Finite Group Databasebeta

← #27384174

Subgroups — #27384174

Order 13568.

1382 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11719949C211yesmaximal, characteristic
2C2×#16810679C211yesmaximal, characteristic, complemented
3#20277120C211yesmaximal, characteristic, complemented
4#20274808C212yesmaximal, complemented
5#20274720C211yesmaximal, characteristic, complemented
6#20274716C211yesmaximal, characteristic, complemented
7#20274712C211yesmaximal, characteristic, complemented
8#20274654C212yesmaximal
9#20274547C212yesmaximal, complemented
10#20274539C212yesmaximal, complemented
11C53×#1803C211yesmaximal, characteristic, complemented
12C22×#11719949C2211yescharacteristic
13C4×#11719949—24no—
14C4×#11719949C418yescomplemented
15C2×Q8×Dic53C2211yescharacteristic
16C2×C4×D212—21no—
17C2×C4×D212C2211yescharacteristic
18C2×#13133179—21no—
19C2×#13133179C2211yescharacteristic
20C2×#13133175—21no—
21C2×#13133175—21no—
22C2×#13133175C2211yescharacteristic
23C2×C42×D53—21no—
24C2×C42×D53C2211yescharacteristic
25#16810679—24no—
26C2×C4⋊#88322C2211yescharacteristic
27C2×#13133141C2212yes—
28C2×#13133139C2212yes—
29C2×#13133137C2212yes—
30C2×#13133129C2212yes—
31C2×#13133127C2212yes—
32#16810493C2211yescharacteristic
33#16810488—21no—
34#16810480—21no—
35#16810480C2212yes—
36C4⋊C4×Dic53—21no—
37C4⋊C4×Dic53C2212yes—
38#16810386C2212yes—
39#16810385C2212yes—
40#16810385C2212yes—
41C2×C4×Q8×C53C2211yescharacteristic
42C2×C4⋊C8×C53C2211yescharacteristic
43C2×#103×C53C2212yes—
44C53×#384C2211yescharacteristic
45C53×#340C2212yes—
46C2×#11719949—22no—
47C2×#11719949—22no—
48C2×#11719949—22no—
49C2×#11719949—22no—
50C2×#11719949—22no—
51C2×#11719949C2×C414yes—
52C22×D212—21no—
53C22×D212C2311yescharacteristic
54C22×C4×D53—21no—
55C22×C4×D53—21no—
56C22×C4×D53—21no—
57C22×C4×D53D411yescharacteristic
58C22×C4×D53D411yescharacteristic
59C22×C4×D53C2311yescharacteristic
60C22×#88322C2312yes—
61Q8×Dic53—22no—
62Q8×Dic53C2×C414yes—
63C4×D212—22no—
64C4×D212—28no—
65C4×D212C2×C414yes—
66#13133179—22no—
67#13133179—28no—
68#13133179C2×C414yes—
69#13133175—28no—
70#13133175—22no—
71#13133175—44no—
72#13133175C2×C414yes—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53C2312yes—
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719921—21no—
79C2×#11719921—22no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—22no—
85C2×C4×Dic53D411yescharacteristic
86C2×C4×Dic53D411yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88C42×D53—28no—
89C42×D53C2×C414yes—
90C42×D53D414yes—
91C4⋊#88322—22no—
92#13133141—24no—
93#13133139—24no—
94#13133137—24no—
95#13133129—24no—
96#13133127—24no—
97#13133124—22no—
98#13133124—22no—
99C22×Q8×C53C2311yescharacteristic
100C22×C8×C53C2312yes—
101C4×Q8×C53—22no—
102C4×Q8×C53C2×C414yes—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53C2312yes—
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C42×C53—21no—
109C2×C42×C53C2311yescharacteristic
110C4⋊C8×C53—22no—
111#103×C53—24no—
112C23×D53—21no—
113C23×D53C2×D411yescharacteristic
114C2×Q8×C53—21no—
115C2×Q8×C53—21no—
116C2×Q8×C53—21no—
117C2×Q8×C53C22×C412yes—
118C2×Q8×C53#4112yes—
119C2×Q8×C53#4112yes—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—22no—
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2411yescharacteristic
126C22×Dic53—21no—
127C22×Dic53—22no—
128C22×Dic53C2×D411yescharacteristic
