Finite Group Databasebeta

← C89×#24723 #36402571

Subgroups — C89×#24723 #36402571

Order 22784.

674 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×#3079C211yesmaximal, characteristic
2C2×#475×C89C211yesmaximal, characteristic, complemented
3C4×#137×C89C211yesmaximal, characteristic
4C89×#2925C211yesmaximal, characteristic
5C89×#2079C212yesmaximal
6C89×#2052C212yesmaximal
7C89×#1997C211yesmaximal, characteristic, complemented
8C89×#1951C212yesmaximal, complemented
9C89×#1950C211yesmaximal, characteristic, complemented
10C89×#1930C212yesmaximal, complemented
11C89×#1801C211yesmaximal, characteristic, complemented
12C2×#137×C89C2211yescharacteristic
13C2×C4⋊Q8×C89C2211yescharacteristic
14C4×#41×C89C2211yescharacteristic
15C2×C4×Q8×C89C2211yescharacteristic
16#477×C89—21no—
17#476×C89—21no—
18#475×C89—21no—
19#475×C89—21no—
20#475×C89—21no—
21#475×C89—22no—
22#474×C89—21no—
23#441×C89—21no—
24#440×C89—21no—
25C4×Q16×C89—21no—
26C4×SD16×C89—21no—
27#433×C89—22no—
28#432×C89—21no—
29#431×C89—21no—
30C2×#108×C89C2212yes—
31#428×C89C2212yes—
32#428×C89C2211yescharacteristic
33C2×#106×C89C2212yes—
34#423×C89C2211yescharacteristic
35#423×C89C2211yescharacteristic
36C2×C4⋊C8×C89C2211yescharacteristic
37#419×C89C2211yescharacteristic
38#417×C89C2212yes—
39#417×C89C2211yescharacteristic
40C2×#103×C89C2212yes—
41C2×#101×C89C2212yes—
42#408×C89C2212yes—
43#405×C89C2212yes—
44C4×M16×C89C2211yescharacteristic
45#343×C89C2212yes—
46#343×C89C2212yes—
47#343×C89C2212yes—
48#341×C89C2212yes—
49C2×#41×C89C2311yescharacteristic
50C22×Q8×C89—21no—
51C22×Q8×C89C2311yescharacteristic
52#137×C89—22no—
53#137×C89—22no—
54C2×Q16×C89—21no—
55C2×SD16×C89—21no—
56C2×M16×C89C2311yescharacteristic
57C2×M16×C89C2312yes—
58C2×M16×C89C2312yes—
59C2×M16×C89C2311yescharacteristic
60C22×C8×C89C2312yes—
61C4⋊Q8×C89—21no—
62C4⋊Q8×C89—21no—
63C4⋊Q8×C89—21no—
64C4⋊Q8×C89—21no—
65C4×Q8×C89—21no—
66C4×Q8×C89—21no—
67C4×Q8×C89—21no—
68C4×Q8×C89—21no—
69C4×Q8×C89D411yescharacteristic
70C4×Q8×C89D411yescharacteristic
71C4×D4×C89—21no—
72C4×D4×C89—21no—
73C4×D4×C89D411yescharacteristic
74C4×D4×C89D411yescharacteristic
75#117×C89—21no—
76#117×C89C2311yescharacteristic
77C2×C4⋊C4×C89—21no—
78C2×C4⋊C4×C89—21no—
79C2×C4⋊C4×C89C2312yes—
80C2×C4⋊C4×C89C2311yescharacteristic
81C2×C42×C89—21no—
82C2×C42×C89—21no—
83C2×C42×C89C2311yescharacteristic
84#108×C89—24no—
85#107×C89—21no—
86#107×C89—22no—
87#106×C89—22no—
88#106×C89—22no—
89#106×C89—21no—
90#106×C89—22no—
91C4⋊C8×C89—21no—
92C4⋊C8×C89—21no—
93C4⋊C8×C89—21no—
94C4⋊C8×C89—21no—
95C4⋊C8×C89D411yescharacteristic
96C4⋊C8×C89D411yescharacteristic
97C4⋊C8×C89D412yes—
98#103×C89—21no—
99#103×C89—22no—
100#103×C89—22no—
101#103×C89—22no—
102#103×C89—21no—
103#103×C89—21no—
104#102×C89—21no—
105#102×C89—22no—
106#101×C89—24no—
107C22⋊C8×C89—22no—
108C22⋊C8×C89—22no—
109#97×C89—22no—
110#97×C89—21no—
111C4×C8×C89—22no—
112C4×C8×C89—21no—
113#41×C89—21no—
114#41×C89—21no—
115#41×C89—21no—
116#41×C89—21no—
