Finite Group Databasebeta

← #27369580

Subgroups — #27369580

Order 13568.

694 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285838C211yesmaximal, characteristic, complemented
2#20284202C211yesmaximal, characteristic, complemented
3#20284170C211yesmaximal, characteristic
4#20274159C212yesmaximal, complemented
5#20274152C212yesmaximal, complemented
6#20274109C212yesmaximal, complemented
7#20274104C212yesmaximal, complemented
8#20273490C211yesmaximal, characteristic, complemented
9#20273390C211yesmaximal, characteristic, complemented
10C2×#16810368C211yesmaximal, characteristic, complemented
11C53×#2988C211yesmaximal, characteristic, complemented
12C2×#13133247—21no—
13C2×#13133237—21no—
14#16811173C2211yescharacteristic
15C2×#13133217C2211yescharacteristic
16#16811028C2212yes—
17#16810970C2212yes—
18#16810937—22no—
19#16810904—21no—
20#16810904—21no—
21#16810845—21no—
22#16810838—21no—
23#16810833—21no—
24#16810833C2211yescharacteristic
25#16810833C2212yes—
26#16810828—21no—
27#16810828C2211yescharacteristic
28#16810828C2212yes—
29#16810823C2211yescharacteristic
30#16810818C2211yescharacteristic
31#16810647C2211yescharacteristic
32C2×C4⋊#88322C2211yescharacteristic
33#16810631—21no—
34#16810617—21no—
35#16810376C2212yes—
36#16810376C2212yes—
37#16810368C2212yes—
38#16810368C2212yes—
39#16810368C2212yes—
40#16810366C2212yes—
41C4×#41×C53C2211yescharacteristic
42C8×Q8×C53C2212yes—
43C8×D4×C53—22no—
44C8×D4×C53C2212yes—
45C53×#431—21no—
46C53×#431C2211yescharacteristic
47C2×C4⋊C8×C53—21no—
48C2×C4⋊C8×C53C2211yescharacteristic
49C2×#11719945—21no—
50C22×Dic106—21no—
51#13133247—44no—
52#13133237—44no—
53C2×#11719925—22no—
54C2×#11719925—21no—
55C2×#11719923—22no—
56C2×#11719923—21no—
57C2×#11719924—21no—
58C2×#11719922—21no—
59C2×#11719917—21no—
60C22×#88322—21no—
61#13133217—21no—
62#13133217—21no—
63#13133217—21no—
64#13133217C2311yescharacteristic
65#13133217C2312yes—
66#13133214C2311yescharacteristic
67#13133205—21no—
68#13133191—21no—
69#13133191—21no—
70C4×D212—21no—
71C4×D212—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74#13133151—22no—
75#13133151—22no—
76C4⋊#88322—21no—
77C4⋊#88322—21no—
78C4⋊#88322C2311yescharacteristic
79C4⋊#88322C2311yescharacteristic
80C4⋊#88322C2312yes—
81#13133140—22no—
82#13133140—22no—
83#13133140—21no—
84#13133140—21no—
85#13133139—22no—
86#13133139—22no—
87#13133137—21no—
88#13133137—21no—
89C2×#41×C53D411yescharacteristic
90C22×C8×C53—22no—
91C22×C8×C53—21no—
92C4×Q8×C53D411yescharacteristic
93C4×Q8×C53C2311yescharacteristic
94C4×D4×C53—21no—
95C4×D4×C53—21no—
96C4×D4×C53D411yescharacteristic
97C4×D4×C53C2311yescharacteristic
98#117×C53—21no—
99#117×C53D411yescharacteristic
100C2×C42×C53—21no—
101C2×C42×C53C2311yescharacteristic
102C4⋊C8×C53—22no—
103C4⋊C8×C53—21no—
104C4⋊C8×C53D411yescharacteristic
105C4⋊C8×C53D411yescharacteristic
