Finite Group Databasebeta

← #27351656

Subgroups — #27351656

Order 13568.

982 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283750C211yesmaximal, characteristic, complemented
2#20277358C211yesmaximal, characteristic, complemented
3#20273287C212yesmaximal, complemented
4#20273271C212yesmaximal, complemented
5#20273258C212yesmaximal, complemented
6#20273248C212yesmaximal, complemented
7#20273240C211yesmaximal, characteristic, complemented
8#20273227C211yesmaximal, characteristic, complemented
9#20273214C211yesmaximal, characteristic, complemented
10#20273201C211yesmaximal, characteristic, complemented
11C53×#3520C211yesmaximal, characteristic
12#16810701—21no—
13#16810695—22no—
14D4×#88322—21no—
15#16810680C2211yescharacteristic
16#16810676C2211yescharacteristic
17#16810587C2212yes—
18C2×C22⋊#88322C2212yes—
19#16810526—21no—
20#16810522—21no—
21#16810516—22no—
22#16810486C2212yes—
23#16810483C2212yes—
24C4×C4⋊Dic53C2211yescharacteristic
25#16810469C2211yescharacteristic
26#16810360C2211yescharacteristic
27#16810360C2211yescharacteristic
28#16810360C2212yes—
29#16810360C2212yes—
30#16810360C2212yes—
31#16810360C2212yes—
32#16810360C2212yes—
33#16810357C2211yescharacteristic
34#16810357C2211yescharacteristic
35#16810357C2212yes—
36C53×#556C411yescharacteristic, complemented
37C53×#554—21no—
38C53×#552C412yescomplemented
39C53×#551—22no—
40D4×Q8×C53C411yescharacteristic, complemented
41C53×#548C412yescomplemented
42C53×#547C411yescharacteristic, complemented
43C53×#547C411yescharacteristic, complemented
44C53×#546—21no—
45C53×#542—21no—
46C53×#539—22no—
47C53×#533C2211yescharacteristic
48C53×#529C2211yescharacteristic
49C53×#525C2212yes—
50C2×C22⋊Q8×C53C2212yes—
51C53×#520—21no—
52C2×C4×D4×C53C2211yescharacteristic
53C2×#11719917—22no—
54C22×#88322—22no—
55C2×C4⋊Dic53—21no—
56C2×C4⋊Dic53—22no—
57C2×C22⋊Dic53—22no—
58C2×C22⋊Dic53—22no—
59C2×C4×Dic53—21no—
60C2×C4×Dic53—21no—
61C4⋊#88322—21no—
62C4⋊#88322—21no—
63C22⋊#88322—22no—
64C22⋊#88322—22no—
65C22⋊#88322C2311yescharacteristic
66C22⋊#88322C2312yes—
67C22⋊#88322C2311yescharacteristic
68#13133128—21no—
69C4×#88322—21no—
70#13133124—21no—
71#13133124—22no—
72#13133124C2311yescharacteristic
73#13133124C2311yescharacteristic
74#13133124C2312yes—
75C2×#41×C53—21no—
76C2×#41×C53C2×C412yes—
77C22×Q8×C53C2×C412yes—
78C22×D4×C53D411yescharacteristic
79C23×C4×C53C2312yes—
80C4⋊Q8×C53—21no—
81C4⋊Q8×C53C2×C411yescharacteristic
82C4⋊Q8×C53C2×C411yescharacteristic
83#126×C53—22no—
84#126×C53—22no—
85#126×C53—21no—
86#125×C53—22no—
87#125×C53—21no—
88#125×C53C2×C411yescharacteristic
89#124×C53—22no—
90#124×C53—21no—
91#124×C53C2×C411yescharacteristic
92#123×C53—22no—
93#123×C53—22no—
94#123×C53—22no—
95#123×C53—22no—
96#123×C53—21no—
97C22⋊Q8×C53—22no—
98C22⋊Q8×C53—21no—
99C22⋊Q8×C53—22no—
100C22⋊Q8×C53—22no—
101C22⋊Q8×C53—22no—
102C22⋊Q8×C53—21no—
103C22⋊Q8×C53C2×C412yes—
104C22⋊Q8×C53C2×C412yes—
105C22⋊Q8×C53C2×C412yes—
106C22⋊Q8×C53C2311yescharacteristic
