Finite Group Databasebeta

← #27369578

Subgroups — #27369578

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#20285833C211yesmaximal, characteristic, complemented
2#20284202C211yesmaximal, characteristic, complemented
3#20283826C211yesmaximal, characteristic, complemented
4#20274157C212yesmaximal, complemented
5#20274150C212yesmaximal, complemented
6#20274107C212yesmaximal, complemented
7#20274102C212yesmaximal, complemented
8#20273489C211yesmaximal, characteristic, complemented
9#20273392C211yesmaximal, characteristic, complemented
10C2×#16810370C211yesmaximal, characteristic, complemented
11C53×#3035C211yesmaximal, characteristic
12C2×#13133247—21no—
13C2×#13133217C2211yescharacteristic
14#16811026C2212yes—
15#16810969C2212yes—
16#16810934—22no—
17#16810903—21no—
18#16810903—21no—
19#16810833—21no—
20#16810833C2212yes—
21#16810828—21no—
22#16810828C2212yes—
23Q8×#88322C2212yes—
24D4×#88322—22no—
25D4×#88322C2212yes—
26#16810676—21no—
27#16810676C2211yescharacteristic
28#16810647C2211yescharacteristic
29C2×C4⋊#88322—21no—
30C2×C4⋊#88322C2211yescharacteristic
31#16810631—21no—
32#16810375C2212yes—
33#16810375C2212yes—
34#16810370C2212yes—
35#16810370C2212yes—
36#16810369C2212yes—
37#16810367C2212yes—
38C2×#135×C53—21no—
39C53×#532C2211yescharacteristic
40C4×#41×C53C2211yescharacteristic
41C53×#464—21no—
42C53×#463—21no—
43C53×#462C2211yescharacteristic
44C53×#461C2211yescharacteristic
45C53×#460C2211yescharacteristic
46C53×#459C2211yescharacteristic
47C2×C4⋊C8×C53C2211yescharacteristic
48C53×#416—21no—
49C22×Dic106—21no—
50#13133247—44no—
51C2×#11719925—22no—
52C2×#11719923—22no—
53C2×#11719917—21no—
54C22×#88322—22no—
55C22×#88322—21no—
56#13133217—21no—
57#13133217—21no—
58#13133217—21no—
59#13133217C2312yes—
60C2×C4⋊Dic53—21no—
61C2×C4⋊Dic53—21no—
62#13133150—22no—
63#13133150—22no—
64C4⋊#88322—21no—
65C4⋊#88322—21no—
66C4⋊#88322—22no—
67C4⋊#88322D411yescharacteristic
68C4⋊#88322D411yescharacteristic
69C4⋊#88322C2312yes—
70C4⋊#88322C2312yes—
71#13133140—22no—
72#13133140—22no—
73#13133138—22no—
74#13133138—22no—
75C22⋊#88322—22no—
76C22⋊#88322D412yes—
77C4×#88322—22no—
78C4×#88322C2312yes—
79C2×#41×C53—21no—
80C2×#41×C53D411yescharacteristic
81#135×C53—44no—
82C2×Q16×C53—21no—
83C2×SD16×C53—21no—
84C2×SD16×C53—21no—
85C2×D8×C53—21no—
86C22×C8×C53—21no—
87C4⋊Q8×C53C2311yescharacteristic
88#127×C53C2311yescharacteristic
89#124×C53—21no—
90#121×C53—21no—
91#121×C53—21no—
92C4×Q8×C53D411yescharacteristic
93C4×Q8×C53C2311yescharacteristic
94C4×D4×C53—21no—
95C4×D4×C53—21no—
96C4×D4×C53—21no—
97C4×D4×C53—21no—
98C4×D4×C53D411yescharacteristic
99C4×D4×C53C2311yescharacteristic
100#117×C53—21no—
101#117×C53D411yescharacteristic
102C2×C42×C53—21no—
103C2×C42×C53C2311yescharacteristic
104C4⋊C8×C53—21no—
105C4⋊C8×C53C2311yescharacteristic
106C4⋊C8×C53C2311yescharacteristic
107#103×C53—21no—
108#103×C53—21no—
109#102×C53—21no—
110#102×C53—21no—
111#41×C53—22no—
112#41×C53—42no—
113#41×C53—42no—
114#41×C53D814yes—
115C2×Q8×C53—21no—
116C2×Q8×C53C2×D411yescharacteristic
117C2×D4×C53—21no—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—41no—
122C2×D4×C53C2×D411yescharacteristic
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—41no—
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53#4111yescharacteristic
130C22×Dic53—41no—
131C2×Dic106—22no—
132C2×Dic106—42no—
133C2×Dic106—42no—
134C2×Dic106—41no—
135Q16×C53—42no—
136SD16×C53—42no—
137SD16×C53—42no—
138D8×C53—42no—
139C2×C8×C53—21no—
140C2×C8×C53—21no—
