Finite Group Databasebeta

← #27361474

Subgroups — #27361474

Order 13568.

1024 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285630C211yesmaximal, characteristic, complemented
2C2×#16810971C211yesmaximal, characteristic, complemented
3#20284195C211yesmaximal, characteristic, complemented
4#20273825C214yesmaximal, complemented
5#20273804C214yesmaximal, complemented
6#20273400C211yesmaximal, characteristic, complemented
7#20273166C211yesmaximal, characteristic, complemented
8#20273097C211yesmaximal, characteristic, complemented
9C53×#3511C211yesmaximal, characteristic, complemented
10C2×#13133242—21no—
11C2×#13133214C2211yescharacteristic
12#16810999—21no—
13#16810971—22no—
14#16810971—21no—
15#16810971C2214yes—
16#16810823—21no—
17#16810823C2212yes—
18#16810818—21no—
19#16810818C2212yes—
20#16810818C2214yes—
21#16810756—22no—
22#16810676C2211yescharacteristic
23#16810647C2211yescharacteristic
24#16810627—21no—
25C2×#13133137—22no—
26C2×C4×#88322C2211yescharacteristic
27#16810369C2214yescomplemented
28Q8⋊#88322C2212yescomplemented
29D4⋊#88322C2214yescomplemented
30D4⋊#88322C2212yescomplemented
31C53×#556C2212yescomplemented
32C53×#555—21no—
33C53×#554—21no—
34C53×#553—22no—
35C53×#548—22no—
36C53×#547C2212yescomplemented
37C53×#542—22no—
38C53×#537—21no—
39C53×#533—21no—
40C53×#532—21no—
41C53×#532C2211yescharacteristic
42C2×C4⋊Q8×C53C2211yescharacteristic
43C53×#520—21no—
44C4×#41×C53C2211yescharacteristic
45C22×D212—21no—
46C22×D212—21no—
47C22×D212—21no—
48#13133242—44no—
49C2×#11719924—22no—
50C2×#11719922—22no—
51C2×#11719922—24no—
52C2×#11719917—21no—
53C22×#88322—21no—
54#13133214—21no—
55#13133214—21no—
56#13133214—21no—
57#13133214C2312yes—
58C4⋊#88322—21no—
59C4⋊#88322C2312yes—
60C4⋊#88322C2312yes—
61#13133137—24no—
62#13133137—22no—
63#13133137—22no—
64#13133137—28no—
65C22⋊#88322—21no—
66C4×#88322—21no—
67C4×#88322—21no—
68C4×#88322C2312yes—
69C2×#41×C53—22no—
70C2×#41×C53—21no—
71C2×#41×C53D411yescharacteristic
72C22×Q8×C53—21no—
73C4⋊Q8×C53—21no—
74C4⋊Q8×C53—21no—
75C4⋊Q8×C53D412yes—
76C4⋊Q8×C53D411yescharacteristic
77C4⋊Q8×C53C2312yes—
78#127×C53D411yescharacteristic
79#126×C53—24no—
80#125×C53—22no—
81#124×C53—24no—
82#124×C53—21no—
83#124×C53—22no—
84#123×C53—24no—
85C22⋊Q8×C53—22no—
86C22⋊Q8×C53—24no—
87C22⋊Q8×C53—22no—
88C22⋊Q8×C53—22no—
89#121×C53—22no—
90#121×C53—22no—
91#121×C53—24no—
92#121×C53—22no—
93#121×C53—21no—
94#121×C53D412yes—
95C4×Q8×C53—22no—
96C4×Q8×C53—22no—
97C4×Q8×C53D411yescharacteristic
98C4×Q8×C53C2311yescharacteristic
99C4×D4×C53—22no—
100C4×D4×C53—24no—
101C4×D4×C53—22no—
102C4×D4×C53—21no—
103C4×D4×C53—21no—
104C4×D4×C53—21no—
105C4×D4×C53D411yescharacteristic
106C4×D4×C53C2311yescharacteristic
107C4×D4×C53C2312yes—
108#117×C53—22no—
109#117×C53—22no—
110#117×C53—21no—
111#117×C53D411yescharacteristic
112C2×C4⋊C4×C53—22no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53D412yes—
115C2×C42×C53—21no—
116C2×C42×C53C2311yescharacteristic
117C23×D53—41no—
118#41×C53—22no—
119#41×C53—22no—
