Finite Group Databasebeta

← C89×#35415 #36413263

Subgroups — C89×#35415 #36413263

Order 22784.

1816 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#2576C213yesmaximal, complemented
2C89×#2553C213yesmaximal, complemented
3C89×#2522C213yesmaximal, complemented
4C89×#2517C213yesmaximal, complemented
5C89×#2494C213yesmaximal, complemented
6C89×#2480C213yesmaximal, complemented
7C89×#2466C213yesmaximal, complemented
8C89×#2418C213yesmaximal
9C89×#2334C213yesmaximal, complemented
10C4×#123×C89C213yesmaximal, complemented
11C89×#2273C211yesmaximal, characteristic, complemented
12C2×#126×C89C2213yes—
13C2×#126×C89C2211yescharacteristic, complemented
14C2×#123×C89C2213yes—
15C2×#120×C89C2211yescharacteristic
16C2×C4×D4×C89C2213yes—
17C22×C22⋊C4×C89C2213yes—
18C22×C42×C89C2211yescharacteristic, complemented
19#401×C89C2213yescomplemented
20#399×C89C2213yes—
21#399×C89C2211yescharacteristic, complemented
22#398×C89C2213yescomplemented
23#397×C89C2213yes—
24#397×C89C2213yes—
25#397×C89C2213yes—
26#397×C89C2213yes—
27#396×C89C2213yes—
28#396×C89C2211yescharacteristic, complemented
29#394×C89C2213yes—
30#394×C89C2213yes—
31#389×C89C2213yescomplemented
32#389×C89C2213yescomplemented
33#389×C89C2213yescomplemented
34#388×C89C2213yes—
35#388×C89C2213yes—
36#388×C89C2213yes—
37#388×C89C2213yes—
38#388×C89C2213yes—
39#388×C89C2213yescomplemented
40#387×C89C2213yescomplemented
41#387×C89C2213yescomplemented
42#387×C89C2213yescomplemented
43#387×C89C2213yescomplemented
44#387×C89C2213yescomplemented
45#387×C89C2213yescomplemented
46#387×C89C2213yescomplemented
47#386×C89C2213yes—
48#386×C89C2213yes—
49#385×C89C2213yes—
50#385×C89C2213yes—
51#385×C89C2213yes—
52#385×C89C2213yescomplemented
53#382×C89C2213yescomplemented
54#382×C89C2213yescomplemented
55#381×C89C2213yescomplemented
56#380×C89C2213yes—
57#380×C89C2213yes—
58#379×C89C2213yes—
59C4×C4⋊C4×C89C2213yescomplemented
60C4×C4⋊C4×C89C2213yescomplemented
61C4×C22⋊C4×C89C2213yes—
62C4×C22⋊C4×C89C2213yes—
63C4×C22⋊C4×C89C2213yes—
64C4×C22⋊C4×C89C2213yes—
65C4×C22⋊C4×C89C2213yes—
66C4×C22⋊C4×C89C2213yescomplemented
67C25×C89C2311yescharacteristic
68C22×D4×C89C2313yes—
69C23×C4×C89C2313yes—
70C23×C4×C89C2313yes—
71C23×C4×C89C2313yes—
72#126×C89—212no—
73#126×C89—42no—
74#123×C89—212no—
75#120×C89—42no—
76C4×D4×C89—46no—
77C2×C4⋊C4×C89C2313yes—
78C2×C4⋊C4×C89C2313yes—
79C2×C4⋊C4×C89C2313yes—
80C2×C4⋊C4×C89C2313yes—
81C2×C4⋊C4×C89C2313yes—
82C2×C4⋊C4×C89C2313yes—
83C2×C4⋊C4×C89C2313yes—
84C2×C4⋊C4×C89C2313yes—
85C2×C4⋊C4×C89C2313yes—
86C2×C4⋊C4×C89C2313yes—
87C2×C4⋊C4×C89C2313yes—
88C2×C22⋊C4×C89—212no—
89C2×C22⋊C4×C89C2313yes—
90C2×C22⋊C4×C89C2313yes—
91C2×C22⋊C4×C89C2313yes—
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×C42×C89—42no—
104C2×C42×C89C2313yes—
