Finite Group Databasebeta

← #27351506

Subgroups — #27351506

Order 13568.

1070 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811154C211yesmaximal, characteristic
2C2×#16810677C211yesmaximal, characteristic, complemented
3#20277117C211yesmaximal, characteristic, complemented
4#20273288C211yesmaximal, characteristic, complemented
5#20273286C211yesmaximal, characteristic, complemented
6#20273283C211yesmaximal, characteristic
7#20273250C214yesmaximal, complemented
8#20273237C214yesmaximal, complemented
9C53×#1519C211yesmaximal, characteristic
10#16811154—21no—
11#16811154—21no—
12#16811154—21no—
13#16811154—21no—
14#16811154C412yescomplemented
15#16811154C412yescomplemented
16#16811154C2212yes—
17#16811154C2212yes—
18C2×#13133210—21no—
19C2×#13133208C411yescharacteristic, complemented
20C2×#13133204C411yescharacteristic, complemented
21C2×#13133198—21no—
22C2×#13133196C2211yescharacteristic
23C2×#13133182C411yescharacteristic, complemented
24C2×D4×Dic53—21no—
25C2×D4×Dic53C411yescharacteristic, complemented
26C2×#13133173—21no—
27C2×#13133173—21no—
28C22×C4×Dic53C2211yescharacteristic
29#16810677—21no—
30#16810677—21no—
31#16810677C2212yescomplemented
32#16810677C2212yescomplemented
33C2×#13133146—21no—
34C2×C22⋊#88322C2211yescharacteristic
35C2×#13133127—21no—
36#16810492—21no—
37#16810487C2212yescomplemented
38#16810483C2211yescharacteristic
39#16810481—21no—
40C4⋊C4×Dic53—21no—
41#16810474C2212yescomplemented
42#16810361C2214yescomplemented
43#16810359C2214yescomplemented
44#16810357C2214yes—
45C2×#121×C53C2211yescharacteristic
46C2×C22⋊C8×C53C2211yescharacteristic
47C53×#380C2211yescharacteristic
48C53×#327C2214yes—
49C22×Dic106C2×C411yescharacteristic
50C23×Dic53—21no—
51C23×Dic53D411yescharacteristic
52C23×Dic53D411yescharacteristic
53C22×#88322—21no—
54#13133210—22no—
55#13133210—22no—
56#13133210—22no—
57#13133210—22no—
58#13133208—21no—
59#13133208—21no—
60#13133208C2×C412yes—
61#13133208C2×C412yes—
62#13133204—21no—
63#13133204—21no—
64#13133204C2×C412yes—
65#13133204C2×C412yes—
66#13133198—22no—
67#13133198—22no—
68#13133198—22no—
69#13133198—22no—
70#13133196—21no—
71#13133196—21no—
72#13133196C2×C412yes—
73#13133196C2312yes—
74#13133182—21no—
75#13133182—21no—
76#13133182C2×C412yes—
77#13133182C2×C412yes—
78D4×Dic53—21no—
79D4×Dic53—21no—
80D4×Dic53—22no—
81D4×Dic53—22no—
82D4×Dic53—22no—
83D4×Dic53—22no—
84D4×Dic53C2×C412yes—
85D4×Dic53C2×C412yes—
86#13133173—22no—
87#13133173—22no—
88#13133173—22no—
89#13133173—22no—
90#13133173—44no—
91C2×C4⋊Dic53—21no—
92C2×C4⋊Dic53—21no—
93C2×C4⋊Dic53C2×C411yescharacteristic
