Finite Group Databasebeta

← #27391045

Subgroups — #27391045

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#20284431C211yesmaximal, characteristic
2#20283614C212yesmaximal, complemented
3C2×#16810745C211yesmaximal, characteristic, complemented
4#20283142C211yesmaximal, characteristic, complemented
5#20275505C211yesmaximal, characteristic, complemented
6#20275387C212yesmaximal, complemented
7#20275213C212yesmaximal, complemented
8#20274766C211yesmaximal, characteristic, complemented
9#20274601C211yesmaximal, characteristic, complemented
10#20274517C212yesmaximal, complemented
11C53×#1976C211yesmaximal, characteristic, complemented
12#16811421C2212yes—
13C2×#13133225C2211yescharacteristic
14C2×#13133224C2212yes—
15C2×#13133223C2211yescharacteristic
16#16811091C2211yescharacteristic
17#16810895C412yescomplemented
18#16810890C412yescomplemented
19#16810883C414yescomplemented
20#16810745—21no—
21#16810745—21no—
22#16810745—21no—
23#16810745—21no—
24#16810742—24no—
25#16810737—24no—
26#16810671C2212yes—
27C2×#13133153C2211yescharacteristic
28C2×#13133153C2211yescharacteristic
29#16810641C2212yes—
30C2×#13133144C2212yes—
31#16810607C2212yes—
32#16810580—21no—
33#16810579C2211yescharacteristic
34#16810579C2212yes—
35#16810577—21no—
36M16×Dic53C2211yescharacteristic
37C4×#11719912—21no—
38C2×#13133127C2211yescharacteristic
39#97×D53—21no—
40#16810393C2212yes—
41#16810393C2212yes—
42#16810382C2212yes—
43C2×#135×C53C2211yescharacteristic
44C53×#427C2211yescharacteristic
45C53×#421C2212yes—
46C53×#410C2212yes—
47C53×#405C2211yescharacteristic
48C2×#11719945C2311yescharacteristic
49#13133248C2×C414yes—
50#13133247—22no—
51#13133243C2×C414yes—
52#13133242—22no—
53#13133237—24no—
54#13133225—22no—
55#13133225D414yes—
56#13133224—22no—
57#13133224—22no—
58#13133224—22no—
59#13133224—22no—
60#13133223—21no—
61#13133223—21no—
62#13133223C2×C412yes—
63#13133223C2×C412yes—
64C2×#11719917—22no—
65C2×#11719917C2312yes—
66C2×#11719917C2311yescharacteristic
67C2×#11719917C2312yes—
68C2×#11719912—21no—
69C2×#11719912—21no—
70C22×#88322—22no—
71C22×#88322C2311yescharacteristic
72C2×C8×D53—21no—
73C2×C8×D53—21no—
74#13133182—21no—
75#13133179—21no—
76#13133177—22no—
77#13133174—21no—
78C2×C4×Dic53C2311yescharacteristic
79C42×D53—21no—
80#13133153—21no—
81#13133153—21no—
82#13133153C2×C412yescomplemented
83#13133153C2×C412yescomplemented
84#13133152—24no—
85#13133145—24no—
86#13133144—22no—
87#13133144—22no—
88#13133136C2×C414yescomplemented
89#13133134C2×C414yescomplemented
90#13133128—21no—
91#13133128—22no—
92#13133127—21no—
93#13133127D411yescharacteristic
94#13133127D411yescharacteristic
95#13133127D411yescharacteristic
96#13133127D411yescharacteristic
97C4×#88322—21no—
98C4×#88322—22no—
99C8×Dic53—21no—
100C2×#41×C53C2312yes—
101#135×C53—21no—
102#135×C53—21no—
103#135×C53—21no—
104#135×C53—21no—
105C2×Q16×C53C2×C411yescharacteristic, complemented
106C2×SD16×C53C2×C412yescomplemented
107C2×D8×C53C2×C411yescharacteristic, complemented
108C2×M16×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110#117×C53C2312yes—
