Finite Group Databasebeta

← #27310733

Subgroups — #27310733

Order 13568.

1978 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133198C211yesmaximal, characteristic
2C2×#16810481C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×Dic53C211yesmaximal, characteristic, complemented
4#20272966C212yesmaximal, complemented
5#20272962C211yesmaximal, characteristic, complemented
6#20272962C211yesmaximal, characteristic, complemented
7#20272958C212yesmaximal, complemented
8#20272956C211yesmaximal, characteristic
9#20272952C212yesmaximal, complemented
10#20272946C212yesmaximal, complemented
11C53×#1438C211yesmaximal, characteristic
12C24×Dic53C2211yescharacteristic
13C2×#13133198—24no—
14C2×#13133198—24no—
15C2×#13133198C418yescomplemented
16C22×#11719919C2211yescharacteristic
17C22×#11719919C2211yescharacteristic
18C22×#11719919C2212yes—
19C22×C22⋊Dic53C2211yescharacteristic
20C22×C22⋊Dic53C2211yescharacteristic
21C22×C22⋊Dic53C2211yescharacteristic
22C22×C22⋊Dic53C2211yescharacteristic
23C22×C22⋊Dic53C2212yes—
24C22×C22⋊Dic53C2211yescharacteristic
25C22×C4×Dic53C2212yes—
26#16810481—24no—
27C22⋊C4×Dic53—24no—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2212yes—
35C23×D4×C53C2211yescharacteristic
36C22×C22⋊C4×C53C2211yescharacteristic
37C22×C22⋊C4×C53C2212yes—
38C22×C22⋊C4×C53C2211yescharacteristic
39C2×#95×C53C2212yes—
40C23×Dic53—42no—
41C23×Dic53—42no—
42C23×Dic53C2×C414yes—
43C23×Dic53D414yes—
44C23×Dic53D414yes—
45C23×Dic53C2312yes—
46C23×Dic53C2311yescharacteristic
47C23×Dic53C2311yescharacteristic
48C23×Dic53C2312yes—
49C23×Dic53C2311yescharacteristic
50C23×Dic53C2311yescharacteristic
51#13133198—232no—
52C2×#11719919—22no—
53C2×#11719919—22no—
54C2×#11719919—22no—
55C2×#11719919—22no—
56C2×#11719919—24no—
57C2×#11719919—42no—
58C2×#11719919—42no—
59C2×#11719919C2×C414yes—
60C2×#11719919C2×C414yes—
61C2×C22⋊Dic53—22no—
62C2×C22⋊Dic53—22no—
63C2×C22⋊Dic53—22no—
64C2×C22⋊Dic53—22no—
65C2×C22⋊Dic53—22no—
66C2×C22⋊Dic53—22no—
67C2×C22⋊Dic53—22no—
68C2×C22⋊Dic53—22no—
69C2×C22⋊Dic53—24no—
70C2×C22⋊Dic53—24no—
71C2×C22⋊Dic53—42no—
72C2×C22⋊Dic53—42no—
73C2×C22⋊Dic53—42no—
74C2×C22⋊Dic53C2×C414yes—
75C2×C22⋊Dic53C2×C414yes—
76C2×C22⋊Dic53C2×C414yes—
77C2×C4×Dic53—24no—
78C2×C4×Dic53—42no—
79C2×C4×Dic53—42no—
80#13133124—24no—
81#13133124—24no—
82#13133124—24no—
83#13133124—24no—
84#13133124—42no—
85#13133124—42no—
86#13133124—42no—
87C25×C53C2311yescharacteristic
88C25×C53C2311yescharacteristic
89C22×D4×C53—22no—
90C22×D4×C53—22no—
91C22×D4×C53—42no—
92C22×D4×C53—42no—
93C22×D4×C53C2×C414yes—
94C23×C4×C53C2312yes—
95C23×C4×C53C2311yescharacteristic
