Finite Group Databasebeta

← C89×#60338 #36438186

Subgroups — C89×#60338 #36438186

Order 22784.

1628 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#3531C211yesmaximal, characteristic
2C89×#2995C212yesmaximal, complemented
3C89×#2994C214yesmaximal, complemented
4C89×#2992C214yesmaximal, complemented
5C89×#2987C211yesmaximal, characteristic, complemented
6C89×#2985C214yesmaximal, complemented
7C89×#2969C214yesmaximal, complemented
8C89×#2967C214yesmaximal, complemented
9C89×#2966C212yesmaximal, complemented
10C89×#2962C214yesmaximal, complemented
11C89×#2960C211yesmaximal, characteristic, complemented
12#564×C89C411yescharacteristic
13#563×C89C414yes—
14#563×C89C412yes—
15#563×C89C414yes—
16#562×C89C414yes—
17#562×C89C411yescharacteristic
18#533×C89C2211yescharacteristic
19#533×C89C2214yes—
20#529×C89C2212yes—
21#529×C89C2214yes—
22#529×C89C2214yes—
23#446×C89C2214yes—
24#446×C89C2212yes—
25#446×C89C2214yes—
26#446×C89C2214yes—
27#446×C89C2214yes—
28#446×C89C2214yes—
29#446×C89C2214yes—
30#446×C89C2214yes—
31C8×Q8×C89C2214yes—
32C8×Q8×C89C2214yes—
33C8×Q8×C89C2212yes—
34#436×C89C2212yes—
35#436×C89C2214yes—
36#436×C89C2214yes—
37#435×C89C2214yes—
38#435×C89C2214yes—
39#435×C89C2214yes—
40#435×C89C2214yes—
41#435×C89C2214yes—
42C8×D4×C89C2212yes—
43C8×D4×C89C2214yes—
44C8×D4×C89C2214yes—
45#433×C89C2214yes—
46#433×C89C2214yes—
47#433×C89C2214yes—
48#433×C89C2214yes—
49#433×C89C2214yes—
50#432×C89C2212yes—
51#432×C89C2214yes—
52#431×C89C2214yes—
53#424×C89C2212yes—
54#424×C89C2214yes—
55#424×C89C2214yes—
56#423×C89C2211yescharacteristic
57C2×C4⋊C8×C89C2214yes—
58#405×C89C2212yes—
59#405×C89C2214yes—
60#405×C89C2214yes—
61C4×M16×C89C2214yes—
62C2×C4×C8×C89C2211yescharacteristic
63C2×M16×C89C2314yes—
64C2×M16×C89C2314yes—
65C2×M16×C89C2312yes—
66C22×C8×C89C2311yescharacteristic
67C22×C8×C89C2311yescharacteristic
68C22×C8×C89C2314yes—
69C22×C8×C89C2314yes—
70C4⋊Q8×C89C2×C412yes—
71C4⋊Q8×C89C2×C414yes—
72C4⋊Q8×C89C2×C414yes—
73#126×C89C2×C414yes—
74#126×C89C2×C414yes—
75#126×C89C2×C414yes—
76#126×C89C2×C414yes—
77#126×C89C2×C414yes—
78#125×C89C2×C412yes—
79#125×C89C2×C414yes—
80#125×C89C2×C414yes—
81#125×C89C2×C411yescharacteristic
82#125×C89C2×C414yes—
83#125×C89C2×C411yescharacteristic
84#125×C89C2×C414yes—
85#124×C89C2×C412yes—
86#124×C89C2×C414yes—
87#124×C89C2×C414yes—
88#123×C89C2×C414yes—
89#123×C89C2×C414yes—
90#123×C89C2×C414yes—
91#123×C89C2×C414yes—
92#123×C89C2×C414yes—
93C22⋊Q8×C89C2×C412yes—
94C22⋊Q8×C89C2×C414yes—
95C22⋊Q8×C89C2×C414yes—
96C22⋊Q8×C89C2×C414yes—
97C22⋊Q8×C89C2×C414yes—
98C22⋊Q8×C89C2×C414yes—
99C22⋊Q8×C89C2×C414yes—
