Finite Group Databasebeta

← C89×#19744 #36397592

Subgroups — C89×#19744 #36397592

Order 22784.

670 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#3231C211yesmaximal, characteristic, complemented
2C89×#3230C211yesmaximal, characteristic, complemented
3C89×#3228C211yesmaximal, characteristic
4C89×#3226C211yesmaximal, characteristic
5C89×#3210C211yesmaximal, characteristic
6C89×#3209C211yesmaximal, characteristic, complemented
7C89×#1723C211yesmaximal, characteristic, complemented
8C89×#1723C211yesmaximal, characteristic, complemented
9C89×#1722C211yesmaximal, characteristic, complemented
10C89×#1720C211yesmaximal, characteristic, complemented
11C89×#1719C211yesmaximal, characteristic, complemented
12C89×#1718C211yesmaximal, characteristic, complemented
13C89×#1713C211yesmaximal, characteristic, complemented
14C89×#1711C211yesmaximal, characteristic, complemented
15C89×#1454C211yesmaximal, characteristic, complemented
16#533×C89C2211yescharacteristic
17#533×C89C2211yescharacteristic
18#532×C89C2211yescharacteristic
19C2×C4⋊Q8×C89C2211yescharacteristic
20#500×C89C2211yescharacteristic
21#500×C89C2211yescharacteristic
22#500×C89C2211yescharacteristic
23#500×C89C2211yescharacteristic
24#500×C89C2211yescharacteristic
25#500×C89C2211yescharacteristic
26#498×C89C2211yescharacteristic
27#498×C89C2211yescharacteristic
28#494×C89C2211yescharacteristic
29#494×C89C2211yescharacteristic
30#494×C89C2211yescharacteristic
31#492×C89C2211yescharacteristic
32#487×C89C2211yescharacteristic
33#487×C89C2211yescharacteristic
34#487×C89C2211yescharacteristic
35#487×C89C2211yescharacteristic
36#487×C89C2211yescharacteristic
37#487×C89C2211yescharacteristic
38#487×C89C2211yescharacteristic
39#486×C89C2211yescharacteristic
40#486×C89C2211yescharacteristic
41#486×C89C2211yescharacteristic
42#486×C89C2211yescharacteristic
43#486×C89C2211yescharacteristic
44#485×C89—22no—
45#484×C89—22no—
46#482×C89—22no—
47#480×C89—22no—
48#479×C89—22no—
49#478×C89—22no—
50#477×C89—22no—
51#474×C89—22no—
52#464×C89—22no—
53#462×C89—22no—
54#452×C89—22no—
55#451×C89—22no—
56#431×C89C2211yescharacteristic
57#431×C89C2211yescharacteristic
58#431×C89C2211yescharacteristic
59#322×C89C2211yescharacteristic
60#322×C89C2211yescharacteristic
61#322×C89C2211yescharacteristic
62C82×C89C2211yescharacteristic
63C2×#41×C89—21no—
64C22×Q8×C89—21no—
65C2×Q16×C89—22no—
66C2×Q16×C89—22no—
67C2×Q16×C89—22no—
68C2×SD16×C89—22no—
69C4⋊Q8×C89—21no—
70C4⋊Q8×C89—21no—
71C4⋊Q8×C89—21no—
72C4⋊Q8×C89C2311yescharacteristic
73C4⋊Q8×C89C2311yescharacteristic
74C4⋊Q8×C89C2311yescharacteristic
75C4⋊Q8×C89C2311yescharacteristic
76C4⋊Q8×C89C2311yescharacteristic
77C4⋊Q8×C89C2311yescharacteristic
78C4⋊Q8×C89C2311yescharacteristic
79#127×C89C2311yescharacteristic
80#125×C89—21no—
81#125×C89—21no—
82#124×C89—21no—
83C22⋊Q8×C89—21no—
84C22⋊Q8×C89—21no—
85C22⋊Q8×C89—21no—
86C22⋊Q8×C89—21no—
87#121×C89—21no—
88#121×C89—21no—
89C4×Q8×C89—21no—
90C4×Q8×C89—21no—
91C4×Q8×C89—21no—
92C4×Q8×C89—21no—
93C4×D4×C89—21no—
94C4×D4×C89—21no—
95#117×C89—21no—
96#117×C89—21no—
97C2×C4⋊C4×C89—21no—
98C2×C4⋊C4×C89—21no—
99C2×C42×C89C2311yescharacteristic
100#107×C89—22no—
101#107×C89—22no—
102#107×C89—22no—
103#107×C89—22no—
104#107×C89—22no—
105#107×C89—22no—
106#106×C89—22no—
107#106×C89—22no—
108C4⋊C8×C89D412yes—
109C4⋊C8×C89D412yes—
110C4⋊C8×C89D412yes—
111#103×C89—22no—
112#103×C89—22no—
113#103×C89—22no—
114#103×C89—22no—
115#103×C89—22no—
116#103×C89—22no—
117#103×C89—22no—
118#102×C89—22no—
119#102×C89—22no—
120#102×C89—22no—
121#102×C89—22no—
122#102×C89—22no—
123C22⋊C8×C89D412yes—
124C22⋊C8×C89D412yes—
125C22⋊C8×C89D412yes—
126C4×C8×C89C2311yescharacteristic
127C4×C8×C89C2311yescharacteristic
128C4×C8×C89C2311yescharacteristic
129C4×C8×C89C2311yescharacteristic
130C4×C8×C89C2311yescharacteristic
131C4×C8×C89C2311yescharacteristic
132#41×C89—82no—
133C2×Q8×C89—21no—
134C2×Q8×C89—21no—
135C2×Q8×C89—21no—
