Finite Group Databasebeta

← #27389183

Subgroups — #27389183

Order 13568.

1270 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811159C211yesmaximal, characteristic
2C2×#16810677C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×Dic53C211yesmaximal, characteristic, complemented
4#20275671C212yesmaximal, complemented
5#20275644C212yesmaximal, complemented
6#20275032C211yesmaximal, characteristic
7#20275030C211yesmaximal, characteristic
8#20275023C211yesmaximal, characteristic, complemented
9#20274547C212yesmaximal, complemented
10#20274539C212yesmaximal, complemented
11C53×#1872C211yesmaximal, characteristic, complemented
12#16811159—24no—
13#16811159C418yescomplemented
14C2×#13133212—21no—
15C2×#13133208C2211yescharacteristic
16C2×#13133206C2211yescharacteristic
17C2×#13133201—21no—
18C2×#13133191C2211yescharacteristic
19C2×Q8×Dic53C2211yescharacteristic
20C2×#13133179C2211yescharacteristic
21C2×#13133177—21no—
22C2×#13133175—21no—
23C22×C4⋊Dic53C2211yescharacteristic
24C22×C4×Dic53—21no—
25C22×C4×Dic53C2211yescharacteristic
26#16810677—24no—
27C2×#13133146C2212yes—
28C2×#13133141C2212yes—
29C2×#13133139C2212yes—
30C2×#13133137C2212yes—
31C2×C22⋊#88322C2211yescharacteristic
32C2×#13133127C2212yes—
33C4⋊C4×Dic53—21no—
34C4⋊C4×Dic53—21no—
35C4⋊C4×Dic53C2212yes—
36C4⋊C4×Dic53C2212yes—
37C2×#13133124—21no—
38#16810386C2212yes—
39#16810385C2212yes—
40#16810385C2212yes—
41C2×C22⋊Q8×C53C2211yescharacteristic
42C22×C4⋊C4×C53C2211yescharacteristic
43C2×#103×C53C2212yes—
44C2×C22⋊C8×C53C2211yescharacteristic
45C53×#340C2212yes—
46C22×D212C2311yescharacteristic
47C22×C22⋊D53—21no—
48C23×Dic53—21no—
49C23×Dic53D411yescharacteristic
50C23×Dic53D411yescharacteristic
51C22×C4×D53—21no—
52C22×#88322C2312yes—
53#13133212—28no—
54#13133208—22no—
55#13133208C2×C414yescomplemented
56#13133206—22no—
57#13133206C2×C414yescomplemented
58#13133201—28no—
59#13133191—22no—
60#13133191C2×C414yes—
61Q8×Dic53—22no—
62Q8×Dic53C2×C414yescomplemented
63#13133179—22no—
64#13133179C2×C414yescomplemented
65#13133177—28no—
66#13133175—28no—
67C2×C4⋊Dic53—24no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2312yes—
73C2×#11719919—21no—
74C2×#11719921—21no—
75C2×#11719921—21no—
76C2×#11719921—21no—
77C2×#11719921—21no—
78C2×C4×Dic53—21no—
79C2×C4×Dic53—22no—
80C2×C4×Dic53—22no—
81C2×C4×Dic53—44no—
82C2×C4×Dic53C2×C414yes—
83C2×C4×Dic53D414yes—
84C2×C4×Dic53D411yescharacteristic
85C2×C4×Dic53D411yescharacteristic
86C2×C4×Dic53C2311yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88#13133146—24no—
89#13133141—24no—
90#13133139—24no—
91#13133137—24no—
92C22⋊#88322—22no—
93#13133127—24no—
94#13133124—22no—
95#13133124—22no—
96C22×Q8×C53C2311yescharacteristic
97C23×C4×C53—21no—
98C23×C4×C53C2311yescharacteristic
99C22×C8×C53C2312yes—
100C22⋊Q8×C53—22no—
101C22⋊Q8×C53C2×C414yes—
102C2×C4⋊C4×C53—24no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C4⋊C4×C53C2312yes—
107C2×C22⋊C4×C53—21no—
108#103×C53—24no—
109C22⋊C8×C53—22no—
110C24×C53C2×D411yescharacteristic
111C23×D53—21no—
112C2×Q8×C53—21no—
113C2×Q8×C53—22no—
114C2×Q8×C53—22no—
115C2×Q8×C53C22×C412yes—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—24no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—22no—
122C22×C4×C53—42no—
