Finite Group Databasebeta

← #27362828

Subgroups — #27362828

Order 13568.

986 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285663C211yesmaximal, characteristic, complemented
2#20284723C211yesmaximal, characteristic, complemented
3#20284512C211yesmaximal, characteristic, complemented
4#20273943C214yesmaximal, complemented
5#20273860C214yesmaximal, complemented
6#20273396C211yesmaximal, characteristic, complemented
7#20273140C211yesmaximal, characteristic, complemented
8#20273106C211yesmaximal, characteristic, complemented
9C2×C53×#555C211yesmaximal, characteristic
10#16811180C2211yescharacteristic
11#16811002C2212yescomplemented
12#16810973C2212yescomplemented
13#16810953—21no—
14#16810923—22no—
15#16810913C2212yescomplemented
16#16810913C2212yescomplemented
17#16810903C2212yescomplemented
18#16810830—22no—
19#16810828C2212yescomplemented
20#16810779—22no—
21#16810774—21no—
22C4×#11719923—22no—
23#16810676C2211yescharacteristic
24#16810676C2211yescharacteristic
25#16810663—21no—
26#16810647C2211yescharacteristic
27#16810628—21no—
28#16810372C2214yescomplemented
29D4⋊#88322C2214yescomplemented
30D4⋊#88322C2214yescomplemented
31C53×#555—22no—
32C53×#555—22no—
33C53×#555—22no—
34C53×#555C2214yes—
35C2×C4⋊Q8×C53C2211yescharacteristic
36C2×#124×C53—22no—
37C2×#121×C53—21no—
38C2×#121×C53—21no—
39C2×C4×D4×C53C2211yescharacteristic
40C2×C4×D4×C53C2211yescharacteristic
41C2×#117×C53—21no—
42C2×#117×C53—21no—
43C2×#11719925—24no—
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—22no—
55#13133148—22no—
56#13133148—22no—
57C4⋊#88322—21no—
58C4⋊#88322—21no—
59C4⋊#88322C2312yes—
60C4⋊#88322C2312yes—
61#13133141—24no—
62#13133140—22no—
63#13133138—22no—
64#13133138—22no—
65#13133138—22no—
66C22⋊#88322—21no—
67C22⋊#88322—21no—
68C4×#88322—21no—
69C4×#88322C2312yes—
70C22×Q8×C53D412yes—
71C22×D4×C53—22no—
72C22×D4×C53D411yescharacteristic
73C22×D4×C53D411yescharacteristic
74C23×C4×C53—21no—
75C23×C4×C53—21no—
76C4⋊Q8×C53—21no—
77C4⋊Q8×C53—21no—
78C4⋊Q8×C53D412yes—
79C4⋊Q8×C53C2312yes—
80#124×C53—28no—
81#124×C53—28no—
82#121×C53—24no—
83#121×C53—24no—
84#121×C53—24no—
85#121×C53—24no—
86C4×D4×C53—21no—
87C4×D4×C53—21no—
88C4×D4×C53—21no—
89C4×D4×C53—21no—
90C4×D4×C53D412yes—
91C4×D4×C53D412yes—
92C4×D4×C53C2312yes—
93C4×D4×C53C2312yes—
94#117×C53—24no—
95#117×C53—24no—
96#117×C53—24no—
97#117×C53—24no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53D411yescharacteristic
101C2×C4⋊C4×C53D411yescharacteristic
102C2×C22⋊C4×C53—22no—
103C2×C22⋊C4×C53—22no—
104C2×C22⋊C4×C53—21no—
105C2×C22⋊C4×C53—21no—
106C2×C42×C53—22no—
107C2×C42×C53C2311yescharacteristic
108C24×C53—21no—
109C24×C53—21no—
110C2×Q8×C53—22no—
111C2×Q8×C53—82no—
112C2×Q8×C53C2×D414yes—
113C2×D4×C53—28no—
114C2×D4×C53—21no—
115C2×D4×C53—21no—
116C2×D4×C53—22no—
117C2×D4×C53—22no—
118C2×D4×C53—42no—
119C2×D4×C53—84no—
120C2×D4×C53C2×D412yes—
121C2×D4×C53C2×D412yes—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—44no—
130C22×C4×C53—42no—
131C22×C4×C53—42no—
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C2×Dic106—21no—
136C2×Dic106—21no—
137C4⋊C4×C53—22no—
138C4⋊C4×C53—22no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—22no—
142C4⋊C4×C53—22no—
143C4⋊C4×C53C2×D412yes—
144C4⋊C4×C53C2×D412yes—
145C22⋊C4×C53—28no—
146C22⋊C4×C53—28no—
147C22⋊C4×C53—22no—
148C22⋊C4×C53—22no—
149C22⋊C4×C53—22no—
150C22⋊C4×C53—22no—
151C42×C53—28no—
152C42×C53C2×D411yescharacteristic
153C42×C53C2×D411yescharacteristic
154C42×C53C2×D411yescharacteristic
