Finite Group Databasebeta

← #27366922

Subgroups — #27366922

Order 13568.

696 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285108C211yesmaximal, characteristic, complemented
2C2×#16810913C211yesmaximal, characteristic, complemented
3D4×#11719917C211yesmaximal, characteristic, complemented
4#20274288C212yesmaximal, complemented
5#20274246C212yesmaximal, complemented
6#20274164C212yesmaximal, complemented
7#20274122C212yesmaximal, complemented
8#20273468C211yesmaximal, characteristic, complemented
9#20273395C211yesmaximal, characteristic, complemented
10#20273392C211yesmaximal, characteristic, complemented
11C53×#3100C211yesmaximal, characteristic
12C22×#11719917—21no—
13C2×#13133217C2211yescharacteristic
14#16810982—22no—
15#16810973C2212yescomplemented
16#16810969C2212yescomplemented
17#16810913—21no—
18#16810913—21no—
19#16810913C2212yes—
20#16810913C2212yes—
21#16810701C2212yes—
22#16810695—22no—
23D4×#88322C2212yes—
24C2×#13133148—21no—
25#16810647C2211yescharacteristic
26C2×C4⋊#88322C2211yescharacteristic
27#16810624—21no—
28#16810624—21no—
29C2×#13133138—21no—
30#16810587—21no—
31C4×#11719917C2211yescharacteristic
32#16810375C2212yescomplemented
33#16810373C2212yescomplemented
34#16810371C2212yescomplemented
35#16810371C2212yescomplemented
36#16810369C2212yescomplemented
37#16810367C2212yescomplemented
38C53×#532C2211yescharacteristic
39C2×C4×D4×C53C2211yescharacteristic
40C53×#480—21no—
41C53×#474C2211yescharacteristic
42C53×#474C2211yescharacteristic
43C53×#459C2211yescharacteristic
44C53×#459C2211yescharacteristic
45C53×#447—21no—
46C4×D8×C53—21no—
47C4×D8×C53—21no—
48C53×#431C2211yescharacteristic
49C22×Dic106—21no—
50C2×#11719917—22no—
51C2×#11719917—21no—
52C2×#11719917—44no—
53C2×#11719917D411yescharacteristic
54C2×#11719917D411yescharacteristic
55C22×#88322—22no—
56C22×#88322—21no—
57#13133217—21no—
58#13133217—21no—
59#13133217—21no—
60#13133217C2312yes—
61C2×C4⋊Dic53—21no—
62C2×C4⋊Dic53—21no—
63#13133148—22no—
64#13133148—22no—
65C4⋊#88322—21no—
66C4⋊#88322—21no—
67C4⋊#88322C2312yes—
68C4⋊#88322C2312yes—
69#13133141—22no—
70#13133141—22no—
71#13133138—22no—
72#13133138—22no—
73#13133138—22no—
74#13133138—22no—
75C22⋊#88322—22no—
76C22⋊#88322—22no—
77#13133128D412yes—
78C4×#88322C2312yes—
79C2×#41×C53—21no—
80C22×D4×C53D411yescharacteristic
81C23×C4×C53—21no—
82C2×D8×C53—21no—
83C2×D8×C53—21no—
84C4⋊Q8×C53C2311yescharacteristic
85#127×C53C2311yescharacteristic
86#124×C53—21no—
87#121×C53—21no—
88#121×C53—21no—
89C4×D4×C53—21no—
90C4×D4×C53—21no—
91C4×D4×C53—21no—
92C4×D4×C53—21no—
93C4×D4×C53D411yescharacteristic
94C4×D4×C53D411yescharacteristic
95C4×D4×C53C2311yescharacteristic
96C4×D4×C53C2311yescharacteristic
97C2×C4⋊C4×C53D411yescharacteristic
98C2×C22⋊C4×C53—21no—
99C2×C42×C53C2311yescharacteristic
100#107×C53—21no—
101#107×C53—21no—
102C4⋊C8×C53C2311yescharacteristic
103C4⋊C8×C53C2311yescharacteristic
