Finite Group Databasebeta

← #27391036

Subgroups — #27391036

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#20283616C212yesmaximal, complemented
3#20283140C211yesmaximal, characteristic, complemented
4#20275614C211yesmaximal, characteristic, complemented
5#20275503C211yesmaximal, characteristic, complemented
6#20275381C212yesmaximal, complemented
7#20275219C212yesmaximal, complemented
8#20274770C211yesmaximal, characteristic, complemented
9#20274599C211yesmaximal, characteristic, complemented
10#20274515C212yesmaximal, complemented
11C2×C53×#444C211yesmaximal, characteristic, complemented
12#16811421C2212yes—
13C2×#13133223C2211yescharacteristic
14C2×#13133223C2211yescharacteristic
15C4×#11719945C2211yescharacteristic
16#16810895C412yescomplemented
17#16810890C412yescomplemented
18#16810883C414yescomplemented
19#16810743—24no—
20#16810738—24no—
21#16810671C2212yes—
22C2×#13133153C2211yescharacteristic
23#16810661C2211yescharacteristic
24#16810640C2212yes—
25#16810639C2212yes—
26#16810607C2212yes—
27#16810597C2212yes—
28#16810580—21no—
29#16810580—21no—
30#16810578C2211yescharacteristic
31#16810578C2211yescharacteristic
32#16810578C2212yes—
33C4×#11719917C2211yescharacteristic
34#16810575—21no—
35#97×D53—21no—
36#16810394C2212yes—
37#16810394C2212yes—
38#16810383C2212yes—
39C2×#135×C53C2211yescharacteristic
40C2×#131×C53C2212yes—
41C53×#444—21no—
42C53×#444—21no—
43C53×#444—21no—
44C53×#444—21no—
45C2×#108×C53C2211yescharacteristic
46C2×#104×C53C2212yes—
47C2×#97×C53C2211yescharacteristic
48C2×#11719945C2311yescharacteristic
49#13133248C2×C414yes—
50#13133247—22no—
51#13133243C2×C414yes—
52#13133242—22no—
53#13133237—24no—
54#13133223—21no—
55#13133223—21no—
56#13133223—22no—
57#13133223C2×C412yes—
58#13133223C2×C412yes—
59#13133223D414yes—
60C2×#11719917C2311yescharacteristic
61C2×#11719917C2312yes—
62C2×#11719917C2311yescharacteristic
63C2×#11719912—21no—
64C2×#11719912—21no—
65C2×C8×D53—21no—
66C2×C8×D53—21no—
67C4×Dic106—21no—
68C4×D212—21no—
69C4×C22⋊D53—22no—
70#13133176C2312yes—
71#13133176C2311yescharacteristic
72#13133174—21no—
73C42×D53—21no—
74#13133153C2×C412yescomplemented
75#13133153C2×C412yescomplemented
76#13133152—24no—
77#13133151—21no—
78#13133149—21no—
79#13133144—24no—
80#13133143—24no—
81#13133136C2×C414yescomplemented
82#13133134C2×C414yescomplemented
83#13133132—24no—
84#13133128—21no—
85#13133127—21no—
86#13133127—21no—
87#13133127—22no—
88C4×#88322—21no—
89C8×Dic53—21no—
90C8×Dic53—21no—
91C8×Dic53—22no—
92C2×#41×C53C2312yes—
93#135×C53—21no—
94#135×C53—21no—
95#135×C53—21no—
96#135×C53—21no—
97C2×Q16×C53C2×C411yescharacteristic, complemented
98C2×SD16×C53C2×C412yescomplemented
99C2×D8×C53C2×C411yescharacteristic, complemented
100#131×C53—22no—
101#131×C53—22no—
102#131×C53—22no—
103#131×C53—22no—
104C2×M16×C53—22no—
105C2×M16×C53C2312yes—
106C22×C8×C53—22no—
107C22×C8×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C2×C42×C53C2311yescharacteristic
110#108×C53—21no—
111#108×C53—21no—
112#104×C53—22no—
113#104×C53—22no—
114#97×C53D411yescharacteristic
115#97×C53D411yescharacteristic
116#97×C53D411yescharacteristic
117#97×C53D411yescharacteristic
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—21no—
126C22×C4×C53—22no—
127C22×C4×C53C2411yescharacteristic
128C2×C22⋊D53—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—22no—
144M16×C53—22no—
145M16×C53—42no—
146M16×C53—42no—
147C2×C8×C53—22no—
148C2×C8×C53—42no—
149C2×C8×C53—42no—
150C2×C8×C53C22×C411yescharacteristic
151C2×C8×C53C22×C411yescharacteristic
152C2×C8×C53C2×D411yescharacteristic
