Finite Group Databasebeta

← #28248660

Subgroups — #28248660

Order 13568.

1878 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283720C218yesmaximal, complemented
2#20283711C218yesmaximal, complemented
3#20283656C211yesmaximal, characteristic, complemented
4#20283650C211yesmaximal, characteristic, complemented
5#20283433C211yesmaximal, characteristic, complemented
6#20283427C211yesmaximal, characteristic, complemented
7#20283419C212yesmaximal, complemented
8#20283410C212yesmaximal, complemented
9#20283359C212yesmaximal, complemented
10#20283353C211yesmaximal, characteristic, complemented
11#20283143C212yesmaximal, complemented
12C2×#16810579C211yesmaximal, characteristic, complemented
13C53×#3455C211yesmaximal, characteristic
14C2×#13133224C2212yes—
15C2×#13133224C2212yes—
16C22×#11719917C2211yescharacteristic
17C23×#88322C2211yescharacteristic
18#16810754C2214yescomplemented
19#16810754C2218yescomplemented
20Q8×#88322C2214yescomplemented
21#16810701C2218yes—
22#16810701C2214yescomplemented
23#16810695C2218yes—
24#16810695C2218yes—
25#16810695C2218yescomplemented
26D4×#88322C2218yes—
27D4×#88322C2214yescomplemented
28#16810684C2218yes—
29#16810684C2218yescomplemented
30#16810676C2218yescomplemented
31#16810650C2212yes—
32#16810650C2212yes—
33#16810647C2211yescharacteristic
34#16810647C2211yescharacteristic
35#16810590C2212yes—
36#16810590C2212yes—
37#16810587C2212yes—
38#16810579C2212yes—
39#16810579C2212yes—
40#16810579C2214yes—
41#16810579C2214yes—
42#16810579C2214yescomplemented
43#16810579C2214yescomplemented
44C4×#11719917C2212yes—
45C4×#11719917C2214yescomplemented
46C2×#13133128C2211yescharacteristic
47C2×#13133128C2211yescharacteristic
48C2×#13133128C2214yescomplemented
49C2×C4×#88322C2211yescharacteristic
50C2×C4×#88322C2211yescharacteristic
51C53×#542C414yes—
52C53×#541C412yes—
53C53×#540C412yes—
54C53×#539C414yes—
55C53×#538C414yes—
56C53×#533C2211yescharacteristic
57C53×#532C2211yescharacteristic
58C53×#529C2218yes—
59C53×#525C2212yes—
60C4×#41×C53C2212yes—
61C2×#117×C53C2211yescharacteristic
62#13133224—28no—
63#13133224—28no—
64C2×#11719917—24no—
65C2×#11719917C2312yes—
66C2×#11719917C2312yes—
67C2×#11719917C2314yes—
68C2×#11719917C2314yes—
69C2×#11719917C2312yes—
70C2×#11719917C2312yes—
71C22×#88322—24no—
72C22×#88322C2311yescharacteristic
73C22×#88322C2311yescharacteristic
74C22×#88322C2314yes—
75C22×#88322C2314yes—
76C22×#88322C2314yes—
77C22×#88322C2311yescharacteristic
78C22×#88322C2311yescharacteristic
79C4⋊#88322C2314yes—
80C4⋊#88322C2318yes—
81C4⋊#88322C2314yes—
