Finite Group Databasebeta

← #28314766

Subgroups — #28314766

Order 13568.

1036 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
1#20291391C211yesmaximal, characteristic, complemented
2C2×#16811626C211yesmaximal, characteristic, complemented
3#20290072C214yesmaximal, complemented
4#20290043C212yesmaximal, complemented
5#20290042C212yesmaximal, complemented
6#20290032C214yesmaximal, complemented
7C2×#16811159C211yesmaximal, characteristic
8C22×#11719940—21no—
9#16811644—21no—
10#16811643—21no—
11#16811640C2212yes—
12#16811639C2212yescomplemented
13#16811626—21no—
14#16811626—21no—
15#16811626—21no—
16#16811626C2212yes—
17#16811626C2212yescomplemented
18C2×#13133284—21no—
19C2×C4⋊#88324C2212yescomplemented
20#16811593—21no—
21C4×#11719940C2212yescomplemented
22Q8⋊#88324C2218yescomplemented
23#16811471C2218yescomplemented
24#16811159—21no—
25#16811159—21no—
26#16811159—21no—
27#16811159—21no—
28#16811159C412yes—
29#16811159C412yes—
30#16811159C2212yes—
31#16811159C2212yes—
32C2×#13133212—21no—
33C2×#13133208C411yescharacteristic
34C2×#13133206C411yescharacteristic
35C2×#13133201—21no—
36C2×#13133191C411yescharacteristic
37C2×Q8×Dic53C2211yescharacteristic
38C2×#13133179C2211yescharacteristic
39C2×#13133177—21no—
40C2×#13133175—21no—
41C22×C4×Dic53C2211yescharacteristic
42C2×C22⋊Q8×C53C411yescharacteristic
43C2×#11719940—22no—
44C2×#11719940—22no—
45C2×#11719940—22no—
46C2×#11719940—44no—
47C22×#88324—22no—
48#13133284—22no—
49#13133284—22no—
50#13133284—22no—
51C4⋊#88324—22no—
52C4⋊#88324—22no—
53C4⋊#88324C2314yes—
54C22⋊#88324—22no—
55C22⋊#88324—22no—
56#13133272—22no—
57C4×#88324C2314yes—
58C22×D212C2×C411yescharacteristic
59C22×C22⋊D53—21no—
60C23×Dic53D411yescharacteristic
61C23×Dic53D411yescharacteristic
62C22×C4×D53—21no—
63#13133212—22no—
64#13133212—22no—
65#13133212—22no—
66#13133212—22no—
67#13133208—21no—
68#13133208—21no—
69#13133208C2×C412yes—
70#13133208C2×C412yes—
71#13133206—21no—
72#13133206—21no—
73#13133206C2×C412yes—
74#13133206C2×C412yes—
75#13133201—22no—
76#13133201—22no—
77#13133201—22no—
78#13133201—22no—
79#13133191—21no—
80#13133191—21no—
81#13133191C2×C412yes—
82#13133191C2×C412yes—
83Q8×Dic53—21no—
84Q8×Dic53—21no—
85Q8×Dic53C2×C412yes—
86Q8×Dic53C2312yes—
87#13133179—21no—
88#13133179—21no—
89#13133179C2×C412yes—
90#13133179C2312yes—
91#13133177—22no—
92#13133177—22no—
93#13133177—22no—
94#13133177—22no—
95#13133175—22no—
96#13133175—22no—
97#13133175—22no—
98#13133175—22no—
99C2×C4⋊Dic53—21no—
100C2×C4⋊Dic53—21no—
101C2×C4⋊Dic53C2×C411yescharacteristic
102C2×#11719919—21no—
103C2×#11719921—21no—
104C2×#11719921—21no—
105C2×#11719921—21no—
106C2×#11719921—21no—
107C2×C4×Dic53—21no—
108C2×C4×Dic53C2×C411yescharacteristic
109C2×C4×Dic53C2×C411yescharacteristic
