Finite Group Databasebeta

← #27305681

Subgroups — #27305681

Order 13568.

1906 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810511C212yesmaximal
2C2×#16810474C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊Dic53C211yesmaximal, characteristic
4C2×C4×C22⋊Dic53C211yesmaximal, characteristic
5#20272952C212yesmaximal
6#20272946C212yesmaximal, complemented
7#20272941C211yesmaximal, characteristic, complemented
8#20272939C211yesmaximal, characteristic, complemented
9#95×Dic53C212yesmaximal, complemented
10#95×Dic53C211yesmaximal, characteristic, complemented
11C53×C22⋊#95C211yesmaximal, characteristic
12C22×#11719919C2212yes—
13C22×C22⋊Dic53C2211yescharacteristic
14C22×C22⋊Dic53C2212yes—
15C22×C22⋊Dic53C2211yescharacteristic
16C22×C4×Dic53C2212yes—
17C22×C4×Dic53C2212yes—
18C22×C4×Dic53C2211yescharacteristic
19C22×C4×Dic53C2211yescharacteristic
20C22×C4×Dic53C2212yes—
21#16810511C4116yescomplemented
22#16810474—24no—
23C4×C22⋊Dic53—24no—
24C4×C22⋊Dic53C418yescomplemented
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2211yescharacteristic
28C2×#13133124C2212yes—
29C2×#13133124C2211yescharacteristic
30C2×#13133124C2211yescharacteristic
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2211yescharacteristic
34C24×C4×C53C2211yescharacteristic
35C22×C22⋊C4×C53C2212yes—
36C2×#95×C53C2211yescharacteristic
37C2×#95×C53C2212yes—
38C2×#95×C53C2211yescharacteristic
39C23×Dic53C2312yes—
40C23×Dic53C2311yescharacteristic
41C23×Dic53C2311yescharacteristic
42C23×Dic53C2311yescharacteristic
43C23×Dic53C2312yes—
44C23×Dic53C2311yescharacteristic
45C2×#11719919—24no—
46C2×#11719919—24no—
47C2×#11719919C2×C418yescomplemented
48C2×C22⋊Dic53—24no—
49C2×C22⋊Dic53—22no—
50C2×C22⋊Dic53—22no—
51C2×C22⋊Dic53—24no—
52C2×C22⋊Dic53C2×C418yes—
53C2×C22⋊Dic53C2×C414yes—
54C2×C22⋊Dic53C2×C418yes—
55C2×C22⋊Dic53C2×C414yes—
56C2×C4×Dic53—22no—
57C2×C4×Dic53—22no—
58C2×C4×Dic53—24no—
59C2×C4×Dic53—24no—
60C2×C4×Dic53—24no—
61C2×C4×Dic53—24no—
62C2×C4×Dic53—42no—
63C2×C4×Dic53—42no—
64C2×C4×Dic53—42no—
65C2×C4×Dic53—42no—
66C2×C4×Dic53—42no—
67C2×C4×Dic53C2×C414yes—
68C2×C4×Dic53C2×C414yes—
69C2×C4×Dic53C2×C418yescomplemented
70#13133124—22no—
71#13133124—24no—
72#13133124—24no—
73#13133124—42no—
74#13133124C2×C414yes—
75#13133124C2×C414yes—
76#13133124C2×C418yescomplemented
77#13133124C2×C418yescomplemented
78#13133124D414yes—
79C25×C53C2311yescharacteristic
80C23×C4×C53—22no—
81C23×C4×C53—24no—
82C23×C4×C53—22no—
83C23×C4×C53—42no—
84C23×C4×C53C2×C414yes—
85C23×C4×C53C2312yes—
86C23×C4×C53C2312yes—
87C23×C4×C53C2311yescharacteristic
88C23×C4×C53C2311yescharacteristic
89C2×C22⋊C4×C53—42no—
90C2×C22⋊C4×C53—42no—
91C2×C22⋊C4×C53C2×C418yes—
92#95×C53—22no—
93#95×C53—24no—
94#95×C53—22no—
95C24×C53—21no—
96C24×C53—22no—
