Finite Group Databasebeta

← #27466096

Subgroups — #27466096

Order 13568.

970 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133188C211yesmaximal, characteristic
2#384×D53C211yesmaximal, characteristic, complemented
3#20276275C212yesmaximal, complemented
4#20276241C212yesmaximal, complemented
5#20275474C211yesmaximal, characteristic, complemented
6#20275471C211yesmaximal, characteristic, complemented
7#20275029C212yesmaximal
8#20274904C212yesmaximal, complemented
9#20274715C211yesmaximal, characteristic, complemented
10#20274692C211yesmaximal, characteristic, complemented
11C53×#1802C211yesmaximal, characteristic, complemented
12C2×#13133188C2211yescharacteristic
13C2×C4×C22⋊D53—21no—
14C2×C4⋊C4×D53—21no—
15C2×C4⋊C4×D53—21no—
16C2×C4⋊C4×D53C2212yes—
17C2×C42×D53C2211yescharacteristic
18C2×#13133151C2212yes—
19C2×#13133150C2212yes—
20C2×C4⋊#88322C2211yescharacteristic
21C2×#13133138C2212yes—
22C2×#13133129C2212yes—
23#16810526C2211yescharacteristic
24#16810519C2211yescharacteristic
25#16810511—21no—
26#16810499—21no—
27#16810493C2211yescharacteristic
28#16810492—21no—
29#16810492—21no—
30#16810492C2212yes—
31#16810480C2211yescharacteristic
32C4×C4⋊Dic53C2211yescharacteristic
33C4×C22⋊Dic53—21no—
34#16810386C2212yes—
35#16810385C2212yes—
36#16810385C2212yes—
37#16810377C2212yes—
38C2×C4×D4×C53C2211yescharacteristic
39C2×C4⋊C8×C53C2211yescharacteristic
40C2×#102×C53C2212yes—
41C53×#384C2211yescharacteristic
42C53×#340C2212yes—
43C22×C22⋊D53—21no—
44C22×C4×D53—22no—
45C22×C4×D53D411yescharacteristic
46C22×C4×D53D411yescharacteristic
47C22×C4×D53C2311yescharacteristic
48C22×#88322C2312yes—
49#13133188—22no—
50#13133188Q814yes—
51C4×C22⋊D53—44no—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53C2311yescharacteristic
55C2×C4⋊Dic53C2312yes—
56C2×C4⋊Dic53C2311yescharacteristic
57C2×#11719919—21no—
58C2×#11719919—22no—
59C2×#11719919—22no—
60C2×#11719919—22no—
61C4⋊C4×D53—42no—
62C4⋊C4×D53—42no—
63C4⋊C4×D53—44no—
64C4⋊C4×D53—44no—
65C2×C22⋊Dic53—21no—
66C2×C22⋊Dic53—21no—
67C2×C22⋊Dic53—21no—
68C2×#11719921—21no—
69C2×C4×Dic53—21no—
70C2×C4×Dic53D411yescharacteristic
71C2×C4×Dic53D411yescharacteristic
72C2×C4×Dic53C2311yescharacteristic
73C42×D53—22no—
74C42×D53—22no—
75#13133151—42no—
76#13133150—42no—
77C4⋊#88322—22no—
78#13133138—42no—
79#13133129—24no—
80#13133124—21no—
81#13133124—21no—
82#13133124—21no—
83C22×D4×C53C2311yescharacteristic
84C23×C4×C53—21no—
85C22×C8×C53C2312yes—
86C4×D4×C53—22no—
87C4×D4×C53—22no—
88C2×C4⋊C4×C53—21no—
89C2×C4⋊C4×C53—21no—
90C2×C4⋊C4×C53C2311yescharacteristic
91C2×C4⋊C4×C53C2312yes—
92C2×C22⋊C4×C53—21no—
93C2×C42×C53C2311yescharacteristic
94C4⋊C8×C53—22no—
95#102×C53—42no—
96C24×C53—21no—
97C23×D53C2×D411yescharacteristic
98C2×D4×C53—21no—
99C2×D4×C53—21no—
100C2×D4×C53—21no—
101C2×D4×C53—21no—
102C2×D4×C53—21no—
103C2×D4×C53C2×Q812yes—
104C22×C4×C53—21no—
105C22×C4×C53—22no—
106C22×C4×C53—42no—
107C22×C4×C53—42no—
108C22×C4×C53—42no—
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53C2411yescharacteristic
112C2×C22⋊D53—24no—
113C2×C22⋊D53—42no—
114C2×C22⋊D53—42no—
115C22×Dic53—21no—
116C22×Dic53—21no—
117C22×Dic53—22no—
