Finite Group Databasebeta

← #27384149

Subgroups — #27384149

Order 13568.

1350 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D53C211yesmaximal, characteristic
2C2×#16810679C211yesmaximal, characteristic
3#20277120C211yesmaximal, characteristic
4#20274800C212yesmaximal
5#20274721C211yesmaximal, characteristic, complemented
6#20274720C211yesmaximal, characteristic, complemented
7#20274717C211yesmaximal, characteristic, complemented
8#20274648C212yesmaximal
9#20274547C212yesmaximal, complemented
10#20274544C212yesmaximal, complemented
11C53×#1803C211yesmaximal, characteristic, complemented
12C22×Q8×D53C2211yescharacteristic
13C2×Q8×Dic53C2211yescharacteristic
14C2×C4×Dic106—21no—
15C2×C4×Dic106C2211yescharacteristic
16C2×#13133182—21no—
17C2×#13133182C2211yescharacteristic
18C4×Q8×D53—24no—
19C4×Q8×D53C418yescomplemented
20C2×#13133175—21no—
21C2×C4⋊C4×D53—21no—
22C2×C4⋊C4×D53C2211yescharacteristic
23C2×C42×D53—21no—
24C2×C42×D53C2211yescharacteristic
25#16810679—24no—
26C2×C4⋊#88322C2211yescharacteristic
27C2×#13133142C2212yes—
28C2×#13133141C2212yes—
29C2×#13133140C2212yes—
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—
46C22×Dic106—21no—
47C22×Dic106C2311yescharacteristic
48C2×Q8×D53—22no—
49C2×Q8×D53—22no—
50C2×Q8×D53—22no—
51C2×Q8×D53—22no—
52C2×Q8×D53—22no—
53C2×Q8×D53C2×C414yes—
54C22×C4×D53—21no—
55C22×C4×D53—21no—
56C22×C4×D53D411yescharacteristic
57C22×C4×D53D411yescharacteristic
58C22×C4×D53C2311yescharacteristic
59C22×#88322C2312yes—
60Q8×Dic53—22no—
61Q8×Dic53C2×C414yes—
62C4×Dic106—22no—
63C4×Dic106—28no—
64C4×Dic106C2×C414yes—
65#13133182—28no—
66#13133182—22no—
67#13133182C2×C414yes—
68#13133175—44no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53C2312yes—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C4⋊C4×D53—28no—
78C4⋊C4×D53—22no—
79C4⋊C4×D53C2×C414yes—
80C2×#11719921—22no—
81C2×C4×Dic53—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#13133142—24no—
93#13133141—24no—
94#13133140—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×D53C2×D411yescharacteristic
113C2×Q8×C53—21no—
114C2×Q8×C53—21no—
115C2×Q8×C53—21no—
116C2×Q8×C53C22×C412yes—
117C2×Q8×C53#4112yes—
118C2×Q8×C53#4112yes—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—22no—
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2411yescharacteristic
125C22×Dic53—21no—
126C22×Dic53—21no—
127C22×Dic53—22no—
128C22×Dic53C2×D411yescharacteristic
129C22×Dic53C2×D411yescharacteristic
130C22×Dic53C2×D411yescharacteristic
131Q8×D53—28no—
132Q8×D53—28no—
133Q8×D53—28no—
134Q8×D53—28no—
135C2×C4×D53—24no—
136C2×C4×D53—24no—
137C2×C4×D53—24no—
138C2×C4×D53—22no—
139C2×C4×D53—22no—
140C2×C4×D53—22no—
141C2×C4×D53—22no—
142C2×C4×D53—21no—
143C2×C4×D53—21no—
144C2×C4×D53—21no—
145C2×C4×D53—21no—
146C2×C4×D53—24no—
147C2×C4×D53—42no—
148C2×C4×D53—42no—
149C2×C4×D53C22⋊C414yes—
150C2×C4×D53C22⋊C414yes—
151C2×C4×D53C22×C412yes—
152C2×C4×D53C2×D412yes—
153C2×Dic106—24no—
154C2×Dic106—24no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—24no—
159C2×Dic106C22×C412yes—