110C2×C4×Dic53C2×C411yescharacteristic
111C2×C4×Dic53D411yescharacteristic
112C2×C4×Dic53D411yescharacteristic
113C2×C4×Dic53D411yescharacteristic
114C2×C4×Dic53D411yescharacteristic
115C2×C4×Dic53D411yescharacteristic
116C2×C4×Dic53D411yescharacteristic
117C2×C4×Dic53C2311yescharacteristic
118C2×C4×Dic53C2311yescharacteristic
119C2×C4×Dic53C2311yescharacteristic
120C22×Q8×C53C2×C411yescharacteristic
121C23×C4×C53C2×C411yescharacteristic
122C22⋊Q8×C53—21no—
123C22⋊Q8×C53—21no—
124C22⋊Q8×C53C2×C412yes—
125C22⋊Q8×C53C2×C412yes—
126C2×C4⋊C4×C53—21no—
127C2×C4⋊C4×C53C2×C411yescharacteristic
128C2×C22⋊C4×C53—21no—
129C24×C53C22⋊C411yescharacteristic
130C23×D53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53—42no—
133C2×Q8×C53C22×C412yes—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—42no—
137C22×C4×C53C22⋊C411yescharacteristic
138C22×C4×C53C22⋊C411yescharacteristic
139C22×C4×C53C22⋊C411yescharacteristic
140C22×C4×C53C22×C411yescharacteristic
141C22×C4×C53C22×C411yescharacteristic
142C22×C4×C53C22×C411yescharacteristic
143C2×C22⋊D53—22no—
144C2×C22⋊D53—22no—
145C2×C22⋊D53—44no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—21no—
149C22×Dic53—42no—
150C22×Dic53C22⋊C411yescharacteristic
151C22×Dic53C22⋊C411yescharacteristic
152C22×Dic53C22⋊C411yescharacteristic
153C22×Dic53C22⋊C411yescharacteristic
154C22×Dic53C22⋊C411yescharacteristic
155C22×Dic53C22⋊C411yescharacteristic
156C22×Dic53C2×D411yescharacteristic
157C22×Dic53C2×D411yescharacteristic
158C22×Dic53C2×D411yescharacteristic
159C22×Dic53C2×D411yescharacteristic
160C22×Dic53C2×D411yescharacteristic
161C22×Dic53C2×D411yescharacteristic
162C2×D212—21no—
163C2×D212—42no—
164C2×D212C22×C412yes—
165C2×C4×D53—22no—
166C2×C4×D53—22no—
167C2×C4×D53—42no—
168C2×C4×D53—42no—
169#11719940—44no—
170#11719940—44no—
171#11719940—44no—
172C2×#88324—24no—
173C2×#88324—24no—
174C2×#88324—44no—
175C4⋊C4×C53—22no—
176C4⋊C4×C53—22no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53C22×C412yes—
179C22⋊C4×C53—22no—
180C22⋊C4×C53—22no—
181#11719921—22no—
182#11719921—22no—
183#11719921—22no—
184#11719921—22no—
185#11719921—22no—
186#11719921—22no—
187#11719921—22no—
188#11719921—22no—
189C4⋊Dic53—21no—
190C4⋊Dic53—22no—
191C4⋊Dic53—22no—
192C4⋊Dic53—22no—
193C4⋊Dic53—22no—
194C4⋊Dic53C22×C412yes—
195#11719919—22no—
196#11719919—22no—
197C4×Dic53—22no—
198C4×Dic53—22no—
199C4×Dic53C22⋊C411yescharacteristic
200C4×Dic53C22⋊C411yescharacteristic
201C4×Dic53C22⋊C411yescharacteristic
202C4×Dic53C22⋊C411yescharacteristic
203C4×Dic53C22⋊C411yescharacteristic
204C4×Dic53C22⋊C411yescharacteristic
205C4×Dic53C22×C411yescharacteristic
206C4×Dic53C22×C411yescharacteristic
207C4×Dic53C22×C411yescharacteristic
208C4×Dic53C2×D411yescharacteristic
209C4×Dic53C2×D411yescharacteristic
