Finite Group Databasebeta

← #17053777

Subgroups — #17053777

Order 4352.

740 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#10587636C211yesmaximal, characteristic
2#14537452C211yesmaximal, characteristic, complemented
3#14537352C211yesmaximal, characteristic, complemented
4#14529496C212yesmaximal, complemented
5#14529386C212yesmaximal, complemented
6#14529109C212yesmaximal
7#14528921C211yesmaximal, characteristic, complemented
8#14528805C211yesmaximal, characteristic, complemented
9#14528653C211yesmaximal, characteristic, complemented
10#14528629C212yesmaximal, complemented
11C17×#1806C211yesmaximal, characteristic, complemented
12C2×#10587636C2211yescharacteristic
13C2×#10587634C2212yes—
14C4×#63252C2211yescharacteristic
15C4×#63231C411yescharacteristic, complemented
16C4×#63231C411yescharacteristic, complemented
17C4×#63230C411yescharacteristic, complemented
18C4×#63230C411yescharacteristic, complemented
19C4×#63229C411yescharacteristic, complemented
20C4×#63229C411yescharacteristic, complemented
21C4×#63228C411yescharacteristic, complemented
22C4×#63228C411yescharacteristic, complemented
23#11762869—22no—
24#11762865—21no—
25#11762861—21no—
26#11762845C2211yescharacteristic
27#11762836—22no—
28#11762836C2212yes—
29#11762833—21no—
30#11762833C2211yescharacteristic
31C2×C4⋊#3622—21no—
32C2×C4⋊#3622C2211yescharacteristic
33C2×#10587476C2212yes—
34#11762805C2211yescharacteristic
35#11762803C2211yescharacteristic
36#11762777C2212yes—
37#11762767C2212yes—
38C2×#10587474C2212yes—
39C2×C4×#3622C2211yescharacteristic
40#11762573C2212yes—
41#11762573C2212yes—
42#11762569C2212yes—
43#11762564C2212yes—
44C4×#41×C17C2211yescharacteristic
45C17×#423C2211yescharacteristic
46C2×C17×#104C2212yes—
47C4×C4⋊C4×C17C2211yescharacteristic
48C17×#339C2212yes—
49C2×#41×C17C2311yescharacteristic
50C2×#63252C2311yescharacteristic
51C2×M16×C17C2312yes—
52C4×Q8×C17C2×C411yescharacteristic
53C4×Q8×C17C2×C411yescharacteristic
54C4×D4×C17—21no—
55C4×D4×C17—21no—
56C4×D4×C17C2×C411yescharacteristic
57C4×D4×C17C2×C411yescharacteristic
58C17×#117—21no—
59C17×#117C2311yescharacteristic
60C2×C4⋊C4×C17—21no—
61C2×C42×C17—21no—
62C2×C42×C17C2312yes—
63C2×C42×C17C2311yescharacteristic
64#10587636—22no—
65#10587636—22no—
66#10587634—24no—
67#10587634—42no—
68C2×#63231C2×C411yescharacteristic
69C2×#63230C2×C411yescharacteristic
70C2×#63229C2×C411yescharacteristic
71C2×#63228C2×C411yescharacteristic
72C4×C22⋊D17—21no—
73C4×C22⋊D17—21no—
74#10587612C2311yescharacteristic
75C2×#63223—22no—
76C2×#63223C2312yes—
77C22×#3622—22no—
78C22×#3622C2311yescharacteristic
79C22×#3622C2311yescharacteristic
80C22×#3622C2312yes—
81C4×D68C2×C411yescharacteristic
82C4×D68C2×C411yescharacteristic
83#10587558—21no—
84C42×D17—21no—
85C4×Dic34C2×C411yescharacteristic
86C4×Dic34C2×C411yescharacteristic
87C17×C4⋊C8—21no—
88C17×C4⋊C8—21no—
89C17×#104—24no—
90C17×#95—21no—
91#10587506C2×C411yescharacteristic
92#10587506C2×C411yescharacteristic
93#10587503C2×C411yescharacteristic
94#10587503C2×C411yescharacteristic
95#10587491—22no—
96#10587491—22no—
97#10587486—22no—
98C8×Dic17—22no—
99#10587481C2×C411yescharacteristic
100#10587481C2×C411yescharacteristic
101#10587480C2×C411yescharacteristic
102#10587480C2×C411yescharacteristic
