Finite Group Databasebeta

← #27310856

Subgroups — #27310856

Order 13568.

2242 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic53C211yesmaximal, characteristic
2C2×#16810537C211yesmaximal, characteristic
3C2×#16810505C211yesmaximal, characteristic, complemented
4C2×#16810495C212yesmaximal, complemented
5#20272979C211yesmaximal, characteristic, complemented
6#20272969C211yesmaximal, characteristic, complemented
7#20272962C212yesmaximal, complemented
8#20272958C212yesmaximal, complemented
9#20272940C211yesmaximal, characteristic, complemented
10#20272937C212yesmaximal, complemented
11C53×#1438C211yesmaximal, characteristic
12C24×Dic53C2212yes—
13C2×D4×Dic53—28no—
14C2×D4×Dic53—24no—
15C22×C4⋊Dic53C2211yescharacteristic
16C22×#11719919C2211yescharacteristic
17C22×#11719919C2211yescharacteristic
18C22×#11719919C2212yes—
19C22×C22⋊Dic53C2212yes—
20C22×C22⋊Dic53C2212yes—
21C22×C22⋊Dic53C2212yes—
22C22×C4×Dic53C2211yescharacteristic
23#16810537C418yescomplemented
24#16810505—24no—
25#16810495—28no—
26C2×#13133124C2211yescharacteristic
27C2×#13133124C2212yes—
28C2×#13133124C2211yescharacteristic
29C2×#13133124C2211yescharacteristic
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2211yescharacteristic
35C23×D4×C53C2211yescharacteristic
36C22×C22⋊C4×C53C2212yes—
37C22×C22⋊C4×C53C2212yes—
38C2×#95×C53C2211yescharacteristic
39C2×#95×C53C2211yescharacteristic
40C23×Dic53—24no—
41C23×Dic53—24no—
42C23×Dic53—42no—
43C23×Dic53—42no—
44C23×Dic53—42no—
45C23×Dic53—42no—
46C23×Dic53D418yes—
47C23×Dic53C2311yescharacteristic
48C23×Dic53C2311yescharacteristic
49C23×Dic53C2312yes—
50C23×Dic53C2311yescharacteristic
51C23×Dic53C2312yes—
52C23×Dic53C2311yescharacteristic
53D4×Dic53—232no—
54C2×C4⋊Dic53—22no—
55C2×C4⋊Dic53—24no—
56C2×#11719919—22no—
57C2×#11719919—22no—
58C2×#11719919—22no—
59C2×#11719919—24no—
60C2×#11719919—24no—
61C2×#11719919—42no—
62C2×#11719919—42no—
63C2×#11719919C2×C414yescomplemented
64C2×C22⋊Dic53—24no—
65C2×C22⋊Dic53—24no—
66C2×C22⋊Dic53—24no—
67C2×C22⋊Dic53—24no—
68C2×C22⋊Dic53—24no—
69C2×C22⋊Dic53—42no—
70C2×C22⋊Dic53—42no—
71C2×C22⋊Dic53C2×C418yescomplemented
72C2×C22⋊Dic53C2×C418yescomplemented
73C2×C4×Dic53—24no—
74C2×C4×Dic53D414yes—
75#13133124—22no—
76#13133124—22no—
77#13133124—22no—
78#13133124—24no—
79#13133124—24no—
80#13133124—42no—
81#13133124—42no—
82#13133124—42no—
83#13133124C2×C414yescomplemented
84C25×C53C2312yes—
85C22×D4×C53—24no—
86C22×D4×C53—24no—
87C22×D4×C53—22no—
88C22×D4×C53—22no—
89C22×D4×C53C2×C414yes—
90C23×C4×C53C2311yescharacteristic
91C23×C4×C53C2311yescharacteristic
92C23×C4×C53C2312yes—
93C23×C4×C53C2311yescharacteristic
94C2×C22⋊C4×C53—24no—
95C2×C22⋊C4×C53—24no—
96C2×C22⋊C4×C53—42no—
97C2×C22⋊C4×C53—42no—
98C2×C22⋊C4×C53—42no—
99C2×C22⋊C4×C53—42no—
100#95×C53—22no—
101#95×C53—22no—
102C24×C53—22no—
103C24×C53—22no—
104C24×C53—22no—
105C24×C53—22no—
106C24×C53—22no—
107C24×C53—22no—
108C24×C53—22no—
109C24×C53—22no—
110C24×C53—22no—
111C24×C53—22no—