160C2×Dic106#4112yes—
161C2×Dic106#4112yes—
162C2×C8×C53—22no—
163C2×C8×C53—22no—
164C2×C8×C53—22no—
165C4⋊C4×C53—24no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—22no—
168C4⋊C4×C53—22no—
169C4⋊C4×C53—42no—
170C4⋊C4×C53—42no—
171C4⋊C4×C53C22×C412yes—
172C42×C53—24no—
173C42×C53C22×C412yes—
174C42×C53C2×D412yes—
175#11719921—44no—
176C4⋊Dic53—21no—
177C4⋊Dic53—24no—
178C4⋊Dic53—22no—
179C4⋊Dic53—22no—
180C4⋊Dic53—42no—
181C4⋊Dic53—42no—
182C4⋊Dic53C22×C412yes—
183#11719919—24no—
184#11719919—24no—
185#11719919—24no—
186C4×Dic53—24no—
187C4×Dic53—24no—
188C4×Dic53—24no—
189C4×Dic53—44no—
190C4×Dic53C22⋊C414yes—
191C4×Dic53C22⋊C414yes—
192C4×Dic53C22×C412yes—
193C4×Dic53C2×D412yes—
194C2×#88322—22no—
195C2×#88322—22no—
196C2×#88322—22no—
197C23×C53C22×D411yescharacteristic
198C22×D53—21no—
199C22×D53—21no—
200C22×D53—21no—
201C22×D53—21no—
202C22×D53C2×C22⋊C412yes—
203C22×D53AGL(1,ℤ/8)12yes—
204C22×D53#13712yes—
205Q8×C53—22no—
206Q8×C53—22no—
207Q8×C53—22no—
208Q8×C53#11714yescomplemented
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—22no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53C2×C22⋊C412yes—
221C2×C4×C53C2×C22⋊C412yes—
222C2×C4×C53#13712yes—
223C2×C4×C53#13712yes—
224C2×C4×C53C23×C411yescharacteristic
225C2×C4×C53C22×D411yescharacteristic
226C2×C4×C53C2×#4111yescharacteristic
227C2×C4×C53C2×#4111yescharacteristic
228C2×C4×C53C2×#4111yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C2×Dic53—22no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—22no—
234C2×Dic53—22no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—22no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53C2×C22⋊C412yes—
246C2×Dic53C2×C22⋊C412yes—
247C2×Dic53C2×C22⋊C412yes—
248C4×D53—216no—
249C4×D53—28no—
250C4×D53—22no—
251C4×D53—22no—
252C4×D53—22no—
253C4×D53—22no—
254C4×D53—28no—
255C4×D53—48no—
256Dic106—216no—
257Dic106—22no—
258Dic106—22no—
259Dic106—22no—
260Dic106#11714yescomplemented
261C8×C53—42no—
262#88322—42no—
263#20740—531nomaximal, Sylow
264C22×C53C22×C22⋊C411yescharacteristic
265C22×C53#52511yescharacteristic
266C22×C53#52511yescharacteristic
267C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C53C2×#13711yescharacteristic
269C22×C53C2×#13711yescharacteristic
270C22×C53C2×#13711yescharacteristic
271C2×D53—22no—
272C2×D53—22no—
273C2×D53—22no—
274C2×D53—22no—
275C2×D53—22no—
276C2×D53—22no—
277C2×D53#41714yes—
278C4×C53—22no—
279C4×C53—24no—
280C4×C53—42no—
281C4×C53—82no—
282C4×C53#41914yes—
283C4×C53C2×#11711yescharacteristic
284C4×C53C2×#11711yescharacteristic
285C4×C53#52511yescharacteristic
286C4×C53#52511yescharacteristic
287Dic53—22no—
288Dic53—22no—
289Dic53—22no—
290Dic53—24no—
291Dic53—42no—
292Dic53—82no—
293C22×C4×Q8—531no—
294C2×#432—531no—
295#2285—531no—
296#1823—532no—
297#1803—531no—
298#1803—531no—
299#1803—531no—
300#1803D5311yescharacteristic, complemented
301#1787—532no—
302#1763—532no—
303#1763—532no—
304C2×C53#228011yescharacteristic
305C2×C53C2×#41911yescharacteristic
306C2×C53C2×#41711yescharacteristic
307C2×C53#311211yescharacteristic
