Finite Group Databasebeta

← #27369584

Subgroups — #27369584

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#20285836C211yesmaximal, characteristic, complemented
2#20284170C211yesmaximal, characteristic
3#20283826C211yesmaximal, characteristic, complemented
4#20274161C212yesmaximal, complemented
5#20274154C212yesmaximal, complemented
6#20274107C212yesmaximal, complemented
7#20274102C212yesmaximal, complemented
8#20273491C211yesmaximal, characteristic, complemented
9#20273393C211yesmaximal, characteristic, complemented
10C2×#16810369C211yesmaximal, characteristic, complemented
11C53×#3039C211yesmaximal, characteristic, complemented
12C2×#13133237—21no—
13#16811173C2211yescharacteristic
14#16811026C2212yes—
15#16810971C2212yes—
16#16810936—22no—
17#16810900—21no—
18#16810900—21no—
19#16810845—21no—
20#16810838—21no—
21#16810833C2211yescharacteristic
22#16810828C2211yescharacteristic
23#16810823C2211yescharacteristic
24#16810818C2211yescharacteristic
25Q8×#88322C2212yes—
26D4×#88322—22no—
27D4×#88322C2212yes—
28#16810676—21no—
29#16810676C2211yescharacteristic
30C2×C4⋊#88322—21no—
31C2×C4⋊#88322C2211yescharacteristic
32C2×C4⋊#88322C2211yescharacteristic
33#16810617—21no—
34#16810375C2212yes—
35#16810375C2212yes—
36#16810370C2212yes—
37#16810369C2212yes—
38#16810369C2212yes—
39#16810366C2212yes—
40C2×#137×C53—21no—
41C2×C4⋊Q8×C53C2211yescharacteristic
42C4×#41×C53C2211yescharacteristic
43C53×#462—21no—
44C53×#462C2212yes—
45C53×#461—21no—
46C53×#461C2212yes—
47C53×#423C2211yescharacteristic
48C53×#419—21no—
49C2×#11719945—21no—
50#13133237—44no—
51C2×#11719925—21no—
52C2×#11719923—21no—
53C2×#11719924—21no—
54C2×#11719922—21no—
55C22×#88322—22no—
56C22×#88322—21no—
57C22×#88322—21no—
58#13133217C2311yescharacteristic
59#13133214C2311yescharacteristic
60#13133205—21no—
61#13133191—21no—
62#13133191—21no—
63C4×D212—21no—
64C4×D212—21no—
65#13133150—22no—
66#13133150—22no—
67C4⋊#88322—21no—
68C4⋊#88322—21no—
69C4⋊#88322—22no—
70C4⋊#88322D411yescharacteristic
71C4⋊#88322D411yescharacteristic
72C4⋊#88322C2311yescharacteristic
73C4⋊#88322C2311yescharacteristic
74C4⋊#88322C2312yes—
75#13133140—21no—
76#13133140—21no—
77#13133137—21no—
78#13133137—21no—
79#13133137—22no—
80#13133137—22no—
81C22⋊#88322—22no—
82C22⋊#88322D412yes—
83C4×#88322—22no—
84C4×#88322C2312yes—
85C2×#41×C53D411yescharacteristic
86C22×Q8×C53—21no—
87#137×C53—44no—
88C2×Q16×C53—22no—
89C2×SD16×C53—22no—
90C2×M16×C53—21no—
91C4⋊Q8×C53—21no—
92C4⋊Q8×C53—21no—
93C4⋊Q8×C53—21no—
94C4⋊Q8×C53C2312yes—
95C4×Q8×C53D411yescharacteristic
96C4×Q8×C53C2311yescharacteristic
97C4×D4×C53—21no—
98C4×D4×C53—21no—
99C4×D4×C53D411yescharacteristic
100C4×D4×C53C2311yescharacteristic
101#117×C53—21no—
102#117×C53D411yescharacteristic
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C42×C53—21no—
106C2×C42×C53C2311yescharacteristic
107C4⋊C8×C53—21no—
108C4⋊C8×C53C2312yes—
109#103×C53—22no—
110#103×C53—22no—
111#41×C53—22no—
112#41×C53D814yes—
113C2×Q8×C53—22no—
114C2×Q8×C53—41no—
115C2×Q8×C53—42no—
116C2×Q8×C53—42no—
117C2×Q8×C53C2×D411yescharacteristic
118C2×D4×C53—21no—
119C2×D4×C53C2×D411yescharacteristic
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—41no—
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53#4111yescharacteristic
127C2×C22⋊D53—41no—
128#11719945—42no—
129#11719945—42no—
130C2×D212—21no—
131C2×D212—21no—
132C2×D212—21no—
133C2×C4×D53—41no—
134C2×Dic106—21no—
135Q16×C53—44no—
136SD16×C53—44no—
137M16×C53—42no—
138C2×C8×C53—22no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—22no—
