Finite Group Databasebeta

← #27362930

Subgroups — #27362930

Order 13568.

1170 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285866C211yesmaximal, characteristic, complemented
2#20284723C211yesmaximal, characteristic, complemented
3#20284512C211yesmaximal, characteristic, complemented
4#20273950C214yesmaximal, complemented
5#20273866C214yesmaximal, complemented
6#20273400C211yesmaximal, characteristic, complemented
7#20273140C211yesmaximal, characteristic, complemented
8#20273106C211yesmaximal, characteristic, complemented
9C2×C53×#547C211yesmaximal, characteristic
10#16811180C2211yescharacteristic
11#16811026C2212yescomplemented
12#16810975C2212yescomplemented
13#16810958—22no—
14#16810953—21no—
15#16810923—22no—
16#16810915—22no—
17#16810913C2212yescomplemented
18#16810913C2212yescomplemented
19#16810903C2212yescomplemented
20#16810828C2212yescomplemented
21#16810774—21no—
22C4×#11719923—22no—
23#16810676C2211yescharacteristic
24#16810676C2211yescharacteristic
25#16810663—21no—
26#16810647C2211yescharacteristic
27#16810628—21no—
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×#11719917—21no—
45#13133217C2311yescharacteristic
46#13133217C2311yescharacteristic
47#13133209—21no—
48#13133196—21no—
49#13133196—21no—
50C4×Dic106—21no—
51C4×Dic106—21no—
52#13133176—21no—
53#13133150—24no—
54#13133150—22no—
55#13133148—22no—
56#13133148—22no—
57C4⋊#88322—21no—
58C4⋊#88322—21no—
59C4⋊#88322C2312yes—
60C4⋊#88322C2312yes—
61#13133140—22no—
62#13133140—24no—
63#13133138—22no—
64#13133138—22no—
65#13133138—22no—
66C22⋊#88322—21no—
67C22⋊#88322—21no—
68C4×#88322—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—22no—
122C2×D4×C53—22no—
123C2×D4×C53—48no—
124C2×D4×C53C2×D412yes—
125C2×D4×C53C2×D412yes—
126C22×C4×C53—28no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—44no—
135C22×C4×C53—48no—
136C22×C4×C53—42no—
137C22×C4×C53—42no—
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C2×Dic106—21no—
142C2×Dic106—21no—
143C4⋊C4×C53—28no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—22no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—22no—
148C4⋊C4×C53—22no—
149C4⋊C4×C53C2×D412yes—
150C4⋊C4×C53C2×D412yes—
151C4⋊C4×C53C2×D414yes—
152C22⋊C4×C53—28no—
153C22⋊C4×C53—22no—
154C22⋊C4×C53—22no—
155C22⋊C4×C53—22no—
156C22⋊C4×C53—22no—
157C42×C53C2×D411yescharacteristic
158C42×C53C2×D411yescharacteristic
159C42×C53C2×D411yescharacteristic
160C42×C53C2411yescharacteristic
161C22⋊Dic53—41no—
162#11719923—44no—
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167#11719919—41no—
168#11719919—41no—
169C4×Dic53—41no—
170#11719917—42no—
171C2×#88322—22no—
172C2×#88322—22no—
173C2×#88322—22no—
174C23×C53—24no—
175C23×C53—22no—
176C23×C53—22no—
177C23×C53—42no—
178C23×C53—42no—
179C23×C53—42no—
180C23×C53—42no—
181C23×C53—42no—
182C23×C53#12011yescharacteristic
183Q8×C53—44no—
184Q8×C53—44no—
185D4×C53—24no—
186D4×C53—22no—
187D4×C53—22no—
188D4×C53—22no—
189D4×C53—416no—
190D4×C53—42no—
191D4×C53AGL(1,ℤ/8)14yes—
192C2×C4×C53—22no—
193C2×C4×C53—22no—
194C2×C4×C53—24no—
195C2×C4×C53—22no—
196C2×C4×C53—41no—
197C2×C4×C53—44no—
198C2×C4×C53—42no—
199C2×C4×C53—44no—
200C2×C4×C53—416no—
201C2×C4×C53—42no—
202C2×C4×C53—42no—
203C2×C4×C53—44no—
