Finite Group Databasebeta

← #17978085

Subgroups — #17978085

Order 4352.

642 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14544886C211yesmaximal, characteristic, complemented
2#14544885C211yesmaximal, characteristic, complemented
3#14544884C211yesmaximal, characteristic, complemented
4#14544858C211yesmaximal, characteristic
5#14544857C211yesmaximal, characteristic
6#14544857C211yesmaximal, characteristic
7#14544844C211yesmaximal, characteristic, complemented
8#14544843C211yesmaximal, characteristic, complemented
9#14544842C211yesmaximal, characteristic, complemented
10#14544731C211yesmaximal, characteristic
11#14544731C211yesmaximal, characteristic
12#14544731C211yesmaximal, characteristic
13#14544717C211yesmaximal, characteristic, complemented
14#14544712C211yesmaximal, characteristic, complemented
15#14531338C211yesmaximal, characteristic
16C22×#63247C2211yescharacteristic
17C2×#10587544C2211yescharacteristic
18C2×#10587544C2211yescharacteristic
19C2×#10587544C2211yescharacteristic
20C2×#10587544C2211yescharacteristic
21C2×#10587544C2211yescharacteristic
22C2×C4⋊#3624C2211yescharacteristic
23C2×C4⋊#3624C2211yescharacteristic
24C2×C4⋊#3624C2211yescharacteristic
25C2×C4⋊#3624C2211yescharacteristic
26C2×C22⋊#3624C2211yescharacteristic
27C2×C22⋊#3624C2211yescharacteristic
28C2×C22⋊#3624C2211yescharacteristic
29C2×#10587542C2211yescharacteristic
30C2×#10587542C2211yescharacteristic
31C2×#10587542C2211yescharacteristic
32#11763675C2211yescharacteristic
33#11763675C2211yescharacteristic
34#11763675C2211yescharacteristic
35#11763675C2211yescharacteristic
36#11763675C2211yescharacteristic
37#11763675C2211yescharacteristic
38#11763675C2211yescharacteristic
39#11763675C2211yescharacteristic
40#11763675C2211yescharacteristic
41#11763675C2211yescharacteristic
42#11763675C2211yescharacteristic
43#11763675C2211yescharacteristic
44C2×#10587565C411yescharacteristic
45#11762703C411yescharacteristic
46#11762697C411yescharacteristic
47#11762693C411yescharacteristic
48#11762692C411yescharacteristic
49#11762690C411yescharacteristic
50#11762676C411yescharacteristic
51C4⋊C4×Dic17C2211yescharacteristic
52#11762660C2211yescharacteristic
53#11762660C2211yescharacteristic
54C22⋊C4×Dic17C2211yescharacteristic
55C22⋊C4×Dic17C2211yescharacteristic
56C22⋊C4×Dic17C2211yescharacteristic
57C42×Dic17C2211yescharacteristic
58C17×#388C411yescharacteristic
59C23×Dic17C2311yescharacteristic
60C2×#63247—41no—
61C2×#63247—41no—
62C2×#63247—41no—
63C22×#3624C2311yescharacteristic
64C22×#3624C2311yescharacteristic
65C22×#3624C2311yescharacteristic
66C22×#3624C2311yescharacteristic
67C22×#3624C2311yescharacteristic
68C22×#3624C2311yescharacteristic
69C22×#3624C2311yescharacteristic
70C22×#3624C2311yescharacteristic