141C2×C8×C53—42no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53C2×D411yescharacteristic
147C4⋊C4×C53C2×D411yescharacteristic
148C22⋊C4×C53—21no—
149C22⋊C4×C53—21no—
150C22⋊C4×C53—21no—
151C22⋊C4×C53—41no—
152C22⋊C4×C53—41no—
153C42×C53—21no—
154C42×C53—21no—
155C42×C53—21no—
156C42×C53C2×D411yescharacteristic
157C42×C53C2×D411yescharacteristic
158C42×C53#4111yescharacteristic
159C42×C53C2411yescharacteristic
160#11719925—44no—
161#11719923—44no—
162C4⋊Dic53—22no—
163C4⋊Dic53—22no—
164C4⋊Dic53—41no—
165C4⋊Dic53—41no—
166#11719917—42no—
167C2×#88322—22no—
168C2×#88322—22no—
169C2×#88322—82no—
170C2×#88322—81no—
171C2×#88322C2×D412yes—
172C2×#88322C2×D412yes—
173C23×C53—21no—
174C23×C53—41no—
175C23×C53#12111yescharacteristic
176Q8×C53—21no—
177Q8×C53—41no—
178Q8×C53—41no—
179Q8×C53C2×D812yes—
180D4×C53—24no—
181D4×C53—21no—
182D4×C53—41no—
183D4×C53—41no—
184D4×C53—41no—
185D4×C53—41no—
186D4×C53—81no—
187D4×C53—82no—
188D4×C53C2×D812yes—
189C2×C4×C53—21no—
190C2×C4×C53—21no—
191C2×C4×C53—21no—
192C2×C4×C53—24no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—41no—
196C2×C4×C53—41no—
197C2×C4×C53—41no—
198C2×C4×C53—41no—
199C2×C4×C53—81no—
200C2×C4×C53—82no—
201C2×C4×C53#12111yescharacteristic
202C2×C4×C53#12111yescharacteristic
203C2×C4×C53#12111yescharacteristic
204C2×C4×C53C2×D812yes—
205C2×C4×C53#13512yes—
206C2×C4×C53C22×D411yescharacteristic
207C2×C4×C53C22×D411yescharacteristic
208C2×C4×C53C2×#4111yescharacteristic
209C2×Dic53—42no—
210C2×Dic53—82no—
211Dic106—42no—
212Dic106—42no—
213Dic106—82no—
214C8×C53—41no—
215C8×C53—41no—
216#88322—22no—
217#88322—22no—
218#88322—42no—
219#88322—42no—
220#19416—531nomaximal, Sylow
221C22×C53—21no—
222C22×C53—21no—
223C22×C53—22no—
224C22×C53—41no—
225C22×C53—41no—
226C22×C53—81no—
227C22×C53—81no—
228C22×C53—81no—
229C22×C53#46612yes—
230C22×C53C2×#12111yescharacteristic
231C4×C53—21no—
232C4×C53—22no—
233C4×C53—21no—
234C4×C53—21no—
235C4×C53—41no—
236C4×C53—41no—
237C4×C53—81no—
238C4×C53#46612yes—
239C4×C53C22×D811yescharacteristic
240C4×C53C2×#13511yescharacteristic
241C4×C53#57711yescharacteristic
242C4×C53#57811yescharacteristic
243Dic53—82no—
244#3226—531no—
245#3039—531no—
246#3035D5311yescharacteristic
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—161no—
258C2×C53#304611yescharacteristic
259C2×C53C2×#46611yescharacteristic
260C2×C53#306911yescharacteristic
261C2×#137—1061no—
262C2×#135—21no—
263#532C2×D5311yescharacteristic
264C2×C4⋊Q8—531no—
265C4×#41C2×D5311yescharacteristic
266#500—532no—
267#486—532no—
268#480—1062no—
269#474—1061no—
270#474—1061no—
271#464—21no—
272#463—21no—
273#462—532no—
274#462—1061no—
275#462C2×D5311yescharacteristic
276#461—532no—
277#461—1061no—
278#461C2×D5311yescharacteristic
279#460C2×D5311yescharacteristic
280#459C2×D5311yescharacteristic
281C8×Q8—532no—
282C8×D4—532no—
283C8×D4—1062no—
284#431—531no—
285#431—1061no—
286#423—531no—
287C2×C4⋊C8—531no—
288C2×C4⋊C8—1061no—
289C2×C4⋊C8C2×D5311yescharacteristic
290#419—1061no—
291#416—21no—
292#335—532no—
293#335—532no—
294#332—532no—
295#332—532no—
296#332—532no—
297#331—532no—
298C53#1941611yescharacteristic, complemented, Sylow
299C2×#41—21no—
300C2×#41C22⋊D5311yescharacteristic
301C22×Q8—1061no—
302#137—2124no—
303#135—44no—
304C2×Q16—21no—
305C2×Q16—1062no—
306C2×SD16—21no—
307C2×SD16—21no—
308C2×SD16—1062no—
309C2×D8—21no—
310C2×M16—1061no—
311C22×C8—21no—
312C22×C8—1062no—
313C22×C8—1061no—
314C4⋊Q8—532no—
315C4⋊Q8—1061no—
316C4⋊Q8—1061no—
317C4⋊Q8—1061no—
318C4⋊Q8C22×D5311yescharacteristic
319#127C22×D5311yescharacteristic