103#10587479C2×C411yescharacteristic
104#10587479C2×C411yescharacteristic
105#10587478C2×C411yescharacteristic
106#10587478C2×C411yescharacteristic
107#10587476—42no—
108C4⋊#3622—22no—
109C4⋊#3622—22no—
110C4⋊#3622—21no—
111C4⋊#3622—21no—
112C4⋊#3622D411yescharacteristic
113C4⋊#3622D411yescharacteristic
114C4⋊#3622D412yes—
115#10587474—42no—
116C4×#3622C2×C411yescharacteristic
117C4×#3622C2×C411yescharacteristic
118C4×#3622C2×C411yescharacteristic
119C4×#3622C2×C411yescharacteristic
120#41×C17—21no—
121#41×C17—21no—
122#41×C17#4112yes—
123#41×C17#4112yes—
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—22no—
131C22×C4×C17#4111yescharacteristic
132C22×C4×C17#4111yescharacteristic
133C22×C4×C17C2411yescharacteristic
134C2×C22⋊D17—21no—
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—
147C4⋊C4×C17—21no—
148C4⋊C4×C17—21no—
149C4⋊C4×C17—42no—
150C4⋊C4×C17C22×C411yescharacteristic
151C4⋊C4×C17C22×C411yescharacteristic
152C22⋊C4×C17—21no—
153C22⋊C4×C17—21no—
154C22⋊C4×C17—21no—
155C42×C17—21no—
156C42×C17—21no—
157C42×C17—21no—
158C42×C17—22no—
159C42×C17—22no—
160C42×C17C22×C411yescharacteristic
161C42×C17C22×C411yescharacteristic
162C42×C17C2×D411yescharacteristic
163C42×C17C2×D411yescharacteristic
164C22⋊Dic17—21no—
165#63231—22no—
166#63230—22no—
167#63229—22no—
168#63228—22no—
169#63227—21no—
170#63227—21no—
171C4⋊Dic17C22×C411yescharacteristic
172C4⋊Dic17C22×C411yescharacteristic
173#63225—21no—
174#63225—21no—
175C4×Dic17—21no—
176C4×Dic17—21no—
177C4×Dic17—21no—
178#63223—22no—
179#63223—22no—
180#63223—44no—
181C2×#3622—22no—
182C2×#3622—21no—
183C2×#3622—21no—
184C2×#3622—21no—
185C2×#3622—22no—
186C2×#3622—21no—
187C2×#3622—22no—
188C2×#3622—44no—
189C2×#3622C22×C411yescharacteristic
190C2×#3622C22×C411yescharacteristic
191C2×#3622C22×C411yescharacteristic
192C2×#3622C22×C411yescharacteristic
193C2×#3622C2×D412yes—
194C2×#3622C2×D412yes—
195#22782—171nomaximal, Sylow
196C23×C17—21no—
197C23×C17C2×#4111yescharacteristic
198C22×D17—21no—
199Q8×C17—21no—
200Q8×C17#11712yes—
201D4×C17—22no—
202D4×C17—22no—
203D4×C17—21no—
204D4×C17#11712yes—
205C2×C4×C17—21no—
206C2×C4×C17—22no—
207C2×C4×C17—22no—
208C2×C4×C17—22no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—41no—
212C2×C4×C17—42no—
213C2×C4×C17—42no—
214C2×C4×C17—42no—
215C2×C4×C17—42no—
216C2×C4×C17—81no—
217C2×C4×C17C4×D411yescharacteristic
218C2×C4×C17C4×D411yescharacteristic
219C2×C4×C17C4×D411yescharacteristic
220C2×C4×C17C4×D411yescharacteristic
221C2×C4×C17C23×C411yescharacteristic
222C2×C4×C17C22×D411yescharacteristic
223C2×C4×C17C2×#4111yescharacteristic
224C2×C4×C17C2×#4111yescharacteristic
225C2×C4×C17H(5,𝔽2)11yescharacteristic
226C2×C4×C172−1+411yescharacteristic
227C22⋊D17—41no—
228C22⋊D17—41no—
229C2×Dic17—21no—
230C2×Dic17—21no—
231C2×Dic17—21no—
232D68—21no—
233D68—21no—
234C4×D17—41no—
235C4×D17—41no—
236C4×D17—42no—
237C4×D17—81no—
238Dic34—21no—
239Dic34—21no—
240C8×C17—42no—
241#3622—21no—
242#3622—21no—
243#3622—21no—
244#3622—21no—
245#3622—22no—
246#3622—22no—
247#3622—42no—
248C4×#135—171no—
249#2925—171no—
250#2906—171no—
251#1938—172no—
252#1922—172no—
253#1876—172no—
254#1837—171no—
255#1806—171no—
256#1806D1711yescharacteristic, complemented
257#1770—171no—
258#1765—172no—
259C22×C17—21no—