136C2×Q8×C89—21no—
137C2×Q8×C89—21no—
138C2×Q8×C89—21no—
139C2×Q8×C89—21no—
140C2×Q8×C89—41no—
141C2×Q8×C89—82no—
142C2×D4×C89—21no—
143C2×D4×C89—21no—
144C2×D4×C89—21no—
145C2×D4×C89—41no—
146C22×C4×C89—41no—
147C22×C4×C89—41no—
148C22×C4×C89C2×D411yescharacteristic
149C22×C4×C89C2×D411yescharacteristic
150C22×C4×C89C2×D411yescharacteristic
151Q16×C89—82no—
152Q16×C89—82no—
153Q16×C89—82no—
154SD16×C89—82no—
155C2×C8×C89C2×D412yes—
156C2×C8×C89C2×D412yes—
157C2×C8×C89C2×D412yes—
158C2×C8×C89C2×D412yes—
159C2×C8×C89C2×D412yes—
160C2×C8×C89C2×D412yes—
161C4⋊C4×C89—21no—
162C4⋊C4×C89—21no—
163C4⋊C4×C89—21no—
164C4⋊C4×C89—21no—
165C4⋊C4×C89—21no—
166C4⋊C4×C89—21no—
167C4⋊C4×C89—21no—
168C4⋊C4×C89—21no—
169C4⋊C4×C89—21no—
170C4⋊C4×C89—21no—
171C4⋊C4×C89—21no—
172C4⋊C4×C89—21no—
173C4⋊C4×C89—21no—
174C4⋊C4×C89—21no—
175C4⋊C4×C89—41no—
176C4⋊C4×C89—41no—
177C4⋊C4×C89—41no—
178C4⋊C4×C89—41no—
179C4⋊C4×C89—41no—
180C4⋊C4×C89—41no—
181C22⋊C4×C89—41no—
182C22⋊C4×C89—41no—
183C22⋊C4×C89—41no—
184C22⋊C4×C89—41no—
185C42×C89—41no—
186C42×C89—41no—
187C42×C89C2×D411yescharacteristic
188C42×C89C2×D411yescharacteristic
189C42×C89C2×D411yescharacteristic
190C42×C89C2411yescharacteristic
191C23×C89—41no—
192C23×C89#12711yescharacteristic
193Q8×C89—42no—
194Q8×C89—42no—
195Q8×C89—42no—
196Q8×C89—42no—
197Q8×C89—42no—
198Q8×C89—42no—
199Q8×C89—42no—
200Q8×C89—82no—
201D4×C89—42no—
202D4×C89—42no—
203D4×C89—42no—
204D4×C89—82no—
205C2×C4×C89—22no—
206C2×C4×C89—22no—
207C2×C4×C89—22no—
208C2×C4×C89—41no—
209C2×C4×C89—41no—
210C2×C4×C89—41no—
211C2×C4×C89—41no—
212C2×C4×C89—41no—
213C2×C4×C89—41no—
214C2×C4×C89—41no—
215C2×C4×C89—82no—
216C2×C4×C89—82no—
217C2×C4×C89#12711yescharacteristic
218C2×C4×C89#12711yescharacteristic
219C2×C4×C89#12711yescharacteristic
220C2×C4×C89C22×D411yescharacteristic
221C2×C4×C89C22×D411yescharacteristic
222C2×C4×C89C22×D411yescharacteristic
223C8×C89—22no—
224C8×C89—22no—
225C8×C89—22no—
226C8×C89—22no—
227C8×C89—22no—
228C8×C89—22no—
229C22×C89—21no—
230C22×C89—21no—
231C22×C89—21no—
232C22×C89—81no—
233C22×C89—81no—
234C22×C89—81no—
235C22×C89C2×#12711yescharacteristic
236C4×C89—21no—
237C4×C89—21no—
238C4×C89—21no—
239C4×C89—81no—
240C4×C89—81no—
241C4×C89—81no—
242C4×C89—81no—
243C4×C89—81no—
244C4×C89—81no—
245C4×C89—81no—
246C4×C89#57612yes—
247C4×C89#57812yes—
248C4×C89#57812yes—
249#19744C8911yesmaximal, characteristic, direct product factor, Sylow
250C2×C89—41no—
251C2×C89—161no—
252C2×C89#315511yescharacteristic
253C2×C89#315611yescharacteristic
254C2×C89#315611yescharacteristic
255#3231C2×C8911yescharacteristic, complemented
256#3230C2×C8911yescharacteristic, complemented
257#3228C2×C8911yescharacteristic
258#3226C2×C8911yescharacteristic
259#3210C2×C8911yescharacteristic
260#3209C2×C8911yescharacteristic, complemented
261#1723C2×C8911yescharacteristic, complemented
262#1723C2×C8911yescharacteristic, complemented
263#1722C2×C8911yescharacteristic, complemented
264#1720C2×C8911yescharacteristic, complemented
265#1719C2×C8911yescharacteristic, complemented
266#1718C2×C8911yescharacteristic, complemented
267#1713C2×C8911yescharacteristic, complemented
268#1711C2×C8911yescharacteristic, complemented
269#1454C2×C8911yescharacteristic, complemented
270C89#1974411yescharacteristic, direct product factor, Sylow
271#533C22×C8911yescharacteristic
272#533C22×C8911yescharacteristic
273#532C22×C8911yescharacteristic
274C2×C4⋊Q8C22×C8911yescharacteristic
275#500C22×C8911yescharacteristic
276#500C22×C8911yescharacteristic
277#500C22×C8911yescharacteristic
278#500C22×C8911yescharacteristic
279#500C22×C8911yescharacteristic
280#500C22×C8911yescharacteristic
281#498C22×C8911yescharacteristic
282#498C22×C8911yescharacteristic
283#494C22×C8911yescharacteristic
284#494C22×C8911yescharacteristic
285#494C22×C8911yescharacteristic
286#492C22×C8911yescharacteristic
287#487C22×C8911yescharacteristic
288#487C22×C8911yescharacteristic
