Finite Group Databasebeta

← C89×#61958 #36439806

Subgroups — C89×#61958 #36439806

Order 22784.

1922 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#3517C211yesmaximal, characteristic, complemented
2C89×#3483C211yesmaximal, characteristic, complemented
3C89×#3392C211yesmaximal, characteristic, complemented
4C89×#3391C211yesmaximal, characteristic, complemented
5C89×#3363C211yesmaximal, characteristic, complemented
6C89×#3362C211yesmaximal, characteristic, complemented
7C89×#3305C213yesmaximal, complemented
8C89×#3303C213yesmaximal, complemented
9C89×#3287C213yesmaximal, complemented
10C89×#3284C213yesmaximal, complemented
11C89×#3247C213yesmaximal, complemented
12C89×#3207C213yesmaximal, complemented
13C89×#3102C213yesmaximal, complemented
14C89×#3035C213yesmaximal, complemented
15C89×#2988C211yesmaximal, characteristic, complemented
16C2×#135×C89C2216yes—
17Q82×C89C2211yescharacteristic
18#557×C89C2211yescharacteristic
19#555×C89—23no—
20#555×C89C2213yes—
21#553×C89—23no—
22#550×C89—21no—
23#550×C89—21no—
24#550×C89C2213yes—
25#550×C89C2213yes—
26#550×C89C2211yescharacteristic
27#550×C89C2211yescharacteristic
28#547×C89C2213yes—
29#540×C89—23no—
30#532×C89—23no—
31#532×C89C2213yes—
32#532×C89C2213yes—
33#529×C89C2213yes—
34#520×C89—21no—
35C4×#41×C89—21no—
36C4×#41×C89C2213yes—
37C4×#41×C89C2211yescharacteristic
38#493×C89C2213yes—
39#492×C89C2213yes—
40#487×C89C2213yes—
41#486×C89C2213yes—
42#485×C89C2213yes—
43#482×C89C2213yes—
44#479×C89C2211yescharacteristic
45#478×C89C2213yes—
46#477×C89C2213yes—
47#476×C89C2213yes—
48#466×C89C2216yes—
49#465×C89C2216yes—
50#464×C89C2213yes—
51#463×C89C2213yes—
52#463×C89C2216yes—
53#462×C89C2213yes—
54#461×C89C2213yes—
55#460×C89C2213yes—
56#460×C89C2213yes—
57#459×C89C2213yes—
58#459×C89C2213yes—
59#452×C89C2213yes—
60#449×C89C2213yes—
61#449×C89C2216yes—
62#449×C89C2216yes—
63C8×Q8×C89C2211yescharacteristic
64C8×Q8×C89C2211yescharacteristic
65C4×Q16×C89C2213yes—
66C4×SD16×C89C2213yes—
67C4×SD16×C89C2213yes—
68C4×D8×C89C2213yes—
69C8×D4×C89C2213yes—
70C8×D4×C89C2213yes—
71#431×C89C2213yes—
72C2×C4⋊C8×C89C2213yes—
73#416×C89C2216yes—
74C2×#41×C89—21no—
75C2×#41×C89—21no—
76C2×#41×C89—23no—
77C2×#41×C89D411yescharacteristic
78C2×#41×C89C2313yes—
79C2×#41×C89C2316yes—
80#135×C89—412no—
81C2×Q16×C89C2313yes—
82C2×SD16×C89C2313yes—
83C2×SD16×C89C2313yes—
84C2×SD16×C89C2316yes—
85C2×D8×C89C2313yes—
86C2×D8×C89C2316yes—
87C22×C8×C89C2316yes—
88C4⋊Q8×C89—23no—
89C4⋊Q8×C89C2313yes—
90C4⋊Q8×C89C2313yes—
91#127×C89—23no—
92#127×C89—23no—