104#102×C53—21no—
105#102×C53—21no—
106#102×C53—21no—
107#102×C53—21no—
108C22⋊C8×C53—21no—
109C4×C8×C53—21no—
110C24×C53—21no—
111#41×C53—42no—
112#41×C53—42no—
113C2×Q8×C53—21no—
114C2×D4×C53—22no—
115C2×D4×C53—21no—
116C2×D4×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—21no—
119C2×D4×C53—41no—
120C2×D4×C53—41no—
121C2×D4×C53C2×D411yescharacteristic
122C2×D4×C53C2×D411yescharacteristic
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—42no—
129C22×C4×C53—42no—
130C22×C4×C53—41no—
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53#4111yescharacteristic
134C22×Dic53—41no—
135C2×Dic106—22no—
136C2×Dic106—41no—
137C2×Dic106—82no—
138D8×C53—42no—
139D8×C53—42no—
140C2×C8×C53—21no—
141C2×C8×C53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53C2×D411yescharacteristic
146C4⋊C4×C53C2×D411yescharacteristic
147C22⋊C4×C53—21no—
148C22⋊C4×C53—21no—
149C22⋊C4×C53—21no—
150C22⋊C4×C53—21no—
151C22⋊C4×C53—41no—
152C22⋊C4×C53—41no—
153C42×C53C2×D411yescharacteristic
154C42×C53C2×D411yescharacteristic
155C42×C53#4111yescharacteristic
156C42×C53C2411yescharacteristic
157C4⋊Dic53—22no—
158C4⋊Dic53—22no—
159C4⋊Dic53—41no—
160C4⋊Dic53—41no—
161#11719917—22no—
162#11719917—22no—
163#11719917—44no—
164#11719917—42no—
165C2×#88322—22no—
166C2×#88322—22no—
167C2×#88322—44no—
168C2×#88322—42no—
169C2×#88322C2×D412yes—
170C2×#88322C2×D412yes—
171C23×C53—21no—
172C23×C53—21no—
173C23×C53—42no—
174C23×C53—42no—
175C23×C53—41no—
176C23×C53—81no—
177C23×C53#12111yescharacteristic
178Q8×C53—41no—
179Q8×C53—41no—
180D4×C53—21no—
181D4×C53—21no—
182D4×C53—21no—
183D4×C53—42no—
184D4×C53—41no—
185D4×C53—41no—
186D4×C53—41no—
187D4×C53—41no—
188D4×C53—81no—
189D4×C53—82no—
190D4×C53#13712yes—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—41no—
196C2×C4×C53—42no—
197C2×C4×C53—41no—
198C2×C4×C53—41no—
199C2×C4×C53—41no—
200C2×C4×C53—41no—
201C2×C4×C53—42no—
202C2×C4×C53—42no—
203C2×C4×C53—41no—
204C2×C4×C53—82no—
205C2×C4×C53#12111yescharacteristic
206C2×C4×C53#12111yescharacteristic
207C2×C4×C53#12111yescharacteristic
208C2×C4×C53#13512yes—
209C2×C4×C53AGL(1,ℤ/8)12yes—
210C2×C4×C53C22×D411yescharacteristic
211C2×C4×C53C22×D411yescharacteristic
212C2×C4×C53C2×#4111yescharacteristic
213C2×Dic53—42no—
214C2×Dic53—82no—
215Dic106—82no—
216Dic106—82no—
217C8×C53—41no—
218C8×C53—41no—
219#88322—22no—
220#88322—22no—
221#88322—42no—
222#88322—42no—
223#19168—531nomaximal, Sylow
224C22×C53—21no—
225C22×C53—21no—
226C22×C53—22no—
227C22×C53—41no—
228C22×C53—41no—
229C22×C53—41no—
230C22×C53—41no—
231C22×C53—41no—
232C22×C53—84no—
233C22×C53—81no—
234C22×C53—81no—
235C22×C53—81no—
236C22×C53#46312yes—
237C22×C53C2×#12111yescharacteristic
238C4×C53—21no—
239C4×C53—21no—
240C4×C53—21no—
241C4×C53—41no—
242C4×C53—41no—
243C4×C53—81no—
244C4×C53#46312yes—
245C4×C53C2×#13511yescharacteristic
246C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
