Finite Group Databasebeta

← #27305479

Subgroups — #27305479

Order 13568.

2782 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133175C211yesmaximal, characteristic
2C22×#13133174C211yesmaximal, characteristic
3C22×C4⋊C4×D53C211yesmaximal, characteristic
4#20272945C211yesmaximal, characteristic, complemented
5#20272944C211yesmaximal, characteristic, complemented
6#20272944C211yesmaximal, characteristic, complemented
7#20272943C214yesmaximal, complemented
8#20272936C214yesmaximal
9C53×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×C4×D53C2212yes—
11C23×C4×D53C2211yescharacteristic
12C2×#13133175—24no—
13C2×#13133175—24no—
14C2×#13133175C418yescomplemented
15C2×#13133174—24no—
16C2×#13133174—24no—
17C2×#13133174C418yescomplemented
18C22×C4⋊Dic53C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20C22×#11719919C2212yes—
21C22×#11719919C2212yes—
22C2×C4⋊C4×D53—24no—
23C2×C4⋊C4×D53—24no—
24C2×C4⋊C4×D53C418yescomplemented
25C22×#11719921C2212yes—
26C22×#11719921C2212yes—
27C22×C4×Dic53C2212yes—
28C22×C4×Dic53C2211yescharacteristic
29C2×#13133124C2214yes—
30C2×#13133124C2214yes—
31C2×#13133124C2214yes—
32C22×C4⋊C4×C53C2211yescharacteristic
33C22×C4⋊C4×C53C2211yescharacteristic
34C22×C42×C53C2211yescharacteristic
35C2×#95×C53C2214yes—
36C24×D53C2311yescharacteristic
37C23×Dic53C2312yes—
38C23×Dic53C2312yes—
39C23×Dic53C2312yes—
40C23×Dic53C2311yescharacteristic
41C22×C4×D53—24no—
42C22×C4×D53—24no—
43C22×C4×D53—44no—
44C22×C4×D53—44no—
45C22×C4×D53C2×C418yes—
46C22×C4×D53C2×C414yes—
47C22×C4×D53C2×C414yes—
48C22×C4×D53C2×C414yes—
49C22×C4×D53D414yes—
50C22×C4×D53D414yes—
51C22×C4×D53D414yes—
52C22×C4×D53Q814yes—
53#13133175—232no—
54#13133174—232no—
55C2×C4⋊Dic53—22no—
56C2×C4⋊Dic53—22no—
57C2×C4⋊Dic53—22no—
58C2×C4⋊Dic53—22no—
59C2×C4⋊Dic53C2×C414yes—
60C2×C4⋊Dic53C2×C414yescomplemented
61C2×#11719919—24no—
62C2×#11719919—24no—
63C2×#11719919—24no—
64C2×#11719919—24no—
65C2×#11719919C2×C418yes—
66C2×#11719919C2×C418yescomplemented
67C4⋊C4×D53—232no—
68C2×#11719921—24no—
69C2×#11719921—24no—
70C2×#11719921—24no—
71C2×#11719921—24no—
72C2×#11719921C2×C418yes—
73C2×#11719921C2×C418yes—
74C2×C4×Dic53—22no—
75C2×C4×Dic53—22no—
76C2×C4×Dic53—24no—
77C2×C4×Dic53—24no—
78C2×C4×Dic53C2×C414yescomplemented
79C2×C4×Dic53C2×C418yescomplemented
80#13133124—44no—
81#13133124—44no—
82#13133124—44no—
83C23×C4×C53C2312yes—
84C23×C4×C53C2312yes—
85C23×C4×C53C2312yes—
86C23×C4×C53C2311yescharacteristic
87C2×C4⋊C4×C53—22no—
88C2×C4⋊C4×C53—22no—
89C2×C4⋊C4×C53—22no—
90C2×C4⋊C4×C53—22no—
91C2×C4⋊C4×C53C2×C414yes—
92C2×C4⋊C4×C53C2×C414yescomplemented
93C2×C42×C53—22no—
94C2×C42×C53—22no—
95C2×C42×C53C2×C414yescomplemented
96#95×C53—44no—
97C24×C53C2411yescharacteristic
98C23×D53—22no—
99C23×D53—22no—
100C23×D53—22no—
101C23×D53—22no—
102C23×D53C22×C412yes—
103C23×D53C22×C412yes—
