Finite Group Databasebeta

← C89×#35481 #36413329

Subgroups — C89×#35481 #36413329

Order 22784.

2066 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#2576C213yesmaximal, complemented
2C89×#2555C213yesmaximal, complemented
3C89×#2518C213yesmaximal, complemented
4C89×#2472C213yesmaximal, complemented
5C89×#2465C213yesmaximal, complemented
6C89×#2459C213yesmaximal, complemented
7C89×#2426C213yesmaximal, complemented
8C89×#2424C213yesmaximal, complemented
9C89×#2333C213yesmaximal, complemented
10C4×#124×C89C213yesmaximal, complemented
11C89×#2273C211yesmaximal, characteristic, complemented
12C2×#124×C89C2213yes—
13C2×#124×C89C2213yescomplemented
14C2×C22⋊Q8×C89C2213yes—
15C2×#121×C89C2213yes—
16C2×#121×C89C2213yes—
17C2×#120×C89C2213yes—
18C2×C4×Q8×C89C2213yescomplemented
19C2×C4×D4×C89C2213yes—
20C22×C22⋊C4×C89C2213yes—
21#401×C89C2211yescharacteristic, complemented
22#397×C89C2213yes—
23#396×C89C2213yes—
24#396×C89C2213yescomplemented
25#395×C89C2213yescomplemented
26#394×C89C2213yes—
27#394×C89C2213yes—
28#394×C89C2213yes—
29#394×C89C2213yes—
30#393×C89C2213yes—
31#392×C89C2213yes—
32#392×C89C2211yescharacteristic, complemented
33#391×C89C2213yescomplemented
34#390×C89C2213yes—
35#390×C89C2213yes—
36#390×C89C2213yescomplemented
37#389×C89C2213yescomplemented
38#388×C89C2213yes—
39#388×C89C2213yes—
40#388×C89C2213yes—
41#388×C89C2213yescomplemented
42#388×C89C2213yescomplemented
43#388×C89C2213yescomplemented
44#388×C89C2213yescomplemented
45#388×C89C2213yescomplemented
46#387×C89C2213yescomplemented
47#387×C89C2213yescomplemented
48#387×C89C2213yescomplemented
49#386×C89C2213yes—
50#386×C89C2213yes—
51#386×C89C2213yes—
52#386×C89C2213yes—
53#386×C89C2213yescomplemented
54#385×C89C2213yes—
55#383×C89C2211yescharacteristic, complemented
56#382×C89C2213yescomplemented
57#381×C89C2213yescomplemented
58#379×C89C2211yescharacteristic
59C4×C22⋊C4×C89C2213yes—
60C4×C22⋊C4×C89C2213yes—
61C4×C22⋊C4×C89C2213yes—
62C4×C22⋊C4×C89C2213yes—
63C4×C22⋊C4×C89C2213yes—
64C4×C22⋊C4×C89C2213yescomplemented
65C4×C22⋊C4×C89C2213yescomplemented
66C43×C89C2211yescharacteristic, complemented
67C25×C89C2311yescharacteristic
68C22×Q8×C89C2313yes—
69C22×D4×C89C2313yes—
70C22×D4×C89C2313yes—
71C22×D4×C89C2313yes—
72C22×D4×C89C2313yes—
73C23×C4×C89C2313yes—
74C23×C4×C89C2313yes—
75C23×C4×C89C2313yes—
76#124×C89—212no—
77#124×C89—46no—
78C22⋊Q8×C89—46no—
79#121×C89—212no—
80#121×C89—46no—
81#120×C89—46no—
82C4×Q8×C89—46no—
83C4×D4×C89—46no—
84C2×C4⋊C4×C89C2313yes—
