Finite Group Databasebeta

← #27368093

Subgroups — #27368093

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#20284508C211yesmaximal, characteristic, complemented
2#20284175C211yesmaximal, characteristic
3C2×#16810702C211yesmaximal, characteristic, complemented
4#20274259C212yesmaximal, complemented
5#20274245C212yesmaximal, complemented
6#20274211C212yesmaximal, complemented
7#20274177C212yesmaximal, complemented
8#20273474C211yesmaximal, characteristic, complemented
9#20273405C211yesmaximal, characteristic, complemented
10#20273392C211yesmaximal, characteristic, complemented
11C53×#3167C211yesmaximal, characteristic, complemented
12C2×#13133242—21no—
13C22×#11719912—21no—
14C2×C4×D212C2211yescharacteristic
15#16810920—22no—
16#16810910C2212yes—
17#16810900C2212yes—
18#16810835—22no—
19#16810825C2212yes—
20#16810815C2212yes—
21#16810702—21no—
22#16810702—21no—
23#16810702C2212yes—
24#16810702C2212yes—
25#16810666—21no—
26#16810647C2211yescharacteristic
27#16810647C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29#16810627—21no—
30#16810624—21no—
31C2×#13133129—21no—
32#16810376C2212yescomplemented
33#16810376C2212yescomplemented
34#16810372C2212yescomplemented
35#16810369C2212yescomplemented
36#16810369C2212yescomplemented
37#16810367C2212yescomplemented
38C53×#533C2211yescharacteristic
39C53×#532C2211yescharacteristic
40C53×#498C2211yescharacteristic, complemented
41C53×#492C2211yescharacteristic, complemented
42C53×#487C2211yescharacteristic, complemented
43C53×#486C2211yescharacteristic, complemented
44C53×#465—21no—
45C53×#465—21no—
46C4×SD16×C53—21no—
47C4×SD16×C53—21no—
48C2×C4×C8×C53C2211yescharacteristic
49C22×D212D411yescharacteristic
50C22×C4×D53—21no—
51#13133242—44no—
52C2×#11719924—22no—
53C2×#11719922—22no—
54C2×#11719917—21no—
55C2×#11719917—21no—
56C2×#11719912—22no—
57C2×#11719912—22no—
58C2×#11719912—44no—
59C22×#88322—21no—
60C4×D212—21no—
61C4×D212—21no—
62C4×D212D412yes—
63C4×D212C2312yes—
64C2×C4⋊Dic53D411yescharacteristic
65C2×#11719921—21no—
66#13133150—22no—
67#13133148—22no—
68C4⋊#88322—21no—
69C4⋊#88322—21no—
70C4⋊#88322—21no—
71C4⋊#88322C2312yes—
72C4⋊#88322C2312yes—
73C4⋊#88322C2312yes—
74#13133141—22no—
75#13133138—22no—
76#13133137—22no—
77#13133137—22no—
78#13133129—22no—
79#13133129—22no—
80C2×#41×C53—21no—
81C2×SD16×C53—21no—
82C2×SD16×C53—21no—
83C22×C8×C53D411yescharacteristic
84C22×C8×C53D411yescharacteristic
85C4⋊Q8×C53C2311yescharacteristic
86C4⋊Q8×C53C2311yescharacteristic
87C4⋊Q8×C53C2311yescharacteristic
88#127×C53C2311yescharacteristic
89#125×C53—21no—
90#124×C53—21no—
91C22⋊Q8×C53—21no—
92C22⋊Q8×C53—21no—
93#121×C53—21no—
94#121×C53—21no—
95C4×Q8×C53—21no—
96C4×Q8×C53—21no—
97C4×D4×C53—21no—
98C4×D4×C53—21no—
99#117×C53—21no—
100C2×C42×C53C2311yescharacteristic
101#106×C53—21no—
102#106×C53—21no—
103#103×C53—21no—
104#103×C53—21no—
105#102×C53—21no—
106#102×C53—21no—