123C22×C4×C53—42no—
124C22×C4×C53C22×C412yes—
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D412yes—
127C22×C4×C53C2411yescharacteristic
128C2×C22⋊D53—24no—
129C2×C22⋊D53—42no—
130C2×C22⋊D53—42no—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—24no—
134C22×Dic53—22no—
135C22×Dic53—21no—
136C22×Dic53—41no—
137C22×Dic53—41no—
138C22×Dic53—42no—
139C22×Dic53—42no—
140C22×Dic53C22⋊C414yes—
141C22×Dic53C22⋊C414yes—
142C22×Dic53C2×D411yescharacteristic
143C22×Dic53C2×D411yescharacteristic
144C22×Dic53C2×D411yescharacteristic
145C22×Dic53C2×D411yescharacteristic
146C2×D212—22no—
147C2×D212—22no—
148C2×D212—21no—
149C2×D212C22×C412yes—
150C2×C4×D53—24no—
151C2×C4×D53—42no—
152C2×C4×D53—42no—
153C2×C8×C53—22no—
154C2×C8×C53—22no—
155C2×C8×C53—22no—
156C4⋊C4×C53—24no—
157C4⋊C4×C53—24no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—22no—
160C4⋊C4×C53C22×C412yes—
161C4⋊C4×C53#4114yes—
162C22⋊C4×C53—24no—
163#11719921—24no—
164#11719921—24no—
165#11719921—24no—
166#11719921—24no—
167C4⋊Dic53—24no—
168C4⋊Dic53—21no—
169C4⋊Dic53—22no—
170C4⋊Dic53—24no—
171C4⋊Dic53—24no—
172C4⋊Dic53C22×C412yes—
173C4⋊Dic53#4114yes—
174#11719919—24no—
175C4×Dic53—24no—
176C4×Dic53—44no—
177C4×Dic53—44no—
178C4×Dic53C22⋊C414yes—
179C4×Dic53C22⋊C414yes—
180C4×Dic53C22×C412yes—
181C4×Dic53C22×C412yes—
182C4×Dic53C2×D412yes—
183C4×Dic53C2×D412yes—
184C2×#88322—22no—
185C2×#88322—22no—
186C2×#88322—22no—
187C23×C53—21no—
188C23×C53—21no—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53C2×C22⋊C412yes—
192C23×C53AGL(1,ℤ/8)12yes—
193C23×C53#13712yes—
194C23×C53C22×D411yescharacteristic
195C22×D53—22no—
196C22×D53—41no—
197C22×D53—41no—
198C22×D53—41no—
199C22×D53—41no—
200C22×D53—41no—
201C22×D53—41no—
202Q8×C53—24no—
203Q8×C53—42no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—24no—
207C2×C4×C53—24no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—24no—
211C2×C4×C53—48no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—42no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—42no—
218C2×C4×C53C2×C22⋊C412yes—
219C2×C4×C53C23×C411yescharacteristic
220C2×C4×C53C22×D411yescharacteristic
221C2×C4×C53C2×#4112yes—
222C2×C4×C53C2×#4112yes—
223C22⋊D53—48no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—28no—
233C2×Dic53—22no—
234C2×Dic53—24no—
235C2×Dic53—44no—
236C2×Dic53—48no—
237C2×Dic53—42no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—42no—
241C2×Dic53C2×C22⋊C412yes—
242C2×Dic53C2×C22⋊C412yes—
243C2×Dic53C2×C22⋊C412yes—
244C2×Dic53C2×C22⋊C412yes—
245D212—24no—
246D212—42no—
247C4×D53—48no—
248C8×C53—42no—
249#88322—42no—
250#21391—531nomaximal, Sylow
251C22×C53—22no—
252C22×C53—22no—
253C22×C53—22no—
254C22×C53—22no—
255C22×C53—22no—
256C22×C53—22no—
257C22×C53#41714yes—
258C22×C53C22×C22⋊C411yescharacteristic
259C22×C53#52511yescharacteristic
260C22×C53#52511yescharacteristic
261C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C53C2×#13711yescharacteristic
263C22×C53#57511yescharacteristic
264C22×C53#57511yescharacteristic
265C2×D53—42no—
266C2×D53—42no—
267C2×D53—42no—
268C2×D53—81no—
269C2×D53—81no—
270C2×D53—81no—
271C2×D53—81no—
272C4×C53—22no—
273C4×C53—44no—
274C4×C53—42no—
275C4×C53—42no—
276C4×C53C4×#4112yes—
277C4×C53#53212yes—