118C22×Dic53C2×D411yescharacteristic
119C22×Dic53C2×D411yescharacteristic
120C22×Dic53C2×D411yescharacteristic
121C2×C4×D53—22no—
122C2×C4×D53—22no—
123C2×C4×D53—22no—
124C2×C4×D53—22no—
125C2×C4×D53—21no—
126C2×C4×D53—21no—
127C2×C4×D53—21no—
128C2×C4×D53—21no—
129C2×C4×D53—22no—
130C2×C4×D53—44no—
131C2×C4×D53—44no—
132C2×C4×D53—44no—
133C2×C4×D53D814yes—
134C2×C4×D53C2×Q812yes—
135C2×C4×D53#4112yes—
136C2×C8×C53—22no—
137C2×C8×C53—22no—
138C2×C8×C53—22no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—22no—
142C4⋊C4×C53—22no—
143C4⋊C4×C53—42no—
144C4⋊C4×C53—42no—
145C22⋊C4×C53—42no—
146C42×C53#4112yes—
147C42×C53#4112yes—
148C22⋊Dic53—24no—
149C22⋊Dic53—42no—
150C22⋊Dic53—42no—
151#11719921—42no—
152C4⋊Dic53—21no—
153C4⋊Dic53—22no—
154C4⋊Dic53—21no—
155C4⋊Dic53—21no—
156C4⋊Dic53—22no—
157C4⋊Dic53—42no—
158C4⋊Dic53—42no—
159C4⋊Dic53C2×Q812yes—
160#11719919—44no—
161#11719919—44no—
162#11719919—42no—
163#11719919—44no—
164C4×Dic53—22no—
165C4×Dic53—22no—
166C4×Dic53—21no—
167C4×Dic53—21no—
168C4×Dic53—42no—
169C2×#88322—22no—
170C2×#88322—22no—
171C2×#88322—22no—
172C23×C53—22no—
173C23×C53—41no—
174C23×C53—41no—
175C23×C53—41no—
176C23×C53—41no—
177C23×C53—41no—
178C23×C53—41no—
179C23×C53C22×D411yescharacteristic
180C22×D53—21no—
181C22×D53—21no—
182C22×D53—21no—
183C22×D53—21no—
184C22×D53C22⋊Q812yes—
185C22×D53C2×D812yes—
186C22×D53#13712yes—
187D4×C53—22no—
188D4×C53—22no—
189D4×C53—41no—
190D4×C53—41no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—41no—
194C2×C4×C53—42no—
195C2×C4×C53—42no—
196C2×C4×C53—41no—
197C2×C4×C53—42no—
198C2×C4×C53—41no—
199C2×C4×C53—84no—
200C2×C4×C53#12312yes—
201C2×C4×C53#12312yes—
202C2×C4×C53C2×D812yes—
203C2×C4×C53#13712yes—
204C2×C4×C53C22×Q811yescharacteristic
205C2×C4×C53C2×#4111yescharacteristic
206C2×C4×C53C2×#4111yescharacteristic
207C2×C4×C53C2×#4111yescharacteristic
208C2×C4×C53H(5,𝔽2)11yescharacteristic
209C2×C4×C53H(5,𝔽2)11yescharacteristic
210C22⋊D53—48no—
211C2×Dic53—21no—
212C2×Dic53—21no—
213C2×Dic53—21no—
214C2×Dic53—21no—
215C2×Dic53—21no—
216C2×Dic53—21no—
217C2×Dic53—22no—
218C2×Dic53—41no—
219C2×Dic53—41no—
220C2×Dic53—42no—
221C2×Dic53—41no—
222C2×Dic53—42no—
223C2×Dic53—41no—
224C2×Dic53—42no—
225C2×Dic53—41no—
226C2×Dic53C22⋊Q812yes—
227C2×Dic53C2×D812yes—
228C2×Dic53AGL(1,ℤ/8)12yes—
229C4×D53—28no—
230C4×D53—22no—
231C4×D53—22no—
232C4×D53—22no—
233C4×D53—22no—
234C4×D53—28no—
235C4×D53—88no—
236C8×C53—42no—
237#88322—42no—
238#27677—531nomaximal, Sylow
239C22×C53—42no—
240C22×C53—42no—
241C22×C53—42no—
242C22×C53—81no—
243C22×C53—81no—
244C22×C53—81no—
245C22×C53—81no—
246C22×C53C2×C22⋊Q811yescharacteristic
247C22×C53C2×#12311yescharacteristic
248C22×C53#53511yescharacteristic
249C22×C53C22×D811yescharacteristic
250C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
251C22×C53C2×#13711yescharacteristic
252C22×C53C2×#13711yescharacteristic
253C2×D53—22no—
254C2×D53—22no—
255C2×D53—22no—
256C2×D53—22no—
257C2×D53—22no—
258C2×D53—22no—
259C2×D53#47414yes—
260C4×C53—22no—
261C4×C53—81no—
262C4×C53—82no—
263C4×C53#48014yes—
264C4×C53#55311yescharacteristic
265C4×C53#55311yescharacteristic
