Finite Group Databasebeta

← #36581205

Subgroups — #36581205

Order 22784.

606 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26023553C211yesmaximal, characteristic
2#26020972C211yesmaximal, characteristic, complemented
3#26020968C211yesmaximal, characteristic, complemented
4#26020865C211yesmaximal, characteristic
5#26020864C211yesmaximal, characteristic
6#26020856C211yesmaximal, characteristic
7#26020817C211yesmaximal, characteristic, complemented
8#26020816C211yesmaximal, characteristic, complemented
9#26020811C211yesmaximal, characteristic, complemented
10#26020436C211yesmaximal, characteristic
11#26020433C211yesmaximal, characteristic
12#26020433C211yesmaximal, characteristic
13#26020405C211yesmaximal, characteristic, complemented
14#26020396C211yesmaximal, characteristic, complemented
15C89×#1940C211yesmaximal, characteristic, complemented
16C22×#13083557C2211yescharacteristic
17C2×C4⋊#10623494C2211yescharacteristic
18C2×#15667941C2211yescharacteristic
19C2×#15667941C2211yescharacteristic
20C2×#15667941C2211yescharacteristic
21C2×#15667941C2211yescharacteristic
22C2×#15667941C2211yescharacteristic
23C2×#15667924C2211yescharacteristic
24C2×#15667924C2211yescharacteristic
25C2×#15667924C2211yescharacteristic
26C2×#15667922C2211yescharacteristic
27C2×C4×#10623494C2211yescharacteristic
28C89⋊#401C411yescharacteristic
29#19151540C411yescharacteristic
30#19151537C411yescharacteristic
31#19151536C411yescharacteristic
32#19151527C411yescharacteristic
33#19151525C411yescharacteristic
34#19151524C411yescharacteristic
35#19151520C411yescharacteristic
36C4×#13083553C2211yescharacteristic
37C4×C4⋊Dic89C2211yescharacteristic
38C4×#13083551C2211yescharacteristic
39C4×#13083551C2211yescharacteristic
40C4×#13083551C2211yescharacteristic
41#19151444C2211yescharacteristic
42#19151353C2211yescharacteristic
43#19151353C2211yescharacteristic
44#19151353C2211yescharacteristic
45#19151353C2211yescharacteristic
46#19151352C2211yescharacteristic
47#19151352C2211yescharacteristic
48#19151352C2211yescharacteristic
49#19151352C2211yescharacteristic
50#19151352C2211yescharacteristic
51#19151352C2211yescharacteristic
52C2×C4⋊C8×C89C2211yescharacteristic
53C2×C4⋊C8×C89C2211yescharacteristic
54C2×C4⋊C8×C89C2211yescharacteristic
55C2×#97×C89C2211yescharacteristic
56C4×C4⋊C4×C89C2211yescharacteristic
57#336×C89C2211yescharacteristic
58#336×C89C2211yescharacteristic
59C22×C4×D89C2311yescharacteristic
60C2×#13083557—41no—
61C2×#13083557—41no—
62C2×#13083557—41no—
63C22×#10623494C2311yescharacteristic
64C22×#10623494C2311yescharacteristic
65C22×#10623494C2311yescharacteristic
66C22×#10623494C2311yescharacteristic
67C2×C4⋊Dic89C2×C411yescharacteristic
68C2×C4⋊Dic89C2×C411yescharacteristic
69C2×#13083553C2×C411yescharacteristic
70C2×#13083553C2×C411yescharacteristic
71C2×#13083551C2×C411yescharacteristic
72C2×#13083551C2×C411yescharacteristic
