Finite Group Databasebeta

← #27391042

Subgroups — #27391042

Order 13568.

868 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284435C211yesmaximal, characteristic
2#20283616C212yesmaximal, complemented
3C2×#16810746C211yesmaximal, characteristic, complemented
4#20275613C211yesmaximal, characteristic, complemented
5#20275505C211yesmaximal, characteristic, complemented
6#20275384C212yesmaximal, complemented
7#20275216C212yesmaximal, complemented
8#20274769C211yesmaximal, characteristic, complemented
9#20274599C211yesmaximal, characteristic, complemented
10#20274516C212yesmaximal, complemented
11C53×#2875C211yesmaximal, characteristic, complemented
12#16811421C2212yes—
13C2×#13133239C2211yescharacteristic
14C2×#13133223C2212yes—
15#16810898C412yescomplemented
16#16810893C412yescomplemented
17#16810886C414yescomplemented
18#16810746—21no—
19#16810746—21no—
20#16810746—21no—
21#16810746—21no—
22#16810743—24no—
23#16810738—24no—
24#16810671C2212yes—
25C2×#13133153C2211yescharacteristic
26C2×#13133153C2211yescharacteristic
27#16810661C2211yescharacteristic
28#16810640C2212yes—
29#16810639C2212yes—
30C2×#13133145C2212yes—
31#16810605C2212yes—
32#16810596C2212yes—
33#16810578C2212yes—
34#16810395C2212yes—
35#16810393C2212yes—
36#16810383C2212yes—
37C2×#131×C53C2211yescharacteristic
38C2×#131×C53C2211yescharacteristic
39C4×#41×C53C2211yescharacteristic
40C53×#405—21no—
41C53×#405—21no—
42C53×#405C2211yescharacteristic
43C53×#405C2211yescharacteristic
44C4×M16×C53—21no—
45C4×M16×C53C2211yescharacteristic
46C2×#97×C53—21no—
47C2×#97×C53C2211yescharacteristic
48C2×#11719945C2312yes—
49#13133248—22no—
50#13133247C2×C414yes—
51#13133243—22no—
52#13133242C2×C414yes—
53#13133239—21no—
54#13133239—21no—
55#13133239—21no—
56#13133239—21no—
57#13133237—24no—
58C2×Dic212C2×C411yescharacteristic, complemented
59C2×#11719913C2×C412yescomplemented
60C2×D424C2×C411yescharacteristic, complemented
61#13133223—22no—
62#13133223—22no—
63#13133223—22no—
64#13133223—22no—
65C2×#11719917C2312yes—
66C2×#11719917C2312yes—
67C2×#11719912—22no—
68C2×C8×D53—22no—
69#13133176C2312yes—
70#13133153—21no—
71#13133153—21no—
72#13133153C2×C412yescomplemented
73#13133153C2×C412yescomplemented
74#13133152—24no—
75#13133151—21no—
76#13133149—21no—
77#13133145—22no—
78#13133145—22no—
79#13133144—24no—
80#13133143—24no—
81#13133135C2×C414yescomplemented
82#13133133C2×C414yescomplemented
83#13133131—24no—
84#13133127—22no—
85C8×Dic53—22no—
86C2×#41×C53C2311yescharacteristic
87#131×C53—21no—
88#131×C53—21no—
89#131×C53—22no—
90#131×C53C2×C412yes—
91#131×C53C2×C412yes—
92#131×C53D414yes—
93C2×M16×C53—21no—
94C2×M16×C53—21no—
95C2×M16×C53C2311yescharacteristic
96C2×M16×C53C2311yescharacteristic
97C22×C8×C53—21no—
98C22×C8×C53—21no—
99C22×C8×C53C2311yescharacteristic
100C22×C8×C53C2311yescharacteristic
101C4×Q8×C53—21no—
102C4×D4×C53—22no—
103C4×D4×C53—21no—
104#117×C53—21no—
105#117×C53C2311yescharacteristic
106C2×C42×C53—21no—
107C2×C42×C53C2311yescharacteristic
108#97×C53—21no—
109#97×C53—21no—
110#97×C53—21no—
111#97×C53D411yescharacteristic
112#97×C53D411yescharacteristic
113#97×C53D411yescharacteristic
114#97×C53D411yescharacteristic