94C2×#11719919—21no—
95C2×#11719919—21no—
96C2×#11719919—21no—
97C2×#11719919—22no—
98C2×#11719919—22no—
99C2×C22⋊Dic53—21no—
100C2×C22⋊Dic53—21no—
101C2×C22⋊Dic53—21no—
102C2×C22⋊Dic53—21no—
103C2×C22⋊Dic53—21no—
104C2×C4×Dic53—21no—
105C2×C4×Dic53—22no—
106C2×C4×Dic53C2×C411yescharacteristic
107C2×C4×Dic53C2×C411yescharacteristic
108C2×C4×Dic53C2×C411yescharacteristic
109C2×C4×Dic53C2×C411yescharacteristic
110C2×C4×Dic53D411yescharacteristic
111C2×C4×Dic53D411yescharacteristic
112C2×C4×Dic53D411yescharacteristic
113C2×C4×Dic53D411yescharacteristic
114C2×C4×Dic53D411yescharacteristic
115C2×C4×Dic53D411yescharacteristic
116C2×C4×Dic53C2311yescharacteristic
117C2×C4×Dic53C2311yescharacteristic
118#13133146—22no—
119#13133146—22no—
120C22⋊#88322—21no—
121C22⋊#88322C2312yes—
122#13133127—22no—
123#13133127—22no—
124#13133124—22no—
125#13133124C2312yes—
126C22×D4×C53—21no—
127C22×D4×C53C2×C411yescharacteristic
128C23×C4×C53C2311yescharacteristic
129C22×C8×C53—21no—
130#121×C53—21no—
131#121×C53—21no—
132#121×C53C2×C412yes—
133#121×C53C2312yes—
134C2×C4⋊C4×C53—21no—
135C2×C4⋊C4×C53C2×C411yescharacteristic
136C2×C22⋊C4×C53—21no—
137C2×C22⋊C4×C53—21no—
138C22⋊C8×C53—21no—
139C22⋊C8×C53C2312yes—
140#95×C53C2312yes—
141C24×C53—21no—
142C24×C53C2×D411yescharacteristic
143C2×D4×C53—22no—
144C2×D4×C53—22no—
145C2×D4×C53—21no—
146C2×D4×C53—42no—
147C2×D4×C53—42no—
148C2×D4×C53—42no—
149C2×D4×C53C22×C412yes—
150C22×C4×C53—21no—
151C22×C4×C53—22no—
152C22×C4×C53—42no—
153C22×C4×C53C22×C411yescharacteristic
154C22×C4×C53C22×C411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2411yescharacteristic
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53—21no—
162C22×Dic53—21no—
163C22×Dic53—22no—
164C22×Dic53—22no—
165C22×Dic53—22no—
166C22×Dic53—42no—
167C22×Dic53—44no—
168C22×Dic53C22⋊C411yescharacteristic
169C22×Dic53C22⋊C411yescharacteristic
170C22×Dic53C22⋊C411yescharacteristic
171C22×Dic53C22⋊C411yescharacteristic
172C22×Dic53C22⋊C411yescharacteristic
173C22×Dic53C22⋊C411yescharacteristic
174C22×Dic53C22⋊C411yescharacteristic
175C22×Dic53C22⋊C411yescharacteristic
176C22×Dic53C2×D411yescharacteristic
177C22×Dic53C2×D411yescharacteristic
178C22×Dic53C2×D411yescharacteristic
179C22×Dic53C2×D411yescharacteristic
180C2×Dic106—21no—
181C2×Dic106—42no—
182C2×Dic106C22×C412yes—
183C2×C8×C53—22no—
184C2×C8×C53—42no—
185C4⋊C4×C53—21no—
186C4⋊C4×C53—42no—
187C4⋊C4×C53C22×C412yes—
188C22⋊C4×C53—22no—
189C22⋊C4×C53—22no—
190C22⋊C4×C53—42no—
191C22⋊Dic53—22no—
192C22⋊Dic53—22no—
193C22⋊Dic53—22no—
194C22⋊Dic53—22no—
195C22⋊Dic53—22no—
196C22⋊Dic53—22no—
