Finite Group Databasebeta

← #27350537

Subgroups — #27350537

Order 13568.

1214 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D53C211yesmaximal, characteristic
2#20283346C211yesmaximal, characteristic
3#20283345C211yesmaximal, characteristic
4#20273163C211yesmaximal, characteristic, complemented
5#20273162C211yesmaximal, characteristic, complemented
6#20273161C211yesmaximal, characteristic, complemented
7#20273142C214yesmaximal, complemented
8#20273108C214yesmaximal, complemented
9C53×#1491C211yesmaximal, characteristic, complemented
10C22×Q8×D53C411yescharacteristic
11C2×Q8×Dic53C411yescharacteristic
12C2×C4×Dic106—21no—
13C2×C4×Dic106C2211yescharacteristic
14C2×#13133182—21no—
15C2×#13133182C411yescharacteristic
16C4×Q8×D53—21no—
17C4×Q8×D53—21no—
18C4×Q8×D53—21no—
19C4×Q8×D53—21no—
20C4×Q8×D53C412yes—
21C4×Q8×D53C412yes—
22C4×Q8×D53C2212yes—
23C4×Q8×D53C2212yes—
24C2×C4⋊C4×D53—21no—
25C2×C4⋊C4×D53C411yescharacteristic
26C2×C42×D53—21no—
27C2×C42×D53C2211yescharacteristic
28#16810680C2211yescharacteristic
29#16810678—21no—
30#16810678—21no—
31#16810678—21no—
32#16810675C2212yes—
33#16810674C2212yes—
34#16810674C2212yes—
35#16810673—21no—
36#16810646—21no—
37#16810645C2212yes—
38#16810586—21no—
39#16810585—21no—
40C4×#11719917C2211yescharacteristic
41#16810575—21no—
42#16810355C2214yescomplemented
43Q8⋊#88322C2214yescomplemented
44#16810353C2214yescomplemented
45C2×C4×Q8×C53C2211yescharacteristic
46C53×#432C2211yescharacteristic
47C4×M16×C53C2211yescharacteristic
48C53×Q8⋊C8C2214yescomplemented
49C22×Dic106—21no—
50C22×Dic106C2×C411yescharacteristic
51C2×Q8×D53—28no—
52C2×Q8×D53—21no—
53C2×Q8×D53—21no—
54C2×Q8×D53C2×C412yes—
55C2×Q8×D53C2×C412yes—
56C22×C4×D53—21no—
57C22×C4×D53—21no—
58C22×C4×D53C2×C411yescharacteristic
59C22×C4×D53D411yescharacteristic
60C22×C4×D53D411yescharacteristic
61C2×#11719917—21no—
62C2×#11719912—22no—
63Q8×Dic53—21no—
64Q8×Dic53—21no—
65Q8×Dic53C2×C412yes—
66Q8×Dic53C2×C412yes—
67C4×Dic106—21no—
68C4×Dic106—21no—
69C4×Dic106—22no—
70C4×Dic106—22no—
71C4×Dic106—22no—
72C4×Dic106—22no—
73C4×Dic106C2×C412yes—
74C4×Dic106C2312yes—
75#13133182—21no—
76#13133182—21no—
77#13133182—22no—
78#13133182—22no—
79#13133182—22no—
80#13133182—22no—
81#13133182C2×C412yes—
82#13133182C2×C412yes—
83C2×C4⋊Dic53—21no—
84C2×C4⋊Dic53C2×C411yescharacteristic
85C2×#11719919—21no—
86C2×#11719919—21no—
87C2×#11719919—21no—
88C4⋊C4×D53—22no—
89C4⋊C4×D53—22no—
90C4⋊C4×D53—21no—
91C4⋊C4×D53—21no—
92C4⋊C4×D53—22no—
93C4⋊C4×D53—22no—
94C4⋊C4×D53C2×C412yes—
95C4⋊C4×D53C2×C412yes—
96C2×C4×Dic53—21no—
97C2×C4×Dic53—21no—
98C2×C4×Dic53—21no—
99C2×C4×Dic53C2×C411yescharacteristic
100C2×C4×Dic53D411yescharacteristic
101C2×C4×Dic53D411yescharacteristic
102C42×D53—22no—
103C42×D53—22no—
104C42×D53—22no—
105C42×D53—22no—
106C42×D53C2×C411yescharacteristic
107C42×D53C2×C411yescharacteristic
108C42×D53D411yescharacteristic