85C2×C4⋊C4×C89C2313yes—
86C2×C4⋊C4×C89C2313yes—
87C2×C4⋊C4×C89C2313yes—
88C2×C4⋊C4×C89C2313yes—
89C2×C4⋊C4×C89C2313yes—
90C2×C4⋊C4×C89C2313yes—
91C2×C22⋊C4×C89—212no—
92C2×C22⋊C4×C89C2313yes—
93C2×C22⋊C4×C89C2313yes—
94C2×C22⋊C4×C89C2313yes—
95C2×C22⋊C4×C89C2313yes—
96C2×C22⋊C4×C89C2313yes—
97C2×C22⋊C4×C89C2313yes—
98C2×C22⋊C4×C89C2313yes—
99C2×C22⋊C4×C89C2313yes—
100C2×C22⋊C4×C89C2313yes—
101C2×C22⋊C4×C89C2313yes—
102C2×C22⋊C4×C89C2313yes—
103C2×C22⋊C4×C89C2313yes—
104C2×C22⋊C4×C89C2313yes—
105C2×C22⋊C4×C89C2313yes—
106C2×C22⋊C4×C89C2313yes—
107C2×C22⋊C4×C89C2313yes—
108C2×C22⋊C4×C89C2313yes—
109C2×C22⋊C4×C89C2313yes—
110C2×C42×C89C2313yes—
111C2×C42×C89C2313yes—
112C2×C42×C89C2311yescharacteristic
113C2×C42×C89C2313yes—
114C2×C42×C89C2311yescharacteristic
115C2×C42×C89C2313yes—
116C2×C42×C89C2313yes—
117#95×C89C2313yes—
118#95×C89C2313yes—
119#95×C89C2311yescharacteristic
120#95×C89C2311yescharacteristic
121#95×C89C2313yes—
122#95×C89C2313yes—
123#95×C89C2313yes—
124#95×C89C2313yes—
125#95×C89C2313yes—
126#95×C89C2313yes—
127#95×C89C2313yes—
128#95×C89C2313yes—
129#95×C89C2313yes—
130#95×C89C2313yes—
131C24×C89—26no—
132C24×C89—43no—
133C24×C89—41no—
134C24×C89C2413yes—
135C24×C89C2413yes—
136C2×Q8×C89—26no—
137C2×Q8×C89—26no—
138C2×Q8×C89—43no—
139C2×D4×C89—26no—
140C2×D4×C89—26no—
141C2×D4×C89—26no—
142C2×D4×C89—26no—
143C2×D4×C89—26no—
144C2×D4×C89—43no—
145C2×D4×C89—43no—
146C2×D4×C89—43no—
147C2×D4×C89—43no—
148C2×D4×C89—43no—
149C2×D4×C89—43no—
150C2×D4×C89—43no—
151C22×C4×C89—26no—
152C22×C4×C89—26no—
153C22×C4×C89—26no—
154C22×C4×C89—43no—
155C22×C4×C89—43no—
156C22×C4×C89—43no—
157C22×C4×C89—43no—
158C22×C4×C89—43no—
159C22×C4×C89—43no—
160C22×C4×C89C2413yes—
161C22×C4×C89C2411yescharacteristic
162C22×C4×C89C2413yes—
163C22×C4×C89C2413yes—
164C22×C4×C89C2413yes—
165C22×C4×C89C2413yes—
166C22×C4×C89C2413yes—
167C22×C4×C89C2413yes—
168C22×C4×C89C2413yes—
169C4⋊C4×C89—26no—
170C4⋊C4×C89—26no—
171C4⋊C4×C89—26no—
172C4⋊C4×C89—43no—
173C4⋊C4×C89—43no—
174C4⋊C4×C89—43no—
175C4⋊C4×C89—43no—
176C22⋊C4×C89—26no—
177C22⋊C4×C89—26no—
178C22⋊C4×C89—26no—
179C22⋊C4×C89—26no—
180C22⋊C4×C89—26no—
181C22⋊C4×C89—26no—
182C22⋊C4×C89—26no—
183C22⋊C4×C89—26no—
184C22⋊C4×C89—26no—
185C22⋊C4×C89—26no—
186C22⋊C4×C89—26no—
187C22⋊C4×C89—43no—
188C22⋊C4×C89—43no—
189C22⋊C4×C89—43no—
190C22⋊C4×C89—43no—
191C22⋊C4×C89—43no—
192C22⋊C4×C89—43no—
193C22⋊C4×C89#41112yes—
194C42×C89—26no—
195C42×C89—26no—
196C42×C89—41no—
197C42×C89—41no—
198C42×C89—43no—