112C24×C53—22no—
113C24×C53—22no—
114C24×C53C22×C414yes—
115C24×C53C2×D414yes—
116C24×C53#4114yes—
117C24×C53C2411yescharacteristic
118C2×D4×C53—28no—
119C2×D4×C53—216no—
120C2×D4×C53—28no—
121C2×D4×C53—216no—
122C2×D4×C53—28no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—21no—
133C22×C4×C53—22no—
134C22×C4×C53—21no—
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53—22no—
138C22×C4×C53—22no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—22no—
142C22×C4×C53C22×C412yes—
143C22×C4×C53C2×D412yes—
144C22×C4×C53#4112yes—
145C22×Dic53—22no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—21no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53—22no—
162C22×Dic53—22no—
163C22×Dic53—22no—
164C22×Dic53—22no—
165C22×Dic53—21no—
166C22×Dic53—21no—
167C22×Dic53—22no—
168C22×Dic53—22no—
169C22×Dic53—22no—
170C22×Dic53—22no—
171C22×Dic53—21no—
172C22×Dic53—22no—
173C22×Dic53—21no—
174C22×Dic53—216no—
175C22×Dic53—48no—
176C22×Dic53—48no—
177C22×Dic53—48no—
178C22×Dic53—48no—
179C22×Dic53—48no—
180C22×Dic53—48no—
181C22×Dic53C22×C414yes—
182C22×Dic53C22×C412yes—
183C22×Dic53C22×C412yes—
184C22×Dic53C2×D412yes—
185C22×Dic53C2×D414yes—
186C22×Dic53C2×D414yes—
187C22×Dic53C2×D412yes—
188C22⋊C4×C53—48no—
189C22⋊C4×C53—48no—
190C22⋊Dic53—216no—
191C22⋊Dic53—216no—
192C22⋊Dic53—48no—
193C4⋊Dic53—28no—
194#11719919—28no—
195#11719919—48no—
196#11719919—44no—
197C4×Dic53—28no—
198C23×C53—24no—
199C23×C53—24no—
200C23×C53—24no—
201C23×C53—24no—
202C23×C53—24no—
203C23×C53—24no—
204C23×C53—24no—
205C23×C53—24no—
206C23×C53—24no—
207C23×C53—24no—
208C23×C53—24no—
209C23×C53—24no—
210C23×C53—24no—
211C23×C53—24no—
212C23×C53—24no—
213C23×C53—24no—
214C23×C53—24no—
215C23×C53—24no—
216C23×C53—42no—
217C23×C53—42no—
218C23×C53—42no—
219C23×C53—42no—
220C23×C53—42no—
221C23×C53—42no—
222C23×C53—42no—
223C23×C53—42no—
224C23×C53—42no—
225C23×C53—42no—
226C23×C53—42no—
227C23×C53—42no—
228C23×C53—42no—
229C23×C53—42no—
230C23×C53—42no—
231C23×C53—42no—
232C23×C53C4×D418yes—
233C23×C53C23×C411yescharacteristic
234C23×C53C22×D412yes—
235C23×C53C22×D411yescharacteristic
236C23×C53C2×#4112yes—
237C23×C53C2×#4111yescharacteristic
238C23×C53H(5,𝔽2)11yescharacteristic
239C23×C53H(5,𝔽2)12yes—
240C23×C53H(5,𝔽2)11yescharacteristic
241C23×C53H(5,𝔽2)11yescharacteristic
242C23×C532−1+412yes—
243C23×C532−1+411yescharacteristic
244D4×C53—232no—
245C2×C4×C53—24no—
246C2×C4×C53—22no—
247C2×C4×C53—22no—
248C2×C4×C53—24no—
249C2×C4×C53—41no—
250C2×C4×C53—42no—
251C2×C4×C53—41no—
252C2×C4×C53—42no—
253C2×C4×C53—42no—
254C2×C4×C53—42no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53—42no—
259C2×C4×C53—41no—
260C2×C4×C53—42no—
261C2×C4×C53—42no—
262C2×C4×C53—42no—
263C2×C4×C53—42no—
264C2×C4×C53—42no—
265C2×C4×C53—42no—
266C2×C4×C53C4×D414yes—
267C2×Dic53—24no—
268C2×Dic53—24no—
269C2×Dic53—22no—
270C2×Dic53—22no—
271C2×Dic53—22no—
272C2×Dic53—22no—
273C2×Dic53—24no—
274C2×Dic53—24no—
275C2×Dic53—24no—
276C2×Dic53—22no—
