Finite Group Databasebeta

← #17980984

Subgroups — #17980984

Order 4352.

1930 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14545714C213yesmaximal, complemented
2#14545711C213yesmaximal, complemented
3#14545706C213yesmaximal, complemented
4#14545702C211yesmaximal, characteristic, complemented
5#14545687C211yesmaximal, characteristic, complemented
6#14545686C211yesmaximal, characteristic, complemented
7#14545681C213yesmaximal, complemented
8#14545626C213yesmaximal, complemented
9#14545625C213yesmaximal, complemented
10#14545595C213yesmaximal, complemented
11#14545590C211yesmaximal, characteristic, complemented
12Q8×#63243C211yesmaximal, characteristic, complemented
13Q8×#63247C211yesmaximal, characteristic, complemented
14D17⋊#446C213yesmaximal, complemented
15#14541993C211yesmaximal, characteristic
16#11763861C2213yes—
17#11763861C2216yes—
18#11763861C2213yes—
19#11763860C2213yes—
20#11763860C2213yes—
21#11763860C2213yes—
22#11763860C2213yes—
23#11763859C2213yes—
24#11763859C2213yes—
25#11763859C2213yes—
26#11763859C2213yes—
27Q8×#3624C2211yescharacteristic
28Q8×#3624C2211yescharacteristic
29Q8×#3624C2211yescharacteristic
30Q8×#3624C2211yescharacteristic
31#11763842C2213yes—
32#11763842C2211yescharacteristic
33#11763841C2216yes—
34#11763841C2213yes—
35#11763841C2213yes—
36#11763839C2216yes—
37#11763839C2213yes—
38#11763839C2213yes—
39#11763837C2213yes—
40#11763837C2213yes—
41#11763837C2213yes—
42#11763837C2213yes—
43#11763832C2216yes—
44#11763831C2216yes—
45#11763831C2213yes—
46#11763830C2213yes—
47#11763828C2213yes—
48#11763828C2213yes—
49#11763827C2213yes—
50#11763826C2213yes—
51#11763813C2213yes—
52#11763812C2213yes—
53#11763810C2213yes—
54#11763810C2216yes—
55C4×#63243C2213yes—
56C4×#63247C2213yes—
57#11763768C2216yes—
58C4×#63242C2213yes—
59C2×#10587694C411yescharacteristic
60#11763291C413yes—
61#11763291C411yescharacteristic
62#11763289C413yes—
63#11763289C413yes—
64#11763285C411yescharacteristic
65#11763285C413yes—
66C4×#63256C2211yescharacteristic
67C4×#63256C2213yes—
68#11763277C2213yes—
69#11763277C2213yes—
70C2×Q8×Dic17C2211yescharacteristic
71C4×Q8×D17C2213yes—
72C4×Q8×D17C2211yescharacteristic
73C17×#519C411yescharacteristic
74C22×Q8×C17C2×C411yescharacteristic
75#10587694—28no—
76C2×#63256C2×C413yes—
77C2×#63256C2×C411yescharacteristic
78C2×Q8×D17C2×C411yescharacteristic
79C2×Q8×D17C2×C413yes—
80C2×#63252C2×C413yes—
81C2×#63252C2×C413yes—
82C2×#63247C2311yescharacteristic
83C2×#63247C2313yes—
84C2×#63243C2311yescharacteristic
85C2×#63243C2313yes—
86C2×#63242C2313yes—
87C2×#63242C2313yes—
88C2×#63242C2316yes—
89C4×Q8×C17C2×C413yes—
90C4×Q8×C17C2×C411yescharacteristic
91C4×Q8×C17C2×C413yes—
92C4×Q8×C17C2×C411yescharacteristic
93C17×#117C2×C413yes—
94C17×#117C2×C413yes—
95Q8×Dic17C2×C413yes—
96Q8×Dic17C2×C411yescharacteristic