117C2×Q8×C89—21no—
118C2×Q8×C89—21no—
119C2×Q8×C89—21no—
120C2×Q8×C89—21no—
121C2×Q8×C89—21no—
122C2×Q8×C89—41no—
123C2×Q8×C89—41no—
124C2×Q8×C89—82no—
125C2×Q8×C89C2×D411yescharacteristic
126C2×Q8×C89#4112yes—
127C2×D4×C89—21no—
128C2×D4×C89C2×D411yescharacteristic
129C22×C4×C89—21no—
130C22×C4×C89—21no—
131C22×C4×C89—21no—
132C22×C4×C89—21no—
133C22×C4×C89—21no—
134C22×C4×C89—22no—
135C22×C4×C89#4111yescharacteristic
136C22×C4×C89#4111yescharacteristic
137C22×C4×C89C2411yescharacteristic
138Q16×C89—24no—
139SD16×C89—24no—
140M16×C89—21no—
141M16×C89—22no—
142M16×C89—22no—
143M16×C89—22no—
144M16×C89—21no—
145M16×C89—21no—
146M16×C89—22no—
147M16×C89#4112yes—
148C2×C8×C89—21no—
149C2×C8×C89—22no—
150C2×C8×C89—22no—
151C2×C8×C89—22no—
152C2×C8×C89—21no—
153C2×C8×C89—22no—
154C2×C8×C89C2×D412yes—
155C2×C8×C89C2×D412yes—
156C4⋊C4×C89—21no—
157C4⋊C4×C89—21no—
158C4⋊C4×C89—21no—
159C4⋊C4×C89—22no—
160C4⋊C4×C89—22no—
161C4⋊C4×C89—21no—
162C4⋊C4×C89—41no—
163C4⋊C4×C89—41no—
164C4⋊C4×C89—41no—
165C4⋊C4×C89—41no—
166C4⋊C4×C89C2×D411yescharacteristic
167C4⋊C4×C89C2×D411yescharacteristic
168C22⋊C4×C89—21no—
169C22⋊C4×C89—21no—
170C22⋊C4×C89—21no—
171C42×C89—21no—
172C42×C89—21no—
173C42×C89—21no—
174C42×C89—41no—
175C42×C89—41no—
176C42×C89C2×D411yescharacteristic
177C42×C89C2×D411yescharacteristic
178C42×C89C2×D411yescharacteristic
179C42×C89C2×D411yescharacteristic
180C23×C89—21no—
181C23×C89C2×#4111yescharacteristic
182Q8×C89—22no—
183Q8×C89—21no—
184Q8×C89—21no—
185Q8×C89—22no—
186Q8×C89—41no—
187Q8×C89—41no—
188Q8×C89—82no—
189Q8×C89—82no—
190D4×C89—21no—
191D4×C89—21no—
192D4×C89—41no—
193D4×C89—41no—
194C2×C4×C89—22no—
195C2×C4×C89—21no—
196C2×C4×C89—21no—
197C2×C4×C89—21no—
198C2×C4×C89—22no—
199C2×C4×C89—41no—
200C2×C4×C89—41no—
201C2×C4×C89—41no—
202C2×C4×C89—41no—
203C2×C4×C89—41no—
204C2×C4×C89—42no—
205C2×C4×C89—41no—
206C2×C4×C89—42no—
207C2×C4×C89—81no—
208C2×C4×C89—81no—
209C2×C4×C89—82no—
210C2×C4×C89#12111yescharacteristic
211C2×C4×C89#12111yescharacteristic
212C2×C4×C89#12111yescharacteristic
213C2×C4×C89#12111yescharacteristic
214C2×C4×C89C22×D411yescharacteristic
215C2×C4×C89C22×D411yescharacteristic
216C2×C4×C89C2×#4111yescharacteristic
217C2×C4×C89C2×#4111yescharacteristic
218C2×C4×C892−1+411yescharacteristic
219C2×C4×C892−1+411yescharacteristic
220C8×C89—22no—
221C8×C89—22no—
222C8×C89—22no—
223C8×C89—42no—
224C8×C89—41no—
225C8×C89—42no—
226C22×C89—21no—
227C22×C89—21no—
228C22×C89—41no—
229C22×C89—41no—
230C22×C89—41no—
231C22×C89C2×#12111yescharacteristic
232C22×C89#55211yescharacteristic
233C22×C89#55611yescharacteristic
234C4×C89—21no—
235C4×C89—21no—
236C4×C89—21no—
237C4×C89—41no—
238C4×C89—42no—
239C4×C89—41no—
240C4×C89—41no—
241C4×C89—82no—
242C4×C89—81no—
243C4×C89#47112yes—
244C4×C89#52511yescharacteristic
245C4×C89#53611yescharacteristic
246C4×C89#54711yescharacteristic
247C4×C89#54811yescharacteristic
248#24723C8911yesmaximal, characteristic, direct product factor, Sylow
