Finite Group Databasebeta

← #27305334

Subgroups — #27305334

Order 13568.

2318 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
1C22×C4⋊C4×D53C211yesmaximal, characteristic
2C2×#16810485C212yesmaximal
3C2×C4×#11719921C211yesmaximal, characteristic
4C2×#16810470C211yesmaximal, characteristic
5#20272950C211yesmaximal, characteristic, complemented
6#20272949C212yesmaximal, complemented
7#20272948C211yesmaximal, characteristic, complemented
8#20272947C211yesmaximal, characteristic, complemented
9#20272946C212yesmaximal, complemented
10#20272936C212yesmaximal
11C53×#1436C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C23×C4×D53C2211yescharacteristic
14C23×C4×D53C2211yescharacteristic
15C22×C4⋊Dic53C2211yescharacteristic
16C22×#11719919C2211yescharacteristic
17C22×#11719919C2211yescharacteristic
18C2×C4⋊C4×D53—24no—
19C2×C4⋊C4×D53—24no—
20C2×C4⋊C4×D53C418yescomplemented
21C22×#11719921C2212yes—
22C22×#11719921C2212yes—
23C22×C4×Dic53C2212yes—
24C22×C4×Dic53C2211yescharacteristic
25#16810485C4116yescomplemented
26C4×#11719921—24no—
27#16810470—24no—
28C2×#13133124C2212yes—
29C2×#13133124C2211yescharacteristic
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2212yes—
35C2×#13133124C2212yes—
36C22×C4⋊C4×C53C2211yescharacteristic
37C22×C42×C53C2211yescharacteristic
38C2×#95×C53C2211yescharacteristic
39C2×#95×C53C2212yes—
40C2×#95×C53C2212yes—
41C24×D53C2311yescharacteristic
42C23×Dic53C2312yes—
43C23×Dic53C2312yes—
44C23×Dic53C2311yescharacteristic
45C23×Dic53C2311yescharacteristic
46C23×Dic53C2311yescharacteristic
47C22×C4×D53—22no—
48C22×C4×D53—22no—
49C22×C4×D53—24no—
50C22×C4×D53—22no—
51C22×C4×D53—22no—
52C22×C4×D53—42no—
53C22×C4×D53—42no—
54C22×C4×D53—42no—
55C22×C4×D53—42no—
56C22×C4×D53C2×C414yes—
57C22×C4×D53C2×C418yes—
58C22×C4×D53C2×C414yes—
59C22×C4×D53C2×C414yes—
60C22×C4×D53D414yes—
61C22×C4×D53D414yes—
62C2×C4⋊Dic53—22no—
63C2×C4⋊Dic53—22no—
64C2×C4⋊Dic53C2×C414yes—
65C2×#11719919—22no—
66C2×#11719919—22no—
67C2×#11719919—22no—
68C2×#11719919—22no—
69C2×#11719919C2×C414yes—
70C2×#11719919C2×C414yes—
71C4⋊C4×D53—232no—
72C2×#11719921—24no—
73C2×#11719921—24no—
74C2×#11719921—24no—
75C2×#11719921—24no—
76C2×#11719921C2×C418yes—
77C2×#11719921C2×C418yes—
78C2×C4×Dic53—22no—
79C2×C4×Dic53—24no—
80C2×C4×Dic53—42no—
81C2×C4×Dic53—42no—
82C2×C4×Dic53—42no—
83#13133124—22no—
84#13133124—24no—
85#13133124—42no—
86#13133124—42no—
87#13133124—42no—
88#13133124C2×C418yescomplemented
89#13133124C2×C418yescomplemented
90#13133124C2×C418yescomplemented
91C23×C4×C53C2312yes—
92C23×C4×C53C2311yescharacteristic
93C23×C4×C53C2312yes—
94C23×C4×C53C2311yescharacteristic
95C23×C4×C53C2311yescharacteristic
96C2×C4⋊C4×C53—22no—
97C2×C4⋊C4×C53—22no—
