Finite Group Databasebeta

← #27279678

Subgroups — #27279678

Order 13568.

1030 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810587C211yesmaximal, characteristic, complemented
2C2×#16810357C212yesmaximal, complemented
3#20272938C211yesmaximal, characteristic, complemented
4#20272593C212yesmaximal, complemented
5C53×#2847C211yesmaximal, characteristic
6C22×#11719917—21no—
7C22×C22⋊Dic53—21no—
8#16810587—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—
26C2×#11719917—24no—
27C2×#11719917—42no—
28C2×#11719917—42no—
29C22×#88322—22no—
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—44no—
59C2×C4⋊C4×C53C2×C411yescharacteristic
60C2×C4⋊C4×C53C2×C411yescharacteristic
61C2×C4⋊C4×C53C2×C411yescharacteristic
62C2×C22⋊C4×C53—21no—
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—
68#95×C53—21no—
69#95×C53—21no—
70#95×C53—21no—
71C24×C53—22no—
72C24×C53—22no—
73C24×C53—21no—
74C24×C53—21no—
75C24×C53—22no—
76C24×C53—22no—
77C24×C53—21no—
78C24×C53—41no—
79C24×C53—41no—
80C24×C53C22⋊C411yescharacteristic
81C24×C53C22⋊C411yescharacteristic
82C24×C53C2×D411yescharacteristic
83C2×D4×C53—22no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—42no—
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—
96C22×C4×C53—22no—
97C22×C4×C53—22no—
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—22no—
101C22×C4×C53—22no—
102C22×C4×C53—22no—
103C22×C4×C53—22no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—21no—
107C22×C4×C53—21no—
108C22×C4×C53—21no—
109C22×C4×C53—21no—
110C22×C4×C53—21no—
111C22×C4×C53—41no—
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—48no—
119C22×C4×C53—48no—
120C22×C4×C53C22⋊C411yescharacteristic
121C22×C4×C53C22⋊C411yescharacteristic
122C22×C4×C53C22×C411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D412yes—
125C22×C4×C53#4112yes—
126C22×Dic53—24no—
127C22×Dic53—42no—
128C22×Dic53—42no—
129C22×Dic53—42no—
130C22×Dic53—42no—
131C22×Dic53—42no—
132C22×Dic53—42no—
133C4⋊C4×C53—22no—
134C4⋊C4×C53—41no—
135C4⋊C4×C53—41no—
136C22⋊C4×C53—42no—
137C22⋊C4×C53—42no—
138C22⋊C4×C53—42no—
139C22⋊C4×C53—42no—
140C22⋊C4×C53—42no—
141C22⋊C4×C53—41no—
142C22⋊C4×C53—41no—
143C22⋊Dic53—48no—
144#11719917—48no—
145C2×#88322—24no—
146C2×#88322—42no—
147C2×#88322—42no—
148C23×C53—21no—
149C23×C53—21no—
150C23×C53—21no—
151C23×C53—21no—
152C23×C53—24no—
153C23×C53—24no—
154C23×C53—21no—
155C23×C53—21no—
156C23×C53—21no—
157C23×C53—21no—
158C23×C53—21no—
159C23×C53—21no—
160C23×C53—21no—
161C23×C53—21no—
162C23×C53—21no—
163C23×C53—21no—
164C23×C53—21no—
165C23×C53—21no—
166C23×C53—44no—
167C23×C53—44no—
168C23×C53—44no—
169C23×C53—44no—
170C23×C53—42no—
171C23×C53—42no—
172C23×C53—42no—
173C23×C53—42no—
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#10412yes—
183C23×C53C2×C22⋊C411yescharacteristic
184C23×C53#12112yes—
185C23×C53AGL(1,ℤ/8)11yescharacteristic
186C23×C53AGL(1,ℤ/8)12yes—
187C23×C53#13711yescharacteristic
188D4×C53—82no—
189D4×C53—84no—
190D4×C53—84no—
191D4×C53—82no—
192D4×C53—82no—
193C2×C4×C53—24no—
194C2×C4×C53—24no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
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—42no—
205C2×C4×C53—42no—
206C2×C4×C53—44no—
