Finite Group Databasebeta

← #27393637

Subgroups — #27393637

Order 13568.

970 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811251C211yesmaximal, characteristic
2C2×#16810684C211yesmaximal, characteristic
3#20275687C211yesmaximal, characteristic, complemented
4#20275683C211yesmaximal, characteristic, complemented
5#20275679C211yesmaximal, characteristic, complemented
6#20275675C211yesmaximal, characteristic, complemented
7#20275672C211yesmaximal, characteristic, complemented
8#20275672C211yesmaximal, characteristic, complemented
9#20275667C211yesmaximal, characteristic, complemented
10#20275663C211yesmaximal, characteristic, complemented
11#20274807C211yesmaximal, characteristic
12#20274807C211yesmaximal, characteristic
13#20274803C211yesmaximal, characteristic
14#20274799C211yesmaximal, characteristic
15C53×#2314C211yesmaximal, characteristic
16#16811251—24no—
17#16811251C418yescomplemented
18C2×#13133217C2211yescharacteristic
19C2×#13133217C2211yescharacteristic
20C2×#13133216—21no—
21C2×#13133209C2211yescharacteristic
22C2×#13133209C2211yescharacteristic
23C2×#13133208—21no—
24C2×#13133194—21no—
25C2×#13133194—21no—
26C2×#13133176C2211yescharacteristic
27C22×C4⋊Dic53C2211yescharacteristic
28#16810684—24no—
29C2×C4⋊#88322C2211yescharacteristic
30C2×C4⋊#88322C2211yescharacteristic
31C2×C22⋊#88322C2211yescharacteristic
32C2×C22⋊#88322C2211yescharacteristic
33C2×#13133128C2211yescharacteristic
34C2×C4×#88322C2211yescharacteristic
35#16810385C2211yescharacteristic
36#16810385C2211yescharacteristic
37#16810385C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51C2×#117×C53C2211yescharacteristic
52C53×#388—21no—
53C53×#388—21no—
54C53×#387—21no—
55C53×#387—21no—
56C53×#384C2211yescharacteristic
57C53×#382C2211yescharacteristic
58C53×#380C2211yescharacteristic
59C53×#380C2211yescharacteristic
60C4×C4⋊C4×C53C2211yescharacteristic
61C4×C4⋊C4×C53C2211yescharacteristic
62C22×Dic106—21no—
63C22×Dic106—21no—
64C23×Dic53—21no—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69#13133217—22no—
70#13133217—22no—
71#13133217C2×C414yes—
72#13133217C2×C414yes—
73#13133216—28no—
74#13133209—22no—
75#13133209—22no—
76#13133209C2×C414yes—
77#13133209C2×C414yes—
78#13133208—28no—
79#13133194—28no—
80#13133194—28no—
81#13133176—22no—
82#13133176C2×C414yes—
83C2×C4⋊Dic53—21no—
84C2×C4⋊Dic53—22no—
85C2×C4⋊Dic53—21no—
86C2×C4⋊Dic53—21no—
87C2×C4⋊Dic53C2×C414yes—
88C2×C4⋊Dic53C2311yescharacteristic
89C2×C4⋊Dic53C2311yescharacteristic
90C2×C4⋊Dic53C2311yescharacteristic
91C2×C4⋊Dic53C2311yescharacteristic
92C2×#11719919—21no—
93C2×#11719919—21no—
94C2×#11719919—21no—
95C2×#11719919—21no—
96C2×C22⋊Dic53—21no—
97C2×C4×Dic53—21no—
98C4⋊#88322—22no—
99C4⋊#88322—22no—
100C22⋊#88322—22no—
101C22⋊#88322—22no—
102#13133128—22no—
103C4×#88322—22no—
104C23×C4×C53C2311yescharacteristic
105#117×C53C2×C414yes—
106#117×C53D414yes—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53D411yescharacteristic
110C2×C4⋊C4×C53D411yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117C2×C22⋊C4×C53D411yescharacteristic
118C2×C22⋊C4×C53D411yescharacteristic
119C2×C42×C53—21no—
120C2×C42×C53—21no—
121C2×C42×C53C2311yescharacteristic
122C2×C42×C53C2311yescharacteristic
123#95×C53—21no—
124#95×C53—21no—
125#95×C53—21no—
126#95×C53—21no—
127#95×C53—21no—
128#95×C53—21no—
129C24×C53C2×D411yescharacteristic
130C22×C4×C53—21no—
131C22×C4×C53—21no—
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×C53C22×C412yes—
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D412yes—
145C22×C4×C53C2411yescharacteristic
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—24no—
151C22×Dic53—81no—
152C22×Dic53—81no—
153C2×Dic106—24no—
154C2×Dic106—24no—
155C2×Dic106—81no—
156C2×Dic106—81no—
157C2×Dic106—81no—
158C2×Dic106—81no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—81no—
168C4⋊C4×C53—81no—
169C4⋊C4×C53C22⋊C414yes—
170C4⋊C4×C53C22⋊C414yes—
171C22⋊C4×C53—81no—
172C22⋊C4×C53—81no—
173C22⋊C4×C53C22⋊C414yes—
174C22⋊C4×C53C22⋊C414yes—