93#127×C89C2313yes—
94#127×C89C2313yes—
95#126×C89—23no—
96#125×C89—23no—
97#125×C89C2311yescharacteristic
98#124×C89—26no—
99#124×C89—23no—
100#124×C89C2313yes—
101#123×C89—23no—
102#123×C89—26no—
103C22⋊Q8×C89C2313yes—
104#121×C89—23no—
105#121×C89—23no—
106#121×C89—23no—
107#121×C89—23no—
108#121×C89—23no—
109#121×C89—23no—
110#121×C89—23no—
111#121×C89—26no—
112#121×C89—26no—
113#121×C89C2316yes—
114#121×C89C2313yes—
115C4×Q8×C89—21no—
116C4×Q8×C89—21no—
117C4×Q8×C89D411yescharacteristic
118C4×Q8×C89C2313yes—
119C4×Q8×C89C2313yes—
120C4×Q8×C89C2311yescharacteristic
121C4×Q8×C89C2311yescharacteristic
122C4×D4×C89—23no—
123C4×D4×C89—23no—
124C4×D4×C89—23no—
125C4×D4×C89—23no—
126C4×D4×C89—23no—
127C4×D4×C89—23no—
128C4×D4×C89—23no—
129C4×D4×C89—23no—
130C4×D4×C89D413yes—
131C4×D4×C89C2313yes—
132C4×D4×C89C2313yes—
133C4×D4×C89C2313yes—
134C4×D4×C89C2313yes—
135#117×C89—23no—
136#117×C89—23no—
137#117×C89—23no—
138#117×C89D413yes—
139#117×C89C2313yes—
140C2×C4⋊C4×C89—23no—
141C2×C42×C89—23no—
142C2×C42×C89C2313yes—
143#107×C89C2311yescharacteristic
144#107×C89C2313yes—
145#106×C89C2311yescharacteristic
146#106×C89C2313yes—
147C4⋊C8×C89C2313yes—
148C4⋊C8×C89C2313yes—
149C4⋊C8×C89C2313yes—
150C4⋊C8×C89C2311yescharacteristic
151#103×C89C2313yes—
152#103×C89C2316yes—
153#103×C89C2311yescharacteristic
154#103×C89C2311yescharacteristic
155#103×C89C2313yes—
156#102×C89C2313yes—
157#102×C89C2316yes—
158#102×C89C2313yes—
159#102×C89C2313yes—
160#102×C89C2313yes—
161C22⋊C8×C89C2313yes—
162C22⋊C8×C89C2313yes—
163C4×C8×C89C2313yes—
164C4×C8×C89C2313yes—
165#41×C89—24no—
166#41×C89—26no—
167#41×C89—43no—
168#41×C89—46no—
169#41×C89—46no—
170#41×C89—44no—
171#41×C89—44no—
172#41×C89—86no—
173C2×Q8×C89—23no—
174C2×Q8×C89C2×D411yescharacteristic
175C2×Q8×C89C2413yes—
176C2×Q8×C89C2411yescharacteristic
177C2×D4×C89—23no—
178C2×D4×C89—23no—
179C2×D4×C89—26no—
180C2×D4×C89—26no—
181C2×D4×C89—23no—
182C2×D4×C89—23no—
183C2×D4×C89—23no—
184C2×D4×C89—23no—
185C2×D4×C89C2×D413yes—
186C2×D4×C89C2413yes—
187C2×D4×C89C2413yes—
188C22×C4×C89—23no—
189C22×C4×C89—23no—
190C22×C4×C89—23no—
191C22×C4×C89—23no—
192C22×C4×C89—26no—
193C22×C4×C89C2×D413yes—
194C22×C4×C89C2×D413yes—
195C22×C4×C89C2413yes—
196Q16×C89—26no—
197SD16×C89—26no—
198SD16×C89—43no—
199SD16×C89—46no—
200D8×C89—43no—
201D8×C89—46no—
202C2×C8×C89—46no—
203C2×C8×C89C2411yescharacteristic
204C2×C8×C89C2411yescharacteristic
205C2×C8×C89C2413yes—
206C2×C8×C89C2413yes—
207C4⋊C4×C89—23no—
208C4⋊C4×C89—23no—
209C4⋊C4×C89—21no—
210C4⋊C4×C89—21no—
