Finite Group Databasebeta

← #27391335

Subgroups — #27391335

Order 13568.

876 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×Dic53C211yesmaximal, characteristic
2#117×Dic53C211yesmaximal, characteristic
3#20275358C211yesmaximal, characteristic, complemented
4#20275351C211yesmaximal, characteristic, complemented
5#20275309C211yesmaximal, characteristic, complemented
6#20275302C211yesmaximal, characteristic, complemented
7#20275028C211yesmaximal, characteristic
8#20274916C211yesmaximal, characteristic
9#20274810C211yesmaximal, characteristic
10#20274615C211yesmaximal, characteristic, complemented
11#20274614C211yesmaximal, characteristic, complemented
12#20274547C211yesmaximal, characteristic, complemented
13#20274547C211yesmaximal, characteristic, complemented
14#20274479C211yesmaximal, characteristic
15C2×C53×#485C211yesmaximal, characteristic
16C2×Q8×Dic53C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18C22×C4×Dic53C2211yescharacteristic
19C2×#13133151C2211yescharacteristic
20C2×#13133149C2211yescharacteristic
21C2×#13133141C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×C22⋊#88322C2211yescharacteristic
24C2×#13133127C2211yescharacteristic
25C2×#13133127C2211yescharacteristic
26#16810530C2211yescharacteristic
27#16810522C2211yescharacteristic
28#16810509—21no—
29#16810498—21no—
30#16810483C2211yescharacteristic
31C4⋊C4×Dic53—21no—
32C4⋊C4×Dic53C2211yescharacteristic
33C4⋊C4×Dic53C2211yescharacteristic
34C4⋊C4×Dic53C2211yescharacteristic
35C4⋊C4×Dic53C2211yescharacteristic
36C4⋊C4×Dic53C2211yescharacteristic
37C22⋊C4×Dic53—21no—
38C22⋊C4×Dic53—21no—
39C22⋊C4×Dic53—21no—
40#16810469—21no—
41C42×Dic53—21no—
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×C22⋊Q8×C53C2211yescharacteristic
53C2×#117×C53C2211yescharacteristic
54C53×#485C418yescomplemented
55C2×#107×C53C2211yescharacteristic
56C2×#106×C53C2211yescharacteristic
57C2×#103×C53C2211yescharacteristic
58C2×#103×C53C2211yescharacteristic
59C2×C22⋊C8×C53C2211yescharacteristic
60C23×Dic53D411yescharacteristic
61C23×Dic53D411yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64Q8×Dic53—22no—
65Q8×Dic53—22no—
66#13133176—22no—
67#13133176—22no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C22⋊Dic53—21no—
75C2×C22⋊Dic53—21no—
76C2×C4×Dic53—22no—
77C2×C4×Dic53—22no—
78C2×C4×Dic53—21no—
79C2×C4×Dic53—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53D411yescharacteristic
86C2×C4×Dic53D411yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88C2×C4×Dic53C2311yescharacteristic
89#13133151—22no—
90#13133149—22no—
91#13133141—22no—
92#13133141—22no—
93C22⋊#88322—22no—
94#13133127—22no—
95#13133127—22no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105C22×Q8×C53C2311yescharacteristic
106C23×C4×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C22⋊Q8×C53—22no—
110C22⋊Q8×C53C2×C414yes—
111#117×C53—22no—
112#117×C53C2×C414yes—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53—21no—
119C2×C42×C53—21no—
120#107×C53C2×C414yescomplemented
121#106×C53C2×C414yescomplemented
122#103×C53C2×C414yescomplemented
123#103×C53C2×C414yescomplemented
124C22⋊C8×C53C2×C414yes—
125C24×C53C2×D411yescharacteristic
126C2×Q8×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53C22×C412yes—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53C22×C412yes—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53#4112yes—
143C22×C4×C53C2411yescharacteristic
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—41no—
152C22×Dic53—41no—
153C22×Dic53—41no—
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53C2×D411yescharacteristic
156C22×Dic53C2×D411yescharacteristic
157C22×Dic53C2×D411yescharacteristic
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53C22×C412yes—
163C2×C8×C53C22×C412yes—
164C4⋊C4×C53—24no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53C22×C412yes—
169C4⋊C4×C53C22×C412yes—
170C4⋊C4×C53C22×C412yes—
171C22⋊C4×C53—24no—
172C22⋊C4×C53—24no—
173C42×C53—24no—
174C22⋊Dic53—24no—