97C24×C53—21no—
98C24×C53—21no—
99C24×C53—21no—
100C24×C53—22no—
101C24×C53—21no—
102C24×C53—22no—
103C24×C53—21no—
104C24×C53C22×C414yes—
105C24×C53C22×C412yes—
106C24×C53C2411yescharacteristic
107C22×C4×C53—21no—
108C22×C4×C53—22no—
109C22×C4×C53—21no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—28no—
125C22×C4×C53—28no—
126C22×C4×C53—44no—
127C22×C4×C53—48no—
128C22×C4×C53—48no—
129C22×C4×C53C22×C414yes—
130C22×C4×C53C22×C414yes—
131C22×C4×C53C22×C412yes—
132C22×C4×C53C22×C412yes—
133C22×C4×C53C2×D412yes—
134C22×C4×C53#4114yes—
135C22×C4×C53#4112yes—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—21no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53C22×C412yes—
161C22×Dic53C22×C414yes—
162C22×Dic53C22×C412yes—
163C22×Dic53C22×C412yes—
164C22×Dic53C22×C414yes—
165C22×Dic53C22×C414yes—
166C22×Dic53C22×C412yes—
167C22×Dic53C22×C414yes—
168C22×Dic53C2×D414yes—
169C22×Dic53#4112yes—
170C22×Dic53#4112yes—
171C22×Dic53#4114yes—
172C22⋊C4×C53—48no—
173C22⋊Dic53—28no—
174C22⋊Dic53—216no—
175C22⋊Dic53C42116yescomplemented
176#11719919—216no—
177C4×Dic53—28no—
178C4×Dic53—44no—
179C4×Dic53—48no—
180C4×Dic53—48no—
181C4×Dic53—48no—
182C23×C53—22no—
183C23×C53—22no—
184C23×C53—22no—
185C23×C53—24no—
186C23×C53—24no—
187C23×C53—24no—
188C23×C53—24no—
189C23×C53—22no—
190C23×C53—22no—
191C23×C53—24no—
192C23×C53—22no—
193C23×C53—24no—
194C23×C53—42no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—42no—
198C23×C53—41no—
199C23×C53—42no—
200C23×C53—42no—
201C23×C53—41no—
202C23×C53—42no—
203C23×C53—42no—
204C23×C53C2×C4214yes—
205C23×C53C23×C411yescharacteristic
206C23×C53C23×C412yes—
207C23×C53C22×D411yescharacteristic
208C23×C53C2×#4111yescharacteristic
209C23×C53C2×#4111yescharacteristic
210C23×C53C2×#4112yes—
211C23×C53C2×#4112yes—
212C23×C53C2×#4111yescharacteristic
213C23×C53H(5,𝔽2)12yes—
214C23×C53H(5,𝔽2)11yescharacteristic
215C23×C532−1+411yescharacteristic
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—24no—
221C2×C4×C53—24no—
222C2×C4×C53—24no—
223C2×C4×C53—24no—
224C2×C4×C53—24no—
225C2×C4×C53—24no—
226C2×C4×C53—24no—
227C2×C4×C53—24no—
228C2×C4×C53—24no—
229C2×C4×C53—24no—
230C2×C4×C53—416no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53—42no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53#11714yes—
240C2×C4×C53C4×D414yes—
241C2×Dic53—22no—
242C2×Dic53—22no—
243C2×Dic53—24no—
244C2×Dic53—24no—
245C2×Dic53—24no—
246C2×Dic53—22no—
247C2×Dic53—24no—
248C2×Dic53—22no—
249C2×Dic53—24no—
250C2×Dic53—22no—
251C2×Dic53—22no—
252C2×Dic53—24no—
253C2×Dic53—24no—
254C2×Dic53—24no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—24no—
259C2×Dic53—22no—
260C2×Dic53—24no—
261C2×Dic53—22no—
262C2×Dic53—24no—
263C2×Dic53—24no—
264C2×Dic53—22no—
265C2×Dic53—22no—