277C2×Dic53—22no—
278C2×Dic53—24no—
279C2×Dic53—24no—
280C2×Dic53—22no—
281C2×Dic53—24no—
282C2×Dic53—24no—
283C2×Dic53—22no—
284C2×Dic53—22no—
285C2×Dic53—432no—
286C2×Dic53—41no—
287C2×Dic53—42no—
288C2×Dic53—41no—
289C2×Dic53—42no—
290C2×Dic53—42no—
291C2×Dic53—42no—
292C2×Dic53—41no—
293C2×Dic53—42no—
294C2×Dic53—42no—
295C2×Dic53—42no—
296C2×Dic53—42no—
297C2×Dic53—42no—
298C2×Dic53—41no—
299C2×Dic53—42no—
300C2×Dic53—42no—
301C2×Dic53—41no—
302C2×Dic53—41no—
303C2×Dic53#12014yes—
304#14300—531nomaximal, Sylow
305C22×C53—28no—
306C22×C53—28no—
307C22×C53—28no—
308C22×C53—44no—
309C22×C53—44no—
310C22×C53—44no—
311C22×C53—44no—
312C22×C53—44no—
313C22×C53—44no—
314C22×C53—44no—
315C22×C53—44no—
316C22×C53—44no—
317C22×C53—44no—
318C22×C53—44no—
319C22×C53—44no—
320C22×C53C2×C4×D412yes—
321C22×C53C2×C4×D411yescharacteristic
322C22×C53#51811yescharacteristic
323C22×C53#52011yescharacteristic
324C22×C53#52012yes—
325C22×C53C2×#12011yescharacteristic
326C22×C53#52512yes—
327C22×C53#52511yescharacteristic
328C22×C53#53412yes—
329C22×C53#53412yes—
330C22×C53#53511yescharacteristic
331C22×C53#53511yescharacteristic
332C22×C53#53611yescharacteristic
333C22×C53#53612yes—
334C22×C53#53712yes—
335C22×C53#53711yescharacteristic
336C22×C53#53812yes—
337C22×C53#53912yes—
338C22×C53#53912yes—
339C22×C53#53912yes—
340C22×C53#54011yescharacteristic
341C22×C53#54212yes—
342C22×C53#54211yescharacteristic
343C4×C53—24no—
344C4×C53—81no—
345C4×C53—82no—
346C4×C53—81no—
347Dic53—24no—
348Dic53—42no—
349Dic53—44no—
350Dic53—44no—
351Dic53—42no—
352Dic53—42no—
353C22×C4×D4—531no—
354C2×#394—531no—
355C2×#387—531no—
356C2×#385—532no—
357#1444—531no—
358#1441—531no—
359#1440—532no—
360#1439—532no—
361#1438D5311yescharacteristic
362#1434—531no—
363C22⋊#95—532no—
364C2×C53—48no—
365C2×C53C4×#12011yescharacteristic
366C2×C53#232212yes—
367C2×C53#232411yescharacteristic
368C2×C53#232512yes—
369C2×C53#232611yescharacteristic
370C2×C53#261311yescharacteristic
371C2×C53#261611yescharacteristic
372C2×C53#263012yes—
373C2×C53#263311yescharacteristic
374C2×C53#263611yescharacteristic
375C2×C53#263812yes—
376C23×D4C2×D5311yescharacteristic
377C24×C4—532no—
378C2×C4×D4—1064no—
379C2×C4×D4—1068no—
380C22×C4⋊C4—531no—
381C22×C4⋊C4—531no—
382C22×C4⋊C4—531no—
383C22×C4⋊C4—532no—
384C22×C22⋊C4—532no—
385C22×C22⋊C4—532no—
386C22×C22⋊C4—532no—
387C22×C22⋊C4C2×D5312yes—
388C22×C22⋊C4C2×D5312yes—
389C22×C42—531no—
390#394—538no—
391#387—1064no—
392#385—1068no—
393C2×#95—531no—
394C2×#95—531no—
395C2×#95—532no—
396C2×#95—531no—
397C2×#95—531no—
398C2×#95—532no—
399C2×#95—532no—
400C2×#95—532no—
401C2×#95—532no—
402C2×#95C2×D5311yescharacteristic
403C2×#95C2×D5311yescharacteristic
404C53#1430011yescharacteristic, complemented, Sylow
405C25C22×D5312yes—
406C22×D4—24no—
407C22×D4—24no—
408C22×D4—22no—
409C22×D4—22no—
410C22×D4C4×D5314yes—
411C23×C4—531no—
412C23×C4—531no—
413C23×C4—532no—
414C23×C4—531no—
415C23×C4—531no—
416C23×C4—532no—
417C23×C4—538no—