199C42×C89—43no—
200C42×C89—43no—
201C23×C89—28no—
202C23×C89—23no—
203C23×C89—23no—
204C23×C89—23no—
205C23×C89—23no—
206C23×C89—23no—
207C23×C89—23no—
208C23×C89—23no—
209C23×C89—23no—
210C23×C89—23no—
211C23×C89—23no—
212C23×C89—23no—
213C23×C89—23no—
214C23×C89—23no—
215C23×C89—86no—
216C23×C89—86no—
217C23×C89C2×#4116yes—
218C23×C89C2511yescharacteristic
219Q8×C89—412no—
220D4×C89—412no—
221D4×C89—86no—
222D4×C89—86no—
223D4×C89—86no—
224C2×C4×C89—23no—
225C2×C4×C89—23no—
226C2×C4×C89—23no—
227C2×C4×C89—23no—
228C2×C4×C89—23no—
229C2×C4×C89—23no—
230C2×C4×C89—23no—
231C2×C4×C89—23no—
232C2×C4×C89—23no—
233C2×C4×C89—23no—
234C2×C4×C89—23no—
235C2×C4×C89—23no—
236C2×C4×C89—23no—
237C2×C4×C89—23no—
238C2×C4×C89—23no—
239C2×C4×C89—23no—
240C2×C4×C89—23no—
241C2×C4×C89—23no—
242C2×C4×C89—23no—
243C2×C4×C89—23no—
244C2×C4×C89—23no—
245C2×C4×C89—23no—
246C2×C4×C89—86no—
247C2×C4×C89—86no—
248C2×C4×C89—86no—
249C2×C4×C89C2×#4116yes—
250C2×C4×C89C2×#4116yes—
251C2×C4×C89H(5,𝔽2)16yes—
252C22×C89—26no—
253C22×C89—26no—
254C22×C89—26no—
255C22×C89—43no—
256C22×C89—43no—
257C22×C89—43no—
258C22×C89—43no—
259C22×C89—43no—
260C22×C89—43no—
261C22×C89—43no—
262C22×C89—43no—
263C22×C89—43no—
264C22×C89—43no—
265C22×C89—43no—
266C22×C89—88no—
267C22×C89C22×#4113yes—
268C22×C89C2×H(5,𝔽2)13yes—
269C22×C89#58511yescharacteristic
270C4×C89—26no—
271C4×C89—26no—
272C4×C89—26no—
273C4×C89—41no—
274C4×C89—43no—
275C4×C89—43no—
276C4×C89—43no—
277C4×C89—43no—
278C4×C89—43no—
279#35481C8911yesmaximal, characteristic, direct product factor, Sylow
280C2×C89—46no—
281C2×C89—83no—
282C2×C89#344411yescharacteristic
283C2×C89#351413yes—
284C2×C89#351913yes—
285#2576C2×C8913yescomplemented
286#2555C2×C8913yescomplemented
287#2518C2×C8913yescomplemented
288#2472C2×C8913yescomplemented
289#2465C2×C8913yescomplemented
290#2459C2×C8913yescomplemented
291#2426C2×C8913yescomplemented
292#2424C2×C8913yescomplemented
293#2333C2×C8913yescomplemented
294C4×#124C2×C8913yescomplemented
295#2273C2×C8911yescharacteristic, complemented
296C89#3548111yescharacteristic, direct product factor, Sylow
297C2×#124C22×C8913yes—
298C2×#124C22×C8913yescomplemented
299C2×C22⋊Q8C22×C8913yes—
300C2×#121C22×C8913yes—
301C2×#121C22×C8913yes—
302C2×#120C22×C8913yes—
303C2×C4×Q8C22×C8913yescomplemented
304C2×C4×D4C22×C8913yes—
305C22×C22⋊C4C22×C8913yes—
306#401C22×C8911yescharacteristic, complemented
307#397C22×C8913yes—
308#396C22×C8913yes—
309#396C22×C8913yescomplemented
310#395C22×C8913yescomplemented
