Finite Group Databasebeta

← #27403539

Subgroups — #27403539

Order 13568.

740 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133237C211yesmaximal, characteristic
2#20283380C211yesmaximal, characteristic, complemented
3#20275425C212yesmaximal, complemented
4#20275311C212yesmaximal, complemented
5#20275042C212yesmaximal
6#20274852C211yesmaximal, characteristic, complemented
7#20274735C211yesmaximal, characteristic, complemented
8#20274732C211yesmaximal, characteristic, complemented
9#20274585C211yesmaximal, characteristic, complemented
10#20274557C212yesmaximal, complemented
11C53×#2906C211yesmaximal, characteristic, complemented
12C2×#13133237C2211yescharacteristic
13C4×#11719945C2211yescharacteristic
14C4×#11719925C411yescharacteristic, complemented
15C4×#11719925C411yescharacteristic, complemented
16C4×#11719924C411yescharacteristic, complemented
17C4×#11719924C411yescharacteristic, complemented
18C4×#11719923C411yescharacteristic, complemented
19C4×#11719923C411yescharacteristic, complemented
20C4×#11719922C411yescharacteristic, complemented
21C4×#11719922C411yescharacteristic, complemented
22#16810683—22no—
23#16810679—21no—
24#16810675—21no—
25#16810659C2211yescharacteristic
26#16810647C2211yescharacteristic
27C2×#13133144C2212yes—
28C2×#13133143C2212yes—
29#16810619C2211yescharacteristic
30#16810617C2211yescharacteristic
31#16810589C2212yes—
32#16810578C2212yes—
33C2×#13133127C2212yes—
34C2×C4×#88322C2211yescharacteristic
35C4×C4⋊Dic53C2211yescharacteristic
36#16810387C2212yes—
37#16810387C2212yes—
38#16810384C2212yes—
39#16810382C2212yes—
40#16810377C2212yes—
41C2×#131×C53C2212yes—
42C4×#41×C53C2211yescharacteristic
43C53×#424—22no—
44C53×#424C2212yes—
45C53×#423—21no—
46C53×#423C2211yescharacteristic
47C2×C4⋊C8×C53—21no—
48C2×C4⋊C8×C53C2211yescharacteristic
49C2×#11719945C2311yescharacteristic
50#13133237—22no—
51#13133237—22no—
52C2×#11719925C2×C411yescharacteristic
53C2×#11719923C2×C411yescharacteristic
54C2×#11719924C2×C411yescharacteristic
55C2×#11719922C2×C411yescharacteristic
56C2×#11719917C2312yes—
57C22×#88322C2311yescharacteristic
58C22×#88322C2311yescharacteristic
59C4×Dic106C2×C411yescharacteristic
60C4×Dic106C2×C411yescharacteristic
61C4×D212C2×C411yescharacteristic
62C4×D212C2×C411yescharacteristic
63C4×C22⋊D53—21no—
64C4×C22⋊D53—21no—
65#13133176C2311yescharacteristic
66#13133174—21no—
67C2×C4⋊Dic53—21no—
68C2×C4×Dic53C2312yes—
69C42×D53—21no—
70#13133150C2×C411yescharacteristic
71#13133150C2×C411yescharacteristic
72#13133148C2×C411yescharacteristic
73#13133148C2×C411yescharacteristic
74C4⋊#88322—21no—
75C4⋊#88322—21no—
76#13133144—24no—
77#13133143—42no—
78#13133141C2×C411yescharacteristic
79#13133141C2×C411yescharacteristic
80#13133140C2×C411yescharacteristic
81#13133140C2×C411yescharacteristic
82#13133138C2×C411yescharacteristic
83#13133138C2×C411yescharacteristic
84#13133137C2×C411yescharacteristic
85#13133137C2×C411yescharacteristic
86#13133129—22no—
87#13133129—22no—
88#13133127—22no—
89#13133127—42no—
90C4×#88322C2×C411yescharacteristic
91C4×#88322C2×C411yescharacteristic
92C4×#88322C2×C411yescharacteristic
93C4×#88322C2×C411yescharacteristic
94C8×Dic53—22no—
95#13133124—21no—
96C2×#41×C53C2311yescharacteristic
97#131×C53—24no—
98#131×C53—42no—
99C2×M16×C53—22no—
100C2×M16×C53C2312yes—
101C22×C8×C53—22no—
102C22×C8×C53C2312yes—
103C4×Q8×C53C2×C411yescharacteristic
104C4×Q8×C53C2×C411yescharacteristic