418C23×C4—1064no—
419C23×C4—1064no—
420C23×C4—2122no—
421C23×C4—2122no—
422C23×C4—2122no—
423C23×C4—2122no—
424C23×C4C22×D5311yescharacteristic
425C23×C4C22×D5311yescharacteristic
426C23×C4C22×D5312yes—
427C23×C4C22×D5311yescharacteristic
428C4×D4—10632no—
429C2×C4⋊C4—534no—
430C2×C4⋊C4—1064no—
431C2×C4⋊C4—1062no—
432C2×C4⋊C4—1062no—
433C2×C4⋊C4—1062no—
434C2×C4⋊C4—1064no—
435C2×C4⋊C4—1064no—
436C2×C4⋊C4—1062no—
437C2×C4⋊C4—2122no—
438C2×C4⋊C4—2122no—
439C2×C22⋊C4—24no—
440C2×C22⋊C4—24no—
441C2×C22⋊C4—42no—
442C2×C22⋊C4—42no—
443C2×C22⋊C4—42no—
444C2×C22⋊C4—42no—
445C2×C22⋊C4—538no—
446C2×C22⋊C4—538no—
447C2×C22⋊C4—1064no—
448C2×C22⋊C4—1064no—
449C2×C22⋊C4—1064no—
450C2×C22⋊C4—1064no—
451C2×C22⋊C4—1064no—
452C2×C22⋊C4—2122no—
453C2×C22⋊C4—2122no—
454C2×C42—534no—
455C2×C42—1064no—
456#95—22no—
457#95—22no—
458#95—534no—
459#95—1064no—
460#95—1064no—
461#95—1062no—
462#95—1062no—
463#95—1062no—
464#95—2122no—
465#95—2122no—
466#95—2122no—
467C24—22no—
468C24—22no—
469C24—22no—
470C24—22no—
471C24—22no—
472C24—22no—
473C24—22no—
474C24—22no—
475C24—22no—
476C24—22no—
477C24—22no—
478C24—22no—
479C24C2×C4×D5314yes—
480C24D4×D5314yes—
481C24#1171994714yes—
482C24C23×D5311yescharacteristic
483C2×D4—28no—
484C2×D4—28no—
485C2×D4—216no—
486C2×D4—216no—
487C2×D4—28no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—21no—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4—22no—
504C22×C4—21no—
505C22×C4—22no—
506C22×C4—21no—
507C22×C4—532no—
508C22×C4—534no—
509C22×C4—534no—
510C22×C4—532no—
511C22×C4—532no—
512C22×C4—532no—
513C22×C4—534no—
514C22×C4—10616no—
515C22×C4—1062no—
516C22×C4—1061no—
517C22×C4—1061no—
518C22×C4—1061no—
519C22×C4—1062no—
520C22×C4—1062no—
521C22×C4—1061no—
522C22×C4—1062no—
523C22×C4—1062no—
524C22×C4—1062no—
525C22×C4—1062no—
526C22×C4—1062no—
527C22×C4—1061no—
528C22×C4—1061no—
529C22×C4—1061no—
530C22×C4—1061no—
531C22×C4—1062no—
532C22×C4—1062no—
533C22×C4—1062no—
534C22×C4—1062no—
535C22×C4—1061no—
536C22×C4—1062no—
537C22×C4—1061no—
538C22×C4—1062no—
539C22×C4—1062no—
540C22×C4—1062no—
541C22×C4—1061no—
542C22×C4—1062no—
543C22×C4—1061no—
544C22×C4—2128no—
545C22×C4—2128no—
546C22×C4—2128no—
547C22×C4—2128no—
548C22×C4—2128no—
549C22×C4—2128no—
550C22×C4C2×C4×D5312yes—
551C22×C4D4×D5312yes—
552C22×C4#1171994712yes—
553C4⋊C4—1068no—
554C4⋊C4—1068no—
555C4⋊C4—2124no—
556C4⋊C4—2128no—
557C22⋊C4—48no—
558C22⋊C4—48no—
559C22⋊C4—10616no—
560C22⋊C4—10616no—
561C22⋊C4—2128no—
562C42—1068no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—24no—
567C23—24no—
568C23—24no—
569C23—24no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—24no—
575C23—24no—
576C23—24no—
577C23—24no—
578C23—24no—
579C23—24no—
580C23—24no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23#1313317718yes—
598C23C22×C4×D5311yescharacteristic
599C23C2×D4×D5312yes—
600C23C2×D4×D5311yescharacteristic
601C23C2×#1171994712yes—
602C23C2×#1171994711yescharacteristic
603C23#1313326211yescharacteristic
