Finite Group Databasebeta

← #27306129

Subgroups — #27306129

Order 13568.

1342 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×#16810488C211yesmaximal, characteristic
2C2×C4⋊C4×Dic53C211yesmaximal, characteristic
3C2×C4×C4⋊Dic53C211yesmaximal, characteristic
4#20272984C212yesmaximal
5#20272976C212yesmaximal
6#20272949C212yesmaximal
7#20272945C211yesmaximal, characteristic
8#20272944C211yesmaximal, characteristic
9#20272944C211yesmaximal, characteristic
10#20272941C212yesmaximal
11C53×C4⋊#95C211yesmaximal, characteristic
12C22×C4⋊Dic53C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14C22×C4⋊Dic53C2211yescharacteristic
15C22×C4⋊Dic53C2211yescharacteristic
16C22×C4×Dic53C2212yes—
17C22×C4×Dic53C2211yescharacteristic
18C22×C4×Dic53C2211yescharacteristic
19#16810488—24no—
20C4⋊C4×Dic53—24no—
21C4×C4⋊Dic53C418yescomplemented
22C2×#13133124C2212yes—
23C2×#13133124C2212yes—
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2211yescharacteristic
32C2×#13133124C2211yescharacteristic
33C22×C4⋊C4×C53C2211yescharacteristic
34C22×C4⋊C4×C53C2211yescharacteristic
35C22×C42×C53C2211yescharacteristic
36C2×#95×C53C2212yes—
37C2×#95×C53C2212yes—
38C23×Dic53C2312yes—
39C23×Dic53C2311yescharacteristic
40C23×Dic53C2311yescharacteristic
41C23×Dic53C2311yescharacteristic
42C23×Dic53C2312yes—
43C23×Dic53C2311yescharacteristic
44C2×C4⋊Dic53—22no—
45C2×C4⋊Dic53—22no—
46C2×C4⋊Dic53—22no—
47C2×C4⋊Dic53—22no—
48C2×C4⋊Dic53—22no—
49C2×C4⋊Dic53—22no—
50C2×C4⋊Dic53—22no—
51C2×C4⋊Dic53—22no—
52C2×C4⋊Dic53—22no—
53C2×C4⋊Dic53C2×C414yes—
54C2×C4⋊Dic53C2×C414yes—
55C2×C4⋊Dic53D414yes—
56C2×C4×Dic53—22no—
57C2×C4×Dic53—22no—
58C2×C4×Dic53—22no—
59C2×C4×Dic53—22no—
60C2×C4×Dic53—24no—
61C2×C4×Dic53—24no—
62C2×C4×Dic53—42no—
63C2×C4×Dic53C2×C414yes—
64C2×C4×Dic53C2×C414yes—
65#13133124—24no—
66#13133124—24no—
67#13133124—24no—
68#13133124—24no—
69#13133124—24no—
70#13133124—42no—
71#13133124—42no—
72#13133124—42no—
73#13133124—42no—
74#13133124C2×C414yes—
75#13133124C2×C414yes—
76C23×C4×C53C2312yes—
77C23×C4×C53C2311yescharacteristic
78C23×C4×C53C2312yes—
79C23×C4×C53C2311yescharacteristic
80C23×C4×C53C2311yescharacteristic
81C2×C4⋊C4×C53—22no—
82C2×C4⋊C4×C53—22no—
83C2×C4⋊C4×C53—22no—
84C2×C4⋊C4×C53—22no—
85C2×C4⋊C4×C53—22no—
86C2×C4⋊C4×C53Q814yes—
87C2×C42×C53—22no—
88C2×C42×C53—22no—
89C2×C42×C53C2×C414yes—
90#95×C53—24no—
91#95×C53—42no—
92C24×C53C2411yescharacteristic
93C22×C4×C53—21no—
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—21no—
97C22×C4×C53—22no—
98C22×C4×C53—22no—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—22no—
102C22×C4×C53—22no—
103C22×C4×C53—21no—
104C22×C4×C53—22no—
105C22×C4×C53—21no—
106C22×C4×C53—22no—
107C22×C4×C53—21no—
108C22×C4×C53—21no—
109C22×C4×C53—22no—
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×C53C22×C412yes—
117C22×C4×C53C22×C412yes—
118C22×C4×C53C22×C412yes—
119C22×C4×C53C2×D412yes—
120C22×C4×C53C2×Q814yes—
121C22×C4×C53C2×Q812yes—
122C22×C4×C53#4112yes—
123C22×C4×C53#4112yes—
124C22×C4×C53#4112yes—
125C22×C4×C53#4112yes—
126C22×C4×C53#4112yes—
127C22×C4×C53#4112yes—
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—22no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—22no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—22no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53—22no—
162C22×Dic53—22no—
163C22×Dic53C22×C412yes—
164C22×Dic53C22×C412yes—
165C22×Dic53C22×C412yes—
166C22×Dic53C22×C412yes—
167C22×Dic53C2×D414yes—
168C22×Dic53#4112yes—