249C2×C89—21no—
250C2×C89—21no—
251C2×C89—81no—
252C2×C89#241511yescharacteristic
253C2×C89C2×#47111yescharacteristic
254C2×C89#306911yescharacteristic
255#3079C2×C8911yescharacteristic
256C2×#475C2×C8911yescharacteristic, complemented
257C4×#137C2×C8911yescharacteristic
258#2925C2×C8911yescharacteristic
259#2079C2×C8912yes—
260#2052C2×C8912yes—
261#1997C2×C8911yescharacteristic, complemented
262#1951C2×C8912yescomplemented
263#1950C2×C8911yescharacteristic, complemented
264#1930C2×C8912yescomplemented
265#1801C2×C8911yescharacteristic, complemented
266C89#2472311yescharacteristic, direct product factor, Sylow
267C2×#137C22×C8911yescharacteristic
268C2×C4⋊Q8C22×C8911yescharacteristic
269C4×#41C22×C8911yescharacteristic
270C2×C4×Q8C22×C8911yescharacteristic
271#477—21no—
272#476—21no—
273#475—21no—
274#475—21no—
275#475—21no—
276#475—22no—
277#474—21no—
278#441—21no—
279#440—21no—
280C4×Q16—21no—
281C4×SD16—21no—
282#433—22no—
283#432—21no—
284#431—21no—
285C2×#108C22×C8912yes—
286#428C22×C8912yes—
287#428C22×C8911yescharacteristic
288C2×#106C22×C8912yes—
289#423C22×C8911yescharacteristic
290#423C22×C8911yescharacteristic
291C2×C4⋊C8C22×C8911yescharacteristic
292#419C22×C8911yescharacteristic
293#417C22×C8912yes—
294#417C22×C8911yescharacteristic
295C2×#103C22×C8912yes—
296C2×#101C22×C8912yes—
297#408C22×C8912yes—
298#405C22×C8912yes—
299C4×M16C22×C8911yescharacteristic
300#343C22×C8912yes—
301#343C22×C8912yes—
302#343C22×C8912yes—
303#341C22×C8912yes—
304C2×#41C23×C8911yescharacteristic
305C22×Q8—21no—
306C22×Q8C23×C8911yescharacteristic
307#137—22no—
308#137—22no—
309C2×Q16—21no—
310C2×SD16—21no—
311C2×M16C23×C8911yescharacteristic
312C2×M16C23×C8912yes—
313C2×M16C23×C8912yes—
314C2×M16C23×C8911yescharacteristic
315C22×C8C23×C8912yes—
316C4⋊Q8—21no—
317C4⋊Q8—21no—
318C4⋊Q8—21no—
319C4⋊Q8—21no—
320C4×Q8—21no—
321C4×Q8—21no—
322C4×Q8—21no—
323C4×Q8—21no—
324C4×Q8D4×C8911yescharacteristic
325C4×Q8D4×C8911yescharacteristic
326C4×D4—21no—
327C4×D4—21no—
328C4×D4D4×C8911yescharacteristic
329C4×D4D4×C8911yescharacteristic
330#117—21no—
331#117C23×C8911yescharacteristic
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4C23×C8911yescharacteristic
335C2×C4⋊C4C23×C8912yes—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42C23×C8911yescharacteristic
339#108—24no—
340#107—21no—
341#107—22no—
342#106—22no—
343#106—22no—
344#106—21no—
345#106—22no—
346C4⋊C8—21no—
347C4⋊C8—21no—
348C4⋊C8—21no—
349C4⋊C8—21no—
350C4⋊C8D4×C8911yescharacteristic
351C4⋊C8D4×C8911yescharacteristic
352C4⋊C8D4×C8912yes—
353#103—21no—
354#103—22no—
355#103—22no—
356#103—22no—
357#103—21no—
358#103—21no—
359#102—21no—
360#102—22no—
361#101—24no—
362C22⋊C8—22no—
363C22⋊C8—22no—
364#97—22no—
365#97—21no—
366C4×C8—22no—
367C4×C8—21no—
368#41—21no—
369#41—21no—
370#41—21no—
371#41—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—41no—
378C2×Q8—41no—
379C2×Q8—82no—
380C2×Q8C2×D4×C8911yescharacteristic
381C2×Q8#41×C8912yes—