289#487C22×C8911yescharacteristic
290#487C22×C8911yescharacteristic
291#487C22×C8911yescharacteristic
292#487C22×C8911yescharacteristic
293#487C22×C8911yescharacteristic
294#486C22×C8911yescharacteristic
295#486C22×C8911yescharacteristic
296#486C22×C8911yescharacteristic
297#486C22×C8911yescharacteristic
298#486C22×C8911yescharacteristic
299#485—22no—
300#484—22no—
301#482—22no—
302#480—22no—
303#479—22no—
304#478—22no—
305#477—22no—
306#474—22no—
307#464—22no—
308#462—22no—
309#452—22no—
310#451—22no—
311#431C22×C8911yescharacteristic
312#431C22×C8911yescharacteristic
313#431C22×C8911yescharacteristic
314#322C22×C8911yescharacteristic
315#322C22×C8911yescharacteristic
316#322C22×C8911yescharacteristic
317C82C22×C8911yescharacteristic
318C2×#41—21no—
319C22×Q8—21no—
320C2×Q16—22no—
321C2×Q16—22no—
322C2×Q16—22no—
323C2×SD16—22no—
324C4⋊Q8—21no—
325C4⋊Q8—21no—
326C4⋊Q8—21no—
327C4⋊Q8C23×C8911yescharacteristic
328C4⋊Q8C23×C8911yescharacteristic
329C4⋊Q8C23×C8911yescharacteristic
330C4⋊Q8C23×C8911yescharacteristic
331C4⋊Q8C23×C8911yescharacteristic
332C4⋊Q8C23×C8911yescharacteristic
333C4⋊Q8C23×C8911yescharacteristic
334#127C23×C8911yescharacteristic
335#125—21no—
336#125—21no—
337#124—21no—
338C22⋊Q8—21no—
339C22⋊Q8—21no—
340C22⋊Q8—21no—
341C22⋊Q8—21no—
342#121—21no—
343#121—21no—
344C4×Q8—21no—
345C4×Q8—21no—
346C4×Q8—21no—
347C4×Q8—21no—
348C4×D4—21no—
349C4×D4—21no—
350#117—21no—
351#117—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C42C23×C8911yescharacteristic
355#107—22no—
356#107—22no—
357#107—22no—
358#107—22no—
359#107—22no—
360#107—22no—
361#106—22no—
362#106—22no—
363C4⋊C8D4×C8912yes—
364C4⋊C8D4×C8912yes—
365C4⋊C8D4×C8912yes—
366#103—22no—
367#103—22no—
368#103—22no—
369#103—22no—
370#103—22no—
371#103—22no—
372#103—22no—
373#102—22no—
374#102—22no—
375#102—22no—
376#102—22no—
377#102—22no—
378C22⋊C8D4×C8912yes—
379C22⋊C8D4×C8912yes—
380C22⋊C8D4×C8912yes—
381C4×C8C23×C8911yescharacteristic
382C4×C8C23×C8911yescharacteristic
383C4×C8C23×C8911yescharacteristic
384C4×C8C23×C8911yescharacteristic
385C4×C8C23×C8911yescharacteristic
386C4×C8C23×C8911yescharacteristic
387#41—82no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—41no—
396C2×Q8—82no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4C2×D4×C8911yescharacteristic
404C22×C4C2×D4×C8911yescharacteristic
405C22×C4C2×D4×C8911yescharacteristic
406Q16—82no—
407Q16—82no—
408Q16—82no—
409SD16—82no—
410C2×C8C2×D4×C8912yes—
411C2×C8C2×D4×C8912yes—
412C2×C8C2×D4×C8912yes—
413C2×C8C2×D4×C8912yes—
414C2×C8C2×D4×C8912yes—
415C2×C8C2×D4×C8912yes—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C42—41no—
441C42—41no—
442C42C2×D4×C8911yescharacteristic
443C42C2×D4×C8911yescharacteristic
444C42C2×D4×C8911yescharacteristic
445C42C24×C8911yescharacteristic
446C23—41no—
447C23#127×C8911yescharacteristic
448Q8—42no—
449Q8—42no—
450Q8—42no—
451Q8—42no—
452Q8—42no—
453Q8—42no—
454Q8—42no—
455Q8—82no—
456D4—42no—
457D4—42no—
458D4—42no—
459D4—82no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4#127×C8911yescharacteristic
473C2×C4#127×C8911yescharacteristic
474C2×C4#127×C8911yescharacteristic
475C2×C4C22×D4×C8911yescharacteristic
476C2×C4C22×D4×C8911yescharacteristic
477C2×C4C22×D4×C8911yescharacteristic
478C8—22no—
479C8—22no—
480C8—22no—
481C8—22no—
482C8—22no—
483C8—22no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22C2×#127×C8911yescharacteristic
491C4—21no—
492C4—21no—
493C4—21no—
494C4—81no—
495C4—81no—
496C4—81no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4—81no—
501C4#576×C8912yes—
502C4#578×C8912yes—
503C4#578×C8912yes—
504C2—41no—
505C2—161no—
506C2C89×#315511yescharacteristic
507C2C89×#315611yescharacteristic
508C2C89×#315611yescharacteristic
subgroupG/H
center, socleC22×C89C2×#127
commutator, Frattini, cosocleC42C24×C89
prime radical#532×C89C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C89 ◁ C2×C42×C89 ◁ G