211C4⋊C4×C89—23no—
212C4⋊C4×C89—23no—
213C4⋊C4×C89—21no—
214C4⋊C4×C89—23no—
215C4⋊C4×C89—23no—
216C4⋊C4×C89C2×D413yes—
217C4⋊C4×C89C2×D413yes—
218C4⋊C4×C89C2411yescharacteristic
219C4⋊C4×C89C2413yes—
220C4⋊C4×C89C2411yescharacteristic
221C4⋊C4×C89C2411yescharacteristic
222C22⋊C4×C89—23no—
223C22⋊C4×C89—23no—
224C22⋊C4×C89—23no—
225C22⋊C4×C89—23no—
226C22⋊C4×C89—26no—
227C22⋊C4×C89—23no—
228C22⋊C4×C89—23no—
229C22⋊C4×C89—23no—
230C22⋊C4×C89—23no—
231C22⋊C4×C89C2×D416yes—
232C42×C89—21no—
233C42×C89—23no—
234C42×C89—23no—
235C42×C89—23no—
236C42×C89C2×D413yes—
237C42×C89C2×D413yes—
238C42×C89C2413yes—
239C42×C89C2411yescharacteristic
240C23×C89—23no—
241C23×C89—23no—
242C23×C89C22×D413yes—
243Q8×C89—21no—
244Q8×C89—23no—
245Q8×C89—23no—
246Q8×C89—83no—
247Q8×C89#13514yescomplemented
248Q8×C89H(5,𝔽2)12yes—
249D4×C89—26no—
250D4×C89—23no—
251D4×C89—23no—
252D4×C89—23no—
253D4×C89—23no—
254D4×C89—46no—
255D4×C89—46no—
256D4×C89—46no—
257D4×C89—83no—
258D4×C89—83no—
259D4×C89—83no—
260D4×C89—86no—
261D4×C89—86no—
262C2×C4×C89—21no—
263C2×C4×C89—26no—
264C2×C4×C89—23no—
265C2×C4×C89—23no—
266C2×C4×C89—21no—
267C2×C4×C89—21no—
268C2×C4×C89—23no—
269C2×C4×C89—23no—
270C2×C4×C89—21no—
271C2×C4×C89—43no—
272C2×C4×C89—46no—
273C2×C4×C89—46no—
274C2×C4×C89—46no—
275C2×C4×C89—46no—
276C2×C4×C89—86no—
277C2×C4×C89C22×D411yescharacteristic
278C2×C4×C89C22×D413yes—
279C2×C4×C89C22×D411yescharacteristic
280C2×C4×C89C22×D413yes—
281C2×C4×C89C22×D413yes—
282C2×C4×C89C2511yescharacteristic
283C8×C89—23no—
284C8×C89—23no—
285C8×C89—21no—
286C8×C89—21no—
287C22×C89—23no—
288C22×C89—23no—
289C22×C89—23no—
290C22×C89—43no—
291C22×C89—43no—
292C22×C89—43no—
293C22×C89—43no—
294C22×C89—43no—
295C22×C89—43no—
296C22×C89C23×D411yescharacteristic
297C4×C89—21no—
298C4×C89—23no—
299C4×C89—23no—
300C4×C89—22no—
301C4×C89—21no—
302C4×C89—43no—
303C4×C89—41no—
304C4×C89—43no—
305C4×C89—41no—
306C4×C89—41no—
307C4×C89C2×#13516yes—
308C4×C89C2×H(5,𝔽2)11yescharacteristic
309C4×C89#58511yescharacteristic
310#61958C8911yesmaximal, characteristic, direct product factor, Sylow
311C2×C89—43no—
312C2×C89—83no—
313C2×C89—83no—
314C2×C89#348811yescharacteristic
315C2×C89C22×#13511yescharacteristic
316C2×C89#358511yescharacteristic
317#3517C2×C8911yescharacteristic, complemented
318#3483C2×C8911yescharacteristic, complemented
319#3392C2×C8911yescharacteristic, complemented
320#3391C2×C8911yescharacteristic, complemented
321#3363C2×C8911yescharacteristic, complemented
322#3362C2×C8911yescharacteristic, complemented
323#3305C2×C8913yescomplemented
