Finite Group Databasebeta

← #27276844

Subgroups — #27276844

Order 13568.

742 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277221C211yesmaximal, characteristic
2#20274693C211yesmaximal, characteristic, complemented
3C2×#16810358C211yesmaximal, characteristic
4C2×#16810356C211yesmaximal, characteristic
5#20272943C211yesmaximal, characteristic, complemented
6#20272593C211yesmaximal, characteristic, complemented
7C53×#1294C211yesmaximal, characteristic, complemented
8C2×#13133191C2211yescharacteristic
9C2×#13133179C411yescharacteristic, complemented
10C2×#13133177—21no—
11C22×#11719919—21no—
12C22×C4×Dic53C2211yescharacteristic
13C2×#13133146—21no—
14C2×C22⋊#88322C2211yescharacteristic
15#16810524C411yescharacteristic, complemented
16#16810519C411yescharacteristic, complemented
17#16810512—21no—
18#16810500—21no—
19C4⋊C4×Dic53C411yescharacteristic, complemented
20#16810470—21no—
21C2×#13133124—21no—
22C2×#13133124C2211yescharacteristic
23#16810377—21no—
24#16810358—22no—
25#16810356—22no—
26C2×C22⋊C8×C53C2211yescharacteristic
27C53×#380C2211yescharacteristic
28C2×#95×C53C2211yescharacteristic
29C22×D212C2×C411yescharacteristic
30C22×C22⋊D53—21no—
31C23×Dic53—21no—
32C23×Dic53D411yescharacteristic
33C23×Dic53D411yescharacteristic
34C22×C4×D53—21no—
35C22×#88322—21no—
36#13133191—21no—
37#13133191—21no—
38#13133191D412yes—
39#13133191Q812yes—
40#13133179—42no—
41#13133177—42no—
42#13133177—42no—
43C2×C4⋊Dic53C2×C411yescharacteristic
44C2×#11719919—24no—
45C2×#11719919—21no—
46C2×#11719919—42no—
47C2×#11719919—42no—
48C2×#11719921—21no—
49C2×#11719921—21no—
50C2×#11719921—21no—
51C2×#11719921—21no—
52C2×C4×Dic53—21no—
53C2×C4×Dic53—42no—
54C2×C4×Dic53C2×C411yescharacteristic
55C2×C4×Dic53C2×C411yescharacteristic
56C2×C4×Dic53D411yescharacteristic
57C2×C4×Dic53D411yescharacteristic
58#13133146—42no—
59C22⋊#88322—21no—
60C22⋊#88322—21no—
61#13133124—21no—
62#13133124—21no—
63#13133124—21no—
64#13133124—21no—
65#13133124—42no—
66C23×C4×C53—21no—
67C23×C4×C53C2311yescharacteristic
68C22×C8×C53—21no—
69C2×C4⋊C4×C53C2×C411yescharacteristic
70C2×C4⋊C4×C53C2×C411yescharacteristic
71C2×C22⋊C4×C53—21no—
72C22⋊C8×C53—21no—
73C22⋊C8×C53—21no—
74#95×C53—21no—
75#95×C53—21no—
76#95×C53—21no—
77C24×C53C2×D411yescharacteristic
78C23×D53—21no—
79C22×C4×C53—22no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—41no—
83C22×C4×C53—41no—
84C22×C4×C53—41no—
85C22×C4×C53—41no—
86C22×C4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53—41no—
89C22×C4×C53—41no—
90C22×C4×C53C22×C411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×Q811yescharacteristic
94C22×C4×C53#4111yescharacteristic
95C22×C4×C53#4111yescharacteristic
96C2×C22⋊D53—22no—
97C2×C22⋊D53—22no—
98C2×C22⋊D53—41no—
99C2×C22⋊D53—41no—
100C2×C22⋊D53—41no—
101C2×C22⋊D53—41no—
102C22×Dic53—21no—
103C22×Dic53—21no—
104C22×Dic53—22no—
105C22×Dic53—22no—
106C22×Dic53—21no—
107C22×Dic53—22no—
108C22×Dic53—41no—
109C22×Dic53—41no—
110C22×Dic53—41no—
111C22×Dic53—41no—
112C22×Dic53—41no—
113C22×Dic53—41no—
114C22×Dic53—41no—
115C22×Dic53—41no—
116C22×Dic53C22⋊C411yescharacteristic
