Finite Group Databasebeta

← #27368097

Subgroups — #27368097

Order 13568.

712 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285274C211yesmaximal, characteristic, complemented
2#20284512C211yesmaximal, characteristic, complemented
3#20284179C211yesmaximal, characteristic
4#20274262C212yesmaximal, complemented
5#20274248C212yesmaximal, complemented
6#20274214C212yesmaximal, complemented
7#20274180C212yesmaximal, complemented
8#20273474C211yesmaximal, characteristic, complemented
9#20273403C211yesmaximal, characteristic, complemented
10#20273390C211yesmaximal, characteristic, complemented
11C2×C53×#436C211yesmaximal, characteristic, complemented
12C2×#13133243—21no—
13#16811180C2211yescharacteristic
14#16811173C2211yescharacteristic
15#16811021C2211yescharacteristic, complemented
16#16810996C2211yescharacteristic, complemented
17#16810974C2211yescharacteristic, complemented
18#16810970C2211yescharacteristic, complemented
19#16810923—22no—
20#16810913C2212yes—
21#16810903C2212yes—
22#16810852—21no—
23#16810852—21no—
24#16810838—22no—
25#16810828C2212yes—
26#16810818C2212yes—
27C4×#11719913—21no—
28C4×#11719913—21no—
29#16810663—21no—
30#16810647C2211yescharacteristic
31#16810647C2211yescharacteristic
32#16810628—21no—
33#16810622—21no—
34#16810376C2212yescomplemented
35#16810376C2212yescomplemented
36#16810371C2212yescomplemented
37#16810370C2212yescomplemented
38#16810368C2212yescomplemented
39#16810366C2212yescomplemented
40C22×M16×C53—21no—
41C2×C4×D4×C53C2211yescharacteristic
42C53×#436—21no—
43C53×#436—21no—
44C53×#436C2212yes—
45C53×#436C2212yes—
46C2×C4⋊C8×C53C2211yescharacteristic
47C2×C22⋊C8×C53—21no—
48C2×C4×C8×C53C2211yescharacteristic
49C2×#11719945—21no—
50#13133243—44no—
51C2×#11719913—21no—
52C2×#11719913—21no—
53C2×#11719923—22no—
54C2×#11719922—22no—
55C2×#11719917—21no—
56C2×#11719917—21no—
57#13133217C2311yescharacteristic
58#13133217C2311yescharacteristic
59#13133217C2311yescharacteristic
60#13133214C2311yescharacteristic
61#13133209—21no—
62#13133205—21no—
63#13133196—21no—
64#13133196—21no—
65#13133191—21no—
66#13133191—21no—
67C4×Dic106—21no—
68C4×Dic106—21no—
69C4×D212—21no—
70C4×D212—21no—
71#13133176—21no—
72#13133150—22no—
73#13133149—21no—
74#13133149—21no—
75#13133148—22no—
76C4⋊#88322—21no—
77C4⋊#88322—21no—
78C4⋊#88322C2312yes—
79C4⋊#88322C2312yes—
80#13133142—21no—
81#13133142—21no—
82#13133140—22no—
83#13133139—21no—
84#13133139—21no—
85#13133138—22no—
86#13133138—22no—
87#13133137—22no—
88C22×D4×C53D411yescharacteristic
89C23×C4×C53—21no—
90C2×M16×C53—22no—
91C2×M16×C53—22no—
92C2×M16×C53—44no—
93C22×C8×C53—21no—
94C22×C8×C53D411yescharacteristic
95C22×C8×C53D411yescharacteristic
96C4×D4×C53—21no—
97C4×D4×C53—21no—
98C4×D4×C53D412yes—
99C4×D4×C53C2312yes—
100C2×C4⋊C4×C53D411yescharacteristic
101C2×C22⋊C4×C53—21no—
102C2×C42×C53C2311yescharacteristic
103C4⋊C8×C53—21no—
104C4⋊C8×C53C2312yes—
105C22⋊C8×C53—22no—
106C22⋊C8×C53—22no—
107C4×C8×C53D411yescharacteristic
108C4×C8×C53D411yescharacteristic