73C2×#13083551C2×C411yescharacteristic
74C2×#13083551C2×C411yescharacteristic
75C2×#13083551C2×C411yescharacteristic
76C2×#13083551C2×C411yescharacteristic
77C2×C4×Dic89C2311yescharacteristic
78C2×C4×Dic89C2311yescharacteristic
79C2×C4×Dic89C2311yescharacteristic
80C4⋊#10623494—41no—
81#15667941—41no—
82#15667941—41no—
83#15667941—41no—
84#15667941—41no—
85#15667941—41no—
86#15667924—41no—
87#15667924—41no—
88#15667924—41no—
89#15667922—41no—
90C4×#10623494—41no—
91#15667919C2×C411yescharacteristic
92#15667919C2×C411yescharacteristic
93#15667919C2×C411yescharacteristic
94#15667919C2×C411yescharacteristic
95#15667919C2×C411yescharacteristic
96#15667919C2×C411yescharacteristic
97#15667919C2×C411yescharacteristic
98#15667919C2×C411yescharacteristic
99#15667919C2×C411yescharacteristic
100#15667919C2×C411yescharacteristic
101#15667919C2×C411yescharacteristic
102#15667919C2×C411yescharacteristic
103#15667919C2×C411yescharacteristic
104#15667919C2×C411yescharacteristic
105C22×C8×C89C2311yescharacteristic
106C22×C8×C89C2311yescharacteristic
107C22×C8×C89C2311yescharacteristic
108C22×C8×C89C2311yescharacteristic
109C2×C4⋊C4×C89C2×C411yescharacteristic
110C2×C4⋊C4×C89C2×C411yescharacteristic
111C2×C42×C89C2311yescharacteristic
112C2×C42×C89C2311yescharacteristic
113C2×C42×C89C2311yescharacteristic
114C4⋊C8×C89—41no—
115C4⋊C8×C89—41no—
116C4⋊C8×C89—41no—
117#97×C89—41no—
118#95×C89C2×C411yescharacteristic
119#95×C89C2×C411yescharacteristic
120C23×D89C22×C411yescharacteristic
121C22×Dic89C22×C411yescharacteristic
122C22×Dic89C22×C411yescharacteristic
123C22×Dic89C22×C411yescharacteristic
124C22×Dic89C22×C411yescharacteristic
125C22×Dic89C22×C411yescharacteristic
126C22×Dic89C22×C411yescharacteristic
127C22×Dic89C22×C411yescharacteristic
128C2×C4×D89—21no—
129C2×C4×D89—21no—
130C2×C4×D89—21no—
131C2×C4×D89—21no—
132C2×C4×D89—21no—
133C2×C4×D89—21no—
134#13083557—44no—
135C2×#10623494—21no—
136C2×#10623494—21no—
137C2×#10623494—21no—
138C2×#10623494—21no—
139C2×#10623494—21no—
140C2×#10623494—21no—
141C2×#10623494—21no—
142C2×#10623494—21no—
143C2×#10623494—21no—
144C2×#10623494—21no—
145C2×#10623494—21no—
146C2×#10623494—21no—
147C4⋊Dic89—22no—
148C4⋊Dic89—41no—
149#13083553—41no—
150#13083553—41no—
151#13083551—22no—
152#13083551—41no—
153#13083551—41no—
154#13083551—41no—
155#13083551—41no—
156#13083551—41no—
157C4×Dic89—21no—
158C4×Dic89—21no—
159C4×Dic89—21no—
160C4×Dic89—21no—
161C4×Dic89—21no—
162C4×Dic89—21no—
163C22×C4×C89C22×C411yescharacteristic
164C22×C4×C89C22×C411yescharacteristic
165C22×C4×C89C22×C411yescharacteristic
166C22×C4×C89C22×C411yescharacteristic
167C22×C4×C89C22×C411yescharacteristic
168C22×C4×C89C22×C411yescharacteristic
169C22×C4×C89C2411yescharacteristic
170C2×C8×C89—21no—
171C2×C8×C89—21no—
172C2×C8×C89—21no—
173C2×C8×C89—21no—
174C2×C8×C89—21no—
175C2×C8×C89—21no—
176C2×C8×C89—21no—
177C2×C8×C89—21no—
178C2×C8×C89—21no—
179C2×C8×C89—21no—