117C22×Dic53C22⋊C411yescharacteristic
118C22×Dic53C22⋊C411yescharacteristic
119C22×Dic53C22⋊C411yescharacteristic
120C22×Dic53D812yes—
121C22×Dic53SD1612yes—
122C2×D212—21no—
123C2×D212—41no—
124C2×D212—41no—
125C2×D212C4⋊C412yescomplemented
126C2×C4×D53—42no—
127C2×C4×D53—42no—
128C2×C4×D53—42no—
129C2×C8×C53—41no—
130C2×C8×C53—41no—
131C2×C8×C53—41no—
132C4⋊C4×C53—41no—
133C4⋊C4×C53—41no—
134C22⋊C4×C53—41no—
135C22⋊C4×C53—41no—
136#11719921—22no—
137#11719921—22no—
138#11719921—42no—
139#11719921—42no—
140#11719921—42no—
141C4⋊Dic53—21no—
142C4⋊Dic53C4⋊C412yescomplemented
143#11719919—48no—
144#11719919—42no—
145C4×Dic53—22no—
146C4×Dic53—22no—
147C4×Dic53—41no—
148C4×Dic53—41no—
149C4×Dic53—42no—
150C2×#88322—41no—
151C2×#88322—41no—
152C2×#88322—41no—
153C23×C53—21no—
154C23×C53—21no—
155C23×C53—21no—
156C23×C53—21no—
157C23×C53C2×C22⋊C411yescharacteristic
158C23×C53#12111yescharacteristic
159C23×C53C22⋊Q811yescharacteristic
160C23×C53C2×D811yescharacteristic
161C23×C53C2×SD1611yescharacteristic
162C23×C53AGL(1,ℤ/8)11yescharacteristic
163C23×C53#13711yescharacteristic
164C22×D53—22no—
165C22×D53—41no—
166C22×D53—41no—
167C22×D53—41no—
168C22×D53—41no—
169C22×D53—41no—
170C22×D53—41no—
171C2×C4×C53—21no—
172C2×C4×C53—21no—
173C2×C4×C53—21no—
174C2×C4×C53—21no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—42no—
180C2×C4×C53—42no—
181C2×C4×C53—42no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53—41no—
185C2×C4×C53—41no—
186C2×C4×C53—41no—
187C2×C4×C53—41no—
188C2×C4×C53—81no—
189C2×C4×C53—81no—
190C2×C4×C53—81no—
191C2×C4×C53—81no—
192C2×C4×C53C2×C4⋊C411yescharacteristic
193C2×C4×C53#11711yescharacteristic
194C2×C4×C53#12111yescharacteristic
195C2×C4×C53C22⋊Q811yescharacteristic
196C22⋊D53—42no—
197C22⋊D53—42no—
198C22⋊D53—42no—
199C22⋊D53—42no—
200C2×Dic53—24no—
201C2×Dic53—24no—
202C2×Dic53—21no—
203C2×Dic53—21no—
204C2×Dic53—21no—
205C2×Dic53—21no—
206C2×Dic53—21no—
207C2×Dic53—21no—
208C2×Dic53—21no—
209C2×Dic53—21no—
210C2×Dic53—22no—
211C2×Dic53—42no—
212C2×Dic53—42no—
213C2×Dic53—41no—
214C2×Dic53—41no—
215C2×Dic53—42no—
216C2×Dic53—42no—
217C2×Dic53—42no—
218C2×Dic53—81no—
219C2×Dic53—81no—
220C2×Dic53—81no—
221C2×Dic53—81no—
222C2×Dic53#9912yes—
223C2×Dic53#10212yes—
224C2×Dic53#10212yes—
225C2×Dic53#10412yes—
226D212—42no—
227D212—42no—
228C4×D53—84no—
229C8×C53—81no—
230#88322—81no—
231#8577—531nomaximal, Sylow
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—21no—
235C22×C53—21no—
236C22×C53—21no—
237C22×C53—21no—
238C22×C53—21no—
239C22×C53—21no—
240C22×C53—21no—
241C22×C53—21no—
242C22×C53—21no—
243C22×C53—21no—
244C22×C53#38011yescharacteristic
245C22×C53C2×#9911yescharacteristic
246C22×C53#41011yescharacteristic
247C22×C53C2×#10211yescharacteristic
248C22×C53#41711yescharacteristic
249C22×C53C2×#10411yescharacteristic
250C22×C53#42111yescharacteristic
251C22×C53#45911yescharacteristic
252C22×C53#46111yescharacteristic
253C22×C53#47411yescharacteristic
254C22×C53#47611yescharacteristic
255C2×D53—42no—
256C2×D53—42no—
257C2×D53—42no—
258C2×D53—81no—
259C2×D53—81no—
260C2×D53—81no—
261C2×D53—81no—
262C4×C53—21no—
