Finite Group Databasebeta

← #27306364

Subgroups — #27306364

Order 13568.

1774 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810520C212yesmaximal
2C2×#16810518C212yesmaximal
3C2×#16810492C212yesmaximal
4C2×#16810492C211yesmaximal, characteristic
5C2×#16810474C211yesmaximal, characteristic
6C2×#16810474C211yesmaximal, characteristic
7#20272980C211yesmaximal, characteristic
8#20272979C211yesmaximal, characteristic
9#20272978C211yesmaximal, characteristic
10#20272943C212yesmaximal
11C53×#1444C211yesmaximal, characteristic
12C22×C4⋊Dic53C2212yes—
13C22×C4⋊Dic53C2211yescharacteristic
14C22×#11719919C2212yes—
15C22×#11719919C2212yes—
16C22×#11719919C2212yes—
17C22×#11719919C2212yes—
18C22×#11719919C2211yescharacteristic
19C22×#11719919C2211yescharacteristic
20C22×#11719919C2212yes—
21C22×C4×Dic53C2211yescharacteristic
22C22×C4×Dic53C2211yescharacteristic
23C22×C4×Dic53C2212yes—
24C22×C4×Dic53C2211yescharacteristic
25#16810520—28no—
26#16810518—28no—
27#16810492—28no—
28#16810492—24no—
29#16810474—24no—
30#16810474C418yescomplemented
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2211yescharacteristic
35C2×#13133124C2211yescharacteristic
36C22×C4⋊C4×C53C2212yes—
37C22×C4⋊C4×C53C2211yescharacteristic
38C2×#95×C53C2212yes—
39C2×#95×C53C2211yescharacteristic
40C2×#95×C53C2211yescharacteristic
41C23×Dic53C2312yes—
42C23×Dic53C2311yescharacteristic
43C23×Dic53C2311yescharacteristic
44C23×Dic53C2312yes—
45C23×Dic53C2311yescharacteristic
46C23×Dic53C2311yescharacteristic
47C2×C4⋊Dic53—24no—
48C2×C4⋊Dic53—22no—
49C2×C4⋊Dic53—24no—
50C2×C4⋊Dic53—42no—
51C2×C4⋊Dic53—42no—
52C2×#11719919—24no—
53C2×#11719919—22no—
54C2×#11719919—22no—
55C2×#11719919—24no—
56C2×#11719919—24no—
57C2×#11719919—24no—
58C2×#11719919—24no—
59C2×#11719919—24no—
60C2×#11719919—24no—
61C2×#11719919—24no—
62C2×#11719919—24no—
63C2×#11719919—22no—
64C2×#11719919—22no—
65C2×#11719919—24no—
66C2×#11719919—42no—
67C2×#11719919—42no—
68C2×#11719919C2×C418yes—
69C2×#11719919D414yes—
70C2×#11719919D418yes—
71C2×#11719919Q814yes—
72C2×#11719919Q818yes—
73C2×C4×Dic53—24no—
74C2×C4×Dic53—24no—
75C2×C4×Dic53—24no—
76C2×C4×Dic53—22no—
77C2×C4×Dic53—22no—
78C2×C4×Dic53—24no—
79C2×C4×Dic53C2×C418yes—
80C2×C4×Dic53C2×C414yes—
81C2×C4×Dic53D414yes—
82C2×C4×Dic53Q814yes—
83#13133124—24no—
84#13133124—22no—
85#13133124—24no—
86#13133124—24no—
87#13133124C2×C414yes—
88C23×C4×C53C2312yes—
89C23×C4×C53C2311yescharacteristic
90C23×C4×C53C2311yescharacteristic
91C23×C4×C53C2312yes—
92C23×C4×C53C2311yescharacteristic
93C2×C4⋊C4×C53—24no—
94C2×C4⋊C4×C53—24no—
95C2×C4⋊C4×C53—22no—
96C2×C4⋊C4×C53—42no—
97C2×C4⋊C4×C53—42no—
98#95×C53—24no—
99#95×C53—22no—
100#95×C53C2×C414yes—
101C24×C53C2411yescharacteristic
102C22×C4×C53—22no—
103C22×C4×C53—22no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—21no—
110C22×C4×C53—21no—
111C22×C4×C53—22no—
112C22×C4×C53—21no—
113C22×C4×C53—21no—
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×C53C22×C412yes—
124C22×C4×C53C22×C414yes—
125C22×C4×C53C2×D412yes—
126C22×C4×C53C2×D412yes—
127C22×C4×C53C2×D414yes—
128C22×C4×C53C2×Q812yes—
129C22×C4×C53C2×Q812yes—