266C2×Dic53—24no—
267C2×Dic53—42no—
268C2×Dic53—42no—
269C2×Dic53—42no—
270C2×Dic53—42no—
271C2×Dic53—42no—
272C2×Dic53—42no—
273C2×Dic53C2×C4214yes—
274C2×Dic53C2×C4214yes—
275C2×Dic53#11714yes—
276C2×Dic53C4×D418yes—
277#13822—531nomaximal, Sylow
278C22×C53—24no—
279C22×C53—24no—
280C22×C53—24no—
281C22×C53—42no—
282C22×C53—42no—
283C22×C53—42no—
284C22×C53—42no—
285C22×C53—44no—
286C22×C53—44no—
287C22×C53—44no—
288C22×C53—44no—
289C22×C53C22×C4211yescharacteristic
290C22×C53C2×#11711yescharacteristic
291C22×C53C2×#11711yescharacteristic
292C22×C53C2×C4×D412yes—
293C22×C53C2×C4×D411yescharacteristic
294C22×C53C4×#4112yes—
295C22×C53C4×#4112yes—
296C22×C53C4×#4112yes—
297C22×C53C4×#4111yescharacteristic
298C22×C53#51811yescharacteristic
299C22×C53#51812yes—
300C22×C53#52011yescharacteristic
301C22×C53#52012yes—
302C22×C53#54611yescharacteristic
303C22×C53#54612yes—
304C22×C53#54811yescharacteristic
305C22×C53#55112yes—
306C22×C53#55211yescharacteristic
307C22×C53#55312yes—
308C22×C53#55312yes—
309C22×C53#55311yescharacteristic
310C22×C53#55612yes—
311C22×C53#55912yes—
312C4×C53—24no—
313C4×C53—24no—
314C4×C53—44no—
315C4×C53—44no—
316Dic53—24no—
317Dic53—28no—
318Dic53—24no—
319Dic53—24no—
320Dic53—42no—
321Dic53—44no—
322C2×#388—532no—
323C2×C4×C4⋊C4—531no—
324C2×C4×C22⋊C4—531no—
325C2×C4×C22⋊C4—531no—
326#1437—532no—
327#1436—532no—
328#1434—531no—
329#1434—531no—
330C22⋊#95D5311yescharacteristic
331C4×#95—532no—
332C4×#95—531no—
333C2×C53—44no—
334C2×C53C4×#11711yescharacteristic
335C2×C53C42×D411yescharacteristic
336C2×C53#227511yescharacteristic
337C2×C53#234211yescharacteristic
338C2×C53#234711yescharacteristic
339C2×C53#235211yescharacteristic
340C2×C53#235711yescharacteristic
341C2×C53#236412yes—
342C2×C53#236712yes—
343C2×C53#237312yes—
344C2×C53#237412yes—
345C24×C4C2×D5311yescharacteristic
346C22×C4⋊C4—532no—
347C22×C22⋊C4—531no—
348C22×C22⋊C4—531no—
349C22×C22⋊C4—532no—
350C22×C22⋊C4C2×D5312yes—
351C22×C42—531no—
352C22×C42—532no—
353C22×C42—531no—
354C22×C42—532no—
355C22×C42—532no—
356#388—5316no—
357C4×C4⋊C4—1064no—
358C4×C22⋊C4—538no—
359C4×C22⋊C4—1064no—
360C2×#95—531no—
361C2×#95—532no—
362C2×#95—531no—
363C2×#95—532no—
364C2×#95—531no—
365C2×#95—532no—
366C2×#95—532no—
367C2×#95—531no—
368C2×#95—532no—
369C2×#95C2×D5311yescharacteristic
370C2×#95C2×D5312yes—
371C2×#95C2×D5311yescharacteristic
372C53#1382211yescharacteristic, complemented, Sylow
373C25C22×D5311yescharacteristic
374C23×C4—22no—
375C23×C4—24no—
376C23×C4—22no—
377C23×C4—42no—
378C23×C4—531no—
379C23×C4—532no—
380C23×C4—531no—
381C23×C4—531no—
382C23×C4—532no—
383C23×C4—531no—
384C23×C4C4×D5314yes—
385C23×C4C22×D5312yes—
386C23×C4C22×D5312yes—
387C23×C4C22×D5311yescharacteristic
388C23×C4C22×D5311yescharacteristic
389C2×C4⋊C4—538no—