311#394C22×C8913yes—
312#394C22×C8913yes—
313#394C22×C8913yes—
314#394C22×C8913yes—
315#393C22×C8913yes—
316#392C22×C8913yes—
317#392C22×C8911yescharacteristic, complemented
318#391C22×C8913yescomplemented
319#390C22×C8913yes—
320#390C22×C8913yes—
321#390C22×C8913yescomplemented
322#389C22×C8913yescomplemented
323#388C22×C8913yes—
324#388C22×C8913yes—
325#388C22×C8913yes—
326#388C22×C8913yescomplemented
327#388C22×C8913yescomplemented
328#388C22×C8913yescomplemented
329#388C22×C8913yescomplemented
330#388C22×C8913yescomplemented
331#387C22×C8913yescomplemented
332#387C22×C8913yescomplemented
333#387C22×C8913yescomplemented
334#386C22×C8913yes—
335#386C22×C8913yes—
336#386C22×C8913yes—
337#386C22×C8913yes—
338#386C22×C8913yescomplemented
339#385C22×C8913yes—
340#383C22×C8911yescharacteristic, complemented
341#382C22×C8913yescomplemented
342#381C22×C8913yescomplemented
343#379C22×C8911yescharacteristic
344C4×C22⋊C4C22×C8913yes—
345C4×C22⋊C4C22×C8913yes—
346C4×C22⋊C4C22×C8913yes—
347C4×C22⋊C4C22×C8913yes—
348C4×C22⋊C4C22×C8913yes—
349C4×C22⋊C4C22×C8913yescomplemented
350C4×C22⋊C4C22×C8913yescomplemented
351C43C22×C8911yescharacteristic, complemented
352C25C23×C8911yescharacteristic
353C22×Q8C23×C8913yes—
354C22×D4C23×C8913yes—
355C22×D4C23×C8913yes—
356C22×D4C23×C8913yes—
357C22×D4C23×C8913yes—
358C23×C4C23×C8913yes—
359C23×C4C23×C8913yes—
360C23×C4C23×C8913yes—
361#124—212no—
362#124—46no—
363C22⋊Q8—46no—
364#121—212no—
365#121—46no—
366#120—46no—
367C4×Q8—46no—
368C4×D4—46no—
369C2×C4⋊C4C23×C8913yes—
370C2×C4⋊C4C23×C8913yes—
371C2×C4⋊C4C23×C8913yes—
372C2×C4⋊C4C23×C8913yes—
373C2×C4⋊C4C23×C8913yes—
374C2×C4⋊C4C23×C8913yes—
375C2×C4⋊C4C23×C8913yes—
376C2×C22⋊C4—212no—
377C2×C22⋊C4C23×C8913yes—
378C2×C22⋊C4C23×C8913yes—
379C2×C22⋊C4C23×C8913yes—
380C2×C22⋊C4C23×C8913yes—
381C2×C22⋊C4C23×C8913yes—
382C2×C22⋊C4C23×C8913yes—
383C2×C22⋊C4C23×C8913yes—
384C2×C22⋊C4C23×C8913yes—
385C2×C22⋊C4C23×C8913yes—
386C2×C22⋊C4C23×C8913yes—
387C2×C22⋊C4C23×C8913yes—
388C2×C22⋊C4C23×C8913yes—
389C2×C22⋊C4C23×C8913yes—
390C2×C22⋊C4C23×C8913yes—
391C2×C22⋊C4C23×C8913yes—
392C2×C22⋊C4C23×C8913yes—
393C2×C22⋊C4C23×C8913yes—
394C2×C22⋊C4C23×C8913yes—
395C2×C42C23×C8913yes—
396C2×C42C23×C8913yes—
397C2×C42C23×C8913yes—
398C2×C42C23×C8913yes—
399C2×C42C23×C8913yes—
400C2×C42C23×C8911yescharacteristic
401C2×C42C23×C8911yescharacteristic
402#95C23×C8913yes—
403#95C23×C8913yes—
404#95C23×C8911yescharacteristic
405#95C23×C8911yescharacteristic
406#95C23×C8913yes—
407#95C23×C8913yes—
