Finite Group Databasebeta

← #27366928

Subgroups — #27366928

Order 13568.

696 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285111C211yesmaximal, characteristic, complemented
2#20284715C211yesmaximal, characteristic
3D4×#11719917C211yesmaximal, characteristic, complemented
4#20274292C212yesmaximal, complemented
5#20274250C212yesmaximal, complemented
6#20274168C212yesmaximal, complemented
7#20274126C212yesmaximal, complemented
8#20273465C211yesmaximal, characteristic, complemented
9#20273396C211yesmaximal, characteristic, complemented
10#20273390C211yesmaximal, characteristic, complemented
11C2×C53×#476C211yesmaximal, characteristic, complemented
12C22×#11719917—21no—
13#16811173C2211yescharacteristic
14#16810985—22no—
15#16810975C2212yescomplemented
16#16810971C2212yescomplemented
17#16810936—21no—
18#16810903C2211yescharacteristic
19#16810903C2211yescharacteristic
20#16810818C2211yescharacteristic
21#16810818C2211yescharacteristic
22#16810757—21no—
23C4×#11719922—21no—
24C4×#11719922—21no—
25#16810701C2212yes—
26#16810695—22no—
27D4×#88322C2212yes—
28#16810676C2211yescharacteristic
29#16810647C2211yescharacteristic
30#16810622—21no—
31#16810622—21no—
32#16810587—21no—
33C4×#11719917C2211yescharacteristic
34#16810375C2212yescomplemented
35#16810373C2212yescomplemented
36#16810372C2212yescomplemented
37#16810372C2212yescomplemented
38#16810368C2212yescomplemented
39#16810366C2212yescomplemented
40C2×C4⋊Q8×C53C2211yescharacteristic
41C2×C4×D4×C53C2211yescharacteristic
42C53×#476—21no—
43C53×#476—21no—
44C53×#476C2212yes—
45C53×#476C2212yes—
46C2×#106×C53—21no—
47C2×C4⋊C8×C53C2211yescharacteristic
48C2×#102×C53—21no—
49C2×#11719945—21no—
50C2×#11719922—21no—
51C2×#11719922—21no—
52C2×#11719917—22no—
53C2×#11719917—21no—
54C2×#11719917—44no—
55C2×#11719917D411yescharacteristic
56C2×#11719917D411yescharacteristic
57C22×#88322—22no—
58#13133217C2311yescharacteristic
59#13133214C2311yescharacteristic
60#13133205—21no—
61#13133191—21no—
62#13133191—21no—
63C4×D212—21no—
64C4×D212—21no—
65#13133150—21no—
66#13133150—21no—
67C4⋊#88322—21no—
68C4⋊#88322C2311yescharacteristic
69C4⋊#88322C2311yescharacteristic
70C4⋊#88322C2312yes—
71#13133140—22no—
72#13133140—22no—
73#13133138—21no—
74#13133138—21no—
75#13133137—21no—
76#13133137—21no—
77#13133137—22no—
78#13133137—22no—
79C22⋊#88322—22no—
80C22⋊#88322—22no—
81C22⋊#88322—21no—
82#13133128D412yes—
83C4×#88322—21no—
84C4×#88322C2312yes—
85C22×Q8×C53—21no—
86C22×D4×C53D411yescharacteristic
87C23×C4×C53—21no—
88C22×C8×C53—21no—
89C4⋊Q8×C53—21no—
90C4⋊Q8×C53—21no—
91C4⋊Q8×C53—21no—
92C4⋊Q8×C53C2312yes—
93C4×D4×C53—21no—
94C4×D4×C53—21no—
95C4×D4×C53D411yescharacteristic
96C4×D4×C53D411yescharacteristic
97C4×D4×C53C2311yescharacteristic
98C4×D4×C53C2311yescharacteristic
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53D411yescharacteristic
102C2×C22⋊C4×C53—21no—