109C42×D53D411yescharacteristic
110C42×D53D411yescharacteristic
111C42×D53D411yescharacteristic
112C42×D53C2311yescharacteristic
113C42×D53C2311yescharacteristic
114C4⋊#88322C2312yes—
115#13133146—22no—
116#13133146—22no—
117#13133146—22no—
118C22⋊#88322—21no—
119#13133129—22no—
120#13133129—22no—
121#13133129—22no—
122#13133128—21no—
123#13133128—21no—
124C4×#88322C2312yes—
125C8×Dic53—22no—
126C22×Q8×C53C2×C411yescharacteristic
127C2×M16×C53—21no—
128C4×Q8×C53—21no—
129C4×Q8×C53—21no—
130C4×Q8×C53C2×C412yes—
131C4×Q8×C53C2312yes—
132C2×C4⋊C4×C53—21no—
133C2×C4⋊C4×C53C2×C411yescharacteristic
134C2×C42×C53—21no—
135C2×C42×C53C2311yescharacteristic
136C4⋊C8×C53C2312yes—
137C22⋊C8×C53—21no—
138#97×C53—21no—
139#97×C53—21no—
140C4×C8×C53C2312yes—
141C23×D53C22⋊C411yescharacteristic
142C2×Q8×C53—24no—
143C2×Q8×C53—21no—
144C2×Q8×C53C22×C412yes—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53C22×C411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53C22⋊C411yescharacteristic
153C22×Dic53C22⋊C411yescharacteristic
154C22×Dic53C22⋊C411yescharacteristic
155Q8×D53—28no—
156Q8×D53—28no—
157Q8×D53—24no—
158Q8×D53—24no—
159Q8×D53—24no—
160Q8×D53M1618yescomplemented
161C2×C4×D53—28no—
162C2×C4×D53—22no—
163C2×C4×D53—22no—
164C2×C4×D53—24no—
165C2×C4×D53—24no—
166C2×C4×D53—22no—
167C2×C4×D53—22no—
168C2×C4×D53—42no—
169C2×C4×D53—44no—
170C2×C4×D53C22⋊C411yescharacteristic
171C2×C4×D53C22⋊C411yescharacteristic
172C2×C4×D53C22⋊C411yescharacteristic
173C2×C4×D53C22⋊C411yescharacteristic
174C2×C4×D53C22⋊C411yescharacteristic
175C2×C4×D53C22⋊C411yescharacteristic
176C2×C4×D53C22×C411yescharacteristic
177C2×C4×D53C22×C411yescharacteristic
178C2×C4×D53C2×D411yescharacteristic
179C2×C4×D53C2×D411yescharacteristic
180C2×C4×D53C2×D411yescharacteristic
181C2×C4×D53C2×D411yescharacteristic
182C2×Dic106—28no—
183C2×Dic106—21no—
184C2×Dic106—22no—
185C2×Dic106—22no—
186C2×Dic106—24no—
187C2×Dic106C22×C412yes—
188M16×C53—42no—
189C2×C8×C53—22no—
190C2×C8×C53—22no—
191C4⋊C4×C53—22no—
192C4⋊C4×C53—21no—
193C4⋊C4×C53—22no—
194C4⋊C4×C53C22×C412yes—
195C42×C53—22no—
196C42×C53—22no—
197C42×C53C22×C411yescharacteristic
198C42×C53C2×D411yescharacteristic
199C42×C53C2×D411yescharacteristic
200C42×C53C2411yescharacteristic
201C4⋊Dic53—21no—
202C4⋊Dic53—22no—
203C4⋊Dic53—22no—
204C4⋊Dic53C22×C412yes—
205#11719919—22no—
206#11719919—22no—
207#11719919—22no—
208#11719919—22no—
209#11719919—22no—
210#11719919—22no—
211C4×Dic53—22no—
212C4×Dic53—22no—
213C4×Dic53—22no—
214C4×Dic53—22no—
215C4×Dic53—22no—
216C4×Dic53—22no—
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×Dic53C2×D411yescharacteristic
226C4×Dic53C2×D411yescharacteristic
227C4×Dic53C2×D411yescharacteristic
228C4×Dic53C2×D411yescharacteristic
229#11719917—42no—
230C2×#88322—22no—
231C2×#88322—22no—
232#11719912—44no—
233C23×C53C2×C22⋊C411yescharacteristic
234C22×D53—22no—
235C22×D53—22no—