111#117×C53C2311yescharacteristic
112#107×C53—21no—
113#106×C53—21no—
114#104×C53—24no—
115#99×C53—24no—
116#97×C53—21no—
117C4×C8×C53—21no—
118#41×C53—22no—
119#41×C53—22no—
120#41×C53—22no—
121#41×C53—22no—
122C2×Q8×C53C22×C412yes—
123C2×D4×C53—22no—
124C2×D4×C53C22×C412yes—
125C22×C4×C53—22no—
126C22×C4×C53C2411yescharacteristic
127C2×C22⋊D53—21no—
128C22×Dic53—21no—
129#11719945—21no—
130#11719945—21no—
131#11719945C22×C412yes—
132#11719945C2×D412yes—
133C2×D212#4111yescharacteristic
134C2×C4×D53—21no—
135C2×C4×D53—22no—
136C2×Dic106#4111yescharacteristic
137Q16×C53—21no—
138Q16×C53—21no—
139SD16×C53—22no—
140SD16×C53—22no—
141D8×C53—21no—
142D8×C53—21no—
143M16×C53—21no—
144M16×C53C2×D412yes—
145C2×C8×C53—21no—
146C2×C8×C53C22×C411yescharacteristic
147C2×C8×C53C22×C411yescharacteristic
148C2×C8×C53C2×D411yescharacteristic
149C2×C8×C53C2×D411yescharacteristic
150C2×C8×C53#4111yescharacteristic
151C2×C8×C53#4111yescharacteristic
152C4⋊C4×C53—21no—
153C4⋊C4×C53—22no—
154C22⋊C4×C53—22no—
155C22⋊C4×C53#4112yes—
156C42×C53—21no—
157C42×C53—22no—
158#11719925—24no—
159#11719924—24no—
160#11719923—24no—
161#11719922—24no—
162#11719921—22no—
163#11719919—22no—
164C4×Dic53—22no—
165C4×Dic53—21no—
166C4×Dic53C2×D411yescharacteristic
167C4×Dic53C2×D411yescharacteristic
168#11719917—21no—
169#11719917—22no—
170#11719917—22no—
171#11719917—22no—
172#11719917—42no—
173#11719917—42no—
174#11719917C22×C412yes—
175#11719917C22×C414yes—
176C2×#88322—21no—
177C2×#88322—22no—
178C2×#88322—22no—
179C2×#88322—42no—
180C2×#88322—42no—
181C2×#88322C2×D411yescharacteristic
182C2×#88322C2×D412yes—
183C2×#88322C2×D411yescharacteristic
184C2×#88322#4112yes—
185#11719912—22no—
186#11719912—22no—
187#11719912—24no—
188C8×D53—22no—
189C8×D53—22no—
190C8×D53—24no—
191C23×C53—22no—
192C23×C53C2×#4111yescharacteristic
193C22×D53—21no—
194Q8×C53—22no—
195Q8×C53—22no—
196D4×C53—22no—
197D4×C53—22no—
198D4×C53—42no—
199D4×C53—42no—
200C2×C4×C53—22no—
201C2×C4×C53—22no—
202C2×C4×C53—42no—
203C2×C4×C53—42no—
204C2×C4×C53—42no—
205C2×C4×C53C23×C411yescharacteristic
206C2×C4×C53C22×D411yescharacteristic
207C2×C4×C53C22×D411yescharacteristic
208C2×C4×C53C2×#4111yescharacteristic
209C2×C4×C53C2×#4111yescharacteristic
210C2×C4×C53C2×#4111yescharacteristic
211C22⋊D53—22no—
212C22⋊D53—22no—
213C2×Dic53—21no—
214C2×Dic53—21no—
215C2×Dic53—21no—
216C2×Dic53—81no—
217D212—21no—
218D212C4×D412yes—
219C4×D53—22no—
220C4×D53—22no—
221C4×D53—82no—
222Dic106—21no—
223Dic106C4×D412yes—
224C8×C53—22no—
225C8×C53C4×D411yescharacteristic
226C8×C53C4×D411yescharacteristic
227C8×C53C4×D411yescharacteristic
228C8×C53C4×D411yescharacteristic
229#88322—22no—
230#88322—24no—
231#88322—42no—
232#88322#12112yes—
233#88322#12112yes—
234#21559—531nomaximal, Sylow
235C22×C53—41no—
236C22×C53—42no—
237C22×C53—82no—
238C22×C53C4×#4111yescharacteristic
239C22×C53#52511yescharacteristic
240C22×C53#53211yescharacteristic
241C2×D53—22no—
242C2×D53—42no—
243C4×C53—42no—
244C4×C53—42no—
245C4×C53—82no—
246C4×C53C2×C4×D411yescharacteristic
247C4×C53C2×C4×D411yescharacteristic