103C2×C42×C53C2311yescharacteristic
104#106×C53—22no—
105#106×C53—22no—
106C4⋊C8×C53—21no—
107C4⋊C8×C53C2312yes—
108#102×C53—22no—
109#102×C53—22no—
110C24×C53—21no—
111C2×Q8×C53—22no—
112C2×Q8×C53—41no—
113C2×Q8×C53—82no—
114C2×D4×C53—22no—
115C2×D4×C53—21no—
116C2×D4×C53—41no—
117C2×D4×C53C2×D411yescharacteristic
118C2×D4×C53C2×D411yescharacteristic
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—42no—
125C22×C4×C53—42no—
126C22×C4×C53—41no—
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53#4111yescharacteristic
130C2×C22⋊D53—41no—
131#11719945—42no—
132#11719945—42no—
133C2×D212—21no—
134C2×D212—21no—
135C2×D212—21no—
136C2×C4×D53—41no—
137C2×Dic106—21no—
138C2×C8×C53—22no—
139C2×C8×C53—42no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—22no—
142C4⋊C4×C53—22no—
143C4⋊C4×C53—41no—
144C4⋊C4×C53—41no—
145C4⋊C4×C53C2×D411yescharacteristic
146C4⋊C4×C53C2×D411yescharacteristic
147C22⋊C4×C53—21no—
148C22⋊C4×C53—21no—
149C22⋊C4×C53—21no—
150C22⋊C4×C53—21no—
151C42×C53C2×D411yescharacteristic
152C42×C53C2×D411yescharacteristic
153C42×C53#4111yescharacteristic
154C42×C53C2411yescharacteristic
155#11719922—42no—
156#11719922—42no—
157#11719921—41no—
158#11719921—41no—
159C4⋊Dic53—21no—
160C4⋊Dic53—21no—
161#11719917—22no—
162#11719917—22no—
163#11719917—44no—
164#11719917—42no—
165C2×#88322—21no—
166C2×#88322—21no—
167C2×#88322—22no—
168C2×#88322—44no—
169C2×#88322C2×D412yes—
170C2×#88322C2×D412yes—
171C23×C53—21no—
172C23×C53—21no—
173C23×C53—42no—
174C23×C53—42no—
175C23×C53—81no—
176C23×C53#12111yescharacteristic
177C22×D53—41no—
178Q8×C53—82no—
179Q8×C53—82no—
180D4×C53—21no—
181D4×C53—21no—
182D4×C53—21no—
183D4×C53—42no—
184D4×C53#13712yes—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—41no—
190C2×C4×C53—42no—
191C2×C4×C53—41no—
192C2×C4×C53—41no—
193C2×C4×C53—41no—
194C2×C4×C53—41no—
195C2×C4×C53—42no—
196C2×C4×C53—42no—
197C2×C4×C53—42no—
198C2×C4×C53—82no—
199C2×C4×C53#12111yescharacteristic
200C2×C4×C53#12111yescharacteristic
201C2×C4×C53#12111yescharacteristic
202C2×C4×C53#13512yes—
203C2×C4×C53AGL(1,ℤ/8)12yes—
204C2×C4×C53C22×D411yescharacteristic
205C2×C4×C53C22×D411yescharacteristic
206C2×C4×C53C2×#4111yescharacteristic
207C22⋊D53—82no—
208C2×Dic53—41no—
209D212—41no—
210D212—41no—
211D212—41no—
212D212—41no—
213D212—81no—
214C4×D53—82no—
215Dic106—41no—
216Dic106—41no—
217C8×C53—42no—
218#88322—22no—
219#88322—22no—
220#88322—42no—
221#88322—41no—
222#88322—41no—
223#19168—531nomaximal, Sylow
224C22×C53—21no—
225C22×C53—21no—
226C22×C53—22no—
227C22×C53—41no—
228C22×C53—41no—
229C22×C53—41no—
230C22×C53—41no—
231C22×C53—41no—
232C22×C53—84no—
233C22×C53#46312yes—
234C22×C53C2×#12111yescharacteristic
235C2×D53—81no—
236C2×D53—81no—
237C2×D53—81no—
238C4×C53—21no—
239C4×C53—21no—
240C4×C53—21no—
241C4×C53—41no—
242C4×C53—41no—
243C4×C53—82no—
244C4×C53#46312yes—
245C4×C53C2×#13511yescharacteristic