97Q8×Dic17C2311yescharacteristic
98Q8×Dic17C2313yes—
99C2×C4⋊Dic17C2×C413yes—
100C2×C4×Dic17C2313yes—
101#10587579C2×C413yes—
102#10587579C2×C413yes—
103#10587579C2313yes—
104#10587579C2313yes—
105#10587578C2×C416yes—
106#10587578C2×C413yes—
107#10587578C2×C413yes—
108C4⋊C4×D17C2×C416yes—
109C4⋊C4×D17C2×C413yes—
110C4⋊C4×D17C2×C413yes—
111#10587573C2×C413yes—
112#10587573C2×C413yes—
113#10587573C2313yes—
114#10587573C2313yes—
115#10587567C2×C413yes—
116#10587567C2×C413yes—
117#10587567C2×C416yes—
118#10587567C2313yes—
119#10587567C2311yescharacteristic
120C4×D68C2×C416yes—
121C4×D68C2×C413yes—
122C4×D68C2×C413yes—
123#10587558C2×C413yes—
124#10587558C2×C416yes—
125#10587558C2×C413yes—
126C42×D17C2313yes—
127C42×D17C2316yes—
128C42×D17C2313yes—
129C4×Dic34C2×C416yes—
130C4×Dic34C2×C413yes—
131C4×Dic34C2×C413yes—
132C22⋊#3624C2311yescharacteristic
133C22⋊#3624C2313yes—
134#10587544C2313yes—
135#10587544C2313yes—
136#10587544C2316yes—
137#10587544C2313yes—
138#10587544C2311yescharacteristic
139#10587543C2313yes—
140#10587543C2313yes—
141#10587543C2311yescharacteristic
142#10587543C2313yes—
143#10587543C2316yes—
144#10587542C2313yes—
145#10587542C2313yes—
146#10587542C2316yes—
147#10587542C2316yes—
148#10587542C2313yes—
149#10587542C2313yes—
150C4⋊#3624C2313yes—
151C4⋊#3624C2313yes—
152C4⋊#3624C2313yes—
153C4⋊#3624C2316yes—
154C4⋊#3624C2313yes—
155C4⋊#3624C2313yes—
156C4⋊#3624C2313yes—
157C4×#3624C2311yescharacteristic
158C4×#3624C2313yes—
159C4×#3624C2313yes—
160C4×#3624C2313yes—
161C4×#3624C2313yes—
162C4×#3624C2313yes—
163C2×Q8×C17—24no—
164C2×Q8×C17C22×C413yes—
165C2×Q8×C17C22×C411yescharacteristic
166C22×C4×C17C22×C413yes—
167C2×C22⋊D17C22×C413yes—
168C2×C22⋊D17C22×C411yescharacteristic
169C22×Dic17C22×C413yes—
170C22×Dic17C2411yescharacteristic
171#63256—212no—
172#63256M1618yescomplemented
173Q8×D17—24no—
174Q8×D17—212no—
175#63252—212no—
176#63252—212no—
177C2×D68C22×C413yes—
178C2×D68C22×C413yes—
179C2×C4×D17C22×C416yes—
180C2×C4×D17C22×C413yes—
181C2×C4×D17C22×C413yes—
182C2×C4×D17C22×C413yes—
183C2×C4×D17C22×C416yes—
184C2×C4×D17C22×C413yes—
185C2×C4×D17C2413yes—
186C2×C4×D17C2411yescharacteristic
187C2×Dic34C22×C413yes—
188C2×Dic34C22×C413yes—
189#63247—22no—
190#63247—43no—
191C2×#3624C2411yescharacteristic
192C2×#3624C2413yes—
193C2×#3624C2413yes—
194C2×#3624C2413yes—
195C2×#3624C2413yes—
196C2×#3624C2411yescharacteristic
197C2×#3624C2411yescharacteristic
198C2×#3624C2411yescharacteristic
199#63243—22no—
200#63243—43no—
201#63242—26no—
202#63242—43no—
203#63242—46no—
204C4⋊C4×C17C22×C413yes—
205C4⋊C4×C17C22×C413yes—
206C4⋊C4×C17C22×C413yes—
207C4⋊C4×C17C22×C413yes—
208C22⋊C4×C17C22×C413yes—
209C22⋊C4×C17C22×C411yescharacteristic
210C42×C17C22×C416yes—
211C42×C17C22×C413yes—
212C42×C17C22×C413yes—
