
26C2×#10587493C2212yes—
27#11762845C2211yescharacteristic
28#11762833C2211yescharacteristic
29C2×#10587508C2212yes—
30C2×#10587496C2212yes—
31#11762805C2211yescharacteristic
32#11762803C2211yescharacteristic
33#11762775C2212yes—
34#11762764C2212yes—
35C2×#10587486C2212yes—
36C2×C4×#3622C2211yescharacteristic
37C4×C4⋊Dic17C2211yescharacteristic
38#11762570C2212yes—
39#11762568C2212yes—
40#11762563C2212yes—
41C4×#41×C17C2211yescharacteristic
42C17×#433—22no—
43C17×#432—21no—
44C17×#431—21no—
45C17×#423C2211yescharacteristic
46C2×C17×C4⋊C8C2211yescharacteristic
47C17×#408C2212yes—
48C17×#405C2212yes—
49C2×#41×C17C2311yescharacteristic
50C2×#63252C2311yescharacteristic
51C2×M16×C17C2312yes—
52C22×C8×C17C2312yes—
53C4×Q8×C17C2×C411yescharacteristic
54C4×Q8×C17C2×C411yescharacteristic
55C4×D4×C17—21no—
56C4×D4×C17—21no—
57C4×D4×C17C2×C411yescharacteristic
58C4×D4×C17C2×C411yescharacteristic
59C17×#117—21no—
60C17×#117C2311yescharacteristic
61C2×C42×C17—21no—
62C2×C42×C17C2311yescharacteristic
63#10587636—22no—
64#10587636—22no—
65C2×#63231C2×C411yescharacteristic
66C2×#63230C2×C411yescharacteristic
67C2×#63229C2×C411yescharacteristic
68C2×#63228C2×C411yescharacteristic
69C4×C22⋊D17—21no—
70C4×C22⋊D17—21no—
71#10587612C2311yescharacteristic
72C2×C4⋊Dic17—21no—
73C2×C4×Dic17C2312yes—
74C2×#63223C2312yes—
75C22×#3622C2311yescharacteristic
76C22×#3622C2311yescharacteristic
77C4×D68C2×C411yescharacteristic
78C4×D68C2×C411yescharacteristic
79#10587558—21no—
80C42×D17—21no—
81C4×Dic34C2×C411yescharacteristic
82C4×Dic34C2×C411yescharacteristic
83C17×C4⋊C8—21no—
84C17×C4⋊C8—21no—
85C17×C4⋊C8D411yescharacteristic
86C17×C4⋊C8D411yescharacteristic
87C17×C4⋊C8D412yes—
88C17×C22⋊C8—22no—
89C17×C22⋊C8—22no—
90C17×#97—22no—
91C4×C8×C17—22no—
92#10587508—42no—
93#10587506C2×C411yescharacteristic
94#10587506C2×C411yescharacteristic
95#10587503C2×C411yescharacteristic
96#10587503C2×C411yescharacteristic
97#10587502—21no—
98#10587496—42no—
99#10587493—24no—
100#10587493—42no—
101#10587491—22no—
102#10587491—22no—
103#10587486—24no—
104#10587486—22no—
105C8×Dic17—22no—
106#10587481C2×C411yescharacteristic
107#10587481C2×C411yescharacteristic
108#10587480C2×C411yescharacteristic
109#10587480C2×C411yescharacteristic
110#10587479C2×C411yescharacteristic
111#10587479C2×C411yescharacteristic
112#10587478C2×C411yescharacteristic
113#10587478C2×C411yescharacteristic
114C4⋊#3622—21no—
115C4⋊#3622—21no—
116C4×#3622C2×C411yescharacteristic
117C4×#3622C2×C411yescharacteristic
118C4×#3622C2×C411yescharacteristic
119C4×#3622C2×C411yescharacteristic
120#41×C17—21no—
121#41×C17—21no—
122#41×C17—21no—
123#41×C17—21no—
124C2×Q8×C17C22×C411yescharacteristic
125C2×D4×C17—21no—
126C2×D4×C17C22×C411yescharacteristic
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17#4111yescharacteristic
131C22×C4×C17#4111yescharacteristic
132C22×C4×C17C2411yescharacteristic
133C2×C22⋊D17—21no—
134C22×Dic17—22no—
135#63252—21no—
136#63252—21no—
137#63252—21no—
138#63252—21no—
139C2×D68C22×C411yescharacteristic
140C2×C4×D17—21no—
141C2×C4×D17—21no—
142C2×C4×D17—21no—
