Finite Group Databasebeta

← #27279658

Subgroups — #27279658

Order 13568.

1062 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊#88322C211yesmaximal, characteristic, complemented
2C2×#16810357C212yesmaximal, complemented
3#20272938C211yesmaximal, characteristic, complemented
4#20272593C212yesmaximal, complemented
5C53×#2847C211yesmaximal, characteristic
6C23×#88322—21no—
7C22×C22⋊Dic53—21no—
8C2×C22⋊#88322—24no—
9C2×C22⋊#88322—21no—
10C2×C22⋊#88322C2212yes—
11C2×#13133124—21no—
12C2×#13133124C2212yes—
13#16810357—24no—
14C24×C4×C53C2211yescharacteristic
15C2×#121×C53C411yescharacteristic, complemented
16C2×#121×C53C411yescharacteristic, complemented
17C2×#121×C53C2211yescharacteristic
18C2×#120×C53—21no—
19C53×#386—21no—
20C53×#386—21no—
21C53×#386—21no—
22C53×#380C411yescharacteristic, complemented
23C53×#380C411yescharacteristic, complemented
24C53×#380C2211yescharacteristic
25C23×Dic53—22no—
26C22×#88322—24no—
27C22×#88322—22no—
28C22×#88322—42no—
29C22×#88322—42no—
30C2×C22⋊Dic53—24no—
31C2×C22⋊Dic53—42no—
32C2×C22⋊Dic53—42no—
33C22⋊#88322—24no—
34C22⋊#88322—22no—
35C22⋊#88322D414yescomplemented
36#13133124—22no—
37#13133124—22no—
38#13133124—42no—
39C25×C53D411yescharacteristic
40C22×D4×C53—21no—
41C22×D4×C53—21no—
42C22×D4×C53—21no—
43C22×D4×C53C2×C411yescharacteristic
44C23×C4×C53—22no—
45C23×C4×C53—22no—
46C23×C4×C53—42no—
47C23×C4×C53—42no—
48C23×C4×C53C2×C411yescharacteristic
49C23×C4×C53C2×C411yescharacteristic
50C23×C4×C53D411yescharacteristic
51C23×C4×C53D411yescharacteristic
52C23×C4×C53D411yescharacteristic
53C23×C4×C53C2311yescharacteristic
54#121×C53—22no—
55#121×C53—22no—
56#121×C53—42no—
57#121×C53—42no—
58#120×C53—42no—
59#120×C53—42no—
60C2×C4⋊C4×C53C2×C411yescharacteristic
61C2×C4⋊C4×C53C2×C411yescharacteristic
62C2×C4⋊C4×C53C2×C411yescharacteristic
63C2×C22⋊C4×C53—21no—
64C2×C22⋊C4×C53—21no—
65C2×C22⋊C4×C53—21no—
66C2×C22⋊C4×C53—21no—
67C2×C22⋊C4×C53—21no—
68C2×C22⋊C4×C53—21no—
69#95×C53—21no—
70#95×C53—21no—
71#95×C53—21no—
72C24×C53—22no—
73C24×C53—22no—
74C24×C53—22no—
75C24×C53—22no—
76C24×C53—21no—
77C24×C53—41no—
78C24×C53—41no—
79C24×C53C22⋊C411yescharacteristic
80C24×C53C22⋊C411yescharacteristic
81C24×C53D812yes—
82C24×C53SD1612yes—
83C24×C53C2×D411yescharacteristic
84C2×D4×C53—22no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C2×D4×C53—42no—
88C2×D4×C53—42no—
89C2×D4×C53—42no—
90C2×D4×C53—42no—
91C2×D4×C53—42no—
92C2×D4×C53—42no—
93C2×D4×C53—42no—
94C2×D4×C53—41no—
95C2×D4×C53—41no—
96C2×D4×C53—42no—
97C22×C4×C53—22no—
98C22×C4×C53—22no—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—22no—
102C22×C4×C53—22no—
103C22×C4×C53—22no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—21no—
108C22×C4×C53—21no—
109C22×C4×C53—21no—
110C22×C4×C53—21no—
111C22×C4×C53—21no—
112C22×C4×C53—41no—