71C2×C4⋊C4×C17C2×C411yescharacteristic
72C2×C22⋊C4×C17C2×C411yescharacteristic
73C2×C22⋊C4×C17C2×C411yescharacteristic
74C2×C22⋊C4×C17C2×C411yescharacteristic
75C2×C42×C17C2×C411yescharacteristic
76C2×C22⋊Dic17C2×C411yescharacteristic
77C2×C22⋊Dic17C2×C411yescharacteristic
78C2×C22⋊Dic17C2×C411yescharacteristic
79C2×C4⋊Dic17C2×C411yescharacteristic
80C2×#63225C2×C411yescharacteristic
81C2×#63225C2×C411yescharacteristic
82C2×#63225C2×C411yescharacteristic
83C2×#63225C2×C411yescharacteristic
84C2×C4×Dic17C2×C411yescharacteristic
85C2×C4×Dic17C2×C411yescharacteristic
86C2×C4×Dic17C2×C411yescharacteristic
87C2×C4×Dic17C2311yescharacteristic
88C2×C4×Dic17C2311yescharacteristic
89C2×C4×Dic17C2311yescharacteristic
90C2×C4×Dic17C2311yescharacteristic
91C2×C4×Dic17C2311yescharacteristic
92C2×C4×Dic17C2311yescharacteristic
93#10587565—24no—
94C22⋊#3624—41no—
95C22⋊#3624—41no—
96C22⋊#3624—41no—
97#10587544—41no—
98#10587544—41no—
99#10587544—41no—
100#10587544—41no—
101#10587544—41no—
102#10587542—41no—
103#10587542—41no—
104#10587542—41no—
105C4⋊#3624—41no—
106C4⋊#3624—41no—
107C4⋊#3624—41no—
108C4⋊#3624—41no—
109C17×#95C2×C411yescharacteristic
110C17×#95C2×C411yescharacteristic
111#10587502C2×C411yescharacteristic
112#10587502C2×C411yescharacteristic
113#10587502C2×C411yescharacteristic
114#10587502C2×C411yescharacteristic
115#10587502C2×C411yescharacteristic
116#10587502C2×C411yescharacteristic
117#10587502C2×C411yescharacteristic
118#10587502C2×C411yescharacteristic
119#10587502C2×C411yescharacteristic
120#10587502C2×C411yescharacteristic
121C24×C17C22×C411yescharacteristic
122C22×C4×C17C22×C411yescharacteristic
123C22×C4×C17C22×C411yescharacteristic
124C22×C4×C17C22×C411yescharacteristic
125C22×C4×C17C22×C411yescharacteristic
126C22×C4×C17C22×C411yescharacteristic
127C22×C4×C17C22×C411yescharacteristic
128C22×Dic17—21no—
129C22×Dic17—21no—
130C22×Dic17—21no—
131C22×Dic17—21no—
132C22×Dic17—21no—
133C22×Dic17—21no—
134C22×Dic17C22×C411yescharacteristic
135C22×Dic17C22×C411yescharacteristic
136C22×Dic17C22×C411yescharacteristic
137C22×Dic17C22×C411yescharacteristic
138C22×Dic17C22×C411yescharacteristic
139C22×Dic17C22×C411yescharacteristic
140C22×Dic17C22×C411yescharacteristic
141C22×Dic17C2411yescharacteristic
142#63247—44no—
143C2×#3624—21no—
144C2×#3624—21no—
145C2×#3624—21no—
146C2×#3624—21no—
147C2×#3624—21no—
148C2×#3624—21no—
149C2×#3624—21no—
150C2×#3624—21no—
151C2×#3624—21no—
152C2×#3624—21no—
153C2×#3624—21no—
154C2×#3624—21no—
155C2×#3624—21no—
156C2×#3624—21no—
157C2×#3624—21no—
158C2×#3624—21no—
159C2×#3624—21no—
160C2×#3624—21no—
161C2×#3624—21no—
162C2×#3624—21no—
163C2×#3624—21no—
164C2×#3624—21no—
165C2×#3624—21no—
166C2×#3624—21no—
167C4⋊C4×C17—41no—
168C22⋊C4×C17—22no—