104C23×D53C22×C412yes—
105C23×D53C2×D412yes—
106C23×D53C2×D412yes—
107C23×D53C2×D412yes—
108C23×D53C2×Q812yes—
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×C53—24no—
117C22×C4×C53—24no—
118C22×C4×C53—22no—
119C22×C4×C53—24no—
120C22×C4×C53—22no—
121C22×C4×C53—24no—
122C22×C4×C53—22no—
123C22×C4×C53C22×C414yes—
124C22×C4×C53C22×C414yes—
125C22×C4×C53C22×C414yes—
126C22×C4×C53C22×C412yes—
127C22×C4×C53C22×C412yes—
128C22×C4×C53C22×C412yes—
129C22×C4×C53C2×D412yes—
130C22×C4×C53C2×D412yes—
131C22×C4×C53C2×D412yes—
132C22×C4×C53C2×Q812yes—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—24no—
142C22×Dic53—24no—
143C22×Dic53—24no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—24no—
147C22×Dic53C22×C414yes—
148C22×Dic53C22×C414yes—
149C22×Dic53C22×C414yes—
150C22×Dic53C22×C412yes—
151C22×Dic53C22×C412yes—
152C22×Dic53C22×C412yes—
153C22×Dic53C2×D412yes—
154C22×Dic53C2×D412yes—
155C22×Dic53C2×D412yes—
156C22×Dic53C2×Q812yes—
157C2×C4×D53—216no—
158C2×C4×D53—416no—
159C2×C4×D53—48no—
160C2×C4×D53—416no—
161C2×C4×D53—48no—
162C2×C4×D53C42116yes—
163C2×C4×D53C22⋊C4116yes—
164C2×C4×D53C22⋊C4116yes—
165C2×C4×D53C22⋊C4116yes—
166C2×C4×D53C4⋊C4116yes—
167C2×C4×D53C4⋊C4116yes—
168C2×C4×D53C4⋊C4116yes—
169C4⋊C4×C53—28no—
170C4⋊C4×C53—28no—
171C42×C53—28no—
172#11719921—216no—
173#11719921—216no—
174C4⋊Dic53—28no—
175C4⋊Dic53—28no—
176#11719919—216no—
177#11719919—216no—
178C4×Dic53—28no—
179C4×Dic53—216no—
180C23×C53C23×C411yescharacteristic
181C23×C53C23×C411yescharacteristic
182C23×C53C23×C411yescharacteristic
183C23×C53C22×D411yescharacteristic
184C23×C53C22×D411yescharacteristic
185C23×C53C22×D411yescharacteristic
186C23×C53C22×Q811yescharacteristic
187C23×C53H(5,𝔽2)12yes—
188C23×C53H(5,𝔽2)12yes—
189C23×C532−1+412yes—
190C23×C532−1+412yes—
191C22×D53—24no—
192C22×D53—24no—
193C22×D53—24no—
194C22×D53—24no—
195C22×D53—24no—
196C22×D53—24no—
197C22×D53—22no—
198C22×D53—24no—
199C22×D53—22no—
200C22×D53—22no—
201C22×D53—24no—
202C22×D53—22no—
203C22×D53—24no—
204C22×D53—24no—
205C22×D53—24no—
206C22×D53—24no—
207C22×D53C2×C4214yes—
208C22×D53C2×C22⋊C414yes—
209C22×D53C2×C22⋊C414yes—
210C22×D53C2×C22⋊C414yes—
211C22×D53C2×C4⋊C414yes—
212C22×D53C2×C4⋊C414yes—
213C22×D53C2×C4⋊C414yes—
214C2×C4×C53—24no—
215C2×C4×C53—24no—
216C2×C4×C53—24no—
217C2×C4×C53—28no—
218C2×C4×C53—28no—
219C2×C4×C53—24no—
220C2×C4×C53—24no—
221C2×C4×C53—42no—
222C2×C4×C53—44no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—44no—
230C2×C4×C53—42no—
231C2×C4×C53C2×C4214yes—
232C2×C4×C53C2×C22⋊C414yes—
233C2×C4×C53C2×C22⋊C414yes—
234C2×C4×C53C2×C22⋊C414yes—
235C2×C4×C53C2×C4⋊C414yes—
236C2×C4×C53C2×C4⋊C414yes—
237C2×C4×C53C2×C4⋊C414yes—
238C2×Dic53—24no—
239C2×Dic53—24no—
240C2×Dic53—24no—
241C2×Dic53—28no—
242C2×Dic53—24no—
243C2×Dic53—24no—
244C2×Dic53—28no—
245C2×Dic53—42no—