120#41×C53—42no—
121#41×C53—42no—
122#41×C53—84no—
123C2×Q8×C53—21no—
124C2×Q8×C53—22no—
125C2×Q8×C53—22no—
126C2×Q8×C53—82no—
127C2×Q8×C53C2×D411yescharacteristic
128C2×D4×C53—21no—
129C2×D4×C53—22no—
130C2×D4×C53—24no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53C2×D411yescharacteristic
134C2×D4×C53C2×D412yes—
135C22×C4×C53—24no—
136C22×C4×C53—22no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—22no—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C2×D212—22no—
145C2×D212—22no—
146C2×D212—22no—
147C2×D212—44no—
148C2×D212—42no—
149C2×D212—42no—
150C2×D212—41no—
151C2×D212—41no—
152C2×D212—41no—
153C2×D212—82no—
154C4⋊C4×C53—24no—
155C4⋊C4×C53—24no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—22no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53C2×D411yescharacteristic
163C4⋊C4×C53C2×D414yes—
164C4⋊C4×C53C2×D411yescharacteristic
165C4⋊C4×C53C2×D412yes—
166C22⋊C4×C53—24no—
167C22⋊C4×C53—24no—
168C22⋊C4×C53—24no—
169C22⋊C4×C53—21no—
170C22⋊C4×C53—21no—
171C22⋊C4×C53—22no—
172C22⋊C4×C53—21no—
173C22⋊C4×C53—22no—
174C42×C53—24no—
175C42×C53—21no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53C2×D411yescharacteristic
179C42×C53C2×D411yescharacteristic
180C42×C53C2×D411yescharacteristic
181C42×C53C2411yescharacteristic
182#11719924—44no—
183#11719922—44no—
184#11719922—84no—
185#11719917—42no—
186C2×#88322—22no—
187C2×#88322—22no—
188C2×#88322—22no—
189C2×#88322—42no—
190C23×C53—21no—
191C23×C53—22no—
192C23×C53#12011yescharacteristic
193C22×D53—42no—
194C22×D53—82no—
195C22×D53—82no—
196C22×D53—82no—
197Q8×C53—21no—
198Q8×C53—21no—
199Q8×C53—44no—
200Q8×C53—42no—
201Q8×C53—82no—
202D4×C53—21no—
203D4×C53—21no—
204D4×C53—24no—
205D4×C53—42no—
206D4×C53—44no—
207D4×C53—42no—
208D4×C53—42no—
209D4×C53—84no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—21no—
213C2×C4×C53—24no—
214C2×C4×C53—22no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—82no—
223C2×C4×C53—84no—
224C2×C4×C53#12011yescharacteristic
225C2×C4×C53#12011yescharacteristic
226C2×C4×C53#12011yescharacteristic
227C2×C4×C53#13512yes—
228C2×C4×C53AGL(1,ℤ/8)12yes—
229C2×C4×C53C22×D411yescharacteristic
230C2×C4×C53C22×D411yescharacteristic
231C2×C4×C53C22×D411yescharacteristic
232D212—44no—
233D212—42no—
234D212—42no—
235D212—44no—
236D212—82no—
237D212—82no—
238D212—82no—
239#88322—42no—
240#88322—42no—
241#88322—42no—
242#18646—531nomaximal, Sylow
243C22×C53—21no—
244C22×C53—21no—
245C22×C53—41no—
246C22×C53—42no—
247C22×C53—41no—
248C22×C53—42no—
249C22×C53—41no—
250C22×C53—42no—
251C22×C53#44912yes—
252C22×C53C2×#12011yescharacteristic
253C2×D53—82no—
254C2×D53—82no—
255C2×D53—82no—
256C2×D53—164no—
257C4×C53—21no—
258C4×C53—21no—
259C4×C53—41no—
260C4×C53—42no—
261C4×C53—41no—
262C4×C53—44no—
263C4×C53—41no—
264C4×C53#44912yes—
265C4×C53#45412yes—
266C4×C53C2×#13511yescharacteristic
267C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
268C4×C53#57612yes—
269#3511D5311yescharacteristic, complemented
270#3205—531no—
271C2×#486—531no—
272#3038—531no—