100C22⋊Q8×C89C2×C414yes—
101#121×C89C2×C412yes—
102#121×C89C2×C414yes—
103#121×C89C2×C414yes—
104C4×Q8×C89C2314yes—
105C4×Q8×C89C2314yes—
106C4×Q8×C89C2312yes—
107C4×D4×C89C2312yes—
108C4×D4×C89C2314yes—
109C4×D4×C89C2314yes—
110#117×C89C2312yes—
111#117×C89C2314yes—
112#117×C89C2314yes—
113C2×C42×C89C2311yescharacteristic
114C2×C42×C89C2314yes—
115C4⋊C8×C89C2314yes—
116C4⋊C8×C89C2314yes—
117C4⋊C8×C89C2314yes—
118C4⋊C8×C89C2314yes—
119C4⋊C8×C89C2314yes—
120C4⋊C8×C89C2314yes—
121C4⋊C8×C89C2312yes—
122C4⋊C8×C89C2314yes—
123C4⋊C8×C89C2314yes—
124C4⋊C8×C89C2314yes—
125C4⋊C8×C89C2312yes—
126C22⋊C8×C89C2314yes—
127C22⋊C8×C89C2314yes—
128C22⋊C8×C89C2314yes—
129C22⋊C8×C89C2314yes—
130C22⋊C8×C89C2314yes—
131#97×C89C2314yes—
132#97×C89C2314yes—
133#97×C89C2314yes—
134#97×C89C2314yes—
135#97×C89C2314yes—
136C4×C8×C89C2312yes—
137C4×C8×C89C2312yes—
138C4×C8×C89C2314yes—
139C4×C8×C89C2314yes—
140C4×C8×C89C2314yes—
141C4×C8×C89C2314yes—
142C2×Q8×C89C22×C414yes—
143C2×Q8×C89C22×C414yes—
144C2×Q8×C89C22×C412yes—
145C2×D4×C89C22×C412yes—
146C2×D4×C89C22×C414yes—
147C2×D4×C89C22×C414yes—
148C22×C4×C89C22×C412yes—
149C22×C4×C89C22×C414yes—
150C22×C4×C89C22×C414yes—
151C22×C4×C89C2411yescharacteristic
152C22×C4×C89C2414yes—
153M16×C89—28no—
154M16×C89—44no—
155M16×C89—42no—
156C2×C8×C89—22no—
157C2×C8×C89—41no—
158C2×C8×C89—44no—
159C2×C8×C89—44no—
160C2×C8×C89C2412yes—
161C2×C8×C89C2414yes—
162C2×C8×C89C2411yescharacteristic
163C2×C8×C89C2414yes—
164C2×C8×C89C2414yes—
165C2×C8×C89C2411yescharacteristic
166C4⋊C4×C89C22×C412yes—
167C4⋊C4×C89C22×C414yes—
168C4⋊C4×C89C22×C414yes—
169C4⋊C4×C89C22×C412yes—
170C4⋊C4×C89C22×C414yes—
171C4⋊C4×C89C22×C414yes—
172C4⋊C4×C89C22×C412yes—
173C4⋊C4×C89C22×C414yes—
174C4⋊C4×C89C22×C414yes—
175C4⋊C4×C89C22×C414yes—
176C4⋊C4×C89C22×C414yes—
177C4⋊C4×C89C22×C412yes—
178C4⋊C4×C89C22×C414yes—
179C4⋊C4×C89C22×C414yes—
180C4⋊C4×C89C22×C414yes—
181C4⋊C4×C89C22×C414yes—
182C4⋊C4×C89C22×C414yes—
183C22⋊C4×C89C22×C414yes—
184C22⋊C4×C89C22×C414yes—
185C22⋊C4×C89C22×C414yes—
186C22⋊C4×C89C22×C414yes—
187C22⋊C4×C89C22×C414yes—
188C22⋊C4×C89C22×C414yes—
189C22⋊C4×C89C22×C414yes—
190C22⋊C4×C89C22×C414yes—
191C22⋊C4×C89C22×C414yes—
192C22⋊C4×C89C22×C414yes—
193C42×C89C22×C411yescharacteristic
194C42×C89C22×C414yes—
195C42×C89C22×C411yescharacteristic
196C42×C89C22×C414yes—
197C42×C89C2412yes—
198C42×C89C2414yes—
199C42×C89C2414yes—
200C23×C89C23×C411yescharacteristic
201C23×C89C23×C414yes—
202Q8×C89—44no—
203Q8×C89—44no—
204Q8×C89—42no—
205D4×C89—42no—
206D4×C89—44no—
207D4×C89—44no—
208C2×C4×C89—21no—
209C2×C4×C89—24no—
210C2×C4×C89—24no—