207C2×C4×C53—44no—
208C2×C4×C53—44no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—41no—
212C2×C4×C53—41no—
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#9912yes—
221C2×C4×C53C4×D412yes—
222C2×C4×C53#12011yescharacteristic
223C2×C4×C53#12311yescharacteristic
224C2×Dic53—44no—
225C2×Dic53—44no—
226C2×Dic53—44no—
227C2×Dic53—82no—
228C2×Dic53—82no—
229C2×Dic53—82no—
230C2×Dic53—82no—
231#88322—44no—
232#9117—531nomaximal, Sylow
233C22×C53—22no—
234C22×C53—22no—
235C22×C53—21no—
236C22×C53—21no—
237C22×C53—22no—
238C22×C53—22no—
239C22×C53—21no—
240C22×C53—21no—
241C22×C53—22no—
242C22×C53—22no—
243C22×C53—22no—
244C22×C53—22no—
245C22×C53—22no—
246C22×C53—22no—
247C22×C53—48no—
248C22×C53—48no—
249C22×C53—41no—
250C22×C53—41no—
251C22×C53—41no—
252C22×C53—41no—
253C22×C53—41no—
254C22×C53—41no—
255C22×C53—41no—
256C22×C53—41no—
257C22×C53—81no—
258C22×C53—81no—
259C22×C53—81no—
260C22×C53—81no—
261C22×C53—81no—
262C22×C53—81no—
263C22×C53—81no—
264C22×C53#38611yescharacteristic
265C22×C53C2×#9911yescharacteristic
266C22×C53C2×#9911yescharacteristic
267C22×C53#41711yescharacteristic
268C22×C53#41811yescharacteristic
269C22×C53C2×#10411yescharacteristic
270C22×C53#42111yescharacteristic
271C22×C53#46812yes—
272C22×C53#46912yes—
273C4×C53—21no—
274C4×C53—22no—
275C4×C53—21no—
276C4×C53—22no—
277C4×C53—41no—
278C4×C53—81no—
279C4×C53—81no—
280C4×C53—81no—
281Dic53—84no—
282C2×#407—531no—
283#2847D5311yescharacteristic
284C2×#327—532no—
285C22⋊#95—531no—
286#1294—532no—
287C2×C53—22no—
288C2×C53—22no—
289C2×C53—42no—
290C2×C53—42no—
291C2×C53—161no—
292C2×C53#150411yescharacteristic
293C2×C53#150711yescharacteristic
294C2×C53#151011yescharacteristic
295C2×C53#151311yescharacteristic
296C2×C53#187111yescharacteristic
297C2×C53#189511yescharacteristic
298C2×C53#189811yescharacteristic
299C24×C4C2×D5311yescharacteristic
300C22×M16—1061no—
301C2×#121Dic5311yescharacteristic, complemented
302C2×#121Dic5311yescharacteristic, complemented
303C2×#121C2×D5311yescharacteristic
304C2×#120—21no—
305C22×C22⋊C4—1061no—
306#407—1064no—
307C2×C22⋊C8—532no—
308C2×C22⋊C8—1061no—
309#386—21no—
310#386—21no—
311#386—21no—
312#380Dic5311yescharacteristic, complemented
313#380Dic5311yescharacteristic, complemented
314#380C2×D5311yescharacteristic
315C2×#95—532no—
316C2×#95—1061no—
317#327—1064no—
318C53#911711yescharacteristic, complemented, Sylow
319C25C22⋊D5311yescharacteristic
320C22×D4—21no—
321C22×D4—21no—
322C22×D4—21no—
323C22×D4C2×Dic5311yescharacteristic
324C23×C4—22no—
325C23×C4—22no—
326C23×C4—42no—
327C23×C4—42no—
328C23×C4—1062no—
329C23×C4C2×Dic5311yescharacteristic
330C23×C4C2×Dic5311yescharacteristic
331C23×C4C22⋊D5311yescharacteristic
332C23×C4C22⋊D5311yescharacteristic
333C23×C4C22⋊D5311yescharacteristic
334C23×C4C22×D5311yescharacteristic
335C2×M16—1064no—
336C2×M16—2122no—
337C2×M16—2122no—
338C22×C8—1062no—
339#121—22no—
340#121—22no—
341#121—42no—
342#121—42no—
343#120—44no—
344C2×C4⋊C4C2×Dic5311yescharacteristic
345C2×C4⋊C4C2×Dic5311yescharacteristic
346C2×C4⋊C4C2×Dic5311yescharacteristic
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—1064no—
354C2×C22⋊C4—2122no—
355C2×C22⋊C4—2122no—
356C22⋊C8—534no—