143C2×Dic34C22×C411yescharacteristic
144M16×C17—22no—
145M16×C17—22no—
146C2×C8×C17—22no—
147C2×C8×C17—22no—
148C2×C8×C17—22no—
149C2×C8×C17C2×D412yes—
150C2×C8×C17C2×D412yes—
151C4⋊C4×C17—21no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17C22×C411yescharacteristic
154C4⋊C4×C17C22×C411yescharacteristic
155C22⋊C4×C17—21no—
156C22⋊C4×C17—21no—
157C22⋊C4×C17—21no—
158C42×C17—21no—
159C42×C17—21no—
160C42×C17—21no—
161C42×C17C22×C411yescharacteristic
162C42×C17C22×C411yescharacteristic
163C42×C17C2×D411yescharacteristic
164C42×C17C2×D411yescharacteristic
165C22⋊Dic17—21no—
166#63231—22no—
167#63230—22no—
168#63229—22no—
169#63228—22no—
170#63227—21no—
171#63227—21no—
172C4⋊Dic17—42no—
173C4⋊Dic17C22×C411yescharacteristic
174C4⋊Dic17C22×C411yescharacteristic
175#63225—21no—
176#63225—21no—
177C4×Dic17—21no—
178C4×Dic17—21no—
179C4×Dic17—21no—
180C4×Dic17—22no—
181C4×Dic17—22no—
182#63223—22no—
183#63223—22no—
184C2×#3622—21no—
185C2×#3622—21no—
186C2×#3622—22no—
187C2×#3622C22×C411yescharacteristic
188C2×#3622C22×C411yescharacteristic
189C2×#3622C22×C411yescharacteristic
190C2×#3622C22×C411yescharacteristic
191C2×#3622#4112yes—
192C2×#3622#4112yes—
193#22803—171nomaximal, Sylow
194C23×C17—21no—
195C23×C17C2×#4111yescharacteristic
196C22×D17—21no—
197Q8×C17—21no—
198Q8×C17—21no—
199D4×C17—21no—
200D4×C17—21no—
201D4×C17—41no—
202D4×C17—41no—
203C2×C4×C17—22no—
204C2×C4×C17—21no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—41no—
208C2×C4×C17—41no—
209C2×C4×C17—41no—
210C2×C4×C17—42no—
211C2×C4×C17—81no—
212C2×C4×C17C4×D411yescharacteristic
213C2×C4×C17C4×D411yescharacteristic
214C2×C4×C17C4×D411yescharacteristic
215C2×C4×C17C4×D411yescharacteristic
216C2×C4×C17C23×C411yescharacteristic
217C2×C4×C17C22×D411yescharacteristic
218C2×C4×C17C2×#4111yescharacteristic
219C2×C4×C17C2×#4111yescharacteristic
220C2×C4×C17H(5,𝔽2)11yescharacteristic
221C2×C4×C172−1+411yescharacteristic
222C22⋊D17—41no—
223C22⋊D17—41no—
224C2×Dic17—21no—
225C2×Dic17—21no—
226C2×Dic17—21no—
227C2×Dic17—42no—
228C2×Dic17—42no—
229C2×Dic17—42no—
230D68—21no—
231D68—21no—
232C4×D17—41no—
233C4×D17—41no—
234C4×D17—42no—
235C4×D17—81no—
236Dic34—21no—
237Dic34—21no—
238C8×C17—22no—
239C8×C17—22no—
240C8×C17—42no—
241#3622—21no—
242#3622—21no—
243#3622—42no—
244#3622#11712yes—
245#3622#11712yes—
246C4×#135—171no—
247#2925—171no—
248#2925D1711yescharacteristic, complemented
249#1975—172no—
250#1929—172no—
251#1929—172no—
252#1848—171no—
253#1806—171no—
254#1806—171no—
255#1773—172no—
256#1770—171no—
257C22×C17—21no—
258C22×C17—21no—
259C22×C17—41no—
260C22×C17—41no—
261C22×C17—41no—
262C22×C17C2×C4×D411yescharacteristic
263C22×C17#55511yescharacteristic
264C22×C17#55611yescharacteristic
265C2×D17—41no—
266C2×D17—41no—
267C2×D17—41no—
268C4×C17—21no—
269C4×C17—21no—
270C4×C17—21no—
271C4×C17—41no—
272C4×C17—41no—
273C4×C17—41no—
274C4×C17#44412yes—
275C4×C17C2×#11711yescharacteristic
276C4×C17#52011yescharacteristic
277C4×C17#54711yescharacteristic
278C4×C17#55011yescharacteristic
279Dic17—41no—
280Dic17—41no—
281Dic17—41no—
282Dic17—82no—