248C4×C53C2×#12111yescharacteristic
249C4×C53#52911yescharacteristic
250Dic53—22no—
251Dic53—41no—
252Dic53—41no—
253#3067—531no—
254#2955—532no—
255C2×#444—531no—
256#2875—531no—
257#1976D5311yescharacteristic, complemented
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—82no—
268C2×C53C4×#12111yescharacteristic
269D53—42no—
270#575—532no—
271C2×#135C2×D5311yescharacteristic
272C2×#131—532no—
273C2×#131—531no—
274C2×#131—531no—
275C4×#41—531no—
276#473—532no—
277#472—532no—
278#471—534no—
279#444—1061no—
280#444—1061no—
281#444—1061no—
282#444—1061no—
283#444—1064no—
284#443—1064no—
285#430—532no—
286C2×#108—531no—
287C2×#108—531no—
288#427C2×D5311yescharacteristic
289#421—532no—
290#421C2×D5312yes—
291C2×#104—532no—
292#413—532no—
293#410C2×D5312yes—
294#405—531no—
295#405—532no—
296#405—1061no—
297#405—1061no—
298#405C2×D5311yescharacteristic
299C4×M16—531no—
300C4×M16—1061no—
301C2×#97—531no—
302C2×#97—1061no—
303#344—532no—
304#344—532no—
305#339—532no—
306C53#2155911yescharacteristic, complemented, Sylow
307C2×#41—531no—
308C2×#41C22×D5312yes—
309#137—534no—
310#137—1062no—
311AGL(1,ℤ/8)—534no—
312AGL(1,ℤ/8)—1062no—
313#135—21no—
314#135—21no—
315#135—21no—
316#135—21no—
317#135—1064no—
318C2×Q16C4×D5311yescharacteristic, complemented
319C2×SD16C4×D5312yescomplemented
320C2×D8C4×D5311yescharacteristic, complemented
321#131—534no—
322#131—532no—
323#131—532no—
324#131—1062no—
325#131—1062no—
326#131—1062no—
327#131—1062no—
328#131—1062no—
329#131—1061no—
330#131—1061no—
331C2×M16—531no—
332C2×M16—532no—
333C2×M16—532no—
334C2×M16—1061no—
335C2×M16—1062no—
336C2×M16—1061no—
337C2×M16C22×D5311yescharacteristic
338C22×C8—531no—
339C22×C8—1061no—
340C22×C8—1062no—
341C22×C8—1061no—
342C22×C8C22×D5311yescharacteristic
343C4×Q8—1061no—
344C4×D4—1061no—
345C4×D4—1062no—
346#117—1061no—
347#117C22×D5312yes—
348#117C22×D5311yescharacteristic
349C2×C42—531no—
350C2×C42—1061no—
351#108—532no—
352#108—532no—
353#108—1064no—
354#108—1061no—
355#108—1061no—
356#107—21no—
357#106—21no—
358#104—24no—
359#104—1064no—
360#104—1062no—
361#104—1062no—
362#101—534no—
363#100—534no—
364#99—24no—
365#97—21no—
366#97—531no—
367#97—531no—
368#97—531no—
369#97—531no—
370#97—1061no—
371#97—1062no—
372#97—1061no—
373C4×C8—21no—
374C4×C8—1061no—
375C4×C8—1062no—
376C4×C8—1061no—
377#41—22no—
378#41—22no—
379#41—22no—
380#41—22no—
381#41—532no—
382#41—532no—
383#41—1061no—
384#41—1061no—
385C2×Q8—531no—
386C2×Q8C2×C4×D5312yes—
387C2×D4—22no—
388C2×D4—531no—
389C2×D4—1061no—
390C2×D4C2×C4×D5312yes—
391C22×C4—22no—
392C22×C4—1062no—
393C22×C4—1061no—
394C22×C4—1061no—
395C22×C4C23×D5311yescharacteristic
396Q16—21no—
397Q16—21no—
398Q16—1064no—
399SD16—22no—
400SD16—22no—
401SD16—1064no—
402SD16—1064no—
403D8—21no—
404D8—21no—
405D8—1064no—
406M16—21no—
407M16—532no—
408M16—534no—
409M16—1064no—
410M16—1062no—
411M16—1061no—
412M16—1062no—
413M16—1062no—
414M16—1062no—
415M16—1062no—
416M16—2122no—
417M16—2122no—
418M16D4×D5312yes—
419C2×C8—21no—
420C2×C8—531no—
421C2×C8—532no—
422C2×C8—531no—
423C2×C8—532no—
424C2×C8—1064no—
425C2×C8—1061no—
