Finite Group Databasebeta

← #27388525

Subgroups — #27388525

Order 13568.

738 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283511C211yesmaximal, characteristic
2#20283365C211yesmaximal, characteristic, complemented
3#20283362C211yesmaximal, characteristic, complemented
4#20275607C211yesmaximal, characteristic
5#20275600C211yesmaximal, characteristic
6#20275465C211yesmaximal, characteristic, complemented
7#20275462C211yesmaximal, characteristic, complemented
8#20275428C211yesmaximal, characteristic, complemented
9#20275424C211yesmaximal, characteristic, complemented
10#20275419C211yesmaximal, characteristic, complemented
11#20274620C211yesmaximal, characteristic, complemented
12#104×Dic53C211yesmaximal, characteristic, complemented
13#20274519C211yesmaximal, characteristic, complemented
14C4×#13133127C211yesmaximal, characteristic, complemented
15C53×#1937C211yesmaximal, characteristic, complemented
16C2×#13133237C2211yescharacteristic
17#41×Dic53C2211yescharacteristic
18#16811091C2211yescharacteristic
19#16810732C412yescomplemented
20#16810727C412yescomplemented
21#16810726C412yescomplemented
22#16810720C412yescomplemented
23#16810683—22no—
24#16810681—22no—
25#16810679—22no—
26#16810677—22no—
27C2×#13133152C2211yescharacteristic
28C2×#13133152C2211yescharacteristic
29#16810659C2211yescharacteristic
30#16810651C2211yescharacteristic
31#16810650C2211yescharacteristic
32#16810649C2211yescharacteristic
33C2×#13133145C2211yescharacteristic
34C2×#13133144C2211yescharacteristic
35C2×#13133143C2211yescharacteristic
36#16810620C2211yescharacteristic
37#16810591C2211yescharacteristic
38#16810590C2211yescharacteristic
39#16810589C2211yescharacteristic
40M16×Dic53C2211yescharacteristic
41M16×Dic53C2211yescharacteristic
42C2×#13133127C2211yescharacteristic
43C2×#13133127C2211yescharacteristic
44C2×#13133127C2211yescharacteristic
45C2×#13133127C2211yescharacteristic
46C2×C4×#88322C2211yescharacteristic
47C42×Dic53C2211yescharacteristic
48#16810388C2211yescharacteristic
49#16810384C2211yescharacteristic
50#16810383C2211yescharacteristic
51#16810382C2211yescharacteristic
52C53×#424C2211yescharacteristic
53C2×#104×C53C2211yescharacteristic
54C53×#416C2211yescharacteristic
55C53×#408C2211yescharacteristic
56C2×C4×C8×C53C2211yescharacteristic
57C53×#341C2211yescharacteristic
58C53×#339C2211yescharacteristic
59C2×#11719945C2311yescharacteristic
60#13133237—22no—
61#13133237—22no—
62C2×#11719925C2×C411yescharacteristic
63C2×#11719923C2×C411yescharacteristic
64C2×#11719924C2×C411yescharacteristic
65C2×#11719922C2×C411yescharacteristic
66C2×#11719917C2311yescharacteristic
67C2×#11719917C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70Q8×Dic53C2×C412yes—
71#13133182C2×C412yes—
72D4×Dic53—22no—
73D4×Dic53C2×C412yes—
74#13133179C2×C412yes—
75#13133177—22no—
76#13133176—21no—
77#13133176C2311yescharacteristic
78#13133174—21no—
79C2×C4×Dic53—21no—
80C2×C4×Dic53—22no—
81C2×C4×Dic53C2311yescharacteristic
82C2×C4×Dic53C2311yescharacteristic
83C42×D53—21no—
84#13133152—22no—
85#13133152—22no—
86#13133150C2×C412yes—
87#13133148C2×C412yes—
88C4⋊#88322—22no—
89#13133146—22no—
90#13133146—22no—
91#13133145—22no—