210C4×Dic53C2×D411yescharacteristic
211C4×Dic53C2×D411yescharacteristic
212C4×Dic53C2×D411yescharacteristic
213C4×Dic53C2×D411yescharacteristic
214C4×Dic53C2411yescharacteristic
215C23×C53—22no—
216C23×C53—22no—
217C23×C53—41no—
218C23×C53C2×C22⋊C411yescharacteristic
219C23×C53C2×C22⋊C411yescharacteristic
220C23×C53C2×C22⋊C411yescharacteristic
221C22×D53—22no—
222C22×D53—42no—
223C22×D53—42no—
224C22×D53—42no—
225Q8×C53—24no—
226Q8×C53—42no—
227C2×C4×C53—22no—
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—42no—
236C2×C4×C53—42no—
237C2×C4×C53C2×C22⋊C411yescharacteristic
238C2×C4×C53C2×C22⋊C411yescharacteristic
239C2×C4×C53C2×C22⋊C411yescharacteristic
240C2×C4×C53C23×C411yescharacteristic
241C22⋊D53—44no—
242C22⋊D53—44no—
243C2×Dic53—22no—
244C2×Dic53—22no—
245C2×Dic53—22no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—22no—
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—42no—
258C2×Dic53C2×C22⋊C411yescharacteristic
259C2×Dic53C2×C22⋊C411yescharacteristic
260C2×Dic53C2×C22⋊C411yescharacteristic
261C2×Dic53C2×C22⋊C411yescharacteristic
262C2×Dic53C2×C22⋊C411yescharacteristic
263C2×Dic53C2×C22⋊C411yescharacteristic
264C2×Dic53AGL(1,ℤ/8)12yes—
265C2×Dic53#13712yes—
266C2×Dic53C22×D411yescharacteristic
267C2×Dic53C22×D411yescharacteristic
268D212—24no—
269D212—42no—
270C4×D53—42no—
271C4×D53—42no—
272C4×D53—44no—
273#88324—44no—
274#88324—44no—
275#17492—531nomaximal, Sylow
276C22×C53—21no—
277C22×C53—21no—
278C22×C53—21no—
279C22×C53—21no—
280C22×C53—21no—
281C22×C53—21no—
282C22×C53—21no—
283C22×C53—44no—
284C22×C53#41712yes—
285C22×C53#42112yes—
286C22×C53C22×C22⋊C411yescharacteristic
287C2×D53—42no—
288C2×D53—42no—
289C2×D53—42no—
290C2×D53—84no—
291C4×C53—21no—
292C4×C53—21no—
293C4×C53—21no—
294C4×C53—42no—
295C4×C53—42no—
296C4×C53#56812yes—
297Dic53—21no—
298Dic53—21no—
299Dic53—21no—
300Dic53—21no—
301Dic53—21no—
302Dic53—21no—
303Dic53—42no—
304Dic53#41712yes—
305Dic53C2×AGL(1,ℤ/8)11yescharacteristic
306Dic53C2×#13711yescharacteristic
307C2×#529—531no—
308#2920—531no—
309C2×#423—531no—
310#1495—534no—
311#1487—532no—
312#1486—532no—
313#1483—534no—
314C2×C53—22no—
315C2×C53—22no—
316C2×C53—41no—
317C2×C53#287911yescharacteristic
318C2×C53C2×#41711yescharacteristic
319C2×C53C2×#42111yescharacteristic
320D53—82no—
321C22×M16—1061no—
322#529—532no—
323#529—532no—
324#529—532no—
325#529—532no—
326#529—1061no—
327#529—1061no—
328#529—1061no—
329#529—1061no—
330C2×#126—1061no—
331C2×#125—531no—
332C2×#124—531no—
333C2×#123—1061no—
334C2×C22⋊Q8#615711yescharacteristic
335C2×#121—531no—
336C2×C4×Q8—531no—
337C2×C4×D4—531no—
338C2×C4×D4—1061no—
339C2×#117—1061no—
340C22×C42—531no—
341#432—1061no—
342#432—1061no—
343#431—532no—
344#431—532no—
345#423—532no—
346#423—532no—
