Finite Group Databasebeta

← #27360288

Subgroups — #27360288

Order 13568.

1178 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285257C211yesmaximal, characteristic, complemented
2#20284814C211yesmaximal, characteristic, complemented
3#20284814C211yesmaximal, characteristic, complemented
4#20273999C214yesmaximal, complemented
5#20273950C214yesmaximal, complemented
6#20273432C211yesmaximal, characteristic, complemented
7#20273158C211yesmaximal, characteristic, complemented
8#20273158C211yesmaximal, characteristic, complemented
9C2×C53×#547C211yesmaximal, characteristic
10#16811180C2211yescharacteristic
11#16811029—22no—
12#16811026C2212yescomplemented
13#16811019C2212yescomplemented
14#16810958—21no—
15#16810958—21no—
16#16810903C2212yescomplemented
17#16810903C2212yescomplemented
18#16810828C2212yescomplemented
19#16810828C2212yescomplemented
20#16810774—21no—
21#16810774—21no—
22#16810723—22no—
23#16810723—22no—
24#16810680C2211yescharacteristic
25#16810680C2211yescharacteristic
26#16810663—22no—
27C4×#11719917C2211yescharacteristic
28#16810369C2214yescomplemented
29D4⋊#88322C2214yescomplemented
30D4⋊#88322C2214yescomplemented
31C22×#41×C53—21no—
32C53×#547—22no—
33C53×#547—22no—
34C53×#547—22no—
35C53×#547C2214yes—
36C2×C4⋊Q8×C53C2211yescharacteristic
37C2×#123×C53—22no—
38C2×C22⋊Q8×C53—21no—
39C2×#121×C53—21no—
40C2×C4×D4×C53C2211yescharacteristic
41C2×C4×D4×C53C2211yescharacteristic
42C22×C4⋊C4×C53—21no—
43C2×#11719923—22no—
44C2×#11719923—22no—
45C2×#11719917—21no—
46#13133217C2311yescharacteristic
47#13133217C2311yescharacteristic
48#13133209—21no—
49#13133196—21no—
50#13133196—21no—
51C4×Dic106—21no—
52C4×Dic106—21no—
53#13133176—21no—
54#13133150—24no—
55#13133150—22no—
56#13133150—22no—
57#13133148—24no—
58C4⋊#88322C2312yes—
59C4⋊#88322C2312yes—
60#13133140—22no—
61#13133140—22no—
62#13133138—22no—
63#13133138—22no—
64C22⋊#88322—21no—
65C22⋊#88322—21no—
66#13133128—21no—
67#13133128—21no—
68#13133128—21no—
69C4×#88322C2312yes—
70C2×#41×C53—24no—
71C2×#41×C53—24no—
72C2×#41×C53—48no—
73C22×Q8×C53—21no—
74C22×D4×C53—22no—
75C22×D4×C53D411yescharacteristic
76C22×D4×C53D411yescharacteristic
77C23×C4×C53—22no—
78C23×C4×C53—21no—
79C23×C4×C53—21no—
80C4⋊Q8×C53—21no—
81C4⋊Q8×C53—21no—
82C4⋊Q8×C53D412yes—
83C4⋊Q8×C53C2312yes—
84#123×C53—28no—
85#123×C53—28no—
86C22⋊Q8×C53—24no—
87C22⋊Q8×C53—24no—
88#121×C53—24no—
89#121×C53—24no—
90C4×D4×C53—21no—
91C4×D4×C53—21no—
92C4×D4×C53—21no—
93C4×D4×C53—21no—
94C4×D4×C53D412yes—
95C4×D4×C53D412yes—
96C4×D4×C53C2312yes—
97C4×D4×C53C2312yes—
98C2×C4⋊C4×C53—22no—
99C2×C4⋊C4×C53—24no—
100C2×C4⋊C4×C53—24no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53D411yescharacteristic
103C2×C4⋊C4×C53D411yescharacteristic
104C2×C4⋊C4×C53D412yes—
105C2×C22⋊C4×C53—22no—
106C2×C22⋊C4×C53—21no—
107C2×C22⋊C4×C53—21no—
108C2×C42×C53C2311yescharacteristic
109C24×C53—21no—
110C24×C53—21no—
111#41×C53—416no—
112#41×C53—48no—
113#41×C53—48no—
114C2×Q8×C53—22no—
115C2×Q8×C53—22no—
116C2×Q8×C53—44no—
117C2×D4×C53—24no—
118C2×D4×C53—28no—
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—42no—
122C2×D4×C53—48no—
123C2×D4×C53C2×D412yes—
124C2×D4×C53C2×D412yes—
125C22×C4×C53—28no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—44no—
134C22×C4×C53—44no—
135C22×C4×C53—48no—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C2×Dic106—21no—
140C2×Dic106—21no—
141C4⋊C4×C53—28no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—22no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—22no—
146C4⋊C4×C53—22no—
147C4⋊C4×C53C2×D412yes—
148C4⋊C4×C53C2×D412yes—