92#13133144—22no—
93#13133143—22no—
94#13133142C2×C412yes—
95#13133139C2×C412yes—
96C22⋊#88322—22no—
97#13133129—22no—
98#13133129—22no—
99#13133127—21no—
100#13133127—21no—
101#13133127—22no—
102#13133127—22no—
103#13133127C2×C412yes—
104#13133127C2×C412yes—
105#13133127D412yes—
106#13133127D412yes—
107C4×#88322—22no—
108C8×Dic53—21no—
109C8×Dic53—21no—
110C2×#41×C53C2311yescharacteristic
111C2×M16×C53C2311yescharacteristic
112C2×M16×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115#117×C53C2311yescharacteristic
116C2×C42×C53C2311yescharacteristic
117C4⋊C8×C53—22no—
118#104×C53—22no—
119#103×C53C2×C412yes—
120#102×C53C2×C412yes—
121C22⋊C8×C53—22no—
122C4×C8×C53—22no—
123#41×C53—21no—
124#41×C53—21no—
125#41×C53—21no—
126#41×C53—21no—
127C2×Q8×C53C22×C411yescharacteristic
128C2×D4×C53—21no—
129C2×D4×C53C22×C411yescharacteristic
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53C2411yescharacteristic
133C2×C22⋊D53—21no—
134C22×Dic53—21no—
135C22×Dic53—22no—
136C22×Dic53#4112yes—
137#11719945—21no—
138#11719945—21no—
139#11719945—21no—
140#11719945—21no—
141C2×D212C22×C411yescharacteristic
142C2×C4×D53—21no—
143C2×C4×D53—22no—
144C2×Dic106C22×C411yescharacteristic
145M16×C53—21no—
146M16×C53—21no—
147M16×C53—21no—
148M16×C53—21no—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53C22×C412yes—
156C2×C8×C53C2×D412yes—
157C4⋊C4×C53C22×C412yes—
158C22⋊C4×C53—21no—
159C42×C53—21no—
160C42×C53—21no—
161C42×C53#4112yes—
162C22⋊Dic53—21no—
163C22⋊Dic53—22no—
164#11719925—22no—
165#11719924—22no—
166#11719923—22no—
167#11719922—22no—
168#11719921—22no—
169C4⋊Dic53—22no—
170C4⋊Dic53C22×C412yes—
171#11719919—22no—
172C4×Dic53—21no—
173C4×Dic53—21no—
174C4×Dic53—22no—
175C4×Dic53—22no—
176C4×Dic53—44no—
177C4×Dic53C22×C412yes—
178C4×Dic53C2×D411yescharacteristic
179C4×Dic53C2×D411yescharacteristic
180C4×Dic53#4112yes—
181#11719917—21no—
182#11719917—21no—
183#11719917—21no—
184#11719917—21no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322C22×C411yescharacteristic
190C2×#88322C22×C411yescharacteristic
191C2×#88322C2×D412yes—
192C2×#88322#4112yes—
193C2×#88322#4112yes—
194C23×C53—21no—
195C23×C53C2×#4111yescharacteristic
196C22×D53—21no—
197Q8×C53—21no—
198Q8×C53—21no—
199D4×C53—21no—
200D4×C53—21no—
201D4×C53—41no—
202D4×C53—41no—
203C2×C4×C53—24no—
204C2×C4×C53—22no—
205C2×C4×C53—22no—
206C2×C4×C53—21no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53C23×C411yescharacteristic
210C2×C4×C53C22×D411yescharacteristic
211C2×C4×C53C2×#4111yescharacteristic
212C2×C4×C53C2×#4111yescharacteristic
213C2×C4×C53C2×#4111yescharacteristic
214C2×C4×C53C2×#4111yescharacteristic
215C22⋊D53—41no—
216C22⋊D53—41no—
217C2×Dic53—21no—
218C2×Dic53—22no—
219C2×Dic53—22no—
220C2×Dic53—42no—
221C2×Dic53—42no—
222C2×Dic53—82no—
223C2×Dic53C4×D412yes—
224C2×Dic53C4×D412yes—
225D212—21no—
226D212—21no—
227C4×D53—41no—
228C4×D53—41no—
229C4×D53—82no—
230Dic106—21no—
231Dic106—21no—
232C8×C53—22no—
233C8×C53—22no—
234C8×C53—41no—
235C8×C53—41no—
236#88322—22no—
237#88322—41no—
238#88322—41no—
239#88322#11712yes—
240#88322#11712yes—
241#21297—531nomaximal, Sylow
242C22×C53—22no—
243C22×C53—42no—
244C22×C53—41no—