324#3303C2×C8913yescomplemented
325#3287C2×C8913yescomplemented
326#3284C2×C8913yescomplemented
327#3247C2×C8913yescomplemented
328#3207C2×C8913yescomplemented
329#3102C2×C8913yescomplemented
330#3035C2×C8913yescomplemented
331#2988C2×C8911yescharacteristic, complemented
332C89#6195811yescharacteristic, direct product factor, Sylow
333C2×#135C22×C8916yes—
334Q82C22×C8911yescharacteristic
335#557C22×C8911yescharacteristic
336#555—23no—
337#555C22×C8913yes—
338#553—23no—
339#550—21no—
340#550—21no—
341#550C22×C8913yes—
342#550C22×C8913yes—
343#550C22×C8911yescharacteristic
344#550C22×C8911yescharacteristic
345#547C22×C8913yes—
346#540—23no—
347#532—23no—
348#532C22×C8913yes—
349#532C22×C8913yes—
350#529C22×C8913yes—
351#520—21no—
352C4×#41—21no—
353C4×#41C22×C8913yes—
354C4×#41C22×C8911yescharacteristic
355#493C22×C8913yes—
356#492C22×C8913yes—
357#487C22×C8913yes—
358#486C22×C8913yes—
359#485C22×C8913yes—
360#482C22×C8913yes—
361#479C22×C8911yescharacteristic
362#478C22×C8913yes—
363#477C22×C8913yes—
364#476C22×C8913yes—
365#466C22×C8916yes—
366#465C22×C8916yes—
367#464C22×C8913yes—
368#463C22×C8913yes—
369#463C22×C8916yes—
370#462C22×C8913yes—
371#461C22×C8913yes—
372#460C22×C8913yes—
373#460C22×C8913yes—
374#459C22×C8913yes—
375#459C22×C8913yes—
376#452C22×C8913yes—
377#449C22×C8913yes—
378#449C22×C8916yes—
379#449C22×C8916yes—
380C8×Q8C22×C8911yescharacteristic
381C8×Q8C22×C8911yescharacteristic
382C4×Q16C22×C8913yes—
383C4×SD16C22×C8913yes—
384C4×SD16C22×C8913yes—
385C4×D8C22×C8913yes—
386C8×D4C22×C8913yes—
387C8×D4C22×C8913yes—
388#431C22×C8913yes—
389C2×C4⋊C8C22×C8913yes—
390#416C22×C8916yes—
391C2×#41—21no—
392C2×#41—21no—
393C2×#41—23no—
394C2×#41D4×C8911yescharacteristic
395C2×#41C23×C8913yes—
396C2×#41C23×C8916yes—
397#135—412no—
398C2×Q16C23×C8913yes—
399C2×SD16C23×C8913yes—
400C2×SD16C23×C8916yes—
401C2×SD16C23×C8913yes—
402C2×D8C23×C8916yes—
403C2×D8C23×C8913yes—
404C22×C8C23×C8916yes—
405C4⋊Q8—23no—
406C4⋊Q8C23×C8913yes—
407C4⋊Q8C23×C8913yes—
408#127—23no—
409#127—23no—
410#127C23×C8913yes—
411#127C23×C8913yes—
412#126—23no—
413#125—23no—
414#125C23×C8911yescharacteristic
415#124—26no—
416#124—23no—
417#124C23×C8913yes—
418#123—23no—
419#123—26no—
420C22⋊Q8C23×C8913yes—
421#121—23no—
422#121—23no—
423#121—23no—
424#121—23no—
425#121—23no—
426#121—23no—
427#121—23no—
428#121—26no—
429#121—26no—
430#121C23×C8913yes—
431#121C23×C8916yes—
432C4×Q8—21no—
433C4×Q8—21no—
434C4×Q8D4×C8911yescharacteristic
435C4×Q8C23×C8911yescharacteristic
436C4×Q8C23×C8913yes—