197C22⋊Dic53—22no—
198C22⋊Dic53—22no—
199C22⋊Dic53—42no—
200C4⋊Dic53—21no—
201C4⋊Dic53—22no—
202C4⋊Dic53—22no—
203C4⋊Dic53—42no—
204C4⋊Dic53C22×C412yes—
205#11719919—22no—
206#11719919—22no—
207#11719919—22no—
208#11719919—22no—
209#11719919—22no—
210#11719919—22no—
211#11719919—44no—
212#11719919—44no—
213C4×Dic53—22no—
214C4×Dic53—22no—
215C4×Dic53—44no—
216C4×Dic53C22⋊C411yescharacteristic
217C4×Dic53C22⋊C411yescharacteristic
218C4×Dic53C22⋊C411yescharacteristic
219C4×Dic53C22⋊C411yescharacteristic
220C4×Dic53C22⋊C411yescharacteristic
221C4×Dic53C22⋊C411yescharacteristic
222C4×Dic53C22⋊C411yescharacteristic
223C4×Dic53C22⋊C411yescharacteristic
224C4×Dic53C22×C411yescharacteristic
225C4×Dic53C22×C411yescharacteristic
226C4×Dic53C22×C411yescharacteristic
227C4×Dic53C22×C411yescharacteristic
228C4×Dic53C2×D411yescharacteristic
229C4×Dic53C2×D411yescharacteristic
230C4×Dic53C2×D411yescharacteristic
231C4×Dic53C2×D411yescharacteristic
232C2×#88322—22no—
233C2×#88322—42no—
234C23×C53—22no—
235C23×C53—41no—
236C23×C53—41no—
237C23×C53—42no—
238C23×C53—42no—
239C23×C53—42no—
240C23×C53C2×C22⋊C411yescharacteristic
241C23×C53C2×C22⋊C411yescharacteristic
242C23×C53AGL(1,ℤ/8)12yes—
243C23×C53#13712yes—
244C23×C53C22×D411yescharacteristic
245D4×C53—42no—
246D4×C53—42no—
247D4×C53—42no—
248D4×C53—44no—
249D4×C53—42no—
250C2×C4×C53—22no—
251C2×C4×C53—22no—
252C2×C4×C53—22no—
253C2×C4×C53—22no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—42no—
258C2×C4×C53—42no—
259C2×C4×C53—42no—
260C2×C4×C53—42no—
261C2×C4×C53C2×C22⋊C411yescharacteristic
262C2×C4×C53C2×C22⋊C411yescharacteristic
263C2×C4×C53C23×C411yescharacteristic
264C2×C4×C53C22×D411yescharacteristic
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—21no—
268C2×Dic53—21no—
269C2×Dic53—22no—
270C2×Dic53—22no—
271C2×Dic53—22no—
272C2×Dic53—22no—
273C2×Dic53—22no—
274C2×Dic53—22no—
275C2×Dic53—22no—
276C2×Dic53—22no—
277C2×Dic53—22no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—44no—
282C2×Dic53—42no—
283C2×Dic53—42no—
284C2×Dic53—44no—
285C2×Dic53—42no—
286C2×Dic53—42no—
287C2×Dic53—42no—
288C2×Dic53C2×C22⋊C411yescharacteristic
289C2×Dic53C2×C22⋊C411yescharacteristic
290C2×Dic53C2×C22⋊C411yescharacteristic
291C2×Dic53C2×C22⋊C411yescharacteristic
292C2×Dic53C2×C22⋊C411yescharacteristic
293C2×Dic53C2×C22⋊C411yescharacteristic
294C2×Dic53C2×C22⋊C411yescharacteristic
295C2×Dic53C2×C22⋊C411yescharacteristic
296Dic106—42no—
297Dic106—42no—
298C8×C53—42no—
299#88322—42no—
300#17581—531nomaximal, Sylow
301C22×C53—21no—
302C22×C53—21no—
303C22×C53—21no—
304C22×C53—21no—