213#63227C22×C411yescharacteristic
214#63227C22×C413yes—
215#63227C22×C413yes—
216#63227C22×C416yes—
217#63227C22×C413yes—
218C4⋊Dic17C22×C416yes—
219C4⋊Dic17C22×C413yes—
220C4⋊Dic17C22×C413yes—
221#63225C22×C416yes—
222#63225C22×C413yes—
223#63225C22×C413yes—
224#63225C22×C413yes—
225#63225C22×C411yescharacteristic
226C4×Dic17C22×C416yes—
227C4×Dic17C22×C413yes—
228C4×Dic17C22×C413yes—
229C4×Dic17C22×C413yes—
230C4×Dic17C22×C413yes—
231C4×Dic17C2413yes—
232C4×Dic17C2411yescharacteristic
233C4×Dic17C2413yes—
234C4×Dic17C2413yes—
235#60343—171nomaximal, Sylow
236C23×C17C23×C411yescharacteristic
237C22×D17C23×C411yescharacteristic
238C22×D17C23×C413yes—
239Q8×C17—26no—
240Q8×C17C2×M1614yes—
241C2×C4×C17—26no—
242C2×C4×C17C23×C411yescharacteristic
243C2×C4×C17C23×C413yes—
244C2×C4×C17C23×C413yes—
245C2×C4×C17C23×C411yescharacteristic
246C2×C4×C17C23×C413yes—
247C2×C4×C17C23×C413yes—
248C22⋊D17—26no—
249C22⋊D17—22no—
250C2×Dic17—21no—
251C2×Dic17—21no—
252C2×Dic17—43no—
253C2×Dic17C23×C413yes—
254C2×Dic17C23×C413yes—
255C2×Dic17C23×C413yes—
256C2×Dic17C23×C411yescharacteristic
257C2×Dic17C23×C411yescharacteristic
258C2×Dic17C2511yescharacteristic
259D68—26no—
260D68C2×M16112yes—
261C4×D17—212no—
262C4×D17—26no—
263C4×D17—23no—
264C4×D17—21no—
265C4×D17—23no—
266C4×D17—21no—
267C4×D17—43no—
268C4×D17—46no—
269C4×D17—43no—
270C4×D17C2×M16112yes—
271Dic34—26no—
272Dic34—26no—
273#3624—23no—
274#3624—21no—
275#3624—21no—
276#3624—21no—
277#3624—23no—
278#3624—23no—
279#3624—23no—
280#36242−1+412yescomplemented
281C4×2−1+4—171no—
282#2990—173no—
283#2990—173no—
284#2989—173no—
285#2988—171no—
286#2984—171no—
287#2984—171no—
288#2983—173no—
289#2970—173no—
290#2970—173no—
291#2965—173no—
292C8×#41—171no—
293Q8×M16—171no—
294Q8×M16—171no—
295C2×#446—173no—
296C22×C17—21no—
297C22×C17—21no—
298C22×C17—21no—
299C22×C17C24×C411yescharacteristic
300C2×D17—23no—
301C2×D17—23no—
302C2×D17—21no—
303C2×D17—21no—
304C2×D17—21no—
305C2×D17C22×M1616yes—
306C4×C17—23no—
307C4×C17—21no—
308C4×C17—23no—
309C4×C17—21no—
310C4×C17—23no—
311C4×C17C22×M1616yes—
312Dic17—21no—
313Dic17—23no—
314Dic17—23no—
315Dic17—23no—
316Dic17C22×M1612yes—
317Dic17C2×2−1+411yescharacteristic
318Dic17#58511yescharacteristic
319C2×2−1+4—171no—
320#520—171no—
321#520—173no—
322#520—173no—
323#520—173no—
324#519—173no—
325#519—171no—
326#519#59511yescharacteristic
327C4×#41—171no—
328C4×#41—173no—
329C4×#41—173no—
330C4×#41—173no—
331C2×C4×Q8—171no—
332C2×C4×Q8—173no—
333C2×C4×Q8—171no—
334#446—173no—
335#446—173no—
336#446—173no—
337#446—173no—
338#446—173no—
339#446—176no—
340#446—173no—
341#446—173no—
342#446—173no—
343#446—173no—
344#446—173no—
345C8×Q8—171no—
346C8×Q8—171no—
347C8×Q8—171no—
348C8×Q8—171no—
349#436—171no—
350#436—173no—
351#435—173no—
352#435—173no—
353#435—176no—
354#435—173no—
