Finite Group Databasebeta

← #28324363

Subgroups — #28324363

Order 13568.

1424 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811653C212yesmaximal, complemented
2#20290889C212yesmaximal, complemented
3#20290695C212yesmaximal, complemented
4#20290691C212yesmaximal, complemented
5#20290607C211yesmaximal, characteristic, complemented
6#20290607C211yesmaximal, characteristic, complemented
7#20290572C211yesmaximal, characteristic, complemented
8#20290442C211yesmaximal, characteristic, complemented
9#20290379C211yesmaximal, characteristic, complemented
10#20290376C211yesmaximal, characteristic, complemented
11C2×C4×#11719949C211yesmaximal, characteristic
12C22×#11719936C2212yes—
13C22×D53⋊C8C2212yes—
14#16811653—28no—
15C2×#13133284C2212yes—
16C2×#13133274C2212yes—
17C2×#13133274C2212yes—
18C2×#13133274C2212yes—
19C2×#13133274C2212yes—
20C2×#13133272C2212yes—
21#16811489C2211yescharacteristic
22#16811489C2211yescharacteristic
23#16811489C2211yescharacteristic
24#16811489C2212yes—
25#16811489C2212yes—
26#16811489C2212yes—
27#16811489C2211yescharacteristic
28#16811489C2212yes—
29C22×#11719949C411yescharacteristic
30C4×#11719949—28no—
31C2×Q8×Dic53C2211yescharacteristic
32C2×C4×D212C411yescharacteristic
33C2×C4×D212C411yescharacteristic
34C2×C4×D212C411yescharacteristic
35C2×#13133179C2211yescharacteristic
36C2×#13133179C2211yescharacteristic
37C2×#13133179C2211yescharacteristic
38C2×#13133175C411yescharacteristic
39C2×#13133175C411yescharacteristic
40C2×#13133175C411yescharacteristic
41C2×C42×D53C2211yescharacteristic
42C2×C42×D53C2211yescharacteristic
43C2×C42×D53C2211yescharacteristic
44C2×C4×Q8×C53C411yescharacteristic
45C2×#11719936—24no—
46C2×#11719936—24no—
47C2×#11719936—42no—
48C2×D53⋊C8—24no—
49C2×D53⋊C8—24no—
50C2×D53⋊C8—42no—
51C22×#88324C2312yes—
52C22×#88324C2312yes—
53C22×#88324C2312yes—
54C22×#88324C2312yes—
55#13133284—24no—
56#13133274—24no—
57#13133274—24no—
58#13133274—24no—
59#13133274—42no—
60#13133272—24no—
61C2×#11719949—24no—
62C2×#11719949—24no—
63C2×#11719949—24no—
64C22×D212C2×C411yescharacteristic
65C22×D212C2×C411yescharacteristic
66C22×D212C2×C411yescharacteristic
67C22×C4×D53C2×C411yescharacteristic
68C22×C4×D53C2×C411yescharacteristic
69C22×C4×D53C2×C411yescharacteristic
70C22×C4×D53C2×C411yescharacteristic
71C22×C4×D53C2×C411yescharacteristic
72C22×C4×D53C2×C411yescharacteristic
73C22×C4×D53C2311yescharacteristic
74C22×C4×D53C2311yescharacteristic
75C22×C4×D53C2311yescharacteristic
76Q8×Dic53—24no—
77C4×D212—24no—
78C4×D212—24no—
79C4×D212—24no—
80#13133179—24no—
81#13133179—24no—
82#13133179D418yes—
83#13133175—24no—
84#13133175—24no—
85#13133175—24no—
86C2×C4⋊Dic53C2×C411yescharacteristic
87C2×C4⋊Dic53C2×C411yescharacteristic
88C2×C4⋊Dic53C2×C411yescharacteristic
89C2×#11719921C2×C411yescharacteristic
90C2×#11719921C2×C411yescharacteristic