96C23×C4×C53C2312yes—
97C2×C22⋊C4×C53—22no—
98C2×C22⋊C4×C53—22no—
99C2×C22⋊C4×C53—24no—
100C2×C22⋊C4×C53—24no—
101C2×C22⋊C4×C53—42no—
102C2×C22⋊C4×C53—42no—
103C2×C22⋊C4×C53—42no—
104#95×C53—42no—
105C24×C53—22no—
106C24×C53—22no—
107C24×C53—22no—
108C24×C53—22no—
109C24×C53—22no—
110C24×C53—22no—
111C24×C53—21no—
112C24×C53—21no—
113C24×C53—22no—
114C24×C53—22no—
115C24×C53—22no—
116C24×C53—22no—
117C24×C53—22no—
118C24×C53—22no—
119C24×C53C22×C412yes—
120C24×C53C22×C412yes—
121C24×C53C2×D412yes—
122C24×C53C2×D412yes—
123C24×C53C2411yescharacteristic
124C2×D4×C53—28no—
125C2×D4×C53—44no—
126C2×D4×C53—44no—
127C2×D4×C53—48no—
128C2×D4×C53—48no—
129C22×C4×C53—22no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53—22no—
138C22×C4×C53—22no—
139C22×C4×C53—22no—
140C22×C4×C53—22no—
141C22×C4×C53—22no—
142C22×C4×C53—22no—
143C22×C4×C53—22no—
144C22×C4×C53—22no—
145C22×C4×C53—22no—
146C22×C4×C53C22×C412yes—
147C22×C4×C53#4114yes—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53—22no—
162C22×Dic53—22no—
163C22×Dic53—22no—
164C22×Dic53—22no—
165C22×Dic53—22no—
166C22×Dic53—22no—
167C22×Dic53—22no—
168C22×Dic53—22no—
169C22×Dic53—22no—
170C22×Dic53—22no—
171C22×Dic53—22no—
172C22×Dic53—22no—
173C22×Dic53—44no—
174C22×Dic53—44no—
175C22×Dic53—48no—
176C22×Dic53—48no—
177C22×Dic53C22⋊C4116yescomplemented
178C22×Dic53C22×C412yes—
179C22×Dic53C22×C412yes—
180C22×Dic53C22×C412yes—
181C22×Dic53C22×C412yes—
182C22×Dic53C2×D412yes—
183C22×Dic53C2×D412yes—
184C22×Dic53C2×D412yes—
185C22×Dic53C2×D412yes—
186C22×Dic53#4114yes—
187C22⋊C4×C53—28no—
188C22⋊C4×C53—48no—
189C22⋊C4×C53—44no—
190C22⋊Dic53—28no—
191C22⋊Dic53—28no—
192C22⋊Dic53—28no—
193C22⋊Dic53—28no—
194C22⋊Dic53—48no—
195C22⋊Dic53—44no—
196#11719919—28no—
197#11719919—28no—
198#11719919—48no—
199C4×Dic53—48no—
200C23×C53—24no—
201C23×C53—22no—
202C23×C53—22no—
203C23×C53—24no—
204C23×C53—22no—
205C23×C53—22no—
206C23×C53—24no—
207C23×C53—24no—
208C23×C53—24no—
209C23×C53—22no—
210C23×C53—22no—
211C23×C53—24no—
212C23×C53—22no—
213C23×C53—24no—
214C23×C53—22no—
215C23×C53—22no—
216C23×C53—24no—
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—41no—
226C23×C53—41no—
227C23×C53—42no—
228C23×C53—42no—
229C23×C53—42no—
230C23×C53—41no—
231C23×C53—41no—
232C23×C53—42no—
233C23×C53—42no—
234C23×C53—42no—
235C23×C53—42no—
236C23×C53—42no—
237C23×C53—42no—
238C23×C53—42no—
239C23×C53—42no—
240C23×C53—42no—
241C23×C53C2×C22⋊C414yes—
242C23×C53C23×C411yescharacteristic
243C23×C53C22×D411yescharacteristic
244C23×C53C22×D411yescharacteristic
245C23×C53C2×#4112yes—
246C23×C53C2×#4112yes—
247C23×C53H(5,𝔽2)12yes—