98C2×C4⋊C4×C53C2×C414yes—
99C2×C42×C53—22no—
100C2×C42×C53—42no—
101#95×C53—22no—
102#95×C53—42no—
103#95×C53C2×C418yescomplemented
104C24×C53C2411yescharacteristic
105C23×D53—22no—
106C23×D53—22no—
107C23×D53—22no—
108C23×D53—22no—
109C23×D53C22×C412yes—
110C23×D53C22×C414yes—
111C23×D53C2×D412yes—
112C23×D53C2×D412yes—
113C23×D53#4114yes—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53C22×C414yes—
136C22×C4×C53C22×C414yes—
137C22×C4×C53C22×C412yes—
138C22×C4×C53C22×C414yes—
139C22×C4×C53C22×C412yes—
140C22×C4×C53C22×C412yes—
141C22×C4×C53C2×D412yes—
142C22×C4×C53C2×D412yes—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
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×Dic53—22no—
164C22×Dic53C22×C414yes—
165C22×Dic53C22×C414yes—
166C22×Dic53C22×C412yes—
167C22×Dic53C22×C414yes—
168C22×Dic53C22×C412yes—
169C22×Dic53C22×C412yes—
170C22×Dic53C2×D412yes—
171C22×Dic53C2×D412yes—
172C2×C4×D53—28no—
173C2×C4×D53—28no—
174C2×C4×D53—216no—
175C2×C4×D53—216no—
176C2×C4×D53—28no—
177C2×C4×D53—48no—
178C2×C4×D53—48no—
179C2×C4×D53—44no—
180C2×C4×D53—44no—
181C2×C4×D53—48no—
182C2×C4×D53—48no—
183C2×C4×D53—44no—
184C2×C4×D53—44no—
185C2×C4×D53C42116yescomplemented
186C2×C4×D53C22⋊C4116yes—
187C4⋊C4×C53—28no—
188C42×C53—44no—
189#11719921—216no—
190#11719921—216no—
191C4⋊Dic53—28no—
192#11719919—28no—
193#11719919—28no—
194C4×Dic53—48no—
195C4×Dic53—44no—
196C23×C53C23×C411yescharacteristic
197C23×C53C23×C412yes—
198C23×C53C22×D411yescharacteristic
199C23×C53C22×D411yescharacteristic
200C23×C53C2×#4112yes—
201C23×C53H(5,𝔽2)12yes—
202C23×C53H(5,𝔽2)12yes—
203C23×C532−1+412yes—
204C23×C532−1+412yes—
205C22×D53—24no—
206C22×D53—24no—
207C22×D53—24no—
208C22×D53—24no—
209C22×D53—24no—
210C22×D53—24no—
211C22×D53—24no—
212C22×D53—22no—
213C22×D53—22no—
214C22×D53—22no—
215C22×D53—24no—
216C22×D53—22no—
217C22×D53—24no—
218C22×D53—24no—
219C22×D53—24no—
220C22×D53—24no—
221C22×D53C2×C4214yes—
222C22×D53C2×C22⋊C414yes—
223C22×D53#11714yes—
224C22×D53C4×D418yes—
225C22×D53C4×D418yes—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—24no—
229C2×C4×C53—24no—
230C2×C4×C53—24no—
231C2×C4×C53—24no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—24no—
236C2×C4×C53—24no—
237C2×C4×C53—24no—
238C2×C4×C53—24no—
239C2×C4×C53—24no—
240C2×C4×C53—24no—
241C2×C4×C53—24no—
242C2×C4×C53—42no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—42no—
246C2×C4×C53—42no—
247C2×C4×C53—42no—
248C2×C4×C53—42no—
249C2×C4×C53—42no—
250C2×C4×C53—42no—
251C2×C4×C53—42no—
252C2×C4×C53—42no—
253C2×C4×C53C2×C4214yes—
254C2×C4×C53C2×C22⋊C414yes—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—24no—
258C2×Dic53—24no—
259C2×Dic53—24no—
260C2×Dic53—22no—
261C2×Dic53—22no—