169C22×Dic53#4112yes—
170C4⋊C4×C53—28no—
171C4⋊C4×C53—44no—
172C42×C53—28no—
173C4⋊Dic53—28no—
174C4⋊Dic53—28no—
175C4⋊Dic53—28no—
176C4⋊Dic53—44no—
177C4×Dic53—28no—
178C4×Dic53—28no—
179C4×Dic53—48no—
180C23×C53C23×C411yescharacteristic
181C23×C53C22×D411yescharacteristic
182C23×C53C22×Q811yescharacteristic
183C23×C53C2×#4111yescharacteristic
184C23×C53C2×#4111yescharacteristic
185C23×C53C2×#4111yescharacteristic
186C23×C53C2×#4111yescharacteristic
187C23×C53C2×#4111yescharacteristic
188C23×C53C2×#4111yescharacteristic
189C23×C53C2×#4111yescharacteristic
190C23×C53C2×#4111yescharacteristic
191C23×C53H(5,𝔽2)11yescharacteristic
192C23×C53H(5,𝔽2)11yescharacteristic
193C23×C532−1+411yescharacteristic
194C23×C532−1+411yescharacteristic
195C2×C4×C53—22no—
196C2×C4×C53—22no—
197C2×C4×C53—22no—
198C2×C4×C53—22no—
199C2×C4×C53—22no—
200C2×C4×C53—22no—
201C2×C4×C53—22no—
202C2×C4×C53—22no—
203C2×C4×C53—22no—
204C2×C4×C53—24no—
205C2×C4×C53—24no—
206C2×C4×C53—24no—
207C2×C4×C53—42no—
208C2×C4×C53—41no—
209C2×C4×C53—42no—
210C2×C4×C53—41no—
211C2×C4×C53—41no—
212C2×C4×C53—41no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53#11714yes—
223C2×C4×C53#11714yes—
224C2×C4×C53C4×D414yes—
225C2×C4×C53C4×Q814yes—
226C2×C4×C53C22⋊Q814yes—
227C2×C4×C53#12314yes—
228C2×C4×C53#12514yes—
229C2×C4×C53#12614yes—
230C2×Dic53—22no—
231C2×Dic53—22no—
232C2×Dic53—22no—
233C2×Dic53—22no—
234C2×Dic53—22no—
235C2×Dic53—22no—
236C2×Dic53—22no—
237C2×Dic53—22no—
238C2×Dic53—22no—
239C2×Dic53—22no—
240C2×Dic53—22no—
241C2×Dic53—22no—
242C2×Dic53—22no—
243C2×Dic53—22no—
244C2×Dic53—22no—
245C2×Dic53—24no—
246C2×Dic53—24no—
247C2×Dic53—24no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—42no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53—42no—
266C2×Dic53—42no—
267C2×Dic53—42no—
268C2×Dic53—42no—
269C2×Dic53—42no—
270C2×Dic53—42no—
271C2×Dic53#11714yes—
272#13885—531nomaximal, Sylow
273C22×C53C2×#11711yescharacteristic
274C22×C53C2×#11711yescharacteristic
275C22×C53C2×#11711yescharacteristic
276C22×C53C2×C4×D411yescharacteristic
277C22×C53C2×C4×Q811yescharacteristic
278C22×C53#52011yescharacteristic
279C22×C53#52011yescharacteristic
280C22×C53C2×C22⋊Q811yescharacteristic
281C22×C53C2×#12311yescharacteristic
282C22×C53C2×#12511yescharacteristic
283C22×C53C2×#12611yescharacteristic
284C22×C53#52911yescharacteristic
285C22×C53#52911yescharacteristic
286C22×C53#52911yescharacteristic
287C22×C53#52911yescharacteristic
288C22×C53#54011yescharacteristic
289C22×C53#54111yescharacteristic
290C22×C53#54211yescharacteristic
291C22×C53#54211yescharacteristic
292C22×C53#54611yescharacteristic
293C22×C53#54711yescharacteristic
294C22×C53#54811yescharacteristic
295C22×C53#55011yescharacteristic
296C22×C53#55211yescharacteristic
297C22×C53#55311yescharacteristic
298C22×C53#55411yescharacteristic
299C22×C53#55411yescharacteristic
300C22×C53#55511yescharacteristic
301C22×C53#55611yescharacteristic
302C22×C53#55611yescharacteristic
303C22×C53#55611yescharacteristic
304C22×C53#55711yescharacteristic
305C22×C53#55711yescharacteristic
306C22×C53#55911yescharacteristic
307C22×C53#55911yescharacteristic
308C4×C53—24no—
309C4×C53—42no—
310C4×C53—44no—
311C4×C53—82no—
312C4×C53#38718yes—
313Dic53—24no—
314Dic53—42no—
315Dic53—42no—
316Dic53—42no—
317Dic53—44no—
318Dic53—82no—
319C2×#382—531no—
320C2×C4×C4⋊C4—531no—
321C2×C4×C4⋊C4—531no—
322#1445—532no—
323#1443—532no—
324#1436—532no—
325C4⋊#95—531no—
326C4⋊#95—531no—
327C4⋊#95—531no—
328C4⋊#95D5311yescharacteristic