236C22×D53—22no—
237C22×D53C2×C22⋊C411yescharacteristic
238C22×D53C2×C22⋊C411yescharacteristic
239Q8×C53—24no—
240Q8×C53—22no—
241Q8×C53C2×M1614yes—
242C2×C4×C53—24no—
243C2×C4×C53—22no—
244C2×C4×C53—22no—
245C2×C4×C53—22no—
246C2×C4×C53—22no—
247C2×C4×C53—22no—
248C2×C4×C53—42no—
249C2×C4×C53C2×C22⋊C411yescharacteristic
250C2×C4×C53C2×C22⋊C411yescharacteristic
251C2×C4×C53C2×C22⋊C411yescharacteristic
252C2×C4×C53C23×C411yescharacteristic
253C2×C4×C53C22×D411yescharacteristic
254C2×C4×C53C22×D411yescharacteristic
255C2×Dic53—24no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—22no—
261C2×Dic53—42no—
262C2×Dic53C2×C22⋊C411yescharacteristic
263C2×Dic53C2×C22⋊C411yescharacteristic
264C2×Dic53C2×C22⋊C411yescharacteristic
265C2×Dic53C2×C22⋊C411yescharacteristic
266C2×Dic53C2×C22⋊C411yescharacteristic
267C2×Dic53C2×C22⋊C411yescharacteristic
268C4×D53—28no—
269C4×D53—28no—
270C4×D53—22no—
271C4×D53—22no—
272C4×D53—22no—
273C4×D53—22no—
274C4×D53—22no—
275C4×D53—22no—
276C4×D53—41no—
277C4×D53—41no—
278C4×D53—41no—
279C4×D53—41no—
280C4×D53—44no—
281C4×D53—44no—
282C4×D53C2×M1614yes—
283C4×D53#13714yes—
284Dic106—28no—
285Dic106—24no—
286Dic106—22no—
287Dic106—28no—
288Dic106C2×M1614yes—
289C8×C53—42no—
290C8×C53—42no—
291#88322—42no—
292#88322—42no—
293#17449—531nomaximal, Sylow
294C22×C53—21no—
295C22×C53—21no—
296C22×C53#42112yes—
297C22×C53C22×C22⋊C411yescharacteristic
298C2×D53—28no—
299C2×D53—21no—
300C2×D53—21no—
301C2×D53—21no—
302C2×D53—21no—
303C2×D53#40712yes—
304C2×D53#41912yes—
305C4×C53—24no—
306C4×C53—21no—
307C4×C53—21no—
308C4×C53—21no—
309C4×C53—21no—
310C4×C53—42no—
311C4×C53C22×M1612yes—
312C4×C53C2×#13712yes—
313Dic53—21no—
314Dic53—24no—
315Dic53—21no—
316Dic53—21no—
317Dic53—21no—
318Dic53—42no—
319Dic53#40712yes—
320Dic53#41912yes—
321C22×C4×Q8—531no—
322#2920—531no—
323#2920—531no—
324#1491—531no—
325#1491—531no—
326#1491—531no—
327#1491D5311yescharacteristic, complemented
328#1487—534no—
329#1479—534no—
330C2×C53—22no—
331C2×C53C2×#40711yescharacteristic
332C2×C53C2×#41911yescharacteristic
333C2×C53C2×#42111yescharacteristic
334D53—22no—
335D53—22no—
336C23×Q8—531no—
337C2×C4×Q8—531no—
338C2×C4×Q8—532no—
339C2×C4×Q8—532no—
340C2×C4×Q8—532no—
341C2×C4×Q8—532no—
342C2×C4×Q8—531no—
343C2×C4×Q8—531no—
344C2×C4×Q8—1061no—
345C2×C4×Q8—1061no—
346C2×C4×Q8—1061no—
347C2×C4×Q8—1061no—
348C2×C4×Q8—1061no—
349C2×C4×Q8—1061no—
350C2×C4×Q8C2×D5311yescharacteristic
351C22×C4⋊C4—531no—
352C22×C4⋊C4—1061no—
353C22×C42—531no—
354C22×C42—1061no—
355#432—531no—
356#432—1061no—
357#432—1061no—
358#432—1061no—
359#432C2×D5311yescharacteristic
360#431—532no—
361#431—532no—
362#431—532no—
363#431—1061no—
364#423—532no—
365#423—1061no—
366#407—1061no—
367#407—1061no—
368C4×M16—531no—
369C4×M16—1061no—
370C4×M16C2×D5311yescharacteristic
371Q8⋊C8—534no—
372Q8⋊C8—534no—
373Q8⋊C8—534no—
374Q8⋊C8C2×D5314yescomplemented