142C4⋊C4×C53—22no—
143C4⋊C4×C53—41no—
144C4⋊C4×C53—41no—
145C4⋊C4×C53C2×D411yescharacteristic
146C4⋊C4×C53C2×D411yescharacteristic
147C22⋊C4×C53—21no—
148C22⋊C4×C53—21no—
149C22⋊C4×C53—21no—
150C42×C53—21no—
151C42×C53—21no—
152C42×C53—21no—
153C42×C53C2×D411yescharacteristic
154C42×C53C2×D411yescharacteristic
155C42×C53#4111yescharacteristic
156C42×C53C2411yescharacteristic
157#11719925—42no—
158#11719924—42no—
159#11719923—42no—
160#11719922—42no—
161#11719921—41no—
162#11719921—41no—
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165C2×#88322—22no—
166C2×#88322—21no—
167C2×#88322—21no—
168C2×#88322—42no—
169C2×#88322—82no—
170C2×#88322—81no—
171C2×#88322C2×D412yes—
172C2×#88322C2×D412yes—
173C23×C53—21no—
174C23×C53#12111yescharacteristic
175C22×D53—41no—
176Q8×C53—21no—
177Q8×C53—42no—
178Q8×C53—42no—
179Q8×C53—82no—
180Q8×C53C2×D812yes—
181D4×C53—24no—
182D4×C53—21no—
183D4×C53C2×D812yes—
184C2×C4×C53—21no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—24no—
188C2×C4×C53—21no—
189C2×C4×C53—21no—
190C2×C4×C53—41no—
191C2×C4×C53—41no—
192C2×C4×C53—41no—
193C2×C4×C53—42no—
194C2×C4×C53—81no—
195C2×C4×C53—82no—
196C2×C4×C53#12111yescharacteristic
197C2×C4×C53#12111yescharacteristic
198C2×C4×C53#12111yescharacteristic
199C2×C4×C53C2×D812yes—
200C2×C4×C53#13512yes—
201C2×C4×C53C22×D411yescharacteristic
202C2×C4×C53C22×D411yescharacteristic
203C2×C4×C53C2×#4111yescharacteristic
204C22⋊D53—82no—
205C2×Dic53—41no—
206D212—41no—
207D212—41no—
208D212—41no—
209D212—41no—
210D212—81no—
211C4×D53—82no—
212Dic106—41no—
213Dic106—41no—
214C8×C53—42no—
215#88322—22no—
216#88322—22no—
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—82no—
238C4×C53#46612yes—
239C4×C53C22×D811yescharacteristic
240C4×C53C2×#13511yescharacteristic
241C4×C53#57711yescharacteristic
242C4×C53#57811yescharacteristic
243Dic53—81no—
244#3226—531no—
245#3039D5311yescharacteristic, complemented
246#3035—531no—
247#2988—531no—
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—21no—
262C2×#135—1061no—
263#532—531no—
264C2×C4⋊Q8C2×D5311yescharacteristic
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—21no—
274#462—531no—
275#462C2×D5312yes—
276#461—21no—
277#461—531no—
278#461C2×D5312yes—
279#460—531no—
280#459—531no—
281C8×Q8—532no—
282C8×D4—532no—
283C8×D4—1062no—
284#431—531no—
285#431—1061no—
286#423C2×D5311yescharacteristic
287C2×C4⋊C8—531no—
288C2×C4⋊C8—531no—
289C2×C4⋊C8—1061no—
290#419—21no—
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×#41C22⋊D5311yescharacteristic
301C22×Q8—21no—
302#137—44no—
303#135—2124no—
304C2×Q16—22no—
305C2×Q16—1061no—
306C2×SD16—22no—
307C2×SD16—1061no—
308C2×SD16—1061no—
309C2×D8—1061no—
310C2×M16—21no—
311C22×C8—1062no—
312C22×C8—1061no—
313C22×C8—1061no—
314C4⋊Q8—21no—
315C4⋊Q8—21no—
316C4⋊Q8—21no—
317C4⋊Q8—531no—
318C4⋊Q8C22×D5312yes—
319#127—531no—
320#124—1061no—
321#121—1061no—
322#121—1061no—
323C4×Q8C22⋊D5311yescharacteristic
324C4×Q8C22×D5311yescharacteristic
325C4×D4—21no—
326C4×D4—21no—
327C4×D4—1061no—
328C4×D4—1061no—
329C4×D4C22⋊D5311yescharacteristic
330C4×D4C22×D5311yescharacteristic
331#117—21no—
332#117C22⋊D5311yescharacteristic
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C42—21no—
336C2×C42C22×D5311yescharacteristic
337#107—1062no—
338#107—1062no—
339C4⋊C8—21no—
340C4⋊C8—531no—
341C4⋊C8—531no—
342C4⋊C8—532no—
343C4⋊C8—531no—
344C4⋊C8—531no—
345C4⋊C8—1062no—
346C4⋊C8—1061no—
347C4⋊C8—1061no—
348C4⋊C8C22×D5312yes—
349#103—22no—
350#103—22no—