355#435—173no—
356#435—176no—
357C8×D4—173no—
358C8×D4—173no—
359C8×D4—173no—
360C8×D4—173no—
361#432—173no—
362#432—176no—
363#432—173no—
364#432—176no—
365#431—173no—
366#431—173no—
367#431—173no—
368#431—173no—
369#423—173no—
370#423—173no—
371C2×C4⋊C8—173no—
372C2×C4⋊C8—176no—
373C4×M16—173no—
374C4×M16—173no—
375C2×#97—176no—
376C2×C4×C8—173no—
377C2×C17—41no—
378C2×C17C4×2−1+411yescharacteristic
379C2×C17C23×M1611yescharacteristic
380C2×C17#357311yescharacteristic
381D17—26no—
382D17—41no—
3832−1+4—348no—
384C2×#41—171no—
385C2×#41—173no—
386C2×#41—173no—
387C2×#41—173no—
388C22×Q8—173no—
389C22×Q8—171no—
390C22×Q8C2×#59511yescharacteristic
391C2×M16—171no—
392C2×M16—171no—
393C2×M16—173no—
394C2×M16—173no—
395C22×C8—173no—
396C22×C8—173no—
397C22×C8—176no—
398C4×Q8—171no—
399C4×Q8—173no—
400C4×Q8—173no—
401C4×Q8—171no—
402C4×Q8—173no—
403C4×Q8—173no—
404C4×Q8—173no—
405C4×Q8—173no—
406C4×Q8—176no—
407C4×Q8—173no—
408C4×Q8—173no—
409C4×Q8C2×#59513yes—
410C4×Q8C2×#59511yescharacteristic
411C4×Q8C2×#59513yes—
412C4×Q8C2×#59511yescharacteristic
413C4×D4—173no—
414C4×D4—173no—
415C4×D4—173no—
416C4×D4—173no—
417C4×D4—173no—
418C4×D4—171no—
419C4×D4—173no—
420C4×D4—173no—
421C4×D4—176no—
422C4×D4—173no—
423C4×D4—173no—
424C4×D4—176no—
425#117—173no—
426#117—173no—
427#117—176no—
428#117—173no—
429#117—173no—
430#117—176no—
431#117C2×#59513yes—
432#117C2×#59513yes—
433C2×C4⋊C4—173no—
434C2×C4⋊C4—173no—
435C2×C4⋊C4—176no—
436C2×C4⋊C4—173no—
437C2×C42—173no—
438C2×C42—173no—
439C2×C42—176no—
440C2×C42—173no—
441C4⋊C8—173no—
442C4⋊C8—173no—
443C4⋊C8—173no—
444C4⋊C8—176no—
445C4⋊C8—173no—
446C4⋊C8—173no—
447C4⋊C8—173no—
448C4⋊C8—173no—
449C4⋊C8—173no—
450C4⋊C8—176no—
451C4⋊C8—171no—
452C4⋊C8—173no—
453C22⋊C8—171no—
454C22⋊C8—173no—
455C22⋊C8—173no—
456C22⋊C8—176no—
457C22⋊C8—173no—
458C22⋊C8—171no—
459C22⋊C8—173no—
460#97—173no—
461#97—173no—
462#97—176no—
463#97—176no—
464#97—173no—
465#97—173no—
466C4×C8—171no—
467C4×C8—173no—
468C4×C8—173no—
469C4×C8—173no—
470C4×C8—173no—
471C4×C8—173no—
472C17#6034311yescharacteristic, complemented, Sylow
473#41—178no—
474#41—3412no—
475#41—3412no—
476#41—3412no—
477C2×Q8—24no—
478C2×Q8—173no—
479C2×Q8—173no—
480C2×Q8—3412no—
481C2×Q8—344no—
482C2×Q8C22×#59513yes—
483C2×Q8C22×#59511yescharacteristic
484C2×D4—171no—
485C2×D4—173no—
486C2×D4—173no—
487C2×D4—173no—
488C22×C4—173no—
489C22×C4—171no—
490C22×C4—173no—
491C22×C4—176no—
492C22×C4—173no—
493C22×C4—173no—
494C22×C4—171no—
495C22×C4—173no—
496C22×C4—176no—
497C22×C4—173no—
498C22×C4C22×#59513yes—
499M16—342no—
500M16—342no—
501M16—683no—
502M16—683no—
503C2×C8—171no—
504C2×C8—171no—
505C2×C8—171no—
506C2×C8—171no—
507C2×C8—173no—
508C2×C8—173no—
509C2×C8—173no—
510C2×C8—173no—
511C2×C8—346no—
512C2×C8—683no—
513C2×C8—686no—
514C4⋊C4—173no—