305C22×C53—21no—
306C22×C53—21no—
307C22×C53—42no—
308C22×C53—42no—
309C22×C53—42no—
310C22×C53—42no—
311C22×C53—42no—
312C22×C53—84no—
313C22×C53#41712yes—
314C22×C53#41712yes—
315C22×C53C22×C22⋊C411yescharacteristic
316C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
317C22×C53C2×#13711yescharacteristic
318C4×C53—21no—
319C4×C53—21no—
320C4×C53—21no—
321C4×C53—42no—
322C4×C53—82no—
323C4×C53#41012yes—
324Dic53—21no—
325Dic53—21no—
326Dic53—21no—
327Dic53—21no—
328Dic53—21no—
329Dic53—21no—
330Dic53—21no—
331Dic53—42no—
332Dic53—42no—
333Dic53—82no—
334Dic53#42112yes—
335C2×#529—531no—
336C2×#432—531no—
337#2285—531no—
338#1520—531no—
339#1520—531no—
340#1519—531no—
341#1519D5311yescharacteristic
342#1512—534no—
343#1509—534no—
344C2×C53—21no—
345C2×C53—22no—
346C2×C53—21no—
347C2×C53—22no—
348C2×C53—82no—
349C2×C53C2×#41011yescharacteristic
350C2×C53C2×#41711yescharacteristic
351C2×C53C2×#42111yescharacteristic
352#529—532no—
353#529—532no—
354#529—532no—
355#529—532no—
356#529—1061no—
357#529—1061no—
358#529—1061no—
359#529—1061no—
360C2×#126—1061no—
361C2×#125—531no—
362C2×#124—531no—
363C2×#123—1061no—
364C2×C22⋊Q8—531no—
365C2×#121C2×D5311yescharacteristic
366C2×C4×Q8—531no—
367C2×C4×D4—531no—
368C2×C4×D4—1061no—
369C2×#117—1061no—
370C2×#117—1061no—
371C22×C42—531no—
372#432—532no—
373#432—532no—
374#432—1061no—
375#432—1061no—
376C2×C4⋊C8—1061no—
377C2×C22⋊C8—531no—
378C2×C22⋊C8C2×D5311yescharacteristic
379C2×#97—1061no—
380#384—1061no—
381#382—532no—
382#380—531no—
383#380—1061no—
384#380C2×D5311yescharacteristic
385C4×C4⋊C4—532no—
386C4×C4⋊C4—1061no—
387#328—534no—
388#328—534no—
389#327—534no—
390#327C2×D5314yes—
391C53#1758111yescharacteristic, complemented, Sylow
392C22×Q8—531no—
393C22×D4—21no—
394C22×D4C4×D5311yescharacteristic
395C23×C4—531no—
396C23×C4—531no—
397C23×C4—1061no—
398C23×C4C22×D5311yescharacteristic
399C22×C8—21no—
400C22×C8—1061no—
401#126—1062no—
402#126—1062no—
403#126—1062no—
404#126—1062no—
405#125—532no—
406#125—532no—
407#125—1061no—
408#125—1061no—
409#124—532no—
410#124—532no—
411#124—1061no—
412#124—1061no—
413#123—1062no—
414#123—1062no—
415#123—1062no—
416#123—1062no—
417C22⋊Q8—532no—
418C22⋊Q8—532no—
419C22⋊Q8—1061no—
420C22⋊Q8—1061no—
421#121—21no—
422#121—21no—
423#121C4×D5312yes—
424#121C22×D5312yes—
425C4×Q8—532no—
426C4×Q8—532no—
427C4×Q8—1061no—
428C4×Q8—1061no—
429C4×D4—532no—
430C4×D4—532no—
431C4×D4—1061no—
432C4×D4—1061no—
433C4×D4—1062no—
434C4×D4—1062no—
435C4×D4—1062no—
436C4×D4—1062no—
437#117—1062no—
438#117—1062no—
439#117—1062no—
440#117—1062no—
