Finite Group Databasebeta

← #27393824

Subgroups — #27393824

Order 13568.

1234 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
1C2×#16811154C211yesmaximal, characteristic
2C2×#16810677C211yesmaximal, characteristic, complemented
3C2×#16810492C211yesmaximal, characteristic, complemented
4#20275682C212yesmaximal, complemented
5#20275530C212yesmaximal, complemented
6#20275033C211yesmaximal, characteristic
7#20275022C211yesmaximal, characteristic, complemented
8#20275020C211yesmaximal, characteristic, complemented
9#20274719C212yesmaximal, complemented
10#20274713C212yesmaximal, complemented
11C53×#1874C211yesmaximal, characteristic
12#16811154—24no—
13#16811154C418yescomplemented
14C2×#13133210—21no—
15C2×#13133208C2211yescharacteristic
16C2×#13133204C2211yescharacteristic
17C2×#13133198—21no—
18C2×#13133196C2211yescharacteristic
19C2×#13133182C2211yescharacteristic
20C2×D4×Dic53—21no—
21C2×D4×Dic53C2211yescharacteristic
22C2×#13133173—21no—
23C22×C4⋊Dic53C2211yescharacteristic
24C22×#11719919—21no—
25C22×#11719919—21no—
26C22×C4×Dic53C2211yescharacteristic
27#16810677—24no—
28C2×#13133146C2212yes—
29C2×#13133142C2212yes—
30C2×#13133140C2212yes—
31C2×#13133138C2212yes—
32C2×C22⋊#88322C2211yescharacteristic
33C2×#13133127C2212yes—
34#16810492—21no—
35#16810492—21no—
36#16810492C2212yes—
37#16810492C2212yes—
38#16810386C2212yes—
39#16810385C2212yes—
40#16810385C2212yes—
41C2×#121×C53C2211yescharacteristic
42C22×C4⋊C4×C53C2211yescharacteristic
43C2×#102×C53C2212yes—
44C2×C22⋊C8×C53C2211yescharacteristic
45C53×#340C2212yes—
46C22×Dic106C2311yescharacteristic
47C23×Dic53—21no—
48C23×Dic53—21no—
49C23×Dic53D411yescharacteristic
50C23×Dic53D411yescharacteristic
51C22×#88322C2312yes—
52#13133210—28no—
53#13133208—22no—
54#13133208C2×C414yescomplemented
55#13133204—22no—
56#13133204C2×C414yescomplemented
57#13133198—28no—
58#13133196—22no—
59#13133196C2×C414yes—
60#13133182—22no—
61#13133182C2×C414yescomplemented
62D4×Dic53—22no—
63D4×Dic53—28no—
64D4×Dic53C2×C414yescomplemented
65#13133173—28no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—42no—
69C2×C4⋊Dic53C2312yes—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×#11719919—21no—
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—22no—
75C2×#11719919—22no—
76C2×#11719919—22no—
77C2×#11719919—22no—
78C2×#11719919—44no—
79C2×#11719919—44no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53C2×C414yes—
86C2×C4×Dic53D414yes—
87C2×C4×Dic53D411yescharacteristic
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53C2311yescharacteristic
90C2×C4×Dic53C2311yescharacteristic
91#13133146—24no—
92#13133142—42no—
93#13133140—42no—
94#13133138—42no—
95C22⋊#88322—22no—
96#13133127—24no—
97C22×D4×C53—21no—
98C22×D4×C53C2311yescharacteristic
99C23×C4×C53—21no—
100C23×C4×C53C2311yescharacteristic
101C22×C8×C53C2312yes—
102#121×C53—22no—
103#121×C53C2×C414yes—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—42no—
106C2×C4⋊C4×C53C2312yes—
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C22⋊C4×C53—21no—
109#102×C53—42no—
110C22⋊C8×C53—22no—
111C24×C53—21no—
112C24×C53C2×D411yescharacteristic
113C2×D4×C53—24no—
114C2×D4×C53—21no—