106C4⋊C8×C53C2312yes—
107C22⋊C8×C53—22no—
108C22⋊C8×C53D412yes—
109C4×C8×C53—22no—
110C4×C8×C53C2312yes—
111#41×C53—22no—
112#41×C53D814yes—
113C2×Q8×C53C2×D411yescharacteristic
114C2×D4×C53—21no—
115C2×D4×C53C2×D411yescharacteristic
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53#4111yescharacteristic
122C2×C22⋊D53—41no—
123C22×Dic53—41no—
124#11719945—42no—
125#11719945—42no—
126C2×D212—21no—
127C2×D212—21no—
128C2×D212—21no—
129C2×C4×D53—41no—
130C2×Dic106—21no—
131C2×Dic106—22no—
132C2×Dic106—42no—
133C2×Dic106—42no—
134C2×Dic106—41no—
135C2×C8×C53—22no—
136C2×C8×C53—82no—
137C2×C8×C53—81no—
138C2×C8×C53C2×D412yes—
139C2×C8×C53C2×D412yes—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53C2×D411yescharacteristic
143C4⋊C4×C53C2×D411yescharacteristic
144C22⋊C4×C53—21no—
145C22⋊C4×C53—21no—
146C22⋊C4×C53—21no—
147C42×C53—21no—
148C42×C53—21no—
149C42×C53—21no—
150C42×C53C2×D411yescharacteristic
151C42×C53C2×D411yescharacteristic
152C42×C53#4111yescharacteristic
153C42×C53C2411yescharacteristic
154#11719925—42no—
155#11719925—44no—
156#11719924—42no—
157#11719923—42no—
158#11719923—44no—
159#11719922—42no—
160#11719921—41no—
161#11719921—41no—
162C4⋊Dic53—22no—
163C4⋊Dic53—21no—
164C4⋊Dic53—22no—
165C4⋊Dic53—21no—
166C4⋊Dic53—41no—
167C4⋊Dic53—41no—
168#11719917—42no—
169C2×#88322—21no—
170C2×#88322—21no—
171C2×#88322—22no—
172C2×#88322—42no—
173C23×C53—21no—
174C23×C53#12111yescharacteristic
175C22×D53—41no—
176Q8×C53—21no—
177Q8×C53C2×D812yes—
178D4×C53—24no—
179D4×C53—21no—
180D4×C53C2×D812yes—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53—21no—
184C2×C4×C53—24no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—41no—
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53—81no—
191C2×C4×C53#12111yescharacteristic
192C2×C4×C53#12111yescharacteristic
193C2×C4×C53#12111yescharacteristic
194C2×C4×C53C2×D812yes—
195C2×C4×C53#13512yes—
196C2×C4×C53C22×D411yescharacteristic
197C2×C4×C53C22×D411yescharacteristic
198C2×C4×C53C2×#4111yescharacteristic
199C22⋊D53—82no—
200C2×Dic53—42no—
201C2×Dic53—41no—
202C2×Dic53—82no—
203D212—41no—
204D212—41no—
205D212—41no—
206D212—41no—
207D212—81no—
208C4×D53—82no—
209Dic106—42no—
210Dic106—41no—
211Dic106—42no—
212Dic106—41no—
213Dic106—82no—
214C8×C53—22no—
215C8×C53—22no—
216C8×C53—42no—
217#88322—41no—
218#88322—41no—
219#88322—42no—
220#19416—531nomaximal, Sylow
221C22×C53—21no—
222C22×C53—21no—
223C22×C53—22no—
224C22×C53—41no—
225C22×C53—41no—
226C22×C53#46612yes—
227C22×C53C2×#12111yescharacteristic
228C2×D53—81no—
229C2×D53—81no—
230C2×D53—81no—
231C4×C53—21no—
232C4×C53—22no—
233C4×C53—21no—
234C4×C53—21no—
235C4×C53—41no—
236C4×C53—41no—
237C4×C53#46612yes—
238C4×C53C22×D811yescharacteristic
239C4×C53C2×#13511yescharacteristic
240C4×C53#57711yescharacteristic
241C4×C53#57811yescharacteristic
242Dic53—82no—
243Dic53—81no—