246C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
247C4×C53C2×#13711yescharacteristic
248C4×C53#57511yescharacteristic
249Dic53—81no—
250#3141—531no—
251#3100—531no—
252C2×#476D5311yescharacteristic, complemented
253D4×M16—531no—
254#1703—532no—
255#1697—532no—
256#1683—532no—
257#1677—532no—
258#1565—531no—
259#1548—531no—
260#1547—531no—
261C2×C53—22no—
262C2×C53—42no—
263C2×C53—81no—
264C2×C53C2×#46311yescharacteristic
265C2×C53#305611yescharacteristic
266C2×C53#306811yescharacteristic
267D53—161no—
268C22×M16—1061no—
269#532—531no—
270C2×C4⋊Q8C2×D5311yescharacteristic
271C2×C4×D4C2×D5311yescharacteristic
272#489—1062no—
273#487—532no—
274#486—532no—
275#480—1061no—
276#476—21no—
277#476—21no—
278#476C2×D5312yes—
279#476C2×D5312yes—
280#474—531no—
281#474—531no—
282#459—531no—
283#459—531no—
284#447—1061no—
285C4×D8—1061no—
286C4×D8—1061no—
287#436—532no—
288#435—1062no—
289C8×D4—532no—
290#431—531no—
291C2×#106—21no—
292#423—531no—
293C2×C4⋊C8C2×D5311yescharacteristic
294#417—1061no—
295#417—1061no—
296C2×#102—21no—
297#407—1061no—
298C4×M16—531no—
299#335—532no—
300#334—532no—
301#333—532no—
302#333—532no—
303#332—532no—
304#331—532no—
305C53#1916811yescharacteristic, complemented, Sylow
306C2×#41—1061no—
307C22×Q8—21no—
308C22×D4C22⋊D5311yescharacteristic
309C23×C4—21no—
310C2×D8—1061no—
311C2×D8—1061no—
312C2×M16—531no—
313C2×M16—531no—
314C2×M16—1062no—
315C2×M16—1061no—
316C2×M16—2124no—
317C22×C8—21no—
318C22×C8—1062no—
319C4⋊Q8—21no—
320C4⋊Q8—21no—
321C4⋊Q8—21no—
322C4⋊Q8—531no—
323C4⋊Q8C22×D5312yes—
324#127—531no—
325#124—1061no—
326#121—1061no—
327#121—1061no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—1061no—
331C4×D4—1061no—
332C4×D4C22⋊D5311yescharacteristic
333C4×D4C22⋊D5311yescharacteristic
334C4×D4C22×D5311yescharacteristic
335C4×D4C22×D5311yescharacteristic
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4C22⋊D5311yescharacteristic
339C2×C22⋊C4—21no—
340C2×C42C22×D5311yescharacteristic
341#107—1061no—
342#107—1061no—
343#106—22no—
344#106—22no—
345C4⋊C8—21no—
346C4⋊C8—532no—
347C4⋊C8—531no—
348C4⋊C8—531no—
349C4⋊C8—1061no—
350C4⋊C8C22×D5312yes—
351#103—1062no—
352#103—1062no—
353#102—22no—
354#102—22no—
355#102—1062no—
356#102—1062no—
357#102—1061no—
358#102—1061no—
359#102—1061no—
360#102—1061no—
361C22⋊C8—1062no—
362C22⋊C8—1062no—
363C22⋊C8—1061no—
364#97—532no—
365C4×C8—532no—
366C4×C8—1061no—
367C24—21no—
368#41—2122no—
369#41—2122no—
370C2×Q8—22no—
371C2×Q8—41no—
372C2×Q8—82no—
373C2×Q8—1061no—
374C2×D4—22no—
375C2×D4—21no—
376C2×D4—41no—
377C2×D4—1061no—
378C2×D4—1061no—
379C2×D4—1061no—
380C2×D4—2121no—
381C2×D4C2×C22⋊D5311yescharacteristic
382C2×D4C2×C22⋊D5311yescharacteristic
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—2121no—
392C22×C4D4×D5311yescharacteristic
393C22×C4#1171994711yescharacteristic
394C22×C4C2×C22⋊D5311yescharacteristic
395D8—2122no—
396D8—2122no—
397M16—1062no—