204C2×C4×C53—42no—
205C2×C4×C53—44no—
206C2×C4×C53#12011yescharacteristic
207C2×C4×C53#12011yescharacteristic
208C2×C4×C53#12011yescharacteristic
209C2×C4×C53#13512yes—
210C2×C4×C53#13712yes—
211C2×C4×C53C22×D411yescharacteristic
212C2×C4×C53C22×D411yescharacteristic
213C2×C4×C53C22×D411yescharacteristic
214C2×Dic53—41no—
215C2×Dic53—41no—
216Dic106—42no—
217Dic106—42no—
218#88322—42no—
219#88322—42no—
220#88322—42no—
221#18783—531nomaximal, Sylow
222C22×C53—21no—
223C22×C53—21no—
224C22×C53—24no—
225C22×C53—48no—
226C22×C53—42no—
227C22×C53—42no—
228C22×C53—42no—
229C22×C53—42no—
230C22×C53—42no—
231C22×C53—84no—
232C22×C53#45212yes—
233C22×C53C2×#12011yescharacteristic
234C4×C53—21no—
235C4×C53—21no—
236C4×C53—21no—
237C4×C53—42no—
238C4×C53—42no—
239C4×C53—44no—
240C4×C53#45212yes—
241C4×C53C2×#13511yescharacteristic
242C4×C53C2×AGL(1,ℤ/8)12yes—
243C4×C53C2×#13711yescharacteristic
244Dic53—81no—
245Dic53—81no—
246C2×#547D5311yescharacteristic
247#3230—531no—
248#3101—531no—
249#3077—531no—
250#1647—534no—
251#1635—534no—
252#1549—531no—
253#1486—531no—
254#1478—531no—
255C2×C53—22no—
256C2×C53—44no—
257C2×C53—82no—
258C2×C53C2×#45211yescharacteristic
259C2×C53#300311yescharacteristic
260C2×C53#302011yescharacteristic
261C22×#41—21no—
262#547—22no—
263#547—22no—
264#547—22no—
265#547C2×D5314yes—
266#533—531no—
267C2×C4⋊Q8C2×D5311yescharacteristic
268C2×#123—22no—
269C2×C22⋊Q8—21no—
270C2×#121—21no—
271C2×C4×D4C2×D5311yescharacteristic
272C2×C4×D4C2×D5311yescharacteristic
273C22×C4⋊C4—21no—
274#500—532no—
275#487—532no—
276#484—1062no—
277#483—1061no—
278#478—1062no—
279#477—1062no—
280#476—532no—
281#476—532no—
282#474—532no—
283#461—532no—
284#450—1061no—
285C4×SD16—1062no—
286#431—531no—
287#431—531no—
288#428—1061no—
289#423—531no—
290#418—1061no—
291#332—534no—
292D4⋊C8—534no—
293D4⋊C8—534no—
294C53#1878311yescharacteristic, complemented, Sylow
295C2×#41—24no—
296C2×#41—24no—
297C2×#41—48no—
298C22×Q8—21no—
299C22×D4—22no—
300C22×D4C22⋊D5311yescharacteristic
301C22×D4C22⋊D5311yescharacteristic
302C23×C4—22no—
303C23×C4—21no—
304C23×C4—21no—
305C2×SD16—1062no—
306C2×M16—1061no—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8—531no—
310C4⋊Q8—531no—
311C4⋊Q8C22⋊D5312yes—
312C4⋊Q8C22×D5312yes—
313#125—1061no—
314#123—28no—
315#123—28no—
316C22⋊Q8—24no—
317C22⋊Q8—24no—
318C22⋊Q8—1061no—
319C22⋊Q8—1061no—
320#121—24no—
321#121—24no—
322C4×Q8—1061no—
323C4×Q8—1061no—
324C4×D4—21no—
325C4×D4—21no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4C22⋊D5312yes—
329C4×D4C22⋊D5312yes—
330C4×D4C22×D5312yes—
331C4×D4C22×D5312yes—
332#117—1061no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4C22⋊D5311yescharacteristic
338C2×C4⋊C4C22⋊D5311yescharacteristic
339C2×C4⋊C4C22⋊D5312yes—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C42C22×D5311yescharacteristic
344#107—1062no—
345#107—1064no—
346#106—1062no—
347#106—1062no—
348C4⋊C8—532no—
349C4⋊C8—532no—
350C4⋊C8—1061no—
351C4⋊C8—1061no—
352#103—1062no—
353#103—1064no—
354#102—1062no—
355#102—1062no—
356#102—1062no—
357C22⋊C8—1061no—
358C22⋊C8—1061no—
359C4×C8—532no—
360C4×C8—1061no—
361C24—21no—
362C24—21no—
363#41—416no—
364#41—48no—
365#41—48no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—44no—