211C2×C4×C89—21no—
212C2×C4×C89—42no—
213C2×C4×C89—44no—
214C2×C4×C89—44no—
215C2×C4×C89C23×C411yescharacteristic
216C2×C4×C89C23×C414yes—
217C2×C4×C89C23×C412yes—
218C2×C4×C89C23×C414yes—
219C2×C4×C89C23×C414yes—
220C2×C4×C89C23×C412yes—
221C2×C4×C89C23×C414yes—
222C2×C4×C89C23×C414yes—
223C2×C4×C89C2511yescharacteristic
224C8×C89—21no—
225C8×C89—24no—
226C8×C89—22no—
227C8×C89—24no—
228C8×C89—24no—
229C8×C892−1+412yes—
230C22×C89—21no—
231C22×C89—24no—
232C22×C89—22no—
233C22×C89—24no—
234C22×C89—24no—
235C22×C89C24×C411yescharacteristic
236C4×C89—21no—
237C4×C89—24no—
238C4×C89—22no—
239C4×C89—24no—
240C4×C89—24no—
241C4×C89—22no—
242C4×C89—24no—
243C4×C89—24no—
244C4×C89C2×2−1+411yescharacteristic
245C4×C89#58511yescharacteristic
246#60338C8911yesmaximal, characteristic, direct product factor, Sylow
247C2×C89—41no—
248C2×C89—44no—
249C2×C89C4×2−1+411yescharacteristic
250C2×C89#357312yes—
251#3531C2×C8911yescharacteristic
252#2995C2×C8912yescomplemented
253#2994C2×C8914yescomplemented
254#2992C2×C8914yescomplemented
255#2987C2×C8911yescharacteristic, complemented
256#2985C2×C8914yescomplemented
257#2969C2×C8914yescomplemented
258#2967C2×C8914yescomplemented
259#2966C2×C8912yescomplemented
260#2962C2×C8914yescomplemented
261#2960C2×C8911yescharacteristic, complemented
262C89#6033811yescharacteristic, direct product factor, Sylow
263#564C4×C8911yescharacteristic
264#563C4×C8914yes—
265#563C4×C8912yes—
266#563C4×C8914yes—
267#562C4×C8914yes—
268#562C4×C8911yescharacteristic
269#533C22×C8911yescharacteristic
270#533C22×C8914yes—
271#529C22×C8912yes—
272#529C22×C8914yes—
273#529C22×C8914yes—
274#446C22×C8914yes—
275#446C22×C8912yes—
276#446C22×C8914yes—
277#446C22×C8914yes—
278#446C22×C8914yes—
279#446C22×C8914yes—
280#446C22×C8914yes—
281#446C22×C8914yes—
282C8×Q8C22×C8914yes—
283C8×Q8C22×C8914yes—
284C8×Q8C22×C8912yes—
285#436C22×C8912yes—
286#436C22×C8914yes—
287#436C22×C8914yes—
288#435C22×C8914yes—
289#435C22×C8914yes—
290#435C22×C8914yes—
291#435C22×C8914yes—
292#435C22×C8914yes—
293C8×D4C22×C8912yes—
294C8×D4C22×C8914yes—
295C8×D4C22×C8914yes—
296#433C22×C8914yes—
297#433C22×C8914yes—
298#433C22×C8914yes—
299#433C22×C8914yes—
300#433C22×C8914yes—
301#432C22×C8912yes—
302#432C22×C8914yes—
303#431C22×C8914yes—
304#424C22×C8912yes—
305#424C22×C8914yes—
306#424C22×C8914yes—
307#423C22×C8911yescharacteristic
308C2×C4⋊C8C22×C8914yes—
309#405C22×C8912yes—
310#405C22×C8914yes—
311#405C22×C8914yes—
312C4×M16C22×C8914yes—
313C2×C4×C8C22×C8911yescharacteristic
314C2×M16C23×C8914yes—
315C2×M16C23×C8914yes—
316C2×M16C23×C8912yes—