382C2×D4—21no—
383C2×D4C2×D4×C8911yescharacteristic
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4#41×C8911yescharacteristic
391C22×C4#41×C8911yescharacteristic
392C22×C4C24×C8911yescharacteristic
393Q16—24no—
394SD16—24no—
395M16—21no—
396M16—22no—
397M16—22no—
398M16—22no—
399M16—21no—
400M16—21no—
401M16—22no—
402M16#41×C8912yes—
403C2×C8—21no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—21no—
408C2×C8—22no—
409C2×C8C2×D4×C8912yes—
410C2×C8C2×D4×C8912yes—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4C2×D4×C8911yescharacteristic
422C4⋊C4C2×D4×C8911yescharacteristic
423C22⋊C4—21no—
424C22⋊C4—21no—
425C22⋊C4—21no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—41no—
430C42—41no—
431C42C2×D4×C8911yescharacteristic
432C42C2×D4×C8911yescharacteristic
433C42C2×D4×C8911yescharacteristic
434C42C2×D4×C8911yescharacteristic
435C23—21no—
436C23C2×#41×C8911yescharacteristic
437Q8—22no—
438Q8—21no—
439Q8—21no—
440Q8—22no—
441Q8—41no—
442Q8—41no—
443Q8—82no—
444Q8—82no—
445D4—21no—
446D4—21no—
447D4—41no—
448D4—41no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—82no—
465C2×C4#121×C8911yescharacteristic
466C2×C4#121×C8911yescharacteristic
467C2×C4#121×C8911yescharacteristic
468C2×C4#121×C8911yescharacteristic
469C2×C4C22×D4×C8911yescharacteristic
470C2×C4C22×D4×C8911yescharacteristic
471C2×C4C2×#41×C8911yescharacteristic
472C2×C4C2×#41×C8911yescharacteristic
473C2×C42−1+4×C8911yescharacteristic
474C2×C42−1+4×C8911yescharacteristic
475C8—22no—
476C8—22no—
477C8—22no—
478C8—42no—
479C8—41no—
480C8—42no—
481C22—21no—
482C22—21no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22C2×#121×C8911yescharacteristic
487C22#552×C8911yescharacteristic
488C22#556×C8911yescharacteristic
489C4—21no—
490C4—21no—
491C4—21no—
492C4—41no—
493C4—41no—
494C4—41no—
495C4—42no—
496C4—82no—
497C4—81no—
498C4#471×C8912yes—
499C4#525×C8911yescharacteristic
500C4#536×C8911yescharacteristic
501C4#547×C8911yescharacteristic
502C4#548×C8911yescharacteristic
503C2—21no—
504C2—21no—
505C2—81no—
506C2C89×#241511yescharacteristic
507C2C2×#471×C8911yescharacteristic
508C2C89×#306911yescharacteristic
subgroupG/H
center, socleC22×C89C2×#121
commutator, Frattini, cosocleC22×C4C24×C89
prime radicalC2×#41×C89C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C89 ◁ C2×C42×C89 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#475×C89C2semidirect
7. C89×#1997C2semidirect
8. C89×#1951C2semidirect
9. C89×#1950C2semidirect
10. C89×#1930C2semidirect
11. C89×#1801C2semidirect
248. #24723C89direct (N × H)
256. C2×#475C2×C89semidirect
261. #1997C2×C89semidirect
262. #1951C2×C89semidirect
263. #1950C2×C89semidirect
264. #1930C2×C89semidirect
265. #1801C2×C89semidirect
266. C89#24723direct (N × H)

Maximal subgroups

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

subgroupindex
C89×#30792
C2×#475×C892
C4×#137×C892
C89×#29252
C89×#20792
C89×#20522
C89×#19972
C89×#19512
C89×#19502
C89×#19302
C89×#18012
#2472389