109C4×C8×C53C2311yescharacteristic
110C4×C8×C53C2311yescharacteristic
111C24×C53—21no—
112C2×D4×C53—21no—
113C2×D4×C53—22no—
114C2×D4×C53—22no—
115C2×D4×C53C2×D412yes—
116C22×C4×C53—21no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—42no—
120C22×C4×C53—42no—
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53#4111yescharacteristic
124C2×C22⋊D53—41no—
125#11719945—42no—
126#11719945—42no—
127C2×D212—21no—
128C2×D212—21no—
129C2×D212—21no—
130C2×C4×D53—41no—
131C2×Dic106—21no—
132C2×Dic106—21no—
133C2×Dic106—21no—
134M16×C53—44no—
135M16×C53—44no—
136C2×C8×C53—22no—
137C2×C8×C53—22no—
138C2×C8×C53—41no—
139C2×C8×C53—42no—
140C2×C8×C53C2×D411yescharacteristic
141C2×C8×C53C2×D411yescharacteristic
142C2×C8×C53C2×D411yescharacteristic
143C2×C8×C53C2×D411yescharacteristic
144C4⋊C4×C53—21no—
145C4⋊C4×C53C2×D412yes—
146C22⋊C4×C53—22no—
147C22⋊C4×C53—22no—
148C42×C53C2×D411yescharacteristic
149C42×C53C2×D411yescharacteristic
150C42×C53#4111yescharacteristic
151C42×C53C2411yescharacteristic
152C22⋊Dic53—41no—
153#11719923—44no—
154#11719922—44no—
155#11719921—41no—
156#11719921—41no—
157C4⋊Dic53—21no—
158C4⋊Dic53—21no—
159C4⋊Dic53—21no—
160C4⋊Dic53—21no—
161C4⋊Dic53—21no—
162C4⋊Dic53—21no—
163#11719919—41no—
164#11719919—41no—
165C4×Dic53—41no—
166#11719917—42no—
167#11719917—42no—
168C2×#88322—22no—
169C2×#88322—22no—
170#11719913—42no—
171#11719913—42no—
172C23×C53—22no—
173C23×C53—42no—
174C23×C53—42no—
175C23×C53—42no—
176C23×C53#12111yescharacteristic
177C22×D53—41no—
178D4×C53—22no—
179D4×C53—22no—
180D4×C53—42no—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53—22no—
184C2×C4×C53—41no—
185C2×C4×C53—44no—
186C2×C4×C53—42no—
187C2×C4×C53—42no—
188C2×C4×C53—42no—
189C2×C4×C53#12111yescharacteristic
190C2×C4×C53#12111yescharacteristic
191C2×C4×C53#12111yescharacteristic
192C2×C4×C53AGL(1,ℤ/8)12yes—
193C2×C4×C53#13712yes—
194C2×C4×C53C22×D411yescharacteristic
195C2×C4×C53C22×D411yescharacteristic
196C2×C4×C53C2×#4111yescharacteristic
197C22⋊D53—82no—
198C2×Dic53—41no—
199C2×Dic53—41no—
200C2×Dic53—41no—
201D212—41no—
202D212—41no—
203D212—41no—
204D212—41no—
205D212—81no—
206C4×D53—82no—
207Dic106—41no—
208Dic106—41no—
209Dic106—41no—
210Dic106—41no—
211Dic106—81no—
212C8×C53—21no—
213C8×C53—21no—
214C8×C53—21no—
215C8×C53—42no—
216C8×C53AGL(1,ℤ/8)12yescomplemented
217#88322—42no—
218#88322—42no—
219#19279—531nomaximal, Sylow
220C22×C53—21no—
221C22×C53—21no—
222C22×C53—42no—
223C22×C53—42no—
224C22×C53—42no—
225C22×C53—84no—
226C22×C53#46812yes—
227C22×C53C2×#12111yescharacteristic
228C2×D53—81no—
229C2×D53—81no—
230C2×D53—81no—
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—21no—
234C4×C53—42no—
235C4×C53#46812yes—
236C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
237C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
238C4×C53C2×#13711yescharacteristic
239C4×C53#57511yescharacteristic
240Dic53—81no—
241Dic53—81no—
242Dic53—81no—
243#3167—531no—
244#3077—531no—
245#3036—531no—
246C2×#436D5311yescharacteristic, complemented