357C22⋊C8—1064no—
358C22⋊C8—1062no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—1062no—
363#95—1062no—
364#95—2122no—
365C24—22no—
366C24—22no—
367C24—21no—
368C24—21no—
369C24—22no—
370C24—22no—
371C24—21no—
372C24—41no—
373C24—41no—
374C24C22⋊Dic5311yescharacteristic
375C24C22⋊Dic5311yescharacteristic
376C24C2×C22⋊D5311yescharacteristic
377C2×D4—22no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—41no—
388C2×D4—42no—
389C2×D4—41no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—1064no—
415C22×C4—2122no—
416C22×C4—2122no—
417C22×C4—2122no—
418C22×C4—2122no—
419C22×C4—2122no—
420C22×C4—2122no—
421C22×C4C22⋊Dic5311yescharacteristic
422C22×C4C22⋊Dic5311yescharacteristic
423C22×C4D4×D5312yes—
424C22×C4#1171994712yes—
425C22×C4C22×Dic5311yescharacteristic
426C22×C4C2×C22⋊D5311yescharacteristic
427M16—2128no—
428C2×C8—1064no—
429C2×C8—2122no—
430C2×C8—2122no—
431C4⋊C4—22no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—2128no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—24no—
447C23—24no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23#1313313212yes—
476C23#1313314412yes—
477C23C2×C22⋊Dic5311yescharacteristic
478C23#1313318912yes—
479C23#1313324211yescharacteristic
480C23#1313324312yes—
481C23#1313324711yescharacteristic
482D4—82no—
483D4—84no—
484D4—84no—
485D4—82no—
486D4—82no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—4242no—
515C2×C4—4242no—
516C2×C4—4242no—
517C2×C4—4242no—
518C2×C4—2124no—
519C2×C4—2124no—
520C2×C4—2124no—
521C2×C4#1313313212yes—
522C2×C4D4×Dic5312yes—
523C2×C4#1313318511yescharacteristic
524C2×C4#1313319811yescharacteristic
525C8—2124no—
526C22—21no—
527C22—21no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—21no—
532C22—21no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—48no—
541C22—48no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22#1681050311yescharacteristic
558C22C2×#1313313211yescharacteristic
559C22C2×#1313313211yescharacteristic
560C22#1681062411yescharacteristic
561C22#1681062811yescharacteristic
562C22C2×#1313314411yescharacteristic
563C22#1681064011yescharacteristic
564C22#1681086712yes—
565C22#1681087412yes—
566C4—21no—
567C4—22no—
568C4—21no—
569C4—22no—
570C4—41no—
571C4—81no—
572C4—81no—
573C4—81no—
574C4—4244no—
575C2—22no—
576C2—22no—
577C2—42no—
578C2—42no—
579C2—161no—
580C2#2027321911yescharacteristic
581C2#2027323111yescharacteristic
582C2#2027324411yescharacteristic
583C2#2027325811yescharacteristic
584C2#2027501811yescharacteristic
585C2#2027515411yescharacteristic
586C2#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#9117
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. C2×#16810587C2semidirect
2. C2×#16810357C2semidirect
3. #20272938C2semidirect
4. #20272593C2semidirect
15. C2×#121×C53C4semidirect
16. C2×#121×C53C4semidirect
22. C53×#380C4semidirect
23. C53×#380C4semidirect
35. C22⋊#88322D4semidirect
301. C2×#121Dic53semidirect
302. C2×#121Dic53semidirect
312. #380Dic53semidirect
313. #380Dic53semidirect
318. C53#9117semidirect

Maximal subgroups

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

subgroupindex
C2×#168105872
C2×#168103572
#202729382
#202725932
C53×#28472
#911753