Finite Group Databasebeta

← C89×#19904 #36397752

Subgroups — C89×#19904 #36397752

Order 22784.

638 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#3358C211yesmaximal, characteristic, complemented
2C89×#3332C211yesmaximal, characteristic, complemented
3C89×#3329C211yesmaximal, characteristic, complemented
4C89×#3303C211yesmaximal, characteristic, complemented
5C89×#1708C211yesmaximal, characteristic, complemented
6C89×#1703C211yesmaximal, characteristic, complemented
7C89×#1701C211yesmaximal, characteristic, complemented
8C89×#1684C211yesmaximal, characteristic, complemented
9C89×#1681C211yesmaximal, characteristic, complemented
10C89×#1654C211yesmaximal, characteristic, complemented
11C89×#1651C211yesmaximal, characteristic, complemented
12C89×#1631C211yesmaximal, characteristic, complemented
13C89×#1624C211yesmaximal, characteristic, complemented
14C89×#1589C211yesmaximal, characteristic, complemented
15C8×SD16×C89C211yesmaximal, characteristic, complemented
16#555×C89C2211yescharacteristic
17#550×C89C2211yescharacteristic
18D4×Q8×C89C2211yescharacteristic
19#547×C89C2211yescharacteristic
20#498×C89C2211yescharacteristic
21#492×C89C2211yescharacteristic
22#487×C89C2211yescharacteristic
23#487×C89C2211yescharacteristic
24#486×C89—21no—
25#486×C89C2211yescharacteristic
26#486×C89C2211yescharacteristic
27#485×C89—21no—
28#485×C89—21no—
29#481×C89—21no—
30#477×C89C2211yescharacteristic
31#476×C89C2211yescharacteristic
32#475×C89C2211yescharacteristic
33#474×C89—21no—
34#474×C89C2211yescharacteristic
35#469×C89—21no—
36#468×C89—21no—
37#465×C89—21no—
38#461×C89C2211yescharacteristic
39#461×C89C2211yescharacteristic
40#459×C89—21no—
41#459×C89C2211yescharacteristic
42#459×C89C2211yescharacteristic
43#449×C89—21no—
44#449×C89—21no—
45#448×C89—21no—
46C8×Q8×C89C2211yescharacteristic
47C4×SD16×C89C2211yescharacteristic
48C4×D8×C89—21no—
49C4×D8×C89C2211yescharacteristic
50#436×C89C2211yescharacteristic
51#436×C89C2211yescharacteristic
52C8×D4×C89C2211yescharacteristic
53#419×C89—21no—
54#417×C89—21no—
55#416×C89—21no—
56#335×C89C2211yescharacteristic
57#334×C89C2211yescharacteristic
58#333×C89C2211yescharacteristic
59#332×C89C2211yescharacteristic
60#332×C89C2211yescharacteristic
61#331×C89C2211yescharacteristic
62Q8⋊C8×C89C2211yescharacteristic
63D4⋊C8×C89C2211yescharacteristic
64D4⋊C8×C89C2211yescharacteristic
65D4⋊C8×C89C2211yescharacteristic
66C82×C89C2211yescharacteristic
67C2×#41×C89—21no—
68C2×#41×C89—21no—
69C22×Q8×C89—21no—
70C2×SD16×C89—21no—
71C2×SD16×C89—21no—
72C2×SD16×C89—41no—
73C2×SD16×C89D411yescharacteristic
74C2×D8×C89—21no—
75C2×D8×C89—21no—
76C2×D8×C89D411yescharacteristic
77C2×M16×C89—21no—
78C2×M16×C89—21no—
79C22×C8×C89—21no—
80C4⋊Q8×C89—21no—
81C4⋊Q8×C89C2311yescharacteristic
82C4⋊Q8×C89C2311yescharacteristic
83C4⋊Q8×C89C2311yescharacteristic
84#127×C89—21no—
85#127×C89C2311yescharacteristic
86#124×C89—41no—
87#123×C89—41no—
88C22⋊Q8×C89—21no—
89C22⋊Q8×C89—21no—
90C22⋊Q8×C89—41no—
91#121×C89—21no—
92#121×C89—21no—
93#121×C89—21no—
94#121×C89—21no—
95#121×C89—41no—
96C4×Q8×C89C2311yescharacteristic
97C4×D4×C89—21no—
98C4×D4×C89—21no—
99C4×D4×C89C2311yescharacteristic
100C4×D4×C89C2311yescharacteristic
101C4×D4×C89C2311yescharacteristic
102#117×C89—21no—
103#117×C89—21no—
104C2×C4⋊C4×C89—21no—
105#107×C89—21no—
106#107×C89—21no—
107#107×C89D411yescharacteristic
108#106×C89—21no—
109#106×C89—21no—
110#106×C89—41no—
111#106×C89D411yescharacteristic