91C2×#11719921C2×C411yescharacteristic
92C2×#11719921C2×C411yescharacteristic
93C2×#11719921C2×C411yescharacteristic
94C2×#11719921C2×C411yescharacteristic
95C2×C4×Dic53C2×C411yescharacteristic
96C2×C4×Dic53C2×C411yescharacteristic
97C2×C4×Dic53C2×C411yescharacteristic
98C2×C4×Dic53C2311yescharacteristic
99C2×C4×Dic53C2311yescharacteristic
100C2×C4×Dic53C2311yescharacteristic
101C2×C4×Dic53C2311yescharacteristic
102C42×D53—24no—
103C42×D53—24no—
104C42×D53—24no—
105C22×Q8×C53C2×C411yescharacteristic
106C4×Q8×C53—24no—
107C2×C4⋊C4×C53C2×C411yescharacteristic
108C2×C4⋊C4×C53C2×C411yescharacteristic
109C2×C4⋊C4×C53C2×C411yescharacteristic
110C2×C42×C53C2×C411yescharacteristic
111C2×C42×C53C2×C411yescharacteristic
112C2×C42×C53C2×C411yescharacteristic
113C23×D53C22×C411yescharacteristic
114C23×D53C22×C411yescharacteristic
115C23×D53C22×C411yescharacteristic
116C2×Q8×C53—22no—
117C2×Q8×C53—22no—
118C2×Q8×C53—22no—
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×C4×C53C22×C411yescharacteristic
126C22×Dic53C22×C411yescharacteristic
127C22×Dic53C22×C411yescharacteristic
128C22×Dic53C22×C411yescharacteristic
129C22×Dic53C22×C411yescharacteristic
130C22×Dic53C2411yescharacteristic
131#11719949—232no—
132C2×D212—22no—
133C2×D212—22no—
134C2×D212—22no—
135C2×D212—22no—
136C2×D212—22no—
137C2×D212—22no—
138C2×D212—22no—
139C2×D212C22⋊C414yes—
140C2×D212M1614yescomplemented
141C2×C4×D53—22no—
142C2×C4×D53—22no—
143C2×C4×D53—22no—
144C2×C4×D53—22no—
145C2×C4×D53—22no—
146C2×C4×D53—22no—
147C2×C4×D53—22no—
148C2×C4×D53—22no—
149C2×C4×D53—22no—
150C2×C4×D53—22no—
151C2×C4×D53—21no—
152C2×C4×D53—21no—
153C2×C4×D53—21no—
154C2×C4×D53—21no—
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—22no—
162C2×C4×D53—22no—
163C2×C4×D53—21no—
164C2×C4×D53—21no—
165C2×C4×D53—41no—
166C2×C4×D53—41no—
167C2×C4×D53—41no—
168C2×C4×D53M1614yescomplemented
169C2×C4×D53C2×D414yes—
170C2×C4×D53C2×D414yes—
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#4114yescomplemented
183#11719936—48no—
184D53⋊C8—48no—
185C4⋊C4×C53—22no—
186C4⋊C4×C53—22no—
187C4⋊C4×C53C22⋊C414yes—
188C42×C53—22no—
189C42×C53—22no—
190C42×C53—22no—
191#11719921—22no—
192#11719921—22no—
193#11719921—22no—
194#11719921—22no—
195#11719921C22⋊C414yes—
196#11719921C22⋊C414yes—
197C4⋊Dic53—22no—
198C4⋊Dic53—22no—
199C4⋊Dic53—22no—
200C4×Dic53—22no—
201C4×Dic53—22no—
202C4×Dic53—22no—
203C4×Dic53—22no—
204C4×Dic53—22no—
205C4×Dic53—22no—
206C4×Dic53C2×D414yes—
207C23×C53C23×C411yescharacteristic
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×D53C2×C22⋊C412yes—
226C22×D53C2×M1612yes—
227C22×D53C2×M1612yes—
228C22×D53C2×M1612yes—
229Q8×C53—28no—
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×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53C2×C22⋊C412yes—
246C2×C4×C53C2×C22⋊C412yes—
247C2×C4×C53C2×C22⋊C412yes—