169C22⋊C4×C17—41no—
170C22⋊C4×C17—41no—
171C42×C17—41no—
172C22⋊Dic17—22no—
173C22⋊Dic17—22no—
174C22⋊Dic17—41no—
175C4⋊Dic17—22no—
176#63225—22no—
177#63225—22no—
178#63225—41no—
179#63225—41no—
180C4×Dic17—21no—
181C4×Dic17—21no—
182C4×Dic17—21no—
183C4×Dic17—21no—
184C4×Dic17—21no—
185C4×Dic17—21no—
186C4×Dic17—21no—
187C4×Dic17—21no—
188C4×Dic17—21no—
189C4×Dic17—21no—
190C4×Dic17—21no—
191C4×Dic17—21no—
192C4×Dic17—41no—
193C4×Dic17—41no—
194C4×Dic17M1614yescomplemented
195#23378—171nomaximal, Sylow
196C23×C17—21no—
197C23×C17—21no—
198C23×C17—21no—
199C23×C17—21no—
200C23×C17—21no—
201C23×C17—21no—
202C23×C17—21no—
203C23×C17C23×C411yescharacteristic
204C2×C4×C17—21no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—21no—
208C2×C4×C17—21no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17C2×M1612yes—
222C2×Dic17—21no—
223C2×Dic17—21no—
224C2×Dic17—21no—
225C2×Dic17—21no—
226C2×Dic17—21no—
227C2×Dic17—21no—
228C2×Dic17—21no—
229C2×Dic17—21no—
230C2×Dic17—21no—
231C2×Dic17—21no—
232C2×Dic17—21no—
233C2×Dic17—21no—
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—21no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17C2×M1612yes—
246C2×Dic17C2×M1612yes—
247C2×Dic17H(5,𝔽2)11yescharacteristic
248C2×Dic17H(5,𝔽2)11yescharacteristic
249C2×Dic17H(5,𝔽2)11yescharacteristic
250C2×Dic172−1+411yescharacteristic
251C2×Dic172−1+411yescharacteristic
252C2×Dic172−1+411yescharacteristic
253#3624—41no—
254#3624—41no—
255#3624—41no—
256#3624—41no—
257#3624—41no—
258#3624—41no—
259#3624—41no—
260#3624—41no—
261C4×#126—171no—
262#1940—171no—
263#1940—171no—
264#1940—171no—
265#1924—171no—
266#1924—171no—
267#1924—171no—
268#1917—171no—
269#1917—171no—
270#1917—171no—
271#1851—171no—
272#1851—171no—
273#1851—171no—
274#1842—171no—
275#1839—171no—
276C22×C17—41no—
277C22×C17—41no—
278C22×C17—41no—
279C22×C17—41no—
280C22×C17—41no—
281C22×C17—41no—
282C22×C17—41no—
283C22×C17#52011yescharacteristic
284C22×C17#52011yescharacteristic
285C22×C17#52011yescharacteristic
286C22×C17C22×M1611yescharacteristic
287C22×C17#56811yescharacteristic
288C22×C17#56811yescharacteristic
289C22×C17#56811yescharacteristic
290C4×C17—22no—
291C4×C17—41no—
292C4×C17—41no—
293C4×C17—41no—
294C4×C17—41no—
295C4×C17—41no—
296Dic17—22no—
297Dic17—22no—
298Dic17—41no—
299Dic17—41no—
300Dic17—41no—
301Dic17—41no—
302Dic17—41no—
303Dic17#56011yescharacteristic
304Dic17#56311yescharacteristic
305Dic17#56311yescharacteristic
306Dic17#56311yescharacteristic
307C22×M16—171no—
308C2×#126—171no—
309C2×C4⋊C8—171no—
310C2×C4⋊C8—171no—
311C2×C4⋊C8—171no—
312C2×C4⋊C8—171no—
313C2×C4⋊C8—171no—
314C2×C4⋊C8—171no—
315C2×C4⋊C8—171no—
316C2×C4⋊C8—171no—
317C2×C4⋊C8—171no—
318C2×C22⋊C8—171no—
319C2×C22⋊C8—171no—
320C2×C22⋊C8—171no—
