Finite Group Databasebeta

← #27413659

Subgroups — #27413659

Order 13568.

874 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810731C211yesmaximal, characteristic, complemented
2C4×#13133196C211yesmaximal, characteristic
3C4×#13133196C211yesmaximal, characteristic
4#20276213C211yesmaximal, characteristic, complemented
5#20276148C211yesmaximal, characteristic
6#20276109C211yesmaximal, characteristic, complemented
7#20276095C211yesmaximal, characteristic, complemented
8#20276038C211yesmaximal, characteristic, complemented
9#20275919C211yesmaximal, characteristic
10#20275912C211yesmaximal, characteristic
11#20275725C211yesmaximal, characteristic
12#20274612C211yesmaximal, characteristic, complemented
13#20274546C211yesmaximal, characteristic, complemented
14#20274546C211yesmaximal, characteristic, complemented
15C2×C53×#432C211yesmaximal, characteristic
16C22×Dic212C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×C4×Dic106C2211yescharacteristic
20C2×C4×Dic106C2211yescharacteristic
21#16810731—24no—
22C2×#13133151C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133149C2211yescharacteristic
25C2×#13133142C2211yescharacteristic
26C2×#13133142C2211yescharacteristic
27C2×#13133142C2211yescharacteristic
28C2×#13133142C2211yescharacteristic
29C2×#13133142C2211yescharacteristic
30C2×#13133142C2211yescharacteristic
31#16810529C2211yescharacteristic
32#16810529C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810521C2211yescharacteristic
35#16810507—21no—
36#16810507—21no—
37#16810496—21no—
38#16810496—21no—
39#16810483C2211yescharacteristic
40#16810483C2211yescharacteristic
41C4×#11719919—21no—
42C4×#11719919—21no—
43C4×C4⋊Dic53C2211yescharacteristic
44C4×C4⋊Dic53C2211yescharacteristic
45C4×C22⋊Dic53—21no—
46C4×C22⋊Dic53—21no—
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53C22×C42×C53C2211yescharacteristic
54C53×#432—24no—
55C2×C4⋊C8×C53C2211yescharacteristic
56C2×C4⋊C8×C53C2211yescharacteristic
57C2×C22⋊C8×C53C2211yescharacteristic
58C2×C22⋊C8×C53C2211yescharacteristic
59C2×#97×C53C2211yescharacteristic
60C2×#97×C53C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62C22×Dic106C2311yescharacteristic
63C2×Dic212—22no—
64C2×Dic212—22no—
65C2×Dic212—22no—
66#13133196—22no—
67#13133196—22no—
68#13133196—22no—
69#13133196—22no—
70C4×Dic106—22no—
71C4×Dic106—22no—
72C4×Dic106—22no—
73C4×Dic106—22no—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53—21no—
89C2×C22⋊Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53—21no—
94#13133151—22no—
95#13133149—22no—
96#13133149—22no—
97#13133142—22no—
98#13133142—22no—
99#13133142—22no—
100#13133142—22no—
101#13133142—22no—
102#13133142—22no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112#13133124—21no—
113C23×C4×C53D411yescharacteristic
114C23×C4×C53D411yescharacteristic
115C23×C4×C53C2311yescharacteristic
116C22×C8×C53C2311yescharacteristic
117C22×C8×C53C2311yescharacteristic
118C22×C8×C53C2311yescharacteristic
119C22×C8×C53C2311yescharacteristic
120C2×C42×C53—22no—
121C2×C42×C53—22no—
122C2×C42×C53D411yescharacteristic
123C2×C42×C53D411yescharacteristic
124C2×C42×C53C2311yescharacteristic
125C2×C42×C53C2311yescharacteristic
126C4⋊C8×C53—22no—
127C4⋊C8×C53—22no—
128C22⋊C8×C53—22no—
129C22⋊C8×C53—22no—
130#97×C53—22no—
131#97×C53D414yes—
132C24×C53C2×D411yescharacteristic
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—41no—
144C22×C4×C53—41no—
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2411yescharacteristic
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106—21no—
162C2×Dic106—21no—
163C2×Dic106—21no—
164C2×Dic106—21no—
165C2×Dic106—21no—
166C2×Dic106—21no—
167C2×Dic106—21no—
168C2×Dic106—21no—
169C2×Dic106#4112yes—
170C2×Dic106#4112yes—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53—21no—
176C2×C8×C53—21no—
177C2×C8×C53—21no—
178C2×C8×C53—21no—
179C2×C8×C53—21no—
180C2×C8×C53—21no—
181C2×C8×C53C2×D412yes—
182C2×C8×C53C2×D412yes—
183C42×C53—22no—
184C42×C53—22no—
185C42×C53—21no—
186C42×C53—21no—