82C4⋊#88322C2318yes—
83C22⋊#88322C2318yes—
84C22⋊#88322C2318yes—
85C22⋊#88322C2318yes—
86#13133128C2314yes—
87#13133128C2314yes—
88#13133128C2314yes—
89#13133128C2312yes—
90#13133128C2312yes—
91#13133128C2314yes—
92C4×#88322C2312yes—
93C4×#88322C2314yes—
94C4×#88322C2314yes—
95C4×#88322C2312yes—
96C4×#88322C2314yes—
97C4×#88322C2314yes—
98C2×#41×C53C2312yes—
99C23×C4×C53C2311yescharacteristic
100C4⋊Q8×C53C2×C411yescharacteristic
101C4⋊Q8×C53C2×C412yes—
102#127×C53C2×C411yescharacteristic
103#126×C53C2×C418yes—
104#126×C53C2×C418yes—
105#125×C53C2×C418yes—
106#125×C53C2×C412yes—
107#124×C53C2×C418yes—
108#124×C53C2×C412yes—
109#123×C53C2×C414yes—
110#123×C53C2×C418yes—
111#123×C53C2×C418yes—
112C22⋊Q8×C53C2×C414yes—
113C22⋊Q8×C53C2×C418yes—
114C22⋊Q8×C53C2×C414yes—
115#121×C53C2×C414yes—
116#121×C53C2×C418yes—
117#121×C53C2×C414yes—
118#120×C53C2×C414yes—
119C4×Q8×C53C2×C414yes—
120C4×Q8×C53C2314yes—
121C4×D4×C53C2×C418yes—
122C4×D4×C53C2×C414yes—
123C4×D4×C53C2318yes—
124C4×D4×C53C2314yes—
125#117×C53C2×C412yes—
126#117×C53C2×C412yes—
127#117×C53C2314yes—
128#117×C53C2312yes—
129#117×C53C2314yes—
130C2×C4⋊C4×C53C2×C412yes—
131C2×C22⋊C4×C53C2×C412yes—
132C2×C42×C53C2311yescharacteristic
133C2×C42×C53C2311yescharacteristic
134C2×C42×C53C2314yes—
135C24×C53C22×C411yescharacteristic
136#41×C53—24no—
137#41×C53—24no—
138C2×Q8×C53C22×C412yes—
139C2×Q8×C53C22×C414yes—
140C2×D4×C53C22×C414yes—
141C2×D4×C53C22×C412yes—
142C2×D4×C53C22×C418yes—
143C2×D4×C53C22×C414yes—
144C22×C4×C53—22no—
145C22×C4×C53—22no—
146C22×C4×C53C22×C411yescharacteristic
147C22×C4×C53C22×C412yes—
148C22×C4×C53C22×C412yes—
149C22×C4×C53C22×C412yes—
150C22×C4×C53C22×C418yes—
151C22×C4×C53C2414yes—
152C22×C4×C53C2412yes—
153C22×C4×C53C2411yescharacteristic
154C4⋊C4×C53C22×C414yes—
155C4⋊C4×C53C22×C414yes—
156C4⋊C4×C53C22×C414yes—
157C4⋊C4×C53C22×C418yes—
158C4⋊C4×C53C22×C414yes—
159C4⋊C4×C53C22×C418yes—
160C4⋊C4×C53C22×C418yes—
161C4⋊C4×C53C22×C414yes—
162C22⋊C4×C53C22×C414yes—
163C22⋊C4×C53C22×C414yes—
164C22⋊C4×C53C22×C414yes—
165C22⋊C4×C53C22×C418yes—
166C22⋊C4×C53C22×C418yes—
167C22⋊C4×C53C22×C418yes—
168C22⋊C4×C53C22×C418yes—
169C42×C53C22×C412yes—
170C42×C53C22×C411yescharacteristic
171C42×C53C22×C411yescharacteristic
172C42×C53C22×C418yes—
173C42×C53C2412yes—
174C42×C53C2412yes—
175C42×C53C2414yes—
176#11719917—24no—
177#11719917—24no—
178#11719917—24no—
179#11719917—24no—
180#11719917—28no—
181#11719917—28no—
182C2×#88322—22no—
183C2×#88322—22no—
184C2×#88322—28no—
185C2×#88322—28no—
186C2×#88322—28no—
187C2×#88322#4114yes—