329#1434—532no—
330C2×C53#228311yescharacteristic
331C2×C53C2×#38711yescharacteristic
332C2×C53#231411yescharacteristic
333C2×C53#235211yescharacteristic
334C2×C53#235411yescharacteristic
335C2×C53#235511yescharacteristic
336C2×C53#235511yescharacteristic
337C2×C53#244711yescharacteristic
338C2×C53#244811yescharacteristic
339C2×C53#245311yescharacteristic
340C2×C53#245411yescharacteristic
341C2×C53#250811yescharacteristic
342C2×C53#250911yescharacteristic
343C2×C53#251311yescharacteristic
344C2×C53#251411yescharacteristic
345C22×C4⋊C4—531no—
346C22×C4⋊C4—531no—
347C22×C4⋊C4—531no—
348C22×C4⋊C4—531no—
349C22×C4⋊C4C2×D5311yescharacteristic
350C22×C4⋊C4C2×D5311yescharacteristic
351C22×C42—531no—
352C22×C42—531no—
353C22×C42—532no—
354C22×C42C2×D5311yescharacteristic
355#382—1064no—
356C4×C4⋊C4—538no—
357C4×C4⋊C4—1064no—
358C2×#95—531no—
359C2×#95—531no—
360C2×#95—532no—
361C2×#95—532no—
362C2×#95—532no—
363C2×#95—532no—
364C2×#95—532no—
365C2×#95—532no—
366C2×#95—532no—
367C2×#95—532no—
368C2×#95—532no—
369C2×#95C2×D5312yes—
370C2×#95C2×D5312yes—
371C53#1388511yescharacteristic, complemented, Sylow
372C23×C4—531no—
373C23×C4—531no—
374C23×C4—531no—
375C23×C4—531no—
376C23×C4—532no—
377C23×C4—532no—
378C23×C4C22×D5312yes—
379C23×C4C22×D5312yes—
380C23×C4C22×D5311yescharacteristic
381C23×C4C22×D5311yescharacteristic
382C23×C4C22×D5311yescharacteristic
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—534no—
389C2×C4⋊C4—534no—
390C2×C4⋊C4—534no—
391C2×C4⋊C4—1062no—
392C2×C4⋊C4—1062no—
393C2×C4⋊C4—1062no—
394C2×C4⋊C4—1062no—
395C2×C4⋊C4—1062no—
396C2×C4⋊C4—1062no—
397C2×C4⋊C4—1062no—
398C2×C4⋊C4—1062no—
399C2×C4⋊C4—1062no—
400C2×C4⋊C4Dic10614yes—
401C2×C42—22no—
402C2×C42—22no—
403C2×C42—534no—
404C2×C42—534no—
405C2×C42—1062no—
406C2×C42—1062no—
407C2×C42—1062no—
408C2×C42—1062no—
409C2×C42—1064no—
410C2×C42—1064no—
411C2×C42—2122no—
412C2×C42C4×D5314yes—
413#95—24no—
414#95—42no—
415#95—534no—
416#95—534no—
417#95—1064no—
418#95—1064no—
419#95—1064no—
420#95—1064no—
421#95—1064no—
422#95—2122no—
423#95—2122no—
424#95—2122no—
425#95—2122no—
426C24C23×D5311yescharacteristic
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
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—532no—
451C22×C4—532no—
452C22×C4—532no—
453C22×C4—532no—
454C22×C4—532no—
455C22×C4—532no—
456C22×C4—534no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1062no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1061no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4—1061no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—1062no—
481C22×C4—1061no—
482C22×C4—1062no—
483C22×C4—1061no—
484C22×C4—1061no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—1062no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4C2×Dic10614yes—
493C22×C4C2×Dic10612yes—
494C22×C4C2×C4×D5312yes—
495C22×C4C2×C4×D5312yes—
496C22×C4C2×C4×D5312yes—
497C22×C4#1171994512yes—
498C22×C4D4×D5312yes—
499C22×C4#1171994712yes—
500C22×C4#1171994712yes—
501C22×C4#1171994712yes—
502C22×C4#1171994912yes—
503C22×C4#1171994912yes—
504C4⋊C4—28no—
505C4⋊C4—44no—
506C4⋊C4—1068no—
507C4⋊C4—1068no—
508C4⋊C4—1068no—
509C4⋊C4—2124no—
510C42—28no—
511C42—1068no—
512C42—1068no—
513C42—2128no—
514C23C22×C4×D5311yescharacteristic
515C23C2×D4×D5311yescharacteristic
516C23C22×Dic10611yescharacteristic
517C23#41×D5311yescharacteristic
518C23#41×D5311yescharacteristic
519C23C2×#1171994711yescharacteristic
520C23C2×#1171994711yescharacteristic
521C23C2×#1171994711yescharacteristic