113C22×C4×C53—41no—
114C22×C4×C53—41no—
115C22×C4×C53—41no—
116C22×C4×C53—41no—
117C22×C4×C53—41no—
118C22×C4×C53—41no—
119C22×C4×C53—48no—
120C22×C4×C53—48no—
121C22×C4×C53C22⋊C411yescharacteristic
122C22×C4×C53C22⋊C411yescharacteristic
123C22×C4×C53C22×C411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D412yes—
126C22×C4×C53#4112yes—
127C22×Dic53—24no—
128C22×Dic53—42no—
129C22×Dic53—42no—
130C22×Dic53—42no—
131C22×Dic53—42no—
132C22×Dic53—42no—
133C22×Dic53—42no—
134C4⋊C4×C53—22no—
135C4⋊C4×C53—41no—
136C4⋊C4×C53—41no—
137C22⋊C4×C53—42no—
138C22⋊C4×C53—42no—
139C22⋊C4×C53—42no—
140C22⋊C4×C53—41no—
141C22⋊C4×C53—41no—
142C22⋊C4×C53—42no—
143C22⋊C4×C53—41no—
144C22⋊C4×C53—41no—
145C22⋊Dic53—48no—
146C2×#88322—24no—
147C2×#88322—48no—
148C2×#88322—42no—
149C2×#88322—42no—
150C23×C53—21no—
151C23×C53—21no—
152C23×C53—21no—
153C23×C53—21no—
154C23×C53—21no—
155C23×C53—21no—
156C23×C53—21no—
157C23×C53—21no—
158C23×C53—24no—
159C23×C53—24no—
160C23×C53—21no—
161C23×C53—21no—
162C23×C53—21no—
163C23×C53—21no—
164C23×C53—21no—
165C23×C53—21no—
166C23×C53—24no—
167C23×C53—24no—
168C23×C53—42no—
169C23×C53—42no—
170C23×C53—44no—
171C23×C53—44no—
172C23×C53—44no—
173C23×C53—44no—
174C23×C53—41no—
175C23×C53—41no—
176C23×C53—41no—
177C23×C53—41no—
178C23×C53—41no—
179C23×C53—41no—
180C23×C53—41no—
181C23×C53#9912yes—
182C23×C53#10212yes—
183C23×C53#10212yes—
184C23×C53#10412yes—
185C23×C53C2×C22⋊C411yescharacteristic
186C23×C53#12112yes—
187C23×C53C2×D811yescharacteristic
188C23×C53C2×SD1611yescharacteristic
189C23×C53#13512yes—
190D4×C53—82no—
191D4×C53—84no—
192D4×C53—84no—
193D4×C53—82no—
194D4×C53—82no—
195C2×C4×C53—24no—
196C2×C4×C53—24no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—42no—
207C2×C4×C53—42no—
208C2×C4×C53—44no—
209C2×C4×C53—44no—
210C2×C4×C53—44no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53#9912yes—
223C2×C4×C53C4×D412yes—
224C2×C4×C53#12011yescharacteristic
225C2×C4×C53#12311yescharacteristic
226C2×Dic53—44no—
227C2×Dic53—44no—
228C2×Dic53—44no—
229C2×Dic53—82no—
230C2×Dic53—82no—
231C2×Dic53—82no—
232C2×Dic53—82no—
233#88322—44no—
234#9113—531nomaximal, Sylow
235C22×C53—22no—
236C22×C53—22no—
237C22×C53—22no—
238C22×C53—22no—
239C22×C53—22no—
240C22×C53—21no—
241C22×C53—21no—
242C22×C53—22no—
243C22×C53—22no—
244C22×C53—22no—
245C22×C53—22no—
246C22×C53—21no—
247C22×C53—22no—
248C22×C53—21no—
249C22×C53—22no—
250C22×C53—22no—
251C22×C53—22no—
252C22×C53—48no—
253C22×C53—48no—
254C22×C53—41no—
255C22×C53—41no—
256C22×C53—41no—
257C22×C53—41no—
258C22×C53—81no—
259C22×C53—81no—
260C22×C53—81no—
261C22×C53—81no—
262C22×C53—81no—
263C22×C53—81no—
264C22×C53—81no—
265C22×C53#32714yescomplemented
266C22×C53#38611yescharacteristic
267C22×C53C2×#9911yescharacteristic
268C22×C53C2×#9911yescharacteristic