107C4×C8×C53D411yescharacteristic
108C4×C8×C53D411yescharacteristic
109C4×C8×C53C2311yescharacteristic
110C4×C8×C53C2311yescharacteristic
111C23×D53—21no—
112#41×C53—42no—
113#41×C53—42no—
114C2×Q8×C53—21no—
115C2×Q8×C53—21no—
116C2×Q8×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—41no—
121C22×C4×C53—41no—
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53#4111yescharacteristic
125C22×Dic53—21no—
126C2×D212—22no—
127C2×D212—22no—
128C2×D212—21no—
129C2×D212C2×D412yes—
130C2×C4×D53—22no—
131C2×C4×D53—22no—
132C2×C4×D53—42no—
133C2×C4×D53—42no—
134SD16×C53—42no—
135SD16×C53—42no—
136C2×C8×C53—22no—
137C2×C8×C53—41no—
138C2×C8×C53C2×D411yescharacteristic
139C2×C8×C53C2×D411yescharacteristic
140C2×C8×C53C2×D411yescharacteristic
141C2×C8×C53C2×D411yescharacteristic
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—41no—
149C4⋊C4×C53—41no—
150C22⋊C4×C53—41no—
151C22⋊C4×C53—41no—
152C22⋊C4×C53—41no—
153C42×C53—41no—
154C42×C53C2×D411yescharacteristic
155C42×C53C2×D411yescharacteristic
156C42×C53#4111yescharacteristic
157C42×C53C2411yescharacteristic
158#11719924—44no—
159#11719922—44no—
160#11719921—22no—
161#11719921—22no—
162C4⋊Dic53—21no—
163C4⋊Dic53C2×D412yes—
164#11719917—42no—
165#11719917—42no—
166C2×#88322—22no—
167C2×#88322—22no—
168C2×#88322—22no—
169C2×#88322—42no—
170#11719912—44no—
171#11719912—44no—
172C23×C53—41no—
173C23×C53#12111yescharacteristic
174C22×D53—22no—
175C22×D53—42no—
176C22×D53—42no—
177C22×D53—42no—
178Q8×C53—41no—
179Q8×C53—41no—
180Q8×C53—41no—
181Q8×C53—41no—
182Q8×C53—81no—
183D4×C53—41no—
184D4×C53—41no—
185D4×C53—41no—
186D4×C53—41no—
187D4×C53—81no—
188D4×C53—82no—
189C2×C4×C53—21no—
190C2×C4×C53—21no—
191C2×C4×C53—41no—
192C2×C4×C53—41no—
193C2×C4×C53—41no—
194C2×C4×C53—41no—
195C2×C4×C53—82no—
196C2×C4×C53#12111yescharacteristic
197C2×C4×C53#12111yescharacteristic
198C2×C4×C53#12111yescharacteristic
199C2×C4×C53AGL(1,ℤ/8)12yes—
200C2×C4×C53#13712yes—
201C2×C4×C53C22×D411yescharacteristic
202C2×C4×C53C22×D411yescharacteristic
203C2×C4×C53C2×#4111yescharacteristic
204C2×Dic53—22no—
205C2×Dic53—42no—
206D212—22no—
207D212—22no—
208D212—42no—
209C4×D53—44no—
210C4×D53—42no—
211C4×D53—42no—
212C8×C53—21no—
213C8×C53—21no—
214C8×C53—21no—
215C8×C53AGL(1,ℤ/8)12yescomplemented
216#88322—42no—
217#88322—42no—
218#88322—42no—
219#19279—531nomaximal, Sylow
220C22×C53—21no—
221C22×C53—21no—
222C22×C53—81no—
223C22×C53—81no—
224C22×C53—81no—
225C22×C53#46812yes—
226C22×C53C2×#12111yescharacteristic
227C2×D53—42no—
228C2×D53—42no—
229C2×D53—42no—
230C2×D53—84no—
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—21no—
234C4×C53—81no—
235C4×C53—81no—
236C4×C53—81no—
237C4×C53#46812yes—
238C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
239C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
240C4×C53C2×#13711yescharacteristic
241C4×C53#57511yescharacteristic
242Dic53—42no—