115C4×C8×C53—21no—
116C4×C8×C53—21no—
117C4×C8×C53—21no—
118#41×C53—21no—
119#41×C53—21no—
120#41×C53C22×C412yes—
121#41×C53C2×D412yes—
122C2×Q8×C53#4111yescharacteristic
123C2×D4×C53—21no—
124C2×D4×C53#4111yescharacteristic
125C22×C4×C53—21no—
126C22×C4×C53—22no—
127C22×C4×C53—21no—
128C22×C4×C53C2411yescharacteristic
129C2×C22⋊D53—22no—
130#11719945—22no—
131#11719945—22no—
132#11719945—22no—
133#11719945—22no—
134C2×D212C22×C412yes—
135C2×C4×D53—22no—
136C2×Dic106C22×C412yes—
137M16×C53—21no—
138M16×C53—21no—
139M16×C53—22no—
140M16×C53—22no—
141M16×C53—24no—
142M16×C53C22×C412yes—
143M16×C53C2×D412yes—
144C2×C8×C53—21no—
145C2×C8×C53—21no—
146C2×C8×C53—22no—
147C2×C8×C53—22no—
148C2×C8×C53—24no—
149C2×C8×C53C22×C411yescharacteristic
150C2×C8×C53C22×C411yescharacteristic
151C2×C8×C53C2×D411yescharacteristic
152C2×C8×C53C2×D411yescharacteristic
153C2×C8×C53C2×D412yes—
154C2×C8×C53C2×D411yescharacteristic
155C2×C8×C53C2×D411yescharacteristic
156C2×C8×C53#4111yescharacteristic
157C2×C8×C53#4111yescharacteristic
158C2×C8×C53#4112yes—
159C4⋊C4×C53—22no—
160C4⋊C4×C53—21no—
161C22⋊C4×C53—22no—
162C22⋊C4×C53#4112yes—
163C42×C53—22no—
164C42×C53—21no—
165C42×C53—21no—
166C42×C53C2×D411yescharacteristic
167C42×C53C2×D411yescharacteristic
168C22⋊Dic53—22no—
169#11719925—24no—
170#11719924—24no—
171#11719923—24no—
172#11719922—24no—
173C4⋊Dic53—22no—
174C4×Dic53—22no—
175#11719917—22no—
176#11719917—22no—
177#11719917—22no—
178#11719917C22×C414yes—
179C2×#88322—22no—
180C2×#88322—22no—
181Dic212—21no—
182Dic212—21no—
183D424—21no—
184D424—21no—
185#11719913—22no—
186#11719913—22no—
187#11719912—42no—
188#11719912—42no—
189C8×D53—42no—
190C8×D53—42no—
191C23×C53—21no—
192C23×C53C2×#4111yescharacteristic
193C22×D53—22no—
194Q8×C53—21no—
195Q8×C53C4×D412yes—
196D4×C53—22no—
197D4×C53—22no—
198D4×C53—21no—
199D4×C53C4×D412yes—
200C2×C4×C53—21no—
201C2×C4×C53—22no—
202C2×C4×C53—22no—
203C2×C4×C53—21no—
204C2×C4×C53—22no—
205C2×C4×C53—21no—
206C2×C4×C53—81no—
207C2×C4×C53—82no—
208C2×C4×C53C23×C411yescharacteristic
209C2×C4×C53C22×D411yescharacteristic
210C2×C4×C53C22×D411yescharacteristic
211C2×C4×C53C2×#4111yescharacteristic
212C2×C4×C53C2×#4111yescharacteristic
213C2×C4×C53C2×#4111yescharacteristic
214C22⋊D53—42no—
215C22⋊D53—42no—
216C2×Dic53—22no—
217C2×Dic53—42no—
218D212—22no—
219D212—22no—
220C4×D53—42no—
221C4×D53—42no—
222Dic106—22no—
223Dic106—22no—
224C8×C53—22no—
225C8×C53—22no—
226C8×C53C4×D411yescharacteristic
227C8×C53C4×D411yescharacteristic
228C8×C53C4×D411yescharacteristic
229C8×C53C4×D411yescharacteristic
230C8×C53#12112yes—
231C8×C53#12112yes—
232#88322—24no—
233#88322—42no—
234#21559—531nomaximal, Sylow
235C22×C53—22no—
236C22×C53—41no—
237C22×C53—42no—
238C22×C53C4×#4111yescharacteristic
239C22×C53#52511yescharacteristic
240C22×C53#53211yescharacteristic
241C2×D53—42no—
242C2×D53—82no—
243C4×C53—22no—
244C4×C53—42no—
245C4×C53—41no—
246C4×C53—41no—
247C4×C53C2×C4×D411yescharacteristic
248C4×C53C2×C4×D411yescharacteristic
249C4×C53C2×#12111yescharacteristic
250C4×C53#52911yescharacteristic
251Dic53—42no—
252Dic53—82no—
253#3067—531no—
254#2955—532no—
255C2×#444—531no—