262C2×Dic53—24no—
263C2×Dic53—22no—
264C2×Dic53—24no—
265C2×Dic53—24no—
266C2×Dic53—24no—
267C2×Dic53—24no—
268C2×Dic53—24no—
269C2×Dic53—24no—
270C2×Dic53—24no—
271C2×Dic53—42no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—42no—
275C2×Dic53—42no—
276C2×Dic53—42no—
277C2×Dic53—42no—
278C2×Dic53—42no—
279C2×Dic53—42no—
280C2×Dic53—42no—
281C2×Dic53—42no—
282C2×Dic53C2×C4214yes—
283C2×Dic53C2×C22⋊C414yes—
284C4×D53—232no—
285C4×D53—416no—
286C4×D53—416no—
287#13776—531nomaximal, Sylow
288C22×C53C22×C4211yescharacteristic
289C22×C53C22×C22⋊C411yescharacteristic
290C22×C53C2×#11711yescharacteristic
291C22×C53C2×C4×D412yes—
292C22×C53C2×C4×D412yes—
293C22×C53#51812yes—
294C22×C53#51912yes—
295C22×C53#52012yes—
296C22×C53#52012yes—
297C22×C53#52012yes—
298C22×C53#52012yes—
299C22×C53#53412yes—
300C22×C53#53411yescharacteristic
301C22×C53#53411yescharacteristic
302C22×C53#53611yescharacteristic
303C22×C53#53612yes—
304C22×C53#53611yescharacteristic
305C22×C53#53912yes—
306C22×C53#53912yes—
307C22×C53#53912yes—
308C22×C53#53912yes—
309C2×D53—24no—
310C2×D53—24no—
311C2×D53—28no—
312C2×D53—28no—
313C2×D53—28no—
314C2×D53—28no—
315C2×D53—28no—
316C2×D53—28no—
317C2×D53C4×C22⋊C418yes—
318C4×C53—24no—
319C4×C53—24no—
320C4×C53—42no—
321C4×C53—44no—
322C4×C53—44no—
323Dic53—24no—
324Dic53—24no—
325Dic53—42no—
326Dic53—44no—
327Dic53—44no—
328C23×C4⋊C4—531no—
329C2×#381—532no—
330C2×C4×C22⋊C4—531no—
331C2×C4×C22⋊C4—531no—
332#1436—531no—
333#1436—532no—
334#1436—532no—
335#1436—531no—
336#1436—531no—
337#1436D5311yescharacteristic, complemented
338C22⋊#95—532no—
339C2×C53C2×C4×C22⋊C411yescharacteristic
340C2×C53#227412yes—
341C2×C53#230811yescharacteristic
342C2×C53#231011yescharacteristic
343C2×C53#231312yes—
344C2×C53#232212yes—
345C2×C53#232212yes—
346C2×C53#232212yes—
347C2×C53#232212yes—
348D53—28no—
349C24×C4—531no—
350C24×C4—531no—
351C24×C4—531no—
352C22×C4⋊C4—531no—
353C22×C4⋊C4—531no—
354C22×C4⋊C4—531no—
355C22×C4⋊C4—538no—
356C22×C4⋊C4—1064no—
357C22×C4⋊C4—1064no—
358C22×C4⋊C4C2×D5311yescharacteristic
359C22×C22⋊C4—532no—
360C22×C22⋊C4—532no—
361C22×C42—532no—
362C22×C42—531no—
363C22×C42C2×D5311yescharacteristic
364#381—5316no—
365C4×C22⋊C4—1064no—
366C4×C22⋊C4—1064no—
367C2×#95—532no—
368C2×#95—532no—
369C2×#95—532no—
370C2×#95—532no—
371C2×#95—531no—
372C2×#95—532no—
373C2×#95—532no—
374C2×#95—532no—
375C2×#95C2×D5311yescharacteristic
376C2×#95C2×D5312yes—
377C2×#95C2×D5312yes—
378C53#1377611yescharacteristic, complemented, Sylow
379C25—531no—
380C23×C4—531no—
381C23×C4—531no—
382C23×C4—534no—
383C23×C4—538no—
384C23×C4—534no—
385C23×C4—534no—
386C23×C4—534no—
387C23×C4—532no—
388C23×C4—532no—
389C23×C4—531no—
390C23×C4—534no—