180C2×C8×C89—21no—
181C2×C8×C89—21no—
182C4⋊C4×C89—22no—
183C4⋊C4×C89—41no—
184C42×C89—21no—
185C42×C89—21no—
186C42×C89—21no—
187C42×C89—21no—
188C42×C89—21no—
189C42×C89—21no—
190C23×C89C23×C411yescharacteristic
191C22×D89—21no—
192C22×D89—21no—
193C22×D89—21no—
194C22×D89—21no—
195C22×D89—21no—
196C22×D89—21no—
197C22×D89—21no—
198C2×C4×C89—21no—
199C2×C4×C89—21no—
200C2×C4×C89—21no—
201C2×C4×C89—21no—
202C2×C4×C89—21no—
203C2×C4×C89—21no—
204C2×C4×C89—21no—
205C2×C4×C89—21no—
206C2×C4×C89—21no—
207C2×C4×C89—21no—
208C2×C4×C89—21no—
209C2×C4×C89—21no—
210C2×C4×C89—21no—
211C2×C4×C89—21no—
212C2×C4×C89—21no—
213C2×C4×C89—21no—
214C2×C4×C89—21no—
215C2×C4×C89#13112yes—
216C2×C4×C89H(5,𝔽2)11yescharacteristic
217C2×C4×C89H(5,𝔽2)11yescharacteristic
218C2×C4×C89H(5,𝔽2)11yescharacteristic
219C2×C4×C892−1+411yescharacteristic
220C2×C4×C892−1+411yescharacteristic
221C2×C4×C892−1+411yescharacteristic
222C2×Dic89—21no—
223C2×Dic89—21no—
224C2×Dic89—21no—
225C2×Dic89—21no—
226C2×Dic89—21no—
227C2×Dic89—21no—
228C2×Dic89—21no—
229C2×Dic89—21no—
230C2×Dic89—21no—
231C2×Dic89—21no—
232C2×Dic89—21no—
233C2×Dic89—21no—
234C2×Dic89—21no—
235C2×Dic89—21no—
236C2×Dic89—21no—
237C2×Dic89—21no—
238C2×Dic89—21no—
239C2×Dic89—21no—
240C2×Dic89—21no—
241C2×Dic89—21no—
242C2×Dic89—21no—
243C4×D89—41no—
244C4×D89—41no—
245C4×D89—41no—
246C4×D89—41no—
247C8×C89—41no—
248C8×C89—41no—
249C8×C89—41no—
250C8×C89—41no—
251#10623494—41no—
252#10623494—41no—
253#10623494—41no—
254#10623494—41no—
255C22×C89#52011yescharacteristic
256C22×C89#52011yescharacteristic
257C22×C89#52011yescharacteristic
258C22×C89C2×#13111yescharacteristic
259C22×C89#56811yescharacteristic
260C22×C89#56811yescharacteristic
261C22×C89#56811yescharacteristic
262C2×D89—41no—
263C2×D89—41no—
264C2×D89—41no—
265C2×D89—41no—
266C2×D89—41no—
267C2×D89—41no—
268C2×D89—41no—
269C4×C89—22no—
270C4×C89—41no—
271C4×C89—41no—
272C4×C89—41no—
273C4×C89—41no—
274C4×C89—41no—
275C4×C89#56011yescharacteristic
276C4×C89#56311yescharacteristic
277C4×C89#56311yescharacteristic
278C4×C89#56311yescharacteristic
279Dic89—41no—
280Dic89—41no—
281Dic89—41no—
282Dic89—41no—
283Dic89—41no—
284Dic89—41no—
285Dic89—41no—
286#23382—891nomaximal, Sylow
287C2×C89#238311yescharacteristic
288C2×C89#299211yescharacteristic
289C2×C89#299311yescharacteristic
290C2×C89#299311yescharacteristic
291C2×C89#299511yescharacteristic
292C2×C89#299511yescharacteristic
293C2×C89#299511yescharacteristic
294D89—81no—
295#2401—891no—
296#1940—891no—
297#1940D8911yescharacteristic, complemented
298#1939—891no—
299#1924—891no—
300#1924—891no—
301#1923—891no—
302#1917—891no—
303#1917—891no—
304#1916—891no—
305#1851—891no—
306#1851—891no—
307#1851—891no—
308#1843—891no—
309#1840—891no—
310C89#2338211yescharacteristic, complemented, Sylow
311C22×M16—891no—
312C2×C4⋊C8—891no—
313C2×C4⋊C8—891no—
314C2×C4⋊C8—891no—
315C2×C4⋊C8—891no—