437C4×Q8C23×C8913yes—
438C4×Q8C23×C8911yescharacteristic
439C4×D4—23no—
440C4×D4—23no—
441C4×D4—23no—
442C4×D4—23no—
443C4×D4—23no—
444C4×D4—23no—
445C4×D4—23no—
446C4×D4—23no—
447C4×D4D4×C8913yes—
448C4×D4C23×C8913yes—
449C4×D4C23×C8913yes—
450C4×D4C23×C8913yes—
451C4×D4C23×C8913yes—
452#117—23no—
453#117—23no—
454#117—23no—
455#117D4×C8913yes—
456#117C23×C8913yes—
457C2×C4⋊C4—23no—
458C2×C42—23no—
459C2×C42C23×C8913yes—
460#107C23×C8911yescharacteristic
461#107C23×C8913yes—
462#106C23×C8911yescharacteristic
463#106C23×C8913yes—
464C4⋊C8C23×C8913yes—
465C4⋊C8C23×C8913yes—
466C4⋊C8C23×C8913yes—
467C4⋊C8C23×C8911yescharacteristic
468#103C23×C8913yes—
469#103C23×C8916yes—
470#103C23×C8911yescharacteristic
471#103C23×C8911yescharacteristic
472#103C23×C8913yes—
473#102C23×C8913yes—
474#102C23×C8916yes—
475#102C23×C8913yes—
476#102C23×C8913yes—
477#102C23×C8913yes—
478C22⋊C8C23×C8913yes—
479C22⋊C8C23×C8913yes—
480C4×C8C23×C8913yes—
481C4×C8C23×C8913yes—
482#41—24no—
483#41—26no—
484#41—43no—
485#41—46no—
486#41—46no—
487#41—44no—
488#41—44no—
489#41—86no—
490C2×Q8—23no—
491C2×Q8C2×D4×C8911yescharacteristic
492C2×Q8C24×C8913yes—
493C2×Q8C24×C8911yescharacteristic
494C2×D4—23no—
495C2×D4—23no—
496C2×D4—26no—
497C2×D4—26no—
498C2×D4—23no—
499C2×D4—23no—
500C2×D4—23no—
501C2×D4—23no—
502C2×D4C2×D4×C8913yes—
503C2×D4C24×C8913yes—
504C2×D4C24×C8913yes—
505C22×C4—23no—
506C22×C4—23no—
507C22×C4—23no—
508C22×C4—23no—
509C22×C4—26no—
510C22×C4C2×D4×C8913yes—
511C22×C4C2×D4×C8913yes—
512C22×C4C24×C8913yes—
513Q16—26no—
514SD16—26no—
515SD16—46no—
516SD16—43no—
517D8—46no—
518D8—43no—
519C2×C8—46no—
520C2×C8C24×C8911yescharacteristic
521C2×C8C24×C8911yescharacteristic
522C2×C8C24×C8913yes—
523C2×C8C24×C8913yes—
524C4⋊C4—23no—
525C4⋊C4—23no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—23no—
529C4⋊C4—23no—
530C4⋊C4—21no—
531C4⋊C4—23no—
532C4⋊C4—23no—
533C4⋊C4C2×D4×C8913yes—
534C4⋊C4C2×D4×C8913yes—
535C4⋊C4C24×C8911yescharacteristic
536C4⋊C4C24×C8913yes—
537C4⋊C4C24×C8911yescharacteristic
538C4⋊C4C24×C8911yescharacteristic
539C22⋊C4—23no—
540C22⋊C4—23no—
541C22⋊C4—23no—
542C22⋊C4—23no—
543C22⋊C4—26no—
544C22⋊C4—23no—
545C22⋊C4—23no—
546C22⋊C4—23no—
547C22⋊C4—23no—
548C22⋊C4C2×D4×C8916yes—
549C42—21no—
550C42—23no—
551C42—23no—
552C42—23no—
553C42C2×D4×C8913yes—
554C42C2×D4×C8913yes—
555C42C24×C8913yes—
556C42C24×C8911yescharacteristic
557C23—23no—
558C23—23no—
559C23C22×D4×C8913yes—
560Q8—21no—
561Q8—23no—
562Q8—23no—
563Q8—83no—
564Q8#135×C8914yescomplemented
565Q8H(5,𝔽2)×C8912yes—
566D4—26no—