426C2×C8—1062no—
427C2×C8—1062no—
428C2×C8—1062no—
429C2×C8—1062no—
430C2×C8—2122no—
431C2×C8—2122no—
432C2×C8C2×C4×D5311yescharacteristic
433C2×C8C2×C4×D5311yescharacteristic
434C2×C8D4×D5311yescharacteristic
435C2×C8D4×D5311yescharacteristic
436C2×C8#1171994711yescharacteristic
437C2×C8#1171994711yescharacteristic
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—1062no—
441C22⋊C4—22no—
442C22⋊C4—1062no—
443C22⋊C4#1171994712yes—
444C42—21no—
445C42—22no—
446C42—531no—
447C42—531no—
448C42—1062no—
449C42—1061no—
450C23—22no—
451C23—1061no—
452C23C2×#1171994711yescharacteristic
453Q8—22no—
454Q8—22no—
455Q8—532no—
456Q8—1061no—
457D4—22no—
458D4—22no—
459D4—42no—
460D4—42no—
461D4—532no—
462D4—1062no—
463D4—1062no—
464D4—1061no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—1061no—
471C2×C4—1062no—
472C2×C4—1062no—
473C2×C4—1061no—
474C2×C4—1061no—
475C2×C4—4242no—
476C2×C4—4241no—
477C2×C4C22×C4×D5311yescharacteristic
478C2×C4C2×D4×D5311yescharacteristic
479C2×C4C2×D4×D5311yescharacteristic
480C2×C4#41×D5311yescharacteristic
481C2×C4#41×D5311yescharacteristic
482C2×C4C2×#1171994711yescharacteristic
483C8—22no—
484C8—532no—
485C8—532no—
486C8—1064no—
487C8—1062no—
488C8—2122no—
489C8#1313317711yescharacteristic
490C8#1313317711yescharacteristic
491C8#1313317711yescharacteristic
492C8#1313317711yescharacteristic
493C22—41no—
494C22—42no—
495C22—82no—
496C22—1062no—
497C22—2122no—
498C22C4×#1171994711yescharacteristic
499C22#1681113311yescharacteristic
500C22#1681117111yescharacteristic
501C4—42no—
502C4—42no—
503C4—82no—
504C4—1062no—
505C4—2121no—
506C4—2121no—
507C4C4×D4×D5311yescharacteristic
508C4C2×#1313317711yescharacteristic
509C4#121×D5311yescharacteristic
510C4#1681115211yescharacteristic
511C2—21no—
512C2—21no—
513C2—21no—
514C2—82no—
515C2—2122no—
516C2#2027723111yescharacteristic
subgroupG/H
centerC4#121×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1976C2
socleC2×C53C4×#121
prime radical#20284431C2
cosocleC22×C4×C53C24
hypercenter#1976D53
hypocenterC53#21559
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1976

Complements

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

NH ≅ G/Ntype
2. #20283614C2semidirect
3. C2×#16810745C2semidirect
4. #20283142C2semidirect
5. #20275505C2semidirect
6. #20275387C2semidirect
7. #20275213C2semidirect
8. #20274766C2semidirect
9. #20274601C2semidirect
10. #20274517C2semidirect
11. C53×#1976C2semidirect
17. #16810895C4semidirect
18. #16810890C4semidirect
19. #16810883C4semidirect
82. #13133153C2×C4semidirect
83. #13133153C2×C4semidirect
88. #13133136C2×C4semidirect
89. #13133134C2×C4semidirect
105. C2×Q16×C53C2×C4semidirect
106. C2×SD16×C53C2×C4semidirect
107. C2×D8×C53C2×C4semidirect
257. #1976D53semidirect
306. C53#21559semidirect
318. C2×Q16C4×D53semidirect
319. C2×SD16C4×D53semidirect
320. C2×D8C4×D53semidirect

Maximal subgroups

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

subgroupindex
#202844312
#202836142
C2×#168107452
#202831422
#202755052
#202753872
#202752132
#202747662
#202746012
#202745172
C53×#19762
#2155953