175C42×C53—81no—
176C42×C53—81no—
177C42×C53C22×C412yes—
178C42×C53C22×C412yes—
179C42×C53C2×D412yes—
180C42×C53C2×D412yes—
181C22⋊Dic53—24no—
182C4⋊Dic53—24no—
183C4⋊Dic53—24no—
184C4⋊Dic53—24no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53C22×C412yes—
190C4⋊Dic53C22×C412yes—
191C4⋊Dic53C22×C412yes—
192C4⋊Dic53C22×C412yes—
193#11719919—24no—
194#11719919—24no—
195#11719919—24no—
196#11719919—24no—
197C4×Dic53—24no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53C2×C22⋊C412yes—
217C23×C53C22×D411yescharacteristic
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53C2×C22⋊C412yes—
245C2×C4×C53C2×C22⋊C412yes—
246C2×C4×C53C2×C22⋊C412yes—
247C2×C4×C53C2×C22⋊C412yes—
248C2×C4×C53C2×C22⋊C412yes—
249C2×C4×C53C23×C411yescharacteristic
250C2×C4×C53C22×D411yescharacteristic
251C2×C4×C532−1+411yescharacteristic
252C2×C4×C532−1+411yescharacteristic
253C2×C4×C532−1+411yescharacteristic
254C2×C4×C532−1+411yescharacteristic
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—22no—
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—84no—
268Dic106—84no—
269Dic106—84no—
270#88322—41no—
271#88322—41no—
272#88322—41no—
273#88322—41no—
274#21843—531nomaximal, Sylow
275C22×C53—22no—
276C22×C53—22no—
277C22×C53—22no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53—41no—
282C22×C53C22×C22⋊C411yescharacteristic
283C22×C53#53611yescharacteristic
284C22×C53#53611yescharacteristic
285C22×C53#57611yescharacteristic
286C22×C53#57711yescharacteristic
287C22×C53#57711yescharacteristic
288C22×C53#57811yescharacteristic
289C4×C53—22no—
290C4×C53—22no—
291C4×C53—22no—
292C4×C53—22no—
293C4×C53—22no—
294C4×C53—81no—
295C4×C53—81no—
296C4×C53—81no—
297C4×C53—81no—
298C4×C53#51911yescharacteristic
299C4×C53#51911yescharacteristic
300C4×C53#53611yescharacteristic
301C4×C53#53611yescharacteristic
302Dic53—42no—
303Dic53—42no—
304Dic53—42no—
305Dic53—42no—
306C2×#544—531no—
307C2×#433—531no—
308#2314D5311yescharacteristic
309#1993—531no—
310#1992—531no—
311#1991—531no—
312#1990—531no—
313#1989—531no—
314#1989—531no—
315#1988—531no—
316#1987—531no—
317#1825—531no—
318#1825—531no—
319#1824—531no—
320#1823—531no—
321C2×C53—42no—
322C2×C53#231011yescharacteristic
323C2×C53#289711yescharacteristic
324C2×C53#289811yescharacteristic
325C2×C53#324111yescharacteristic
326C2×C53#324211yescharacteristic
327C2×C53#324311yescharacteristic
328C2×C53#324411yescharacteristic
329#544—538no—
330#544—1064no—
331C2×C4⋊Q8—531no—
332C2×C4⋊Q8—531no—
333C2×C4⋊Q8—1061no—
334C2×#125—531no—
335C2×#125—531no—
336C2×#125—1061no—
337C2×C22⋊Q8—1061no—
338C2×C22⋊Q8—1061no—
339C2×#117—531no—
340C2×#117C2×D5311yescharacteristic
341C22×C4⋊C4—531no—
342#433—1064no—
343C2×C4⋊C8—531no—
344C2×C4⋊C8—531no—
345C2×C22⋊C8—531no—
346C2×C22⋊C8—531no—
347C2×#97—531no—
348C2×C4×C8—531no—
349#388—21no—
350#388—21no—
351#387—21no—
352#387—21no—
353#384C2×D5311yescharacteristic
354#382C2×D5311yescharacteristic
355#380C2×D5311yescharacteristic
356#380C2×D5311yescharacteristic
357C4×C4⋊C4C2×D5311yescharacteristic
358C4×C4⋊C4C2×D5311yescharacteristic
359#340—531no—
360#340—531no—
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#340—531no—
370#340—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#340—531no—
375C53#2184311yescharacteristic, complemented, Sylow
376C22×Q8—1061no—
377C22×Q8—1061no—
378C23×C4—1061no—
379C23×C4C22×D5311yescharacteristic
380C22×C8—531no—
381C22×C8—531no—
382C22×C8—531no—
383C22×C8—531no—
384C4⋊Q8—534no—
385C4⋊Q8—534no—
386C4⋊Q8—1068no—
387C4⋊Q8—1062no—
388C4⋊Q8—1062no—
389#125—534no—
390#125—534no—
391#125—1068no—
392#125—1062no—
393#125—1062no—
394C22⋊Q8—1068no—
395C22⋊Q8—1068no—
396#117—534no—
397#117—1062no—
398#117C4×D5314yes—
399#117D21214yes—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—534no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—1062no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4C22⋊D5311yescharacteristic
416C2×C4⋊C4C22⋊D5311yescharacteristic
417C2×C4⋊C4C22×D5311yescharacteristic