105C4×D4×C53—21no—
106C4×D4×C53—21no—
107C4×D4×C53C2×C411yescharacteristic
108C4×D4×C53C2×C411yescharacteristic
109#117×C53—21no—
110#117×C53C2311yescharacteristic
111C2×C42×C53—21no—
112C2×C42×C53C2311yescharacteristic
113C4⋊C8×C53—22no—
114C4⋊C8×C53—22no—
115C4⋊C8×C53—21no—
116C4⋊C8×C53—21no—
117C4⋊C8×C53D411yescharacteristic
118C4⋊C8×C53D411yescharacteristic
119C4⋊C8×C53D412yes—
120#41×C53—21no—
121#41×C53—21no—
122#41×C53#4112yes—
123#41×C53#4112yes—
124C2×Q8×C53C22×C411yescharacteristic
125C2×D4×C53—21no—
126C2×D4×C53C22×C411yescharacteristic
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53#4111yescharacteristic
131C22×C4×C53#4111yescharacteristic
132C22×C4×C53C2411yescharacteristic
133C2×C22⋊D53—21no—
134C22×Dic53—22no—
135#11719945—21no—
136#11719945—21no—
137#11719945—21no—
138#11719945—21no—
139C2×D212C22×C411yescharacteristic
140C2×C4×D53—21no—
141C2×C4×D53—21no—
142C2×C4×D53—21no—
143C2×Dic106C22×C411yescharacteristic
144M16×C53—22no—
145M16×C53—22no—
146M16×C53—44no—
147C2×C8×C53—22no—
148C2×C8×C53—22no—
149C2×C8×C53—22no—
150C2×C8×C53—44no—
151C2×C8×C53C2×D412yes—
152C2×C8×C53C2×D412yes—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53C22×C411yescharacteristic
156C4⋊C4×C53C22×C411yescharacteristic
157C22⋊C4×C53—21no—
158C22⋊C4×C53—21no—
159C22⋊C4×C53—21no—
160C42×C53—21no—
161C42×C53—21no—
162C42×C53—21no—
163C42×C53C22×C411yescharacteristic
164C42×C53C22×C411yescharacteristic
165C42×C53C2×D411yescharacteristic
166C42×C53C2×D411yescharacteristic
167C22⋊Dic53—21no—
168#11719925—22no—
169#11719924—22no—
170#11719923—22no—
171#11719922—22no—
172#11719921—21no—
173#11719921—21no—
174C4⋊Dic53—42no—
175C4⋊Dic53C22×C411yescharacteristic
176C4⋊Dic53C22×C411yescharacteristic
177#11719919—21no—
178#11719919—21no—
179C4×Dic53—21no—
180C4×Dic53—21no—
181C4×Dic53—21no—
182C4×Dic53—22no—
183C4×Dic53—22no—
184#11719917—22no—
185#11719917—22no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—22no—
191C2×#88322C22×C411yescharacteristic
192C2×#88322C22×C411yescharacteristic
193C2×#88322C22×C411yescharacteristic
194C2×#88322C22×C411yescharacteristic
195C23×C53—21no—
196C23×C53C2×#4111yescharacteristic
197C22×D53—21no—
198Q8×C53—21no—
199Q8×C53#11712yes—
200D4×C53—22no—
201D4×C53—22no—
202D4×C53—21no—
203D4×C53#11712yes—
204C2×C4×C53—21no—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—41no—
211C2×C4×C53—42no—
212C2×C4×C53—81no—
213C2×C4×C53C4×D411yescharacteristic
214C2×C4×C53C4×D411yescharacteristic
215C2×C4×C53C4×D411yescharacteristic
216C2×C4×C53C4×D411yescharacteristic
217C2×C4×C53C23×C411yescharacteristic
218C2×C4×C53C22×D411yescharacteristic
219C2×C4×C53C2×#4111yescharacteristic
220C2×C4×C53C2×#4111yescharacteristic
221C2×C4×C53H(5,𝔽2)11yescharacteristic
222C2×C4×C532−1+411yescharacteristic
223C22⋊D53—41no—
224C22⋊D53—41no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—42no—
229C2×Dic53—42no—
230C2×Dic53—42no—
231D212—21no—
232D212—21no—
233C4×D53—41no—
234C4×D53—41no—
235C4×D53—42no—
236C4×D53—81no—
237Dic106—21no—
238Dic106—21no—
239C8×C53—22no—
240C8×C53—22no—
241C8×C53—42no—
242#88322—21no—
243#88322—21no—
244#88322—21no—
245#88322—21no—
246#88322—42no—
247#22782—531nomaximal, Sylow
248C22×C53—21no—
249C22×C53—21no—
250C22×C53—22no—
251C22×C53—41no—
252C22×C53—41no—
253C22×C53C2×C4×D411yescharacteristic
254C22×C53#55211yescharacteristic
255C22×C53#55311yescharacteristic
256C2×D53—41no—
257C2×D53—41no—
258C2×D53—41no—
