Finite Group Databasebeta

← #28319241

Subgroups — #28319241

Order 13568.

1342 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811648C212yesmaximal, complemented
2#20290607C211yesmaximal, characteristic, complemented
3#20290607C211yesmaximal, characteristic, complemented
4#20290574C212yesmaximal, complemented
5#20290572C211yesmaximal, characteristic, complemented
6#20290570C211yesmaximal, characteristic, complemented
7#20290540C212yesmaximal
8#20290444C212yesmaximal, complemented
9#20290442C211yesmaximal, characteristic, complemented
10C4×#13133274C211yesmaximal, characteristic, complemented
11C2×C4×#11719949C211yesmaximal, characteristic
12#16811648—28no—
13C2×#13133284C2212yes—
14C2×C4⋊#88324C2212yes—
15C2×#13133274C2212yes—
16C2×#13133274C2212yes—
17C2×#13133274C2212yes—
18C2×#13133274C2212yes—
19C2×#13133274C2212yes—
20C2×#13133274C2212yes—
21C2×#13133272C2212yes—
22C2×C4×#88324C2212yes—
23#16811489C2211yescharacteristic
24#16811489C2211yescharacteristic
25#16811489C2211yescharacteristic
26#16811489C2211yescharacteristic
27#16811489C2212yes—
28#16811489C2212yes—
29C22×#11719949C2211yescharacteristic
30C4×#11719949—28no—
31C2×Q8×Dic53C411yescharacteristic
32C2×C4×D212C411yescharacteristic
33C2×C4×D212C411yescharacteristic
34C2×C4×D212C411yescharacteristic
35C2×#13133179C411yescharacteristic
36C2×#13133179C411yescharacteristic
37C2×#13133179C411yescharacteristic
38C2×#13133175C2211yescharacteristic
39C2×#13133175C2211yescharacteristic
40C2×#13133175C2211yescharacteristic
41C2×C42×D53C2211yescharacteristic
42C2×C42×D53C2211yescharacteristic
43C2×C42×D53C2211yescharacteristic
44C2×C4×Q8×C53C411yescharacteristic
45C22×#88324C2312yes—
46C22×#88324C2312yes—
47C22×#88324C2312yes—
48C22×#88324C2312yes—
49#13133284—24no—
50C4⋊#88324—24no—
51#13133274—24no—
52#13133274—24no—
53#13133274—24no—
54#13133274—24no—
55#13133274—24no—
56#13133274—24no—
57#13133272—24no—
58C4×#88324—24no—
59C2×#11719949—24no—
60C2×#11719949—24no—
61C2×#11719949—22no—
62C2×#11719949—22no—
63C22×D212C2×C411yescharacteristic
64C22×D212C2×C411yescharacteristic
65C22×D212C2×C411yescharacteristic
66C22×C4×D53C2×C411yescharacteristic
67C22×C4×D53C2×C411yescharacteristic
68C22×C4×D53C2×C411yescharacteristic
69C22×C4×D53C2×C411yescharacteristic
70C22×C4×D53C2×C411yescharacteristic
71C22×C4×D53C2×C411yescharacteristic
72C22×C4×D53C2311yescharacteristic
73C22×C4×D53C2311yescharacteristic
74C22×C4×D53C2311yescharacteristic
75Q8×Dic53—24no—
76C4×D212—24no—
77C4×D212—24no—
78C4×D212—24no—
79#13133179—24no—
80#13133179—24no—
81#13133179—24no—
82#13133175—24no—
83#13133175—24no—
84#13133175D418yes—
85C2×C4⋊Dic53C2×C411yescharacteristic
86C2×C4⋊Dic53C2×C411yescharacteristic
87C2×C4⋊Dic53C2×C411yescharacteristic
88C2×#11719921C2×C411yescharacteristic
89C2×#11719921C2×C411yescharacteristic
90C2×#11719921C2×C411yescharacteristic
91C2×#11719921C2×C411yescharacteristic
92C2×#11719921C2×C411yescharacteristic