256#2875D5311yescharacteristic, complemented
257#1976—531no—
258#1953—531no—
259#1932—532no—
260#1906—532no—
261#1814—531no—
262#1773—531no—
263#1757—532no—
264C2×C53—21no—
265C2×C53—21no—
266C2×C53—21no—
267C2×C53—42no—
268C2×C53C4×#12111yescharacteristic
269D53—82no—
270#575—532no—
271C2×#135—531no—
272C2×#131—532no—
273C2×#131C2×D5311yescharacteristic
274C2×#131C2×D5311yescharacteristic
275C4×#41C2×D5311yescharacteristic
276#473—532no—
277#472—532no—
278#471—534no—
279#444—1064no—
280#444—1061no—
281#444—1061no—
282#444—1061no—
283#444—1061no—
284#443—1064no—
285#430—532no—
286C2×#108—531no—
287C2×#108—531no—
288#427—531no—
289#421—532no—
290#421—532no—
291C2×#104—532no—
292#413—532no—
293#410—532no—
294#405—21no—
295#405—21no—
296#405—532no—
297#405C2×D5311yescharacteristic
298#405C2×D5311yescharacteristic
299C4×M16—21no—
300C4×M16C2×D5311yescharacteristic
301C2×#97—21no—
302C2×#97C2×D5311yescharacteristic
303#344—532no—
304#344—532no—
305#339—532no—
306C53#2155911yescharacteristic, complemented, Sylow
307C2×#41—532no—
308C2×#41C22×D5311yescharacteristic
309#137—534no—
310#137—1062no—
311AGL(1,ℤ/8)—534no—
312AGL(1,ℤ/8)—1062no—
313#135—1064no—
314#135—1061no—
315#135—1061no—
316#135—1061no—
317#135—1061no—
318C2×Q16—531no—
319C2×SD16—532no—
320C2×D8—531no—
321#131—21no—
322#131—21no—
323#131—22no—
324#131—1062no—
325#131—1062no—
326#131—1062no—
327#131—1062no—
328#131C4×D5312yes—
329#131C4×D5312yes—
330#131D21214yes—
331C2×M16—21no—
332C2×M16—21no—
333C2×M16—532no—
334C2×M16—532no—
335C2×M16—1062no—
336C2×M16C22×D5311yescharacteristic
337C2×M16C22×D5311yescharacteristic
338C22×C8—21no—
339C22×C8—21no—
340C22×C8—1062no—
341C22×C8C22×D5311yescharacteristic
342C22×C8C22×D5311yescharacteristic
343C4×Q8—21no—
344C4×D4—21no—
345C4×D4—22no—
346#117—21no—
347#117—532no—
348#117C22×D5311yescharacteristic
349C2×C42—21no—
350C2×C42C22×D5311yescharacteristic
351#108—532no—
352#108—532no—
353#108—1064no—
354#108—1061no—
355#108—1061no—
356#107—1061no—
357#106—1061no—
358#104—1064no—
359#104—1064no—
360#104—1062no—
361#104—1062no—
362#101—534no—
363#100—534no—
364#99—1064no—
365#97—21no—
366#97—21no—
367#97—21no—
368#97—1062no—
369#97C22⋊D5311yescharacteristic
370#97C22⋊D5311yescharacteristic
371#97C22⋊D5311yescharacteristic
372#97C22⋊D5311yescharacteristic
373C4×C8—21no—
374C4×C8—21no—
375C4×C8—21no—
376C4×C8—1062no—
377#41—21no—
378#41—21no—
379#41—1062no—
380#41—1062no—
381#41—1062no—
382#41—1062no—
383#41C2×C4×D5312yes—
384#41C2×D21212yes—
385C2×Q8—532no—
386C2×Q8#1171994511yescharacteristic
387C2×D4—21no—
388C2×D4—532no—
389C2×D4—1062no—
390C2×D4#1171994511yescharacteristic
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—1062no—
395C22×C4C23×D5311yescharacteristic
396Q16—1064no—
397Q16—1061no—
398Q16—1061no—
399SD16—1064no—
400SD16—1064no—
401SD16—1062no—
402SD16—1062no—
403D8—1064no—
404D8—1061no—
405D8—1061no—
406M16—21no—
407M16—21no—
408M16—22no—
409M16—22no—
410M16—24no—
411M16—534no—
412M16—1062no—
413M16—1062no—
414M16—1062no—
415M16—2122no—
416M16—2122no—
417M16C2×C4×D5312yes—
418M16C2×D21212yes—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—24no—
424C2×C8—1062no—