266C4×C53#55411yescharacteristic
267C4×C53#55411yescharacteristic
268Dic53—22no—
269Dic53—22no—
270Dic53—22no—
271Dic53—42no—
272Dic53—82no—
273Dic53—81no—
274C4×#121—531no—
275C2×#384—531no—
276#2099—532no—
277#2093—532no—
278#1947—531no—
279#1947—531no—
280#1874—532no—
281#1852—532no—
282#1802—531no—
283#1802D5311yescharacteristic, complemented
284#1797—531no—
285C2×C53—82no—
286C2×C53#248711yescharacteristic
287C2×C53C2×#47411yescharacteristic
288C2×C53#307911yescharacteristic
289C2×C53C2×#48011yescharacteristic
290C2×C53#309211yescharacteristic
291C2×C53#320511yescharacteristic
292C2×C53#321811yescharacteristic
293D53—24no—
294C2×#121—531no—
295C2×C4×D4—1061no—
296C2×C4×D4C2×D5311yescharacteristic
297C22×C4⋊C4—532no—
298C22×C4⋊C4—1061no—
299C22×C4⋊C4—1061no—
300C22×C42—531no—
301C2×#107—532no—
302C2×#107—532no—
303C2×C4⋊C8—531no—
304C2×C4⋊C8C2×D5311yescharacteristic
305C2×#102—532no—
306C2×#102C2×D5312yes—
307C2×C22⋊C8—532no—
308#390—531no—
309#389—531no—
310#388—1061no—
311#386—1061no—
312#384—532no—
313#384—531no—
314#384—1061no—
315#384—1061no—
316#384C2×D5311yescharacteristic
317#380—531no—
318C4×C4⋊C4—531no—
319C4×C22⋊C4—1061no—
320#340—532no—
321#340—532no—
322#340—532no—
323#340C2×D5312yes—
324#336—532no—
325C53#2767711yescharacteristic, complemented, Sylow
326C22×D4—1061no—
327C22×D4C22×D5311yescharacteristic
328C23×C4—21no—
329C23×C4—531no—
330C23×C4—531no—
331C23×C4—531no—
332C23×C4—1062no—
333C22×C8—532no—
334C22×C8C22×D5312yes—
335#121—534no—
336#121—1062no—
337C4×D4—22no—
338C4×D4—22no—
339C4×D4—2124no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—531no—
343C2×C4⋊C4—532no—
344C2×C4⋊C4—531no—
345C2×C4⋊C4—1062no—
346C2×C4⋊C4—1062no—
347C2×C4⋊C4—1062no—
348C2×C4⋊C4—1061no—
349C2×C4⋊C4—1061no—
350C2×C4⋊C4—1061no—
351C2×C4⋊C4—2122no—
352C2×C4⋊C4—2122no—
353C2×C4⋊C4—2124no—
354C2×C4⋊C4—2124no—
355C2×C4⋊C4C22×D5312yes—
356C2×C4⋊C4C22×D5311yescharacteristic
357C2×C22⋊C4—21no—
358C2×C22⋊C4—1061no—
359C2×C22⋊C4—1061no—
360C2×C22⋊C4—1061no—
361C2×C22⋊C4—1061no—
362C2×C42—531no—
363C2×C42—531no—
364C2×C42—531no—
365C2×C42—1062no—
366C2×C42—1062no—
367C2×C42—1061no—
368C2×C42C22×D5311yescharacteristic
369#107—2122no—
370#107—2122no—
371C4⋊C8—22no—
372C4⋊C8—1062no—
373#102—42no—
374#102—2122no—
375C22⋊C8—1064no—
376#95—1061no—
377#95—1061no—
378#95—1061no—
379C24—21no—
380C24—531no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—1064no—
387C2×D4—2122no—
388C2×D4—2122no—
389C2×D4Q8×D5312yes—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—531no—
396C22×C4—531no—
397C22×C4—531no—
398C22×C4—534no—
399C22×C4—532no—
400C22×C4—532no—
401C22×C4—1062no—
402C22×C4—1062no—
403C22×C4—1062no—
404C22×C4—1062no—
405C22×C4—1062no—
406C22×C4—1061no—
407C22×C4—1061no—
408C22×C4—1061no—
409C22×C4—1061no—
410C22×C4—1061no—
411C22×C4—1061no—
412C22×C4—1062no—
413C22×C4—2124no—
414C22×C4—2124no—
415C22×C4—2124no—
416C22×C4D4×D5311yescharacteristic
417C22×C4D4×D5311yescharacteristic
418C22×C4C23×D5311yescharacteristic
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—1062no—
423C2×C8—1062no—
424C2×C8—1062no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—532no—