351#103—1061no—
352#103—1061no—
353#102—1062no—
354#102—1062no—
355#102—1061no—
356#102—1061no—
357C22⋊C8—532no—
358C22⋊C8—1062no—
359C4×C8—532no—
360C4×C8—1062no—
361#41—22no—
362#41—2122no—
363#41—2122no—
364#41#1171992214yes—
365C2×Q8—22no—
366C2×Q8—41no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—1061no—
370C2×Q8C2×C22⋊D5311yescharacteristic
371C2×D4—21no—
372C2×D4—1061no—
373C2×D4—1061no—
374C2×D4—1061no—
375C2×D4—2121no—
376C2×D4C2×C22⋊D5311yescharacteristic
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—41no—
381C22×C4—2121no—
382C22×C4D4×D5311yescharacteristic
383C22×C4#1171994711yescharacteristic
384C22×C4C2×C22⋊D5311yescharacteristic
385Q16—44no—
386Q16—2122no—
387SD16—44no—
388SD16—2122no—
389SD16—2122no—
390D8—2122no—
391M16—42no—
392C2×C8—22no—
393C2×C8—532no—
394C2×C8—532no—
395C2×C8—1062no—
396C2×C8—1061no—
397C2×C8—1061no—
398C2×C8—4242no—
399C2×C8—4241no—
400C2×C8—2122no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—1061no—
408C4⋊C4—1061no—
409C4⋊C4C2×C22⋊D5311yescharacteristic
410C4⋊C4C2×C22⋊D5311yescharacteristic
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—21no—
414C22⋊C4—2121no—
415C22⋊C4—2121no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42D4×D5311yescharacteristic
420C42#1171994711yescharacteristic
421C42C2×C22⋊D5311yescharacteristic
422C42C23×D5311yescharacteristic
423C23—21no—
424C23—2121no—
425C23#1313318911yescharacteristic
426Q8—21no—
427Q8—42no—
428Q8—42no—
429Q8—82no—
430Q8—2121no—
431Q8—2121no—
432Q8C2×#1171992212yes—
433D4—24no—
434D4—21no—
435D4—4241no—
436D4—4242no—
437D4—2121no—
438D4—2121no—
439D4—2121no—
440D4—2121no—
441D4C2×#1171992212yes—
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—42no—
452C2×C4—81no—
453C2×C4—82no—
454C2×C4—4242no—
455C2×C4—2121no—
456C2×C4#1313318911yescharacteristic
457C2×C4#1313318911yescharacteristic
458C2×C4#1313318911yescharacteristic
459C2×C4C2×#1171992212yes—
460C2×C4#1313323712yes—
461C2×C4C2×D4×D5311yescharacteristic
462C2×C4C22×C22⋊D5311yescharacteristic
463C2×C4C2×#1171994711yescharacteristic
464C8—42no—
465C8—1062no—
466C8—1062no—
467C8—2121no—
468C8—2121no—
469C8—2122no—
470C22—21no—
471C22—21no—
472C22—22no—
473C22—41no—
474C22—41no—
475C22—4241no—
476C22—4241no—
477C22—4241no—
478C22#1681085712yes—
479C22C2×#1313318911yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—22no—
484C4—41no—
485C4—41no—
486C4—82no—
487C4—4241no—
488C4#1681085712yes—
489C4C22×#1171992211yescharacteristic
490C4C2×#1313323711yescharacteristic
491C4#1681143211yescharacteristic
492C4#1681143911yescharacteristic
493C2—22no—
494C2—42no—
495C2—8481no—
496C2#2028426911yescharacteristic
497C2C2×#1681085711yescharacteristic
498C2#2028444911yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3039C2
socleC22×C53C2×#121
prime radical#20284170C2
hypercenter#3039D53
hypocenterC53#19416
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3039

Complements

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

NH ≅ G/Ntype
1. #20285836C2semidirect
3. #20283826C2semidirect
4. #20274161C2semidirect
5. #20274154C2semidirect
6. #20274107C2semidirect
7. #20274102C2semidirect
8. #20273491C2semidirect
9. #20273393C2semidirect
10. C2×#16810369C2semidirect
11. C53×#3039C2semidirect
245. #3039D53semidirect
298. C53#19416semidirect

Maximal subgroups

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

subgroupindex
#202858362
#202841702
#202838262
#202741612
#202741542
#202741072
#202741022
#202734912
#202733932
C2×#168103692
C53×#30392
#1941653