283C2×#135—171no—
284C4×#41—171no—
285C4×#41C2×D1711yescharacteristic
286C4×Q16—171no—
287C4×Q16—171no—
288C4×SD16—171no—
289C4×SD16—171no—
290C4×SD16—171no—
291C4×SD16—171no—
292C4×D8—171no—
293C4×D8—171no—
294#433—22no—
295#433—342no—
296#432—21no—
297#432—341no—
298#431—21no—
299#431—341no—
300C2×#108—172no—
301C2×#108—172no—
302#427—171no—
303#423—171no—
304#423C2×D1711yescharacteristic
305C2×C4⋊C8C2×D1711yescharacteristic
306C2×#104—172no—
307C2×#104—172no—
308#416—171no—
309#416—171no—
310#408—172no—
311#408C2×D1712yes—
312#405—172no—
313#405C2×D1712yes—
314C2×#97—172no—
315C2×C4×C8—171no—
316C4×C4⋊C4—171no—
317#339—172no—
318#339—172no—
319#336—172no—
320C2×C17—21no—
321C2×C17—21no—
322C2×C17—81no—
323C2×C17#235411yescharacteristic
324C2×C17C2×#44411yescharacteristic
325C2×C17#295711yescharacteristic
326D17—81no—
327C2×#41—171no—
328C2×#41C22×D1711yescharacteristic
329#135—342no—
330#135—342no—
331C2×Q16—171no—
332C2×SD16—171no—
333C2×SD16—171no—
334C2×D8—171no—
335C2×M16—172no—
336C2×M16C22×D1712yes—
337C22×C8—171no—
338C22×C8—171no—
339C22×C8C22×D1712yes—
340C4×Q8—171no—
341C4×Q8—171no—
342C4×Q8C4×D1711yescharacteristic
343C4×Q8C4×D1711yescharacteristic
344C4×D4—21no—
345C4×D4—21no—
346C4×D4—171no—
347C4×D4—171no—
348C4×D4—341no—
349C4×D4—341no—
350C4×D4C4×D1711yescharacteristic
351C4×D4C4×D1711yescharacteristic
352#117—21no—
353#117—171no—
354#117—341no—
355#117C22×D1711yescharacteristic
356C2×C4⋊C4—341no—
357C2×C42—21no—
358C2×C42—172no—
359C2×C42—341no—
360C2×C42C22×D1711yescharacteristic
361#108—344no—
362#108—682no—
363#107—171no—
364#107—171no—
365#106—171no—
366#106—171no—
367C4⋊C8—21no—
368C4⋊C8—21no—
369C4⋊C8—341no—
370C4⋊C8—341no—
371C4⋊C8C22⋊D1711yescharacteristic
372C4⋊C8C22⋊D1711yescharacteristic
373C4⋊C8C22⋊D1712yes—
374#104—682no—
375#104—682no—
376#103—171no—
377#103—171no—
378#103—171no—
379#103—171no—
380#102—171no—
381#102—171no—
382#102—171no—
383#102—171no—
384C22⋊C8—22no—
385C22⋊C8—22no—
386C22⋊C8—342no—
387C22⋊C8—342no—
388#97—22no—
389#97—344no—
390#97—342no—
391C4×C8—22no—
392C4×C8—171no—
393C4×C8—171no—
394C4×C8—171no—
395C4×C8—171no—
396C4×C8—342no—
397#95—341no—
398C17#2280311yescharacteristic, complemented, Sylow
399#41—21no—
400#41—21no—
401#41—21no—
402#41—21no—
403#41—341no—
404#41—341no—
405#41—341no—
406#41—341no—
407C2×Q8—171no—
408C2×Q8C2×C4×D1711yescharacteristic
409C2×D4—21no—
410C2×D4—171no—
411C2×D4—341no—
412C2×D4C2×C4×D1711yescharacteristic
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—341no—
417C22×C4—341no—
418C22×C4—341no—
419C22×C4—342no—
420C22×C4#6325211yescharacteristic
421C22×C4#6325211yescharacteristic
422C22×C4C23×D1711yescharacteristic
423Q16—342no—
424SD16—342no—
425SD16—342no—
426D8—342no—
427M16—22no—
428M16—22no—
429M16—342no—
430M16—342no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—171no—
435C2×C8—171no—
436C2×C8—172no—
437C2×C8—172no—
438C2×C8—171no—
439C2×C8—171no—
440C2×C8—341no—
441C2×C8—341no—
442C2×C8—342no—
443C2×C8C2×C22⋊D1712yes—
444C2×C8C2×C22⋊D1712yes—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—171no—
