Finite Group Databasebeta

← #17980012

Subgroups — #17980012

Order 4352.

1950 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14545630C211yesmaximal, characteristic, complemented
2#14545628C211yesmaximal, characteristic, complemented
3#14545617C213yesmaximal, complemented
4#14545616C213yesmaximal, complemented
5#14545600C213yesmaximal, complemented
6#14545598C213yesmaximal, complemented
7#14545590C211yesmaximal, characteristic, complemented
8#41×#3624C211yesmaximal, characteristic, complemented
9Q8×#63247C211yesmaximal, characteristic, complemented
10C2×Q8×#3624C211yesmaximal, characteristic, complemented
11#14545526C213yesmaximal, complemented
12#14545507C213yesmaximal, complemented
13#14545479C213yesmaximal, complemented
14C4×#10587678C213yesmaximal, complemented
15#14542374C211yesmaximal, characteristic
16C2×#10587678C2213yes—
17C2×#10587678C2213yes—
18#11763860C2213yes—
19#11763860C2213yes—
20Q8×#3624C2211yescharacteristic
21Q8×#3624C2213yes—
22Q8×#3624C2213yes—
23Q8×#3624C2211yescharacteristic
24Q8×#3624C2211yescharacteristic
25Q8×#3624C2211yescharacteristic
26#11763844C2213yes—
27#11763844C2213yes—
28#11763843C2213yes—
29#11763843C2213yes—
30#11763842C2216yes—
31#11763841C2216yes—
32#11763838C2216yes—
33#11763837C2213yes—
34#11763837C2213yes—
35#11763836C2216yes—
36D4×#3624C2213yes—
37D4×#3624C2213yes—
38#11763834C2216yes—
39#11763833C2216yes—
40#11763831C2213yes—
41#11763830C2213yes—
42#11763828C2213yes—
43#11763825C2213yes—
44#11763812C2213yes—
45C2×C4⋊#3624C2213yes—
46#11763782C2216yes—
47#11763774C2213yes—
48#11763774C2213yes—
49#11763773C2213yes—
50#11763773C2213yes—
51#11763773C2211yescharacteristic
52#11763773C2213yes—
53C4×#63243C2213yes—
54C4×#63247C2213yes—
55C4×#63247C2213yes—
56C4×#63242C2213yes—
57C2×C4×#3624C2213yes—
58C2×C4×#3624C2213yes—
59#11763541C413yes—
60#11763535C411yescharacteristic
61#11763530C413yes—
62#11763511C413yes—
63#11763505C413yes—
64#11763478C411yescharacteristic
65#11763470C411yescharacteristic
66#11763354C2213yes—
67#11763345C2213yes—
68#11763338C2213yes—
69#41×Dic17C2211yescharacteristic
70C4×#63256C2211yescharacteristic
71C4×#63254C2213yes—
72C2×Q8×Dic17C2211yescharacteristic
73C17×#548C411yescharacteristic
74C2×#41×C17C2×C411yescharacteristic
75C22×Q8×C17C2×C411yescharacteristic
76C2×#63256C2×C411yescharacteristic
77C2×#63254C2313yes—
78C2×#63247C2313yes—
79C2×#63247C2313yes—
80C2×#63247C2313yes—
81C2×#63247C2311yescharacteristic
82C22×#3624C2313yes—
83C22×#3624C2313yes—
84C22×#3624C2313yes—
85C22×#3624C2311yescharacteristic
86#10587678—212no—
87#10587678—212no—
88C2×#63243C2313yes—
89C2×#63243C2313yes—
90C2×#63242C2313yes—
91C2×#63242C2313yes—
92C17×#124C2×C413yes—
93C17×C22⋊Q8C2×C413yes—
94C17×#121C2×C413yes—
95C4×Q8×C17C2×C411yescharacteristic
96C4×D4×C17C2×C413yes—
97#10587633C2×C413yes—
98#10587632C2×C413yes—