155C42×C53C2411yescharacteristic
156C22⋊Dic53—41no—
157#11719925—84no—
158#11719923—44no—
159C4⋊Dic53—21no—
160C4⋊Dic53—21no—
161C4⋊Dic53—21no—
162C4⋊Dic53—21no—
163#11719919—41no—
164#11719919—41no—
165C4×Dic53—41no—
166#11719917—42no—
167C2×#88322—22no—
168C2×#88322—22no—
169C2×#88322—22no—
170C23×C53—22no—
171C23×C53—22no—
172C23×C53—42no—
173C23×C53—42no—
174C23×C53—42no—
175C23×C53—42no—
176C23×C53—42no—
177C23×C53—42no—
178C23×C53#12011yescharacteristic
179Q8×C53—44no—
180Q8×C53—82no—
181D4×C53—22no—
182D4×C53—22no—
183D4×C53—22no—
184D4×C53—22no—
185D4×C53—42no—
186D4×C53—42no—
187D4×C53—88no—
188C2×C4×C53—22no—
189C2×C4×C53—22no—
190C2×C4×C53—22no—
191C2×C4×C53—24no—
192C2×C4×C53—41no—
193C2×C4×C53—44no—
194C2×C4×C53—42no—
195C2×C4×C53—44no—
196C2×C4×C53—42no—
197C2×C4×C53—42no—
198C2×C4×C53—42no—
199C2×C4×C53—44no—
200C2×C4×C53—44no—
201C2×C4×C53#12011yescharacteristic
202C2×C4×C53#12011yescharacteristic
203C2×C4×C53#12011yescharacteristic
204C2×C4×C53#13512yes—
205C2×C4×C53#13712yes—
206C2×C4×C53C22×D411yescharacteristic
207C2×C4×C53C22×D411yescharacteristic
208C2×C4×C53C22×D411yescharacteristic
209C2×Dic53—41no—
210C2×Dic53—41no—
211Dic106—42no—
212Dic106—42no—
213#88322—42no—
214#88322—42no—
215#88322—42no—
216#18773—531nomaximal, Sylow
217C22×C53—21no—
218C22×C53—21no—
219C22×C53—42no—
220C22×C53—42no—
221C22×C53—42no—
222C22×C53—42no—
223C22×C53—42no—
224C22×C53—42no—
225C22×C53—84no—
226C22×C53—84no—
227C22×C53#45212yes—
228C22×C53C2×#12011yescharacteristic
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—21no—
232C4×C53—42no—
233C4×C53—42no—
234C4×C53—44no—
235C4×C53#45212yes—
236C4×C53C2×#13511yescharacteristic
237C4×C53C2×#13711yescharacteristic
238C4×C53#57512yes—
239Dic53—81no—
240Dic53—81no—
241C2×#555D5311yescharacteristic
242#3209—531no—
243#3101—531no—
244#3077—531no—
245#1646—534no—
246#1634—534no—
247#1548—531no—
248#1486—531no—
249#1478—531no—
250C2×C53—22no—
251C2×C53—82no—
252C2×C53—82no—
253C2×C53C2×#45211yescharacteristic
254C2×C53#300511yescharacteristic
255C2×C53#301911yescharacteristic
256#555—22no—
257#555—22no—
258#555—22no—
259#555C2×D5314yes—
260#533—531no—
261C2×C4⋊Q8C2×D5311yescharacteristic
262C2×#124—22no—
263C2×#121—21no—
264C2×#121—21no—
265C2×C4×D4C2×D5311yescharacteristic
266C2×C4×D4C2×D5311yescharacteristic
267C2×#117—21no—
268C2×#117—21no—
269#494—532no—
270#487—532no—
271#483—1061no—
272#478—1062no—
273#476—532no—
274#476—532no—
275#474—532no—
276#462—1062no—
277#461—532no—
278#451—1062no—
279#450—1061no—
280C4×SD16—1062no—
281#431—531no—
282#431—531no—
283#428—1061no—
284#423—531no—
285#418—1061no—
286#333—534no—
287D4⋊C8—534no—
288D4⋊C8—534no—
289C53#1877311yescharacteristic, complemented, Sylow
290C22×Q8C22⋊D5312yes—
291C22×D4—22no—
292C22×D4C22⋊D5311yescharacteristic
293C22×D4C22⋊D5311yescharacteristic
294C23×C4—21no—
295C23×C4—21no—
296C2×Q16—1064no—
297C2×SD16—1062no—
298C2×M16—1061no—
299C4⋊Q8—21no—
300C4⋊Q8—21no—
301C4⋊Q8—531no—
302C4⋊Q8—531no—
303C4⋊Q8C22⋊D5312yes—
304C4⋊Q8C22×D5312yes—
305#125—1061no—
306#124—28no—
307#124—28no—
308C22⋊Q8—1061no—
309C22⋊Q8—1061no—
310#121—24no—
311#121—24no—
312#121—24no—
313#121—24no—
314C4×Q8—1061no—
315C4×Q8—1061no—
316C4×D4—21no—
317C4×D4—21no—
318C4×D4—21no—
319C4×D4—21no—
320C4×D4C22⋊D5312yes—
321C4×D4C22⋊D5312yes—
322C4×D4C22×D5312yes—
323C4×D4C22×D5312yes—
324#117—24no—
325#117—24no—
326#117—24no—
327#117—24no—
328#117—1061no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4C22⋊D5311yescharacteristic