247C4×C53C2×#13711yescharacteristic
248C4×C53#57511yescharacteristic
249Dic53—82no—
250#3141—531no—
251#3100D5311yescharacteristic
252C2×#476—531no—
253D4×M16—531no—
254#1703—532no—
255#1697—532no—
256#1683—532no—
257#1677—532no—
258#1565—531no—
259#1548—531no—
260#1547—531no—
261C2×C53—22no—
262C2×C53—42no—
263C2×C53—81no—
264C2×C53—161no—
265C2×C53C2×#46311yescharacteristic
266C2×C53#305611yescharacteristic
267C2×C53#306811yescharacteristic
268C22×M16—1061no—
269#532C2×D5311yescharacteristic
270C2×C4⋊Q8—531no—
271C2×C4×D4C2×D5311yescharacteristic
272#489—1062no—
273#487—532no—
274#486—532no—
275#480—21no—
276#476—532no—
277#476—532no—
278#476—1061no—
279#476—1061no—
280#474C2×D5311yescharacteristic
281#474C2×D5311yescharacteristic
282#459C2×D5311yescharacteristic
283#459C2×D5311yescharacteristic
284#447—21no—
285C4×D8—21no—
286C4×D8—21no—
287#436—532no—
288#435—1062no—
289C8×D4—532no—
290#431C2×D5311yescharacteristic
291C2×#106—1061no—
292#423—531no—
293C2×C4⋊C8—531no—
294#417—1061no—
295#417—1061no—
296C2×#102—1061no—
297#407—1061no—
298C4×M16—531no—
299#335—532no—
300#334—532no—
301#333—532no—
302#333—532no—
303#332—532no—
304#331—532no—
305C53#1916811yescharacteristic, complemented, Sylow
306C2×#41—21no—
307C22×Q8—1061no—
308C22×D4C22⋊D5311yescharacteristic
309C23×C4—21no—
310C2×D8—21no—
311C2×D8—21no—
312C2×M16—531no—
313C2×M16—531no—
314C2×M16—1062no—
315C2×M16—1061no—
316C2×M16—2124no—
317C22×C8—1062no—
318C22×C8—1061no—
319C4⋊Q8—532no—
320C4⋊Q8—1061no—
321C4⋊Q8—1061no—
322C4⋊Q8—1061no—
323C4⋊Q8C22×D5311yescharacteristic
324#127C22×D5311yescharacteristic
325#124—21no—
326#121—21no—
327#121—21no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—21no—
331C4×D4—21no—
332C4×D4C22⋊D5311yescharacteristic
333C4×D4C22⋊D5311yescharacteristic
334C4×D4C22×D5311yescharacteristic
335C4×D4C22×D5311yescharacteristic
336C2×C4⋊C4—1061no—
337C2×C4⋊C4—1061no—
338C2×C4⋊C4C22⋊D5311yescharacteristic
339C2×C22⋊C4—21no—
340C2×C42C22×D5311yescharacteristic
341#107—21no—
342#107—21no—
343#106—1062no—
344#106—1062no—
345C4⋊C8—532no—
346C4⋊C8—532no—
347C4⋊C8—1061no—
348C4⋊C8—1061no—
349C4⋊C8C22×D5311yescharacteristic
350C4⋊C8C22×D5311yescharacteristic
351#103—1062no—
352#103—1062no—
353#102—21no—
354#102—21no—
355#102—21no—
356#102—21no—
357#102—1062no—
358#102—1062no—
359#102—1062no—
360#102—1062no—
361C22⋊C8—21no—
362C22⋊C8—1062no—
363C22⋊C8—1062no—
364#97—532no—
365C4×C8—21no—
366C4×C8—532no—
367C24—21no—
368#41—42no—
369#41—42no—
370C2×Q8—21no—
371C2×Q8—1062no—
372C2×Q8—4242no—
373C2×Q8—2121no—
374C2×D4—22no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4C2×C22⋊D5311yescharacteristic
382C2×D4C2×C22⋊D5311yescharacteristic
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—2121no—
392C22×C4D4×D5311yescharacteristic
393C22×C4#1171994711yescharacteristic
394C22×C4C2×C22⋊D5311yescharacteristic
395D8—42no—
396D8—42no—