273#1628—534no—
274#1625—534no—
275#1549—531no—
276#1492—531no—
277#1476—531no—
278C2×C53—22no—
279C2×C53—81no—
280C2×C53—82no—
281C2×C53C2×#44911yescharacteristic
282C2×C53#300811yescharacteristic
283C2×C53C2×#45411yescharacteristic
284D53—162no—
285C2×AGL(1,ℤ/8)—1061no—
286#556C2×D5312yescomplemented
287#555—21no—
288#554—21no—
289#553—22no—
290#548—22no—
291#547C2×D5312yescomplemented
292#542—22no—
293#537—21no—
294#533—21no—
295#532—21no—
296#532C2×D5311yescharacteristic
297C2×C4⋊Q8C2×D5311yescharacteristic
298C2×#127—531no—
299#520—21no—
300C4×#41C2×D5311yescharacteristic
301#493—1061no—
302#486—534no—
303#486—1062no—
304#486—1061no—
305#460—532no—
306#460—1061no—
307#459—532no—
308#459—534no—
309#459—1061no—
310#447—1062no—
311#431—531no—
312#423—531no—
313#418—1061no—
314C2×#102—1062no—
315C2×C4×C8—531no—
316#332—534no—
317Q8⋊C8—532no—
318D4⋊C8—532no—
319D4⋊C8—534no—
320C53#1864611yescharacteristic, complemented, Sylow
321C2×#41—22no—
322C2×#41—21no—
323C2×#41C22⋊D5311yescharacteristic
324C22×Q8—21no—
325C22×D4—1061no—
326C22×D4—1061no—
327C22×D4—1061no—
328AGL(1,ℤ/8)—2124no—
329C2×SD16—1062no—
330C2×D8—1064no—
331C2×D8—1062no—
332C2×M16—1061no—
333C22×C8—1061no—
334C4⋊Q8—21no—
335C4⋊Q8—21no—
336C4⋊Q8C22⋊D5312yes—
337C4⋊Q8C22⋊D5311yescharacteristic
338C4⋊Q8C22×D5312yes—
339#127—532no—
340#127—1061no—
341#127—1061no—
342#127—1061no—
343#127C22⋊D5311yescharacteristic
344#126—24no—
345#125—22no—
346#124—24no—
347#124—22no—
348#124—21no—
349#123—24no—
350C22⋊Q8—22no—
351C22⋊Q8—24no—
352C22⋊Q8—22no—
353C22⋊Q8—22no—
354#121—22no—
355#121—22no—
356#121—24no—
357#121—22no—
358#121—21no—
359#121C22⋊D5312yes—
360C4×Q8—22no—
361C4×Q8—22no—
362C4×Q8C22⋊D5311yescharacteristic
363C4×Q8C22×D5311yescharacteristic
364C4×D4—22no—
365C4×D4—24no—
366C4×D4—22no—
367C4×D4—21no—
368C4×D4—21no—
369C4×D4—21no—
370C4×D4C22⋊D5311yescharacteristic
371C4×D4C22×D5311yescharacteristic
372C4×D4C22×D5312yes—
373#117—22no—
374#117—22no—
375#117—21no—
376#117C22⋊D5311yescharacteristic
377C2×C4⋊C4—21no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4C22⋊D5312yes—
380C2×C42—21no—
381C2×C42C22×D5311yescharacteristic
382C4⋊C8—532no—
383C4⋊C8—532no—
384C4⋊C8—1061no—
385#102—1068no—
386#102—1062no—
387#102—1062no—
388#102—1064no—
389C22⋊C8—1061no—
390C4×C8—532no—
391C4×C8—1061no—
392C4×C8—1061no—
393C24—2121no—
394#41—22no—
395#41—22no—
396#41—42no—
397#41—42no—
398#41—84no—
399C2×Q8—21no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—82no—
403C2×Q8C2×C22⋊D5311yescharacteristic
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—24no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—1062no—
410C2×D4—1062no—
411C2×D4—1062no—
412C2×D4—4242no—
413C2×D4—2124no—
414C2×D4—2121no—
415C2×D4—2121no—
416C2×D4—2121no—
417C2×D4—2122no—
418C2×D4—2122no—
419C2×D4C2×C22⋊D5311yescharacteristic
420C2×D4C2×C22⋊D5312yes—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4C2×C22⋊D5311yescharacteristic
428C22×C4C2×C22⋊D5311yescharacteristic
429C22×C4C2×C22⋊D5311yescharacteristic