269C22×C53C2×#10211yescharacteristic
270C22×C53#41611yescharacteristic
271C22×C53C2×#10411yescharacteristic
272C22×C53#42111yescharacteristic
273C22×C53#46512yes—
274C22×C53#46612yes—
275C4×C53—21no—
276C4×C53—22no—
277C4×C53—21no—
278C4×C53—22no—
279C4×C53—41no—
280C4×C53—81no—
281C4×C53—81no—
282C4×C53—81no—
283Dic53—84no—
284C22×C22⋊C8—531no—
285#2847D5311yescharacteristic
286C2×#327—532no—
287C22⋊#95—531no—
288#1294—532no—
289C2×C53—22no—
290C2×C53—24no—
291C2×C53—22no—
292C2×C53—42no—
293C2×C53—161no—
294C2×C53C2×#32711yescharacteristic
295C2×C53#150111yescharacteristic
296C2×C53#151611yescharacteristic
297C2×C53#151911yescharacteristic
298C2×C53#187111yescharacteristic
299C2×C53#189311yescharacteristic
300C2×C53#189811yescharacteristic
301C24×C4C2×D5311yescharacteristic
302C23×C8—1061no—
303C2×#121Dic5311yescharacteristic, complemented
304C2×#121Dic5311yescharacteristic, complemented
305C2×#121C2×D5311yescharacteristic
306C2×#120—21no—
307C22×C22⋊C4—1061no—
308C2×C22⋊C8—532no—
309C2×C22⋊C8—1064no—
310C2×C22⋊C8—1061no—
311#386—21no—
312#386—21no—
313#386—21no—
314#380Dic5311yescharacteristic, complemented
315#380Dic5311yescharacteristic, complemented
316#380C2×D5311yescharacteristic
317C2×#95—532no—
318C2×#95—1061no—
319#327—1064no—
320C53#911311yescharacteristic, complemented, Sylow
321C25C22⋊D5311yescharacteristic
322C22×D4—21no—
323C22×D4—21no—
324C22×D4—21no—
325C22×D4C2×Dic5311yescharacteristic
326C23×C4—22no—
327C23×C4—22no—
328C23×C4—42no—
329C23×C4—42no—
330C23×C4—1062no—
331C23×C4C2×Dic5311yescharacteristic
332C23×C4C2×Dic5311yescharacteristic
333C23×C4C22⋊D5311yescharacteristic
334C23×C4C22⋊D5311yescharacteristic
335C23×C4C22⋊D5311yescharacteristic
336C23×C4C22×D5311yescharacteristic
337C22×C8—1064no—
338C22×C8—1062no—
339C22×C8—2122no—
340C22×C8—2122no—
341#121—22no—
342#121—22no—
343#121—42no—
344#121—42no—
345#120—42no—
346#120—42no—
347C2×C4⋊C4C2×Dic5311yescharacteristic
348C2×C4⋊C4C2×Dic5311yescharacteristic
349C2×C4⋊C4C2×Dic5311yescharacteristic
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—1064no—
357C2×C22⋊C4—2122no—
358C2×C22⋊C4—2122no—
359C22⋊C8—534no—
360C22⋊C8—1064no—
361C22⋊C8—1062no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—1062no—
366#95—1062no—
367#95—2122no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—21no—
373C24—41no—
374C24—41no—
375C24#1171992212yes—
376C24#1171992312yes—
377C24C22⋊Dic5311yescharacteristic
378C24C22⋊Dic5311yescharacteristic
379C24C2×C22⋊D5311yescharacteristic
380C2×D4—22no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—42no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—1064no—
418C22×C4—2122no—
419C22×C4—2122no—
420C22×C4—2122no—
421C22×C4—2122no—
422C22×C4—2122no—
423C22×C4—2122no—
424C22×C4C22⋊Dic5311yescharacteristic