99Q8×Dic17C2×C411yescharacteristic
100Q8×Dic17C2×C411yescharacteristic
101Q8×Dic17C2×C413yes—
102Q8×Dic17C2311yescharacteristic
103Q8×Dic17C2313yes—
104#10587626C2×C413yes—
105#10587625C2×C416yes—
106#10587624C2×C413yes—
107D4×Dic17C2×C413yes—
108D4×Dic17C2×C413yes—
109D4×Dic17C2313yes—
110#10587616C2×C413yes—
111#10587616C2×C413yes—
112C4×C22⋊D17C2×C413yes—
113C4×C22⋊D17C2×C413yes—
114#10587612C2×C413yes—
115C2×C4⋊Dic17C2×C413yes—
116C2×C4×Dic17C2313yes—
117C2×C4×Dic17C2313yes—
118#10587584C2×C416yes—
119#10587579C2313yes—
120#10587578C2×C413yes—
121#10587578C2313yes—
122#10587576C2×C413yes—
123#10587574C2×C413yes—
124#10587573C2×C413yes—
125#10587572C2×C416yes—
126#10587571C2×C416yes—
127#10587569C2×C416yes—
128#10587567C2×C413yes—
129#10587567C2×C413yes—
130#10587567C2316yes—
131#10587565C2×C416yes—
132#10587563C2311yescharacteristic
133#10587563C2313yes—
134C4×D68C2×C413yes—
135C42×D17C2313yes—
136#10587556C2×C413yes—
137C4×Dic34C2×C413yes—
138C22⋊#3624C2316yes—
139C22⋊#3624C2316yes—
140#10587544C2316yes—
141#10587543C2316yes—
142#10587543C2316yes—
143#10587542C2311yescharacteristic
144#10587542C2313yes—
145#10587542C2313yes—
146#10587542C2313yes—
147#10587542C2311yescharacteristic
148#10587542C2313yes—
149C4⋊#3624C2313yes—
150C4⋊#3624C2313yes—
151C4⋊#3624C2316yes—
152C4⋊#3624C2313yes—
153C4⋊#3624C2313yes—
154C4×#3624C2311yescharacteristic
155C4×#3624C2313yes—
156C4×#3624C2313yes—
157C4×#3624C2313yes—
158C4×#3624C2313yes—
159C4×#3624C2313yes—
160C4×#3624C2313yes—
161C4×#3624C2311yescharacteristic
162C4×#3624C2313yes—
163C4×#3624C2313yes—
164#41×C17—24no—
165C2×Q8×C17—24no—
166C2×Q8×C17C22×C413yes—
167C2×Q8×C17C22×C411yescharacteristic
168C2×D4×C17C22×C413yes—
169C2×D4×C17C22×C413yes—
170C22×C4×C17C22×C413yes—
171C22×C4×C17C22×C413yes—
172C2×C22⋊D17C22×C413yes—
173C2×C22⋊D17C22×C413yes—
174C2×C22⋊D17C22×C416yes—
175C22×Dic17C22×C413yes—
176C22×Dic17C22×C413yes—
177C22×Dic17C2413yes—
178C22×Dic17C2411yescharacteristic
179#63256—24no—
180#63254—26no—
181#63254—26no—
182C2×D68C22×C413yes—
183C2×C4×D17C22×C413yes—
184C2×C4×D17C22×C413yes—
185C2×C4×D17C2413yes—
186C2×Dic34C22×C413yes—
187#63247—26no—
188#63247—26no—
189#63247—26no—
190#63247#4114yescomplemented
191C2×#3624—26no—
192C2×#3624—26no—
193C2×#3624—26no—
194C2×#3624#4114yescomplemented
195C2×#3624C2413yes—
196C2×#3624C2413yes—
197C2×#3624C2413yes—
198C2×#3624C2413yes—
199C2×#3624C2411yescharacteristic
200C2×#3624C2411yescharacteristic
201C2×#3624C2411yescharacteristic
202C2×#3624C2411yescharacteristic
203#63243—26no—
204#63243—26no—
205#63242—26no—
206#63242—26no—
207C4⋊C4×C17C22×C413yes—
208C4⋊C4×C17C22×C413yes—
209C22⋊C4×C17C22×C416yes—
210C22⋊C4×C17C22×C413yes—
211C22⋊C4×C17C22×C411yescharacteristic
212C42×C17C22×C413yes—
213C22⋊Dic17C22×C413yes—
214C22⋊Dic17C22×C411yescharacteristic
215C22⋊Dic17C22×C416yes—