259C4×C53—22no—
260C4×C53—21no—
261C4×C53—21no—
262C4×C53—21no—
263C4×C53—41no—
264C4×C53—41no—
265C4×C53#44412yes—
266C4×C53C2×#11711yescharacteristic
267C4×C53#52011yescharacteristic
268C4×C53#54611yescharacteristic
269C4×C53#54811yescharacteristic
270Dic53—41no—
271Dic53—41no—
272Dic53—41no—
273Dic53—82no—
274C4×#135—531no—
275#2925—531no—
276#2906D5311yescharacteristic, complemented
277#1938—532no—
278#1922—532no—
279#1876—532no—
280#1837—531no—
281#1806—531no—
282#1806—531no—
283#1770—531no—
284#1765—532no—
285C2×C53—21no—
286C2×C53—21no—
287C2×C53—42no—
288C2×C53#235211yescharacteristic
289C2×C53C2×#44411yescharacteristic
290C2×C53#295711yescharacteristic
291D53—81no—
292C2×#135—531no—
293C2×#131C2×D5312yes—
294C4×#41—531no—
295C4×#41C2×D5311yescharacteristic
296C4×Q16—531no—
297C4×Q16—531no—
298C4×SD16—531no—
299C4×SD16—531no—
300C4×SD16—531no—
301C4×SD16—531no—
302C4×D8—531no—
303C4×D8—531no—
304#433—1062no—
305#432—1061no—
306#431—1061no—
307#427—531no—
308#424—22no—
309#424C2×D5312yes—
310#423—21no—
311#423—531no—
312#423C2×D5311yescharacteristic
313C2×C4⋊C8—21no—
314C2×C4⋊C8C2×D5311yescharacteristic
315C2×#104—532no—
316C2×#104—532no—
317#416—531no—
318#416—531no—
319#408—532no—
320#405—532no—
321C2×#97—532no—
322C2×C4×C8—531no—
323C4×C4⋊C4—531no—
324#341—532no—
325#341—532no—
326#339—532no—
327#339—532no—
328#336—532no—
329C53#2278211yescharacteristic, complemented, Sylow
330C2×#41—531no—
331C2×#41C22×D5311yescharacteristic
332#135—1062no—
333#135—1062no—
334C2×Q16—531no—
335C2×SD16—531no—
336C2×SD16—531no—
337C2×D8—531no—
338#131—24no—
339#131—42no—
340C2×M16—22no—
341C2×M16—532no—
342C2×M16C22×D5312yes—
343C22×C8—22no—
344C22×C8—531no—
345C22×C8—531no—
346C22×C8C22×D5312yes—
347C4×Q8—531no—
348C4×Q8—531no—
349C4×Q8C4×D5311yescharacteristic
350C4×Q8C4×D5311yescharacteristic
351C4×D4—21no—
352C4×D4—21no—
353C4×D4—531no—
354C4×D4—531no—
355C4×D4—1061no—
356C4×D4—1061no—
357C4×D4C4×D5311yescharacteristic
358C4×D4C4×D5311yescharacteristic
359#117—21no—
360#117—531no—
361#117—1061no—
362#117C22×D5311yescharacteristic
363C2×C4⋊C4—1061no—
364C2×C42—21no—
365C2×C42—532no—
366C2×C42—1061no—
367C2×C42C22×D5311yescharacteristic
368#107—531no—
369#107—531no—
370#106—531no—
371#106—531no—
372C4⋊C8—22no—
373C4⋊C8—22no—
374C4⋊C8—21no—
375C4⋊C8—21no—
376C4⋊C8—1061no—
377C4⋊C8—1061no—
378C4⋊C8C22⋊D5311yescharacteristic
379C4⋊C8C22⋊D5311yescharacteristic
380C4⋊C8C22⋊D5312yes—
381#104—1064no—
382#104—2122no—
383#103—531no—
384#103—531no—
385#103—531no—
386#103—531no—
387#102—531no—
388#102—531no—
389#102—531no—
390#102—531no—
391C22⋊C8—1062no—
392C22⋊C8—1062no—
393#97—1062no—
394#97—2122no—
395C4×C8—531no—
396C4×C8—531no—
397C4×C8—531no—
398C4×C8—531no—
399C4×C8—1062no—
400#95—1061no—
401#41—21no—
402#41—21no—
403#41—1061no—
404#41—1061no—
405#41—1061no—
406#41—1061no—
407#41#1171994712yes—
408#41#1171994912yes—
409C2×Q8—531no—
410C2×Q8C2×C4×D5311yescharacteristic
411C2×D4—21no—
412C2×D4—531no—
413C2×D4—1061no—
414C2×D4C2×C4×D5311yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—1061no—
419C22×C4—1061no—
420C22×C4—1061no—
421C22×C4—1062no—
422C22×C4#1171994511yescharacteristic
423C22×C4#1171994511yescharacteristic
424C22×C4C23×D5311yescharacteristic
425Q16—1062no—
426SD16—1062no—
427SD16—1062no—
428D8—1062no—
429M16—22no—
430M16—22no—
431M16—44no—
432M16—1062no—
433M16—1062no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—44no—
438C2×C8—531no—