308C2×C53#311211yescharacteristic
309C2×C53#311311yescharacteristic
310C2×C53#311311yescharacteristic
311D53—24no—
312C23×Q8—531no—
313C2×C4×Q8—531no—
314C2×C4×Q8—538no—
315C2×C4×Q8—531no—
316C2×C4×Q8—531no—
317C2×C4×Q8—1061no—
318C2×C4×Q8—1061no—
319C2×C4×Q8—1064no—
320C2×C4×Q8C2×D5311yescharacteristic
321C2×#117—1061no—
322C22×C4⋊C4—531no—
323C22×C4⋊C4—1061no—
324C22×C42—531no—
325C22×C42—1061no—
326#432—1064no—
327C2×C4⋊C8—531no—
328C2×C4⋊C8C2×D5311yescharacteristic
329C2×#103—532no—
330C2×#103—532no—
331C2×#103—532no—
332C2×#103C2×D5312yes—
333C2×C22⋊C8—532no—
334C2×#97—532no—
335#384—531no—
336#384C2×D5311yescharacteristic
337#382—1061no—
338#380—532no—
339#380—1061no—
340C4×C4⋊C4—532no—
341C4×C4⋊C4—1061no—
342#340—532no—
343#340—532no—
344#340—532no—
345#340C2×D5312yes—
346C53#2074011yescharacteristic, complemented, Sylow
347C22×Q8—534no—
348C22×Q8—531no—
349C22×Q8—1061no—
350C22×Q8—1062no—
351C22×Q8—1062no—
352C22×Q8—1062no—
353C22×Q8—1062no—
354C22×Q8—1062no—
355C22×Q8C22×D5311yescharacteristic
356C23×C4—531no—
357C23×C4—531no—
358C23×C4—531no—
359C23×C4—1061no—
360C23×C4—1061no—
361C22×C8—532no—
362C22×C8C22×D5312yes—
363C4×Q8—22no—
364C4×Q8—534no—
365C4×Q8—534no—
366C4×Q8—534no—
367C4×Q8—1068no—
368C4×Q8—1068no—
369C4×Q8—1062no—
370C4×Q8—1062no—
371C4×Q8—1062no—
372C4×Q8C4×D5314yes—
373#117—2124no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—534no—
379C2×C4⋊C4—532no—
380C2×C4⋊C4—1062no—
381C2×C4⋊C4—1068no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4C22×D5312yes—
389C2×C4⋊C4C22×D5311yescharacteristic
390C2×C22⋊C4—1062no—
391C2×C42—21no—
392C2×C42—531no—
393C2×C42—531no—
394C2×C42—531no—
395C2×C42—534no—
396C2×C42—534no—
397C2×C42—1068no—
398C2×C42—1061no—
399C2×C42—1061no—
400C2×C42—1061no—
401C2×C42—1062no—
402C2×C42C22×D5311yescharacteristic
403C4⋊C8—22no—
404C4⋊C8—1062no—
405#103—24no—
406#103—1064no—
407#103—1064no—
408#103—1064no—
409C22⋊C8—1064no—
410#97—1064no—
411#95—1062no—
412#95—1062no—
413C24—531no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—532no—
418C2×Q8—532no—
419C2×Q8—532no—
420C2×Q8—1064no—
421C2×Q8—1064no—
422C2×Q8—1064no—
423C2×Q8—1068no—
424C2×Q8—1068no—
425C2×Q8—1068no—
426C2×Q8—1068no—
427C2×Q8—1061no—
428C2×Q8—1061no—
429C2×Q8—1061no—
430C2×Q8C2×C4×D5312yes—
431C2×Q8#1171994712yes—
432C2×Q8#1171994912yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—531no—
437C22×C4—531no—
438C22×C4—531no—
439C22×C4—534no—
440C22×C4—534no—
441C22×C4—532no—
442C22×C4—532no—
443C22×C4—1064no—
444C22×C4—1062no—
445C22×C4—1062no—
446C22×C4—1064no—
447C22×C4—1064no—
448C22×C4—1064no—
449C22×C4—1061no—
450C22×C4—1062no—
451C22×C4—1062no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1062no—
458C22×C4—2122no—
459C22×C4—2122no—
460C22×C4D4×D5311yescharacteristic
461C22×C4D4×D5311yescharacteristic
462C22×C4C23×D5311yescharacteristic
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—1062no—
467C2×C8—1062no—
468C2×C8—1062no—
469C4⋊C4—24no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—532no—
476C4⋊C4—1064no—