397M16—1062no—
398M16—1062no—
399M16—2124no—
400M16—2122no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—532no—
404C2×C8—532no—
405C2×C8—1062no—
406C2×C8—1062no—
407C2×C8—2124no—
408C2×C8—2122no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—1062no—
413C4⋊C4—1062no—
414C4⋊C4—2121no—
415C4⋊C4—2121no—
416C4⋊C4C2×C22⋊D5311yescharacteristic
417C4⋊C4C2×C22⋊D5311yescharacteristic
418C22⋊C4—21no—
419C22⋊C4—21no—
420C22⋊C4—21no—
421C22⋊C4—21no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C42D4×D5311yescharacteristic
425C42#1171994711yescharacteristic
426C42C2×C22⋊D5311yescharacteristic
427C42C23×D5311yescharacteristic
428C23—21no—
429C23—21no—
430C23—42no—
431C23—42no—
432C23—41no—
433C23—81no—
434C23#1313318911yescharacteristic
435Q8—41no—
436Q8—41no—
437Q8—4242no—
438Q8—4242no—
439D4—21no—
440D4—21no—
441D4—21no—
442D4—42no—
443D4—41no—
444D4—41no—
445D4—41no—
446D4—41no—
447D4—81no—
448D4—82no—
449D4#1313324712yes—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—82no—
464C2×C4—4242no—
465C2×C4—2122no—
466C2×C4#1313318911yescharacteristic
467C2×C4#1313318911yescharacteristic
468C2×C4#1313318911yescharacteristic
469C2×C4#1313323612yes—
470C2×C4#1313324112yes—
471C2×C4C2×D4×D5311yescharacteristic
472C2×C4C22×C22⋊D5311yescharacteristic
473C2×C4C2×#1171994711yescharacteristic
474C8—41no—
475C8—41no—
476C8—1062no—
477C8—1062no—
478C8—2122no—
479C8—2122no—
480C22—21no—
481C22—21no—
482C22—22no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—84no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22#1681083612yes—
493C22C2×#1313318911yescharacteristic
494C4—21no—
495C4—21no—
496C4—21no—
497C4—41no—
498C4—41no—
499C4—81no—
500C4—4242no—
501C4#1681083612yes—
502C4C2×#1313323611yescharacteristic
503C4C2×#1313324111yescharacteristic
504C4C2×#1313324711yescharacteristic
505C4#1681142111yescharacteristic
506C2—22no—
507C2—42no—
508C2—81no—
509C2—161no—
510C2C2×#1681083611yescharacteristic
511C2#2028434711yescharacteristic
512C2#2028444511yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3100C2
socleC22×C53C2×#121
hypercenter#3100D53
hypocenterC53#19168
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. #20285108C2semidirect
2. C2×#16810913C2semidirect
3. D4×#11719917C2semidirect
4. #20274288C2semidirect
5. #20274246C2semidirect
6. #20274164C2semidirect
7. #20274122C2semidirect
8. #20273468C2semidirect
9. #20273395C2semidirect
10. #20273392C2semidirect
15. #16810973C22semidirect
16. #16810969C22semidirect
32. #16810375C22semidirect
33. #16810373C22semidirect
34. #16810371C22semidirect
35. #16810371C22semidirect
36. #16810369C22semidirect
37. #16810367C22semidirect
305. C53#19168semidirect

Maximal subgroups

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

subgroupindex
#202851082
C2×#168109132
D4×#117199172
#202742882
#202742462
#202741642
#202741222
#202734682
#202733952
#202733922
C53×#31002
#1916853