175C22⋊Dic53—42no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—42no—
183C4⋊Dic53—42no—
184C4⋊Dic53#4112yes—
185C4⋊Dic53#4112yes—
186C4×Dic53—24no—
187C4×Dic53—21no—
188C4×Dic53—21no—
189C4×Dic53—22no—
190C4×Dic53—22no—
191C4×Dic53—42no—
192C4×Dic53—42no—
193C4×Dic53—42no—
194C4×Dic53—42no—
195C4×Dic53—42no—
196C4×Dic53—42no—
197C4×Dic53#4112yes—
198C4×Dic53#4112yes—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53C4×D412yes—
212C23×C53C22×D411yescharacteristic
213Q8×C53—22no—
214Q8×C53—22no—
215Q8×C53—41no—
216Q8×C53—41no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53C4×D412yes—
236C2×C4×C53C23×C411yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—22no—
250C2×Dic53—22no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—44no—
254C2×Dic53—44no—
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#12312yes—
266C2×Dic53#12312yes—
267C2×Dic53#13712yes—
268C8×C53—22no—
269C8×C53—22no—
270#88322—41no—
271#88322—41no—
272#21583—531nomaximal, Sylow
273C22×C53—22no—
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53C2×C4×D411yescharacteristic
281C22×C53C2×#12311yescharacteristic
282C22×C53#52511yescharacteristic
283C22×C53C2×#13711yescharacteristic
284C22×C53#57511yescharacteristic
285C22×C53#57511yescharacteristic
286C22×C53#57511yescharacteristic
287C4×C53—22no—
288C4×C53—42no—
289C4×C53—42no—
290C4×C53—42no—
291C4×C53—42no—
292C4×C53C4×#4111yescharacteristic
293C4×C53C4×#4111yescharacteristic
294C4×C53#52911yescharacteristic
295C4×C53#52911yescharacteristic
296Dic53—22no—
297Dic53—22no—
298Dic53—22no—
299Dic53—41no—
300Dic53—42no—
301Dic53—42no—
302Dic53—81no—
303Dic53—81no—
304C2×#485D5311yescharacteristic
305C4×C22⋊Q8—531no—
306C4×#117—531no—
307#1928—531no—
308#1927—531no—
309#1921—531no—
310#1920—531no—
311#1873—531no—
312#1854—531no—
313#1825—531no—
314#1776—531no—
315#1776—531no—
316#1763—531no—
317#1763—531no—
318#1749—531no—
319C2×C53—42no—
320C2×C53C4×#12311yescharacteristic
321C2×C53#294311yescharacteristic
322C2×C53C4×#13711yescharacteristic
323C2×C53#309311yescharacteristic
324C2×C53#309511yescharacteristic
325C2×C53#311111yescharacteristic
326C2×C53#311611yescharacteristic
327C2×C22⋊Q8C2×D5311yescharacteristic
328C2×C4×Q8—531no—
329C2×#117—531no—
330C2×#117C2×D5311yescharacteristic
331C22×C42—531no—
332#485Dic5318yescomplemented
333C2×#107—531no—
334C2×#107C2×D5311yescharacteristic
335C2×#106—531no—
336C2×#106C2×D5311yescharacteristic
337C2×#103—531no—
338C2×#103—531no—
339C2×#103C2×D5311yescharacteristic
340C2×#103C2×D5311yescharacteristic
341C2×C22⋊C8—531no—
342C2×C22⋊C8C2×D5311yescharacteristic
343C2×#97—531no—
344C2×#97—531no—
345#391—531no—
346#389—531no—
347#387—1061no—
348#385—1061no—
349#380—531no—
350C4×C4⋊C4—531no—
351C4×C4⋊C4—531no—
352C4×C4⋊C4—531no—
353C4×C4⋊C4—531no—
354C4×C4⋊C4—531no—
355C4×C4⋊C4—1061no—
356C4×C22⋊C4—1061no—
357C4×C22⋊C4—1061no—
358C4×C22⋊C4—1061no—
359#376—1061no—
360C43—1061no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#336—531no—
370#336—531no—
371C53#2158311yescharacteristic, complemented, Sylow
372C22×Q8C22×D5311yescharacteristic
373C23×C4—531no—
374C23×C4—531no—
375C23×C4C22×D5311yescharacteristic
376C22×C8—531no—
377C22×C8—531no—
378C22×C8C22×D5311yescharacteristic
379C22×C8C22×D5311yescharacteristic
380C22⋊Q8—22no—
381C22⋊Q8C2×Dic5314yes—
382C4×Q8—1062no—
383C4×Q8—1062no—
384#117—22no—
385#117—1062no—
386#117—1062no—
387#117C2×Dic5314yes—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4C22×D5311yescharacteristic
396C2×C4⋊C4C22×D5311yescharacteristic
397C2×C4⋊C4C22×D5311yescharacteristic
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—1061no—
401C2×C22⋊C4—1061no—
402C2×C42—21no—
403C2×C42—531no—
404C2×C42—531no—
405C2×C42—531no—
406C2×C42—531no—
407C2×C42—1062no—
408C2×C42—1062no—
409C2×C42—1061no—
410C2×C42—1061no—
411C2×C42—1061no—
412C2×C42—1061no—
413C2×C42—1061no—
414C2×C42—1061no—
415C2×C42—1061no—
416#107—1062no—
417#107C2×Dic5314yescomplemented
418#106—1062no—
