Finite Group Databasebeta

← #27393645

Subgroups — #27393645

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
2#20277324C211yesmaximal, characteristic
3#20275688C211yesmaximal, characteristic, complemented
4#20275684C211yesmaximal, characteristic, complemented
5#20275680C211yesmaximal, characteristic, complemented
6#20275676C211yesmaximal, characteristic, complemented
7#20275670C211yesmaximal, characteristic, complemented
8#20275670C211yesmaximal, characteristic, complemented
9#20275666C211yesmaximal, characteristic, complemented
10#20275662C211yesmaximal, characteristic, complemented
11#20274811C211yesmaximal, characteristic
12#20274811C211yesmaximal, characteristic
13#20274806C211yesmaximal, characteristic
14#20274802C211yesmaximal, characteristic
15C2×C53×#433C211yesmaximal, 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#16810516—21no—
29#16810516—21no—
30#16810509—21no—
31#16810509—21no—
32#16810493C2211yescharacteristic
33#16810489C2211yescharacteristic
34#16810483C2211yescharacteristic
35#16810483C2211yescharacteristic
36C4×C4⋊Dic53C2211yescharacteristic
37C4×C4⋊Dic53C2211yescharacteristic
38#16810386C2211yescharacteristic
39#16810386C2211yescharacteristic
40#16810386C2211yescharacteristic
41#16810386C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53#16810386C2211yescharacteristic
54C2×#117×C53C2211yescharacteristic
55C53×#433—24no—
56C2×C4⋊C8×C53C2211yescharacteristic
57C2×C4⋊C8×C53C2211yescharacteristic
58C2×C22⋊C8×C53C2211yescharacteristic
59C2×C22⋊C8×C53C2211yescharacteristic
60C2×#97×C53C2211yescharacteristic
61C2×C4×C8×C53C2211yescharacteristic
62C22×Dic106—21no—
63C22×Dic106—21no—
64C23×Dic53—21no—
65#13133217—22no—
66#13133217—22no—
67#13133217C2×C414yes—
68#13133217C2×C414yes—
69#13133216—28no—
70#13133209—22no—
71#13133209—22no—
72#13133209C2×C414yes—
73#13133209C2×C414yes—
74#13133208—28no—
75#13133194—28no—
76#13133194—28no—
77#13133176—22no—
78#13133176C2×C414yes—
79C2×C4⋊Dic53—21no—
80C2×C4⋊Dic53—22no—
81C2×C4⋊Dic53—21no—
82C2×C4⋊Dic53—21no—
83C2×C4⋊Dic53—21no—
84C2×C4⋊Dic53—21no—
85C2×C4⋊Dic53C2×C414yes—
86C2×C4⋊Dic53C2311yescharacteristic
87C2×C4⋊Dic53C2311yescharacteristic
88C2×C4⋊Dic53C2311yescharacteristic
89C2×C4⋊Dic53C2311yescharacteristic
90C2×C4⋊Dic53C2311yescharacteristic
91C2×C4⋊Dic53C2311yescharacteristic
92C2×C4⋊Dic53C2311yescharacteristic
93C2×C4⋊Dic53C2311yescharacteristic
94C2×#11719919—21no—
95C2×#11719919—21no—
96C2×#11719919—21no—
97C2×#11719919—21no—
98C2×C22⋊Dic53—21no—
99C2×C22⋊Dic53—21no—
100C2×C22⋊Dic53—21no—
101C2×C4×Dic53—21no—
102C2×C4×Dic53—21no—
103C2×C4×Dic53—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110C23×C4×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115#117×C53C2×C414yes—
116#117×C53D414yes—
117C2×C4⋊C4×C53D411yescharacteristic
118C2×C4⋊C4×C53D411yescharacteristic
119C2×C22⋊C4×C53D411yescharacteristic
120C2×C22⋊C4×C53D411yescharacteristic
121C2×C42×C53C2311yescharacteristic
122C2×C42×C53C2311yescharacteristic
123C4⋊C8×C53—22no—
124C4⋊C8×C53—22no—
125C22⋊C8×C53—22no—
126C22⋊C8×C53—22no—
127#97×C53—22no—
128C4×C8×C53—22no—
129C24×C53C2×D411yescharacteristic
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53C22×C412yes—
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D412yes—
141C22×C4×C53C2411yescharacteristic