320#124—21no—
321#121—21no—
322#121—21no—
323C4×Q8C22⋊D5311yescharacteristic
324C4×Q8C22×D5311yescharacteristic
325C4×D4—21no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4—21no—
329C4×D4C22⋊D5311yescharacteristic
330C4×D4C22×D5311yescharacteristic
331#117—21no—
332#117C22⋊D5311yescharacteristic
333C2×C4⋊C4—1061no—
334C2×C4⋊C4—1061no—
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—532no—
344C4⋊C8—1062no—
345C4⋊C8—1061no—
346C4⋊C8—1061no—
347C4⋊C8C22×D5311yescharacteristic
348C4⋊C8C22×D5311yescharacteristic
349#103—21no—
350#103—21no—
351#103—1062no—
352#103—1062no—
353#102—21no—
354#102—21no—
355#102—1062no—
356#102—1062no—
357C22⋊C8—532no—
358C22⋊C8—1062no—
359C4×C8—532no—
360C4×C8—1062no—
361#41—22no—
362#41—42no—
363#41—42no—
364#41#1171992214yes—
365C2×Q8—21no—
366C2×Q8—1062no—
367C2×Q8—2122no—
368C2×Q8—2122no—
369C2×Q8—2121no—
370C2×Q8C2×C22⋊D5311yescharacteristic
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—41no—
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—42no—
386Q16—2124no—
387SD16—42no—
388SD16—42no—
389SD16—2124no—
390D8—42no—
391M16—2122no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—42no—
395C2×C8—532no—
396C2×C8—532no—
397C2×C8—1062no—
398C2×C8—1062no—
399C2×C8—4242no—
400C2×C8—4241no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—1062no—
406C4⋊C4—1062no—
407C4⋊C4—2121no—
408C4⋊C4—2121no—
409C4⋊C4C2×C22⋊D5311yescharacteristic
410C4⋊C4C2×C22⋊D5311yescharacteristic
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—21no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42D4×D5311yescharacteristic
420C42#1171994711yescharacteristic
421C42C2×C22⋊D5311yescharacteristic
422C42C23×D5311yescharacteristic
423C23—21no—
424C23—41no—
425C23#1313318911yescharacteristic
426Q8—21no—
427Q8—41no—
428Q8—41no—
429Q8—4242no—
430Q8—2122no—
431Q8—2122no—
432Q8C2×#1171992212yes—
433D4—24no—
434D4—21no—
435D4—41no—
436D4—41no—
437D4—41no—
438D4—41no—
439D4—81no—
440D4—82no—
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—41no—
452C2×C4—81no—
453C2×C4—82no—
454C2×C4—4242no—
455C2×C4—2122no—
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—41no—
465C8—41no—
466C8—1062no—
467C8—1062no—
468C8—2122no—
469C8—2122no—
470C22—21no—
471C22—21no—
472C22—22no—
473C22—41no—
474C22—41no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22#1681085712yes—
479C22C2×#1313318911yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—22no—
484C4—41no—
485C4—41no—
486C4—81no—
487C4—4242no—
488C4#1681085712yes—
489C4C22×#1171992211yescharacteristic
490C4C2×#1313323711yescharacteristic
491C4#1681143011yescharacteristic
492C4#1681143711yescharacteristic
493C2—22no—
494C2—42no—
495C2—161no—
496C2#2028426311yescharacteristic
497C2C2×#1681085711yescharacteristic
498C2#2028445411yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3035C2
socleC22×C53C2×#121
hypercenter#3035D53
hypocenterC53#19416
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3035

Complements

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

NH ≅ G/Ntype
1. #20285833C2semidirect
2. #20284202C2semidirect
3. #20283826C2semidirect
4. #20274157C2semidirect
5. #20274150C2semidirect
6. #20274107C2semidirect
7. #20274102C2semidirect
8. #20273489C2semidirect
9. #20273392C2semidirect
10. C2×#16810370C2semidirect
298. C53#19416semidirect

Maximal subgroups

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

subgroupindex
#202858332
#202842022
#202838262
#202741572
#202741502
#202741072
#202741022
#202734892
#202733922
C2×#168103702
C53×#30352
#1941653