130C22×C4×C53C2×Q814yes—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—22no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53C22×C414yes—
151C22×Dic53C22×C412yes—
152C22×Dic53C22×C412yes—
153C22×Dic53C2×D412yes—
154C22×Dic53C2×D414yes—
155C22×Dic53C2×D412yes—
156C22×Dic53C2×D412yes—
157C22×Dic53C2×D414yes—
158C22×Dic53C2×D412yes—
159C22×Dic53C2×Q812yes—
160C22×Dic53C2×Q814yes—
161C22×Dic53C2×Q812yes—
162C22×Dic53C2×Q812yes—
163C22×Dic53C2×Q814yes—
164C22×Dic53C2×Q812yes—
165C22×Dic53#4112yes—
166C22×Dic53#4112yes—
167C4⋊C4×C53—48no—
168C4⋊C4×C53—44no—
169C4⋊Dic53—48no—
170C4⋊Dic53—44no—
171#11719919—28no—
172#11719919—28no—
173#11719919—48no—
174#11719919—48no—
175#11719919—48no—
176#11719919—48no—
177#11719919C4⋊C4132yescomplemented
178C4×Dic53—216no—
179C4×Dic53—44no—
180C4×Dic53—44no—
181C4×Dic53C4⋊C4116yescomplemented
182C23×C53C23×C411yescharacteristic
183C23×C53C22×D411yescharacteristic
184C23×C53C22×D412yes—
185C23×C53C22×D411yescharacteristic
186C23×C53C22×Q811yescharacteristic
187C23×C53C22×Q812yes—
188C23×C53C22×Q811yescharacteristic
189C23×C53C2×#4111yescharacteristic
190C23×C53C2×#4111yescharacteristic
191C23×C53H(5,𝔽2)12yes—
192C23×C532−1+412yes—
193C2×C4×C53—24no—
194C2×C4×C53—24no—
195C2×C4×C53—24no—
196C2×C4×C53—22no—
197C2×C4×C53—22no—
198C2×C4×C53—24no—
199C2×C4×C53—24no—
200C2×C4×C53—24no—
201C2×C4×C53—24no—
202C2×C4×C53—24no—
203C2×C4×C53—22no—
204C2×C4×C53—22no—
205C2×C4×C53—24no—
206C2×C4×C53—24no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53—41no—
210C2×C4×C53—41no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
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×C53C2×C4⋊C414yes—
221C2×C4×C53C2×C4⋊C418yes—
222C2×Dic53—24no—
223C2×Dic53—24no—
224C2×Dic53—24no—
225C2×Dic53—24no—
226C2×Dic53—24no—
227C2×Dic53—24no—
228C2×Dic53—22no—
229C2×Dic53—22no—
230C2×Dic53—24no—
231C2×Dic53—24no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—42no—
237C2×Dic53—42no—
238C2×Dic53—42no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53C2×C4⋊C418yes—
246C2×Dic53C2×C4⋊C414yes—
247C2×Dic53C2×C4⋊C418yes—
248C2×Dic53C2×C4⋊C414yes—
249C2×Dic53C4×D414yes—
250C2×Dic53C4×Q814yes—
251C2×Dic53#12114yes—
252C2×Dic53C22⋊Q814yes—
253C2×Dic53#12514yes—
254C2×Dic53#12714yes—
255C2×Dic53C4⋊Q814yes—
256C2×Dic53C4⋊Q818yes—
257C2×Dic53C4⋊Q814yes—
258#13910—531nomaximal, Sylow
259C22×C53C22×C4⋊C411yescharacteristic
260C22×C53C22×C4⋊C412yes—
261C22×C53C2×C4×D411yescharacteristic
262C22×C53C2×C4×Q811yescharacteristic
263C22×C53#52012yes—
264C22×C53C2×#12111yescharacteristic
265C22×C53C2×C22⋊Q811yescharacteristic
266C22×C53C2×#12511yescharacteristic
267C22×C53C2×#12711yescharacteristic
268C22×C53C2×C4⋊Q811yescharacteristic
269C22×C53C2×C4⋊Q812yes—
270C22×C53C2×C4⋊Q811yescharacteristic
271C22×C53#53511yescharacteristic
272C22×C53#53611yescharacteristic
273C22×C53#54412yes—
274C22×C53D4212yes—
275C22×C53#54712yes—
276C22×C53D4×Q812yes—
277C22×C53D4×Q812yes—
278C22×C53#55012yes—
279C22×C53#55412yes—
280C22×C53#55712yes—
281C22×C53Q8212yes—
282C4×C53—24no—
283C4×C53—28no—
284C4×C53—42no—
285C4×C53—42no—
286C4×C53—82no—
287Dic53—28no—
288Dic53—42no—