441#117—2124no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—531no—
444C2×C4⋊C4—1061no—
445C2×C4⋊C4—1061no—
446C2×C4⋊C4—1061no—
447C2×C4⋊C4—1061no—
448C2×C4⋊C4—1062no—
449C2×C4⋊C4—1062no—
450C2×C4⋊C4—1061no—
451C2×C4⋊C4C4×D5311yescharacteristic
452C2×C22⋊C4—21no—
453C2×C22⋊C4—21no—
454C2×C22⋊C4—1061no—
455C2×C22⋊C4—1061no—
456C2×C22⋊C4—1061no—
457C2×C22⋊C4—1061no—
458C2×C22⋊C4—1061no—
459C2×C42—531no—
460C2×C42—531no—
461C2×C42—531no—
462C2×C42—531no—
463C2×C42—531no—
464C2×C42—531no—
465C2×C42—531no—
466C2×C42—531no—
467C2×C42—531no—
468C2×C42—531no—
469C2×C42—531no—
470C2×C42—531no—
471C2×C42—1061no—
472C2×C42—1062no—
473C4⋊C8—1062no—
474C4⋊C8—1062no—
475C22⋊C8—21no—
476C22⋊C8—532no—
477C22⋊C8—1061no—
478C22⋊C8C22×D5312yes—
479#97—1062no—
480#97—1062no—
481#95—532no—
482#95—1062no—
483#95C22×D5312yes—
484C24—21no—
485C24D4×D5311yescharacteristic
486C2×Q8—532no—
487C2×Q8—1061no—
488C2×Q8—2122no—
489C2×D4—22no—
490C2×D4—22no—
491C2×D4—21no—
492C2×D4—42no—
493C2×D4—42no—
494C2×D4—42no—
495C2×D4C2×C4×D5312yes—
496C22×C4—21no—
497C22×C4—22no—
498C22×C4—42no—
499C22×C4—531no—
500C22×C4—531no—
501C22×C4—531no—
502C22×C4—531no—
503C22×C4—531no—
504C22×C4—531no—
505C22×C4—531no—
506C22×C4—531no—
507C22×C4—531no—
508C22×C4—531no—
509C22×C4—531no—
510C22×C4—531no—
511C22×C4—1062no—
512C22×C4—1062no—
513C22×C4—1062no—
514C22×C4—1062no—
515C22×C4—1061no—
516C22×C4—1061no—
517C22×C4—1062no—
518C22×C4—2122no—
519C22×C4—2124no—
520C22×C4C2×C4×D5311yescharacteristic
521C22×C4C2×C4×D5311yescharacteristic
522C22×C4D4×D5311yescharacteristic
523C22×C4D4×D5311yescharacteristic
524C22×C4D4×D5311yescharacteristic
525C22×C4C23×D5311yescharacteristic
526C2×C8—22no—
527C2×C8—42no—
528C2×C8—1062no—
529C2×C8—2122no—
530C4⋊C4—21no—
531C4⋊C4—42no—
532C4⋊C4—532no—
533C4⋊C4—1062no—
534C4⋊C4—1062no—
535C4⋊C4—1062no—
536C4⋊C4—1062no—
537C4⋊C4—1062no—
538C4⋊C4—1062no—
539C4⋊C4—1061no—
540C4⋊C4—1062no—
541C4⋊C4—1062no—
542C4⋊C4—2122no—
543C4⋊C4—2124no—
544C4⋊C4—2124no—
545C4⋊C4C2×C4×D5312yes—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—42no—
549C22⋊C4—1062no—
550C22⋊C4—1062no—
551C22⋊C4—1062no—
552C22⋊C4—1062no—
553C22⋊C4—1062no—
554C22⋊C4—1062no—
555C22⋊C4—1062no—
556C22⋊C4—1062no—
557C22⋊C4—2122no—
558C42—531no—
559C42—531no—
560C42—531no—
561C42—531no—
562C42—531no—
563C42—531no—
564C42—531no—
565C42—531no—
566C42—531no—
567C42—531no—
568C42—531no—
569C42—531no—
570C42—531no—
571C42—531no—
572C42—531no—
573C42—531no—
574C42—1062no—
575C42—1062no—
576C42—2124no—
577C23—22no—
578C23—41no—
579C23—41no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23C22⋊C4×D5311yescharacteristic