247#1699—532no—
248#1697—532no—
249#1691—532no—
250#1685—532no—
251#1567—531no—
252#1550—531no—
253#1547—531no—
254C2×C53—22no—
255C2×C53—82no—
256C2×C53#303611yescharacteristic
257C2×C53C2×#46811yescharacteristic
258C2×C53#306711yescharacteristic
259D53—161no—
260C2×AGL(1,ℤ/8)—1061no—
261C22×M16—21no—
262#533—531no—
263#532—531no—
264C2×C4×D4C2×D5311yescharacteristic
265#498—531no—
266#492—531no—
267#487—531no—
268#486—531no—
269#478—1062no—
270#476—532no—
271#474—532no—
272#465—1061no—
273#465—1061no—
274#463—1062no—
275#461—532no—
276#459—532no—
277C4×SD16—1061no—
278C4×SD16—1061no—
279#436—21no—
280#436—21no—
281#436C2×D5312yes—
282#436C2×D5312yes—
283#428—1061no—
284#423—531no—
285#423—531no—
286C2×C4⋊C8C2×D5311yescharacteristic
287#418—1061no—
288#417—1061no—
289C2×C22⋊C8—21no—
290C2×C4×C8C2×D5311yescharacteristic
291#335—532no—
292#335—532no—
293#333—532no—
294#332—532no—
295#332—532no—
296#331—532no—
297C53#1927911yescharacteristic, complemented, Sylow
298C2×#41—1061no—
299C22×D4D21211yescharacteristic
300C23×C4—21no—
301AGL(1,ℤ/8)—2124no—
302C2×SD16—1062no—
303C2×SD16—1061no—
304C2×SD16—1061no—
305C2×D8—1062no—
306C2×M16—22no—
307C2×M16—22no—
308C2×M16—44no—
309C2×M16—1061no—
310C2×M16—1061no—
311C22×C8—21no—
312C22×C8C22⋊D5311yescharacteristic
313C22×C8C22⋊D5311yescharacteristic
314C4⋊Q8—531no—
315C4⋊Q8—531no—
316C4⋊Q8—531no—
317#127—531no—
318#125—1061no—
319#124—1061no—
320C22⋊Q8—1061no—
321C22⋊Q8—1061no—
322#121—1061no—
323#121—1061no—
324C4×Q8—1061no—
325C4×Q8—1061no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4—1061no—
329C4×D4—1061no—
330C4×D4D21212yes—
331C4×D4C22×D5312yes—
332#117—1061no—
333C2×C4⋊C4D21211yescharacteristic
334C2×C22⋊C4—21no—
335C2×C42C22×D5311yescharacteristic
336#107—1062no—
337#106—1061no—
338#106—1061no—
339#106—1062no—
340C4⋊C8—21no—
341C4⋊C8—532no—
342C4⋊C8—532no—
343C4⋊C8—1061no—
344C4⋊C8—1061no—
345C4⋊C8C22×D5312yes—
346#103—1061no—
347#103—1061no—
348#103—1062no—
349#102—1062no—
350#102—1062no—
351#102—1061no—
352#102—1061no—
353#102—1062no—
354C22⋊C8—22no—
355C22⋊C8—22no—
356C4×C8C22⋊D5311yescharacteristic
357C4×C8C22⋊D5311yescharacteristic
358C4×C8C22×D5311yescharacteristic
359C4×C8C22×D5311yescharacteristic
360C24—21no—
361#41—2122no—
362#41—2122no—
363C2×Q8—1061no—
364C2×Q8—1061no—
365C2×Q8—1061no—
366C2×D4—21no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—1061no—
370C2×D4—1061no—
371C2×D4—1061no—
372C2×D4—2121no—
373C2×D4C2×D21212yes—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—2121no—
380C22×C4C2×D21211yescharacteristic
381C22×C4#1171994511yescharacteristic
382C22×C4C2×C22⋊D5311yescharacteristic
383SD16—2122no—
384SD16—2124no—
385SD16—2122no—
386D8—2124no—
387M16—44no—
388M16—44no—
389M16—2122no—
390M16—2122no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—41no—
394C2×C8—42no—
395C2×C8—1062no—
396C2×C8—1062no—
397C2×C8C2×C22⋊D5311yescharacteristic
398C2×C8C2×C22⋊D5311yescharacteristic
399C2×C8C2×C22⋊D5311yescharacteristic
400C2×C8C2×C22⋊D5311yescharacteristic