188C2×#88322#4114yes—
189C2×#88322C2412yes—
190C2×#88322C2414yes—
191C2×#88322C2412yes—
192C2×#88322C2414yes—
193C2×#88322C2411yescharacteristic
194C2×#88322C2411yescharacteristic
195C2×#88322C2412yes—
196C23×C53—22no—
197C23×C53—42no—
198C23×C53C23×C414yes—
199C23×C53C23×C412yes—
200C23×C53C23×C411yescharacteristic
201Q8×C53—24no—
202Q8×C53—44no—
203D4×C53—28no—
204D4×C53—24no—
205D4×C53—44no—
206D4×C53—48no—
207C2×C4×C53—22no—
208C2×C4×C53—24no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—24no—
212C2×C4×C53—24no—
213C2×C4×C53—42no—
214C2×C4×C53—48no—
215C2×C4×C53—42no—
216C2×C4×C53C23×C414yes—
217C2×C4×C53C23×C412yes—
218C2×C4×C53C23×C412yes—
219C2×C4×C53C23×C418yes—
220C2×C4×C53C23×C412yes—
221C2×C4×C53C23×C414yes—
222C2×C4×C53C23×C411yescharacteristic
223C2×C4×C53C2×#4112yes—
224C2×C4×C53C2×#4112yes—
225C2×C4×C53C2511yescharacteristic
226#88322—22no—
227#88322—22no—
228#88322—24no—
229#88322—24no—
230#88322C2×#4114yes—
231#88322C2×#4112yes—
232#88322C2×#4112yes—
233#60209—531nomaximal, Sylow
234C22×C53—22no—
235C22×C53—22no—
236C22×C53—24no—
237C22×C53—28no—
238C22×C53—24no—
239C22×C53—44no—
240C22×C53C4×#4112yes—
241C22×C53C24×C411yescharacteristic
242C4×C53—22no—
243C4×C53—24no—
244C4×C53—24no—
245C4×C53—22no—
246C4×C53—28no—
247C4×C53—22no—
248C4×C53C4×#4112yes—
249C4×C53C22×#4111yescharacteristic
250C4×C53C22×#4111yescharacteristic
251#3455D5311yescharacteristic
252#2969—538no—
253#2968—538no—
254#2961—531no—
255#2960—531no—
256#2932—531no—
257#2931—531no—
258#2930—532no—
259#2929—532no—
260#2922—532no—
261#2921—531no—
262#2875—532no—
263C2×#405—531no—
264C2×C53—22no—
265C2×C53—44no—
266C2×C53—42no—
267C2×C53C2×C4×#4111yescharacteristic
268C2×C53#357312yes—
269C2×#131—532no—
270C2×#131—532no—
271C22×M16—531no—
272C23×C8—531no—
273#542Dic5314yes—
274#541Dic5312yes—
275#540Dic5312yes—
276#539Dic5314yes—
277#538Dic5314yes—
278#533C2×D5311yescharacteristic
279#532C2×D5311yescharacteristic
280#529C2×D5318yes—
281#525C2×D5312yes—
282C4×#41C2×D5312yes—
283C2×#117C2×D5311yescharacteristic
284#446—534no—
285#446—538no—
286C8×Q8—534no—
287#436—538no—
288#436—534no—
289#435—538no—
290#435—538no—
291#435—538no—
292C8×D4—538no—
293C8×D4—534no—
294#433—538no—
295#433—538no—
296#431—538no—
297#424—532no—
298#424—532no—
299#423—531no—
300#423—531no—
301#408—532no—
302#408—532no—
303#407—532no—
304#405—534no—
305#405—534no—
306#405—532no—
307#405—532no—
308#405—534no—
309#405—534no—
310C4×M16—532no—
311C4×M16—534no—
312C2×#97—531no—
313C2×#97—531no—
314C2×#97—534no—
315C2×C4×C8—531no—
316C2×C4×C8—531no—