317C22×C8C23×C8911yescharacteristic
318C22×C8C23×C8911yescharacteristic
319C22×C8C23×C8914yes—
320C22×C8C23×C8914yes—
321C4⋊Q8C2×C4×C8912yes—
322C4⋊Q8C2×C4×C8914yes—
323C4⋊Q8C2×C4×C8914yes—
324#126C2×C4×C8914yes—
325#126C2×C4×C8914yes—
326#126C2×C4×C8914yes—
327#126C2×C4×C8914yes—
328#126C2×C4×C8914yes—
329#125C2×C4×C8912yes—
330#125C2×C4×C8914yes—
331#125C2×C4×C8914yes—
332#125C2×C4×C8911yescharacteristic
333#125C2×C4×C8914yes—
334#125C2×C4×C8911yescharacteristic
335#125C2×C4×C8914yes—
336#124C2×C4×C8912yes—
337#124C2×C4×C8914yes—
338#124C2×C4×C8914yes—
339#123C2×C4×C8914yes—
340#123C2×C4×C8914yes—
341#123C2×C4×C8914yes—
342#123C2×C4×C8914yes—
343#123C2×C4×C8914yes—
344C22⋊Q8C2×C4×C8912yes—
345C22⋊Q8C2×C4×C8914yes—
346C22⋊Q8C2×C4×C8914yes—
347C22⋊Q8C2×C4×C8914yes—
348C22⋊Q8C2×C4×C8914yes—
349C22⋊Q8C2×C4×C8914yes—
350C22⋊Q8C2×C4×C8914yes—
351C22⋊Q8C2×C4×C8914yes—
352#121C2×C4×C8912yes—
353#121C2×C4×C8914yes—
354#121C2×C4×C8914yes—
355C4×Q8C23×C8912yes—
356C4×Q8C23×C8914yes—
357C4×Q8C23×C8914yes—
358C4×D4C23×C8914yes—
359C4×D4C23×C8912yes—
360C4×D4C23×C8914yes—
361#117C23×C8914yes—
362#117C23×C8912yes—
363#117C23×C8914yes—
364C2×C42C23×C8911yescharacteristic
365C2×C42C23×C8914yes—
366C4⋊C8C23×C8914yes—
367C4⋊C8C23×C8914yes—
368C4⋊C8C23×C8914yes—
369C4⋊C8C23×C8914yes—
370C4⋊C8C23×C8914yes—
371C4⋊C8C23×C8914yes—
372C4⋊C8C23×C8912yes—
373C4⋊C8C23×C8914yes—
374C4⋊C8C23×C8914yes—
375C4⋊C8C23×C8914yes—
376C4⋊C8C23×C8912yes—
377C22⋊C8C23×C8914yes—
378C22⋊C8C23×C8914yes—
379C22⋊C8C23×C8914yes—
380C22⋊C8C23×C8914yes—
381C22⋊C8C23×C8914yes—
382#97C23×C8914yes—
383#97C23×C8914yes—
384#97C23×C8914yes—
385#97C23×C8914yes—
386#97C23×C8914yes—
387C4×C8C23×C8912yes—
388C4×C8C23×C8912yes—
389C4×C8C23×C8914yes—
390C4×C8C23×C8914yes—
391C4×C8C23×C8914yes—
392C4×C8C23×C8914yes—
393C2×Q8C22×C4×C8912yes—
394C2×Q8C22×C4×C8914yes—
395C2×Q8C22×C4×C8914yes—
396C2×D4C22×C4×C8914yes—
397C2×D4C22×C4×C8912yes—
398C2×D4C22×C4×C8914yes—
399C22×C4C22×C4×C8912yes—
400C22×C4C22×C4×C8914yes—
401C22×C4C22×C4×C8914yes—
402C22×C4C24×C8911yescharacteristic
403C22×C4C24×C8914yes—
404M16—28no—
405M16—44no—
406M16—42no—
407C2×C8—22no—
408C2×C8—41no—
409C2×C8—44no—
410C2×C8—44no—
411C2×C8C24×C8912yes—
412C2×C8C24×C8914yes—
413C2×C8C24×C8911yescharacteristic
414C2×C8C24×C8914yes—
415C2×C8C24×C8914yes—
416C2×C8C24×C8911yescharacteristic
417C4⋊C4C22×C4×C8914yes—
418C4⋊C4C22×C4×C8912yes—
419C4⋊C4C22×C4×C8914yes—
420C4⋊C4C22×C4×C8914yes—
421C4⋊C4C22×C4×C8912yes—
422C4⋊C4C22×C4×C8914yes—
423C4⋊C4C22×C4×C8914yes—
424C4⋊C4C22×C4×C8912yes—
425C4⋊C4C22×C4×C8914yes—
426C4⋊C4C22×C4×C8912yes—
427C4⋊C4C22×C4×C8914yes—