408#95C23×C8913yes—
409#95C23×C8913yes—
410#95C23×C8913yes—
411#95C23×C8913yes—
412#95C23×C8913yes—
413#95C23×C8913yes—
414#95C23×C8913yes—
415#95C23×C8913yes—
416C24—26no—
417C24—43no—
418C24—41no—
419C24C24×C8913yes—
420C24C24×C8913yes—
421C2×Q8—26no—
422C2×Q8—26no—
423C2×Q8—43no—
424C2×D4—26no—
425C2×D4—26no—
426C2×D4—26no—
427C2×D4—26no—
428C2×D4—26no—
429C2×D4—43no—
430C2×D4—43no—
431C2×D4—43no—
432C2×D4—43no—
433C2×D4—43no—
434C2×D4—43no—
435C2×D4—43no—
436C22×C4—26no—
437C22×C4—26no—
438C22×C4—26no—
439C22×C4—43no—
440C22×C4—43no—
441C22×C4—43no—
442C22×C4—43no—
443C22×C4—43no—
444C22×C4—43no—
445C22×C4C24×C8913yes—
446C22×C4C24×C8911yescharacteristic
447C22×C4C24×C8913yes—
448C22×C4C24×C8913yes—
449C22×C4C24×C8913yes—
450C22×C4C24×C8913yes—
451C22×C4C24×C8913yes—
452C22×C4C24×C8913yes—
453C22×C4C24×C8913yes—
454C4⋊C4—26no—
455C4⋊C4—26no—
456C4⋊C4—26no—
457C4⋊C4—43no—
458C4⋊C4—43no—
459C4⋊C4—43no—
460C4⋊C4—43no—
461C22⋊C4—26no—
462C22⋊C4—26no—
463C22⋊C4—26no—
464C22⋊C4—26no—
465C22⋊C4—26no—
466C22⋊C4—26no—
467C22⋊C4—26no—
468C22⋊C4—26no—
469C22⋊C4—26no—
470C22⋊C4—26no—
471C22⋊C4—26no—
472C22⋊C4—43no—
473C22⋊C4—43no—
474C22⋊C4—43no—
475C22⋊C4—43no—
476C22⋊C4—43no—
477C22⋊C4—43no—
478C22⋊C4#41×C89112yes—
479C42—26no—
480C42—26no—
481C42—43no—
482C42—43no—
483C42—43no—
484C42—41no—
485C42—41no—
486C23—28no—
487C23—23no—
488C23—23no—
489C23—23no—
490C23—23no—
491C23—23no—
492C23—23no—
493C23—23no—
494C23—23no—
495C23—23no—
496C23—23no—
497C23—23no—
498C23—23no—
499C23—23no—
500C23—86no—
501C23—86no—
502C23C2×#41×C8916yes—
503C23C25×C8911yescharacteristic
504Q8—412no—
505D4—412no—
506D4—86no—
507D4—86no—
508D4—86no—
509C2×C4—23no—
510C2×C4—23no—
511C2×C4—23no—
512C2×C4—23no—
513C2×C4—23no—
514C2×C4—23no—
515C2×C4—23no—
516C2×C4—23no—
517C2×C4—23no—
518C2×C4—23no—
519C2×C4—23no—
520C2×C4—23no—
521C2×C4—23no—
522C2×C4—23no—
523C2×C4—23no—
524C2×C4—23no—
525C2×C4—23no—
526C2×C4—23no—
527C2×C4—23no—
528C2×C4—23no—
529C2×C4—23no—
530C2×C4—23no—
531C2×C4—86no—
532C2×C4—86no—
533C2×C4—86no—
534C2×C4C2×#41×C8916yes—
535C2×C4C2×#41×C8916yes—
536C2×C4H(5,𝔽2)×C8916yes—
537C22—26no—
538C22—26no—
539C22—26no—
540C22—43no—
541C22—43no—
542C22—43no—
543C22—43no—
544C22—43no—
545C22—43no—
546C22—43no—
547C22—43no—
548C22—43no—
549C22—43no—
550C22—43no—
551C22—88no—
552C22C22×#41×C8913yes—
553C22C2×H(5,𝔽2)×C8913yes—
554C22#585×C8911yescharacteristic
555C4—26no—
556C4—26no—
557C4—26no—
558C4—41no—
559C4—43no—
560C4—43no—
561C4—43no—
562C4—43no—