604C23#1313326212yes—
605C23#1313326211yescharacteristic
606C23#1313326311yescharacteristic
607C23#1313326412yes—
608C23#1313326411yescharacteristic
609D4—232no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—534no—
632C2×C4—1064no—
633C2×C4—1064no—
634C2×C4—1062no—
635C2×C4—1062no—
636C2×C4—1064no—
637C2×C4—1064no—
638C2×C4—1062no—
639C2×C4—1062no—
640C2×C4—1062no—
641C2×C4—1064no—
642C2×C4—1062no—
643C2×C4—1064no—
644C2×C4—1062no—
645C2×C4—1064no—
646C2×C4—1064no—
647C2×C4—1064no—
648C2×C4—1062no—
649C2×C4—1062no—
650C2×C4—21232no—
651C2×C4—2121no—
652C2×C4—2121no—
653C2×C4—2122no—
654C2×C4—2122no—
655C2×C4—2122no—
656C2×C4—2122no—
657C2×C4—2122no—
658C2×C4—2121no—
659C2×C4—2122no—
660C2×C4—2122no—
661C2×C4—2122no—
662C2×C4—2122no—
663C2×C4—2121no—
664C2×C4—2122no—
665C2×C4—2122no—
666C2×C4—2121no—
667C2×C4—2121no—
668C2×C4#1313317714yes—
669C22—28no—
670C22—28no—
671C22—28no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—44no—
682C22—44no—
683C22—44no—
684C22C2×#1313317712yes—
685C22C2×#1313317711yescharacteristic
686C22#1681109511yescharacteristic
687C22#1681110211yescharacteristic
688C22#1681110212yes—
689C22#120×D5311yescharacteristic
690C22#1681112911yescharacteristic
691C22#1681112912yes—
692C22#1681118112yes—
693C22#1681118312yes—
694C22#1681118611yescharacteristic
695C22#1681118911yescharacteristic
696C22#1681119311yescharacteristic
697C22#1681119512yes—
698C22#1681120012yes—
699C22#1681120211yescharacteristic
700C22#1681120812yes—
701C22#1681121712yes—
702C22#1681121812yes—
703C22#1681121812yes—
704C22#1681122711yescharacteristic
705C22#1681124012yes—
706C22#1681124211yescharacteristic
707C4—24no—
708C4—81no—
709C4—81no—
710C4—82no—
711C4—1064no—
712C4—2124no—
713C4—2122no—
714C4—2122no—
715C4—2124no—
716C4—2122no—
717C2—48no—
718C2#2027721411yescharacteristic
719C2#2027738912yes—
720C2#2027741111yescharacteristic
721C2#2027741612yes—
722C2#2027742511yescharacteristic
723C2#2028042611yescharacteristic
724C2#2028044311yescharacteristic
725C2#2028055112yes—
726C2#2028058211yescharacteristic
727C2#2028060911yescharacteristic
728C2#2028063112yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1438C2
socle, cosocleC24×C53C24
prime radicalC23×D4×C53C22
hypercenter#1438D53
hypocenterC53#14300
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1438

Complements

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

NH ≅ G/Ntype
3. C2×#16810505C2semidirect
4. C2×#16810495C2semidirect
5. #20272979C2semidirect
6. #20272969C2semidirect
7. #20272962C2semidirect
8. #20272958C2semidirect
9. #20272940C2semidirect
10. #20272937C2semidirect
23. #16810537C4semidirect
63. C2×#11719919C2×C4semidirect
71. C2×C22⋊Dic53C2×C4semidirect
72. C2×C22⋊Dic53C2×C4semidirect
83. #13133124C2×C4semidirect
404. C53#14300semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Dic532
C2×#168105372
C2×#168105052
C2×#168104952
#202729792
#202729692
#202729622
#202729582
#202729402
#202729372
C53×#14382
#1430053