398M16—1062no—
399M16—2124no—
400M16—2122no—
401C2×C8—22no—
402C2×C8—42no—
403C2×C8—532no—
404C2×C8—532no—
405C2×C8—1062no—
406C2×C8—1061no—
407C2×C8—1061no—
408C2×C8—2124no—
409C4⋊C4—21no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—1061no—
415C4⋊C4—1061no—
416C4⋊C4C2×C22⋊D5311yescharacteristic
417C4⋊C4C2×C22⋊D5311yescharacteristic
418C22⋊C4—21no—
419C22⋊C4—21no—
420C22⋊C4—21no—
421C22⋊C4—21no—
422C22⋊C4—2121no—
423C22⋊C4—2121no—
424C42D4×D5311yescharacteristic
425C42#1171994711yescharacteristic
426C42C2×C22⋊D5311yescharacteristic
427C42C23×D5311yescharacteristic
428C23—21no—
429C23—21no—
430C23—42no—
431C23—42no—
432C23—81no—
433C23—2121no—
434C23#1313318911yescharacteristic
435Q8—82no—
436Q8—82no—
437Q8—2121no—
438Q8—2121no—
439D4—21no—
440D4—21no—
441D4—21no—
442D4—42no—
443D4—4241no—
444D4—4242no—
445D4—2121no—
446D4—2121no—
447D4—2121no—
448D4—2121no—
449D4#1313324812yes—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—82no—
464C2×C4—4242no—
465C2×C4—2121no—
466C2×C4#1313318911yescharacteristic
467C2×C4#1313318911yescharacteristic
468C2×C4#1313318911yescharacteristic
469C2×C4#1313323812yes—
470C2×C4#1313324412yes—
471C2×C4C2×D4×D5311yescharacteristic
472C2×C4C22×C22⋊D5311yescharacteristic
473C2×C4C2×#1171994711yescharacteristic
474C8—42no—
475C8—1062no—
476C8—1062no—
477C8—2122no—
478C8—2121no—
479C8—2121no—
480C22—21no—
481C22—21no—
482C22—22no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—84no—
489C22—4241no—
490C22—4241no—
491C22—4241no—
492C22#1681084012yes—
493C22C2×#1313318911yescharacteristic
494C4—21no—
495C4—21no—
496C4—21no—
497C4—41no—
498C4—41no—
499C4—82no—
500C4—4241no—
501C4#1681084012yes—
502C4C2×#1313323811yescharacteristic
503C4C2×#1313324411yescharacteristic
504C4C2×#1313324811yescharacteristic
505C4#1681142111yescharacteristic
506C2—22no—
507C2—42no—
508C2—81no—
509C2—8481no—
510C2C2×#1681084011yescharacteristic
511C2#2028434111yescharacteristic
512C2#2028444011yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#476C2
socleC22×C53C2×#121
prime radical#20284715C2
hypercenterC2×#476D53
hypocenterC53#19168
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. #20285111C2semidirect
3. D4×#11719917C2semidirect
4. #20274292C2semidirect
5. #20274250C2semidirect
6. #20274168C2semidirect
7. #20274126C2semidirect
8. #20273465C2semidirect
9. #20273396C2semidirect
10. #20273390C2semidirect
11. C2×C53×#476C2semidirect
15. #16810975C22semidirect
16. #16810971C22semidirect
34. #16810375C22semidirect
35. #16810373C22semidirect
36. #16810372C22semidirect
37. #16810372C22semidirect
38. #16810368C22semidirect
39. #16810366C22semidirect
252. C2×#476D53semidirect
305. C53#19168semidirect

Maximal subgroups

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

subgroupindex
#202851112
#202847152
D4×#117199172
#202742922
#202742502
#202741682
#202741262
#202734652
#202733962
#202733902
C2×C53×#4762
#1916853