115C2×D4×C53—22no—
116C2×D4×C53—22no—
117C2×D4×C53—42no—
118C2×D4×C53—42no—
119C2×D4×C53C22×C412yes—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—21no—
123C22×C4×C53—22no—
124C22×C4×C53—42no—
125C22×C4×C53—42no—
126C22×C4×C53—42no—
127C22×C4×C53C22×C412yes—
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D412yes—
130C22×C4×C53C2411yescharacteristic
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—24no—
138C22×Dic53—22no—
139C22×Dic53—42no—
140C22×Dic53—42no—
141C22×Dic53—42no—
142C22×Dic53—42no—
143C22×Dic53—42no—
144C22×Dic53C22⋊C414yes—
145C22×Dic53C22⋊C414yes—
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C2×Dic106—22no—
151C2×Dic106—22no—
152C2×Dic106—21no—
153C2×Dic106C22×C412yes—
154C2×C8×C53—22no—
155C2×C8×C53—22no—
156C2×C8×C53—22no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—22no—
159C4⋊C4×C53—22no—
160C4⋊C4×C53—42no—
161C4⋊C4×C53C22×C412yes—
162C22⋊C4×C53—24no—
163C22⋊Dic53—24no—
164C22⋊Dic53—24no—
165C22⋊Dic53—24no—
166C22⋊Dic53—24no—
167C4⋊Dic53—21no—
168C4⋊Dic53—24no—
169C4⋊Dic53—22no—
170C4⋊Dic53—22no—
171C4⋊Dic53—42no—
172C4⋊Dic53C22×C412yes—
173#11719919—24no—
174#11719919—24no—
175#11719919—24no—
176#11719919—44no—
177#11719919—44no—
178#11719919—44no—
179#11719919—44no—
180C4×Dic53—24no—
181C4×Dic53C22⋊C414yes—
182C4×Dic53C22⋊C414yes—
183C4×Dic53C22×C412yes—
184C4×Dic53C22×C412yes—
185C4×Dic53C2×D412yes—
186C4×Dic53C2×D412yes—
187C2×#88322—22no—
188C2×#88322—22no—
189C2×#88322—22no—
190C23×C53—21no—
191C23×C53—21no—
192C23×C53—21no—
193C23×C53—21no—
194C23×C53—22no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53C2×C22⋊C412yes—
202C23×C53AGL(1,ℤ/8)12yes—
203C23×C53#13712yes—
204C23×C53C22×D411yescharacteristic
205D4×C53—24no—
206D4×C53—48no—
207D4×C53—42no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—24no—
211C2×C4×C53—24no—
212C2×C4×C53—22no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53—42no—
218C2×C4×C53—84no—
219C2×C4×C53C2×C22⋊C412yes—
220C2×C4×C53C23×C411yescharacteristic
221C2×C4×C53C22×D411yescharacteristic
222C2×C4×C53H(5,𝔽2)12yes—
223C2×C4×C532−1+412yes—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—28no—
231C2×Dic53—28no—
232C2×Dic53—22no—
233C2×Dic53—22no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—48no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53—84no—
243C2×Dic53C2×C22⋊C412yes—
244C2×Dic53C2×C22⋊C412yes—
245C2×Dic53C2×C22⋊C412yes—
246C2×Dic53C2×C22⋊C412yes—
247C2×Dic53AGL(1,ℤ/8)12yes—
248C2×Dic53#13712yes—
249Dic106—24no—
250Dic106—42no—
251C8×C53—42no—
252#88322—42no—
253#21867—531nomaximal, Sylow
254C22×C53—22no—
255C22×C53—22no—
256C22×C53—22no—
257C22×C53—22no—
258C22×C53—22no—
259C22×C53—22no—
260C22×C53—42no—
261C22×C53—42no—
262C22×C53—42no—
263C22×C53—81no—
264C22×C53—81no—
265C22×C53—81no—
266C22×C53—81no—
267C22×C53#41714yes—
268C22×C53C22×C22⋊C411yescharacteristic
269C22×C53#53511yescharacteristic
270C22×C53#53611yescharacteristic
271C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
272C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
273C22×C53C2×#13711yescharacteristic
274C22×C53C2×#13711yescharacteristic
275C4×C53—22no—
276C4×C53—42no—
277C4×C53—82no—