390C2×C4⋊C4—1064no—
391C2×C4⋊C4—1064no—
392C2×C22⋊C4—42no—
393C2×C22⋊C4—42no—
394C2×C22⋊C4—534no—
395C2×C22⋊C4—538no—
396C2×C22⋊C4—534no—
397C2×C22⋊C4—538no—
398C2×C22⋊C4—1062no—
399C2×C22⋊C4—1064no—
400C2×C22⋊C4—1062no—
401C2×C22⋊C4—1064no—
402C2×C22⋊C4C4×D5318yes—
403C2×C42—534no—
404C2×C42—534no—
405C2×C42—538no—
406C2×C42—1062no—
407C2×C42—1062no—
408C2×C42—1064no—
409C2×C42—1064no—
410C2×C42—1064no—
411C2×C42—1064no—
412C2×C42—2122no—
413C2×C42—2122no—
414C2×C42—2122no—
415C2×C42—2122no—
416C2×C42—2122no—
417#95—22no—
418#95—24no—
419#95—22no—
420#95—534no—
421#95—534no—
422#95—538no—
423#95—538no—
424#95—534no—
425#95—1064no—
426#95—1064no—
427#95—1062no—
428#95—2122no—
429C24—21no—
430C24—21no—
431C24—21no—
432C24—22no—
433C24—22no—
434C24—21no—
435C24—21no—
436C24—22no—
437C24—21no—
438C24C2×C4×D5314yes—
439C24C2×C4×D5312yes—
440C24C23×D5311yescharacteristic
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—28no—
459C22×C4—28no—
460C22×C4—44no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—532no—
464C22×C4—532no—
465C22×C4—534no—
466C22×C4—532no—
467C22×C4—534no—
468C22×C4—534no—
469C22×C4—534no—
470C22×C4—532no—
471C22×C4—534no—
472C22×C4—532no—
473C22×C4—534no—
474C22×C4—532no—
475C22×C4—1061no—
476C22×C4—1061no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1061no—
480C22×C4—1062no—
481C22×C4—1061no—
482C22×C4—1061no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1061no—
486C22×C4—1062no—
487C22×C4—1061no—
488C22×C4—1061no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1061no—
493C22×C4—1061no—
494C22×C4—1062no—
495C22×C4—1062no—
496C22×C4—1062no—
497C22×C4—1062no—
498C22×C4—1062no—
499C22×C4C2×C4×D5314yes—
500C22×C4C2×C4×D5314yes—
501C22×C4C2×C4×D5312yes—
502C22×C4C2×C4×D5312yes—
503C22×C4D4×D5312yes—
504C22×C4#1171994714yes—
505C22×C4#1171994712yes—
506C4⋊C4—10616no—
507C22⋊C4—48no—
508C22⋊C4—5316no—
509C22⋊C4—1068no—
510C22⋊C4—10616no—
511C42—1068no—
512C42—2124no—
513C42—2128no—
514C42—2128no—
515C42—2128no—
516C23—22no—
517C23—22no—
518C23—24no—
519C23—22no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—22no—
524C23—24no—
525C23—22no—
526C23—24no—
527C23—22no—
528C23—41no—
529C23—41no—
530C23—42no—
531C23—41no—
532C23—42no—
533C23—42no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23C42×D5314yes—
539C23C22×C4×D5311yescharacteristic
540C23C22×C4×D5312yes—
541C23C2×D4×D5311yescharacteristic
542C23#41×D5311yescharacteristic
543C23#41×D5311yescharacteristic
544C23#41×D5312yes—
545C23C2×#1171994711yescharacteristic
546C23C2×#1171994712yes—
547C23#1313326211yescharacteristic
548C23#1313326212yes—
549C23#1313326511yescharacteristic
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—416no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—534no—