93C2×#11719921C2×C411yescharacteristic
94C2×C4×Dic53C2×C411yescharacteristic
95C2×C4×Dic53C2×C411yescharacteristic
96C2×C4×Dic53C2×C411yescharacteristic
97C2×C4×Dic53C2311yescharacteristic
98C2×C4×Dic53C2311yescharacteristic
99C2×C4×Dic53C2311yescharacteristic
100C2×C4×Dic53C2311yescharacteristic
101C42×D53—24no—
102C42×D53—24no—
103C42×D53—24no—
104C22×Q8×C53C2×C411yescharacteristic
105C4×Q8×C53—24no—
106C2×C4⋊C4×C53C2×C411yescharacteristic
107C2×C4⋊C4×C53C2×C411yescharacteristic
108C2×C4⋊C4×C53C2×C411yescharacteristic
109C2×C42×C53C2×C411yescharacteristic
110C2×C42×C53C2×C411yescharacteristic
111C2×C42×C53C2×C411yescharacteristic
112C23×D53C22×C411yescharacteristic
113C23×D53C22×C411yescharacteristic
114C23×D53C22×C411yescharacteristic
115C2×Q8×C53—22no—
116C2×Q8×C53—22no—
117C2×Q8×C53—22no—
118C22×C4×C53C22×C411yescharacteristic
119C22×C4×C53C22×C411yescharacteristic
120C22×C4×C53C22×C411yescharacteristic
121C22×C4×C53C22×C411yescharacteristic
122C22×C4×C53C22×C411yescharacteristic
123C22×C4×C53C22×C411yescharacteristic
124C22×C4×C53C22×C411yescharacteristic
125C22×Dic53C22×C411yescharacteristic
126C22×Dic53C22×C411yescharacteristic
127C22×Dic53C22×C411yescharacteristic
128C22×Dic53C22×C411yescharacteristic
129C22×Dic53C2411yescharacteristic
130#11719949—216no—
131#11719949—216no—
132C2×D212—22no—
133C2×D212—22no—
134C2×D212—22no—
135C2×D212—22no—
136C2×D212—22no—
137C2×D212—22no—
138C2×D212—22no—
139C2×D212—22no—
140C2×D212—22no—
141C2×C4×D53—22no—
142C2×C4×D53—22no—
143C2×C4×D53—22no—
144C2×C4×D53—21no—
145C2×C4×D53—21no—
146C2×C4×D53—22no—
147C2×C4×D53—21no—
148C2×C4×D53—21no—
149C2×C4×D53—22no—
150C2×C4×D53—22no—
151C2×C4×D53—22no—
152C2×C4×D53—22no—
153C2×C4×D53—22no—
154C2×C4×D53—22no—
155C2×C4×D53—22no—
156C2×C4×D53—22no—
157C2×C4×D53—22no—
158C2×C4×D53—22no—
159C2×C4×D53—22no—
160C2×C4×D53—22no—
161C2×C4×D53—41no—
162C2×C4×D53—41no—
163C2×C4×D53—41no—
164C2×C4×D53—41no—
165C2×C4×D53—41no—
166C2×C4×D53—41no—
167C2×C4×D53C2×D414yes—
168C2×C4×D53#4114yes—
169C2×C4×D53#4112yes—
170C2×C4×D53#4112yes—
171C2×#88324—22no—
172C2×#88324—22no—
173C2×#88324—22no—
174C2×#88324—22no—
175C2×#88324—22no—
176C2×#88324—22no—
177C2×#88324—22no—
178C2×#88324—22no—
179C2×#88324—22no—
180C2×#88324—22no—
181C2×#88324—22no—
182C2×#88324—22no—
183C4⋊C4×C53—22no—
184C4⋊C4×C53—22no—
185C4⋊C4×C53C22⋊C414yes—
186C42×C53—22no—
187C42×C53—22no—
188C42×C53—22no—
189#11719921—22no—
190#11719921—22no—
191#11719921—22no—
192#11719921—22no—
193#11719921C22⋊C414yes—
194#11719921C22⋊C414yes—
195C4⋊Dic53—22no—
196C4⋊Dic53—22no—
197C4⋊Dic53C22⋊C414yes—
198C4×Dic53—22no—
199C4×Dic53—22no—
200C4×Dic53—22no—
201C4×Dic53—22no—
202C4×Dic53—22no—
203C4×Dic53C2×D414yes—
204C4×Dic53C2×D414yes—
205C23×C53C23×C411yescharacteristic
206C22×D53—21no—
207C22×D53—21no—
208C22×D53—21no—
209C22×D53—21no—
210C22×D53—21no—
211C22×D53—21no—
212C22×D53—21no—
213C22×D53—21no—