347#423—1061no—
348#423—1061no—
349#423—1061no—
350C2×C4⋊C8—532no—
351C2×C4⋊C8—1061no—
352#407—1061no—
353C4×M16—532no—
354Q8⋊C8—538no—
355D4⋊C8—538no—
356C53#1749211yescharacteristic, complemented, Sylow
357C22×Q8C2×#615711yescharacteristic
358C22×D4—531no—
359C22×D4—1061no—
360C23×C4—531no—
361C23×C4—531no—
362C23×C4—1061no—
363C23×C4C2×#615711yescharacteristic
364C2×M16—1062no—
365C2×M16—1062no—
366C2×M16—1062no—
367C2×M16—2124no—
368C22×C8—1062no—
369#126—1062no—
370#126—1062no—
371#126—1062no—
372#126—1062no—
373#125—532no—
374#125—532no—
375#125—1061no—
376#125—1061no—
377#124—532no—
378#124—532no—
379#124—1061no—
380#124—1061no—
381#123—1062no—
382#123—1062no—
383#123—1062no—
384#123—1062no—
385C22⋊Q8—21no—
386C22⋊Q8—21no—
387C22⋊Q8C2×#615712yes—
388C22⋊Q8C2×#615712yes—
389#121—532no—
390#121—532no—
391#121—1061no—
392#121—1061no—
393C4×Q8—532no—
394C4×Q8—532no—
395C4×Q8—1061no—
396C4×Q8—1061no—
397C4×D4—532no—
398C4×D4—532no—
399C4×D4—1061no—
400C4×D4—1061no—
401C4×D4—1062no—
402C4×D4—1062no—
403C4×D4—1062no—
404C4×D4—1062no—
405#117—1062no—
406#117—1062no—
407#117—1062no—
408#117—1062no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4C2×#615711yescharacteristic
415C2×C22⋊C4—21no—
416C2×C22⋊C4—1061no—
417C2×C22⋊C4—1061no—
418C2×C22⋊C4—1061no—
419C2×C22⋊C4—1061no—
420C2×C42—531no—
421C2×C42—531no—
422C2×C42—531no—
423C2×C42—531no—
424C2×C42—531no—
425C2×C42—531no—
426C2×C42—531no—
427C2×C42—531no—
428C2×C42—531no—
429C2×C42—531no—
430C2×C42—531no—
431C2×C42—531no—
432C2×C42—1061no—
433C4⋊C8—534no—
434C4⋊C8—1062no—
435C4⋊C8—1062no—
436C4⋊C8—1062no—
437C4⋊C8—1062no—
438C4⋊C8—1062no—
439C22⋊C8—1062no—
440C22⋊C8—1062no—
441#97—1062no—
442C4×C8—534no—
443C24—1061no—
444C24C22⋊#615711yescharacteristic
445C2×Q8—21no—
446C2×Q8—42no—
447C2×Q8C22×#615712yes—
448C2×D4—532no—
449C2×D4—1062no—
450C2×D4—1062no—
451C2×D4—1061no—
452C2×D4—2122no—
453C2×D4—2124no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—42no—
457C22×C4—531no—
458C22×C4—531no—
459C22×C4—531no—
460C22×C4—531no—
461C22×C4—531no—
462C22×C4—531no—
463C22×C4—531no—
464C22×C4—531no—
465C22×C4—531no—
466C22×C4—531no—
467C22×C4—531no—
468C22×C4—531no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1061no—
474C22×C4—2122no—
475C22×C4—2122no—
476C22×C4—2122no—
477C22×C4C22⋊#615711yescharacteristic
478C22×C4C22⋊#615711yescharacteristic
479C22×C4C22⋊#615711yescharacteristic
480C22×C4C22×#615711yescharacteristic
481C22×C4C22×#615711yescharacteristic
482C22×C4C22×#615711yescharacteristic
483M16—2124no—
484M16—2124no—
485M16—2124no—
486C2×C8—1064no—
487C2×C8—1064no—
488C2×C8—2124no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—532no—
493C4⋊C4—1062no—
494C4⋊C4—1062no—
495C4⋊C4—1062no—
496C4⋊C4—1062no—
497C4⋊C4—1062no—
498C4⋊C4—1062no—