244#3226—531no—
245#3039—531no—
246#3035—531no—
247#2988D5311yescharacteristic, complemented
248#1682—532no—
249#1681—532no—
250#1674—532no—
251#1673—532no—
252#1571—531no—
253#1547—531no—
254C2×#332—531no—
255C2×C53—22no—
256C2×C53—42no—
257C2×C53#304611yescharacteristic
258C2×C53C2×#46611yescharacteristic
259C2×C53#306911yescharacteristic
260D53—161no—
261C2×#137—1061no—
262C2×#135—1061no—
263#532—531no—
264C2×C4⋊Q8—531no—
265C4×#41C2×D5311yescharacteristic
266#500—532no—
267#486—532no—
268#480—1062no—
269#474—1061no—
270#474—1061no—
271#464—1061no—
272#463—1061no—
273#462—531no—
274#462—532no—
275#462—1061no—
276#461—531no—
277#461—532no—
278#461—1061no—
279#460—531no—
280#459—531no—
281C8×Q8C2×D5312yes—
282C8×D4—22no—
283C8×D4C2×D5312yes—
284#431—21no—
285#431C2×D5311yescharacteristic
286#423—531no—
287C2×C4⋊C8—21no—
288C2×C4⋊C8—531no—
289C2×C4⋊C8C2×D5311yescharacteristic
290#419—1061no—
291#416—1061no—
292#335—532no—
293#335—532no—
294#332—532no—
295#332—532no—
296#332—532no—
297#331—532no—
298C53#1941611yescharacteristic, complemented, Sylow
299C2×#41—1061no—
300C2×#41D21211yescharacteristic
301C22×Q8—1061no—
302#137—2124no—
303#135—2124no—
304C2×Q16—1061no—
305C2×Q16—1062no—
306C2×SD16—1061no—
307C2×SD16—1061no—
308C2×SD16—1062no—
309C2×D8—1061no—
310C2×M16—1061no—
311C22×C8—21no—
312C22×C8—22no—
313C22×C8—1061no—
314C4⋊Q8—532no—
315C4⋊Q8—531no—
316C4⋊Q8—1061no—
317C4⋊Q8—1061no—
318C4⋊Q8—1061no—
319#127—531no—
320#124—1061no—
321#121—1061no—
322#121—1061no—
323C4×Q8D21211yescharacteristic
324C4×Q8C22×D5311yescharacteristic
325C4×D4—21no—
326C4×D4—21no—
327C4×D4—1061no—
328C4×D4—1061no—
329C4×D4D21211yescharacteristic
330C4×D4C22×D5311yescharacteristic
331#117—21no—
332#117D21211yescharacteristic
333C2×C4⋊C4—1061no—
334C2×C4⋊C4—1061no—
335C2×C42—21no—
336C2×C42C22×D5311yescharacteristic
337#107—1062no—
338#107—1062no—
339C4⋊C8—22no—
340C4⋊C8—21no—
341C4⋊C8—532no—
342C4⋊C8—531no—
343C4⋊C8—531no—
344C4⋊C8—1061no—
345C4⋊C8—1061no—
346C4⋊C8C22⋊D5311yescharacteristic
347C4⋊C8C22⋊D5311yescharacteristic
348C4⋊C8C22×D5312yes—
349#103—1061no—
350#103—1061no—
351#103—1062no—
352#103—1062no—
353#102—1062no—
354#102—1062no—
355#102—1061no—
356#102—1061no—
357C22⋊C8—22no—
358C22⋊C8C22⋊D5312yes—
359C4×C8—22no—
360C4×C8C22×D5312yes—
361#41—22no—
362#41—2122no—
363#41—2122no—
364#41D42414yes—
365C2×Q8—1061no—
366C2×Q8—1062no—
367C2×Q8—2122no—
368C2×Q8—2122no—
369C2×Q8—2121no—
370C2×Q8C2×D21211yescharacteristic
371C2×D4—21no—
372C2×D4—1061no—
373C2×D4—1061no—
374C2×D4—1061no—
375C2×D4—2121no—
376C2×D4C2×D21211yescharacteristic
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—2121no—
381C22×C4—2121no—
382C22×C4C2×D21211yescharacteristic
383C22×C4#1171994511yescharacteristic
384C22×C4C2×C22⋊D5311yescharacteristic
385Q16—2122no—
386Q16—2124no—
387SD16—2122no—
388SD16—2122no—
389SD16—2124no—