216#63227C22×C416yes—
217#63227C22×C413yes—
218#63227C22×C413yes—
219#63227C22×C416yes—
220C4⋊Dic17C22×C413yes—
221C4⋊Dic17C22×C413yes—
222C4⋊Dic17C22×C413yes—
223C4⋊Dic17C22×C413yes—
224C4⋊Dic17C22×C416yes—
225#63225C22×C411yescharacteristic
226#63225C22×C413yes—
227#63225C22×C416yes—
228#63225C22×C413yes—
229#63225C22×C411yescharacteristic
230C4×Dic17C22×C413yes—
231C4×Dic17C22×C413yes—
232C4×Dic17C22×C413yes—
233C4×Dic17C22×C413yes—
234C4×Dic17C22×C413yes—
235C4×Dic17C2413yes—
236C4×Dic17C2411yescharacteristic
237C4×Dic17C2411yescharacteristic
238C4×Dic17C2413yes—
239#60218—171nomaximal, Sylow
240C23×C17C23×C413yes—
241C23×C17C23×C411yescharacteristic
242C22×D17C23×C413yes—
243Q8×C17—26no—
244Q8×C17#13114yescomplemented
245D4×C17—26no—
246D4×C17—26no—
247C2×C4×C17—26no—
248C2×C4×C17—26no—
249C2×C4×C17C23×C411yescharacteristic
250C2×C4×C17C23×C413yes—
251C2×C4×C17C23×C413yes—
252C2×C4×C17C23×C411yescharacteristic
253C2×C4×C17C23×C413yes—
254C22⋊D17—26no—
255C22⋊D17—26no—
256C22⋊D17—46no—
257C2×Dic17—23no—
258C2×Dic17—23no—
259C2×Dic17—43no—
260C2×Dic17—43no—
261C2×Dic17C23×C411yescharacteristic
262C2×Dic17C23×C413yes—
263C2×Dic17C23×C413yes—
264C2×Dic17C23×C411yescharacteristic
265C2×Dic17C23×C413yes—
266C2×Dic17C23×C411yescharacteristic
267C2×Dic17C2×#4112yes—
268C2×Dic17C2×#4112yes—
269C2×Dic17C2511yescharacteristic
270D68—26no—
271C4×D17—26no—
272C4×D17—23no—
273C4×D17—23no—
274C4×D17—43no—
275Dic34—26no—
276#3624—23no—
277#3624—23no—
278#3624—23no—
279#3624—23no—
280#3624C2×#4112yes—
281#3624C2×#4112yes—
282#3624C2×#4112yes—
283#3624C2×#4112yes—
284#3480—171no—
285#2971—171no—
286#2971—171no—
287#2969—173no—
288#2969—173no—
289#2966—173no—
290#2966—173no—
291C8×#41—171no—
292C8×#41—171no—
293Q8×M16—171no—
294C2×C8×Q8—171no—
295#2935—173no—
296#2931—173no—
297#2925—173no—
298C4×#131—173no—
299C22×C17—21no—
300C22×C17—21no—
301C22×C17—23no—
302C22×C17—23no—
303C22×C17—23no—
304C22×C17C4×#4112yes—
305C22×C17C24×C411yescharacteristic
306C2×D17—23no—
307C2×D17—23no—
308C2×D17—23no—
309C4×C17—23no—
310C4×C17—21no—
311C4×C17—23no—
312C4×C17—21no—
313C4×C17C2×#13116yes—
314Dic17—23no—
315Dic17—21no—
316Dic17—23no—
317Dic17—21no—
318Dic17—23no—
319Dic17C4×#4112yes—
320Dic17C22×#4111yescharacteristic
321Dic17C22×#4111yescharacteristic
322C2×#131—173no—
323C2×#131—173no—
324#559—173no—
325#557—171no—
326#556—173no—
327#553—173no—
328#553—173no—
329#550—171no—
330#548—171no—
331#548#59511yescharacteristic
332#532—173no—
333#529—173no—
334#529—173no—
335C4×#41—171no—
336C4×#41—171no—
337C4×#41—173no—
338C2×C4×Q8—171no—
339#446—173no—
340#446—173no—
341C8×Q8—171no—
342C8×Q8—173no—
343C8×Q8—173no—
344C8×Q8—171no—
345C8×Q8—171no—
346C8×Q8—171no—
347#436—176no—
348#436—173no—
349#436—173no—
350#436—173no—
351#436—173no—
352#435—176no—
353#435—176no—
354C8×D4—173no—