418C2×C4⋊C4C22×D5311yescharacteristic
419C2×C4⋊C4C22×D5311yescharacteristic
420C2×C4⋊C4C22×D5311yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4C22⋊D5311yescharacteristic
425C2×C22⋊C4C22⋊D5311yescharacteristic
426C2×C42—21no—
427C2×C42—21no—
428C2×C42—1061no—
429C2×C42C22×D5311yescharacteristic
430C2×C42C22×D5311yescharacteristic
431C4⋊C8—1062no—
432C4⋊C8—1062no—
433C22⋊C8—1062no—
434C22⋊C8—1062no—
435#97—1062no—
436C4×C8—1062no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—21no—
442#95—21no—
443C24C2×C22⋊D5311yescharacteristic
444C2×Q8—1064no—
445C2×Q8—1064no—
446C2×Q8—4241no—
447C2×Q8—4241no—
448C2×Q8—4241no—
449C2×Q8—4241no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—1064no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—4241no—
464C22×C4—4241no—
465C22×C4C2×C4×D5312yes—
466C22×C4C2×D21212yes—
467C22×C4C2×C22⋊D5311yescharacteristic
468C22×C4C2×C22⋊D5311yescharacteristic
469C22×C4C2×C22⋊D5311yescharacteristic
470C22×C4C2×C22⋊D5311yescharacteristic
471C22×C4C2×C22⋊D5311yescharacteristic
472C22×C4C23×D5311yescharacteristic
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—81no—
494C4⋊C4—81no—
495C4⋊C4—532no—
496C4⋊C4—532no—
497C4⋊C4—532no—
498C4⋊C4—532no—
499C4⋊C4—1064no—
500C4⋊C4—1064no—
501C4⋊C4—1064no—
502C4⋊C4—1064no—
503C4⋊C4—1064no—
504C4⋊C4—1064no—
505C4⋊C4—1064no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4#1171992114yes—
511C4⋊C4#1171992114yes—
512C22⋊C4—81no—
513C22⋊C4—81no—
514C22⋊C4—1064no—
515C22⋊C4#1171992114yes—
516C22⋊C4#1171992114yes—
517C42—81no—
518C42—81no—
519C42—1064no—
520C42C2×C4×D5312yes—
521C42C2×C4×D5312yes—
522C42C2×D21212yes—
523C42C2×D21212yes—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23C2×#1171992112yes—
531C23C22×C22⋊D5311yescharacteristic
532Q8—4244no—
533Q8—4244no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—1062no—
561C2×C4—1062no—
562C2×C4—1062no—
563C2×C4—1062no—
564C2×C4—4244no—
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×C4C2×#1171992112yes—
574C2×C4C2×#1171992112yes—
575C2×C4C2×#1171992112yes—
576C2×C4C2×#1171992112yes—
577C2×C4C2×#1171992112yes—
578C2×C4C22×C4×D5311yescharacteristic
579C2×C4C22×D21211yescharacteristic
580C2×C4#1313326411yescharacteristic
581C2×C4#1313326411yescharacteristic
582C2×C4#1313326411yescharacteristic
583C2×C4#1313326411yescharacteristic
584C8—2121no—
585C8—2121no—
586C8—2121no—
587C8—2121no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22C22×#1171992111yescharacteristic
596C22#1681119411yescharacteristic
597C22#1681119411yescharacteristic
598C22#1681142611yescharacteristic
599C22#1681143111yescharacteristic
600C22#1681143411yescharacteristic
601C22#1681143811yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—22no—
607C4—81no—
608C4—81no—
609C4—81no—
610C4—81no—
611C4—2122no—
612C4—2122no—
613C4—2122no—
614C4—2122no—
615C4#1681110011yescharacteristic
616C4#1681110011yescharacteristic
617C4#1681119911yescharacteristic
618C4#1681119911yescharacteristic
619C2—42no—
620C2#2027729911yescharacteristic
621C2#2028323811yescharacteristic
622C2#2028324511yescharacteristic
623C2#2028596211yescharacteristic
624C2#2028597711yescharacteristic
625C2#2028599211yescharacteristic
626C2#2028600711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2314C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
cosocleC22×C4×C53C24
hypercenter#2314D53
hypocenterC53#21843
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2314

Complements

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

NH ≅ G/Ntype
3. #20275687C2semidirect
4. #20275683C2semidirect
5. #20275679C2semidirect
6. #20275675C2semidirect
7. #20275672C2semidirect
8. #20275672C2semidirect
9. #20275667C2semidirect
10. #20275663C2semidirect
17. #16811251C4semidirect
375. C53#21843semidirect

Maximal subgroups

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

subgroupindex
C2×#168112512
C2×#168106842
#202756872
#202756832
#202756792
#202756752
#202756722
#202756672
#202756632
#202748072
#202748032
#202747992
C53×#23142
#2184353