499C4⋊C4—1061no—
500C4⋊C4C22×#615712yes—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—1062no—
504C22⋊C4—1062no—
505C22⋊C4—1062no—
506C22⋊C4—1062no—
507C22⋊C4—1062no—
508C22⋊C4—1062no—
509C22⋊C4—1062no—
510C22⋊C4—1062no—
511C42—531no—
512C42—531no—
513C42—531no—
514C42—531no—
515C42—531no—
516C42—531no—
517C42—531no—
518C42—531no—
519C42—531no—
520C42—531no—
521C42—531no—
522C42—531no—
523C42—531no—
524C42—531no—
525C42—531no—
526C42—531no—
527C42—1062no—
528C42—1062no—
529C23—22no—
530C23—22no—
531C23—41no—
532C23—1062no—
533C23—2122no—
534C23—2122no—
535C23—2122no—
536C23C2×C22⋊#615711yescharacteristic
537C23C2×C22⋊#615711yescharacteristic
538C23C2×C22⋊#615711yescharacteristic
539Q8—24no—
540Q8—42no—
541D4—1064no—
542D4—2122no—
543D4—2124no—
544D4—2124no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—532no—
556C2×C4—532no—
557C2×C4—531no—
558C2×C4—531no—
559C2×C4—531no—
560C2×C4—531no—
561C2×C4—531no—
562C2×C4—531no—
563C2×C4—531no—
564C2×C4—531no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—1061no—
569C2×C4—1062no—
570C2×C4—1062no—
571C2×C4—1062no—
572C2×C4—1062no—
573C2×C4—1062no—
574C2×C4—1062no—
575C2×C4—2122no—
576C2×C4—2122no—
577C2×C4—2124no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2122no—
583C2×C4C2×C22⋊#615711yescharacteristic
584C2×C4C2×C22⋊#615711yescharacteristic
585C2×C4C2×C22⋊#615711yescharacteristic
586C2×C4C23×#615711yescharacteristic
587C8—2124no—
588C8—2124no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—44no—
597C22—4244no—
598C22—2122no—
599C22—2122no—
600C22—2122no—
601C22#1681161412yes—
602C22#1681162212yes—
603C22C22×C22⋊#615711yescharacteristic
604C4—21no—
605C4—21no—
606C4—21no—
607C4—42no—
608C4—42no—
609C4—531no—
610C4—531no—
611C4—532no—
612C4—1061no—
613C4—1061no—
614C4—1061no—
615C4—1061no—
616C4—1061no—
617C4—1061no—
618C4—2122no—
619C4#1681170112yes—
620C2—22no—
621C2—22no—
622C2—41no—
623C2—4242no—
624C2#2029129911yescharacteristic
625C2C2×#1681161411yescharacteristic
626C2C2×#1681162211yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC2×C22⋊Q8×C53C4
socleC22×C53C22×C22⋊C4
prime radicalC2×#13133191C4
cosocleC4×Dic53C24
hypercenterC2×C22⋊Q8#6157
hypocenterC53#17492
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #20291391C2semidirect
2. C2×#16811626C2semidirect
3. #20290072C2semidirect
4. #20290043C2semidirect
5. #20290042C2semidirect
6. #20290032C2semidirect
12. #16811639C22semidirect
17. #16811626C22semidirect
19. C2×C4⋊#88324C22semidirect
21. C4×#11719940C22semidirect
22. Q8⋊#88324C22semidirect
23. #16811471C22semidirect
356. C53#17492semidirect

Maximal subgroups

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

subgroupindex
#202913912
C2×#168116262
#202900722
#202900432
#202900422
#202900322
C2×#168111592
#1749253