391C23×C4—1062no—
392C23×C4—1062no—
393C23×C4—1064no—
394C23×C4—1062no—
395C23×C4—1062no—
396C23×C4—2122no—
397C23×C4—2122no—
398C23×C4—2122no—
399C23×C4—2122no—
400C23×C4C22×D5312yes—
401C23×C4C22×D5312yes—
402C23×C4C22×D5311yescharacteristic
403C23×C4C22×D5311yescharacteristic
404C23×C4C22×D5311yescharacteristic
405C2×C4⋊C4—22no—
406C2×C4⋊C4—22no—
407C2×C4⋊C4—534no—
408C2×C4⋊C4—534no—
409C2×C4⋊C4—534no—
410C2×C4⋊C4—10632no—
411C2×C4⋊C4—1062no—
412C2×C4⋊C4—1062no—
413C2×C4⋊C4—1062no—
414C2×C4⋊C4—1062no—
415C2×C4⋊C4—1062no—
416C2×C4⋊C4—1062no—
417C2×C4⋊C4C4×D5314yes—
418C2×C22⋊C4—538no—
419C2×C22⋊C4—538no—
420C2×C22⋊C4—1064no—
421C2×C22⋊C4—1064no—
422C2×C22⋊C4—1064no—
423C2×C22⋊C4—1064no—
424C2×C42—22no—
425C2×C42—42no—
426C2×C42—1064no—
427C2×C42—1062no—
428C2×C42—2122no—
429C2×C42—2122no—
430C2×C42—2122no—
431#95—22no—
432#95—42no—
433#95—538no—
434#95—538no—
435#95—538no—
436#95—1064no—
437#95—1062no—
438#95—2122no—
439#95—2122no—
440#95—2122no—
441#95C4×D5318yescomplemented
442C24—534no—
443C24—532no—
444C24—532no—
445C24—532no—
446C24—534no—
447C24—1062no—
448C24—1062no—
449C24—1062no—
450C24—1062no—
451C24C23×D5311yescharacteristic
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—534no—
474C22×C4—532no—
475C22×C4—532no—
476C22×C4—532no—
477C22×C4—532no—
478C22×C4—5316no—
479C22×C4—5316no—
480C22×C4—534no—
481C22×C4—534no—
482C22×C4—532no—
483C22×C4—1068no—
484C22×C4—1068no—
485C22×C4—10616no—
486C22×C4—10616no—
487C22×C4—1068no—
488C22×C4—1061no—
489C22×C4—1061no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4—1062no—
494C22×C4—1062no—
495C22×C4—1061no—
496C22×C4—1061no—
497C22×C4—1062no—
498C22×C4—1062no—
499C22×C4—1062no—
500C22×C4—1062no—
501C22×C4—1062no—
502C22×C4—1062no—
503C22×C4—1062no—
504C22×C4—1062no—
505C22×C4—1062no—
506C22×C4—1062no—
507C22×C4—1062no—
508C22×C4—1062no—
509C22×C4—2128no—
510C22×C4—2128no—
511C22×C4—2124no—
512C22×C4—2124no—
513C22×C4—2128no—
514C22×C4—2128no—
515C22×C4—2124no—
516C22×C4—2124no—
517C22×C4C2×C4×D5314yes—
518C22×C4C2×C4×D5314yes—
519C22×C4C2×C4×D5312yes—
520C22×C4C2×C4×D5314yes—
521C22×C4C2×C4×D5312yes—
522C22×C4C2×C4×D5312yes—
523C22×C4D4×D5312yes—
524C22×C4D4×D5312yes—
525C4⋊C4—28no—
526C4⋊C4—1068no—
527C4⋊C4—1068no—
528C4⋊C4—1068no—
529C22⋊C4—10616no—
530C22⋊C4—10616no—
531C42—44no—
532C42—2124no—
533C42—2128no—
534C23—534no—
535C23—534no—
536C23—534no—
537C23—538no—
538C23—538no—
539C23—1064no—
540C23—1064no—
541C23—1064no—
542C23—1064no—
543C23—1064no—
544C23—1064no—
545C23—1064no—
546C23—1062no—
547C23—1062no—
548C23—1062no—
549C23—1064no—
550C23—1062no—
551C23—1064no—
552C23—1064no—
553C23—1064no—
554C23—1064no—
555C23C22×C4×D5311yescharacteristic
556C23C22×C4×D5312yes—