214C22×D53—21no—
215C22×D53—21no—
216C22×D53—21no—
217C22×D53—21no—
218C22×D53—21no—
219C22×D53—21no—
220C22×D53—21no—
221C22×D53—21no—
222C22×D53—21no—
223C22×D53—21no—
224C22×D53C2×C22⋊C412yes—
225C22×D53#11712yes—
226C22×D53#11712yes—
227Q8×C53—28no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53C2×C22⋊C412yes—
244C2×C4×C53C2×C22⋊C412yes—
245C2×C4×C53C2×C22⋊C412yes—
246C2×C4×C53#11712yes—
247C2×C4×C53#11712yes—
248C2×C4×C53#13112yes—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53C2×C22⋊C412yes—
259C2×Dic53C2×C22⋊C412yes—
260C2×Dic53#13112yes—
261C2×Dic53C22×D412yes—
262C2×Dic53C2×#4112yes—
263C2×Dic53C2×#4111yescharacteristic
264C2×Dic53C2×#4111yescharacteristic
265D212—28no—
266D212—28no—
267D212—28no—
268C4×D53—24no—
269C4×D53—24no—
270C4×D53—24no—
271C4×D53—24no—
272C4×D53—24no—
273C4×D53—24no—
274C4×D53—44no—
275C4×D53—44no—
276C4×D53—44no—
277C4×D53—44no—
278C4×D53—44no—
279C4×D53—44no—
280#88324—42no—
281#88324—42no—
282#88324—42no—
283#88324—42no—
284#20720—531nomaximal, Sylow
285C22×C53C22×C22⋊C411yescharacteristic
286C22×C53C2×#11711yescharacteristic
287C22×C53C2×#11711yescharacteristic
288C22×C53C2×#13111yescharacteristic
289C22×C53C2×#13111yescharacteristic
290C22×C53#56811yescharacteristic
291C22×C53#56811yescharacteristic
292C2×D53—22no—
293C2×D53—22no—
294C2×D53—22no—
295C2×D53—22no—
296C2×D53—22no—
297C2×D53—22no—
298C2×D53—22no—
299C2×D53—22no—
300C2×D53—22no—
301C2×D53—41no—
302C2×D53—41no—
303C2×D53—41no—
304C2×D53—41no—
305C2×D53—41no—
306C2×D53—41no—
307C2×D53—41no—
308C2×D53—41no—
309C2×D53—41no—
310C2×D53—41no—
311C2×D53—41no—
312C2×D53—41no—
313C4×C53—22no—
314C4×C53—22no—
315C4×C53—22no—
316C4×C53—22no—
317C4×C53—22no—
318C4×C53—22no—
319C4×C53—41no—
320Dic53—22no—
321Dic53—22no—
322Dic53—22no—
323Dic53—41no—
324Dic53#52512yes—
325Dic53#52512yes—
326C2×C4×#41—531no—
327C2×#433—532no—
328#1817—531no—
329#1817—531no—
330#1798—532no—
331#1797—531no—
332#1797—531no—
333#1780—532no—
334#1750—532no—
335#1749—531no—
336C4×C22⋊C8—531no—
337C2×C53#228011yescharacteristic
338C2×C53#288011yescharacteristic
339C2×C53#288011yescharacteristic
340C2×C53#292211yescharacteristic
341C2×C53#292311yescharacteristic
342C2×C53#292511yescharacteristic
343C2×C53#292511yescharacteristic
344D53—42no—
345D53—42no—
346D53—42no—
347C22×#41—531no—
348C4×#41—1068no—
349C2×C4×Q8—531no—
350C2×C4×Q8#615711yescharacteristic
351C2×C4×D4—531no—
352C2×C4×D4—531no—
353C2×C4×D4—531no—
354C2×C4×D4—531no—
355C2×C4×D4—531no—
356C2×C4×D4—531no—
357C2×#117—531no—
358C2×#117—531no—
359C2×#117—531no—
360C22×C42—531no—
361C22×C42—531no—
362C22×C42—531no—
363#433—1068no—
364C2×C4⋊C8—532no—
365C2×C4⋊C8—532no—
366C2×C22⋊C8—532no—
367C2×C22⋊C8—532no—
368C2×C22⋊C8—532no—
369C2×C22⋊C8—532no—
370C2×C22⋊C8—532no—
371C2×C22⋊C8—532no—
372C2×#97—532no—
373C2×C4×C8—532no—
374#336—531no—
375#336—531no—
376#336—531no—
377#336—531no—
378#336—532no—
379#336—532no—