425C2×C8—1062no—
426C2×C8—2122no—
427C2×C8—2122no—
428C2×C8C2×C4×D5311yescharacteristic
429C2×C8C2×C4×D5311yescharacteristic
430C2×C8C2×D21212yes—
431C2×C8#1171994511yescharacteristic
432C2×C8#1171994511yescharacteristic
433C2×C8#1171994512yes—
434C2×C8C2×C22⋊D5311yescharacteristic
435C2×C8C2×C22⋊D5311yescharacteristic
436C2×C8C2×C22⋊D5311yescharacteristic
437C2×C8C2×C22⋊D5311yescharacteristic
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—1062no—
441C22⋊C4—22no—
442C22⋊C4—1062no—
443C22⋊C4#1171994512yes—
444C42—22no—
445C42—21no—
446C42—21no—
447C42—1062no—
448C42C2×C22⋊D5311yescharacteristic
449C42C2×C22⋊D5311yescharacteristic
450C23—21no—
451C23—1062no—
452C23C2×#1171994511yescharacteristic
453Q8—21no—
454Q8—1062no—
455Q8—1062no—
456Q8C4×D21212yes—
457D4—22no—
458D4—22no—
459D4—21no—
460D4—1062no—
461D4—1062no—
462D4—2122no—
463D4—2122no—
464D4C4×D21212yes—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—81no—
472C2×C4—82no—
473C2×C4—1062no—
474C2×C4—2122no—
475C2×C4—2122no—
476C2×C4—2122no—
477C2×C4C22×C4×D5311yescharacteristic
478C2×C4C22×C22⋊D5311yescharacteristic
479C2×C4C22×D21211yescharacteristic
480C2×C4C2×#1171994511yescharacteristic
481C2×C4C2×#1171994511yescharacteristic
482C2×C4C2×#1171994511yescharacteristic
483C8—22no—
484C8—22no—
485C8—1064no—
486C8—2122no—
487C8C4×C22⋊D5311yescharacteristic
488C8C4×C22⋊D5311yescharacteristic
489C8C4×C22⋊D5311yescharacteristic
490C8C4×C22⋊D5311yescharacteristic
491C8#1313319112yes—
492C8#1313319112yes—
493C22—22no—
494C22—41no—
495C22—42no—
496C22—4242no—
497C22—2122no—
498C22C4×#1171994511yescharacteristic
499C22#1681113411yescharacteristic
500C22#1681117311yescharacteristic
501C4—22no—
502C4—41no—
503C4—42no—
504C4—41no—
505C4—4242no—
506C4—2122no—
507C4C2×C4×C22⋊D5311yescharacteristic
508C4C2×C4×D21211yescharacteristic
509C4C2×#1313319111yescharacteristic
510C4#1681115711yescharacteristic
511C2—21no—
512C2—21no—
513C2—21no—
514C2—42no—
515C2—4242no—
516C2C4×#1313319111yescharacteristic
subgroupG/H
centerC4C2×#13133191
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2875C2
socleC2×C53C4×#121
prime radical#20284435C2
cosocleC22×C4×C53C24
hypercenter#2875D53
hypocenterC53#21559
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C8 ◁ #2875

Complements

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

NH ≅ G/Ntype
2. #20283616C2semidirect
3. C2×#16810746C2semidirect
4. #20275613C2semidirect
5. #20275505C2semidirect
6. #20275384C2semidirect
7. #20275216C2semidirect
8. #20274769C2semidirect
9. #20274599C2semidirect
10. #20274516C2semidirect
11. C53×#2875C2semidirect
15. #16810898C4semidirect
16. #16810893C4semidirect
17. #16810886C4semidirect
58. C2×Dic212C2×C4semidirect
59. C2×#11719913C2×C4semidirect
60. C2×D424C2×C4semidirect
72. #13133153C2×C4semidirect
73. #13133153C2×C4semidirect
81. #13133135C2×C4semidirect
82. #13133133C2×C4semidirect
256. #2875D53semidirect
306. C53#21559semidirect

Maximal subgroups

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

subgroupindex
#202844352
#202836162
C2×#168107462
#202756132
#202755052
#202753842
#202752162
#202747692
#202745992
#202745162
C53×#28752
#2155953