567D4—23no—
568D4—23no—
569D4—23no—
570D4—23no—
571D4—46no—
572D4—46no—
573D4—46no—
574D4—83no—
575D4—83no—
576D4—83no—
577D4—86no—
578D4—86no—
579C2×C4—21no—
580C2×C4—26no—
581C2×C4—23no—
582C2×C4—23no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—23no—
586C2×C4—23no—
587C2×C4—21no—
588C2×C4—43no—
589C2×C4—46no—
590C2×C4—46no—
591C2×C4—46no—
592C2×C4—46no—
593C2×C4—86no—
594C2×C4C22×D4×C8911yescharacteristic
595C2×C4C22×D4×C8913yes—
596C2×C4C22×D4×C8911yescharacteristic
597C2×C4C22×D4×C8913yes—
598C2×C4C22×D4×C8913yes—
599C2×C4C25×C8911yescharacteristic
600C8—23no—
601C8—23no—
602C8—21no—
603C8—21no—
604C22—23no—
605C22—23no—
606C22—23no—
607C22—43no—
608C22—43no—
609C22—43no—
610C22—43no—
611C22—43no—
612C22—43no—
613C22C23×D4×C8911yescharacteristic
614C4—21no—
615C4—21no—
616C4—23no—
617C4—23no—
618C4—22no—
619C4—41no—
620C4—41no—
621C4—43no—
622C4—41no—
623C4—43no—
624C4C2×#135×C8916yes—
625C4C2×H(5,𝔽2)×C8911yescharacteristic
626C4#585×C8911yescharacteristic
627C2—43no—
628C2—83no—
629C2—83no—
630C2C89×#348811yescharacteristic
631C2C22×#135×C8911yescharacteristic
632C2C89×#358511yescharacteristic
subgroupG/H
center, socleC22×C89C23×D4
commutator, Frattini, cosocleC2×C4C25×C89
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C89 ◁ C4×#41×C89 ◁ G

Complements

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

NH ≅ G/Ntype
1. C89×#3517C2semidirect
2. C89×#3483C2semidirect
3. C89×#3392C2semidirect
4. C89×#3391C2semidirect
5. C89×#3363C2semidirect
6. C89×#3362C2semidirect
7. C89×#3305C2semidirect
8. C89×#3303C2semidirect
9. C89×#3287C2semidirect
10. C89×#3284C2semidirect
11. C89×#3247C2semidirect
12. C89×#3207C2semidirect
13. C89×#3102C2semidirect
14. C89×#3035C2semidirect
15. C89×#2988C2semidirect
247. Q8×C89#135semidirect
310. #61958C89direct (N × H)
317. #3517C2×C89semidirect
318. #3483C2×C89semidirect
319. #3392C2×C89semidirect
320. #3391C2×C89semidirect
321. #3363C2×C89semidirect
322. #3362C2×C89semidirect
323. #3305C2×C89semidirect
324. #3303C2×C89semidirect
325. #3287C2×C89semidirect
326. #3284C2×C89semidirect
327. #3247C2×C89semidirect
328. #3207C2×C89semidirect
329. #3102C2×C89semidirect
330. #3035C2×C89semidirect
331. #2988C2×C89semidirect
332. C89#61958direct (N × H)
564. Q8#135×C89semidirect

Maximal subgroups

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

subgroupindex
C89×#35172
C89×#34832
C89×#33922
C89×#33912
C89×#33632
C89×#33622
C89×#33052
C89×#33032
C89×#32872
C89×#32842
C89×#32472
C89×#32072
C89×#31022
C89×#30352
C89×#29882
#6195889