278C4×C53#52012yes—
279C4×C53#53712yes—
280Dic53—22no—
281Dic53—22no—
282Dic53—22no—
283Dic53—42no—
284Dic53—42no—
285Dic53—82no—
286Dic53#41714yes—
287C2×#529—531no—
288C2×#432—531no—
289C2×#384—531no—
290#1992—532no—
291#1959—532no—
292#1874—531no—
293#1874D5311yescharacteristic
294#1872—531no—
295#1872—531no—
296#1803—532no—
297#1802—532no—
298C2×C53—24no—
299C2×C53—82no—
300C2×C53#231111yescharacteristic
301C2×C53C2×#41711yescharacteristic
302C2×C53C2×#41711yescharacteristic
303C2×C53#321711yescharacteristic
304C2×C53#321811yescharacteristic
305C2×C53#323811yescharacteristic
306C2×C53#323911yescharacteristic
307#529—538no—
308#529—1064no—
309C2×#126—1061no—
310C2×#125—531no—
311C2×#124—531no—
312C2×#123—1061no—
313C2×C22⋊Q8—531no—
314C2×#121C2×D5311yescharacteristic
315C2×C4×Q8—531no—
316C2×C4×D4—531no—
317C2×C4×D4—1061no—
318C2×#117—1061no—
319C22×C4⋊C4—531no—
320C22×C4⋊C4—1061no—
321C22×C4⋊C4—1061no—
322C22×C4⋊C4C2×D5311yescharacteristic
323C22×C42—531no—
324#432—1064no—
325C2×C4⋊C8—532no—
326C2×#103—532no—
327C2×#103—532no—
328C2×#102—532no—
329C2×#102C2×D5312yes—
330C2×C22⋊C8—531no—
331C2×C22⋊C8C2×D5311yescharacteristic
332C2×#97—532no—
333#384—532no—
334#384—532no—
335#384—1061no—
336#384—1061no—
337#340—532no—
338#340—532no—
339#340—532no—
340#340C2×D5312yes—
341C53#2186711yescharacteristic, complemented, Sylow
342C22×Q8—531no—
343C22×D4—21no—
344C22×D4C22×D5311yescharacteristic
345C23×C4—21no—
346C23×C4—531no—
347C23×C4—531no—
348C23×C4—1061no—
349C23×C4—1061no—
350C23×C4C22×D5311yescharacteristic
351C22×C8—532no—
352C22×C8C22×D5312yes—
353#126—1068no—
354#125—534no—
355#125—1062no—
356#124—534no—
357#124—1062no—
358#123—1068no—
359C22⋊Q8—534no—
360C22⋊Q8—1062no—
361#121—22no—
362#121C4×D5314yes—
363C4×Q8—534no—
364C4×Q8—1062no—
365C4×D4—534no—
366C4×D4—1062no—
367C4×D4—1068no—
368#117—1068no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—42no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—532no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4—1062no—
378C2×C4⋊C4—1062no—
379C2×C4⋊C4—1062no—
380C2×C4⋊C4—1062no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—2122no—
383C2×C4⋊C4—2124no—
384C2×C4⋊C4—2124no—
385C2×C4⋊C4C22×D5312yes—
386C2×C4⋊C4C22×D5311yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—1061no—
389C2×C22⋊C4—1061no—
390C2×C22⋊C4—1061no—
391C2×C22⋊C4—1061no—
392C2×C42—534no—
393C2×C42—531no—
394C2×C42—531no—
395C2×C42—534no—
396C2×C42—531no—
397C2×C42—531no—
398C2×C42—1061no—
399C4⋊C8—1064no—
400#103—2122no—
401#103—2122no—
402#102—42no—
403#102—2122no—
404C22⋊C8—22no—
405C22⋊C8—1062no—
406#97—1064no—
407C24—21no—
408C24D4×D5311yescharacteristic
409C2×Q8—532no—
410C2×Q8—1061no—
411C2×Q8—1062no—
412C2×Q8—1062no—
413C2×D4—24no—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4C2×C4×D5312yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—531no—
428C22×C4—534no—
429C22×C4—534no—
430C22×C4—531no—
431C22×C4—531no—
432C22×C4—531no—
433C22×C4—1062no—
434C22×C4—1062no—
435C22×C4—1064no—
436C22×C4—1062no—
437C22×C4—1062no—
438C22×C4—1061no—
439C22×C4—1061no—
440C22×C4—1062no—
441C22×C4—2122no—
442C22×C4—2122no—
443C22×C4—2122no—
444C22×C4—2122no—