355C8×D4—176no—
356C8×D4—173no—
357C8×D4—173no—
358C8×D4—173no—
359#433—176no—
360#433—176no—
361#432—173no—
362#432—173no—
363#431—173no—
364#431—173no—
365#423—173no—
366C2×C4⋊C8—173no—
367#408—176no—
368#405—171no—
369#405—173no—
370#405—173no—
371#405—173no—
372#405—173no—
373#405—173no—
374C4×M16—173no—
375C4×M16—173no—
376C4×M16—173no—
377C2×C4×C8—173no—
378C2×C4×C8—173no—
379C2×C4×C8—173no—
380C2×C17—22no—
381C2×C17—43no—
382C2×C17C2×C4×#4111yescharacteristic
383C2×C17C22×#13111yescharacteristic
384C2×C17#357311yescharacteristic
385D17—43no—
386C2×#41—171no—
387C2×#41—173no—
388C2×#41C2×#59511yescharacteristic
389C22×Q8C2×#59511yescharacteristic
390#131—3412no—
391#131—3412no—
392C2×M16—171no—
393C2×M16—173no—
394C2×M16—173no—
395C2×M16—173no—
396C2×M16—173no—
397C2×M16—173no—
398C22×C8—171no—
399C22×C8—173no—
400C22×C8—173no—
401C22×C8—173no—
402C22×C8—173no—
403C22×C8—173no—
404C4⋊Q8—173no—
405#127—173no—
406#126—176no—
407#126—176no—
408#125—173no—
409#125—173no—
410#124—173no—
411#124—176no—
412#124C2×#59513yes—
413#123—176no—
414#123—176no—
415C22⋊Q8—173no—
416C22⋊Q8C2×#59513yes—
417#121—173no—
418#121—176no—
419#121—173no—
420#121—173no—
421#121C2×#59513yes—
422C4×Q8—171no—
423C4×Q8—173no—
424C4×Q8—171no—
425C4×Q8—173no—
426C4×Q8—171no—
427C4×Q8—173no—
428C4×Q8—173no—
429C4×Q8C2×#59511yescharacteristic
430C4×D4—173no—
431C4×D4—173no—
432C4×D4—173no—
433C4×D4—173no—
434C4×D4—173no—
435C4×D4—173no—
436C4×D4—173no—
437C4×D4—173no—
438C4×D4—176no—
439C4×D4—173no—
440C4×D4C2×#59513yes—
441#117—173no—
442#117—173no—
443#117—173no—
444#117—171no—
445#117—173no—
446C2×C4⋊C4—173no—
447C2×C42—173no—
448C2×C42—173no—
449C2×C42—173no—
450C4⋊C8—173no—
451C4⋊C8—173no—
452C4⋊C8—176no—
453C4⋊C8—173no—
454C4⋊C8—173no—
455C4⋊C8—176no—
456C22⋊C8—176no—
457C22⋊C8—176no—
458C22⋊C8—176no—
459C22⋊C8—176no—
460#97—171no—
461#97—171no—
462#97—173no—
463#97—173no—
464#97—173no—
465#97—173no—
466C4×C8—171no—
467C4×C8—171no—
468C4×C8—173no—
469C4×C8—173no—
470C4×C8—173no—
471C4×C8—173no—
472C4×C8—173no—
473C4×C8—173no—
474C4×C8—173no—
475C4×C8—173no—
476C17#6021811yescharacteristic, complemented, Sylow
477#41—24no—
478#41—346no—
479#41—346no—
480#41—344no—
481C2×Q8—24no—
482C2×Q8—173no—
483C2×Q8C22×#59513yes—
484C2×Q8C22×#59511yescharacteristic
485C2×D4—173no—
486C2×D4—173no—
487C2×D4—176no—
488C2×D4—173no—
489C2×D4C22×#59513yes—
490C2×D4C22×#59513yes—
491C22×C4—171no—
492C22×C4—173no—
493C22×C4—173no—
494C22×C4—173no—
495C22×C4—173no—
496C22×C4—173no—
497C22×C4—173no—
498C22×C4C22×#59513yes—
499C22×C4C22×#59513yes—
500M16—174no—
501M16—346no—
502M16—346no—
503M16—346no—
504M16—346no—
505M16—346no—
506C2×C8—174no—
507C2×C8—171no—
508C2×C8—171no—
509C2×C8—171no—
510C2×C8—171no—
511C2×C8—173no—
512C2×C8—173no—
513C2×C8—173no—
514C2×C8—173no—
515C2×C8—346no—
516C2×C8—346no—