401C4⋊C4—21no—
402C4⋊C4—1061no—
403C4⋊C4—1061no—
404C4⋊C4—1061no—
405C4⋊C4—1061no—
406C4⋊C4—1061no—
407C4⋊C4—1061no—
408C4⋊C4—2121no—
409C4⋊C4—2121no—
410C4⋊C4C2×D21212yes—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—2121no—
414C22⋊C4—2121no—
415C22⋊C4—2121no—
416C42—2121no—
417C42C2×D21211yescharacteristic
418C42#1171994511yescharacteristic
419C42C2×C22⋊D5311yescharacteristic
420C42C23×D5311yescharacteristic
421C23—22no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—2121no—
426C23#1313319111yescharacteristic
427Q8—4241no—
428Q8—2121no—
429Q8—2121no—
430Q8—2121no—
431Q8—2121no—
432D4—22no—
433D4—22no—
434D4—42no—
435D4—4241no—
436D4—4242no—
437D4—2121no—
438D4—2121no—
439D4—2121no—
440D4—2121no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—41no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—4242no—
450C2×C4—2121no—
451C2×C4—2121no—
452C2×C4—2121no—
453C2×C4#1313319111yescharacteristic
454C2×C4#1313319111yescharacteristic
455C2×C4#1313319111yescharacteristic
456C2×C4#1313324512yes—
457C2×C4#1313325012yes—
458C2×C4C22×C22⋊D5311yescharacteristic
459C2×C4C22×D21211yescharacteristic
460C2×C4C2×#1171994511yescharacteristic
461C8—21no—
462C8—21no—
463C8—21no—
464C8—42no—
465C8—2122no—
466C8—2122no—
467C8#1313324312yescomplemented
468C22—21no—
469C22—21no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—84no—
474C22—4241no—
475C22—4241no—
476C22—4241no—
477C22#1681086912yes—
478C22C2×#1313319111yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—42no—
483C4—4241no—
484C4—4241no—
485C4—4241no—
486C4#1681086912yes—
487C4C2×#1313324311yescharacteristic
488C4C2×#1313324511yescharacteristic
489C4C2×#1313325011yescharacteristic
490C4#1681142111yescharacteristic
491C2—22no—
492C2—82no—
493C2—8481no—
494C2#2028417911yescharacteristic
495C2C2×#1681086911yescharacteristic
496C2#2028443511yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#436C2
socleC22×C53C2×#121
prime radical#20284179C2
hypercenterC2×#436D53
hypocenterC53#19279
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#436

Complements

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

NH ≅ G/Ntype
1. #20285274C2semidirect
2. #20284512C2semidirect
4. #20274262C2semidirect
5. #20274248C2semidirect
6. #20274214C2semidirect
7. #20274180C2semidirect
8. #20273474C2semidirect
9. #20273403C2semidirect
10. #20273390C2semidirect
11. C2×C53×#436C2semidirect
15. #16811021C22semidirect
16. #16810996C22semidirect
17. #16810974C22semidirect
18. #16810970C22semidirect
34. #16810376C22semidirect
35. #16810376C22semidirect
36. #16810371C22semidirect
37. #16810370C22semidirect
38. #16810368C22semidirect
39. #16810366C22semidirect
216. C8×C53AGL(1,ℤ/8)semidirect
246. C2×#436D53semidirect
297. C53#19279semidirect
467. C8#13133243semidirect

Maximal subgroups

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

subgroupindex
#202852742
#202845122
#202841792
#202742622
#202742482
#202742142
#202741802
#202734742
#202734032
#202733902
C2×C53×#4362
#1927953