445C22×C4—2122no—
446C22×C4C2×C4×D5312yes—
447C22×C4D4×D5311yescharacteristic
448C22×C4D4×D5312yes—
449C22×C4C23×D5311yescharacteristic
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—1062no—
454C2×C8—1062no—
455C2×C8—1062no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—42no—
460C4⋊C4—532no—
461C4⋊C4—1064no—
462C4⋊C4—1064no—
463C4⋊C4—1064no—
464C4⋊C4—1064no—
465C4⋊C4—1061no—
466C4⋊C4—1062no—
467C4⋊C4—1062no—
468C4⋊C4—2122no—
469C4⋊C4—2124no—
470C4⋊C4—2124no—
471C4⋊C4—2124no—
472C4⋊C4—2124no—
473C4⋊C4C2×C4×D5312yes—
474C22⋊C4—24no—
475C22⋊C4—1064no—
476C22⋊C4—1064no—
477C22⋊C4—1064no—
478C22⋊C4—1064no—
479C42—534no—
480C42—534no—
481C42—532no—
482C42—532no—
483C42—532no—
484C42—532no—
485C42—1064no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23C22⋊C4×D5312yes—
498C23#1313324112yes—
499C23#1313324612yes—
500C23C2×D4×D5311yescharacteristic
501Q8—1064no—
502Q8—2122no—
503D4—24no—
504D4—48no—
505D4—42no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—84no—
517C2×C4—532no—
518C2×C4—532no—
519C2×C4—532no—
520C2×C4—532no—
521C2×C4—532no—
522C2×C4—532no—
523C2×C4—1068no—
524C2×C4—1068no—
525C2×C4—1061no—
526C2×C4—1061no—
527C2×C4—1061no—
528C2×C4—1061no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—1062no—
532C2×C4—1062no—
533C2×C4—4244no—
534C2×C4—2128no—
535C2×C4—2122no—
536C2×C4—2122no—
537C2×C4—2122no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4C22⋊C4×D5312yes—
543C2×C4C22×C4×D5311yescharacteristic
544C2×C4C2×D4×D5311yescharacteristic
545C2×C4#1313326212yes—
546C2×C4#1313326412yes—
547C8—42no—
548C8—2122no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22#1681062114yes—
563C22C2×C22⋊C4×D5311yescharacteristic
564C22#1681118811yescharacteristic
565C22#1681119511yescharacteristic
566C22AGL(1,ℤ/8)×D5311yescharacteristic
567C22C2×#1313324111yescharacteristic
568C22#137×D5311yescharacteristic
569C22C2×#1313324611yescharacteristic
570C4—22no—
571C4—42no—
572C4—82no—
573C4—534no—
574C4—1062no—
575C4—1062no—
576C4—1062no—
577C4—4242no—
578C4—2122no—
579C4—2122no—
580C4#1681110212yes—
581C4#1681120012yes—
582C2—24no—
583C2—82no—
584C2#2027730211yescharacteristic
585C2#417×D5311yescharacteristic
586C2C2×#1681062111yescharacteristic
587C2#2028573111yescharacteristic
588C2#2028574011yescharacteristic
589C2#2028593411yescharacteristic
590C2#2028594311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1874C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
cosocleC22×C4×C53C24
hypercenter#1874D53
hypocenterC53#21867
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
2. C2×#16810677C2semidirect
3. C2×#16810492C2semidirect
4. #20275682C2semidirect
5. #20275530C2semidirect
7. #20275022C2semidirect
8. #20275020C2semidirect
9. #20274719C2semidirect
10. #20274713C2semidirect
13. #16811154C4semidirect
54. #13133208C2×C4semidirect
56. #13133204C2×C4semidirect
61. #13133182C2×C4semidirect
64. D4×Dic53C2×C4semidirect
341. C53#21867semidirect

Maximal subgroups

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

subgroupindex
C2×#168111542
C2×#168106772
C2×#168104922
#202756822
#202755302
#202750332
#202750222
#202750202
#202747192
#202747132
C53×#18742
#2186753