375C53#1744911yescharacteristic, complemented, Sylow
376C22×Q8—532no—
377C22×Q8—532no—
378C22×Q8—531no—
379C22×Q8—1061no—
380C22×Q8—1061no—
381C22×Q8—1061no—
382C22×Q8—1068no—
383C22×Q8C4×D5311yescharacteristic
384C23×C4—531no—
385C23×C4—531no—
386C23×C4—531no—
387C23×C4—1061no—
388C23×C4—1061no—
389C2×M16—21no—
390C2×M16—1062no—
391C2×M16—1061no—
392C4×Q8—21no—
393C4×Q8—21no—
394C4×Q8—532no—
395C4×Q8—532no—
396C4×Q8—532no—
397C4×Q8—532no—
398C4×Q8—532no—
399C4×Q8—532no—
400C4×Q8—1062no—
401C4×Q8—1062no—
402C4×Q8—1062no—
403C4×Q8—1062no—
404C4×Q8—1062no—
405C4×Q8—1062no—
406C4×Q8—1061no—
407C4×Q8—1061no—
408C4×Q8—1062no—
409C4×Q8—1062no—
410C4×Q8—1061no—
411C4×Q8—1061no—
412C4×Q8—1061no—
413C4×Q8—1061no—
414C4×Q8C4×D5312yes—
415C4×Q8C22×D5312yes—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—531no—
418C2×C4⋊C4—532no—
419C2×C4⋊C4—532no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4—1062no—
423C2×C4⋊C4—1062no—
424C2×C4⋊C4—1062no—
425C2×C4⋊C4—1062no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4C4×D5311yescharacteristic
431C2×C42—21no—
432C2×C42—531no—
433C2×C42—531no—
434C2×C42—531no—
435C2×C42—531no—
436C2×C42—531no—
437C2×C42—531no—
438C2×C42—531no—
439C2×C42—531no—
440C2×C42—531no—
441C2×C42—531no—
442C2×C42—531no—
443C2×C42—1062no—
444C2×C42—1062no—
445C2×C42—1062no—
446C2×C42—1062no—
447C2×C42—1061no—
448C2×C42—1061no—
449C2×C42—1061no—
450C2×C42C22×D5311yescharacteristic
451C4⋊C8—532no—
452C4⋊C8—1062no—
453C4⋊C8—1062no—
454C4⋊C8—1062no—
455C4⋊C8C22×D5312yes—
456C22⋊C8—21no—
457C22⋊C8—1062no—
458C22⋊C8—1062no—
459C22⋊C8—1062no—
460C22⋊C8—1061no—
461#97—21no—
462#97—21no—
463#97—1061no—
464#97—1061no—
465C4×C8—532no—
466C4×C8—1062no—
467C4×C8C22×D5312yes—
468C24—531no—
469C2×Q8—24no—
470C2×Q8—21no—
471C2×Q8—538no—
472C2×Q8—532no—
473C2×Q8—1062no—
474C2×Q8—1062no—
475C2×Q8—1064no—
476C2×Q8—1068no—
477C2×Q8—1064no—
478C2×Q8—1064no—
479C2×Q8—1068no—
480C2×Q8—1068no—
481C2×Q8—1064no—
482C2×Q8—1061no—
483C2×Q8C2×C4×D5312yes—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—531no—
487C22×C4—531no—
488C22×C4—531no—
489C22×C4—531no—
490C22×C4—531no—
491C22×C4—531no—
492C22×C4—531no—
493C22×C4—531no—
494C22×C4—531no—
495C22×C4—531no—
496C22×C4—531no—
497C22×C4—531no—
498C22×C4—531no—
499C22×C4—531no—
500C22×C4—531no—
501C22×C4—1064no—
502C22×C4—1064no—
503C22×C4—1062no—
504C22×C4—1062no—
505C22×C4—1068no—
506C22×C4—1062no—
507C22×C4—1062no—
508C22×C4—1061no—
509C22×C4—1061no—
510C22×C4—2122no—
511C22×C4—2124no—
512C22×C4C2×C4×D5311yescharacteristic
513C22×C4D4×D5311yescharacteristic
514C22×C4D4×D5311yescharacteristic
515M16—42no—
516M16—2124no—
517M16—2122no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—1062no—
521C2×C8—1062no—
522C4⋊C4—22no—
523C4⋊C4—21no—
524C4⋊C4—22no—
525C4⋊C4—532no—
526C4⋊C4—1062no—
527C4⋊C4—1062no—
528C4⋊C4—1062no—
529C4⋊C4—1062no—
530C4⋊C4—1062no—
531C4⋊C4—1062no—