153C2×C8×C53C2×D411yescharacteristic
154C2×C8×C53C2×D412yes—
155C2×C8×C53C2×D411yescharacteristic
156C2×C8×C53C2×D411yescharacteristic
157C2×C8×C53#4111yescharacteristic
158C2×C8×C53#4111yescharacteristic
159C2×C8×C53#4112yes—
160C42×C53—21no—
161C42×C53C2×D411yescharacteristic
162C42×C53C2×D411yescharacteristic
163C22⋊Dic53—22no—
164C22⋊Dic53#4112yes—
165#11719925—24no—
166#11719924—24no—
167#11719923—24no—
168#11719922—24no—
169#11719921—22no—
170C4⋊Dic53—21no—
171C4⋊Dic53—22no—
172#11719919—22no—
173C4×Dic53—22no—
174C4×Dic53—21no—
175C4×Dic53—22no—
176#11719917—21no—
177#11719917—21no—
178#11719917—22no—
179#11719917C22×C412yes—
180#11719917C22×C414yes—
181#11719917C2×D412yes—
182C2×#88322—21no—
183C2×#88322—21no—
184C2×#88322—22no—
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—21no—
202C2×C4×C53—21no—
203C2×C4×C53—42no—
204C2×C4×C53—42no—
205C2×C4×C53—81no—
206C2×C4×C53C23×C411yescharacteristic
207C2×C4×C53C22×D411yescharacteristic
208C2×C4×C53C22×D411yescharacteristic
209C2×C4×C53C2×#4111yescharacteristic
210C2×C4×C53C2×#4111yescharacteristic
211C2×C4×C53C2×#4111yescharacteristic
212C22⋊D53—22no—
213C22⋊D53—22no—
214C2×Dic53—21no—
215C2×Dic53—22no—
216C2×Dic53—42no—
217D212—21no—
218D212C4×D412yes—
219C4×D53—22no—
220C4×D53—22no—
221C4×D53—82no—
222Dic106—21no—
223Dic106C4×D412yes—
224C8×C53—42no—
225C8×C53C4×D411yescharacteristic
226C8×C53C4×D411yescharacteristic
227C8×C53C4×D411yescharacteristic
228C8×C53C4×D411yescharacteristic
229C8×C53#12112yes—
230C8×C53#12112yes—
231#88322—22no—
232#88322—24no—
233#88322—22no—
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—41no—
244C4×C53—41no—
245C4×C53—42no—
246C4×C53C2×C4×D411yescharacteristic
247C4×C53C2×C4×D411yescharacteristic
248C4×C53C2×#12111yescharacteristic
249C4×C53#52911yescharacteristic
250Dic53—22no—
251Dic53—42no—
252Dic53—82no—
253#3067—531no—
254#2955—532no—
255C2×#444D5311yescharacteristic, complemented
256#2875—531no—
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—82no—
268C2×C53C4×#12111yescharacteristic
269D53—42no—
270#575—532no—
271C2×#135C2×D5311yescharacteristic
272C2×#131—531no—
273C2×#131—531no—
274C2×#131C2×D5312yes—
275C4×#41—531no—
276#473—532no—
277#472—532no—
278#471—534no—
279#444—21no—
280#444—21no—
281#444—21no—
282#444—21no—
283#444—1064no—
284#443—1064no—
285#430—532no—
286C2×#108—531no—
287C2×#108C2×D5311yescharacteristic
288#427—531no—
289#421—532no—
290#421—532no—
291C2×#104C2×D5312yes—
292#413—532no—
293#410—532no—
294#405—531no—
295#405—531no—
296#405—532no—
297#405—1061no—
298#405—1061no—
299C4×M16—531no—
300C4×M16—1061no—
301C2×#97—1061no—
302C2×#97C2×D5311yescharacteristic
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—22no—
322#131—22no—
323#131—22no—
324#131—22no—
325#131—532no—
326#131—532no—
327#131—534no—
328#131—1061no—
329#131—1061no—
330#131—1062no—
331C2×M16—22no—
332C2×M16—531no—
333C2×M16—531no—
334C2×M16—532no—
335C2×M16—1061no—
336C2×M16—1061no—
337C2×M16C22×D5312yes—
338C22×C8—22no—
339C22×C8—1061no—
340C22×C8—1061no—
341C22×C8C22×D5311yescharacteristic
342C22×C8C22×D5311yescharacteristic
343C4×Q8—1061no—
344C4×D4—1061no—
345C4×D4—1062no—
346#117—531no—
347#117—532no—
348#117—1061no—
349C2×C42—1061no—
350C2×C42C22×D5311yescharacteristic
351#108—21no—
352#108—21no—
353#108—532no—
354#108—532no—
355#108—1064no—
356#107—1061no—
357#106—1061no—
358#104—22no—
359#104—22no—
360#104—1064no—
361#104—1064no—
362#101—534no—
363#100—534no—
364#99—1064no—
365#97—1062no—
366#97—1061no—
367#97—1061no—
368#97—1061no—
369#97C22⋊D5311yescharacteristic