517C2×C8—346no—
518C2×C8—346no—
519C2×C8—346no—
520C4⋊C4—173no—
521C4⋊C4—173no—
522C4⋊C4—176no—
523C4⋊C4—171no—
524C4⋊C4—171no—
525C4⋊C4—173no—
526C4⋊C4—173no—
527C4⋊C4—173no—
528C4⋊C4—173no—
529C4⋊C4—176no—
530C4⋊C4C22×#59513yes—
531C4⋊C4C22×#59513yes—
532C22⋊C4—171no—
533C22⋊C4—173no—
534C22⋊C4—176no—
535C22⋊C4—173no—
536C22⋊C4—176no—
537C22⋊C4—173no—
538C22⋊C4—176no—
539C22⋊C4C22×#59511yescharacteristic
540C22⋊C4C22×#59513yes—
541C22⋊C4C22×#59516yes—
542C42—171no—
543C42—171no—
544C42—173no—
545C42—173no—
546C42—173no—
547C42—173no—
548C42—173no—
549C42—173no—
550C42—173no—
551C42C22×#59513yes—
552C23—173no—
553C23C23×#59513yes—
554C23C23×#59511yescharacteristic
555Q8—26no—
556Q8—346no—
557Q8#1058767814yescomplemented
558D4—26no—
559D4—26no—
560D4—346no—
561D4—346no—
562D4—346no—
563D4—686no—
564C2×C4—26no—
565C2×C4—26no—
566C2×C4—172no—
567C2×C4—172no—
568C2×C4—171no—
569C2×C4—171no—
570C2×C4—171no—
571C2×C4—171no—
572C2×C4—173no—
573C2×C4—173no—
574C2×C4—173no—
575C2×C4—343no—
576C2×C4—343no—
577C2×C4—343no—
578C2×C4—343no—
579C2×C4—346no—
580C2×C4—683no—
581C2×C4—683no—
582C2×C4—683no—
583C2×C4C23×#59511yescharacteristic
584C2×C4C23×#59513yes—
585C2×C4C23×#59511yescharacteristic
586C2×C4C23×#59513yes—
587C2×C4C23×#59513yes—
588C8—172no—
589C8—172no—
590C8—172no—
591C8—172no—
592C8—343no—
593C8—343no—
594C8—343no—
595C8—343no—
596C22—21no—
597C22—21no—
598C22—23no—
599C22—23no—
600C22—23no—
601C22—343no—
602C22—343no—
603C22—343no—
604C22#41×#59512yes—
605C22C24×#59511yescharacteristic
606C4—23no—
607C4—21no—
608C4—23no—
609C4—21no—
610C4—171no—
611C4—171no—
612C4—172no—
613C4—343no—
614C4—341no—
615C4—343no—
616C4—341no—
617C4—343no—
618C4C2×#1058767816yes—
619C2—22no—
620C2—43no—
621C2—683no—
622C2C2×#41×#59511yescharacteristic
623C2C22×#1058767811yescharacteristic
624C2#1454578011yescharacteristic
subgroupG/H
center, FrattiniC22C24×#595
commutator, socleC22×C17C24×C4
FittingC17×#548C4
prime radical#14542374C2
cosocleC2×Dic17C25
hypercenter#548#595
hypocenterC17#60218
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #14545630C2semidirect
2. #14545628C2semidirect
3. #14545617C2semidirect
4. #14545616C2semidirect
5. #14545600C2semidirect
6. #14545598C2semidirect
7. #14545590C2semidirect
8. #41×#3624C2semidirect
9. Q8×#63247C2semidirect
10. C2×Q8×#3624C2semidirect
11. #14545526C2semidirect
12. #14545507C2semidirect
13. #14545479C2semidirect
14. C4×#10587678C2semidirect
190. #63247#41semidirect
194. C2×#3624#41semidirect
244. Q8×C17#131semidirect
476. C17#60218semidirect
557. Q8#10587678semidirect

Maximal subgroups

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

subgroupindex
#145456302
#145456282
#145456172
#145456162
#145456002
#145455982
#145455902
#41×#36242
Q8×#632472
C2×Q8×#36242
#145455262
#145455072
#145454792
C4×#105876782
#145423742
#6021817