187C42×C53C2×D412yes—
188C42×C53#4112yes—
189C22⋊Dic53—24no—
190C22⋊Dic53—24no—
191C22⋊Dic53—42no—
192C22⋊Dic53—42no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53#4112yes—
204C4⋊Dic53#4112yes—
205#11719919—24no—
206#11719919—24no—
207#11719919—42no—
208#11719919—42no—
209#11719919—42no—
210#11719919—42no—
211C4×Dic53—42no—
212C4×Dic53—42no—
213C4×Dic53—42no—
214C4×Dic53—42no—
215Dic212—28no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53#13712yes—
223C23×C53C22×D411yescharacteristic
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—44no—
237C2×C4×C53—44no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53#12112yes—
243C2×C4×C53#12112yes—
244C2×C4×C53#13712yes—
245C2×C4×C53C22×D411yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—22no—
254C2×Dic53—22no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275Dic106—22no—
276Dic106—22no—
277Dic106—22no—
278Dic106—22no—
279Dic106—41no—
280Dic106—41no—
281Dic106—41no—
282Dic106—41no—
283C8×C53—22no—
284C8×C53—22no—
285C8×C53—41no—
286C8×C53—41no—
287#23709—531nomaximal, Sylow
288C22×C53—22no—
289C22×C53—22no—
290C22×C53—22no—
291C22×C53—41no—
292C22×C53—41no—
293C22×C53—41no—
294C22×C53—41no—
295C22×C53C2×#12111yescharacteristic
296C22×C53#52511yescharacteristic
297C22×C53#52511yescharacteristic
298C22×C53C2×#13711yescharacteristic
299C22×C53C2×#13711yescharacteristic
300C22×C53#57511yescharacteristic
301C22×C53#57511yescharacteristic
302C4×C53—22no—
303C4×C53—22no—
304C4×C53—22no—
305C4×C53—41no—
306C4×C53—41no—
307C4×C53#52511yescharacteristic
308C4×C53#52911yescharacteristic
309C4×C53#52911yescharacteristic
310C4×C53#53211yescharacteristic
311Dic53—42no—
312Dic53—42no—
313Dic53—42no—
314Dic53—42no—
315Dic53—81no—
316Dic53—81no—
317Dic53—81no—
318Dic53—81no—
319C2×#441—531no—
320C2×#432D5311yescharacteristic
321C4×C22⋊Q8—531no—
322C4×C22⋊Q8—531no—
323#2088—531no—
324#2077—531no—
325#2068—531no—
326#2066—531no—
327#2056—531no—
328#2036—531no—
329#2035—531no—
330#2001—531no—
331#1776—531no—
332#1763—531no—
333#1763—531no—
334C2×C53—42no—
335C2×C53#239211yescharacteristic
336C2×C53#305611yescharacteristic
337C2×C53#305611yescharacteristic
338C2×C53#311111yescharacteristic
339C2×C53#311311yescharacteristic
340C2×C53#311611yescharacteristic
341C2×C53#311711yescharacteristic
342C22×Q16—531no—
343C2×C22⋊Q8—531no—
344C2×C22⋊Q8—531no—
345C2×C4×Q8—531no—
346C2×C4×Q8—531no—
347C22×C42C2×D5311yescharacteristic
348#441—1064no—
349#432—24no—
350C2×#107—531no—
351C2×#106—531no—
352C2×#106—531no—
353C2×C4⋊C8C2×D5311yescharacteristic
354C2×C4⋊C8C2×D5311yescharacteristic
355C2×#103—531no—
356C2×#103—531no—
357C2×#103—531no—
358C2×#103—531no—
359C2×#103—531no—
360C2×#103—531no—
361C2×C22⋊C8C2×D5311yescharacteristic
362C2×C22⋊C8C2×D5311yescharacteristic
363C2×#97C2×D5311yescharacteristic
364C2×#97C2×D5311yescharacteristic
365#391—531no—
366#391—531no—
367#389—531no—
368#389—531no—
369#387—1061no—
370#387—1061no—
371#385—1061no—
372#385—1061no—
373#380—531no—
374#380—531no—
375C4×C4⋊C4—531no—
376C4×C4⋊C4—531no—
377C4×C4⋊C4—1061no—
378C4×C4⋊C4—1061no—
379C4×C22⋊C4—1061no—
380C4×C22⋊C4—1061no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387C53#2370911yescharacteristic, complemented, Sylow
388C22×Q8—531no—
389C22×Q8—531no—
390C23×C4D21211yescharacteristic
391C23×C4D21211yescharacteristic
392C23×C4C22×D5311yescharacteristic
393C2×Q16—1062no—
394C2×Q16—1062no—
395C2×Q16—1062no—
396C22×C8C22×D5311yescharacteristic
397C22×C8C22×D5311yescharacteristic
398C22×C8C22×D5311yescharacteristic
399C22×C8C22×D5311yescharacteristic
400C22⋊Q8—1062no—
401C22⋊Q8—1062no—
402C22⋊Q8—1062no—
403C22⋊Q8—1062no—
404C4×Q8—1062no—
405C4×Q8—1062no—
406C4×Q8—1062no—
407C4×Q8—1062no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—1061no—
420C2×C22⋊C4—1061no—
421C2×C22⋊C4—1061no—
422C2×C22⋊C4—1061no—
423C2×C22⋊C4—1061no—
424C2×C42—22no—
425C2×C42—22no—
426C2×C42—1061no—
427C2×C42—1061no—
428C2×C42—1061no—
429C2×C42—1061no—
430C2×C42D21211yescharacteristic
431C2×C42D21211yescharacteristic