260C22×C17—21no—
261C22×C17—22no—
262C22×C17—41no—
263C22×C17—41no—
264C22×C17C2×C4×D411yescharacteristic
265C22×C17#55211yescharacteristic
266C22×C17#55311yescharacteristic
267C2×D17—41no—
268C2×D17—41no—
269C2×D17—41no—
270C4×C17—22no—
271C4×C17—21no—
272C4×C17—21no—
273C4×C17—21no—
274C4×C17—41no—
275C4×C17—41no—
276C4×C17—82no—
277C4×C17#44412yes—
278C4×C17C2×#11711yescharacteristic
279C4×C17#52011yescharacteristic
280C4×C17#54611yescharacteristic
281C4×C17#54811yescharacteristic
282Dic17—41no—
283Dic17—41no—
284Dic17—41no—
285C2×#135—171no—
286C2×#131—172no—
287C4×#41—171no—
288C4×#41C2×D1711yescharacteristic
289C4×Q16—171no—
290C4×Q16—171no—
291C4×SD16—171no—
292C4×SD16—171no—
293C4×SD16—171no—
294C4×SD16—171no—
295C4×D8—171no—
296C4×D8—171no—
297#433—342no—
298#432—341no—
299#431—341no—
300#427—171no—
301#424—172no—
302#424—342no—
303#423—171no—
304#423—341no—
305#423C2×D1711yescharacteristic
306C2×C4⋊C8—171no—
307C2×C4⋊C8—341no—
308C2×#104—172no—
309C2×#104C2×D1712yes—
310#416—171no—
311#416—171no—
312#408—172no—
313#405—172no—
314C2×#97—172no—
315C2×C4×C8—171no—
316C4×C4⋊C4C2×D1711yescharacteristic
317#341—172no—
318#341—172no—
319#339—172no—
320#339C2×D1712yes—
321#336—172no—
322C2×C17—21no—
323C2×C17—21no—
324C2×C17—42no—
325C2×C17#235211yescharacteristic
326C2×C17C2×#44411yescharacteristic
327C2×C17#295711yescharacteristic
328D17—81no—
329C2×#41—171no—
330C2×#41C22×D1711yescharacteristic
331#135—342no—
332#135—342no—
333C2×Q16—171no—
334C2×SD16—171no—
335C2×SD16—171no—
336C2×D8—171no—
337#131—344no—
338#131—682no—
339C2×M16—172no—
340C2×M16—342no—
341C2×M16C22×D1712yes—
342C22×C8—172no—
343C22×C8—171no—
344C22×C8—171no—
345C22×C8—342no—
346C4×Q8—171no—
347C4×Q8—171no—
348C4×Q8C4×D1711yescharacteristic
349C4×Q8C4×D1711yescharacteristic
350C4×D4—21no—
351C4×D4—21no—
352C4×D4—171no—
353C4×D4—171no—
354C4×D4—341no—
355C4×D4—341no—
356C4×D4C4×D1711yescharacteristic
357C4×D4C4×D1711yescharacteristic
358#117—21no—
359#117—171no—
360#117—341no—
361#117C22×D1711yescharacteristic
362C2×C4⋊C4—21no—
363C2×C42—21no—
364C2×C42—341no—
365C2×C42C22×D1712yes—
366C2×C42C22×D1711yescharacteristic
367#107—171no—
368#107—171no—
369#106—171no—
370#106—171no—
371C4⋊C8—21no—
372C4⋊C8—21no—
373C4⋊C8—171no—
374C4⋊C8—171no—
375C4⋊C8—172no—
376C4⋊C8—342no—
377C4⋊C8—342no—
378C4⋊C8—341no—
379C4⋊C8—341no—
380#104—24no—
381#104—682no—
382#103—171no—
383#103—171no—
384#103—171no—
385#103—171no—
386#102—171no—
387#102—171no—
388#102—171no—
389#102—171no—
390C22⋊C8—342no—
391C22⋊C8—342no—
392#97—342no—
393#97—682no—
394C4×C8—171no—
395C4×C8—171no—
396C4×C8—171no—
397C4×C8—171no—
398C4×C8—342no—
399#95—21no—
400C17#2278211yescharacteristic, complemented, Sylow
401#41—21no—
402#41—21no—
403#41—341no—
404#41—341no—
405#41—341no—
406#41—341no—
407#41#6325412yes—
408#41#6325612yes—
409C2×Q8—171no—
410C2×Q8C2×C4×D1711yescharacteristic
411C2×D4—21no—
412C2×D4—171no—
413C2×D4—341no—
414C2×D4C2×C4×D1711yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—341no—
420C22×C4—341no—
421C22×C4—341no—
422C22×C4#6325411yescharacteristic
423C22×C4#6325411yescharacteristic
424C22×C4C23×D1711yescharacteristic
425Q16—342no—
426SD16—342no—
427SD16—342no—
428D8—342no—
429M16—22no—
430M16—22no—
431M16—342no—
432M16—342no—