149C4⋊C4×C53C2×D414yes—
150C22⋊C4×C53—28no—
151C22⋊C4×C53—22no—
152C22⋊C4×C53—22no—
153C22⋊C4×C53—22no—
154C22⋊C4×C53—22no—
155C42×C53C2×D411yescharacteristic
156C42×C53C2×D411yescharacteristic
157C42×C53C2×D411yescharacteristic
158C42×C53C2411yescharacteristic
159C22⋊Dic53—41no—
160#11719923—44no—
161#11719923—44no—
162C4⋊Dic53—21no—
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166#11719919—41no—
167#11719919—41no—
168C4×Dic53—41no—
169#11719917—24no—
170C2×#88322—22no—
171C2×#88322—22no—
172C2×#88322—22no—
173C23×C53—24no—
174C23×C53—22no—
175C23×C53—22no—
176C23×C53—42no—
177C23×C53—42no—
178C23×C53—42no—
179C23×C53—42no—
180C23×C53—42no—
181C23×C53#12011yescharacteristic
182Q8×C53—44no—
183Q8×C53—44no—
184D4×C53—24no—
185D4×C53—22no—
186D4×C53—22no—
187D4×C53—22no—
188D4×C53—42no—
189D4×C53—416no—
190D4×C53AGL(1,ℤ/8)14yes—
191C2×C4×C53—22no—
192C2×C4×C53—22no—
193C2×C4×C53—22no—
194C2×C4×C53—22no—
195C2×C4×C53—24no—
196C2×C4×C53—22no—
197C2×C4×C53—44no—
198C2×C4×C53—44no—
199C2×C4×C53—42no—
200C2×C4×C53—42no—
201C2×C4×C53—44no—
202C2×C4×C53—416no—
203C2×C4×C53—44no—
204C2×C4×C53—44no—
205C2×C4×C53#12011yescharacteristic
206C2×C4×C53#12011yescharacteristic
207C2×C4×C53#12011yescharacteristic
208C2×C4×C53C22×D411yescharacteristic
209C2×C4×C53C22×D411yescharacteristic
210C2×C4×C53C22×D411yescharacteristic
211C2×Dic53—41no—
212C2×Dic53—41no—
213Dic106—42no—
214Dic106—42no—
215#88322—24no—
216#88322—42no—
217#88322—42no—
218#18539—531nomaximal, Sylow
219C22×C53—21no—
220C22×C53—21no—
221C22×C53—24no—
222C22×C53—48no—
223C22×C53—42no—
224C22×C53—42no—
225C22×C53—42no—
226C22×C53—42no—
227C22×C53—42no—
228C22×C53—84no—
229C22×C53#45612yes—
230C22×C53C2×#12011yescharacteristic
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—21no—
234C4×C53—21no—
235C4×C53—42no—
236C4×C53—44no—
237C4×C53—42no—
238C4×C53C2×AGL(1,ℤ/8)12yes—
239C4×C53#57512yes—
240Dic53—81no—
241Dic53—81no—
242C2×#547D5311yescharacteristic
243#3165—531no—
244#3110—531no—
245#3110—531no—
246#1654—534no—
247#1647—534no—
248#1556—531no—
249#1490—531no—
250#1490—531no—
251C2×C53—22no—
252C2×C53—44no—
253C2×C53—82no—
254C2×C53#300311yescharacteristic
255C2×C53#300511yescharacteristic
256C2×C53C2×#45611yescharacteristic
257C22×#41—21no—
258#547—22no—
259#547—22no—
260#547—22no—
261#547C2×D5314yes—
262#533—531no—
263C2×C4⋊Q8C2×D5311yescharacteristic
264C2×#123—22no—
265C2×C22⋊Q8—21no—
266C2×#121—21no—
267C2×C4×D4C2×D5311yescharacteristic
268C2×C4×D4C2×D5311yescharacteristic
269C22×C4⋊C4—21no—
270#501—1062no—
271#500—532no—
272#498—532no—
273#484—1061no—
274#484—1061no—
275#474—532no—
276#474—532no—
277#461—532no—
278#461—532no—
279#450—1061no—
280#450—1061no—
281#440—1062no—
282#440—1062no—
283#432—531no—
284#432—531no—
285#428—1062no—
286C4×M16—531no—
287#332—534no—
288D4⋊C8—534no—
289D4⋊C8—534no—
290C53#1853911yescharacteristic, complemented, Sylow
291C2×#41—24no—
292C2×#41—24no—
293C2×#41—48no—
294C22×Q8—21no—
295C22×D4—22no—
296C22×D4C22⋊D5311yescharacteristic
297C22×D4C22⋊D5311yescharacteristic
298C23×C4—22no—
299C23×C4—21no—
300C23×C4—21no—
301C2×SD16—1062no—
302C2×SD16—1062no—
303C2×M16—1061no—
304C4⋊Q8—21no—
305C4⋊Q8—21no—
306C4⋊Q8—531no—
307C4⋊Q8—531no—
308C4⋊Q8C22⋊D5312yes—
309C4⋊Q8C22×D5312yes—
310#125—1061no—
311#123—28no—
312#123—28no—
313C22⋊Q8—24no—
314C22⋊Q8—24no—
315C22⋊Q8—1061no—
316C22⋊Q8—1061no—
317#121—24no—
318#121—24no—
319C4×Q8—1061no—
320C4×Q8—1061no—
321C4×D4—21no—
322C4×D4—21no—
323C4×D4—21no—
324C4×D4—21no—
325C4×D4C22⋊D5312yes—
326C4×D4C22⋊D5312yes—
327C4×D4C22×D5312yes—