129C22×Dic53C2×D411yescharacteristic
130C22×Dic53C2×D411yescharacteristic
131#11719949—28no—
132#11719949—28no—
133#11719949—28no—
134#11719949—28no—
135C2×D212—24no—
136C2×D212—24no—
137C2×D212—24no—
138C2×D212—21no—
139C2×D212—21no—
140C2×D212—21no—
141C2×D212C22×C412yes—
142C2×D212#4112yes—
143C2×D212#4112yes—
144C2×C4×D53—24no—
145C2×C4×D53—24no—
146C2×C4×D53—22no—
147C2×C4×D53—22no—
148C2×C4×D53—21no—
149C2×C4×D53—21no—
150C2×C4×D53—21no—
151C2×C4×D53—21no—
152C2×C4×D53—24no—
153C2×C4×D53—24no—
154C2×C4×D53—24no—
155C2×C4×D53—41no—
156C2×C4×D53—41no—
157C2×C4×D53—42no—
158C2×C4×D53—42no—
159C2×C4×D53—42no—
160C2×C4×D53—42no—
161C2×C4×D53C22⋊C414yes—
162C2×C4×D53C22⋊C414yes—
163C2×C4×D53C22×C412yes—
164C2×C4×D53C2×D412yes—
165C2×C8×C53—22no—
166C2×C8×C53—22no—
167C2×C8×C53—22no—
168C4⋊C4×C53—24no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—22no—
171C4⋊C4×C53—22no—
172C4⋊C4×C53—42no—
173C4⋊C4×C53—42no—
174C4⋊C4×C53C22×C412yes—
175C42×C53—24no—
176C42×C53C22×C412yes—
177C42×C53C2×D412yes—
178#11719921—24no—
179#11719921—24no—
180#11719921—24no—
181#11719921—44no—
182C4⋊Dic53—21no—
183C4⋊Dic53—24no—
184C4⋊Dic53—22no—
185C4⋊Dic53—22no—
186C4⋊Dic53—42no—
187C4⋊Dic53—42no—
188C4⋊Dic53C22×C412yes—
189C4×Dic53—24no—
190C4×Dic53—24no—
191C4×Dic53—44no—
192C4×Dic53C22⋊C414yes—
193C4×Dic53C22⋊C414yes—
194C4×Dic53C22×C412yes—
195C4×Dic53C2×D412yes—
196C2×#88322—22no—
197C2×#88322—22no—
198C2×#88322—22no—
199C23×C53C22×D411yescharacteristic
200C22×D53—22no—
201C22×D53—21no—
202C22×D53—21no—
203C22×D53—22no—
204C22×D53—22no—
205C22×D53—21no—
206C22×D53—21no—
207C22×D53—21no—
208C22×D53—21no—
209C22×D53—41no—
210C22×D53—41no—
211C22×D53—41no—
212C22×D53—41no—
213C22×D53C2×C22⋊C412yes—
214Q8×C53—22no—
215Q8×C53—22no—
216Q8×C53—22no—
217Q8×C53#11714yescomplemented
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—22no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53C2×C22⋊C412yes—
230C2×C4×C53C2×C22⋊C412yes—
231C2×C4×C53AGL(1,ℤ/8)12yes—
232C2×C4×C53#13712yes—
233C2×C4×C53C23×C411yescharacteristic
234C2×C4×C53C22×D411yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—22no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53C2×C22⋊C412yes—
252C2×Dic53C2×C22⋊C412yes—
253C2×Dic53C2×C22⋊C412yes—
254D212—216no—
255D212—22no—
256D212—22no—
257D212—22no—
258D212#11714yescomplemented
259C4×D53—216no—
260C4×D53—22no—
261C4×D53—22no—
262C4×D53—22no—
263C4×D53—22no—
264C4×D53—28no—
265C4×D53—44no—
266C4×D53—48no—
267C4×D53—48no—
268C8×C53—42no—
269#88322—42no—
270#20743—531nomaximal, Sylow
271C22×C53C22×C22⋊C411yescharacteristic
272C22×C53#52511yescharacteristic
273C22×C53#52511yescharacteristic
274C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C53C2×#13711yescharacteristic
276C22×C53#57511yescharacteristic
277C22×C53#57511yescharacteristic
278C2×D53—22no—
279C2×D53—22no—
280C2×D53—22no—
281C2×D53—24no—
282C2×D53—41no—
283C2×D53—41no—
284C2×D53—42no—
285C2×D53—42no—
286C2×D53—41no—
287C2×D53—41no—
288C2×D53—42no—
289C2×D53—42no—
290C2×D53—42no—
291C2×D53—42no—
292C4×C53—22no—
293C4×C53—24no—
294C4×C53—42no—
295C4×C53—82no—
296C4×C53#41714yes—
297C4×C53C2×#11711yescharacteristic
298C4×C53C2×#11711yescharacteristic
299C4×C53#52511yescharacteristic
300C4×C53#52511yescharacteristic
301Dic53—22no—
302Dic53—22no—
303Dic53—22no—
304Dic53—42no—
305Dic53—82no—
306C2×C4×#41—531no—