246C2×Dic53—44no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—44no—
255C2×Dic53C2×C4214yes—
256C2×Dic53C2×C22⋊C414yes—
257C2×Dic53C2×C22⋊C414yes—
258C2×Dic53C2×C22⋊C414yes—
259C2×Dic53C2×C4⋊C414yes—
260C2×Dic53C2×C4⋊C414yes—
261C2×Dic53C2×C4⋊C414yes—
262C4×D53—432no—
263C4×D53#95164yescomplemented
264#13795—531nomaximal, Sylow
265C22×C53C22×C4211yescharacteristic
266C22×C53C22×C22⋊C411yescharacteristic
267C22×C53C22×C22⋊C411yescharacteristic
268C22×C53C22×C22⋊C411yescharacteristic
269C22×C53C22×C4⋊C411yescharacteristic
270C22×C53C22×C4⋊C411yescharacteristic
271C22×C53C22×C4⋊C411yescharacteristic
272C22×C53#51812yes—
273C22×C53#51912yes—
274C22×C53#52012yes—
275C22×C53#52012yes—
276C22×C53#52012yes—
277C22×C53#52012yes—
278C22×C53#53511yescharacteristic
279C22×C53#53512yes—
280C22×C53#53511yescharacteristic
281C22×C53#53611yescharacteristic
282C22×C53#53612yes—
283C22×C53#53611yescharacteristic
284C22×C53#53712yes—
285C22×C53#53712yes—
286C22×C53#54412yes—
287C22×C53#54412yes—
288C2×D53—24no—
289C2×D53—24no—
290C2×D53—28no—
291C2×D53—24no—
292C2×D53—24no—
293C2×D53—28no—
294C2×D53—24no—
295C2×D53—28no—
296C2×D53—24no—
297C2×D53—28no—
298C2×D53C2×#9518yes—
299C4×C53—44no—
300C4×C53—44no—
301C4×C53—44no—
302C4×C53C2×#9518yes—
303Dic53—44no—
304Dic53—44no—
305Dic53—44no—
306Dic53C2×#9518yes—
307C22×#117—531no—
308C22×#117—531no—
309C23×C4⋊C4—531no—
310C4⋊#95—531no—
311C4⋊#95—534no—
312C4⋊#95—531no—
313C4⋊#95—531no—
314C4⋊#95D5311yescharacteristic, complemented
315C22⋊#95—534no—
316C2×C53C22×#9511yescharacteristic
317C2×C53#227412yes—
318C2×C53#230911yescharacteristic
319C2×C53#231011yescharacteristic
320C2×C53#231112yes—
321C2×C53#231111yescharacteristic
322C2×C53#231111yescharacteristic
323C2×C53#231612yes—
324C2×C53#231711yescharacteristic
325C2×C53#231712yes—
326C2×C53#231711yescharacteristic
327D53—28no—
328C24×C4—531no—
329C24×C4—532no—
330C2×#117—538no—
331C2×#117—538no—
332C2×#117—1064no—
333C2×#117—1064no—
334C2×#117—1064no—
335C2×#117—1064no—
336C22×C4⋊C4—532no—
337C22×C4⋊C4—532no—
338C22×C4⋊C4—531no—
339C22×C4⋊C4—531no—
340C22×C4⋊C4—538no—
341C22×C4⋊C4—1064no—
342C22×C4⋊C4—1064no—
343C22×C4⋊C4C2×D5311yescharacteristic
344C22×C4⋊C4C2×D5311yescharacteristic
345C22×C22⋊C4—532no—
346C22×C22⋊C4—532no—
347C22×C42—532no—
348C22×C42—531no—
349C22×C42C2×D5311yescharacteristic
350C2×#95—534no—
351C2×#95—534no—
352C2×#95—534no—
353C2×#95C2×D5314yes—
354C53#1379511yescharacteristic, complemented, Sylow
355C25—531no—
356C23×C4—531no—
357C23×C4—534no—
358C23×C4—534no—
359C23×C4—534no—
360C23×C4—534no—
361C23×C4—534no—
362C23×C4—534no—
363C23×C4—534no—
364C23×C4—532no—
365C23×C4—532no—
366C23×C4—532no—
367C23×C4—538no—
368C23×C4—1064no—
369C23×C4—1064no—
370C23×C4—2124no—
371C23×C4—2124no—
372C23×C4C22×D5312yes—
373C23×C4C22×D5312yes—
374C23×C4C22×D5312yes—