584C23C22⋊C4×D5311yescharacteristic
585C23#1313324112yes—
586C23#1313324612yes—
587C23C2×D4×D5311yescharacteristic
588Q8—2122no—
589Q8—2122no—
590D4—42no—
591D4—44no—
592D4—42no—
593D4—42no—
594D4—42no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—531no—
607C2×C4—531no—
608C2×C4—531no—
609C2×C4—531no—
610C2×C4—531no—
611C2×C4—531no—
612C2×C4—531no—
613C2×C4—531no—
614C2×C4—1062no—
615C2×C4—1062no—
616C2×C4—1062no—
617C2×C4—1062no—
618C2×C4—1062no—
619C2×C4—1062no—
620C2×C4—1062no—
621C2×C4—1061no—
622C2×C4—1061no—
623C2×C4—1061no—
624C2×C4—1061no—
625C2×C4—1062no—
626C2×C4—1062no—
627C2×C4—2124no—
628C2×C4—2121no—
629C2×C4—2121no—
630C2×C4—2121no—
631C2×C4—2124no—
632C2×C4—2122no—
633C2×C4—2122no—
634C2×C4—2122no—
635C2×C4—2122no—
636C2×C4—2122no—
637C2×C4C22⋊C4×D5311yescharacteristic
638C2×C4C22⋊C4×D5311yescharacteristic
639C2×C4C22×C4×D5311yescharacteristic
640C2×C4C2×D4×D5311yescharacteristic
641C8—42no—
642C8—2122no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—84no—
655C22#1681062112yes—
656C22#1681062112yes—
657C22C2×C22⋊C4×D5311yescharacteristic
658C22C2×#1313324111yescharacteristic
659C22C2×#1313324611yescharacteristic
660C4—21no—
661C4—21no—
662C4—21no—
663C4—42no—
664C4—82no—
665C4—532no—
666C4—1061no—
667C4—1061no—
668C4—1061no—
669C4—1061no—
670C4—1061no—
671C4—1061no—
672C4—1061no—
673C4—4242no—
674C4—2122no—
675C4—2122no—
676C4#1681059512yes—
677C2—21no—
678C2—22no—
679C2—22no—
680C2—21no—
681C2—82no—
682C2C2×#1681059511yescharacteristic
683C2C2×#1681062111yescharacteristic
684C2#421×D5311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1519C2
socleC22×C53C22×C22⋊C4
prime radicalC2×#121×C53C22
cosocleC22×C4×C53C24
hypercenter#1519D53
hypocenterC53#17581
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1519

Complements

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

NH ≅ G/Ntype
2. C2×#16810677C2semidirect
3. #20277117C2semidirect
4. #20273288C2semidirect
5. #20273286C2semidirect
7. #20273250C2semidirect
8. #20273237C2semidirect
14. #16811154C4semidirect
15. #16811154C4semidirect
19. C2×#13133208C4semidirect
20. C2×#13133204C4semidirect
23. C2×#13133182C4semidirect
25. C2×D4×Dic53C4semidirect
31. #16810677C22semidirect
32. #16810677C22semidirect
37. #16810487C22semidirect
41. #16810474C22semidirect
42. #16810361C22semidirect
43. #16810359C22semidirect
391. C53#17581semidirect

Maximal subgroups

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

subgroupindex
C2×#168111542
C2×#168106772
#202771172
#202732882
#202732862
#202732832
#202732502
#202732372
C53×#15192
#1758153