430SD16—2124no—
431D8—4244no—
432D8—2124no—
433M16—2122no—
434C2×C8—1062no—
435C2×C8—1062no—
436C2×C8—1062no—
437C2×C8—2122no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4C2×C22⋊D5311yescharacteristic
447C4⋊C4C2×C22⋊D5314yes—
448C4⋊C4C2×C22⋊D5311yescharacteristic
449C4⋊C4C2×C22⋊D5312yes—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—24no—
453C22⋊C4—21no—
454C22⋊C4—21no—
455C22⋊C4—22no—
456C22⋊C4—21no—
457C22⋊C4—22no—
458C42—24no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42C2×C22⋊D5311yescharacteristic
463C42C2×C22⋊D5311yescharacteristic
464C42C2×C22⋊D5311yescharacteristic
465C42C23×D5311yescharacteristic
466C23—21no—
467C23—22no—
468C23—4242no—
469C23—4242no—
470C23—4242no—
471C23—2122no—
472C23#1313318611yescharacteristic
473Q8—21no—
474Q8—21no—
475Q8—42no—
476Q8—44no—
477Q8—82no—
478D4—21no—
479D4—21no—
480D4—24no—
481D4—42no—
482D4—44no—
483D4—42no—
484D4—42no—
485D4—84no—
486D4—4242no—
487D4—4242no—
488D4—4242no—
489D4—2124no—
490D4—2122no—
491D4—2122no—
492D4—2124no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—24no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—82no—
506C2×C4—84no—
507C2×C4#1313318611yescharacteristic
508C2×C4#1313318611yescharacteristic
509C2×C4#1313318611yescharacteristic
510C2×C4#1313323712yes—
511C2×C4#1313324212yes—
512C2×C4C22×C22⋊D5311yescharacteristic
513C2×C4C22×C22⋊D5311yescharacteristic
514C2×C4C22×C22⋊D5311yescharacteristic
515C8—2122no—
516C8—2122no—
517C8—2122no—
518C22—21no—
519C22—21no—
520C22—41no—
521C22—42no—
522C22—41no—
523C22—42no—
524C22—41no—
525C22—42no—
526C22—8484no—
527C22—4242no—
528C22—4242no—
529C22—4242no—
530C22#1681076812yes—
531C22C2×#1313318611yescharacteristic
532C4—21no—
533C4—21no—
534C4—41no—
535C4—42no—
536C4—41no—
537C4—44no—
538C4—41no—
539C4#1681076812yes—
540C4#1681079612yes—
541C4C2×#1313323711yescharacteristic
542C4C2×#1313324211yescharacteristic
543C4#1681142612yes—
544C2—22no—
545C2—81no—
546C2—82no—
547C2—8482no—
548C2C2×#1681076811yescharacteristic
549C2#2028398311yescharacteristic
550C2C2×#1681079611yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3511C2
socleC22×C53C2×#120
hypercenter#3511D53
hypocenterC53#18646
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3511

Complements

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

NH ≅ G/Ntype
1. #20285630C2semidirect
2. C2×#16810971C2semidirect
3. #20284195C2semidirect
4. #20273825C2semidirect
5. #20273804C2semidirect
6. #20273400C2semidirect
7. #20273166C2semidirect
8. #20273097C2semidirect
9. C53×#3511C2semidirect
27. #16810369C22semidirect
28. Q8⋊#88322C22semidirect
29. D4⋊#88322C22semidirect
30. D4⋊#88322C22semidirect
31. C53×#556C22semidirect
36. C53×#547C22semidirect
269. #3511D53semidirect
286. #556C2×D53semidirect
291. #547C2×D53semidirect
320. C53#18646semidirect

Maximal subgroups

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

subgroupindex
#202856302
C2×#168109712
#202841952
#202738252
#202738042
#202734002
#202731662
#202730972
C53×#35112
#1864653