532C4⋊C4—1061no—
533C4⋊C4—1062no—
534C4⋊C4—1062no—
535C4⋊C4C2×C4×D5312yes—
536C42—22no—
537C42—22no—
538C42—531no—
539C42—531no—
540C42—531no—
541C42—531no—
542C42—531no—
543C42—531no—
544C42—531no—
545C42—531no—
546C42—531no—
547C42—531no—
548C42—531no—
549C42—531no—
550C42—1062no—
551C42—1062no—
552C42—1062no—
553C42—1062no—
554C42—1062no—
555C42—1062no—
556C42C2×C4×D5311yescharacteristic
557C42D4×D5311yescharacteristic
558C42D4×D5311yescharacteristic
559C42C23×D5311yescharacteristic
560C23—531no—
561C23—531no—
562C23—1062no—
563C23—1062no—
564C23—1062no—
565C23C22⋊C4×D5311yescharacteristic
566Q8—24no—
567Q8—22no—
568Q8—534no—
569Q8—1068no—
570Q8—1068no—
571Q8—1064no—
572Q8—1062no—
573Q8M16×D5314yes—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—42no—
581C2×C4—531no—
582C2×C4—531no—
583C2×C4—534no—
584C2×C4—534no—
585C2×C4—531no—
586C2×C4—531no—
587C2×C4—531no—
588C2×C4—531no—
589C2×C4—1062no—
590C2×C4—1062no—
591C2×C4—1062no—
592C2×C4—1062no—
593C2×C4—1068no—
594C2×C4—1068no—
595C2×C4—1062no—
596C2×C4—1062no—
597C2×C4—1064no—
598C2×C4—1062no—
599C2×C4—1062no—
600C2×C4—1062no—
601C2×C4—1062no—
602C2×C4—1062no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4—2124no—
608C2×C4—2124no—
609C2×C4—2122no—
610C2×C4C22⋊C4×D5311yescharacteristic
611C2×C4C22⋊C4×D5311yescharacteristic
612C2×C4C22⋊C4×D5311yescharacteristic
613C2×C4C22×C4×D5311yescharacteristic
614C2×C4C2×D4×D5311yescharacteristic
615C2×C4C2×D4×D5311yescharacteristic
616C8—42no—
617C8—42no—
618C8—2122no—
619C8—2122no—
620C22—21no—
621C22—21no—
622C22—532no—
623C22—532no—
624C22—1068no—
625C22—1061no—
626C22—1061no—
627C22—1061no—
628C22—1061no—
629C22#1681063812yes—
630C22C2×C22⋊C4×D5311yescharacteristic
631C4—24no—
632C4—21no—
633C4—21no—
634C4—21no—
635C4—21no—
636C4—42no—
637C4—532no—
638C4—532no—
639C4—1061no—
640C4—1064no—
641C4—1061no—
642C4—1061no—
643C4—1061no—
644C4—2122no—
645C4C2×M16×D5312yes—
646C4#137×D5312yes—
647C2—22no—
648C2—1062no—
649C2—1062no—
650C2#407×D5311yescharacteristic
651C2#419×D5311yescharacteristic
652C2C2×#1681063811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1491C2
socleC22×C53C22×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC42×C53C24
hypercenter#1491D53
hypocenterC53#17449
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1491

Complements

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

NH ≅ G/Ntype
4. #20273163C2semidirect
5. #20273162C2semidirect
6. #20273161C2semidirect
7. #20273142C2semidirect
8. #20273108C2semidirect
9. C53×#1491C2semidirect
42. #16810355C22semidirect
43. Q8⋊#88322C22semidirect
44. #16810353C22semidirect
48. C53×Q8⋊C8C22semidirect
160. Q8×D53M16semidirect
327. #1491D53semidirect
374. Q8⋊C8C2×D53semidirect
375. C53#17449semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D532
#202833462
#202833452
#202731632
#202731622
#202731612
#202731422
#202731082
C53×#14912
#1744953