289Dic53—42no—
290Dic53—82no—
291Dic53#38418yescomplemented
292Dic53#38918yescomplemented
293C2×#389—532no—
294C2×#389—532no—
295C2×#384—532no—
296C2×#384—531no—
297C2×C4×C4⋊C4—531no—
298C2×C4×C4⋊C4—531no—
299#1444—531no—
300#1444—531no—
301#1444—531no—
302#1444D5311yescharacteristic
303C4⋊#95—532no—
304C2×C53C2×#38411yescharacteristic
305C2×C53C2×#38911yescharacteristic
306C2×C53#231711yescharacteristic
307C2×C53D4×C4⋊C412yes—
308C2×C53Q8×C4⋊C412yes—
309C2×C53#254111yescharacteristic
310C2×C53#254211yescharacteristic
311C2×C53#254611yescharacteristic
312C2×C53#255112yes—
313C2×C53#255212yes—
314C2×C53#256711yescharacteristic
315C22×C4⋊C4—532no—
316C22×C4⋊C4—531no—
317C22×C4⋊C4—531no—
318C22×C4⋊C4—532no—
319C22×C4⋊C4—532no—
320C22×C4⋊C4—532no—
321C22×C4⋊C4—532no—
322C22×C4⋊C4—532no—
323C22×C4⋊C4—531no—
324C22×C4⋊C4C2×D5312yes—
325C22×C4⋊C4C2×D5311yescharacteristic
326C22×C42—531no—
327C22×C42—532no—
328C22×C42—531no—
329C22×C42—531no—
330#389—1068no—
331#389—1068no—
332#384—1064no—
333#384—1068no—
334C4×C4⋊C4—538no—
335C4×C4⋊C4—1064no—
336C2×#95—531no—
337C2×#95—532no—
338C2×#95—532no—
339C2×#95—532no—
340C2×#95—531no—
341C2×#95C2×D5312yes—
342C2×#95C2×D5311yescharacteristic
343C2×#95C2×D5311yescharacteristic
344C53#1391011yescharacteristic, complemented, Sylow
345C23×C4—531no—
346C23×C4—531no—
347C23×C4—532no—
348C23×C4—531no—
349C23×C4—531no—
350C23×C4—532no—
351C23×C4C22×D5312yes—
352C23×C4C22×D5311yescharacteristic
353C23×C4C22×D5311yescharacteristic
354C23×C4C22×D5312yes—
355C23×C4C22×D5311yescharacteristic
356C2×C4⋊C4—24no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—42no—
360C2×C4⋊C4—42no—
361C2×C4⋊C4—538no—
362C2×C4⋊C4—538no—
363C2×C4⋊C4—538no—
364C2×C4⋊C4—534no—
365C2×C4⋊C4—534no—
366C2×C4⋊C4—1062no—
367C2×C4⋊C4—1062no—
368C2×C4⋊C4—1062no—
369C2×C4⋊C4—1062no—
370C2×C4⋊C4—1064no—
371C2×C4⋊C4—1064no—
372C2×C4⋊C4—1064no—
373C2×C4⋊C4—1064no—
374C2×C4⋊C4—1064no—
375C2×C4⋊C4—1064no—
376C2×C4⋊C4—1064no—
377C2×C4⋊C4—1064no—
378C2×C4⋊C4—1064no—
379C2×C4⋊C4—1064no—
380C2×C4⋊C4—1064no—
381C2×C4⋊C4—1062no—
382C2×C4⋊C4—1064no—
383C2×C4⋊C4—2122no—
384C2×C4⋊C4—2122no—
385C2×C4⋊C4—2122no—
386C2×C4⋊C4—2122no—
387C2×C42—534no—
388C2×C42—534no—
389C2×C42—534no—
390C2×C42—538no—
391C2×C42—1064no—
392C2×C42—1064no—
393C2×C42—1062no—
394C2×C42—1062no—
395C2×C42—1064no—
396C2×C42—1064no—
397#95—22no—
398#95—24no—
399#95—534no—
400#95—1064no—
401#95—1064no—
402#95—1062no—
403#95—1064no—
404#95C4×D5314yes—
405C24C23×D5311yescharacteristic
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—532no—
428C22×C4—532no—
429C22×C4—532no—
430C22×C4—532no—
431C22×C4—534no—
432C22×C4—532no—
433C22×C4—532no—
434C22×C4—532no—
435C22×C4—534no—
436C22×C4—532no—
437C22×C4—532no—
438C22×C4—532no—
439C22×C4—534no—
440C22×C4—534no—
441C22×C4—534no—
442C22×C4—532no—
443C22×C4—532no—
444C22×C4—1062no—
445C22×C4—1062no—
446C22×C4—1062no—
447C22×C4—1062no—
448C22×C4—1062no—
449C22×C4—1061no—
450C22×C4—1061no—
451C22×C4—1062no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1062no—
455C22×C4—1062no—
456C22×C4—1062no—
457C22×C4—1062no—