419#106C2×Dic5314yescomplemented
420#103—1062no—
421#103—1062no—
422#103C2×Dic5314yescomplemented
423#103C2×Dic5314yescomplemented
424C22⋊C8—1062no—
425C22⋊C8C2×Dic5314yes—
426#97—1062no—
427#97—1062no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437C24D4×D5311yescharacteristic
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8C22×Dic5312yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—531no—
453C22×C4—531no—
454C22×C4—531no—
455C22×C4—531no—
456C22×C4—1062no—
457C22×C4—1062no—
458C22×C4—1062no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—2121no—
464C22×C4—2121no—
465C22×C4—2121no—
466C22×C4D4×D5311yescharacteristic
467C22×C4#1171994712yes—
468C22×C4C22×Dic5312yes—
469C22×C4C23×D5311yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8C22×Dic5312yes—
481C2×C8C22×Dic5312yes—
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—532no—
487C4⋊C4—532no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—2122no—
495C4⋊C4—2122no—
496C4⋊C4C22×Dic5312yes—
497C4⋊C4C22×Dic5312yes—
498C4⋊C4C22×Dic5312yes—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—1064no—
502C22⋊C4—2122no—
503C42—24no—
504C42—532no—
505C42—532no—
506C42—1062no—
507C42—1062no—
508C42—1064no—
509C42—1061no—
510C42—1061no—
511C42—2122no—
512C42—2122no—
513C42—2122no—
514C42—2122no—
515C42—2122no—
516C42—2122no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23D4×Dic5312yes—
524C23C2×D4×D5311yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—41no—
528Q8—41no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—532no—
548C2×C4—532no—
549C2×C4—532no—
550C2×C4—1061no—
551C2×C4—1061no—
552C2×C4—1061no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—1062no—
558C2×C4—1062no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—2124no—
562C2×C4—2124no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4D4×Dic5312yes—
574C2×C4C23×Dic5311yescharacteristic
575C2×C4#41×D5311yescharacteristic
576C2×C4#41×D5311yescharacteristic
577C2×C4#41×D5311yescharacteristic
578C2×C4#41×D5311yescharacteristic
579C2×C4C2×#1171994711yescharacteristic
580C8—22no—
581C8—22no—
582C8—2121no—
583C8—2121no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22C2×D4×Dic5311yescharacteristic
592C22#123×D5311yescharacteristic
593C22#1681113111yescharacteristic
594C22#137×D5311yescharacteristic
595C22#1681142011yescharacteristic
596C22#1681142011yescharacteristic
597C22#1681142211yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—1062no—
604C4—1062no—
605C4—1062no—
606C4—4241no—
607C4—4241no—
608C4—2122no—
609C4—2122no—
610C4—2121no—
611C4#41×Dic5311yescharacteristic
612C4#41×Dic5311yescharacteristic
613C4#1681115011yescharacteristic
614C4#1681115011yescharacteristic
615C2—42no—
616C2#123×Dic5311yescharacteristic
617C2#2028351611yescharacteristic
618C2#137×Dic5311yescharacteristic
619C2#2028464711yescharacteristic
620C2#2028466211yescharacteristic
621C2#2028482711yescharacteristic
622C2#2028487211yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#485C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
cosocleC22×C4×C53C24
hypercenterC2×#485D53
hypocenterC53#21583
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#485

Complements

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

NH ≅ G/Ntype
3. #20275358C2semidirect
4. #20275351C2semidirect
5. #20275309C2semidirect
6. #20275302C2semidirect
10. #20274615C2semidirect
11. #20274614C2semidirect
12. #20274547C2semidirect
13. #20274547C2semidirect
54. C53×#485C4semidirect
120. #107×C53C2×C4semidirect
121. #106×C53C2×C4semidirect
122. #103×C53C2×C4semidirect
123. #103×C53C2×C4semidirect
332. #485Dic53semidirect
371. C53#21583semidirect
417. #107C2×Dic53semidirect
419. #106C2×Dic53semidirect
422. #103C2×Dic53semidirect
423. #103C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×Dic532
#117×Dic532
#202753582
#202753512
#202753092
#202753022
#202750282
#202749162
#202748102
#202746152
#202746142
#202745472
#202744792
C2×C53×#4852
#2158353