248C2×C4×C53C2×M1612yes—
249C2×C4×C53C2×M1612yes—
250C2×C4×C53#13112yes—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53C2×C22⋊C412yes—
262C2×Dic53C2×M1612yes—
263C2×Dic53C22×D412yes—
264C2×Dic53C2×#4112yes—
265C2×Dic53H(5,𝔽2)11yescharacteristic
266C2×Dic532−1+411yescharacteristic
267D212—28no—
268D212—28no—
269D212—28no—
270C4×D53—28no—
271C4×D53—28no—
272C4×D53—28no—
273C4×D53—24no—
274C4×D53—24no—
275C4×D53—24no—
276C4×D53—24no—
277C4×D53—44no—
278C4×D53—44no—
279C4×D53—44no—
280C4×D53—42no—
281C4×D53—42no—
282#88324—24no—
283#88324—42no—
284#88324—42no—
285#88324—42no—
286#22471—531nomaximal, Sylow
287C22×C53C22×C22⋊C411yescharacteristic
288C22×C53C4×#4111yescharacteristic
289C22×C53#52011yescharacteristic
290C22×C53C22×M1611yescharacteristic
291C22×C53C22×M1611yescharacteristic
292C22×C53C2×#13111yescharacteristic
293C22×C53#56811yescharacteristic
294C2×D53—22no—
295C2×D53—22no—
296C2×D53—22no—
297C2×D53—22no—
298C2×D53—22no—
299C2×D53—22no—
300C2×D53—22no—
301C2×D53—22no—
302C2×D53—22no—
303C2×D53—22no—
304C2×D53—22no—
305C2×D53—22no—
306C2×D53—41no—
307C2×D53—41no—
308C2×D53—41no—
309C2×D53—41no—
310C2×D53—41no—
311C2×D53—41no—
312C2×D53—41no—
313C2×D53—41no—
314C2×D53#40714yes—
315C4×C53—22no—
316C4×C53—22no—
317C4×C53—22no—
318C4×C53—22no—
319C4×C53—22no—
320C4×C53—22no—
321C4×C53—41no—
322Dic53—22no—
323Dic53—22no—
324Dic53—41no—
325Dic53—41no—
326Dic53#54612yes—
327Dic53#54712yes—
328C2×C4×#41—531no—
329C2×#435—532no—
330#1985—532no—
331#1870—532no—
332#1869—532no—
333#1817—531no—
334#1817—531no—
335#1797—531no—
336#1749—531no—
337#1728—531no—
338C2×#336—531no—
339C2×C53#233911yescharacteristic
340C2×C53C2×#40711yescharacteristic
341C2×C53#288011yescharacteristic
342C2×C53#296511yescharacteristic
343C2×C53#297011yescharacteristic
344C2×C53#298911yescharacteristic
345C2×C53#299011yescharacteristic
346D53—24no—
347D53—42no—
348D53—42no—
349C22×#41—531no—
350C22×M16—532no—
351C23×C8—532no—
352C4×#41—1068no—
353C2×C4×Q8—531no—
354C2×C4×Q8#615711yescharacteristic
355C2×C4×D4—531no—
356C2×C4×D4—531no—
357C2×C4×D4—531no—
358C2×C4×D4—531no—
359C2×C4×D4—531no—
360C2×C4×D4—531no—
361C2×#117—531no—
362C2×#117—531no—
363C2×#117—531no—
364C22×C42—531no—
365C22×C42—531no—
366C22×C42—531no—
367#435—1068no—
368C2×C4⋊C8—532no—
369C2×C22⋊C8—532no—
370C2×C22⋊C8—532no—
371C2×C22⋊C8—532no—
372C2×C22⋊C8—532no—
373C2×#97—532no—
374#336—531no—
375#336—531no—
376#336—531no—
377#336—532no—
378#336—532no—
379#336—532no—
380#336—532no—
381#336—531no—
382C53#2247111yescharacteristic, complemented, Sylow
383C2×#41—1064no—
384C2×#41—1064no—
385C2×#41—1064no—
386C22×Q8C2×#615711yescharacteristic
387C22×D4—531no—
388C22×D4—531no—
389C22×D4—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—
398C23×C4—531no—
399C2×M16—1064no—
400C2×M16—1064no—
401C2×M16—2122no—
402C22×C8—532no—
403C22×C8—532no—