328C4×D4C22×D5312yes—
329#117—1061no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4C22⋊D5311yescharacteristic
335C2×C4⋊C4C22⋊D5311yescharacteristic
336C2×C4⋊C4C22⋊D5312yes—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C42C22×D5311yescharacteristic
341#107—1064no—
342#107—1062no—
343#107—1062no—
344#106—1064no—
345C4⋊C8—532no—
346C4⋊C8—532no—
347#103—1062no—
348#103—1062no—
349#102—1062no—
350#102—1062no—
351C22⋊C8—1061no—
352C22⋊C8—1061no—
353#97—1061no—
354#97—1061no—
355#97—1061no—
356C4×C8—532no—
357C24—21no—
358C24—21no—
359#41—416no—
360#41—48no—
361#41—48no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—44no—
365C2×Q8—1061no—
366C2×Q8—1061no—
367C2×D4—24no—
368C2×D4—28no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—42no—
372C2×D4—48no—
373C2×D4C2×C22⋊D5312yes—
374C2×D4C2×C22⋊D5312yes—
375C22×C4—28no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—48no—
386C22×C4C2×C22⋊D5311yescharacteristic
387C22×C4C2×C22⋊D5311yescharacteristic
388C22×C4C2×C22⋊D5311yescharacteristic
389SD16—2124no—
390SD16—2124no—
391M16—1064no—
392C2×C8—1062no—
393C2×C8—1062no—
394C2×C8—1062no—
395C4⋊C4—28no—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—1061no—
402C4⋊C4—1061no—
403C4⋊C4—1061no—
404C4⋊C4—1061no—
405C4⋊C4—2121no—
406C4⋊C4—2121no—
407C4⋊C4C2×C22⋊D5312yes—
408C4⋊C4C2×C22⋊D5312yes—
409C4⋊C4C2×C22⋊D5314yes—
410C22⋊C4—28no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—22no—
415C22⋊C4—2121no—
416C42—2121no—
417C42C2×C22⋊D5311yescharacteristic
418C42C2×C22⋊D5311yescharacteristic
419C42C2×C22⋊D5311yescharacteristic
420C42C23×D5311yescharacteristic
421C23—24no—
422C23—22no—
423C23—22no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23#1313318611yescharacteristic
430Q8—44no—
431Q8—44no—
432Q8—2122no—
433Q8—2122no—
434D4—24no—
435D4—22no—
436D4—22no—
437D4—22no—
438D4—42no—
439D4—416no—
440D4#1313324314yes—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—416no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—2121no—
456C2×C4—2121no—
457C2×C4#1313318611yescharacteristic
458C2×C4#1313318611yescharacteristic
459C2×C4#1313318611yescharacteristic
460C2×C4C22×C22⋊D5311yescharacteristic
461C2×C4C22×C22⋊D5311yescharacteristic
462C2×C4C22×C22⋊D5311yescharacteristic
463C8—1064no—
464C8—2122no—
465C8—2122no—
466C22—21no—
467C22—21no—
468C22—24no—
469C22—48no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—84no—
476C22#1681080612yes—
477C22C2×#1313318611yescharacteristic
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—42no—
483C4—44no—
484C4—42no—
485C4—4241no—
486C4—4241no—
487C4C2×#1313324312yes—
488C4#1681142112yes—
489C2—22no—
490C2—44no—
491C2—82no—
492C2#2028394211yescharacteristic
493C2#2028395711yescharacteristic
494C2C2×#1681080611yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC2×C53×#547C2
socleC22×C53C2×#120
hypercenterC2×#547D53
hypocenterC53#18539
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. #20285257C2semidirect
2. #20284814C2semidirect
3. #20284814C2semidirect
4. #20273999C2semidirect
5. #20273950C2semidirect
6. #20273432C2semidirect
7. #20273158C2semidirect
8. #20273158C2semidirect
12. #16811026C22semidirect
13. #16811019C22semidirect
16. #16810903C22semidirect
17. #16810903C22semidirect
18. #16810828C22semidirect
19. #16810828C22semidirect
28. #16810369C22semidirect
29. D4⋊#88322C22semidirect
30. D4⋊#88322C22semidirect
290. C53#18539semidirect

Maximal subgroups

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

subgroupindex
#202852572
#202848142
#202739992
#202739502
#202734322
#202731582
C2×C53×#5472
#1853953