245C22×C53C2×C4×D411yescharacteristic
246C22×C53#52911yescharacteristic
247C22×C53#52911yescharacteristic
248C2×D53—42no—
249C2×D53—41no—
250C4×C53—24no—
251C4×C53—42no—
252C4×C53—41no—
253C4×C53C2×#11711yescharacteristic
254C4×C53C4×#4111yescharacteristic
255C4×C53#53211yescharacteristic
256C4×C53#53211yescharacteristic
257Dic53—22no—
258Dic53—22no—
259Dic53—42no—
260Dic53—42no—
261Dic53—41no—
262#2943—531no—
263#2923—531no—
264#2923—531no—
265#1975—531no—
266#1974—531no—
267#1946—531no—
268#1945—531no—
269#1938—531no—
270#1938—531no—
271#1937—531no—
272#1937D5311yescharacteristic, complemented
273#1778—531no—
274C4×#104—531no—
275C4×#104—531no—
276C4×#97—531no—
277C2×C53—21no—
278C2×C53—21no—
279C2×C53—81no—
280C2×C53#229311yescharacteristic
281C2×C53#295612yes—
282D53—81no—
283C2×#135—531no—
284C4×#41—531no—
285C4×#41—531no—
286#442—532no—
287#441—532no—
288#440—532no—
289#440—532no—
290#433—1062no—
291#433—1062no—
292#432—1062no—
293#432—1062no—
294C2×#108—531no—
295C2×#108—531no—
296#427—531no—
297#424—531no—
298#424—531no—
299#424—531no—
300#424C2×D5311yescharacteristic
301C2×#104—531no—
302C2×#104—531no—
303C2×#104—531no—
304C2×#104C2×D5311yescharacteristic
305#416—531no—
306#416C2×D5311yescharacteristic
307#408—531no—
308#408—531no—
309#408—531no—
310#408C2×D5311yescharacteristic
311C4×M16—531no—
312C4×M16—531no—
313C2×#97—531no—
314C2×#97—531no—
315C2×#97—531no—
316C2×#97—531no—
317C2×C4×C8—531no—
318C2×C4×C8C2×D5311yescharacteristic
319C43—531no—
320#341—531no—
321#341C2×D5311yescharacteristic
322#339—531no—
323#339—531no—
324#339—531no—
325#339C2×D5311yescharacteristic
326C53#2129711yescharacteristic, complemented, Sylow
327C2×#41—531no—
328C2×#41C22×D5311yescharacteristic
329#135—1062no—
330#135—1062no—
331C2×Q16—531no—
332C2×SD16—531no—
333C2×SD16—531no—
334C2×D8—531no—
335C2×M16—531no—
336C2×M16—531no—
337C2×M16C22×D5311yescharacteristic
338C2×M16C22×D5311yescharacteristic
339C22×C8—531no—
340C22×C8—531no—
341C22×C8C22×D5311yescharacteristic
342C22×C8C22×D5311yescharacteristic
343C4×Q8—532no—
344C4×Q8—532no—
345C4×D4—532no—
346C4×D4—532no—
347C4×D4—1062no—
348C4×D4—1062no—
349#117—531no—
350#117—1061no—
351#117—1061no—
352#117C22×D5311yescharacteristic
353C2×C42—531no—
354C2×C42—531no—
355C2×C42—1061no—
356C2×C42—1061no—
357C2×C42—1062no—
358C2×C42C22×D5311yescharacteristic
359#108—1062no—
360#108—1062no—
361#107—532no—
362#106—532no—
363C4⋊C8—22no—
364C4⋊C8—1062no—
365C4⋊C8—1062no—
366C4⋊C8—1062no—
367#104—22no—
368#104—1062no—
369#104—1062no—
370#104—1062no—
371#103—532no—
372#103C4×D5312yes—
373#102—532no—
374#102C4×D5312yes—
375C22⋊C8—22no—
376C22⋊C8—1062no—
377C22⋊C8—1062no—
378C22⋊C8—1062no—
379#97—532no—
380#97—532no—
381#97—532no—
382#97—532no—
383#97—1062no—
384#97—1061no—
385#97—1061no—
386#97—1062no—
387C4×C8—22no—
388C4×C8—1062no—
389C4×C8—1061no—
390C4×C8—1061no—
391#41—21no—
392#41—21no—
393#41—21no—
394#41—21no—
395#41—1061no—
396#41—1061no—
397#41—1061no—
398#41—1061no—
399C2×Q8—531no—
400C2×Q8C2×C4×D5311yescharacteristic
401C2×D4—21no—
402C2×D4—531no—
403C2×D4—1061no—
404C2×D4C2×C4×D5311yescharacteristic
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—532no—
408C22×C4—1062no—