112C4⋊C8×C89—21no—
113C4⋊C8×C89C2311yescharacteristic
114C4⋊C8×C89C2311yescharacteristic
115C4⋊C8×C89C2311yescharacteristic
116C4⋊C8×C89C2311yescharacteristic
117#103×C89—21no—
118#103×C89—21no—
119#103×C89—21no—
120#103×C89—21no—
121#103×C89—21no—
122#103×C89—21no—
123#103×C89—21no—
124#103×C89—21no—
125#103×C89D411yescharacteristic
126#102×C89—21no—
127#102×C89—21no—
128#102×C89—21no—
129#102×C89—21no—
130#102×C89—21no—
131#102×C89—21no—
132#102×C89—21no—
133#102×C89—21no—
134#102×C89D411yescharacteristic
135#102×C89D411yescharacteristic
136#102×C89D411yescharacteristic
137C22⋊C8×C89—21no—
138C22⋊C8×C89—21no—
139C22⋊C8×C89—21no—
140C4×C8×C89—21no—
141C4×C8×C89C2311yescharacteristic
142C4×C8×C89C2311yescharacteristic
143C4×C8×C89C2311yescharacteristic
144#41×C89—42no—
145#41×C89—42no—
146#41×C89—81no—
147#41×C89—81no—
148C2×Q8×C89—21no—
149C2×Q8×C89—21no—
150C2×Q8×C89—21no—
151C2×Q8×C89—41no—
152C2×Q8×C89—42no—
153C2×Q8×C89—81no—
154C2×Q8×C89C2×D411yescharacteristic
155C2×D4×C89—21no—
156C2×D4×C89—21no—
157C2×D4×C89—21no—
158C2×D4×C89—41no—
159C2×D4×C89—41no—
160C2×D4×C89—41no—
161C2×D4×C89—41no—
162C2×D4×C89C2×D411yescharacteristic
163C2×D4×C89C2×D411yescharacteristic
164C2×D4×C89C2×D411yescharacteristic
165C22×C4×C89—21no—
166C22×C4×C89—21no—
167C22×C4×C89—21no—
168C22×C4×C89—41no—
169C22×C4×C89—41no—
170C22×C4×C89—41no—
171SD16×C89—22no—
172SD16×C89—82no—
173SD16×C89—81no—
174SD16×C89—81no—
175D8×C89—41no—
176D8×C89—42no—
177D8×C89—81no—
178M16×C89—81no—
179M16×C89—81no—
180C2×C8×C89—21no—
181C2×C8×C89—21no—
182C2×C8×C89—21no—
183C2×C8×C89—21no—
184C2×C8×C89—21no—
185C2×C8×C89—81no—
186C2×C8×C89C2×D411yescharacteristic
187C2×C8×C89C2×D411yescharacteristic
188C2×C8×C89C2×D411yescharacteristic
189C2×C8×C89C2×D411yescharacteristic
190C4⋊C4×C89—21no—
191C4⋊C4×C89—21no—
192C4⋊C4×C89—21no—
193C4⋊C4×C89—21no—
194C4⋊C4×C89—21no—
195C4⋊C4×C89—21no—
196C4⋊C4×C89—21no—
197C4⋊C4×C89—41no—
198C4⋊C4×C89—41no—
199C4⋊C4×C89C2×D411yescharacteristic
200C4⋊C4×C89C2×D411yescharacteristic
201C4⋊C4×C89C2×D411yescharacteristic
202C4⋊C4×C89C2×D411yescharacteristic
203C22⋊C4×C89—21no—
204C22⋊C4×C89—21no—
205C22⋊C4×C89—21no—
206C22⋊C4×C89—41no—
207C22⋊C4×C89—41no—
208C22⋊C4×C89—41no—
209C22⋊C4×C89—41no—
210C22⋊C4×C89—41no—
211C42×C89—21no—
212C42×C89—41no—
213C42×C89C2411yescharacteristic
214C23×C89—21no—
215C23×C89—21no—
216C23×C89—21no—
217C23×C89—41no—
218Q8×C89—21no—
219Q8×C89—41no—
220Q8×C89—41no—
221Q8×C89—41no—
222Q8×C89—41no—
223Q8×C89—82no—
224Q8×C89—81no—
225Q8×C89#13512yes—
226D4×C89—21no—
227D4×C89—21no—
228D4×C89—21no—
229D4×C89—21no—
230D4×C89—21no—
231D4×C89—21no—
232D4×C89—41no—
233D4×C89—41no—
234D4×C89—41no—
235D4×C89—41no—
236D4×C89—82no—
237D4×C89—81no—
238D4×C89—82no—
239D4×C89—161no—
240D4×C89—161no—
241C2×C4×C89—21no—
242C2×C4×C89—21no—
243C2×C4×C89—21no—
244C2×C4×C89—21no—
245C2×C4×C89—21no—
246C2×C4×C89—41no—
247C2×C4×C89—41no—
248C2×C4×C89—41no—
249C2×C4×C89—41no—
250C2×C4×C89—41no—
251C2×C4×C89—41no—
252C2×C4×C89—41no—
253C2×C4×C89—41no—
254C2×C4×C89—41no—
255C2×C4×C89—82no—
256C2×C4×C89—82no—
257C2×C4×C89—82no—
258C2×C4×C89C22×D411yescharacteristic
259C2×C4×C89C22×D411yescharacteristic
260C2×C4×C89H(5,𝔽2)11yescharacteristic
261C8×C89—22no—
262C8×C89—21no—
263C8×C89—21no—
264C8×C89—21no—
265C8×C89—21no—
266C8×C89—41no—
267C8×C89—41no—
268C8×C89—41no—
269C8×C89—41no—
270C22×C89—41no—
271C22×C89—41no—
272C22×C89—41no—
273C22×C89—41no—
274C22×C89—41no—