433M16—684no—
434C2×C8—22no—
435C2×C8—171no—
436C2×C8—171no—
437C2×C8—171no—
438C2×C8—171no—
439C2×C8—172no—
440C2×C8—172no—
441C2×C8—341no—
442C2×C8—342no—
443C2×C8—341no—
444C2×C8—341no—
445C2×C8—342no—
446C2×C8—341no—
447C2×C8—342no—
448C2×C8—684no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—42no—
452C4⋊C4—171no—
453C4⋊C4—171no—
454C4⋊C4—341no—
455C4⋊C4—341no—
456C4⋊C4C2×C4×D1711yescharacteristic
457C4⋊C4C2×C4×D1711yescharacteristic
458C22⋊C4—21no—
459C22⋊C4—21no—
460C22⋊C4—21no—
461C22⋊C4—341no—
462C22⋊C4—341no—
463C22⋊C4—341no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—22no—
468C42—22no—
469C42—341no—
470C42—341no—
471C42—341no—
472C42C2×C4×D1711yescharacteristic
473C42C2×C4×D1711yescharacteristic
474C42D4×D1711yescharacteristic
475C42D4×D1711yescharacteristic
476C23—21no—
477C23—341no—
478C23C2×#6325411yescharacteristic
479Q8—21no—
480Q8—341no—
481Q8—341no—
482Q8#1058757812yes—
483D4—22no—
484D4—22no—
485D4—21no—
486D4—341no—
487D4—341no—
488D4—681no—
489D4—681no—
490D4#1058757812yes—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—81no—
503C2×C4—341no—
504C2×C4—341no—
505C2×C4—341no—
506C2×C4—681no—
507C2×C4—682no—
508C2×C4—681no—
509C2×C4—1361no—
510C2×C4#1058756711yescharacteristic
511C2×C4#1058756711yescharacteristic
512C2×C4#1058756711yescharacteristic
513C2×C4#1058756711yescharacteristic
514C2×C4C22×C4×D1711yescharacteristic
515C2×C4C2×D4×D1711yescharacteristic
516C2×C4C2×#6325411yescharacteristic
517C2×C4C2×#6325611yescharacteristic
518C2×C4#1058769611yescharacteristic
519C2×C4#1058769711yescharacteristic
520C8—42no—
521C8—341no—
522C8—341no—
523C8—341no—
524C8—341no—
525C8—342no—
526C8—342no—
527C8—682no—
528C22—21no—
529C22—21no—
530C22—22no—
531C22—41no—
532C22—41no—
533C22—681no—
534C22—681no—
535C22—681no—
536C22C2×#1058756711yescharacteristic
537C22#1176350111yescharacteristic
538C22#1176351111yescharacteristic
539C4—22no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—41no—
544C4—41no—
545C4—82no—
546C4—681no—
547C4—681no—
548C4—681no—
549C4#1176292812yes—
550C4C2×#1058757811yescharacteristic
551C4#1176328911yescharacteristic
552C4#1176345011yescharacteristic
553C4#1176346511yescharacteristic
554C2—21no—
555C2—21no—
556C2—42no—
557C2—1361no—
558C2#1453171511yescharacteristic
559C2C2×#1176292811yescharacteristic
560C2#1453769711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1806C2
socleC22×C17C2×C4×D4
prime radicalC2×#10587636C22
cosocleC22×C4×C17C24
hypercenter#1806D17
hypocenterC17#22782
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
2. #14537452C2semidirect
3. #14537352C2semidirect
4. #14529496C2semidirect
5. #14529386C2semidirect
7. #14528921C2semidirect
8. #14528805C2semidirect
9. #14528653C2semidirect
10. #14528629C2semidirect
11. C17×#1806C2semidirect
15. C4×#63231C4semidirect
16. C4×#63231C4semidirect
17. C4×#63230C4semidirect
18. C4×#63230C4semidirect
19. C4×#63229C4semidirect
20. C4×#63229C4semidirect
21. C4×#63228C4semidirect
22. C4×#63228C4semidirect
256. #1806D17semidirect
400. C17#22782semidirect

Maximal subgroups

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

subgroupindex
C4×#105876362
#145374522
#145373522
#145294962
#145293862
#145291092
#145289212
#145288052
#145286532
#145286292
C17×#18062
#2278217