428C4⋊C4C22×C4×C8914yes—
429C4⋊C4C22×C4×C8914yes—
430C4⋊C4C22×C4×C8914yes—
431C4⋊C4C22×C4×C8914yes—
432C4⋊C4C22×C4×C8914yes—
433C4⋊C4C22×C4×C8914yes—
434C22⋊C4C22×C4×C8914yes—
435C22⋊C4C22×C4×C8914yes—
436C22⋊C4C22×C4×C8914yes—
437C22⋊C4C22×C4×C8914yes—
438C22⋊C4C22×C4×C8914yes—
439C22⋊C4C22×C4×C8914yes—
440C22⋊C4C22×C4×C8914yes—
441C22⋊C4C22×C4×C8914yes—
442C22⋊C4C22×C4×C8914yes—
443C22⋊C4C22×C4×C8914yes—
444C42C22×C4×C8911yescharacteristic
445C42C22×C4×C8914yes—
446C42C22×C4×C8911yescharacteristic
447C42C22×C4×C8914yes—
448C42C24×C8912yes—
449C42C24×C8914yes—
450C42C24×C8914yes—
451C23C23×C4×C8911yescharacteristic
452C23C23×C4×C8914yes—
453Q8—42no—
454Q8—44no—
455Q8—44no—
456D4—44no—
457D4—42no—
458D4—44no—
459C2×C4—21no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—21no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4C23×C4×C8911yescharacteristic
467C2×C4C23×C4×C8914yes—
468C2×C4C23×C4×C8912yes—
469C2×C4C23×C4×C8914yes—
470C2×C4C23×C4×C8914yes—
471C2×C4C23×C4×C8912yes—
472C2×C4C23×C4×C8914yes—
473C2×C4C23×C4×C8914yes—
474C2×C4C25×C8911yescharacteristic
475C8—21no—
476C8—24no—
477C8—22no—
478C8—24no—
479C8—24no—
480C82−1+4×C8912yes—
481C22—21no—
482C22—24no—
483C22—22no—
484C22—24no—
485C22—24no—
486C22C24×C4×C8911yescharacteristic
487C4—21no—
488C4—24no—
489C4—22no—
490C4—24no—
491C4—24no—
492C4—22no—
493C4—24no—
494C4—24no—
495C4C2×2−1+4×C8911yescharacteristic
496C4#585×C8911yescharacteristic
497C2—41no—
498C2—44no—
499C2C4×2−1+4×C8911yescharacteristic
500C2C89×#357312yes—
subgroupG/H
centerC2×C4×C89C25
commutatorC22C24×C4×C89
Frattini, cosocleC2×C4C25×C89
socleC22×C89C24×C4
prime radicalC89×#3531C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C89 ◁ G

Complements

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

NH ≅ G/Ntype
2. C89×#2995C2semidirect
3. C89×#2994C2semidirect
4. C89×#2992C2semidirect
5. C89×#2987C2semidirect
6. C89×#2985C2semidirect
7. C89×#2969C2semidirect
8. C89×#2967C2semidirect
9. C89×#2966C2semidirect
10. C89×#2962C2semidirect
11. C89×#2960C2semidirect
246. #60338C89direct (N × H)
252. #2995C2×C89semidirect
253. #2994C2×C89semidirect
254. #2992C2×C89semidirect
255. #2987C2×C89semidirect
256. #2985C2×C89semidirect
257. #2969C2×C89semidirect
258. #2967C2×C89semidirect
259. #2966C2×C89semidirect
260. #2962C2×C89semidirect
261. #2960C2×C89semidirect
262. C89#60338direct (N × H)

Maximal subgroups

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

subgroupindex
C89×#35312
C89×#29952
C89×#29942
C89×#29922
C89×#29872
C89×#29852
C89×#29692
C89×#29672
C89×#29662
C89×#29622
C89×#29602
#6033889