107C22⋊Q8×C53C2312yes—
108#121×C53—22no—
109#121×C53—21no—
110#121×C53C2×C412yes—
111#121×C53C2311yescharacteristic
112#120×C53—22no—
113C4×Q8×C53—21no—
114C4×Q8×C53—21no—
115C4×Q8×C53C2×C411yescharacteristic
116C4×Q8×C53C2×C412yes—
117C4×D4×C53—22no—
118C4×D4×C53—21no—
119C4×D4×C53—22no—
120C4×D4×C53—22no—
121C4×D4×C53—21no—
122C4×D4×C53C2×C411yescharacteristic
123C4×D4×C53C2×C411yescharacteristic
124C4×D4×C53C2×C411yescharacteristic
125C4×D4×C53C2×C412yes—
126C4×D4×C53D411yescharacteristic
127C4×D4×C53D412yes—
128C4×D4×C53D411yescharacteristic
129#117×C53—22no—
130#117×C53—21no—
131#117×C53—21no—
132#117×C53C2×C412yes—
133C2×C4⋊C4×C53—21no—
134C2×C4⋊C4×C53—22no—
135C2×C4⋊C4×C53C2×C412yes—
136C2×C4⋊C4×C53D411yescharacteristic
137C2×C22⋊C4×C53—22no—
138C2×C22⋊C4×C53D412yes—
139C2×C42×C53C2311yescharacteristic
140C24×C53C2×D412yes—
141#41×C53—44no—
142#41×C53—82no—
143C2×Q8×C53—21no—
144C2×Q8×C53—22no—
145C2×Q8×C53—82no—
146C2×Q8×C53C22×C411yescharacteristic
147C2×Q8×C53C22×C412yes—
148C2×D4×C53—22no—
149C2×D4×C53—21no—
150C2×D4×C53—22no—
151C2×D4×C53—42no—
152C2×D4×C53C22⋊C411yescharacteristic
153C2×D4×C53C22⋊C411yescharacteristic
154C2×D4×C53C22⋊C412yes—
155C2×D4×C53C22×C411yescharacteristic
156C22×C4×C53—22no—
157C22×C4×C53—22no—
158C22×C4×C53—21no—
159C22×C4×C53—22no—
160C22×C4×C53—82no—
161C22×C4×C53C22×C412yes—
162C22×C4×C53C22×C412yes—
163C22×C4×C53C2×D411yescharacteristic
164C22×C4×C53C2×D411yescharacteristic
165C22×C4×C53C2×D412yes—
166C22×C4×C53C2×D412yes—
167C22×C4×C53C2×D412yes—
168C22×C4×C53C2411yescharacteristic
169C22×Dic53—21no—
170C22×Dic53—21no—
171C22×Dic53—22no—
172C4⋊C4×C53—22no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—22no—
175C4⋊C4×C53—22no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—22no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—22no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53—21no—
184C4⋊C4×C53—22no—
185C4⋊C4×C53C22⋊C411yescharacteristic
186C4⋊C4×C53C22⋊C412yes—
187C4⋊C4×C53C22⋊C411yescharacteristic
188C4⋊C4×C53C22×C411yescharacteristic
189C4⋊C4×C53C22×C412yes—
190C4⋊C4×C53C22×C411yescharacteristic
191C22⋊C4×C53—22no—
192C22⋊C4×C53—22no—
193C22⋊C4×C53—22no—
194C22⋊C4×C53—22no—
195C22⋊C4×C53—21no—
196C22⋊C4×C53—22no—
197C22⋊C4×C53—21no—
198C22⋊C4×C53—21no—
199C22⋊C4×C53—22no—
200C22⋊C4×C53—21no—
201C22⋊C4×C53—21no—
202C22⋊C4×C53C22⋊C412yes—
203C22⋊C4×C53C22⋊C412yes—
204C22⋊C4×C53C22⋊C412yes—
205C22⋊C4×C53C22⋊C412yes—
206C42×C53—22no—
207C42×C53—22no—
208C42×C53—21no—
209C42×C53C22×C411yescharacteristic
210C42×C53C22×C411yescharacteristic
211C42×C53C2×D411yescharacteristic
212C42×C53C2×D411yescharacteristic
213C22⋊Dic53—82no—
214C22⋊Dic53—82no—
215C4⋊Dic53—42no—
216C4⋊Dic53—82no—
217C4×Dic53—42no—
218C4×Dic53—81no—
219#11719917—82no—
220C2×#88322—21no—
221C2×#88322—21no—
222C2×#88322—22no—
223C2×#88322—82no—
224C23×C53—22no—
225C23×C53—22no—
226C23×C53—21no—
227C23×C53—42no—
228C23×C53—42no—
229C23×C53C2×C22⋊C412yes—
230C23×C53C2×C22⋊C412yes—