263C4×C53—21no—
264C4×C53—21no—
265C4×C53—21no—
266C4×C53—82no—
267C4×C53—81no—
268C4×C53—81no—
269Dic53—22no—
270Dic53—22no—
271Dic53—22no—
272Dic53—22no—
273Dic53—42no—
274Dic53—82no—
275C4×#121—531no—
276#1797—531no—
277C2×#327—531no—
278C2×#327—531no—
279C4⋊#95—531no—
280#1294—531no—
281#1294D5311yescharacteristic, complemented
282C2×C53—21no—
283C2×C53—21no—
284C2×C53—21no—
285C2×C53—21no—
286C2×C53#150211yescharacteristic
287C2×C53#150411yescharacteristic
288C2×C53#151211yescharacteristic
289C2×C53#151911yescharacteristic
290C2×C53#179911yescharacteristic
291C2×C53#180211yescharacteristic
292C2×C53#180611yescharacteristic
293D53—82no—
294C2×#121—531no—
295C2×C4×D4—531no—
296C2×C4×D4—1061no—
297C22×C4⋊C4—1061no—
298C22×C42—531no—
299C2×C4⋊C8—1061no—
300C2×C22⋊C8—531no—
301C2×C22⋊C8C2×D5311yescharacteristic
302#390—531no—
303#389—531no—
304#388—1061no—
305#386—1061no—
306#380C2×D5311yescharacteristic
307C4×C4⋊C4—531no—
308C4×C22⋊C4—1061no—
309C2×#95—531no—
310C2×#95—1061no—
311C2×#95C2×D5311yescharacteristic
312#336—1061no—
313#327—1062no—
314#327—1062no—
315C53#857711yescharacteristic, complemented, Sylow
316C22×D4—531no—
317C22×D4—1061no—
318C23×C4—21no—
319C23×C4—531no—
320C23×C4—531no—
321C23×C4—1061no—
322C23×C4—1061no—
323C23×C4C22×D5311yescharacteristic
324C22×C8—21no—
325C22×C8—1061no—
326#121—532no—
327#121—532no—
328#121—1061no—
329#121—1061no—
330C4×D4—2122no—
331C4×D4—2122no—
332C4×D4—2122no—
333C2×C4⋊C4—531no—
334C2×C4⋊C4—1061no—
335C2×C4⋊C4—1064no—
336C2×C4⋊C4—2122no—
337C2×C4⋊C4—2122no—
338C2×C4⋊C4C4×D5311yescharacteristic
339C2×C4⋊C4C4×D5311yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—1061no—
342C2×C22⋊C4—1061no—
343C2×C22⋊C4—1061no—
344C2×C22⋊C4—1061no—
345C2×C42—531no—
346C2×C42—531no—
347C2×C42—531no—
348C2×C42—531no—
349C2×C42—1061no—
350C2×C42—2122no—
351C4⋊C8—2122no—
352C22⋊C8—21no—
353C22⋊C8—21no—
354C22⋊C8—1061no—
355C22⋊C8—1061no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—1061no—
360#95—1061no—
361#95—1061no—
362#95—1061no—
363#95—2122no—
364C24—1061no—
365C24D4×D5311yescharacteristic
366C2×D4—532no—
367C2×D4—1062no—
368C2×D4—1062no—
369C2×D4—1061no—
370C2×D4—2121no—
371C2×D4—2121no—
372C2×D4—2121no—
373C2×D4—2121no—
374C2×D4—2121no—
375C2×D4—2121no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—531no—
388C22×C4—531no—
389C22×C4—531no—
390C22×C4—531no—
391C22×C4—532no—
392C22×C4—532no—
393C22×C4—1062no—
394C22×C4—1062no—
395C22×C4—1061no—
396C22×C4—1061no—
397C22×C4—1061no—
398C22×C4—1062no—
399C22×C4—2122no—
400C22×C4—2122no—
401C22×C4—2122no—
402C22×C4—2121no—
403C22×C4—2121no—
404C22×C4—2121no—
405C22×C4—2121no—
406C22×C4—2121no—
407C22×C4—2121no—
408C22×C4—2121no—
409C22×C4—2121no—
410C22×C4C2×C4×D5311yescharacteristic
411C22×C4#1171994511yescharacteristic
412C22×C4#1171994511yescharacteristic
413C22×C4D4×D5311yescharacteristic
414C22×C4D4×D5311yescharacteristic
415C22×C4Q8×D5311yescharacteristic
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—2121no—
420C2×C8—2121no—
421C2×C8—2121no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—532no—
425C4⋊C4—1061no—