275C22×C89—41no—
276C22×C89—41no—
277C22×C89—41no—
278C22×C89—41no—
279C22×C89—81no—
280C22×C89—81no—
281C22×C89—81no—
282C22×C89D4211yescharacteristic
283C4×C89—22no—
284C4×C89—21no—
285C4×C89—41no—
286C4×C89—41no—
287C4×C89—41no—
288C4×C89—41no—
289C4×C89—81no—
290C4×C89—81no—
291C4×C89—81no—
292C4×C89C2×#13511yescharacteristic
293C4×C89#57511yescharacteristic
294C4×C89#57611yescharacteristic
295C4×C89#57711yescharacteristic
296#19904C8911yesmaximal, characteristic, direct product factor, Sylow
297C2×C89—81no—
298C2×C89—81no—
299C2×C89—81no—
300C2×C89—161no—
301C2×C89#327511yescharacteristic
302C2×C89#328011yescharacteristic
303C2×C89#329111yescharacteristic
304#3358C2×C8911yescharacteristic, complemented
305#3332C2×C8911yescharacteristic, complemented
306#3329C2×C8911yescharacteristic, complemented
307#3303C2×C8911yescharacteristic, complemented
308#1708C2×C8911yescharacteristic, complemented
309#1703C2×C8911yescharacteristic, complemented
310#1701C2×C8911yescharacteristic, complemented
311#1684C2×C8911yescharacteristic, complemented
312#1681C2×C8911yescharacteristic, complemented
313#1654C2×C8911yescharacteristic, complemented
314#1651C2×C8911yescharacteristic, complemented
315#1631C2×C8911yescharacteristic, complemented
316#1624C2×C8911yescharacteristic, complemented
317#1589C2×C8911yescharacteristic, complemented
318C8×SD16C2×C8911yescharacteristic, complemented
319C89#1990411yescharacteristic, direct product factor, Sylow
320#555C22×C8911yescharacteristic
321#550C22×C8911yescharacteristic
322D4×Q8C22×C8911yescharacteristic
323#547C22×C8911yescharacteristic
324#498C22×C8911yescharacteristic
325#492C22×C8911yescharacteristic
326#487C22×C8911yescharacteristic
327#487C22×C8911yescharacteristic
328#486—21no—
329#486C22×C8911yescharacteristic
330#486C22×C8911yescharacteristic
331#485—21no—
332#485—21no—
333#481—21no—
334#477C22×C8911yescharacteristic
335#476C22×C8911yescharacteristic
336#475C22×C8911yescharacteristic
337#474—21no—
338#474C22×C8911yescharacteristic
339#469—21no—
340#468—21no—
341#465—21no—
342#461C22×C8911yescharacteristic
343#461C22×C8911yescharacteristic
344#459—21no—
345#459C22×C8911yescharacteristic
346#459C22×C8911yescharacteristic
347#449—21no—
348#449—21no—
349#448—21no—
350C8×Q8C22×C8911yescharacteristic
351C4×SD16C22×C8911yescharacteristic
352C4×D8—21no—
353C4×D8C22×C8911yescharacteristic
354#436C22×C8911yescharacteristic
355#436C22×C8911yescharacteristic
356C8×D4C22×C8911yescharacteristic
357#419—21no—
358#417—21no—
359#416—21no—
360#335C22×C8911yescharacteristic
361#334C22×C8911yescharacteristic
362#333C22×C8911yescharacteristic
363#332C22×C8911yescharacteristic
364#332C22×C8911yescharacteristic
365#331C22×C8911yescharacteristic
366Q8⋊C8C22×C8911yescharacteristic
367D4⋊C8C22×C8911yescharacteristic
368D4⋊C8C22×C8911yescharacteristic
369D4⋊C8C22×C8911yescharacteristic
370C82C22×C8911yescharacteristic
371C2×#41—21no—
372C2×#41—21no—
373C22×Q8—21no—
374C2×SD16—21no—
375C2×SD16—21no—
376C2×SD16—41no—
377C2×SD16D4×C8911yescharacteristic
378C2×D8—21no—
379C2×D8—21no—
380C2×D8D4×C8911yescharacteristic
381C2×M16—21no—
382C2×M16—21no—
383C22×C8—21no—
384C4⋊Q8—21no—
385C4⋊Q8C23×C8911yescharacteristic
386C4⋊Q8C23×C8911yescharacteristic
387C4⋊Q8C23×C8911yescharacteristic
388#127—21no—
389#127C23×C8911yescharacteristic
390#124—41no—
391#123—41no—
392C22⋊Q8—21no—
393C22⋊Q8—21no—
394C22⋊Q8—41no—
395#121—21no—
396#121—21no—
397#121—21no—