439C2×C8—531no—
440C2×C8—531no—
441C2×C8—531no—
442C2×C8—1061no—
443C2×C8—1061no—
444C2×C8—1061no—
445C2×C8—1061no—
446C2×C8—1062no—
447C2×C8C2×C22⋊D5312yes—
448C2×C8C2×C22⋊D5312yes—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—531no—
452C4⋊C4—531no—
453C4⋊C4—1061no—
454C4⋊C4—1061no—
455C4⋊C4—2122no—
456C4⋊C4C2×C4×D5311yescharacteristic
457C4⋊C4C2×C4×D5311yescharacteristic
458C22⋊C4—21no—
459C22⋊C4—21no—
460C22⋊C4—21no—
461C22⋊C4—1061no—
462C22⋊C4—1061no—
463C22⋊C4—1061no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—1061no—
468C42—1061no—
469C42—1061no—
470C42—1062no—
471C42—1062no—
472C42C2×C4×D5311yescharacteristic
473C42C2×C4×D5311yescharacteristic
474C42C2×C22⋊D5311yescharacteristic
475C42C2×C22⋊D5311yescharacteristic
476C23—21no—
477C23—1061no—
478C23C2×#1171994511yescharacteristic
479Q8—21no—
480Q8—1061no—
481Q8—1061no—
482Q8#1313317512yes—
483D4—22no—
484D4—22no—
485D4—21no—
486D4—1061no—
487D4—1061no—
488D4—2121no—
489D4—2121no—
490D4#1313317512yes—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—81no—
500C2×C4—1061no—
501C2×C4—1061no—
502C2×C4—1061no—
503C2×C4—4241no—
504C2×C4—2122no—
505C2×C4—2122no—
506C2×C4—2122no—
507C2×C4—2121no—
508C2×C4—2121no—
509C2×C4—2122no—
510C2×C4C4×C22⋊D5311yescharacteristic
511C2×C4C4×C22⋊D5311yescharacteristic
512C2×C4C4×C22⋊D5311yescharacteristic
513C2×C4C4×C22⋊D5311yescharacteristic
514C2×C4C22×C4×D5311yescharacteristic
515C2×C4C22×C22⋊D5311yescharacteristic
516C2×C4C2×#1171994711yescharacteristic
517C2×C4C2×#1171994911yescharacteristic
518C2×C4#1313326211yescharacteristic
519C2×C4#1313326511yescharacteristic
520C8—22no—
521C8—22no—
522C8—42no—
523C8—1061no—
524C8—1061no—
525C8—1061no—
526C8—1061no—
527C8—2122no—
528C22—21no—
529C22—21no—
530C22—22no—
531C22—41no—
532C22—41no—
533C22—2121no—
534C22—2121no—
535C22—2121no—
536C22C2×C4×C22⋊D5311yescharacteristic
537C22#1681130811yescharacteristic
538C22#1681132111yescharacteristic
539C4—22no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—41no—
544C4—41no—
545C4—4242no—
546C4—2121no—
547C4—2121no—
548C4—2121no—
549C4#1681074312yes—
550C4C2×#1313317511yescharacteristic
551C4#1681110411yescharacteristic
552C4#1681126711yescharacteristic
553C4#1681128411yescharacteristic
554C2—21no—
555C2—21no—
556C2—42no—
557C2—4241no—
558C2#2027764011yescharacteristic
559C2C2×#1681074311yescharacteristic
560C2#2028363011yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2906C2
socleC22×C53C2×C4×D4
prime radicalC2×#13133237C22
cosocleC22×C4×C53C24
hypercenter#2906D53
hypocenterC53#22782
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ #2906

Complements

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

NH ≅ G/Ntype
2. #20283380C2semidirect
3. #20275425C2semidirect
4. #20275311C2semidirect
6. #20274852C2semidirect
7. #20274735C2semidirect
8. #20274732C2semidirect
9. #20274585C2semidirect
10. #20274557C2semidirect
11. C53×#2906C2semidirect
14. C4×#11719925C4semidirect
15. C4×#11719925C4semidirect
16. C4×#11719924C4semidirect
17. C4×#11719924C4semidirect
18. C4×#11719923C4semidirect
19. C4×#11719923C4semidirect
20. C4×#11719922C4semidirect
21. C4×#11719922C4semidirect
276. #2906D53semidirect
329. C53#22782semidirect

Maximal subgroups

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

subgroupindex
C4×#131332372
#202833802
#202754252
#202753112
#202750422
#202748522
#202747352
#202747322
#202745852
#202745572
C53×#29062
#2278253