278Dic53—22no—
279Dic53—22no—
280Dic53—22no—
281Dic53—22no—
282Dic53—44no—
283Dic53—42no—
284C2×#529—531no—
285C2×#432—531no—
286C2×C4×C4⋊C4—531no—
287#1989—532no—
288#1983—532no—
289#1874—531no—
290#1874—531no—
291#1872—531no—
292#1872D5311yescharacteristic, complemented
293#1763—532no—
294#1762—532no—
295C2×C53—24no—
296C2×C53#229711yescharacteristic
297C2×C53C2×#41711yescharacteristic
298C2×C53#289611yescharacteristic
299C2×C53#311411yescharacteristic
300C2×C53#311411yescharacteristic
301C2×C53#311711yescharacteristic
302C2×C53#311711yescharacteristic
303D53—82no—
304#529—538no—
305#529—1064no—
306C2×#126—1061no—
307C2×#125—531no—
308C2×#124—531no—
309C2×#123—1061no—
310C2×C22⋊Q8C2×D5311yescharacteristic
311C2×#121—531no—
312C2×C4×Q8—531no—
313C2×C4×D4—531no—
314C2×C4×D4—1061no—
315C2×#117—1061no—
316C22×C4⋊C4—531no—
317C22×C4⋊C4C2×D5311yescharacteristic
318C22×C42—531no—
319C22×C42—1061no—
320#432—1064no—
321C2×C4⋊C8—532no—
322C2×#103—532no—
323C2×#103C2×D5312yes—
324C2×#102—532no—
325C2×#102—532no—
326C2×C22⋊C8—531no—
327C2×C22⋊C8C2×D5311yescharacteristic
328C2×#97—532no—
329C4×C4⋊C4—532no—
330C4×C4⋊C4—532no—
331C4×C4⋊C4—1061no—
332C4×C4⋊C4—1061no—
333C2×#95—1061no—
334#340—532no—
335#340—532no—
336#340—532no—
337#340C2×D5312yes—
338C53#2139111yescharacteristic, complemented, Sylow
339C22×Q8C22×D5311yescharacteristic
340C22×D4—531no—
341C22×D4—1061no—
342C23×C4—21no—
343C23×C4—531no—
344C23×C4—531no—
345C23×C4—1061no—
346C23×C4—1061no—
347C23×C4C22×D5311yescharacteristic
348C22×C8—532no—
349C22×C8C22×D5312yes—
350#126—1068no—
351#125—534no—
352#125—1062no—
353#124—534no—
354#124—1062no—
355#123—1068no—
356C22⋊Q8—22no—
357C22⋊Q8C4×D5314yes—
358#121—534no—
359#121—1062no—
360C4×Q8—534no—
361C4×Q8—1062no—
362C4×D4—534no—
363C4×D4—1062no—
364C4×D4—1068no—
365#117—1068no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—24no—
369C2×C4⋊C4—532no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—1064no—
372C2×C4⋊C4—1061no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4C22×D5311yescharacteristic
377C2×C4⋊C4C22×D5312yes—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—1061no—
380C2×C22⋊C4—1061no—
381C2×C22⋊C4—1061no—
382C2×C22⋊C4—1061no—
383C2×C42—534no—
384C2×C42—531no—
385C2×C42—531no—
386C2×C42—534no—
387C2×C42—531no—
388C2×C42—531no—
389C2×C42—1061no—
390C2×C42—1062no—
391C2×C42—1062no—
392C2×C42—2124no—
393C4⋊C8—1064no—
394#103—24no—
395#103—1064no—
396#102—1064no—
397#102—1064no—
398C22⋊C8—22no—
399C22⋊C8—1062no—
400#97—1064no—
401#95—1062no—
402#95—1062no—
403C24—1061no—
404C24D4×D5311yescharacteristic
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8C2×C4×D5312yes—
409C2×D4—532no—
410C2×D4—1064no—
411C2×D4—1061no—
412C2×D4—1062no—
413C2×D4—1062no—
414C2×D4—2122no—
415C2×D4—2122no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—534no—
425C22×C4—531no—
426C22×C4—531no—
427C22×C4—531no—
428C22×C4—531no—
429C22×C4—534no—
430C22×C4—1064no—
431C22×C4—1062no—
432C22×C4—1062no—
433C22×C4—1064no—
434C22×C4—1062no—
435C22×C4—1061no—
436C22×C4—2122no—
437C22×C4—2122no—
438C22×C4—2121no—
439C22×C4—2121no—
440C22×C4—2122no—
441C22×C4—2122no—
442C22×C4C2×C4×D5312yes—
443C22×C4D4×D5311yescharacteristic
444C22×C4D4×D5312yes—
445C22×C4C23×D5311yescharacteristic