105C2×C42×C89C2313yes—
106C2×C42×C89C2311yescharacteristic
107C2×C42×C89C2313yes—
108C2×C42×C89C2311yescharacteristic
109C2×C42×C89C2313yes—
110C2×C42×C89C2313yes—
111#95×C89C2313yes—
112#95×C89C2313yes—
113#95×C89C2311yescharacteristic
114#95×C89C2311yescharacteristic
115#95×C89C2313yes—
116#95×C89C2313yes—
117#95×C89C2313yes—
118#95×C89C2313yes—
119#95×C89C2313yes—
120#95×C89C2313yes—
121#95×C89C2313yes—
122#95×C89C2313yes—
123#95×C89C2313yes—
124#95×C89C2313yes—
125#95×C89C2313yes—
126#95×C89C2313yes—
127#95×C89C2313yes—
128#95×C89C2313yes—
129C24×C89—26no—
130C24×C89—41no—
131C24×C89—43no—
132C24×C89C2413yes—
133C24×C89C2411yescharacteristic
134C2×D4×C89—26no—
135C2×D4×C89—26no—
136C2×D4×C89—43no—
137C22×C4×C89—26no—
138C22×C4×C89—26no—
139C22×C4×C89—26no—
140C22×C4×C89—26no—
141C22×C4×C89—43no—
142C22×C4×C89—43no—
143C22×C4×C89—43no—
144C22×C4×C89—43no—
145C22×C4×C89—43no—
146C22×C4×C89C2413yes—
147C22×C4×C89C2413yes—
148C22×C4×C89C2413yes—
149C22×C4×C89C2413yes—
150C22×C4×C89C2413yes—
151C22×C4×C89C2413yes—
152C22×C4×C89C2413yes—
153C22×C4×C89C2413yes—
154C22×C4×C89C2413yes—
155C4⋊C4×C89—26no—
156C4⋊C4×C89—26no—
157C4⋊C4×C89—26no—
158C4⋊C4×C89—26no—
159C4⋊C4×C89—26no—
160C4⋊C4×C89—43no—
161C4⋊C4×C89—43no—
162C4⋊C4×C89—43no—
163C4⋊C4×C89—43no—
164C4⋊C4×C89—43no—
165C4⋊C4×C89—43no—
166C22⋊C4×C89—26no—
167C22⋊C4×C89—26no—
168C22⋊C4×C89—26no—
169C22⋊C4×C89—26no—
170C22⋊C4×C89—26no—
171C22⋊C4×C89—26no—
172C22⋊C4×C89—26no—
173C22⋊C4×C89—26no—
174C22⋊C4×C89—43no—
175C22⋊C4×C89—43no—
176C22⋊C4×C89—43no—
177C22⋊C4×C89—43no—
178C22⋊C4×C89—43no—
179C22⋊C4×C89#41112yes—
180C42×C89—26no—
181C42×C89—41no—
182C42×C89—43no—
183C42×C89—43no—
184C42×C89—43no—
185C42×C89—43no—
186C42×C89—41no—
187C23×C89—28no—
188C23×C89—23no—
189C23×C89—23no—
190C23×C89—23no—
191C23×C89—23no—
192C23×C89—23no—
193C23×C89—23no—
194C23×C89—23no—
195C23×C89—23no—
196C23×C89—86no—
197C23×C89—86no—
198C23×C89C2×#4116yes—
199C23×C89H(5,𝔽2)12yes—
200C23×C89C2511yescharacteristic
201D4×C89—412no—
202C2×C4×C89—23no—
203C2×C4×C89—23no—
204C2×C4×C89—23no—
205C2×C4×C89—23no—
206C2×C4×C89—23no—
207C2×C4×C89—23no—
208C2×C4×C89—23no—
209C2×C4×C89—23no—
210C2×C4×C89—23no—
211C2×C4×C89—23no—
212C2×C4×C89—23no—
213C2×C4×C89—23no—
214C2×C4×C89—23no—
215C2×C4×C89—23no—
216C2×C4×C89—23no—
217C2×C4×C89—23no—
218C2×C4×C89—23no—
219C2×C4×C89—23no—
220C2×C4×C89—23no—
221C2×C4×C89—23no—
222C2×C4×C89—23no—
223C2×C4×C89—23no—
224C2×C4×C89—23no—
225C2×C4×C89—23no—
226C2×C4×C89—23no—
227C2×C4×C89—412no—
228C2×C4×C89—86no—
229C2×C4×C89—86no—
230C2×C4×C89C2×#4116yes—
231C2×C4×C89C2×#4116yes—
232C22×C89—26no—
233C22×C89—26no—
234C22×C89—26no—
235C22×C89—26no—
236C22×C89—41no—
237C22×C89—43no—
238C22×C89—43no—
239C22×C89—43no—
240C22×C89—43no—