448C4⋊C4—171no—
449C4⋊C4—341no—
450C4⋊C4—341no—
451C4⋊C4—682no—
452C4⋊C4C2×C4×D1711yescharacteristic
453C4⋊C4C2×C4×D1711yescharacteristic
454C22⋊C4—21no—
455C22⋊C4—21no—
456C22⋊C4—21no—
457C22⋊C4—341no—
458C22⋊C4—341no—
459C22⋊C4—341no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—341no—
464C42—341no—
465C42—341no—
466C42—342no—
467C42—342no—
468C42C2×C4×D1711yescharacteristic
469C42C2×C4×D1711yescharacteristic
470C42C2×C22⋊D1711yescharacteristic
471C42C2×C22⋊D1711yescharacteristic
472C23—21no—
473C23—341no—
474C23C2×#6325211yescharacteristic
475Q8—21no—
476Q8—21no—
477Q8—341no—
478Q8—341no—
479D4—21no—
480D4—21no—
481D4—41no—
482D4—41no—
483D4—341no—
484D4—341no—
485D4—681no—
486D4—681no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—81no—
496C2×C4—341no—
497C2×C4—341no—
498C2×C4—341no—
499C2×C4—681no—
500C2×C4—681no—
501C2×C4—682no—
502C2×C4—682no—
503C2×C4—682no—
504C2×C4—682no—
505C2×C4—1361no—
506C2×C4C4×C22⋊D1711yescharacteristic
507C2×C4C4×C22⋊D1711yescharacteristic
508C2×C4C4×C22⋊D1711yescharacteristic
509C2×C4C4×C22⋊D1711yescharacteristic
510C2×C4C22×C4×D1711yescharacteristic
511C2×C4#41×D1711yescharacteristic
512C2×C4#41×D1711yescharacteristic
513C2×C4#1058769611yescharacteristic
514C2×C4#1058769711yescharacteristic
515C2×C4C22×C22⋊D1711yescharacteristic
516C8—22no—
517C8—22no—
518C8—42no—
519C8—172no—
520C8—172no—
521C8—341no—
522C8—341no—
523C8—682no—
524C22—21no—
525C22—21no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—681no—
530C22—681no—
531C22—681no—
532C22C2×C4×C22⋊D1711yescharacteristic
533C22#1176352211yescharacteristic
534C22#1176352711yescharacteristic
535C4—21no—
536C4—21no—
537C4—21no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—681no—
542C4—681no—
543C4—681no—
544C4—1362no—
545C4#1176292912yes—
546C4#117×D1711yescharacteristic
547C4#1176328911yescharacteristic
548C4#1176345811yescharacteristic
549C4#1176347811yescharacteristic
550C2—21no—
551C2—21no—
552C2—81no—
553C2—1361no—
554C2#1453173411yescharacteristic
555C2C2×#1176292911yescharacteristic
556C2#1453769911yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2925C2
socleC22×C17C2×C4×D4
prime radicalC2×#10587636C22
cosocleC22×C4×C17C24
hypercenter#2925D17
hypocenterC17#22803
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ #2925

Complements

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

NH ≅ G/Ntype
2. #14537449C2semidirect
4. #14529432C2semidirect
5. #14529429C2semidirect
6. #14528955C2semidirect
7. #14528804C2semidirect
8. #14528801C2semidirect
9. #14528669C2semidirect
10. #14528654C2semidirect
11. C17×#2925C2semidirect
14. C4×#63231C4semidirect
15. C4×#63231C4semidirect
16. C4×#63230C4semidirect
17. C4×#63230C4semidirect
18. C4×#63229C4semidirect
19. C4×#63229C4semidirect
20. C4×#63228C4semidirect
21. C4×#63228C4semidirect
248. #2925D17semidirect
398. C17#22803semidirect

Maximal subgroups

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

subgroupindex
C4×#105876362
#145374492
#145296762
#145294322
#145294292
#145289552
#145288042
#145288012
#145286692
#145286542
C17×#29252
#2280317