231C23×C53C22×D411yescharacteristic
232Q8×C53—41no—
233Q8×C53—42no—
234Q8×C53—42no—
235Q8×C53—82no—
236D4×C53—22no—
237D4×C53—22no—
238D4×C53—42no—
239D4×C53—41no—
240D4×C53—44no—
241D4×C53—41no—
242D4×C53—41no—
243D4×C53—82no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—22no—
247C2×C4×C53—21no—
248C2×C4×C53—22no—
249C2×C4×C53—21no—
250C2×C4×C53—22no—
251C2×C4×C53—21no—
252C2×C4×C53—41no—
253C2×C4×C53—42no—
254C2×C4×C53—42no—
255C2×C4×C53—42no—
256C2×C4×C53—42no—
257C2×C4×C53—42no—
258C2×C4×C53—44no—
259C2×C4×C53—42no—
260C2×C4×C53—82no—
261C2×C4×C53—82no—
262C2×C4×C53C2×C22⋊C412yes—
263C2×C4×C53C2×C22⋊C412yes—
264C2×C4×C53C2×C22⋊C411yescharacteristic
265C2×C4×C53C2×C22⋊C411yescharacteristic
266C2×C4×C53C2×C22⋊C411yescharacteristic
267C2×C4×C53C2×C22⋊C411yescharacteristic
268C2×C4×C53C23×C411yescharacteristic
269C2×C4×C53C22×D411yescharacteristic
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—42no—
273C2×Dic53—41no—
274C2×Dic53—42no—
275C2×Dic53—41no—
276C2×Dic53—42no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279#88322—41no—
280#88322—42no—
281#88322—41no—
282#17600—531nomaximal, Sylow
283C22×C53—21no—
284C22×C53—21no—
285C22×C53—22no—
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—22no—
289C22×C53—22no—
290C22×C53—22no—
291C22×C53—41no—
292C22×C53—41no—
293C22×C53—41no—
294C22×C53—82no—
295C22×C53—82no—
296C22×C53C22×C22⋊C411yescharacteristic
297C22×C53#57711yescharacteristic
298C22×C53#57811yescharacteristic
299C4×C53—21no—
300C4×C53—21no—
301C4×C53—22no—
302C4×C53—22no—
303C4×C53—21no—
304C4×C53—21no—
305C4×C53—21no—
306C4×C53—42no—
307C4×C53—41no—
308C4×C53—41no—
309C4×C53—41no—
310C4×C53—42no—
311C4×C53#42112yes—
312Dic53—81no—
313Dic53—82no—
314Dic53—81no—
315#3520D5311yescharacteristic
316#2974—531no—
317#2319—531no—
318#1520—532no—
319#1516—532no—
320#1513—532no—
321#1511—532no—
322#1509—531no—
323#1506—531no—
324#1503—531no—
325#1500—531no—
326C2×C53—21no—
327C2×C53—21no—
328C2×C53—42no—
329C2×C53—42no—
330C2×C53—81no—
331C2×C53#288311yescharacteristic
332C2×C53#289811yescharacteristic
333C2×C53C2×#42111yescharacteristic
334#556Dic5311yescharacteristic, complemented
335#554—21no—
336#552Dic5312yescomplemented
337#551—22no—
338D4×Q8Dic5311yescharacteristic, complemented
339#548Dic5312yescomplemented
340#547Dic5311yescharacteristic, complemented
341#547Dic5311yescharacteristic, complemented
342#546—21no—
343#542—21no—
344#539—22no—
345#533C2×D5311yescharacteristic
346#529C2×D5311yescharacteristic
347#525C2×D5312yes—
348C2×C22⋊Q8C2×D5312yes—
349#520—21no—
350C2×C4×D4C2×D5311yescharacteristic
351#436—1061no—
352#435—1062no—
353C8×D4—1061no—
354#432—531no—
355#431—531no—
356#407—532no—
357C2×C22⋊C8—532no—
358#390—1061no—
359#389—1061no—
360#388—1062no—
361#381—532no—
362#380—532no—
363C4×C4⋊C4—531no—
364#376—531no—
365#328—531no—
366#328—531no—
367#328—532no—
368#328—532no—
369#328—532no—
370#328—532no—
371#328—532no—
372#327—531no—
373#327—531no—
374#327—532no—