241C22×C89—43no—
242C22×C89—88no—
243C22×C89C22×#4113yes—
244C22×C89C2×H(5,𝔽2)11yescharacteristic
245C22×C89#58513yes—
246C4×C89—26no—
247C4×C89—26no—
248C4×C89—43no—
249C4×C89—43no—
250C4×C89—43no—
251C4×C89—43no—
252C4×C89—43no—
253C4×C89—43no—
254C4×C89—43no—
255#35415C8911yesmaximal, characteristic, direct product factor, Sylow
256C2×C89—42no—
257C2×C89—46no—
258C2×C89#344411yescharacteristic
259C2×C89#351313yes—
260C2×C89#352113yes—
261#2576C2×C8913yescomplemented
262#2553C2×C8913yescomplemented
263#2522C2×C8913yescomplemented
264#2517C2×C8913yescomplemented
265#2494C2×C8913yescomplemented
266#2480C2×C8913yescomplemented
267#2466C2×C8913yescomplemented
268#2418C2×C8913yes—
269#2334C2×C8913yescomplemented
270C4×#123C2×C8913yescomplemented
271#2273C2×C8911yescharacteristic, complemented
272C89#3541511yescharacteristic, direct product factor, Sylow
273C2×#126C22×C8913yes—
274C2×#126C22×C8911yescharacteristic, complemented
275C2×#123C22×C8913yes—
276C2×#120C22×C8911yescharacteristic
277C2×C4×D4C22×C8913yes—
278C22×C22⋊C4C22×C8913yes—
279C22×C42C22×C8911yescharacteristic, complemented
280#401C22×C8913yescomplemented
281#399C22×C8913yes—
282#399C22×C8911yescharacteristic, complemented
283#398C22×C8913yescomplemented
284#397C22×C8913yes—
285#397C22×C8913yes—
286#397C22×C8913yes—
287#397C22×C8913yes—
288#396C22×C8913yes—
289#396C22×C8911yescharacteristic, complemented
290#394C22×C8913yes—
291#394C22×C8913yes—
292#389C22×C8913yescomplemented
293#389C22×C8913yescomplemented
294#389C22×C8913yescomplemented
295#388C22×C8913yes—
296#388C22×C8913yes—
297#388C22×C8913yes—
298#388C22×C8913yes—
299#388C22×C8913yes—
300#388C22×C8913yescomplemented
301#387C22×C8913yescomplemented
302#387C22×C8913yescomplemented
303#387C22×C8913yescomplemented
304#387C22×C8913yescomplemented
305#387C22×C8913yescomplemented
306#387C22×C8913yescomplemented
307#387C22×C8913yescomplemented
308#386C22×C8913yes—
309#386C22×C8913yes—
310#385C22×C8913yes—
311#385C22×C8913yes—
312#385C22×C8913yes—
313#385C22×C8913yescomplemented
314#382C22×C8913yescomplemented
315#382C22×C8913yescomplemented
316#381C22×C8913yescomplemented
317#380C22×C8913yes—
318#380C22×C8913yes—
319#379C22×C8913yes—
320C4×C4⋊C4C22×C8913yescomplemented
321C4×C4⋊C4C22×C8913yescomplemented
322C4×C22⋊C4C22×C8913yes—
323C4×C22⋊C4C22×C8913yes—
324C4×C22⋊C4C22×C8913yes—
325C4×C22⋊C4C22×C8913yes—
326C4×C22⋊C4C22×C8913yes—
327C4×C22⋊C4C22×C8913yescomplemented
328C25C23×C8911yescharacteristic
329C22×D4C23×C8913yes—
330C23×C4C23×C8913yes—
331C23×C4C23×C8913yes—
332C23×C4C23×C8913yes—
333#126—212no—
334#126—42no—
335#123—212no—
336#120—42no—
337C4×D4—46no—
338C2×C4⋊C4C23×C8913yes—
339C2×C4⋊C4C23×C8913yes—
340C2×C4⋊C4C23×C8913yes—
341C2×C4⋊C4C23×C8913yes—
342C2×C4⋊C4C23×C8913yes—