375C23×C4C22×D5311yescharacteristic
376#117—10632no—
377#117—10632no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—538no—
383C2×C4⋊C4—534no—
384C2×C4⋊C4—538no—
385C2×C4⋊C4—534no—
386C2×C4⋊C4—10632no—
387C2×C4⋊C4—1064no—
388C2×C4⋊C4—1064no—
389C2×C4⋊C4—1064no—
390C2×C4⋊C4—1064no—
391C2×C4⋊C4—1062no—
392C2×C4⋊C4—1062no—
393C2×C4⋊C4—1062no—
394C2×C4⋊C4—1062no—
395C2×C4⋊C4C4×D5314yes—
396C2×C4⋊C4C4×D5314yescomplemented
397C2×C22⋊C4—538no—
398C2×C22⋊C4—538no—
399C2×C22⋊C4—1064no—
400C2×C22⋊C4—1064no—
401C2×C22⋊C4—1064no—
402C2×C22⋊C4—1064no—
403C2×C42—22no—
404C2×C42—22no—
405C2×C42—538no—
406C2×C42—534no—
407C2×C42—1064no—
408C2×C42—1064no—
409C2×C42—1062no—
410C2×C42—1062no—
411C2×C42C4×D5314yescomplemented
412#95—44no—
413#95—2124no—
414#95—2124no—
415#95—2124no—
416C24—532no—
417C24—532no—
418C24—532no—
419C24—532no—
420C24—532no—
421C24—532no—
422C24—532no—
423C24—1062no—
424C24—1062no—
425C24—1062no—
426C24—1062no—
427C24C23×D5311yescharacteristic
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—532no—
443C22×C4—532no—
444C22×C4—532no—
445C22×C4—532no—
446C22×C4—532no—
447C22×C4—532no—
448C22×C4—532no—
449C22×C4—5316no—
450C22×C4—5316no—
451C22×C4—5316no—
452C22×C4—5316no—
453C22×C4—5316no—
454C22×C4—5316no—
455C22×C4—5316no—
456C22×C4—534no—
457C22×C4—534no—
458C22×C4—534no—
459C22×C4—10616no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1062no—
465C22×C4—1062no—
466C22×C4—1062no—
467C22×C4—1064no—
468C22×C4—1062no—
469C22×C4—1064no—
470C22×C4—1064no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1064no—
474C22×C4—21216no—
475C22×C4—2128no—
476C22×C4—21216no—
477C22×C4—2128no—
478C22×C4C2×C4×D5314yes—
479C22×C4C2×C4×D5314yes—
480C22×C4C2×C4×D5314yes—
481C22×C4C2×C4×D5312yes—
482C22×C4C2×C4×D5312yes—
483C22×C4C2×C4×D5312yes—
484C22×C4D4×D5312yes—
485C22×C4D4×D5312yes—
486C22×C4D4×D5312yes—
487C22×C4Q8×D5312yes—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—10616no—
491C4⋊C4—10616no—
492C4⋊C4—1068no—
493C4⋊C4—1068no—
494C22⋊C4—10616no—
495C22⋊C4—10616no—
496C42—28no—
497C42—10616no—
498C42—1068no—
499C23—534no—
500C23—534no—
501C23—534no—
502C23—534no—
503C23—534no—
504C23—534no—
505C23—534no—
506C23—1064no—
507C23—1064no—
508C23—1064no—
509C23—1064no—
510C23—1064no—
511C23—1064no—
512C23—1062no—
513C23—1064no—
514C23—1062no—
515C23—1062no—
516C23—1064no—
517C23—1062no—
518C23—1064no—
519C23—1064no—
520C23—1064no—
521C23—1064no—
522C23C22×C4×D5311yescharacteristic
523C23C22×C4×D5311yescharacteristic
524C23C22×C4×D5311yescharacteristic
525C23C2×D4×D5311yescharacteristic
526C23C2×D4×D5311yescharacteristic
527C23C2×D4×D5311yescharacteristic
528C23C2×Q8×D5311yescharacteristic
529C23#1313326212yes—
530C23#1313326312yes—
531C23#1313326412yes—
532C23#1313326512yes—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—534no—