458C22×C4—1062no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4C2×C4×D5312yes—
464C22×C4C2×C4×D5314yes—
465C22×C4D4×D5312yes—
466C22×C4D4×D5312yes—
467C22×C4D4×D5314yes—
468C22×C4Q8×D5312yes—
469C22×C4Q8×D5312yes—
470C22×C4Q8×D5314yes—
471C4⋊C4—48no—
472C4⋊C4—44no—
473C4⋊C4—5332no—
474C4⋊C4—1068no—
475C4⋊C4—1068no—
476C4⋊C4—2128no—
477C4⋊C4—2128no—
478C4⋊C4—2124no—
479C4⋊C4—2128no—
480C4⋊C4—2128no—
481C4⋊C4—2128no—
482C42—5316no—
483C42—10616no—
484C42—2124no—
485C42—2124no—
486C23C22×C4×D5311yescharacteristic
487C23C2×D4×D5312yes—
488C23C2×D4×D5311yescharacteristic
489C23C2×D4×D5311yescharacteristic
490C23C2×Q8×D5311yescharacteristic
491C23C2×Q8×D5312yes—
492C23C2×Q8×D5311yescharacteristic
493C23#41×D5311yescharacteristic
494C23#41×D5311yescharacteristic
495C23#1313326212yes—
496C23#1313326412yes—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—534no—
525C2×C4—534no—
526C2×C4—538no—
527C2×C4—534no—
528C2×C4—534no—
529C2×C4—534no—
530C2×C4—534no—
531C2×C4—534no—
532C2×C4—534no—
533C2×C4—534no—
534C2×C4—538no—
535C2×C4—534no—
536C2×C4—538no—
537C2×C4—1064no—
538C2×C4—1064no—
539C2×C4—1064no—
540C2×C4—1064no—
541C2×C4—1064no—
542C2×C4—1064no—
543C2×C4—1064no—
544C2×C4—1062no—
545C2×C4—1062no—
546C2×C4—1064no—
547C2×C4—2122no—
548C2×C4—2122no—
549C2×C4—2122no—
550C2×C4—2122no—
551C2×C4—2122no—
552C2×C4—2122no—
553C2×C4—2122no—
554C2×C4—2122no—
555C2×C4—2122no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4C4⋊C4×D5314yes—
561C2×C4C4⋊C4×D5318yes—
562C22C2×C4⋊C4×D5311yescharacteristic
563C22C2×C4⋊C4×D5312yes—
564C22C4×D4×D5311yescharacteristic
565C22C4×Q8×D5311yescharacteristic
566C22#1681110212yes—
567C22#121×D5311yescharacteristic
568C22C22⋊Q8×D5311yescharacteristic
569C22#125×D5311yescharacteristic
570C22#127×D5311yescharacteristic
571C22C4⋊Q8×D5311yescharacteristic
572C22C4⋊Q8×D5312yes—
573C22C4⋊Q8×D5311yescharacteristic
574C22#1681118811yescharacteristic
575C22#1681119511yescharacteristic
576C22#1681125212yes—
577C22#1681125512yes—
578C22#1681127112yes—
579C22#1681128612yes—
580C22#1681128612yes—
581C22#1681129112yes—
582C22#1681132712yes—
583C22#1681135012yes—
584C22#1681135312yes—
585C4—24no—
586C4—28no—
587C4—42no—
588C4—42no—
589C4—82no—
590C4—538no—
591C4—538no—
592C4—1068no—
593C4—4242no—
594C4—2122no—
595C4—2122no—
596C2#384×D5311yescharacteristic
597C2#389×D5311yescharacteristic
598C2#2027733411yescharacteristic
599C2#2027761312yes—
600C2#2027762912yes—
601C2#2027965611yescharacteristic
602C2#2027966311yescharacteristic
603C2#2027969911yescharacteristic
604C2#2027974012yes—
605C2#2027974512yes—
606C2#2027991911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1444C2
socle, prime radical, cosocleC24×C53C24
hypercenter#1444D53
hypocenterC53#13910
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
30. #16810474C4semidirect
177. #11719919C4⋊C4semidirect
181. C4×Dic53C4⋊C4semidirect
291. Dic53#384semidirect
292. Dic53#389semidirect
344. C53#13910semidirect

Maximal subgroups

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

subgroupindex
C2×#168105202
C2×#168105182
C2×#168104922
C2×#168104742
#202729802
#202729792
#202729782
#202729432
C53×#14442
#1391053