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—1062no—
450C2×C8—1062no—
451C2×C8—1062no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—532no—
457C4⋊C4—534no—
458C4⋊C4—1064no—
459C4⋊C4—1064no—
460C4⋊C4—1064no—
461C4⋊C4—1064no—
462C4⋊C4—1061no—
463C4⋊C4—1062no—
464C4⋊C4C2×C4×D5312yes—
465C4⋊C4#1171994714yes—
466C22⋊C4—24no—
467C22⋊C4—1064no—
468C22⋊C4—1064no—
469C22⋊C4—1064no—
470C22⋊C4—1064no—
471C42—534no—
472C42—534no—
473C42—532no—
474C42—532no—
475C42—532no—
476C42—532no—
477C42—1064no—
478C42—2124no—
479C42—2124no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—1062no—
485C23—2121no—
486C23—2121no—
487C23—2121no—
488C23—2121no—
489C23—2121no—
490C23—2121no—
491C23C22⋊C4×D5312yes—
492C23#1313324412yes—
493C23#1313324912yes—
494C23C2×D4×D5311yescharacteristic
495Q8—24no—
496Q8—42no—
497D4—1064no—
498D4—2128no—
499D4—2122no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—48no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—532no—
515C2×C4—532no—
516C2×C4—532no—
517C2×C4—532no—
518C2×C4—1068no—
519C2×C4—1061no—
520C2×C4—1061no—
521C2×C4—1061no—
522C2×C4—1061no—
523C2×C4—1061no—
524C2×C4—1061no—
525C2×C4—1061no—
526C2×C4—1061no—
527C2×C4—1062no—
528C2×C4—1064no—
529C2×C4—2128no—
530C2×C4—2124no—
531C2×C4—2128no—
532C2×C4—2122no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2122no—
536C2×C4C22⋊C4×D5312yes—
537C2×C4C22×C4×D5311yescharacteristic
538C2×C4C2×D4×D5311yescharacteristic
539C2×C4#41×D5312yes—
540C2×C4C2×#1171994712yes—
541C8—42no—
542C8—2122no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—4241no—
550C22—4241no—
551C22—4241no—
552C22—4241no—
553C22—2122no—
554C22—2122no—
555C22—2122no—
556C22#1681062314yes—
557C22C2×C22⋊C4×D5311yescharacteristic
558C22#1681112911yescharacteristic
559C22#1681113111yescharacteristic
560C22C2×#1313324411yescharacteristic
561C22C2×#1313324911yescharacteristic
562C22#1681141911yescharacteristic
563C22#1681142211yescharacteristic
564C4—22no—
565C4—42no—
566C4—44no—
567C4—42no—
568C4—1062no—
569C4—1062no—
570C4—1062no—
571C4—1062no—
572C4—2122no—
573C4—2124no—
574C4C4×#1171994712yes—
575C4#1681116812yes—
576C2—24no—
577C2—4242no—
578C2#2027720011yescharacteristic
579C2C2×#1681062311yescharacteristic
580C2#2028323111yescharacteristic
581C2#2028485411yescharacteristic
582C2#2028485611yescharacteristic
583C2#2028488711yescharacteristic
584C2#2028488911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1872C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
cosocleC22×C4×C53C24
hypercenter#1872D53
hypocenterC53#21391
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
2. C2×#16810677C2semidirect
3. C2×C4⋊C4×Dic53C2semidirect
4. #20275671C2semidirect
5. #20275644C2semidirect
8. #20275023C2semidirect
9. #20274547C2semidirect
10. #20274539C2semidirect
11. C53×#1872C2semidirect
13. #16811159C4semidirect
55. #13133208C2×C4semidirect
57. #13133206C2×C4semidirect
62. Q8×Dic53C2×C4semidirect
64. #13133179C2×C4semidirect
292. #1872D53semidirect
338. C53#21391semidirect

Maximal subgroups

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

subgroupindex
C2×#168111592
C2×#168106772
C2×C4⋊C4×Dic532
#202756712
#202756442
#202750322
#202750302
#202750232
#202745472
#202745392
C53×#18722
#2139153