398#121—21no—
399#121—41no—
400C4×Q8C23×C8911yescharacteristic
401C4×D4—21no—
402C4×D4—21no—
403C4×D4C23×C8911yescharacteristic
404C4×D4C23×C8911yescharacteristic
405C4×D4C23×C8911yescharacteristic
406#117—21no—
407#117—21no—
408C2×C4⋊C4—21no—
409#107—21no—
410#107—21no—
411#107D4×C8911yescharacteristic
412#106—21no—
413#106—21no—
414#106—41no—
415#106D4×C8911yescharacteristic
416C4⋊C8—21no—
417C4⋊C8C23×C8911yescharacteristic
418C4⋊C8C23×C8911yescharacteristic
419C4⋊C8C23×C8911yescharacteristic
420C4⋊C8C23×C8911yescharacteristic
421#103—21no—
422#103—21no—
423#103—21no—
424#103—21no—
425#103—21no—
426#103—21no—
427#103—21no—
428#103—21no—
429#103D4×C8911yescharacteristic
430#102—21no—
431#102—21no—
432#102—21no—
433#102—21no—
434#102—21no—
435#102—21no—
436#102—21no—
437#102—21no—
438#102D4×C8911yescharacteristic
439#102D4×C8911yescharacteristic
440#102D4×C8911yescharacteristic
441C22⋊C8—21no—
442C22⋊C8—21no—
443C22⋊C8—21no—
444C4×C8—21no—
445C4×C8C23×C8911yescharacteristic
446C4×C8C23×C8911yescharacteristic
447C4×C8C23×C8911yescharacteristic
448#41—42no—
449#41—42no—
450#41—81no—
451#41—81no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—41no—
456C2×Q8—42no—
457C2×Q8—81no—
458C2×Q8C2×D4×C8911yescharacteristic
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—41no—
463C2×D4—41no—
464C2×D4—41no—
465C2×D4—41no—
466C2×D4C2×D4×C8911yescharacteristic
467C2×D4C2×D4×C8911yescharacteristic
468C2×D4C2×D4×C8911yescharacteristic
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475SD16—22no—
476SD16—82no—
477SD16—81no—
478SD16—81no—
479D8—41no—
480D8—42no—
481D8—81no—
482M16—81no—
483M16—81no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—81no—
490C2×C8C2×D4×C8911yescharacteristic
491C2×C8C2×D4×C8911yescharacteristic
492C2×C8C2×D4×C8911yescharacteristic
493C2×C8C2×D4×C8911yescharacteristic
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4C2×D4×C8911yescharacteristic
504C4⋊C4C2×D4×C8911yescharacteristic
505C4⋊C4C2×D4×C8911yescharacteristic
506C4⋊C4C2×D4×C8911yescharacteristic
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—41no—
511C22⋊C4—41no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C42—21no—
516C42—41no—
517C42C24×C8911yescharacteristic
518C23—21no—
519C23—21no—
520C23—21no—
521C23—41no—
522Q8—21no—
523Q8—41no—
524Q8—41no—
525Q8—41no—
526Q8—41no—
527Q8—82no—
528Q8—81no—
529Q8#135×C8912yes—
530D4—21no—
531D4—21no—
532D4—21no—
533D4—21no—
534D4—21no—
535D4—21no—
536D4—41no—
537D4—41no—
538D4—41no—
539D4—41no—
540D4—82no—
541D4—81no—
542D4—82no—
543D4—161no—
544D4—161no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4C22×D4×C8911yescharacteristic
563C2×C4C22×D4×C8911yescharacteristic
564C2×C4H(5,𝔽2)×C8911yescharacteristic
565C8—22no—
566C8—21no—
567C8—21no—
568C8—21no—
569C8—21no—
570C8—41no—
571C8—41no—
572C8—41no—
573C8—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22D42×C8911yescharacteristic
587C4—22no—
588C4—21no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—81no—
594C4—81no—
595C4—81no—
596C4C2×#135×C8911yescharacteristic
597C4#575×C8911yescharacteristic
598C4#576×C8911yescharacteristic
599C4#577×C8911yescharacteristic
600C2—81no—
601C2—81no—
602C2—81no—
603C2—161no—
604C2C89×#327511yescharacteristic
605C2C89×#328011yescharacteristic
606C2C89×#329111yescharacteristic
subgroupG/H
center, socleC22×C89D42
commutator, Frattini, cosocleC42C24×C89
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C89 ◁ C42×C89 ◁ G