Complements

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

NH ≅ G/Ntype
1. C89×#3231C2semidirect
2. C89×#3230C2semidirect
6. C89×#3209C2semidirect
7. C89×#1723C2semidirect
8. C89×#1723C2semidirect
9. C89×#1722C2semidirect
10. C89×#1720C2semidirect
11. C89×#1719C2semidirect
12. C89×#1718C2semidirect
13. C89×#1713C2semidirect
14. C89×#1711C2semidirect
15. C89×#1454C2semidirect
249. #19744C89direct (N × H)
255. #3231C2×C89semidirect
256. #3230C2×C89semidirect
260. #3209C2×C89semidirect
261. #1723C2×C89semidirect
262. #1723C2×C89semidirect
263. #1722C2×C89semidirect
264. #1720C2×C89semidirect
265. #1719C2×C89semidirect
266. #1718C2×C89semidirect
267. #1713C2×C89semidirect
268. #1711C2×C89semidirect
269. #1454C2×C89semidirect
270. C89#19744direct (N × H)

Maximal subgroups

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

subgroupindex
C89×#32312
C89×#32302
C89×#32282
C89×#32262
C89×#32102
C89×#32092
C89×#17232
C89×#17222
C89×#17202
C89×#17192
C89×#17182
C89×#17132
C89×#17112
C89×#14542
#1974489