390D8—2122no—
391M16—2122no—
392C2×C8—22no—
393C2×C8—82no—
394C2×C8—81no—
395C2×C8—1061no—
396C2×C8—1061no—
397C2×C8—1062no—
398C2×C8—2122no—
399C2×C8C2×C22⋊D5312yes—
400C2×C8C2×C22⋊D5312yes—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—1062no—
404C4⋊C4—1061no—
405C4⋊C4—1062no—
406C4⋊C4—1061no—
407C4⋊C4—2121no—
408C4⋊C4—2121no—
409C4⋊C4C2×D21211yescharacteristic
410C4⋊C4C2×D21211yescharacteristic
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—21no—
414C22⋊C4—2121no—
415C22⋊C4—2121no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42C2×D21211yescharacteristic
420C42#1171994511yescharacteristic
421C42C2×C22⋊D5311yescharacteristic
422C42C23×D5311yescharacteristic
423C23—21no—
424C23—2121no—
425C23#1313319111yescharacteristic
426Q8—21no—
427Q8—4242no—
428Q8—2122no—
429Q8—2121no—
430Q8—2122no—
431Q8—2121no—
432Q8C2×D42412yes—
433D4—24no—
434D4—21no—
435D4—4241no—
436D4—4242no—
437D4—2121no—
438D4—2121no—
439D4—2121no—
440D4—2121no—
441D4C2×D42412yes—
442C2×C4—21no—
443C2×C4—24no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—81no—
452C2×C4—4242no—
453C2×C4—4242no—
454C2×C4—2121no—
455C2×C4—2122no—
456C2×C4#1313319111yescharacteristic
457C2×C4#1313319111yescharacteristic
458C2×C4#1313319111yescharacteristic
459C2×C4C2×D42412yes—
460C2×C4#1313323912yes—
461C2×C4C22×C22⋊D5311yescharacteristic
462C2×C4C22×D21211yescharacteristic
463C2×C4C2×#1171994511yescharacteristic
464C8—22no—
465C8—22no—
466C8—42no—
467C8—2121no—
468C8—2121no—
469C8—2122no—
470C22—21no—
471C22—21no—
472C22—22no—
473C22—41no—
474C22—41no—
475C22—4241no—
476C22—4241no—
477C22—4241no—
478C22#1681085912yes—
479C22C2×#1313319111yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—22no—
484C4—41no—
485C4—41no—
486C4—4241no—
487C4—4242no—
488C4#1681085912yes—
489C4C22×D42411yescharacteristic
490C4C2×#1313323911yescharacteristic
491C4#1681143411yescharacteristic
492C4#1681143811yescharacteristic
493C2—22no—
494C2—42no—
495C2—8481no—
496C2#2028426511yescharacteristic
497C2C2×#1681085911yescharacteristic
498C2#2028445211yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2988C2
socleC22×C53C2×#121
prime radical#20284170C2
hypercenter#2988D53
hypocenterC53#19416
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2988

Complements

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

NH ≅ G/Ntype
1. #20285838C2semidirect
2. #20284202C2semidirect
4. #20274159C2semidirect
5. #20274152C2semidirect
6. #20274109C2semidirect
7. #20274104C2semidirect
8. #20273490C2semidirect
9. #20273390C2semidirect
10. C2×#16810368C2semidirect
11. C53×#2988C2semidirect
247. #2988D53semidirect
298. C53#19416semidirect

Maximal subgroups

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

subgroupindex
#202858382
#202842022
#202841702
#202741592
#202741522
#202741092
#202741042
#202734902
#202733902
C2×#168103682
C53×#29882
#1941653