425C22×C4C22⋊Dic5311yescharacteristic
426C22×C4D4×D5312yes—
427C22×C4#1171994712yes—
428C22×C4C22×Dic5311yescharacteristic
429C22×C4C2×C22⋊D5311yescharacteristic
430C2×C8—1064no—
431C2×C8—2128no—
432C2×C8—2122no—
433C2×C8—2122no—
434C4⋊C4—22no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—42no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—2128no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23#1313313212yes—
478C23#1313313812yes—
479C23#1313313812yes—
480C23#1313314412yes—
481C23C2×C22⋊Dic5311yescharacteristic
482C23#1313318912yes—
483C23C2×#1171992211yescharacteristic
484C23C2×#1171992311yescharacteristic
485C23#1313323712yes—
486D4—82no—
487D4—84no—
488D4—82no—
489D4—82no—
490D4—84no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—4242no—
519C2×C4—4242no—
520C2×C4—4242no—
521C2×C4—4242no—
522C2×C4—2124no—
523C2×C4—2124no—
524C2×C4—2124no—
525C2×C4#1313313212yes—
526C2×C4D4×Dic5312yes—
527C2×C4#1313318511yescharacteristic
528C2×C4#1313319811yescharacteristic
529C8—2124no—
530C22—21no—
531C22—21no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—21no—
539C22—22no—
540C22—21no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—48no—
548C22—48no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22#1681035714yescomplemented
561C22#1681050311yescharacteristic
562C22C2×#1313313211yescharacteristic
563C22C2×#1313313211yescharacteristic
564C22C2×#1313313811yescharacteristic
565C22#1681061911yescharacteristic
566C22C2×#1313314411yescharacteristic
567C22#1681064011yescharacteristic
568C22#1681085012yes—
569C22#1681085712yes—
570C4—21no—
571C4—22no—
572C4—21no—
573C4—22no—
574C4—41no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4—4244no—
579C2—22no—
580C2—22no—
581C2—24no—
582C2—42no—
583C2—161no—
584C2C2×#1681035711yescharacteristic
585C2#2027320611yescharacteristic
586C2#2027327111yescharacteristic
587C2#2027328311yescharacteristic
588C2#2027501811yescharacteristic
589C2#2027514411yescharacteristic
590C2#2027516911yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
Fitting, prime radicalC53×#2847C2
socleC23×C53C2×C22⋊C4
cosocleC23×C4×C53C23
hypercenter#2847D53
hypocenterC53#9113
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2847

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊#88322C2semidirect
2. C2×#16810357C2semidirect
3. #20272938C2semidirect
4. #20272593C2semidirect
15. C2×#121×C53C4semidirect
16. C2×#121×C53C4semidirect
22. C53×#380C4semidirect
23. C53×#380C4semidirect
35. C22⋊#88322D4semidirect
265. C22×C53#327semidirect
303. C2×#121Dic53semidirect
304. C2×#121Dic53semidirect
314. #380Dic53semidirect
315. #380Dic53semidirect
320. C53#9113semidirect
560. C22#16810357semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊#883222
C2×#168103572
#202729382
#202725932
C53×#28472
#911353