370#97C22⋊D5311yescharacteristic
371#97C22⋊D5311yescharacteristic
372#97C22⋊D5311yescharacteristic
373C4×C8—1062no—
374C4×C8—1061no—
375C4×C8—1061no—
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—21no—
392C22×C4—22no—
393C22×C4—1062no—
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—22no—
407M16—22no—
408M16—42no—
409M16—42no—
410M16—532no—
411M16—534no—
412M16—532no—
413M16—1062no—
414M16—1061no—
415M16—1061no—
416M16—1062no—
417M16—1062no—
418M16—1064no—
419C2×C8—22no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—1062no—
423C2×C8—1061no—
424C2×C8—1061no—
425C2×C8—1062no—
426C2×C8—1062no—
427C2×C8—1064no—
428C2×C8C2×C4×D5311yescharacteristic
429C2×C8C2×C4×D5311yescharacteristic
430C2×C8#1171994511yescharacteristic
431C2×C8#1171994511yescharacteristic
432C2×C8D4×D5312yes—
433C2×C8#1171994712yes—
434C2×C8C2×C22⋊D5311yescharacteristic
435C2×C8C2×C22⋊D5311yescharacteristic
436C2×C8C2×C22⋊D5311yescharacteristic
437C2×C8C2×C22⋊D5311yescharacteristic
438C4⋊C4—1062no—
439C4⋊C4—1061no—
440C4⋊C4—1062no—
441C22⋊C4—532no—
442C22⋊C4—1062no—
443C22⋊C4—1062no—
444C42—21no—
445C42—1062no—
446C42—1061no—
447C42—1062no—
448C42C2×C22⋊D5311yescharacteristic
449C42C2×C22⋊D5311yescharacteristic
450C23—22no—
451C23—1061no—
452C23#41×D5311yescharacteristic
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—21no—
467C2×C4—21no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—81no—
471C2×C4—1061no—
472C2×C4—1062no—
473C2×C4—1062no—
474C2×C4—1062no—
475C2×C4—4242no—
476C2×C4—2122no—
477C2×C4C22×C4×D5311yescharacteristic
478C2×C4C2×D4×D5311yescharacteristic
479C2×C4C22×C22⋊D5311yescharacteristic
480C2×C4#41×D5311yescharacteristic
481C2×C4C2×#1171994711yescharacteristic
482C2×C4C2×#1171994511yescharacteristic
483C8—42no—
484C8—1064no—
485C8—1062no—
486C8—1062no—
487C8C4×C22⋊D5311yescharacteristic
488C8C4×C22⋊D5311yescharacteristic
489C8C4×C22⋊D5311yescharacteristic
490C8C4×C22⋊D5311yescharacteristic
491C8#1313318812yes—
492C8#1313318812yes—
493C22—41no—
494C22—42no—
495C22—82no—
496C22—1062no—
497C22—2122no—
498C22C4×#1171994711yescharacteristic
499C22#1681113011yescharacteristic
500C22#1681117011yescharacteristic
501C4—41no—
502C4—41no—
503C4—42no—
504C4—1062no—
505C4—4242no—
506C4—2122no—
507C4C4×D4×D5311yescharacteristic
508C4C2×C4×C22⋊D5311yescharacteristic
509C4C2×#1313318811yescharacteristic
510C4#1681115511yescharacteristic
511C2—21no—
512C2—21no—
513C2—21no—
514C2—82no—
515C2—2122no—
516C2C4×#1313318811yescharacteristic
subgroupG/H
centerC4C2×#13133188
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#444C2
socleC2×C53C4×#121
prime radical#20284431C2
cosocleC22×C4×C53C24
hypercenterC2×#444D53
hypocenterC53#21559
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C8 ◁ C2×#444

Complements

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

NH ≅ G/Ntype
2. #20283616C2semidirect
3. #20283140C2semidirect
4. #20275614C2semidirect
5. #20275503C2semidirect
6. #20275381C2semidirect
7. #20275219C2semidirect
8. #20274770C2semidirect
9. #20274599C2semidirect
10. #20274515C2semidirect
11. C2×C53×#444C2semidirect
16. #16810895C4semidirect
17. #16810890C4semidirect
18. #16810883C4semidirect
74. #13133153C2×C4semidirect
75. #13133153C2×C4semidirect
81. #13133136C2×C4semidirect
82. #13133134C2×C4semidirect
97. C2×Q16×C53C2×C4semidirect
98. C2×SD16×C53C2×C4semidirect
99. C2×D8×C53C2×C4semidirect
255. C2×#444D53semidirect
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
#202836162
#202831402
#202756142
#202755032
#202753812
#202752192
#202747702
#202745992
#202745152
C2×C53×#4442
#2155953