380C53#2072011yescharacteristic, complemented, Sylow
381C2×#41—1064no—
382C2×#41—1064no—
383C2×#41—1062no—
384C2×#41—1062no—
385C22×Q8C2×#615711yescharacteristic
386C22×D4—531no—
387C22×D4—531no—
388C22×D4—531no—
389C23×C4—531no—
390C23×C4—531no—
391C23×C4—531no—
392C23×C4—531no—
393C23×C4—531no—
394C23×C4—531no—
395C23×C4—531no—
396C23×C4—531no—
397C23×C4—531no—
398C22×C8—532no—
399C22×C8—532no—
400C22×C8—532no—
401C22×C8—532no—
402C4×Q8—24no—
403C4×Q8—1064no—
404C4×D4—1064no—
405C4×D4—1064no—
406C4×D4—1064no—
407C4×D4—1064no—
408C4×D4—1064no—
409C4×D4—1064no—
410#117—538no—
411#117—1064no—
412#117—1064no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—531no—
415C2×C4⋊C4—531no—
416C2×C4⋊C4C2×#615711yescharacteristic
417C2×C4⋊C4C2×#615711yescharacteristic
418C2×C4⋊C4C2×#615711yescharacteristic
419C2×C22⋊C4—531no—
420C2×C22⋊C4—531no—
421C2×C22⋊C4—531no—
422C2×C22⋊C4—531no—
423C2×C22⋊C4—531no—
424C2×C22⋊C4—531no—
425C2×C42—531no—
426C2×C42—531no—
427C2×C42—531no—
428C2×C42—531no—
429C2×C42—531no—
430C2×C42—531no—
431C2×C42—531no—
432C2×C42—1064no—
433C2×C42—1064no—
434C2×C42—1064no—
435C2×C42C2×#615711yescharacteristic
436C2×C42C2×#615711yescharacteristic
437C2×C42C2×#615711yescharacteristic
438C4⋊C8—1064no—
439C4⋊C8—1064no—
440C22⋊C8—1064no—
441C22⋊C8—1064no—
442C22⋊C8—1064no—
443C22⋊C8—1064no—
444C22⋊C8—1064no—
445C22⋊C8—1064no—
446#97—1064no—
447C4×C8—1064no—
448C24—531no—
449C24—531no—
450C24—531no—
451#41—10616no—
452#41—10616no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×D4—1062no—
457C2×D4—1062no—
458C2×D4—1062no—
459C2×D4—1062no—
460C2×D4—1062no—
461C2×D4—1062no—
462C2×D4—1062no—
463C2×D4—1062no—
464C2×D4—1062no—
465C22×C4—531no—
466C22×C4—531no—
467C22×C4—531no—
468C22×C4—531no—
469C22×C4—534no—
470C22×C4—534no—
471C22×C4—532no—
472C22×C4—532no—
473C22×C4—531no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1062no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—1062no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4—1062no—
494C22×C4—2121no—
495C22×C4—2121no—
496C22×C4—2121no—
497C22×C4—2121no—
498C22×C4—2121no—
499C22×C4—2121no—
500C22×C4C22×#615711yescharacteristic
501C22×C4C22×#615711yescharacteristic
502C22×C4C22×#615711yescharacteristic
503C22×C4C22×#615711yescharacteristic
504C22×C4C22×#615711yescharacteristic
505C22×C4C22×#615711yescharacteristic
506C22×C4C22×#615711yescharacteristic
507C2×C8—1062no—
508C2×C8—1062no—
509C2×C8—1062no—
510C2×C8—1062no—
511C2×C8—1062no—
512C2×C8—1062no—
513C2×C8—1062no—
514C2×C8—1062no—
515C2×C8—1062no—
516C2×C8—1062no—
517C2×C8—1062no—
518C2×C8—1062no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—534no—
522C4⋊C4—1062no—
523C4⋊C4—1062no—
524C4⋊C4C22⋊#615714yes—
525C22⋊C4—534no—
526C22⋊C4—534no—
527C22⋊C4—1062no—
528C22⋊C4—1062no—
529C22⋊C4—1062no—
530C22⋊C4—1062no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—534no—
535C42—534no—
536C42—1062no—
537C42—1062no—