557C23C2×D4×D5311yescharacteristic
558C23C2×D4×D5311yescharacteristic
559C23#41×D5312yes—
560C23#1313326212yes—
561C23#1313326312yes—
562C23#1313326412yes—
563C23#1313326512yes—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—534no—
592C2×C4—534no—
593C2×C4—10632no—
594C2×C4—1062no—
595C2×C4—1062no—
596C2×C4—1064no—
597C2×C4—1064no—
598C2×C4—1064no—
599C2×C4—1064no—
600C2×C4—1062no—
601C2×C4—1062no—
602C2×C4—1062no—
603C2×C4—1064no—
604C2×C4—1064no—
605C2×C4—1064no—
606C2×C4—1064no—
607C2×C4—1064no—
608C2×C4—1064no—
609C2×C4—1064no—
610C2×C4—21216no—
611C2×C4—21216no—
612C2×C4—2122no—
613C2×C4—2122no—
614C2×C4—2121no—
615C2×C4—2122no—
616C2×C4—2121no—
617C2×C4—2122no—
618C2×C4—2122no—
619C2×C4—2122no—
620C2×C4—2122no—
621C2×C4—2122no—
622C2×C4—2122no—
623C2×C4C42×D5314yes—
624C2×C4C22⋊C4×D5314yes—
625C22—538no—
626C22—1064no—
627C22—1064no—
628C22—1068no—
629C22—1068no—
630C22—1068no—
631C22—1068no—
632C22—1068no—
633C22—1068no—
634C22C2×C42×D5311yescharacteristic
635C22C2×C22⋊C4×D5311yescharacteristic
636C22#117×D5311yescharacteristic
637C22C4×D4×D5312yes—
638C22C4×D4×D5312yes—
639C22#1681109512yes—
640C22#1681109912yes—
641C22#1681110212yes—
642C22#1681110312yes—
643C22#1681110412yes—
644C22#1681110512yes—
645C22#1681118112yes—
646C22#1681118311yescharacteristic
647C22#1681118411yescharacteristic
648C22#1681119311yescharacteristic
649C22#1681119612yes—
650C22#1681119811yescharacteristic
651C22#1681121412yes—
652C22#1681121612yes—
653C22#1681121912yes—
654C22#1681122112yes—
655C4—24no—
656C4—24no—
657C4—42no—
658C4—44no—
659C4—44no—
660C4—1064no—
661C4—1064no—
662C4—2122no—
663C4—2124no—
664C4—2124no—
665C2—1068no—
666C2C4×C22⋊C4×D5311yescharacteristic
667C2#2027706912yes—
668C2#2027728911yescharacteristic
669C2#2027729811yescharacteristic
670C2#2027731412yes—
671C2#2027737912yes—
672C2#2027738212yes—
673C2#2027738512yes—
674C2#2027738812yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC22×C53C22×C42
FittingC53×#1436C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenter#1436D53
hypocenterC53#13776
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
5. #20272950C2semidirect
6. #20272949C2semidirect
7. #20272948C2semidirect
8. #20272947C2semidirect
9. #20272946C2semidirect
11. C53×#1436C2semidirect
20. C2×C4⋊C4×D53C4semidirect
25. #16810485C4semidirect
88. #13133124C2×C4semidirect
89. #13133124C2×C4semidirect
90. #13133124C2×C4semidirect
103. #95×C53C2×C4semidirect
185. C2×C4×D53C42semidirect
337. #1436D53semidirect
378. C53#13776semidirect
441. #95C4×D53semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D532
C2×#168104852
C2×C4×#117199212
C2×#168104702
#202729502
#202729492
#202729482
#202729472
#202729462
#202729362
C53×#14362
#1377653