321C2×#97—171no—
322C2×#97—171no—
323C2×#97—171no—
324#388—171no—
325#388—171no—
326#388#59511yescharacteristic
327#387—171no—
328#387—171no—
329#387—171no—
330#383—171no—
331C4×C4⋊C4—171no—
332C4×C4⋊C4—171no—
333C4×C4⋊C4—171no—
334C4×C22⋊C4—171no—
335C4×C22⋊C4—171no—
336C4×C22⋊C4—171no—
337C43—171no—
338#336—171no—
339#336—171no—
340#336—171no—
341#336—171no—
342#336—171no—
343#336—171no—
344#336—171no—
345#336—171no—
346#336—171no—
347#336—171no—
348#336—171no—
349#336—171no—
350C2×C17—81no—
351C2×C17#238311yescharacteristic
352C2×C17#298911yescharacteristic
353C2×C17#298911yescharacteristic
354C2×C17#298911yescharacteristic
355C2×C17#299511yescharacteristic
356C2×C17#299511yescharacteristic
357C2×C17#299511yescharacteristic
358C23×C4—171no—
359C2×M16—681no—
360C2×M16—681no—
361C2×M16—681no—
362C22×C8—171no—
363C22×C8—171no—
364C22×C8—171no—
365C22×C8—171no—
366C22×C8—171no—
367C22×C8—171no—
368C22×C8—171no—
369C22×C8—171no—
370#126—344no—
371C2×C4⋊C4—171no—
372C2×C4⋊C4—171no—
373C2×C4⋊C4—171no—
374C2×C4⋊C4—171no—
375C2×C4⋊C4—171no—
376C2×C4⋊C4C2×#59511yescharacteristic
377C2×C22⋊C4—171no—
378C2×C22⋊C4—171no—
379C2×C22⋊C4—171no—
380C2×C22⋊C4C2×#59511yescharacteristic
381C2×C22⋊C4C2×#59511yescharacteristic
382C2×C22⋊C4C2×#59511yescharacteristic
383C2×C42—171no—
384C2×C42—171no—
385C2×C42—171no—
386C2×C42—171no—
387C2×C42—171no—
388C2×C42—171no—
389C2×C42—171no—
390C2×C42—171no—
391C2×C42—171no—
392C2×C42C2×#59511yescharacteristic
393C4⋊C8—681no—
394C4⋊C8—681no—
395C4⋊C8—681no—
396C4⋊C8—681no—
397C4⋊C8—681no—
398C4⋊C8—681no—
399C4⋊C8—681no—
400C4⋊C8—681no—
401C4⋊C8—681no—
402C22⋊C8—681no—
403C22⋊C8—681no—
404C22⋊C8—681no—
405#97—681no—
406#97—681no—
407#97—681no—
408#95—171no—
409#95—171no—
410#95—171no—
411#95—171no—
412#95—171no—
413#95—171no—
414#95—171no—
415#95—171no—
416#95—171no—
417#95—171no—
418#95C2×#59511yescharacteristic
419#95C2×#59511yescharacteristic
420C17#2337811yescharacteristic, complemented, Sylow
421C24C22×#59511yescharacteristic
422C22×C4—171no—
423C22×C4—171no—
424C22×C4—171no—
425C22×C4—171no—
426C22×C4—171no—
427C22×C4—171no—
428C22×C4—171no—
429C22×C4—171no—
430C22×C4—341no—
431C22×C4—341no—
432C22×C4—341no—
433C22×C4—341no—
434C22×C4—341no—
435C22×C4—341no—
436C22×C4C22×#59511yescharacteristic
437C22×C4C22×#59511yescharacteristic
438C22×C4C22×#59511yescharacteristic
439C22×C4C22×#59511yescharacteristic
440C22×C4C22×#59511yescharacteristic
441C22×C4C22×#59511yescharacteristic
442M16—684no—
443C2×C8—341no—
444C2×C8—341no—
445C2×C8—341no—
446C2×C8—341no—
447C2×C8—341no—
448C2×C8—341no—
449C2×C8—341no—
450C2×C8—341no—
451C2×C8—341no—
452C2×C8—341no—
453C2×C8—341no—
454C2×C8—341no—
455C2×C8—341no—
456C2×C8—341no—
457C2×C8—341no—
458C2×C8—341no—
459C2×C8—341no—