409C22×C4—1062no—
410C22×C4—1061no—
411C22×C4—1061no—
412C22×C4C23×D5311yescharacteristic
413Q16—1062no—
414SD16—1062no—
415SD16—1062no—
416D8—1062no—
417M16—21no—
418M16—21no—
419M16—21no—
420M16—21no—
421M16—1061no—
422M16—1061no—
423M16—1061no—
424M16—1061no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—531no—
432C2×C8—531no—
433C2×C8—532no—
434C2×C8—532no—
435C2×C8—532no—
436C2×C8—1061no—
437C2×C8—1061no—
438C2×C8—1061no—
439C2×C8—1061no—
440C2×C8C2×C4×D5312yes—
441C2×C8D4×D5312yes—
442C4⋊C4—532no—
443C4⋊C4—1062no—
444C4⋊C4—1062no—
445C4⋊C4C2×C4×D5312yes—
446C22⋊C4—21no—
447C22⋊C4—1062no—
448C22⋊C4—1062no—
449C22⋊C4—1061no—
450C42—21no—
451C42—21no—
452C42—532no—
453C42—531no—
454C42—531no—
455C42—532no—
456C42—1062no—
457C42—1062no—
458C42—1061no—
459C42—1061no—
460C42—2124no—
461C42#1171994712yes—
462C23—21no—
463C23—1061no—
464C23#41×D5311yescharacteristic
465Q8—21no—
466Q8—21no—
467Q8—1061no—
468Q8—1061no—
469D4—21no—
470D4—21no—
471D4—41no—
472D4—41no—
473D4—1061no—
474D4—1061no—
475D4—2121no—
476D4—2121no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—532no—
484C2×C4—532no—
485C2×C4—1062no—
486C2×C4—1061no—
487C2×C4—1062no—
488C2×C4—4242no—
489C2×C4—4242no—
490C2×C4—2122no—
491C2×C4—2122no—
492C2×C4—2121no—
493C2×C4—2121no—
494C2×C4C22×C4×D5311yescharacteristic
495C2×C4C2×D4×D5311yescharacteristic
496C2×C4#41×D5311yescharacteristic
497C2×C4#41×D5311yescharacteristic
498C2×C4#41×D5311yescharacteristic
499C2×C4C2×#1171994711yescharacteristic
500C8—22no—
501C8—22no—
502C8—41no—
503C8—41no—
504C8—532no—
505C8—532no—
506C8—1062no—
507C8—2121no—
508C8—2121no—
509C22—22no—
510C22—41no—
511C22—42no—
512C22—2121no—
513C22—2122no—
514C22C4×D4×D5311yescharacteristic
515C22#1681115011yescharacteristic
516C22#1681115611yescharacteristic
517C4—24no—
518C4—41no—
519C4—42no—
520C4—1062no—
521C4—1062no—
522C4—2122no—
523C4—2121no—
524C4—2122no—
525C4#117×D5311yescharacteristic
526C4C4×#1171994711yescharacteristic
527C4#1681116811yescharacteristic
528C4#1681117211yescharacteristic
529C2—21no—
530C2—21no—
531C2—81no—
532C2—4241no—
533C2#2027717511yescharacteristic
534C2#2028362012yes—
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1937C2
socleC22×C53C2×C4×D4
prime radicalC2×#13133237C22
cosocleC22×C4×C53C24
hypercenter#1937D53
hypocenterC53#21297
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1937

Complements

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

NH ≅ G/Ntype
2. #20283365C2semidirect
3. #20283362C2semidirect
6. #20275465C2semidirect
7. #20275462C2semidirect
8. #20275428C2semidirect
9. #20275424C2semidirect
10. #20275419C2semidirect
11. #20274620C2semidirect
12. #104×Dic53C2semidirect
13. #20274519C2semidirect
14. C4×#13133127C2semidirect
15. C53×#1937C2semidirect
19. #16810732C4semidirect
20. #16810727C4semidirect
21. #16810726C4semidirect
22. #16810720C4semidirect
272. #1937D53semidirect
326. C53#21297semidirect

Maximal subgroups

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

subgroupindex
#202835112
#202833652
#202833622
#202756072
#202756002
#202754652
#202754622
#202754282
#202754242
#202754192
#202746202
#104×Dic532
#202745192
C4×#131331272
C53×#19372
#2129753