243#3167D5311yescharacteristic, complemented
244#3077—531no—
245#3036—531no—
246C2×#436—531no—
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—161no—
256C2×C53#303611yescharacteristic
257C2×C53C2×#46811yescharacteristic
258C2×C53#306711yescharacteristic
259D53—82no—
260C2×AGL(1,ℤ/8)—1061no—
261C22×M16—1061no—
262#533C2×D5311yescharacteristic
263#532C2×D5311yescharacteristic
264C2×C4×D4—531no—
265#498C2×D5311yescharacteristic, complemented
266#492C2×D5311yescharacteristic, complemented
267#487C2×D5311yescharacteristic, complemented
268#486C2×D5311yescharacteristic, complemented
269#478—1062no—
270#476—532no—
271#474—532no—
272#465—21no—
273#465—21no—
274#463—1062no—
275#461—532no—
276#459—532no—
277C4×SD16—21no—
278C4×SD16—21no—
279#436—532no—
280#436—532no—
281#436—1061no—
282#436—1061no—
283#428—1061no—
284#423—531no—
285#423—531no—
286C2×C4⋊C8—531no—
287#418—1061no—
288#417—1061no—
289C2×C22⋊C8—1061no—
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—21no—
299C22×D4—531no—
300C23×C4—1061no—
301AGL(1,ℤ/8)—2124no—
302C2×SD16—21no—
303C2×SD16—21no—
304C2×SD16—1062no—
305C2×D8—1062no—
306C2×M16—1061no—
307C2×M16—1061no—
308C2×M16—1062no—
309C2×M16—1062no—
310C2×M16—2124no—
311C22×C8—1061no—
312C22×C8C22⋊D5311yescharacteristic
313C22×C8C22⋊D5311yescharacteristic
314C4⋊Q8C22×D5311yescharacteristic
315C4⋊Q8C22×D5311yescharacteristic
316C4⋊Q8C22×D5311yescharacteristic
317#127C22×D5311yescharacteristic
318#125—21no—
319#124—21no—
320C22⋊Q8—21no—
321C22⋊Q8—21no—
322#121—21no—
323#121—21no—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4—532no—
329C4×D4—532no—
330C4×D4—1061no—
331C4×D4—1061no—
332#117—21no—
333C2×C4⋊C4—531no—
334C2×C22⋊C4—1061no—
335C2×C42C22×D5311yescharacteristic
336#107—1062no—
337#106—21no—
338#106—21no—
339#106—1062no—
340C4⋊C8—532no—
341C4⋊C8—532no—
342C4⋊C8—532no—
343C4⋊C8—1061no—
344C4⋊C8—1061no—
345C4⋊C8—1061no—
346#103—21no—
347#103—21no—
348#103—1062no—
349#102—21no—
350#102—21no—
351#102—1062no—
352#102—1062no—
353#102—1062no—
354C22⋊C8—1062no—
355C22⋊C8—1062no—
356C4×C8C22⋊D5311yescharacteristic
357C4×C8C22⋊D5311yescharacteristic
358C4×C8C22×D5311yescharacteristic
359C4×C8C22×D5311yescharacteristic
360C24—1061no—
361#41—42no—
362#41—42no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—41no—
370C2×D4—532no—
371C2×D4—1061no—
372C2×D4—1062no—
373C2×D4—1062no—
374C22×C4—41no—
375C22×C4—1061no—
376C22×C4—1062no—
377C22×C4—1062no—
378C22×C4—2122no—
379C22×C4—2122no—
380C22×C4D4×D5311yescharacteristic
381C22×C4#1171994711yescharacteristic
382C22×C4C2×C22⋊D5311yescharacteristic
383SD16—42no—
384SD16—42no—
385SD16—2124no—
386D8—2124no—
387M16—2124no—
388M16—2122no—
389M16—2122no—
390M16—2124no—
391C2×C8—22no—
392C2×C8—41no—
393C2×C8—1062no—
394C2×C8—1062no—
395C2×C8—1062no—
396C2×C8—2122no—
397C2×C8C2×C22⋊D5311yescharacteristic
398C2×C8C2×C22⋊D5311yescharacteristic
399C2×C8C2×C22⋊D5311yescharacteristic