Complements

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

NH ≅ G/Ntype
1. C89×#3358C2semidirect
2. C89×#3332C2semidirect
3. C89×#3329C2semidirect
4. C89×#3303C2semidirect
5. C89×#1708C2semidirect
6. C89×#1703C2semidirect
7. C89×#1701C2semidirect
8. C89×#1684C2semidirect
9. C89×#1681C2semidirect
10. C89×#1654C2semidirect
11. C89×#1651C2semidirect
12. C89×#1631C2semidirect
13. C89×#1624C2semidirect
14. C89×#1589C2semidirect
15. C8×SD16×C89C2semidirect
296. #19904C89direct (N × H)
304. #3358C2×C89semidirect
305. #3332C2×C89semidirect
306. #3329C2×C89semidirect
307. #3303C2×C89semidirect
308. #1708C2×C89semidirect
309. #1703C2×C89semidirect
310. #1701C2×C89semidirect
311. #1684C2×C89semidirect
312. #1681C2×C89semidirect
313. #1654C2×C89semidirect
314. #1651C2×C89semidirect
315. #1631C2×C89semidirect
316. #1624C2×C89semidirect
317. #1589C2×C89semidirect
318. C8×SD16C2×C89semidirect
319. C89#19904direct (N × H)

Maximal subgroups

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

subgroupindex
C89×#33582
C89×#33322
C89×#33292
C89×#33032
C89×#17082
C89×#17032
C89×#17012
C89×#16842
C89×#16812
C89×#16542
C89×#16512
C89×#16312
C89×#16242
C89×#15892
C8×SD16×C892
#1990489