404C22×C8—532no—
405C22×C8—532no—
406C22×C8—1064no—
407C22×C8—1064no—
408C22×C8—2122no—
409C4×Q8—24no—
410C4×Q8—1064no—
411C4×D4—538no—
412C4×D4—1064no—
413C4×D4—1064no—
414C4×D4—1064no—
415C4×D4—1064no—
416C4×D4—1064no—
417#117—1064no—
418#117—1064no—
419#117—1064no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—531no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4C2×#615711yescharacteristic
424C2×C4⋊C4C2×#615711yescharacteristic
425C2×C4⋊C4C2×#615711yescharacteristic
426C2×C22⋊C4—531no—
427C2×C22⋊C4—531no—
428C2×C22⋊C4—531no—
429C2×C22⋊C4—531no—
430C2×C22⋊C4—531no—
431C2×C22⋊C4—531no—
432C2×C42—531no—
433C2×C42—531no—
434C2×C42—531no—
435C2×C42—531no—
436C2×C42—531no—
437C2×C42—531no—
438C2×C42—531no—
439C2×C42—1064no—
440C2×C42—1064no—
441C2×C42—1064no—
442C2×C42C2×#615711yescharacteristic
443C2×C42C2×#615711yescharacteristic
444C2×C42C2×#615711yescharacteristic
445C4⋊C8—1064no—
446C22⋊C8—1064no—
447C22⋊C8—1064no—
448C22⋊C8—1064no—
449C22⋊C8—2122no—
450#97—1064no—
451C24—531no—
452C24—531no—
453C24—531no—
454#41—10632no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×D4—534no—
459C2×D4—534no—
460C2×D4—1062no—
461C2×D4—1062no—
462C2×D4—1062no—
463C2×D4—1062no—
464C2×D4—1062no—
465C2×D4—1062no—
466C2×D4—1062no—
467C22×C4—531no—
468C22×C4—531no—
469C22×C4—531no—
470C22×C4—534no—
471C22×C4—534no—
472C22×C4—534no—
473C22×C4—531no—
474C22×C4—531no—
475C22×C4—1062no—
476C22×C4—1061no—
477C22×C4—1062no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1061no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—1061no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4—1062no—
494C22×C4—1062no—
495C22×C4—1062no—
496C22×C4—1062no—
497C22×C4—1062no—
498C22×C4—1062no—
499C22×C4—2121no—
500C22×C4—2121no—
501C22×C4—2121no—
502C22×C4C22×#615711yescharacteristic
503C22×C4C22×#615711yescharacteristic
504C22×C4C22×#615711yescharacteristic
505C22×C4C22×#615711yescharacteristic
506C22×C4C22×#615711yescharacteristic
507C22×C4C22×#615711yescharacteristic
508C22×C4C22×#615711yescharacteristic
509M16—2128no—
510C2×C8—534no—
511C2×C8—1062no—
512C2×C8—1062no—
513C2×C8—1062no—
514C2×C8—1062no—
515C2×C8—1062no—
516C2×C8—1062no—
517C2×C8—1062no—
518C2×C8—1062no—
519C2×C8—1062no—
520C2×C8—1062no—
521C2×C8—1062no—
522C2×C8—2128no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—1062no—
526C4⋊C4—1062no—
527C4⋊C4—1062no—
528C4⋊C4C22⋊#615714yes—
529C22⋊C4—534no—
530C22⋊C4—534no—
531C22⋊C4—1062no—
532C22⋊C4—1062no—
533C22⋊C4—1062no—
534C22⋊C4—1062no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—534no—
539C42—1062no—
540C42—1062no—
541C42—1062no—
542C42—1062no—
543C42—1062no—
544C42—1062no—
545C23—532no—
546C23—532no—
547C23—532no—
548C23—532no—
549C23—532no—
550C23—1061no—
551C23—1061no—
552C23—1061no—
553C23—1061no—
554C23—1061no—
555C23—1061no—
556C23—1061no—
557C23—1061no—