317C53#6020911yescharacteristic, complemented, Sylow
318C2×#41C22×D5312yes—
319C23×C4C22×D5311yescharacteristic
320#131—1068no—
321#131—1068no—
322C2×M16—532no—
323C2×M16—532no—
324C2×M16—532no—
325C2×M16—532no—
326C2×M16—534no—
327C2×M16—534no—
328C2×M16—1064no—
329C22×C8—531no—
330C22×C8—531no—
331C22×C8—531no—
332C22×C8—534no—
333C22×C8—531no—
334C22×C8—534no—
335C22×C8—534no—
336C22×C8—1064no—
337C4⋊Q8C2×Dic5311yescharacteristic
338C4⋊Q8C2×Dic5312yes—
339#127C2×Dic5311yescharacteristic
340#126C2×Dic5318yes—
341#126C2×Dic5318yes—
342#125C2×Dic5318yes—
343#125C2×Dic5312yes—
344#124C2×Dic5318yes—
345#124C2×Dic5312yes—
346#123C2×Dic5314yes—
347#123C2×Dic5318yes—
348#123C2×Dic5318yes—
349C22⋊Q8C2×Dic5314yes—
350C22⋊Q8C2×Dic5318yes—
351C22⋊Q8C2×Dic5314yes—
352#121C2×Dic5314yes—
353#121C2×Dic5318yes—
354#121C2×Dic5314yes—
355#120C2×Dic5314yes—
356C4×Q8C2×Dic5314yes—
357C4×Q8C22×D5314yes—
358C4×D4C2×Dic5318yes—
359C4×D4C2×Dic5314yes—
360C4×D4C22×D5318yes—
361C4×D4C22×D5314yes—
362#117C2×Dic5312yes—
363#117C2×Dic5312yes—
364#117C22×D5314yes—
365#117C22×D5312yes—
366#117C22×D5314yes—
367C2×C4⋊C4C2×Dic5312yes—
368C2×C22⋊C4C2×Dic5312yes—
369C2×C42C22×D5311yescharacteristic
370C2×C42C22×D5311yescharacteristic
371C2×C42C22×D5314yes—
372C4⋊C8—534no—
373C4⋊C8—534no—
374C4⋊C8—538no—
375C4⋊C8—538no—
376C22⋊C8—538no—
377C22⋊C8—538no—
378C22⋊C8—538no—
379#97—532no—
380#97—532no—
381#97—534no—
382#97—534no—
383#97—534no—
384#97—534no—
385C4×C8—534no—
386C4×C8—532no—
387C4×C8—532no—
388C4×C8—534no—
389C4×C8—534no—
390C4×C8—534no—
391C24C22×Dic5311yescharacteristic
392#41—24no—
393#41—24no—
394C2×Q8C22×Dic5312yes—
395C2×Q8C22×Dic5314yes—
396C2×D4C22×Dic5314yes—
397C2×D4C22×Dic5312yes—
398C2×D4C22×Dic5318yes—
399C2×D4C22×Dic5314yes—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4C22×Dic5311yescharacteristic
403C22×C4C22×Dic5312yes—
404C22×C4C22×Dic5312yes—
405C22×C4C22×Dic5312yes—
406C22×C4C22×Dic5318yes—
407C22×C4C23×D5314yes—
408C22×C4C23×D5312yes—
409C22×C4C23×D5311yescharacteristic
410M16—1064no—
411M16—1064no—
412M16—1068no—
413M16—1068no—
414M16—1064no—
415M16—1064no—
416C2×C8—534no—
417C2×C8—534no—
418C2×C8—531no—
419C2×C8—531no—
420C2×C8—532no—
421C2×C8—532no—
422C2×C8—532no—
423C2×C8—534no—
424C2×C8—534no—
425C2×C8—1068no—
426C2×C8—1068no—
427C2×C8—1068no—
428C2×C8—1062no—
429C2×C8—1062no—
430C4⋊C4C22×Dic5314yes—
431C4⋊C4C22×Dic5314yes—
432C4⋊C4C22×Dic5314yes—
433C4⋊C4C22×Dic5318yes—
434C4⋊C4C22×Dic5314yes—
435C4⋊C4C22×Dic5318yes—
436C4⋊C4C22×Dic5318yes—
437C4⋊C4C22×Dic5314yes—
438C22⋊C4C22×Dic5314yes—