400C2×C8C2×C22⋊D5311yescharacteristic
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—532no—
410C4⋊C4—1061no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—1062no—
415C22⋊C4—1062no—
416C42—41no—
417C42D4×D5311yescharacteristic
418C42#1171994711yescharacteristic
419C42C2×C22⋊D5311yescharacteristic
420C42C23×D5311yescharacteristic
421C23—41no—
422C23—1062no—
423C23—2122no—
424C23—2122no—
425C23—2122no—
426C23#1313318811yescharacteristic
427Q8—41no—
428Q8—41no—
429Q8—41no—
430Q8—41no—
431Q8—81no—
432D4—41no—
433D4—41no—
434D4—41no—
435D4—41no—
436D4—81no—
437D4—82no—
438D4—1062no—
439D4—1062no—
440D4—2122no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—82no—
448C2×C4—1062no—
449C2×C4—2124no—
450C2×C4—2122no—
451C2×C4—2122no—
452C2×C4—2122no—
453C2×C4#1313318811yescharacteristic
454C2×C4#1313318811yescharacteristic
455C2×C4#1313318811yescharacteristic
456C2×C4#1313324412yes—
457C2×C4#1313324612yes—
458C2×C4C2×D4×D5311yescharacteristic
459C2×C4C22×C22⋊D5311yescharacteristic
460C2×C4C2×#1171994711yescharacteristic
461C8—21no—
462C8—21no—
463C8—21no—
464C8—2122no—
465C8—2122no—
466C8—2122no—
467C8#1313324212yescomplemented
468C22—21no—
469C22—21no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—4244no—
474C22—2122no—
475C22—2122no—
476C22—2122no—
477C22#1681086612yes—
478C22C2×#1313318811yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—81no—
483C4—81no—
484C4—81no—
485C4—2122no—
486C4#1681086612yes—
487C4C2×#1313324211yescharacteristic
488C4C2×#1313324411yescharacteristic
489C4C2×#1313324611yescharacteristic
490C4#1681142111yescharacteristic
491C2—22no—
492C2—161no—
493C2—4242no—
494C2#2028417511yescharacteristic
495C2C2×#1681086611yescharacteristic
496C2#2028443111yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3167C2
socleC22×C53C2×#121
prime radical#20284175C2
hypercenter#3167D53
hypocenterC53#19279
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #20284508C2semidirect
3. C2×#16810702C2semidirect
4. #20274259C2semidirect
5. #20274245C2semidirect
6. #20274211C2semidirect
7. #20274177C2semidirect
8. #20273474C2semidirect
9. #20273405C2semidirect
10. #20273392C2semidirect
11. C53×#3167C2semidirect
32. #16810376C22semidirect
33. #16810376C22semidirect
34. #16810372C22semidirect
35. #16810369C22semidirect
36. #16810369C22semidirect
37. #16810367C22semidirect
40. C53×#498C22semidirect
41. C53×#492C22semidirect
42. C53×#487C22semidirect
43. C53×#486C22semidirect
215. C8×C53AGL(1,ℤ/8)semidirect
243. #3167D53semidirect
265. #498C2×D53semidirect
266. #492C2×D53semidirect
267. #487C2×D53semidirect
268. #486C2×D53semidirect
297. C53#19279semidirect
467. C8#13133242semidirect

Maximal subgroups

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

subgroupindex
#202845082
#202841752
C2×#168107022
#202742592
#202742452
#202742112
#202741772
#202734742
#202734052
#202733922
C53×#31672
#1927953