432C4⋊C4—1061no—
433C4⋊C4—1061no—
434C4⋊C4—1062no—
435C4⋊C4—1062no—
436C4⋊C4—1061no—
437C4⋊C4—2122no—
438C4⋊C4—2124no—
439C4⋊C4—2124no—
440C4⋊C4—2122no—
441C4⋊C4—2124no—
442C4⋊C4—2122no—
443C22⋊C4—42no—
444C22⋊C4—1064no—
445C22⋊C4—2122no—
446C22⋊C4—2122no—
447C22⋊C4—2122no—
448C42—1062no—
449C42—1062no—
450C42—1061no—
451C42—1061no—
452C42—2122no—
453C42#1171994512yes—
454C42#1171994712yes—
455C23—22no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—532no—
463C23—532no—
464C23—532no—
465C23—1061no—
466C23—1061no—
467C23—1061no—
468C23—1061no—
469C23C2×D4×D5311yescharacteristic
470D4—22no—
471D4—22no—
472D4—41no—
473D4—41no—
474D4—2128no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—84no—
484C2×C4—532no—
485C2×C4—532no—
486C2×C4—532no—
487C2×C4—1068no—
488C2×C4—1061no—
489C2×C4—1061no—
490C2×C4—1061no—
491C2×C4—1061no—
492C2×C4—1061no—
493C2×C4—1061no—
494C2×C4—1062no—
495C2×C4—1062no—
496C2×C4—1062no—
497C2×C4—1062no—
498C2×C4—1068no—
499C2×C4—1062no—
500C2×C4—4248no—
501C2×C4—2121no—
502C2×C4—2122no—
503C2×C4—2121no—
504C2×C4—2122no—
505C2×C4—2122no—
506C2×C4—2121no—
507C2×C4—2121no—
508C2×C4—2121no—
509C2×C4#1313319912yes—
510C2×C4#1313319912yes—
511C2×C4D8×D5312yes—
512C2×C4#1313324612yes—
513C2×C4C2×Q8×D5311yescharacteristic
514C2×C4#41×D5311yescharacteristic
515C2×C4C2×#1171994711yescharacteristic
516C2×C4C2×#1171994511yescharacteristic
517C2×C4#1313326211yescharacteristic
518C2×C4#1313326311yescharacteristic
519C8—42no—
520C8—2122no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22—534no—
529C22—1062no—
530C22—1062no—
531C22—1062no—
532C22—1062no—
533C22—1062no—
534C22—1062no—
535C22C22⋊Q8×D5311yescharacteristic
536C22C2×#1313319911yescharacteristic
537C22#1681118911yescharacteristic
538C22C2×D8×D5311yescharacteristic
539C22AGL(1,ℤ/8)×D5311yescharacteristic
540C22#137×D5311yescharacteristic
541C22C2×#1313324611yescharacteristic
542C4—22no—
543C4—81no—
544C4—82no—
545C4—1062no—
546C4—1062no—
547C4—1062no—
548C4—4242no—
549C4—4241no—
550C4—2122no—
551C4#1681093514yes—
552C4#1681131711yescharacteristic
553C4#1681132011yescharacteristic
554C4#1681132811yescharacteristic
555C4#1681133111yescharacteristic
556C2—82no—
557C2—1064no—
558C2#2027902811yescharacteristic
559C2#474×D5311yescharacteristic
560C2#2028452711yescharacteristic
561C2C2×#1681093511yescharacteristic
562C2#2028463611yescharacteristic
563C2#2028563211yescharacteristic
564C2#2028574111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1802C2
socleC23×C53C22×D4
prime radicalC2×#13133188C22
hypercenter#1802D53
hypocenterC53#27677
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. #384×D53C2semidirect
3. #20276275C2semidirect
4. #20276241C2semidirect
5. #20275474C2semidirect
6. #20275471C2semidirect
8. #20274904C2semidirect
9. #20274715C2semidirect
10. #20274692C2semidirect
11. C53×#1802C2semidirect
283. #1802D53semidirect
325. C53#27677semidirect

Maximal subgroups

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

subgroupindex
C4×#131331882
#384×D532
#202762752
#202762412
#202754742
#202754712
#202750292
#202749042
#202747152
#202746922
C53×#18022
#2767753