375C53#1760011yescharacteristic, complemented, Sylow
376C2×#41—21no—
377C2×#41C2×Dic5312yes—
378C22×Q8C2×Dic5312yes—
379C22×D4C22⋊D5311yescharacteristic
380C23×C4C22×D5312yes—
381C2×M16—1062no—
382C22×C8—1062no—
383C4⋊Q8—21no—
384C4⋊Q8C2×Dic5311yescharacteristic
385C4⋊Q8C2×Dic5311yescharacteristic
386#126—22no—
387#126—22no—
388#126—21no—
389#125—22no—
390#125—21no—
391#125C2×Dic5311yescharacteristic
392#124—21no—
393#124—22no—
394#124C2×Dic5311yescharacteristic
395#123—22no—
396#123—22no—
397#123—21no—
398#123—22no—
399#123—22no—
400C22⋊Q8—22no—
401C22⋊Q8—22no—
402C22⋊Q8—21no—
403C22⋊Q8—22no—
404C22⋊Q8—21no—
405C22⋊Q8—22no—
406C22⋊Q8C2×Dic5312yes—
407C22⋊Q8C2×Dic5312yes—
408C22⋊Q8C2×Dic5312yes—
409C22⋊Q8C22×D5311yescharacteristic
410C22⋊Q8C22×D5312yes—
411#121—22no—
412#121—21no—
413#121C2×Dic5312yes—
414#121C22×D5311yescharacteristic
415#120—22no—
416C4×Q8—21no—
417C4×Q8—21no—
418C4×Q8C2×Dic5311yescharacteristic
419C4×Q8C2×Dic5312yes—
420C4×D4—21no—
421C4×D4—22no—
422C4×D4—22no—
423C4×D4—21no—
424C4×D4—22no—
425C4×D4C2×Dic5311yescharacteristic
426C4×D4C2×Dic5311yescharacteristic
427C4×D4C2×Dic5311yescharacteristic
428C4×D4C2×Dic5312yes—
429C4×D4C22⋊D5311yescharacteristic
430C4×D4C22⋊D5312yes—
431C4×D4C22⋊D5311yescharacteristic
432#117—21no—
433#117—22no—
434#117—21no—
435#117C2×Dic5312yes—
436C2×C4⋊C4—22no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—1062no—
439C2×C4⋊C4—1061no—
440C2×C4⋊C4C2×Dic5312yes—
441C2×C4⋊C4C22⋊D5311yescharacteristic
442C2×C22⋊C4—22no—
443C2×C22⋊C4—1062no—
444C2×C22⋊C4—1062no—
445C2×C22⋊C4C22⋊D5312yes—
446C2×C42—1061no—
447C2×C42—1061no—
448C2×C42C22×D5311yescharacteristic
449C4⋊C8—1061no—
450C4⋊C8—1061no—
451C22⋊C8—531no—
452C22⋊C8—532no—
453C22⋊C8—531no—
454C22⋊C8—1062no—
455C22⋊C8—1062no—
456#97—1061no—
457C4×C8—1061no—
458#95—531no—
459#95—531no—
460#95—532no—
461#95—1061no—
462#95—1062no—
463C24C2×C22⋊D5312yes—
464#41—44no—
465#41—82no—
466C2×Q8—21no—
467C2×Q8—22no—
468C2×Q8—82no—
469C2×Q8C22×Dic5311yescharacteristic
470C2×Q8C22×Dic5312yes—
471C2×D4—22no—
472C2×D4—21no—
473C2×D4—22no—
474C2×D4—42no—
475C2×D4C22⋊Dic5311yescharacteristic
476C2×D4C22⋊Dic5311yescharacteristic
477C2×D4C22⋊Dic5312yes—
478C2×D4C22×Dic5311yescharacteristic
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—82no—
484C22×C4—1061no—
485C22×C4—1061no—
486C22×C4—1062no—
487C22×C4C22×Dic5312yes—
488C22×C4C22×Dic5312yes—
489C22×C4C2×C22⋊D5311yescharacteristic
490C22×C4C2×C22⋊D5311yescharacteristic
491C22×C4C2×C22⋊D5312yes—
492C22×C4C2×C22⋊D5312yes—
493C22×C4C2×C22⋊D5312yes—
494C22×C4C23×D5311yescharacteristic
495M16—4242no—
496C2×C8—1061no—
497C2×C8—1061no—
498C2×C8—1062no—
499C2×C8—4242no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—4242no—
514C4⋊C4—2122no—
515C4⋊C4C22⋊Dic5311yescharacteristic
516C4⋊C4C22⋊Dic5312yes—
517C4⋊C4C22⋊Dic5311yescharacteristic
518C4⋊C4C22×Dic5311yescharacteristic
519C4⋊C4C22×Dic5312yes—
520C4⋊C4C22×Dic5311yescharacteristic