248C23×C53H(5,𝔽2)12yes—
249C23×C53H(5,𝔽2)12yes—
250C23×C532−1+412yes—
251D4×C53—416no—
252C2×C4×C53—22no—
253C2×C4×C53—22no—
254C2×C4×C53—22no—
255C2×C4×C53—24no—
256C2×C4×C53—24no—
257C2×C4×C53—24no—
258C2×C4×C53—42no—
259C2×C4×C53—42no—
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×C53—42no—
267C2×C4×C53—42no—
268C2×C4×C53—42no—
269C2×C4×C53—42no—
270C2×C4×C53—42no—
271C2×Dic53—22no—
272C2×Dic53—22no—
273C2×Dic53—22no—
274C2×Dic53—22no—
275C2×Dic53—24no—
276C2×Dic53—24no—
277C2×Dic53—24no—
278C2×Dic53—24no—
279C2×Dic53—22no—
280C2×Dic53—22no—
281C2×Dic53—24no—
282C2×Dic53—24no—
283C2×Dic53—24no—
284C2×Dic53—24no—
285C2×Dic53—24no—
286C2×Dic53—416no—
287C2×Dic53—41no—
288C2×Dic53—41no—
289C2×Dic53—42no—
290C2×Dic53—42no—
291C2×Dic53—41no—
292C2×Dic53—41no—
293C2×Dic53—42no—
294C2×Dic53—42no—
295C2×Dic53—42no—
296C2×Dic53—42no—
297C2×Dic53—42no—
298C2×Dic53—42no—
299C2×Dic53—42no—
300C2×Dic53—42no—
301C2×Dic53—42no—
302C2×Dic53—42no—
303C2×Dic53—42no—
304C2×Dic53C2×C22⋊C414yes—
305C2×Dic53C2×C22⋊C414yes—
306#14287—531nomaximal, Sylow
307C22×C53—24no—
308C22×C53—24no—
309C22×C53—24no—
310C22×C53—44no—
311C22×C53—44no—
312C22×C53—44no—
313C22×C53—44no—
314C22×C53—44no—
315C22×C53—44no—
316C22×C53—42no—
317C22×C53—42no—
318C22×C53—44no—
319C22×C53—44no—
320C22×C53—42no—
321C22×C53—82no—
322C22×C53—82no—
323C22×C53—82no—
324C22×C53—82no—
325C22×C53C22×C22⋊C411yescharacteristic
326C22×C53C4×#4112yes—
327C22×C53#51811yescharacteristic
328C22×C53#51811yescharacteristic
329C22×C53#52012yes—
330C22×C53#52511yescharacteristic
331C22×C53#52511yescharacteristic
332C22×C53#53212yes—
333C22×C53#53412yes—
334C22×C53#53412yes—
335C22×C53#53511yescharacteristic
336C22×C53#53611yescharacteristic
337C22×C53#53712yes—
338C22×C53#53812yes—
339C22×C53#53812yes—
340C22×C53#53912yes—
341C22×C53#53912yes—
342C22×C53#54012yes—
343C22×C53#54012yes—
344C22×C53#54212yes—
345C22×C53#54212yes—
346C4×C53—42no—
347C4×C53—44no—
348C4×C53—82no—
349Dic53—24no—
350Dic53—24no—
351Dic53—42no—
352Dic53—44no—
353Dic53—42no—
354Dic53—82no—
355C22×#123—531no—
356C2×#380—531no—
357C2×C4×C22⋊C4—531no—
358#1441—532no—
359#1440—531no—
360#1440—531no—
361#1439—532no—
362#1438—531no—
363#1438D5311yescharacteristic
364#1437—532no—
365#1436—532no—
366C2×C53—44no—
367C2×C53—82no—
368C2×C53#229711yescharacteristic
369C2×C53#230811yescharacteristic
370C2×C53#231111yescharacteristic
371C2×C53#233512yes—
372C2×C53#233512yes—
373C2×C53#263012yes—
374C2×C53#263012yes—
375C2×C53#266112yes—
376C2×C53#266312yes—
377C23×D4C2×D5311yescharacteristic
378C24×C4—531no—
379C2×#123—538no—
380C2×#123—1064no—