316C2×C4⋊C8—891no—
317C2×C4⋊C8—891no—
318C2×C4⋊C8C2×D8911yescharacteristic
319C2×C4⋊C8C2×D8911yescharacteristic
320C2×C4⋊C8C2×D8911yescharacteristic
321C2×C22⋊C8—891no—
322C2×C22⋊C8—891no—
323C2×C22⋊C8—891no—
324C2×#97—891no—
325C2×#97C2×D8911yescharacteristic
326C2×C4×C8—891no—
327#401—891no—
328#400—891no—
329#400—891no—
330#400—891no—
331#397—891no—
332#397—891no—
333#397—891no—
334#396—891no—
335C4×C4⋊C4—891no—
336C4×C4⋊C4—891no—
337C4×C4⋊C4C2×D8911yescharacteristic
338C4×C22⋊C4—891no—
339C4×C22⋊C4—891no—
340C4×C22⋊C4—891no—
341#376—891no—
342#336—891no—
343#336—891no—
344#336—891no—
345#336—891no—
346#336—891no—
347#336—891no—
348#336—891no—
349#336—891no—
350#336—891no—
351#336—891no—
352#336C2×D8911yescharacteristic
353#336C2×D8911yescharacteristic
354C23×C4—891no—
355C2×M16—3561no—
356C2×M16—3561no—
357C2×M16—3561no—
358C22×C8—891no—
359C22×C8—891no—
360C22×C8—891no—
361C22×C8—891no—
362C22×C8C22×D8911yescharacteristic
363C22×C8C22×D8911yescharacteristic
364C22×C8C22×D8911yescharacteristic
365C22×C8C22×D8911yescharacteristic
366C2×C4⋊C4—891no—
367C2×C4⋊C4—891no—
368C2×C4⋊C4—891no—
369C2×C4⋊C4—891no—
370C2×C4⋊C4C4×D8911yescharacteristic
371C2×C4⋊C4C4×D8911yescharacteristic
372C2×C22⋊C4—891no—
373C2×C22⋊C4—891no—
374C2×C22⋊C4—891no—
375C2×C22⋊C4—891no—
376C2×C22⋊C4—891no—
377C2×C22⋊C4—891no—
378C2×C42—891no—
379C2×C42—891no—
380C2×C42—891no—
381C2×C42C22×D8911yescharacteristic
382C2×C42C22×D8911yescharacteristic
383C2×C42C22×D8911yescharacteristic
384C4⋊C8—41no—
385C4⋊C8—41no—
386C4⋊C8—41no—
387C4⋊C8—3561no—
388C4⋊C8—3561no—
389C4⋊C8—3561no—
390C4⋊C8—3561no—
391C4⋊C8—3561no—
392C4⋊C8—3561no—
393C22⋊C8—3561no—
394C22⋊C8—3561no—
395C22⋊C8—3561no—
396#97—41no—
397#97—3561no—
398C4×C8—3561no—
399#95—891no—
400#95—891no—
401#95—891no—
402#95—891no—
403#95—891no—
404#95—891no—
405#95—891no—
406#95—891no—
407#95—891no—
408#95—891no—
409#95—891no—
410#95—891no—
411#95—891no—
412#95—891no—
413#95C4×D8911yescharacteristic
414#95C4×D8911yescharacteristic
415C24—891no—
416C22×C4—891no—
417C22×C4—891no—
418C22×C4—891no—
419C22×C4—891no—
420C22×C4—891no—
421C22×C4—891no—
422C22×C4—891no—
423C22×C4—1781no—
424C22×C4—1781no—
425C22×C4—1781no—
426C22×C4—1781no—
427C22×C4—1781no—
428C22×C4—1781no—
429C22×C4C2×C4×D8911yescharacteristic
430C22×C4C2×C4×D8911yescharacteristic
431C22×C4C2×C4×D8911yescharacteristic
432C22×C4C2×C4×D8911yescharacteristic
433C22×C4C2×C4×D8911yescharacteristic
434C22×C4C2×C4×D8911yescharacteristic
435C22×C4C23×D8911yescharacteristic
436M16—3564no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—1781no—
450C2×C8—1781no—
451C2×C8—1781no—
452C2×C8—1781no—
453C2×C8—1781no—
454C2×C8—1781no—
455C2×C8—1781no—
456C2×C8—1781no—
457C2×C8—1781no—
458C2×C8—1781no—
459C2×C8—1781no—
460C2×C8—1781no—
461C4⋊C4—22no—
462C4⋊C4—41no—
463C4⋊C4—3561no—
464C4⋊C4—3561no—
465C4⋊C4—3561no—
466C4⋊C4—1782no—
467C22⋊C4—3561no—
468C22⋊C4—3561no—
469C22⋊C4—3561no—