332C2×C4⋊C4C22⋊D5311yescharacteristic
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C42—22no—
338C2×C42C22×D5311yescharacteristic
339#107—1062no—
340#106—1062no—
341#106—1062no—
342C4⋊C8—532no—
343C4⋊C8—532no—
344C4⋊C8—1061no—
345C4⋊C8—1061no—
346#103—1064no—
347#103—1062no—
348#102—1062no—
349#102—1062no—
350#102—1062no—
351C22⋊C8—1061no—
352C22⋊C8—1061no—
353C4×C8—532no—
354C4×C8—1061no—
355C24—21no—
356C24—21no—
357C2×Q8—22no—
358C2×Q8—82no—
359C2×Q8—1061no—
360C2×Q8—1061no—
361C2×Q8C2×C22⋊D5314yes—
362C2×D4—28no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—42no—
368C2×D4—84no—
369C2×D4C2×C22⋊D5312yes—
370C2×D4C2×C22⋊D5312yes—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—44no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4C2×C22⋊D5311yescharacteristic
382C22×C4C2×C22⋊D5311yescharacteristic
383C22×C4C2×C22⋊D5311yescharacteristic
384Q16—4244no—
385SD16—2124no—
386M16—2122no—
387C2×C8—1062no—
388C2×C8—1062no—
389C2×C8—1062no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—1061no—
397C4⋊C4—1061no—
398C4⋊C4—1061no—
399C4⋊C4—1061no—
400C4⋊C4—2121no—
401C4⋊C4—2121no—
402C4⋊C4C2×C22⋊D5312yes—
403C4⋊C4C2×C22⋊D5312yes—
404C22⋊C4—28no—
405C22⋊C4—28no—
406C22⋊C4—22no—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—2121no—
411C42—28no—
412C42—2121no—
413C42C2×C22⋊D5311yescharacteristic
414C42C2×C22⋊D5311yescharacteristic
415C42C2×C22⋊D5311yescharacteristic
416C42C23×D5311yescharacteristic
417C23—22no—
418C23—22no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23#1313318611yescharacteristic
426Q8—44no—
427Q8—82no—
428Q8—2122no—
429Q8—2122no—
430D4—22no—
431D4—22no—
432D4—22no—
433D4—22no—
434D4—42no—
435D4—42no—
436D4—88no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—41no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—2121no—
451C2×C4—2121no—
452C2×C4#1313318611yescharacteristic
453C2×C4#1313318611yescharacteristic
454C2×C4#1313318611yescharacteristic
455C2×C4#1313323712yes—
456C2×C4#1313324712yes—
457C2×C4C22×C22⋊D5311yescharacteristic
458C2×C4C22×C22⋊D5311yescharacteristic
459C2×C4C22×C22⋊D5311yescharacteristic
460C8—2122no—
461C8—2122no—
462C8—2122no—
463C22—21no—
464C22—21no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—84no—
472C22—84no—
473C22#1681078512yes—
474C22C2×#1313318611yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—42no—
479C4—42no—
480C4—44no—
481C4—4241no—
482C4—4241no—
483C4#1681078512yes—
484C4C2×#1313323711yescharacteristic
485C4C2×#1313324711yescharacteristic
486C4#1681142112yes—
487C2—22no—
488C2—82no—
489C2—82no—
490C2C2×#1681078511yescharacteristic
491C2#2028395711yescharacteristic
492C2#2028407211yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC2×C53×#555C2
socleC22×C53C2×#120
hypercenterC2×#555D53
hypocenterC53#18773
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. #20285663C2semidirect
2. #20284723C2semidirect
3. #20284512C2semidirect
4. #20273943C2semidirect
5. #20273860C2semidirect
6. #20273396C2semidirect
7. #20273140C2semidirect
8. #20273106C2semidirect
11. #16811002C22semidirect
12. #16810973C22semidirect
15. #16810913C22semidirect
16. #16810913C22semidirect
17. #16810903C22semidirect
19. #16810828C22semidirect
28. #16810372C22semidirect
29. D4⋊#88322C22semidirect
30. D4⋊#88322C22semidirect
289. C53#18773semidirect

Maximal subgroups

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

subgroupindex
#202856632
#202847232
#202845122
#202739432
#202738602
#202733962
#202731402
#202731062
C2×C53×#5552
#1877353