343C2×C4⋊C4C23×C8913yes—
344C2×C4⋊C4C23×C8913yes—
345C2×C4⋊C4C23×C8913yes—
346C2×C4⋊C4C23×C8913yes—
347C2×C4⋊C4C23×C8913yes—
348C2×C4⋊C4C23×C8913yes—
349C2×C22⋊C4—212no—
350C2×C22⋊C4C23×C8913yes—
351C2×C22⋊C4C23×C8913yes—
352C2×C22⋊C4C23×C8913yes—
353C2×C22⋊C4C23×C8913yes—
354C2×C22⋊C4C23×C8913yes—
355C2×C22⋊C4C23×C8913yes—
356C2×C22⋊C4C23×C8913yes—
357C2×C22⋊C4C23×C8913yes—
358C2×C22⋊C4C23×C8913yes—
359C2×C22⋊C4C23×C8913yes—
360C2×C22⋊C4C23×C8913yes—
361C2×C22⋊C4C23×C8913yes—
362C2×C22⋊C4C23×C8913yes—
363C2×C22⋊C4C23×C8913yes—
364C2×C42—42no—
365C2×C42C23×C8913yes—
366C2×C42C23×C8913yes—
367C2×C42C23×C8911yescharacteristic
368C2×C42C23×C8913yes—
369C2×C42C23×C8911yescharacteristic
370C2×C42C23×C8913yes—
371C2×C42C23×C8913yes—
372#95C23×C8913yes—
373#95C23×C8913yes—
374#95C23×C8911yescharacteristic
375#95C23×C8911yescharacteristic
376#95C23×C8913yes—
377#95C23×C8913yes—
378#95C23×C8913yes—
379#95C23×C8913yes—
380#95C23×C8913yes—
381#95C23×C8913yes—
382#95C23×C8913yes—
383#95C23×C8913yes—
384#95C23×C8913yes—
385#95C23×C8913yes—
386#95C23×C8913yes—
387#95C23×C8913yes—
388#95C23×C8913yes—
389#95C23×C8913yes—
390C24—26no—
391C24—41no—
392C24—43no—
393C24C24×C8913yes—
394C24C24×C8911yescharacteristic
395C2×D4—26no—
396C2×D4—26no—
397C2×D4—43no—
398C22×C4—26no—
399C22×C4—26no—
400C22×C4—26no—
401C22×C4—26no—
402C22×C4—43no—
403C22×C4—43no—
404C22×C4—43no—
405C22×C4—43no—
406C22×C4—43no—
407C22×C4C24×C8913yes—
408C22×C4C24×C8913yes—
409C22×C4C24×C8913yes—
410C22×C4C24×C8913yes—
411C22×C4C24×C8913yes—
412C22×C4C24×C8913yes—
413C22×C4C24×C8913yes—
414C22×C4C24×C8913yes—
415C22×C4C24×C8913yes—
416C4⋊C4—26no—
417C4⋊C4—26no—
418C4⋊C4—26no—
419C4⋊C4—26no—
420C4⋊C4—26no—
421C4⋊C4—43no—
422C4⋊C4—43no—
423C4⋊C4—43no—
424C4⋊C4—43no—
425C4⋊C4—43no—
426C4⋊C4—43no—
427C22⋊C4—26no—
428C22⋊C4—26no—
429C22⋊C4—26no—
430C22⋊C4—26no—
431C22⋊C4—26no—
432C22⋊C4—26no—
433C22⋊C4—26no—
434C22⋊C4—26no—
435C22⋊C4—43no—
436C22⋊C4—43no—
437C22⋊C4—43no—
438C22⋊C4—43no—
439C22⋊C4—43no—
440C22⋊C4#41×C89112yes—
441C42—26no—
442C42—41no—
443C42—43no—
444C42—43no—
445C42—41no—
446C42—43no—
447C42—43no—
448C23—28no—
449C23—23no—
450C23—23no—
451C23—23no—
452C23—23no—
453C23—23no—
454C23—23no—
455C23—23no—
456C23—23no—
457C23—86no—
458C23—86no—
459C23C2×#41×C8916yes—
460C23H(5,𝔽2)×C8912yes—
461C23C25×C8911yescharacteristic
462D4—412no—
463C2×C4—23no—
464C2×C4—23no—
465C2×C4—23no—
466C2×C4—23no—
467C2×C4—23no—
468C2×C4—23no—
469C2×C4—23no—
470C2×C4—23no—
471C2×C4—23no—
472C2×C4—23no—
473C2×C4—23no—
474C2×C4—23no—
475C2×C4—23no—
476C2×C4—23no—
477C2×C4—23no—
478C2×C4—23no—
479C2×C4—23no—
480C2×C4—23no—
481C2×C4—23no—
482C2×C4—23no—
483C2×C4—23no—
484C2×C4—23no—
485C2×C4—23no—
486C2×C4—23no—
487C2×C4—23no—
488C2×C4—412no—
489C2×C4—86no—