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—24no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—81no—
152C22×Dic53—81no—
153C2×Dic106—24no—
154C2×Dic106—24no—
155C2×Dic106—81no—
156C2×Dic106—81no—
157C2×Dic106—81no—
158C2×Dic106—81no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C4⋊C4×C53C22⋊C414yes—
172C4⋊C4×C53C22⋊C414yes—
173C22⋊C4×C53C22⋊C414yes—
174C22⋊C4×C53C22⋊C414yes—
175C42×C53C22×C412yes—
176C42×C53C22×C412yes—
177C42×C53C2×D412yes—
178C42×C53C2×D412yes—
179C22⋊Dic53—24no—
180C22⋊Dic53—81no—
181C22⋊Dic53—81no—
182C4⋊Dic53—24no—
183C4⋊Dic53—24no—
184C4⋊Dic53—24no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—81no—
198C4⋊Dic53—81no—
199C4⋊Dic53C22×C412yes—
200C4⋊Dic53C22×C412yes—
201C4⋊Dic53C22×C412yes—
202C4⋊Dic53C22×C412yes—
203#11719919—24no—
204#11719919—24no—
205#11719919—24no—
206#11719919—24no—
207C4×Dic53—24no—
208C4×Dic53—81no—
209C4×Dic53—81no—
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×C53C2×C22⋊C412yes—
233C2×C4×C53C2×C22⋊C412yes—
234C2×C4×C53C2×C22⋊C412yes—
235C2×C4×C53C2×C22⋊C412yes—
236C2×C4×C53C2×C22⋊C412yes—
237C2×C4×C53C23×C411yescharacteristic
238C2×C4×C53C22×D411yescharacteristic
239C2×C4×C532−1+411yescharacteristic
240C2×C4×C532−1+411yescharacteristic
241C2×C4×C532−1+411yescharacteristic
242C2×C4×C532−1+411yescharacteristic
243C2×Dic53—22no—
244C2×Dic53—22no—
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
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—84no—
268Dic106—84no—
269Dic106—84no—
270C8×C53—41no—
271C8×C53—41no—
272C8×C53—41no—
273C8×C53—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#51911yescharacteristic
295C4×C53#51911yescharacteristic
296C4×C53#53611yescharacteristic
297C4×C53#53611yescharacteristic
298Dic53—42no—
299Dic53—42no—
300Dic53—42no—
301Dic53—42no—
302Dic53—81no—
303Dic53—81no—
304Dic53—81no—
305Dic53—81no—
306C2×#544—531no—
307C2×#433D5311yescharacteristic
308#2314—531no—
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—24no—
343C2×C4⋊C8C2×D5311yescharacteristic
344C2×C4⋊C8C2×D5311yescharacteristic
345C2×C22⋊C8C2×D5311yescharacteristic
346C2×C22⋊C8C2×D5311yescharacteristic
347C2×#97C2×D5311yescharacteristic
348C2×C4×C8C2×D5311yescharacteristic
349#388—1061no—
350#388—1061no—
351#387—1061no—
352#387—1061no—
353#384—531no—
354#382—531no—
355#380—531no—
356#380—531no—
357C4×C4⋊C4—531no—
358C4×C4⋊C4—531no—
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×C8C22×D5311yescharacteristic
381C22×C8C22×D5311yescharacteristic
382C22×C8C22×D5311yescharacteristic
383C22×C8C22×D5311yescharacteristic
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#117C22⋊D5314yes—
400C2×C4⋊C4—534no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—531no—
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⋊C4—1061no—
416C2×C4⋊C4—1062no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4D21211yescharacteristic
420C2×C4⋊C4D21211yescharacteristic
421C2×C22⋊C4—1061no—
422C2×C22⋊C4—1061no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4D21211yescharacteristic
425C2×C22⋊C4D21211yescharacteristic
426C2×C42—1061no—
427C2×C42—1061no—
428C2×C42—1061no—
429C2×C42C22×D5311yescharacteristic
430C2×C42C22×D5311yescharacteristic
431C4⋊C8—22no—
432C4⋊C8—22no—
433C22⋊C8—22no—
434C22⋊C8—22no—
435#97—22no—
436C4×C8—22no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442#95—1061no—