574C2×C4—534no—
575C2×C4—534no—
576C2×C4—538no—
577C2×C4—1062no—
578C2×C4—1064no—
579C2×C4—1062no—
580C2×C4—1062no—
581C2×C4—1064no—
582C2×C4—1064no—
583C2×C4—1062no—
584C2×C4—1064no—
585C2×C4—1062no—
586C2×C4—1062no—
587C2×C4—1064no—
588C2×C4—1064no—
589C2×C4—1062no—
590C2×C4—1064no—
591C2×C4—1064no—
592C2×C4—1062no—
593C2×C4—1064no—
594C2×C4—1062no—
595C2×C4—1062no—
596C2×C4—1064no—
597C2×C4—1062no—
598C2×C4—1064no—
599C2×C4—1064no—
600C2×C4—1062no—
601C2×C4—1062no—
602C2×C4—1064no—
603C2×C4—2122no—
604C2×C4—2122no—
605C2×C4—2122no—
606C2×C4—2122no—
607C2×C4—2122no—
608C2×C4—2122no—
609C2×C4#1313317314yes—
610C2×C4#1313317714yes—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—44no—
615C22—44no—
616C22—42no—
617C22—44no—
618C22—44no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22C2×C42×D5311yescharacteristic
623C22#117×D5311yescharacteristic
624C22C2×#1313317311yescharacteristic
625C22C4×D4×D5312yes—
626C22C2×#1313317711yescharacteristic
627C22C4×#1171994712yes—
628C22C4×#1171994712yes—
629C22C4×#1171994712yes—
630C22#1681109111yescharacteristic
631C22#1681109612yes—
632C22#1681109611yescharacteristic
633C22#1681110412yes—
634C22#1681110411yescharacteristic
635C22#1681126012yes—
636C22#1681127011yescharacteristic
637C22#1681128311yescharacteristic
638C22#1681130412yes—
639C22#1681130711yescharacteristic
640C22#1681131812yes—
641C22#1681132012yes—
642C22#1681132011yescharacteristic
643C22#1681134412yes—
644C22#1681135712yes—
645C4—24no—
646C4—24no—
647C4—44no—
648C4—44no—
649C4—1068no—
650C4—1064no—
651C4—1064no—
652C4—1064no—
653C4—2124no—
654C4—2122no—
655C2—44no—
656C2C4×#1313317311yescharacteristic
657C2C4×#1313317711yescharacteristic
658C2#2027707311yescharacteristic
659C2#2027756611yescharacteristic
660C2#2027761011yescharacteristic
661C2#2027763711yescharacteristic
662C2#2027768311yescharacteristic
663C2#2027776412yes—
664C2#2027778312yes—
665C2#2027783012yes—
666C2#2027784812yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC22×C53C22×C42
FittingC53×C22⋊#95C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenterC22⋊#95D53
hypocenterC53#13822
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
2. C2×#16810474C2semidirect
6. #20272946C2semidirect
7. #20272941C2semidirect
8. #20272939C2semidirect
9. #95×Dic53C2semidirect
10. #95×Dic53C2semidirect
21. #16810511C4semidirect
24. C4×C22⋊Dic53C4semidirect
47. C2×#11719919C2×C4semidirect
69. C2×C4×Dic53C2×C4semidirect
76. #13133124C2×C4semidirect
77. #13133124C2×C4semidirect
175. C22⋊Dic53C42semidirect
372. C53#13822semidirect

Maximal subgroups

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

subgroupindex
C2×#168105112
C2×#168104742
C2×C4×C22⋊Dic532
#202729522
#202729462
#202729412
#202729392
#95×Dic532
C53×C22⋊#952
#1382253