490C2×C4—86no—
491C2×C4C2×#41×C8916yes—
492C2×C4C2×#41×C8916yes—
493C22—26no—
494C22—26no—
495C22—26no—
496C22—26no—
497C22—41no—
498C22—43no—
499C22—43no—
500C22—43no—
501C22—43no—
502C22—43no—
503C22—88no—
504C22C22×#41×C8913yes—
505C22C2×H(5,𝔽2)×C8911yescharacteristic
506C22#585×C8913yes—
507C4—26no—
508C4—26no—
509C4—43no—
510C4—43no—
511C4—43no—
512C4—43no—
513C4—43no—
514C4—43no—
515C4—43no—
516C2—42no—
517C2—46no—
518C2C89×#344411yescharacteristic
519C2C89×#351313yes—
520C2C89×#352113yes—
subgroupG/H
center, socleC23×C89C25
commutator, Frattini, cosocleC23C25×C89
prime radicalC2×#120×C89C22
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×#2553C2semidirect
3. C89×#2522C2semidirect
4. C89×#2517C2semidirect
5. C89×#2494C2semidirect
6. C89×#2480C2semidirect
7. C89×#2466C2semidirect
9. C89×#2334C2semidirect
10. C4×#123×C89C2semidirect
11. C89×#2273C2semidirect
13. C2×#126×C89C22semidirect
18. C22×C42×C89C22semidirect
19. #401×C89C22semidirect
21. #399×C89C22semidirect
22. #398×C89C22semidirect
28. #396×C89C22semidirect
31. #389×C89C22semidirect
32. #389×C89C22semidirect
33. #389×C89C22semidirect
39. #388×C89C22semidirect
40. #387×C89C22semidirect
41. #387×C89C22semidirect
42. #387×C89C22semidirect
43. #387×C89C22semidirect
44. #387×C89C22semidirect
45. #387×C89C22semidirect
46. #387×C89C22semidirect
52. #385×C89C22semidirect
53. #382×C89C22semidirect
54. #382×C89C22semidirect
55. #381×C89C22semidirect
59. C4×C4⋊C4×C89C22semidirect
60. C4×C4⋊C4×C89C22semidirect
66. C4×C22⋊C4×C89C22semidirect
255. #35415C89direct (N × H)
261. #2576C2×C89semidirect
262. #2553C2×C89semidirect
263. #2522C2×C89semidirect
264. #2517C2×C89semidirect
265. #2494C2×C89semidirect
266. #2480C2×C89semidirect
267. #2466C2×C89semidirect
269. #2334C2×C89semidirect
270. C4×#123C2×C89semidirect
271. #2273C2×C89semidirect
272. C89#35415direct (N × H)
274. C2×#126C22×C89semidirect
279. C22×C42C22×C89semidirect
280. #401C22×C89semidirect
282. #399C22×C89semidirect
283. #398C22×C89semidirect
289. #396C22×C89semidirect
292. #389C22×C89semidirect
293. #389C22×C89semidirect
294. #389C22×C89semidirect
300. #388C22×C89semidirect
301. #387C22×C89semidirect
302. #387C22×C89semidirect
303. #387C22×C89semidirect
304. #387C22×C89semidirect
305. #387C22×C89semidirect
306. #387C22×C89semidirect
307. #387C22×C89semidirect
313. #385C22×C89semidirect
314. #382C22×C89semidirect
315. #382C22×C89semidirect
316. #381C22×C89semidirect
320. C4×C4⋊C4C22×C89semidirect
321. C4×C4⋊C4C22×C89semidirect
327. C4×C22⋊C4C22×C89semidirect

Maximal subgroups

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

subgroupindex
C89×#25762
C89×#25532
C89×#25222
C89×#25172
C89×#24942
C89×#24802
C89×#24662
C89×#24182
C89×#23342
C4×#123×C892
C89×#22732
#3541589