515C4⋊C4—176no—
516C4⋊C4—173no—
517C4⋊C4—176no—
518C4⋊C4—173no—
519C4⋊C4—173no—
520C4⋊C4—171no—
521C4⋊C4—173no—
522C4⋊C4C22×#59513yes—
523C4⋊C4C22×#59513yes—
524C4⋊C4C22×#59513yes—
525C4⋊C4C22×#59513yes—
526C22⋊C4—173no—
527C22⋊C4—171no—
528C22⋊C4—176no—
529C22⋊C4—173no—
530C22⋊C4—173no—
531C22⋊C4C22×#59513yes—
532C22⋊C4C22×#59511yescharacteristic
533C42—171no—
534C42—173no—
535C42—173no—
536C42—173no—
537C42—173no—
538C42—173no—
539C42—173no—
540C42—176no—
541C42—173no—
542C42C22×#59516yes—
543C42C22×#59513yes—
544C42C22×#59513yes—
545C23—173no—
546C23—171no—
547C23C23×#59511yescharacteristic
548Q8—26no—
549Q8—346no—
550Q8—346no—
551Q8D17⋊M1614yes—
552D4—1712no—
553D4—346no—
554D4—342no—
555D4—346no—
556C2×C4—26no—
557C2×C4—171no—
558C2×C4—171no—
559C2×C4—1712no—
560C2×C4—171no—
561C2×C4—173no—
562C2×C4—173no—
563C2×C4—173no—
564C2×C4—343no—
565C2×C4—343no—
566C2×C4—341no—
567C2×C4—341no—
568C2×C4—3412no—
569C2×C4—341no—
570C2×C4—341no—
571C2×C4—346no—
572C2×C4—683no—
573C2×C4—686no—
574C2×C4—683no—
575C2×C4—683no—
576C2×C4C23×#59513yes—
577C2×C4C23×#59511yescharacteristic
578C2×C4C23×#59513yes—
579C2×C4C23×#59511yescharacteristic
580C2×C4C23×#59513yes—
581C2×C4C23×#59513yes—
582C8—172no—
583C8—343no—
584C8—341no—
585C8—343no—
586C8—343no—
587C8—343no—
588C8—341no—
589C8—341no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—176no—
594C22—343no—
595C22—343no—
596C22—341no—
597C22—341no—
598C22—341no—
599C22C24×#59511yescharacteristic
600C4—23no—
601C4—21no—
602C4—23no—
603C4—21no—
604C4—23no—
605C4—171no—
606C4—171no—
607C4—172no—
608C4—341no—
609C4—343no—
610C4—343no—
611C4—343no—
612C4C2×D17⋊M1616yes—
613C2—41no—
614C2—346no—
615C2—681no—
616C22−1+4×#59511yescharacteristic
617C2C22×D17⋊M1611yescharacteristic
618C2#1454578011yescharacteristic
subgroupG/H
center, FrattiniC22C24×#595
commutator, socleC22×C17C24×C4
FittingC17×#519C4
prime radicalC2×#10587694C4
cosocleC2×Dic17C25
hypercenter#519#595
hypocenterC17#60343
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ #519

Complements

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

NH ≅ G/Ntype
1. #14545714C2semidirect
2. #14545711C2semidirect
3. #14545706C2semidirect
4. #14545702C2semidirect
5. #14545687C2semidirect
6. #14545686C2semidirect
7. #14545681C2semidirect
8. #14545626C2semidirect
9. #14545625C2semidirect
10. #14545595C2semidirect
11. #14545590C2semidirect
12. Q8×#63243C2semidirect
13. Q8×#63247C2semidirect
14. D17⋊#446C2semidirect
172. #63256M16semidirect
280. #36242−1+4semidirect
472. C17#60343semidirect

Maximal subgroups

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

subgroupindex
#145457142
#145457112
#145457062
#145457022
#145456872
#145456862
#145456812
#145456262
#145456252
#145455952
#145455902
Q8×#632432
Q8×#632472
D17⋊#4462
#145419932
#6034317