426C4⋊C4—2122no—
427C4⋊C4—2128no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—1062no—
431C22⋊C4—1062no—
432C22⋊C4—2122no—
433C22⋊C4—2122no—
434C22⋊C4—2122no—
435C42—1062no—
436C42—1062no—
437C42—2121no—
438C42—2121no—
439C42—2122no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—1062no—
445C23—2121no—
446C23—2121no—
447C23—2121no—
448C23—2121no—
449C23—2121no—
450C23—2121no—
451C23C22⋊C4×D5311yescharacteristic
452C23#1313319011yescharacteristic
453C23#1313319511yescharacteristic
454C23D8×D5311yescharacteristic
455C23SD16×D5311yescharacteristic
456C23#1313324411yescharacteristic
457C23#1313324911yescharacteristic
458D4—2122no—
459D4—2122no—
460D4—2122no—
461D4—2122no—
462D4—2122no—
463D4—2122no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—532no—
486C2×C4—532no—
487C2×C4—532no—
488C2×C4—532no—
489C2×C4—1064no—
490C2×C4—1064no—
491C2×C4—1061no—
492C2×C4—1061no—
493C2×C4—1061no—
494C2×C4—1061no—
495C2×C4—1061no—
496C2×C4—1061no—
497C2×C4—1061no—
498C2×C4—1061no—
499C2×C4—1062no—
500C2×C4—4244no—
501C2×C4—4241no—
502C2×C4—4241no—
503C2×C4—4241no—
504C2×C4—4241no—
505C2×C4—2122no—
506C2×C4—2122no—
507C2×C4—2122no—
508C2×C4—2122no—
509C2×C4—2122no—
510C2×C4—2121no—
511C2×C4—2121no—
512C2×C4C4⋊C4×D5311yescharacteristic
513C2×C4#1313317411yescharacteristic
514C2×C4#1313319011yescharacteristic
515C2×C4#1313319511yescharacteristic
516C8—81no—
517C8—4241no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—4241no—
531C22—4241no—
532C22—4241no—
533C22—4241no—
534C22—2122no—
535C22—2122no—
536C22—2122no—
537C22#1681048211yescharacteristic
538C22#99×D5311yescharacteristic
539C22#1681059511yescharacteristic
540C22#102×D5311yescharacteristic
541C22#1681062311yescharacteristic
542C22#104×D5311yescharacteristic
543C22#1681063811yescharacteristic
544C22#1681081711yescharacteristic
545C22#1681082711yescharacteristic
546C22#1681090211yescharacteristic
547C22#1681091211yescharacteristic
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—82no—
553C4—81no—
554C4—81no—
555C4—1062no—
556C4—1062no—
557C4—1062no—
558C4—1062no—
559C4—4242no—
560C4—2122no—
561C2—21no—
562C2—21no—
563C2—21no—
564C2—21no—
565C2—4242no—
566C2#2027320811yescharacteristic
567C2#2027321811yescharacteristic
568C2#2027325211yescharacteristic
569C2#2027328111yescharacteristic
570C2#2027470211yescharacteristic
571C2#2027471411yescharacteristic
572C2#2027473411yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1294C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133191C22
cosocleC23×C4×C53C23
hypercenter#1294D53
hypocenterC53#8577
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
2. #20274693C2semidirect
5. #20272943C2semidirect
6. #20272593C2semidirect
7. C53×#1294C2semidirect
9. C2×#13133179C4semidirect
15. #16810524C4semidirect
16. #16810519C4semidirect
19. C4⋊C4×Dic53C4semidirect
125. C2×D212C4⋊C4semidirect
142. C4⋊Dic53C4⋊C4semidirect
281. #1294D53semidirect
315. C53#8577semidirect

Maximal subgroups

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

subgroupindex
#202772212
#202746932
C2×#168103582
C2×#168103562
#202729432
#202725932
C53×#12942
#857753