522C23C2×#1171994511yescharacteristic
523C23C2×#1171994911yescharacteristic
524C23C2×#1171994911yescharacteristic
525C23#1313326211yescharacteristic
526C23#1313326311yescharacteristic
527C23#1313326411yescharacteristic
528C23#1313326511yescharacteristic
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—534no—
557C2×C4—1062no—
558C2×C4—1062no—
559C2×C4—1062no—
560C2×C4—1062no—
561C2×C4—1062no—
562C2×C4—1062no—
563C2×C4—1064no—
564C2×C4—1064no—
565C2×C4—1062no—
566C2×C4—1062no—
567C2×C4—1062no—
568C2×C4—1062no—
569C2×C4—1062no—
570C2×C4—1062no—
571C2×C4—1062no—
572C2×C4—1062no—
573C2×C4—1062no—
574C2×C4—1064no—
575C2×C4—2122no—
576C2×C4—2122no—
577C2×C4—2122no—
578C2×C4—2122no—
579C2×C4—2122no—
580C2×C4—2122no—
581C2×C4—2122no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2122no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2122no—
596C2×C4—2122no—
597C2×C4—2122no—
598C2×C4#1313317314yes—
599C2×C4#1313317514yes—
600C2×C4#1313317914yes—
601C2×C4C4×Dic10614yes—
602C2×C4#1313319414yes—
603C2×C4#1313319914yes—
604C2×C4#1313320814yes—
605C2×C4#1313321214yes—
606C22#117×D5311yescharacteristic
607C22C2×#1313317311yescharacteristic
608C22C2×#1313317511yescharacteristic
609C22C2×#1313317911yescharacteristic
610C22C2×C4×Dic10611yescharacteristic
611C22#1681110311yescharacteristic
612C22#1681110411yescharacteristic
613C22C2×#1313319411yescharacteristic
614C22C2×#1313319911yescharacteristic
615C22C2×#1313320811yescharacteristic
616C22C2×#1313321211yescharacteristic
617C22#1681115411yescharacteristic
618C22#1681115411yescharacteristic
619C22#1681115911yescharacteristic
620C22#1681115911yescharacteristic
621C22#1681122611yescharacteristic
622C22#1681123311yescharacteristic
623C22#1681124011yescharacteristic
624C22#1681124611yescharacteristic
625C22#1681126611yescharacteristic
626C22#1681127611yescharacteristic
627C22#1681128111yescharacteristic
628C22#1681129311yescharacteristic
629C22#1681131111yescharacteristic
630C22#1681132611yescharacteristic
631C22#1681133011yescharacteristic
632C22#1681133211yescharacteristic
633C22#1681133911yescharacteristic
634C22#1681134311yescharacteristic
635C22#1681134511yescharacteristic
636C22#1681134711yescharacteristic
637C22#1681134911yescharacteristic
638C22#1681135211yescharacteristic
639C22#1681135611yescharacteristic
640C22#1681135811yescharacteristic
641C4—24no—
642C4—44no—
643C4—42no—
644C4—82no—
645C4—1064no—
646C4—4242no—
647C4—2122no—
648C4—2122no—
649C4—2122no—
650C4—2124no—
651C4#1681050618yes—
652C2#2027711011yescharacteristic
653C2C2×#1681050611yescharacteristic
654C2#2027732211yescharacteristic
655C2#2027764111yescharacteristic
656C2#2027766611yescharacteristic
657C2#2027767011yescharacteristic
658C2#2027767311yescharacteristic
659C2#2027850511yescharacteristic
660C2#2027851611yescharacteristic
661C2#2027857911yescharacteristic
662C2#2027859011yescharacteristic
663C2#2027925611yescharacteristic
664C2#2027926711yescharacteristic
665C2#2027931511yescharacteristic
666C2#2027932611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×C4⋊#95C2
socle, prime radical, cosocleC24×C53C24
hypercenterC4⋊#95D53
hypocenterC53#13885
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
21. C4×C4⋊Dic53C4semidirect
371. C53#13885semidirect

Maximal subgroups

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

subgroupindex
C2×#168104882
C2×C4⋊C4×Dic532
C2×C4×C4⋊Dic532
#202729842
#202729762
#202729492
#202729452
#202729442
#202729412
C53×C4⋊#952
#1388553