558C23—1061no—
559C23—1061no—
560C23—1061no—
561C23—1061no—
562C23—1061no—
563C23—1061no—
564C23—1061no—
565C23—1061no—
566C23C23×#615711yescharacteristic
567Q8—28no—
568D4—1068no—
569D4—1068no—
570D4—1068no—
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—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—532no—
587C2×C4—532no—
588C2×C4—531no—
589C2×C4—531no—
590C2×C4—532no—
591C2×C4—532no—
592C2×C4—1064no—
593C2×C4—1064no—
594C2×C4—1064no—
595C2×C4—1064no—
596C2×C4—1068no—
597C2×C4—1068no—
598C2×C4—1068no—
599C2×C4—1061no—
600C2×C4—1061no—
601C2×C4—1061no—
602C2×C4—1061no—
603C2×C4—1061no—
604C2×C4—1061no—
605C2×C4—1061no—
606C2×C4—1061no—
607C2×C4—1061no—
608C2×C4—1061no—
609C2×C4—2124no—
610C2×C4—2124no—
611C2×C4—2124no—
612C2×C4—2122no—
613C2×C4—2122no—
614C2×C4C2×C22⋊#615712yes—
615C2×C4C2×C22⋊#615712yes—
616C2×C4C2×C22⋊#615712yes—
617C2×C4D53⋊M1612yes—
618C2×C4D53⋊M1612yes—
619C2×C4#1313330512yes—
620C8—1064no—
621C8—2122no—
622C8—2122no—
623C8—2122no—
624C22—534no—
625C22—1062no—
626C22—1062no—
627C22—1062no—
628C22—1062no—
629C22—1062no—
630C22—1062no—
631C22—1062no—
632C22—1062no—
633C22—1062no—
634C22—1062no—
635C22—1062no—
636C22—1062no—
637C22—2121no—
638C22—2121no—
639C22—2121no—
640C22—2121no—
641C22—2121no—
642C22—2121no—
643C22—2121no—
644C22—2121no—
645C22C22×C22⋊#615711yescharacteristic
646C22#41×#615711yescharacteristic
647C22#1681169211yescharacteristic
648C22C2×D53⋊M1611yescharacteristic
649C22C2×D53⋊M1611yescharacteristic
650C22C2×#1313330511yescharacteristic
651C22#1681170311yescharacteristic
652C4—22no—
653C4—22no—
654C4—22no—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—41no—
659C4—532no—
660C4—532no—
661C4—1062no—
662C4—1062no—
663C4—2121no—
664C4—2121no—
665C2—1064no—
666C2—2122no—
667C2—2122no—
668C2#2029121811yescharacteristic
669C2D53⋊#40711yescharacteristic
670C2#2029130411yescharacteristic
671C2#2029152311yescharacteristic
672C2#2029155711yescharacteristic
673C2#2029163911yescharacteristic
674C2#2029164511yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6157
commutator, socleC23×C53C23×C4
FittingC2×C4×Q8×C53C4
prime radicalC22×#11719949C4
cosocleC22×Dic53C24
hypercenterC2×C4×Q8#6157
hypocenterC53#22471
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×#16811653C2semidirect
2. #20290889C2semidirect
3. #20290695C2semidirect
4. #20290691C2semidirect
5. #20290607C2semidirect
6. #20290607C2semidirect
7. #20290572C2semidirect
8. #20290442C2semidirect
9. #20290379C2semidirect
10. #20290376C2semidirect
140. C2×D212M16semidirect
168. C2×C4×D53M16semidirect
182. C2×#88324#41semidirect
382. C53#22471semidirect

Maximal subgroups

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

subgroupindex
C2×#168116532
#202908892
#202906952
#202906912
#202906072
#202905722
#202904422
#202903792
#202903762
C2×C4×#117199492
#2247153