538C42—1062no—
539C42—1062no—
540C42—1062no—
541C23—532no—
542C23—532no—
543C23—532no—
544C23—1061no—
545C23—1061no—
546C23—1061no—
547C23—1061no—
548C23—1061no—
549C23—1061no—
550C23—1061no—
551C23—1061no—
552C23—1061no—
553C23—1061no—
554C23—1061no—
555C23—1061no—
556C23—1061no—
557C23—1061no—
558C23—1061no—
559C23—1061no—
560C23—1061no—
561C23—1061no—
562C23C23×#615711yescharacteristic
563Q8—28no—
564D4—1068no—
565D4—1068no—
566D4—1068no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—532no—
583C2×C4—532no—
584C2×C4—531no—
585C2×C4—531no—
586C2×C4—532no—
587C2×C4—532no—
588C2×C4—532no—
589C2×C4—1064no—
590C2×C4—1064no—
591C2×C4—1064no—
592C2×C4—1064no—
593C2×C4—1064no—
594C2×C4—1064no—
595C2×C4—1061no—
596C2×C4—1061no—
597C2×C4—1061no—
598C2×C4—1061no—
599C2×C4—1061no—
600C2×C4—1061no—
601C2×C4—1061no—
602C2×C4—1061no—
603C2×C4—1061no—
604C2×C4—2124no—
605C2×C4—2124no—
606C2×C4—2124no—
607C2×C4—2124no—
608C2×C4—2124no—
609C2×C4—2124no—
610C2×C4C2×C22⋊#615712yes—
611C2×C4C2×C22⋊#615712yes—
612C2×C4C2×C22⋊#615712yes—
613C2×C4#1313329612yes—
614C2×C4#1313329612yes—
615C2×C4#1313330512yes—
616C8—2122no—
617C8—2122no—
618C8—2122no—
619C8—2122no—
620C22—1062no—
621C22—1062no—
622C22—1062no—
623C22—1062no—
624C22—1062no—
625C22—1062no—
626C22—1062no—
627C22—1062no—
628C22—1062no—
629C22—2121no—
630C22—2121no—
631C22—2121no—
632C22—2121no—
633C22—2121no—
634C22—2121no—
635C22—2121no—
636C22—2121no—
637C22—2121no—
638C22—2121no—
639C22—2121no—
640C22—2121no—
641C22C22×C22⋊#615711yescharacteristic
642C22C2×#1313329611yescharacteristic
643C22C2×#1313329611yescharacteristic
644C22D53⋊#13111yescharacteristic
645C22C2×#1313330511yescharacteristic
646C22#1681170211yescharacteristic
647C22#1681170311yescharacteristic
648C4—22no—
649C4—22no—
650C4—22no—
651C4—22no—
652C4—22no—
653C4—22no—
654C4—41no—
655C4—532no—
656C4—532no—
657C4—1062no—
658C4—1062no—
659C4—1062no—
660C4—2121no—
661C2—2122no—
662C2—2122no—
663C2—2122no—
664C2#2029115911yescharacteristic
665C2#2029130411yescharacteristic
666C2#2029130511yescharacteristic
667C2#2029140011yescharacteristic
668C2#2029140311yescharacteristic
669C2#2029141411yescharacteristic
670C2#2029141511yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6157
commutator, socleC23×C53C23×C4
FittingC2×C4×Q8×C53C4
prime radicalC22×#11719949C22
cosocleC22×Dic53C24
hypercenterC2×C4×Q8#6157
hypocenterC53#20720
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16811648C2semidirect
2. #20290607C2semidirect
3. #20290607C2semidirect
4. #20290574C2semidirect
5. #20290572C2semidirect
6. #20290570C2semidirect
8. #20290444C2semidirect
9. #20290442C2semidirect
10. C4×#13133274C2semidirect
380. C53#20720semidirect

Maximal subgroups

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

subgroupindex
C2×#168116482
#202906072
#202905742
#202905722
#202905702
#202905402
#202904442
#202904422
C4×#131332742
C2×C4×#117199492
#2072053