439C22⋊C4C22×Dic5314yes—
440C22⋊C4C22×Dic5314yes—
441C22⋊C4C22×Dic5318yes—
442C22⋊C4C22×Dic5318yes—
443C22⋊C4C22×Dic5318yes—
444C22⋊C4C22×Dic5318yes—
445C42C22×Dic5312yes—
446C42C22×Dic5311yescharacteristic
447C42C22×Dic5311yescharacteristic
448C42C22×Dic5318yes—
449C42C23×D5312yes—
450C42C23×D5312yes—
451C42C23×D5314yes—
452C23—22no—
453C23—42no—
454C23C23×Dic5312yes—
455C23C23×Dic5314yes—
456C23C23×Dic5311yescharacteristic
457Q8—24no—
458Q8—44no—
459D4—28no—
460D4—24no—
461D4—44no—
462D4—48no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—42no—
470C2×C4—48no—
471C2×C4—42no—
472C2×C4C23×Dic5314yes—
473C2×C4C23×Dic5314yes—
474C2×C4C23×Dic5312yes—
475C2×C4C23×Dic5318yes—
476C2×C4C23×Dic5312yes—
477C2×C4C23×Dic5312yes—
478C2×C4C23×Dic5311yescharacteristic
479C2×C4#41×D5312yes—
480C2×C4#41×D5312yes—
481C2×C4C24×D5311yescharacteristic
482C8—534no—
483C8—532no—
484C8—532no—
485C8—1062no—
486C8—1064no—
487C8—1064no—
488C8—1062no—
489C22—22no—
490C22—28no—
491C22—24no—
492C22—22no—
493C22—24no—
494C22—44no—
495C22#41×Dic5312yes—
496C22C24×Dic5311yescharacteristic
497C4—24no—
498C4—22no—
499C4—28no—
500C4—22no—
501C4—22no—
502C4—24no—
503C4#41×Dic5312yes—
504C4C2×#41×D5311yescharacteristic
505C4C2×#41×D5311yescharacteristic
506C2—22no—
507C2—42no—
508C2—44no—
509C2C2×#41×Dic5311yescharacteristic
510C2#2028942712yes—
subgroupG/H
center, FrattiniC2×C4C24×D53
commutator, socleC22×C53C24×C4
Fitting, prime radicalC53×#3455C2
cosocleC2×C4×C53C25
hypercenter#3455D53
hypocenterC53#60209
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ #3455

Complements

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

NH ≅ G/Ntype
1. #20283720C2semidirect
2. #20283711C2semidirect
3. #20283656C2semidirect
4. #20283650C2semidirect
5. #20283433C2semidirect
6. #20283427C2semidirect
7. #20283419C2semidirect
8. #20283410C2semidirect
9. #20283359C2semidirect
10. #20283353C2semidirect
11. #20283143C2semidirect
12. C2×#16810579C2semidirect
18. #16810754C22semidirect
19. #16810754C22semidirect
20. Q8×#88322C22semidirect
22. #16810701C22semidirect
25. #16810695C22semidirect
27. D4×#88322C22semidirect
29. #16810684C22semidirect
30. #16810676C22semidirect
42. #16810579C22semidirect
43. #16810579C22semidirect
45. C4×#11719917C22semidirect
48. C2×#13133128C22semidirect
317. C53#60209semidirect

Maximal subgroups

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

subgroupindex
#202837202
#202837112
#202836562
#202836502
#202834332
#202834272
#202834192
#202834102
#202833592
#202833532
#202831432
C2×#168105792
C53×#34552
#6020953