551C2×C4—534no—
552C2×C4—534no—
553C2×C4—534no—
554C2×C4—534no—
555C2×C4—534no—
556C2×C4—534no—
557C2×C4—5364no—
558C2×C4—1064no—
559C2×C4—1064no—
560C2×C4—1064no—
561C2×C4—1068no—
562C2×C4—1064no—
563C2×C4—1064no—
564C2×C4—1068no—
565C2×C4—21232no—
566C2×C4—2122no—
567C2×C4—2122no—
568C2×C4—2124no—
569C2×C4—2122no—
570C2×C4—2122no—
571C2×C4—2122no—
572C2×C4—2122no—
573C2×C4—2122no—
574C2×C4—2122no—
575C2×C4—2124no—
576C2×C4C42×D5314yes—
577C2×C4C22⋊C4×D5314yes—
578C2×C4C22⋊C4×D5314yes—
579C2×C4C22⋊C4×D5314yes—
580C2×C4C4⋊C4×D5314yes—
581C2×C4C4⋊C4×D5314yes—
582C2×C4C4⋊C4×D5314yes—
583C22—538no—
584C22—1064no—
585C22—1064no—
586C22—1068no—
587C22—1064no—
588C22—1064no—
589C22—1068no—
590C22—1068no—
591C22—1064no—
592C22—1068no—
593C22—1064no—
594C22C2×C42×D5311yescharacteristic
595C22C2×C22⋊C4×D5311yescharacteristic
596C22C2×C22⋊C4×D5311yescharacteristic
597C22C2×C22⋊C4×D5311yescharacteristic
598C22C2×C4⋊C4×D5311yescharacteristic
599C22C2×C4⋊C4×D5311yescharacteristic
600C22C2×C4⋊C4×D5311yescharacteristic
601C22#1681109512yes—
602C22#1681109912yes—
603C22#1681110212yes—
604C22#1681110312yes—
605C22#1681110412yes—
606C22#1681110512yes—
607C22#1681118611yescharacteristic
608C22#1681118912yes—
609C22#1681119111yescharacteristic
610C22#1681119311yescharacteristic
611C22#1681119612yes—
612C22#1681119811yescharacteristic
613C22#1681120212yes—
614C22#1681120312yes—
615C22#1681125012yes—
616C22#1681125312yes—
617C4—44no—
618C4—44no—
619C4—44no—
620C4—538no—
621C4—2124no—
622C4—2124no—
623C4—2124no—
624C4#95×D5318yes—
625C2—1068no—
626C2C2×#95×D5311yescharacteristic
627C2#2027706912yes—
628C2#2027729411yescharacteristic
629C2#2027729811yescharacteristic
630C2#2027730412yes—
631C2#2027730511yescharacteristic
632C2#2027730611yescharacteristic
633C2#2027733012yes—
634C2#2027733611yescharacteristic
635C2#2027733712yes—
636C2#2027734011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC22×C53C22×C42
FittingC53×C4⋊#95C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenterC4⋊#95D53
hypocenterC53#13795
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
4. #20272945C2semidirect
5. #20272944C2semidirect
6. #20272944C2semidirect
7. #20272943C2semidirect
9. C53×C4⋊#95C2semidirect
14. C2×#13133175C4semidirect
17. C2×#13133174C4semidirect
24. C2×C4⋊C4×D53C4semidirect
60. C2×C4⋊Dic53C2×C4semidirect
66. C2×#11719919C2×C4semidirect
78. C2×C4×Dic53C2×C4semidirect
79. C2×C4×Dic53C2×C4semidirect
92. C2×C4⋊C4×C53C2×C4semidirect
95. C2×C42×C53C2×C4semidirect
263. C4×D53#95semidirect
314. C4⋊#95D53semidirect
354. C53#13795semidirect
396. C2×C4⋊C4C4×D53semidirect
411. C2×C42C4×D53semidirect

Maximal subgroups

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

subgroupindex
C22×#131331752
C22×#131331742
C22×C4⋊C4×D532
#202729452
#202729442
#202729432
#202729362
C53×C4⋊#952
#1379553