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—21no—
526C22⋊C4—22no—
527C22⋊C4—21no—
528C22⋊C4—21no—
529C22⋊C4—22no—
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—4242no—
533C22⋊C4—4242no—
534C22⋊C4C22⋊Dic5312yes—
535C22⋊C4C22⋊Dic5312yes—
536C22⋊C4C22⋊Dic5312yes—
537C22⋊C4C22⋊Dic5312yes—
538C42—22no—
539C42—22no—
540C42—21no—
541C42—4241no—
542C42—2122no—
543C42C22×Dic5311yescharacteristic
544C42C22×Dic5311yescharacteristic
545C42C2×C22⋊D5311yescharacteristic
546C42C2×C22⋊D5311yescharacteristic
547C23—22no—
548C23—22no—
549C23—21no—
550C23—42no—
551C23—42no—
552C23C2×C22⋊Dic5312yes—
553C23C2×C22⋊Dic5312yes—
554C23C22×C22⋊D5311yescharacteristic
555Q8—41no—
556Q8—42no—
557Q8—42no—
558Q8—82no—
559D4—22no—
560D4—22no—
561D4—42no—
562D4—41no—
563D4—44no—
564D4—41no—
565D4—41no—
566D4—82no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2122no—
589C2×C4—2121no—
590C2×C4—2122no—
591C2×C4—2122no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4C2×C22⋊Dic5311yescharacteristic
595C2×C4C2×C22⋊Dic5312yes—
596C2×C4C2×C22⋊Dic5311yescharacteristic
597C2×C4C2×C22⋊Dic5311yescharacteristic
598C2×C4C2×C22⋊Dic5312yes—
599C2×C4C2×C22⋊Dic5311yescharacteristic
600C2×C4C23×Dic5311yescharacteristic
601C2×C4C22×C22⋊D5311yescharacteristic
602C8—2121no—
603C8—2122no—
604C8—2121no—
605C22—21no—
606C22—21no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—82no—
617C22—82no—
618C22C22×C22⋊Dic5311yescharacteristic
619C22#1681143111yescharacteristic
620C22#1681143811yescharacteristic
621C4—21no—
622C4—21no—
623C4—22no—
624C4—22no—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—42no—
629C4—41no—
630C4—41no—
631C4—41no—
632C4—42no—
633C4—4241no—
634C4—4242no—
635C4—4241no—
636C4#1681064012yes—
637C2—21no—
638C2—21no—
639C2—42no—
640C2—42no—
641C2—81no—
642C2#2028317611yescharacteristic
643C2#2028324811yescharacteristic
644C2C2×#1681064011yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
Fitting, prime radicalC53×#3520C2
socleC22×C53C22×C22⋊C4
cosocleC22×C4×C53C24
hypercenter#3520D53
hypocenterC53#17600
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #3520

Complements

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

NH ≅ G/Ntype
1. #20283750C2semidirect
2. #20277358C2semidirect
3. #20273287C2semidirect
4. #20273271C2semidirect
5. #20273258C2semidirect
6. #20273248C2semidirect
7. #20273240C2semidirect
8. #20273227C2semidirect
9. #20273214C2semidirect
10. #20273201C2semidirect
36. C53×#556C4semidirect
38. C53×#552C4semidirect
40. D4×Q8×C53C4semidirect
41. C53×#548C4semidirect
42. C53×#547C4semidirect
43. C53×#547C4semidirect
334. #556Dic53semidirect
336. #552Dic53semidirect
338. D4×Q8Dic53semidirect
339. #548Dic53semidirect
340. #547Dic53semidirect
341. #547Dic53semidirect
375. C53#17600semidirect

Maximal subgroups

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

subgroupindex
#202837502
#202773582
#202732872
#202732712
#202732582
#202732482
#202732402
#202732272
#202732142
#202732012
C53×#35202
#1760053