307C2×#432—531no—
308#2285—531no—
309#1825—532no—
310#1803—531no—
311#1803D5311yescharacteristic, complemented
312#1802—531no—
313#1802—531no—
314#1788—532no—
315#1763—532no—
316#1762—532no—
317C2×C53#228011yescharacteristic
318C2×C53C2×#41711yescharacteristic
319C2×C53#289611yescharacteristic
320C2×C53#311511yescharacteristic
321C2×C53#311511yescharacteristic
322C2×C53#311611yescharacteristic
323C2×C53#311611yescharacteristic
324D53—42no—
325D53—44no—
326C22×#41—531no—
327C4×#41—538no—
328C4×#41—1064no—
329C2×C4×Q8—531no—
330C2×C4×Q8C2×D5311yescharacteristic
331C2×C4×D4—531no—
332C2×C4×D4—531no—
333C2×C4×D4—1061no—
334C2×C4×D4—1061no—
335C2×#117—531no—
336C2×#117—1061no—
337C2×#117—1061no—
338C22×C42—531no—
339C22×C42—1061no—
340#432—1064no—
341C2×C4⋊C8—531no—
342C2×C4⋊C8C2×D5311yescharacteristic
343C2×#103—532no—
344C2×#103C2×D5312yes—
345C2×#102—532no—
346C2×#102—532no—
347C2×C22⋊C8—532no—
348C2×#97—532no—
349#384—531no—
350#384C2×D5311yescharacteristic
351#382—1061no—
352#380—532no—
353#380—1061no—
354C4×C4⋊C4—532no—
355C4×C4⋊C4—1061no—
356#340—532no—
357#340—532no—
358#340—532no—
359#340C2×D5312yes—
360C53#2074311yescharacteristic, complemented, Sylow
361C2×#41—534no—
362C2×#41—1062no—
363C2×#41—1062no—
364C2×#41—1062no—
365C2×#41—1062no—
366C2×#41—1062no—
367C22×Q8C22×D5311yescharacteristic
368C22×D4—531no—
369C22×D4—1061no—
370C23×C4—531no—
371C23×C4—531no—
372C23×C4—531no—
373C23×C4—1061no—
374C23×C4—1061no—
375C23×C4—1061no—
376C22×C8—532no—
377C22×C8C22×D5312yes—
378C4×Q8—22no—
379C4×Q8—534no—
380C4×Q8—1062no—
381C4×Q8C4×D5314yes—
382C4×D4—534no—
383C4×D4—534no—
384C4×D4—1068no—
385C4×D4—1062no—
386C4×D4—1068no—
387C4×D4—1062no—
388#117—534no—
389#117—1068no—
390#117—1062no—
391#117—2124no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—532no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4C22×D5312yes—
401C2×C4⋊C4C22×D5311yescharacteristic
402C2×C22⋊C4—1061no—
403C2×C22⋊C4—1061no—
404C2×C22⋊C4—1061no—
405C2×C22⋊C4—1062no—
406C2×C42—21no—
407C2×C42—531no—
408C2×C42—531no—
409C2×C42—531no—
410C2×C42—534no—
411C2×C42—534no—
412C2×C42—1068no—
413C2×C42—1061no—
414C2×C42—1061no—
415C2×C42—1062no—
416C2×C42C22×D5311yescharacteristic
417C4⋊C8—22no—
418C4⋊C8—1062no—
419#103—24no—
420#103—1064no—
421#102—1064no—
422#102—1064no—
423C22⋊C8—1064no—
424#97—1064no—
425#95—1062no—
426#95—1062no—
427C24—531no—
428C24—1061no—
429#41—1068no—
430#41—1068no—
431#41—1068no—
432#41—1068no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8C2×C4×D5312yes—
437C2×Q8#1171994712yes—
438C2×Q8#1171994912yes—
439C2×D4—532no—
440C2×D4—532no—
441C2×D4—532no—
442C2×D4—1064no—
443C2×D4—1064no—
444C2×D4—1064no—
445C2×D4—1061no—
446C2×D4—1061no—
447C2×D4—1061no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—531no—
452C22×C4—531no—
453C22×C4—531no—
454C22×C4—534no—
455C22×C4—532no—
456C22×C4—532no—
457C22×C4—534no—
458C22×C4—1064no—
459C22×C4—1064no—
460C22×C4—1064no—
461C22×C4—1064no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1064no—
470C22×C4—1062no—
471C22×C4—2121no—
472C22×C4—2121no—
473C22×C4—2122no—
474C22×C4—2122no—
475C22×C4—2122no—
476C22×C4—2122no—
477C22×C4D4×D5311yescharacteristic
478C22×C4D4×D5311yescharacteristic
479C22×C4C23×D5311yescharacteristic
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—1062no—
484C2×C8—1062no—
485C2×C8—1062no—