432C2×C42C22×D5311yescharacteristic
433C2×C42C22×D5311yescharacteristic
434#107—1062no—
435#106—1062no—
436#106—1062no—
437C4⋊C8—22no—
438C4⋊C8—22no—
439#103—1062no—
440#103—1062no—
441#103—1062no—
442#103—1062no—
443#103—1062no—
444#103—1062no—
445C22⋊C8—22no—
446C22⋊C8—22no—
447#97—22no—
448#97C22⋊D5314yes—
449#95—1061no—
450#95—1061no—
451#95—1061no—
452#95—1061no—
453#95—1061no—
454#95—1061no—
455#95—1061no—
456#95—1061no—
457#95—1061no—
458#95—1061no—
459C24C2×D21211yescharacteristic
460C2×Q8—532no—
461C2×Q8—532no—
462C2×Q8—1061no—
463C2×Q8—1061no—
464C2×Q8—1061no—
465C2×Q8—1061no—
466C2×Q8—1061no—
467C2×Q8—1061no—
468C2×Q8—1061no—
469C2×Q8—1061no—
470C2×Q8—1061no—
471C2×Q8—1061no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—1061no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4—1061no—
488C22×C4—1061no—
489C22×C4—1061no—
490C22×C4—1061no—
491C22×C4—1061no—
492C22×C4C2×D21211yescharacteristic
493C22×C4C2×D21211yescharacteristic
494C22×C4C2×D21211yescharacteristic
495C22×C4C2×D21211yescharacteristic
496C22×C4C2×D21211yescharacteristic
497C22×C4C23×D5311yescharacteristic
498Q16—1068no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8C2×C22⋊D5312yes—
510C2×C8C2×C22⋊D5312yes—
511C4⋊C4—532no—
512C4⋊C4—532no—
513C4⋊C4—1064no—
514C4⋊C4—1064no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—1061no—
524C4⋊C4—1061no—
525C4⋊C4—2122no—
526C4⋊C4—2122no—
527C4⋊C4—2122no—
528C4⋊C4—2122no—
529C22⋊C4—1064no—
530C22⋊C4—1064no—
531C22⋊C4—2122no—
532C22⋊C4—2122no—
533C42—22no—
534C42—22no—
535C42—21no—
536C42—21no—
537C42—2122no—
538C42—2122no—
539C42—2122no—
540C42—2122no—
541C42#1171994512yes—
542C42C2×C22⋊D5312yes—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23#1313325012yes—
550C23C22×D21211yescharacteristic
551Q8—1062no—
552Q8—1062no—
553Q8—1062no—
554Q8—1062no—
555Q8—2121no—
556Q8—2121no—
557Q8—2121no—
558Q8—2121no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—1062no—
578C2×C4—1062no—
579C2×C4—1062no—
580C2×C4—1062no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4#1313319112yes—
602C2×C4#1313319112yes—
603C2×C4#1313325012yes—
604C2×C4C22×C22⋊D5311yescharacteristic
605C2×C4#41×D5311yescharacteristic
606C2×C4#41×D5311yescharacteristic
607C2×C4#41×D5311yescharacteristic
608C2×C4#41×D5311yescharacteristic
609C2×C4C2×#1171994511yescharacteristic
610C8—22no—
611C8—22no—
612C8—41no—
613C8—41no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22C2×#1313319111yescharacteristic
622C22#1681113511yescharacteristic
623C22#1681113511yescharacteristic
624C22C2×#1313325011yescharacteristic
625C22C2×#1313325011yescharacteristic
626C22#1681142311yescharacteristic
627C22#1681142311yescharacteristic
628C4—22no—
629C4—22no—
630C4—22no—
631C4—41no—
632C4—41no—
633C4—4241no—
634C4—4241no—
635C4—4241no—
636C4—4241no—
637C4—2122no—
638C4—2122no—
639C4—2122no—
640C4—2122no—
641C4#1681113011yescharacteristic
642C4#1681115111yescharacteristic
643C4#1681115111yescharacteristic
644C4#1681116911yescharacteristic
645C2—42no—
646C2#2027795111yescharacteristic
647C2#2028434511yescharacteristic
648C2#2028434511yescharacteristic
649C2#2028482611yescharacteristic
650C2#2028484611yescharacteristic
651C2#2028487811yescharacteristic
652C2#2028488811yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#432C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenterC2×#432D53
hypocenterC53#23709
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#432

Complements

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

NH ≅ G/Ntype
1. C2×#16810731C2semidirect
4. #20276213C2semidirect
6. #20276109C2semidirect
7. #20276095C2semidirect
8. #20276038C2semidirect
12. #20274612C2semidirect
13. #20274546C2semidirect
14. #20274546C2semidirect
387. C53#23709semidirect

Maximal subgroups

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

subgroupindex
C2×#168107312
C4×#131331962
#202762132
#202761482
#202761092
#202760952
#202760382
#202759192
#202759122
#202757252
#202746122
#202745462
C2×C53×#4322
#2370953