477C4⋊C4—1064no—
478C4⋊C4—1064no—
479C4⋊C4—1064no—
480C4⋊C4—1061no—
481C4⋊C4—1062no—
482C4⋊C4—1062no—
483C4⋊C4—2122no—
484C4⋊C4—2122no—
485C4⋊C4C2×C4×D5312yes—
486C22⋊C4—2124no—
487C42—24no—
488C42—534no—
489C42—534no—
490C42—532no—
491C42—532no—
492C42—1064no—
493C42—1064no—
494C42—1064no—
495C42—2124no—
496C42C2×C4×D5312yes—
497C42D4×D5312yes—
498C23—532no—
499C23—532no—
500C23—532no—
501C23—1061no—
502C23—1061no—
503C23—1061no—
504C23—1061no—
505C23C2×D4×D5311yescharacteristic
506Q8—22no—
507Q8—22no—
508Q8—22no—
509Q8—534no—
510Q8—10616no—
511Q8—1062no—
512Q8—1062no—
513Q8—1062no—
514Q8#1313317514yescomplemented
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—532no—
527C2×C4—532no—
528C2×C4—532no—
529C2×C4—1068no—
530C2×C4—10616no—
531C2×C4—1062no—
532C2×C4—1061no—
533C2×C4—1062no—
534C2×C4—1061no—
535C2×C4—1062no—
536C2×C4—1061no—
537C2×C4—1061no—
538C2×C4—1061no—
539C2×C4—1061no—
540C2×C4—1062no—
541C2×C4—1062no—
542C2×C4—1062no—
543C2×C4—1062no—
544C2×C4—1068no—
545C2×C4—1062no—
546C2×C4—2128no—
547C2×C4—2122no—
548C2×C4—2122no—
549C2×C4—2122no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4C22⋊C4×D5312yes—
553C2×C4C22⋊C4×D5312yes—
554C2×C4#1313324612yes—
555C2×C4#1313324912yes—
556C2×C4C22×C4×D5311yescharacteristic
557C2×C4C2×D4×D5311yescharacteristic
558C2×C4#41×D5311yescharacteristic
559C2×C4#41×D5311yescharacteristic
560C2×C4C2×#1171994711yescharacteristic
561C2×C4C2×#1171994911yescharacteristic
562C8—42no—
563C8—2122no—
564C22—534no—
565C22—1062no—
566C22—1062no—
567C22—1062no—
568C22—1062no—
569C22—1062no—
570C22—1062no—
571C22C2×C22⋊C4×D5311yescharacteristic
572C22#1681113311yescharacteristic
573C22#1681113611yescharacteristic
574C22AGL(1,ℤ/8)×D5311yescharacteristic
575C22#137×D5311yescharacteristic
576C22C2×#1313324611yescharacteristic
577C22C2×#1313324911yescharacteristic
578C4—22no—
579C4—24no—
580C4—42no—
581C4—82no—
582C4—1062no—
583C4—1062no—
584C4—1062no—
585C4—1064no—
586C4—4242no—
587C4—2122no—
588C4#1681063014yes—
589C4#117×D5311yescharacteristic
590C4C2×#1313317511yescharacteristic
591C4#1681113311yescharacteristic
592C4#1681113611yescharacteristic
593C2—1064no—
594C2#2027709711yescharacteristic
595C2C2×#1681063011yescharacteristic
596C2#417×D5311yescharacteristic
597C2#2028483611yescharacteristic
598C2#2028483911yescharacteristic
599C2#2028484411yescharacteristic
600C2#2028484711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1803C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1803D53
hypocenterC53#20740
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
5. #20274721C2semidirect
6. #20274720C2semidirect
7. #20274717C2semidirect
9. #20274547C2semidirect
10. #20274544C2semidirect
11. C53×#1803C2semidirect
19. C4×Q8×D53C4semidirect
208. Q8×C53#117semidirect
260. Dic106#117semidirect
300. #1803D53semidirect
346. C53#20740semidirect
514. Q8#13133175semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D532
C2×#168106792
#202771202
#202748002
#202747212
#202747202
#202747172
#202746482
#202745472
#202745442
C53×#18032
#2074053