369C2×Q8—1061no—
370C2×Q8—1061no—
371C2×D4—24no—
372C2×D4—28no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—48no—
378C2×D4C2×C22⋊D5312yes—
379C2×D4C2×C22⋊D5312yes—
380C22×C4—28no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—44no—
389C22×C4—48no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4C2×C22⋊D5311yescharacteristic
393C22×C4C2×C22⋊D5311yescharacteristic
394C22×C4C2×C22⋊D5311yescharacteristic
395SD16—2124no—
396M16—2122no—
397C2×C8—1062no—
398C2×C8—1062no—
399C2×C8—1062no—
400C4⋊C4—28no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—21no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—1061no—
407C4⋊C4—1061no—
408C4⋊C4—1061no—
409C4⋊C4—1061no—
410C4⋊C4—2121no—
411C4⋊C4—2121no—
412C4⋊C4C2×C22⋊D5312yes—
413C4⋊C4C2×C22⋊D5314yes—
414C4⋊C4C2×C22⋊D5312yes—
415C22⋊C4—28no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—2121no—
421C42—2121no—
422C42C2×C22⋊D5311yescharacteristic
423C42C2×C22⋊D5311yescharacteristic
424C42C2×C22⋊D5311yescharacteristic
425C42C23×D5311yescharacteristic
426C23—24no—
427C23—22no—
428C23—22no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23#1313318611yescharacteristic
435Q8—44no—
436Q8—44no—
437Q8—2122no—
438Q8—2122no—
439D4—24no—
440D4—22no—
441D4—22no—
442D4—22no—
443D4—416no—
444D4—42no—
445D4#1313324314yes—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—41no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—416no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—2121no—
461C2×C4—2121no—
462C2×C4#1313318611yescharacteristic
463C2×C4#1313318611yescharacteristic
464C2×C4#1313318611yescharacteristic
465C2×C4#1313323712yes—
466C2×C4#1313324712yes—
467C2×C4C22×C22⋊D5311yescharacteristic
468C2×C4C22×C22⋊D5311yescharacteristic
469C2×C4C22×C22⋊D5311yescharacteristic
470C8—2122no—
471C8—2122no—
472C8—2122no—
473C22—21no—
474C22—21no—
475C22—24no—
476C22—48no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—84no—
483C22#1681078512yes—
484C22C2×#1313318611yescharacteristic
485C4—21no—
486C4—21no—
487C4—21no—
488C4—42no—
489C4—42no—
490C4—44no—
491C4—4241no—
492C4—4241no—
493C4#1681078512yes—
494C4C2×#1313323711yescharacteristic
495C4C2×#1313324312yes—
496C4C2×#1313324711yescharacteristic
497C2—22no—
498C2—44no—
499C2—82no—
500C2C2×#1681078511yescharacteristic
501C2#2028394211yescharacteristic
502C2#2028407911yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC2×C53×#547C2
socleC22×C53C2×#120
hypercenterC2×#547D53
hypocenterC53#18783
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. #20285866C2semidirect
2. #20284723C2semidirect
3. #20284512C2semidirect
4. #20273950C2semidirect
5. #20273866C2semidirect
6. #20273400C2semidirect
7. #20273140C2semidirect
8. #20273106C2semidirect
11. #16811026C22semidirect
12. #16810975C22semidirect
17. #16810913C22semidirect
18. #16810913C22semidirect
19. #16810903C22semidirect
20. #16810828C22semidirect
28. #16810369C22semidirect
29. D4⋊#88322C22semidirect
30. D4⋊#88322C22semidirect
294. C53#18783semidirect

Maximal subgroups

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

subgroupindex
#202858662
#202847232
#202845122
#202739502
#202738662
#202734002
#202731402
#202731062
C2×C53×#5472
#1878353