460C2×C8—341no—
461C2×C8—341no—
462C2×C8—341no—
463C2×C8—341no—
464C2×C8—341no—
465C2×C8—341no—
466C2×C8—341no—
467C4⋊C4—41no—
468C4⋊C4—342no—
469C4⋊C4—342no—
470C4⋊C4—342no—
471C4⋊C4—681no—
472C4⋊C4—681no—
473C22⋊C4—22no—
474C22⋊C4—41no—
475C22⋊C4—41no—
476C22⋊C4—342no—
477C22⋊C4—342no—
478C22⋊C4—681no—
479C42—41no—
480C42—174no—
481C42—341no—
482C42—341no—
483C42—341no—
484C42—341no—
485C42—341no—
486C42—341no—
487C42—341no—
488C42—341no—
489C42—341no—
490C42—341no—
491C42—341no—
492C42—341no—
493C42—681no—
494C42—681no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23C23×#59511yescharacteristic
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—171no—
521C2×C4—171no—
522C2×C4—171no—
523C2×C4—171no—
524C2×C4—171no—
525C2×C4—171no—
526C2×C4—172no—
527C2×C4—172no—
528C2×C4—341no—
529C2×C4—341no—
530C2×C4—341no—
531C2×C4—341no—
532C2×C4—341no—
533C2×C4—341no—
534C2×C4—341no—
535C2×C4—341no—
536C2×C4—341no—
537C2×C4—341no—
538C2×C4—341no—
539C2×C4—341no—
540C2×C4—341no—
541C2×C4—341no—
542C2×C4—341no—
543C2×C4—341no—
544C2×C4—341no—
545C2×C4—341no—
546C2×C4—341no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—681no—
551C2×C4D17⋊M1612yes—
552C8—681no—
553C8—681no—
554C8—681no—
555C8—681no—
556C8—681no—
557C8—681no—
558C8—681no—
559C8—681no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22#1176387811yescharacteristic
568C22#1176387811yescharacteristic
569C22#1176387811yescharacteristic
570C22C2×D17⋊M1611yescharacteristic
571C22#1176388711yescharacteristic
572C22#1176388711yescharacteristic
573C22#1176388811yescharacteristic
574C4—22no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—171no—
581C4—171no—
582C4—171no—
583C4—171no—
584C4—342no—
585C4—342no—
586C4—681no—
587C4—681no—
588C4—681no—
589C4—681no—
590C4—681no—
591C2—81no—
592C2#1454533111yescharacteristic
593C2#1454570411yescharacteristic
594C2#1454570611yescharacteristic
595C2#1454570711yescharacteristic
596C2#1454573711yescharacteristic
597C2#1454573811yescharacteristic
598C2#1454574011yescharacteristic
subgroupG/H
center, FrattiniC23C23×#595
commutator, socleC23×C17C23×C4
FittingC17×#388C4
prime radicalC24×C17C22×C4
cosocleC22×Dic17C24
hypercenter#388#595
hypocenterC17#23378
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. #14544886C2semidirect
2. #14544885C2semidirect
3. #14544884C2semidirect
7. #14544844C2semidirect
8. #14544843C2semidirect
9. #14544842C2semidirect
13. #14544717C2semidirect
14. #14544712C2semidirect
194. C4×Dic17M16semidirect
420. C17#23378semidirect

Maximal subgroups

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

subgroupindex
#145448862
#145448852
#145448842
#145448582
#145448572
#145448442
#145448432
#145448422
#145447312
#145447172
#145447122
#145313382
#2337817