470C22⋊C4—3561no—
471C22⋊C4—3561no—
472C22⋊C4—1782no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—1781no—
480C42—1781no—
481C42—1781no—
482C42—1781no—
483C42—1781no—
484C42—1781no—
485C23—1781no—
486C23—1781no—
487C23—1781no—
488C23—1781no—
489C23—1781no—
490C23—1781no—
491C23—1781no—
492C23C22×C4×D8911yescharacteristic
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—3561no—
511C2×C4—3561no—
512C2×C4—3561no—
513C2×C4—3561no—
514C2×C4—1781no—
515C2×C4—1781no—
516C2×C4—1781no—
517C2×C4—1781no—
518C2×C4—1781no—
519C2×C4—1781no—
520C2×C4—1781no—
521C2×C4—1781no—
522C2×C4—1781no—
523C2×C4—1781no—
524C2×C4—1781no—
525C2×C4—1781no—
526C2×C4—1781no—
527C2×C4—1781no—
528C2×C4—1781no—
529C2×C4—1781no—
530C2×C4—1781no—
531C2×C4—1781no—
532C2×C4—1781no—
533C2×C4—1781no—
534C2×C4—1781no—
535C2×C4#1566802012yes—
536C2×C4#1566805711yescharacteristic
537C2×C4#1566805711yescharacteristic
538C2×C4#1566805811yescharacteristic
539C2×C4#1566805911yescharacteristic
540C2×C4#1566806011yescharacteristic
541C2×C4#1566806011yescharacteristic
542C8—41no—
543C8—41no—
544C8—41no—
545C8—41no—
546C8—3561no—
547C8—3561no—
548C8—3561no—
549C8—3561no—
550C22—3561no—
551C22—3561no—
552C22—3561no—
553C22—3561no—
554C22—3561no—
555C22—3561no—
556C22—3561no—
557C22#1915207711yescharacteristic
558C22#1915207911yescharacteristic
559C22#1915208011yescharacteristic
560C22C2×#1566802011yescharacteristic
561C22#1915236311yescharacteristic
562C22#1915236311yescharacteristic
563C22#1915236411yescharacteristic
564C4—22no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—3561no—
571C4—3561no—
572C4—3561no—
573C4—3561no—
574C4—3561no—
575C4—3561no—
576C4—3561no—
577C4#1915233611yescharacteristic
578C4#1915234711yescharacteristic
579C4#1915234911yescharacteristic
580C4#1915235211yescharacteristic
581C2—7121no—
582C2#2602344311yescharacteristic
583C2#2602939211yescharacteristic
584C2#2602939711yescharacteristic
585C2#2602940411yescharacteristic
586C2#2602941811yescharacteristic
587C2#2602942011yescharacteristic
588C2#2602942111yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D89
commutator, socleC23×C89C23×C4
FittingC89×#1940C2
prime radicalC23×D89C22×C4
cosocleC22×C4×C89C24
hypercenter#1940D89
hypocenterC89#23382
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C22×C4 ◁ #1940

Complements

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

NH ≅ G/Ntype
2. #26020972C2semidirect
3. #26020968C2semidirect
7. #26020817C2semidirect
8. #26020816C2semidirect
9. #26020811C2semidirect
13. #26020405C2semidirect
14. #26020396C2semidirect
15. C89×#1940C2semidirect
297. #1940D89semidirect
310. C89#23382semidirect

Maximal subgroups

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

subgroupindex
#260235532
#260209722
#260209682
#260208652
#260208642
#260208562
#260208172
#260208162
#260208112
#260204362
#260204332
#260204052
#260203962
C89×#19402
#2338289