486C4⋊C4—24no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—532no—
493C4⋊C4—1064no—
494C4⋊C4—1061no—
495C4⋊C4—1062no—
496C4⋊C4—1062no—
497C4⋊C4—2122no—
498C4⋊C4—2122no—
499C4⋊C4C2×C4×D5312yes—
500C22⋊C4—1064no—
501C22⋊C4—1064no—
502C22⋊C4—1064no—
503C22⋊C4—2124no—
504C42—24no—
505C42—534no—
506C42—534no—
507C42—532no—
508C42—532no—
509C42—1064no—
510C42—1064no—
511C42—2124no—
512C42C2×C4×D5312yes—
513C42D4×D5312yes—
514C23—532no—
515C23—1062no—
516C23—1062no—
517C23—1061no—
518C23—1062no—
519C23—1061no—
520C23—1061no—
521C23—1061no—
522C23—1061no—
523C23—1061no—
524C23—2121no—
525C23—2121no—
526C23—2121no—
527C23—2121no—
528C23C2×D4×D5311yescharacteristic
529Q8—22no—
530Q8—22no—
531Q8—22no—
532Q8#1313317514yescomplemented
533D4—534no—
534D4—10616no—
535D4—1062no—
536D4—1062no—
537D4—1062no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—532no—
550C2×C4—532no—
551C2×C4—532no—
552C2×C4—10616no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—1061no—
558C2×C4—1061no—
559C2×C4—1062no—
560C2×C4—1062no—
561C2×C4—1062no—
562C2×C4—1068no—
563C2×C4—1062no—
564C2×C4—1062no—
565C2×C4—2124no—
566C2×C4—2128no—
567C2×C4—2122no—
568C2×C4—2122no—
569C2×C4—2122no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2128no—
573C2×C4C22⋊C4×D5312yes—
574C2×C4C22⋊C4×D5312yes—
575C2×C4#1313324412yes—
576C2×C4#1313324912yes—
577C2×C4C22×C4×D5311yescharacteristic
578C2×C4C2×D4×D5311yescharacteristic
579C2×C4#41×D5311yescharacteristic
580C2×C4#41×D5311yescharacteristic
581C2×C4C2×#1171994711yescharacteristic
582C2×C4C2×#1171994911yescharacteristic
583C8—42no—
584C8—2122no—
585C22—1062no—
586C22—1062no—
587C22—1064no—
588C22—1062no—
589C22—2122no—
590C22—2122no—
591C22—2121no—
592C22—2122no—
593C22—2121no—
594C22—2122no—
595C22—2121no—
596C22—2122no—
597C22—2121no—
598C22—2122no—
599C22C2×C22⋊C4×D5311yescharacteristic
600C22#1681113311yescharacteristic
601C22#1681113611yescharacteristic
602C22C2×#1313324411yescharacteristic
603C22C2×#1313324911yescharacteristic
604C22#1681141911yescharacteristic
605C22#1681142211yescharacteristic
606C4—22no—
607C4—24no—
608C4—42no—
609C4—82no—
610C4—1062no—
611C4—1062no—
612C4—1062no—
613C4—4242no—
614C4—2122no—
615C4#1681062314yes—
616C4#117×D5311yescharacteristic
617C4C2×#1313317511yescharacteristic
618C4#1681113311yescharacteristic
619C4#1681113611yescharacteristic
620C2—2122no—
621C2—2124no—
622C2#2027709711yescharacteristic
623C2C2×#1681062311yescharacteristic
624C2#2028323111yescharacteristic
625C2#2028486511yescharacteristic
626C2#2028486911yescharacteristic
627C2#2028487611yescharacteristic
628C2#2028488011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1803C2
socleC23×C53C22×D4
prime radicalC22×#11719949C22
cosocleC22×C4×C53C24
hypercenter#1803D53
hypocenterC53#20743
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. C2×#16810679C2semidirect
3. #20277120C2semidirect
4. #20274808C2semidirect
5. #20274720C2semidirect
6. #20274716C2semidirect
7. #20274712C2semidirect
9. #20274547C2semidirect
10. #20274539C2semidirect
11. C53×#1803C2semidirect
14. C4×#11719949C4semidirect
217. Q8×C53#117semidirect
258. D212#117semidirect
311. #1803D53semidirect
360. C53#20743semidirect
532. Q8#13133175semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117199492
C2×#168106792
#202771202
#202748082
#202747202
#202747162
#202747122
#202746542
#202745472
#202745392
C53×#18032
#2074353