563C4—43no—
564C2—46no—
565C2—83no—
566C2C89×#344411yescharacteristic
567C2C89×#351413yes—
568C2C89×#351913yes—
subgroupG/H
center, socleC23×C89C25
commutator, Frattini, cosocleC23C25×C89
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C89 ◁ G

Complements

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

NH ≅ G/Ntype
1. C89×#2576C2semidirect
2. C89×#2555C2semidirect
3. C89×#2518C2semidirect
4. C89×#2472C2semidirect
5. C89×#2465C2semidirect
6. C89×#2459C2semidirect
7. C89×#2426C2semidirect
8. C89×#2424C2semidirect
9. C89×#2333C2semidirect
10. C4×#124×C89C2semidirect
11. C89×#2273C2semidirect
13. C2×#124×C89C22semidirect
18. C2×C4×Q8×C89C22semidirect
21. #401×C89C22semidirect
24. #396×C89C22semidirect
25. #395×C89C22semidirect
32. #392×C89C22semidirect
33. #391×C89C22semidirect
36. #390×C89C22semidirect
37. #389×C89C22semidirect
41. #388×C89C22semidirect
42. #388×C89C22semidirect
43. #388×C89C22semidirect
44. #388×C89C22semidirect
45. #388×C89C22semidirect
46. #387×C89C22semidirect
47. #387×C89C22semidirect
48. #387×C89C22semidirect
53. #386×C89C22semidirect
55. #383×C89C22semidirect
56. #382×C89C22semidirect
57. #381×C89C22semidirect
64. C4×C22⋊C4×C89C22semidirect
65. C4×C22⋊C4×C89C22semidirect
66. C43×C89C22semidirect
279. #35481C89direct (N × H)
285. #2576C2×C89semidirect
286. #2555C2×C89semidirect
287. #2518C2×C89semidirect
288. #2472C2×C89semidirect
289. #2465C2×C89semidirect
290. #2459C2×C89semidirect
291. #2426C2×C89semidirect
292. #2424C2×C89semidirect
293. #2333C2×C89semidirect
294. C4×#124C2×C89semidirect
295. #2273C2×C89semidirect
296. C89#35481direct (N × H)
298. C2×#124C22×C89semidirect
303. C2×C4×Q8C22×C89semidirect
306. #401C22×C89semidirect
309. #396C22×C89semidirect
310. #395C22×C89semidirect
317. #392C22×C89semidirect
318. #391C22×C89semidirect
321. #390C22×C89semidirect
322. #389C22×C89semidirect
326. #388C22×C89semidirect
327. #388C22×C89semidirect
328. #388C22×C89semidirect
329. #388C22×C89semidirect
330. #388C22×C89semidirect
331. #387C22×C89semidirect
332. #387C22×C89semidirect
333. #387C22×C89semidirect
338. #386C22×C89semidirect
340. #383C22×C89semidirect
341. #382C22×C89semidirect
342. #381C22×C89semidirect
349. C4×C22⋊C4C22×C89semidirect
350. C4×C22⋊C4C22×C89semidirect
351. C43C22×C89semidirect

Maximal subgroups

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

subgroupindex
C89×#25762
C89×#25552
C89×#25182
C89×#24722
C89×#24652
C89×#24592
C89×#24262
C89×#24242
C89×#23332
C4×#124×C892
C89×#22732
#3548189