443C24C2×D21211yescharacteristic
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—1064no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—4241no—
464C22×C4—4241no—
465C22×C4C2×C4×D5312yes—
466C22×C4C2×D21211yescharacteristic
467C22×C4C2×D21211yescharacteristic
468C22×C4C2×D21211yescharacteristic
469C22×C4C2×D21211yescharacteristic
470C22×C4C2×D21211yescharacteristic
471C22×C4C2×C22⋊D5312yes—
472C22×C4C23×D5311yescharacteristic
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C4⋊C4—532no—
486C4⋊C4—532no—
487C4⋊C4—532no—
488C4⋊C4—532no—
489C4⋊C4—1064no—
490C4⋊C4—1064no—
491C4⋊C4—1064no—
492C4⋊C4—1064no—
493C4⋊C4—1064no—
494C4⋊C4—1064no—
495C4⋊C4—1064no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—4241no—
509C4⋊C4—4241no—
510C4⋊C4#1171992114yes—
511C4⋊C4#1171992114yes—
512C22⋊C4—1064no—
513C22⋊C4—4241no—
514C22⋊C4—4241no—
515C22⋊C4#1171992114yes—
516C22⋊C4#1171992114yes—
517C42—1064no—
518C42—4241no—
519C42—4241no—
520C42C2×C4×D5312yes—
521C42C2×C4×D5312yes—
522C42C2×C22⋊D5312yes—
523C42C2×C22⋊D5312yes—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23C2×#1171992112yes—
531C23C22×D21211yescharacteristic
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—1062no—
549C2×C4—1062no—
550C2×C4—1062no—
551C2×C4—1062no—
552C2×C4—4244no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
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×C4C2×#1171992112yes—
574C2×C4C2×#1171992112yes—
575C2×C4C2×#1171992112yes—
576C2×C4C2×#1171992112yes—
577C2×C4C2×#1171992112yes—
578C2×C4C22×C4×D5311yescharacteristic
579C2×C4C22×C22⋊D5311yescharacteristic
580C2×C4#1313326411yescharacteristic
581C2×C4#1313326411yescharacteristic
582C2×C4#1313326411yescharacteristic
583C2×C4#1313326411yescharacteristic
584C8—41no—
585C8—41no—
586C8—41no—
587C8—41no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22C22×#1171992111yescharacteristic
596C22#1681119911yescharacteristic
597C22#1681119911yescharacteristic
598C22#1681142811yescharacteristic
599C22#1681143511yescharacteristic
600C22#1681143511yescharacteristic
601C22#1681144011yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—22no—
607C4—4241no—
608C4—4241no—
609C4—4241no—
610C4—4241no—
611C4—2122no—
612C4—2122no—
613C4—2122no—
614C4—2122no—
615C4#1681110011yescharacteristic
616C4#1681110011yescharacteristic
617C4#1681119411yescharacteristic
618C4#1681119411yescharacteristic
619C2—42no—
620C2#2027729911yescharacteristic
621C2#2028324211yescharacteristic
622C2#2028324911yescharacteristic
623C2#2028597011yescharacteristic
624C2#2028598511yescharacteristic
625C2#2028600011yescharacteristic
626C2#2028601511yescharacteristic
subgroupG/H
centerC23C22×D212
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#433C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
cosocleC22×C4×C53C24
hypercenterC2×#433D53
hypocenterC53#21843
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
3. #20275688C2semidirect
4. #20275684C2semidirect
5. #20275680C2semidirect
6. #20275676C2semidirect
7. #20275670C2semidirect
8. #20275670C2semidirect
9. #20275666C2semidirect
10. #20275662C2semidirect
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
#202773242
#202756882
#202756842
#202756802
#202756762
#202756702
#202756662
#202756622
#202748112
#202748062
#202748022
C2×C53×#4332
#2184353