381C2×#123—1064no—
382C22×C4⋊C4—531no—
383C22×C4⋊C4—531no—
384C22×C4⋊C4—532no—
385C22×C22⋊C4—531no—
386C22×C22⋊C4—531no—
387C22×C22⋊C4—531no—
388C22×C22⋊C4—531no—
389C22×C22⋊C4—532no—
390C22×C22⋊C4—531no—
391C22×C22⋊C4C2×D5311yescharacteristic
392C22×C22⋊C4C2×D5312yes—
393C22×C22⋊C4C2×D5311yescharacteristic
394C22×C42—532no—
395#380—1064no—
396C4×C22⋊C4—1064no—
397C2×#95—532no—
398C2×#95—532no—
399C2×#95—532no—
400C2×#95—532no—
401C2×#95—532no—
402C2×#95—532no—
403C2×#95—532no—
404C2×#95C2×D5312yes—
405C53#1428711yescharacteristic, complemented, Sylow
406C25C22×D5311yescharacteristic
407C25C22×D5311yescharacteristic
408C22×D4—22no—
409C22×D4—22no—
410C22×D4—42no—
411C22×D4—42no—
412C22×D4C4×D5314yes—
413C23×C4—531no—
414C23×C4—531no—
415C23×C4—534no—
416C23×C4—534no—
417C23×C4—534no—
418C23×C4—531no—
419C23×C4—531no—
420C23×C4—532no—
421C23×C4—532no—
422C23×C4—2122no—
423C23×C4—2122no—
424C23×C4C22×D5312yes—
425C23×C4C22×D5311yescharacteristic
426C23×C4C22×D5312yes—
427#123—10632no—
428C2×C4⋊C4—534no—
429C2×C4⋊C4—534no—
430C2×C4⋊C4—1062no—
431C2×C4⋊C4—1062no—
432C2×C4⋊C4—1062no—
433C2×C4⋊C4—1062no—
434C2×C4⋊C4—1064no—
435C2×C4⋊C4—2122no—
436C2×C4⋊C4—2122no—
437C2×C22⋊C4—22no—
438C2×C22⋊C4—24no—
439C2×C22⋊C4—22no—
440C2×C22⋊C4—24no—
441C2×C22⋊C4—42no—
442C2×C22⋊C4—42no—
443C2×C22⋊C4—42no—
444C2×C22⋊C4—534no—
445C2×C22⋊C4—534no—
446C2×C22⋊C4—534no—
447C2×C22⋊C4—1062no—
448C2×C22⋊C4—1062no—
449C2×C22⋊C4—1062no—
450C2×C22⋊C4—1062no—
451C2×C22⋊C4—1064no—
452C2×C22⋊C4—1062no—
453C2×C22⋊C4—1062no—
454C2×C22⋊C4—1062no—
455C2×C22⋊C4—1064no—
456C2×C22⋊C4—1062no—
457C2×C22⋊C4—2122no—
458C2×C22⋊C4—2122no—
459C2×C22⋊C4—2122no—
460C2×C42—1064no—
461C2×C42—2122no—
462C2×C42—2122no—
463#95—42no—
464#95—1064no—
465#95—1064no—
466#95—1064no—
467#95—1064no—
468#95—2122no—
469#95—2122no—
470#95—2122no—
471C24—22no—
472C24—22no—
473C24—22no—
474C24—22no—
475C24—22no—
476C24—22no—
477C24—21no—
478C24—21no—
479C24—22no—
480C24—22no—
481C24—22no—
482C24—22no—
483C24—22no—
484C24—22no—
485C24C2×C4×D5312yes—
486C24C2×C4×D5312yes—
487C24D4×D5312yes—
488C24D4×D5312yes—
489C24C23×D5311yescharacteristic
490C2×D4—28no—
491C2×D4—44no—
492C2×D4—44no—
493C2×D4—48no—
494C2×D4—48no—
495C22×C4—22no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—532no—
513C22×C4—532no—
514C22×C4—532no—
515C22×C4—532no—
516C22×C4—532no—
517C22×C4—532no—
518C22×C4—5316no—
519C22×C4—532no—
520C22×C4—532no—
521C22×C4—534no—
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—1062no—
536C22×C4—1062no—
537C22×C4—1062no—
538C22×C4—1062no—
539C22×C4—1062no—
540C22×C4—1062no—
541C22×C4—1062no—
542C22×C4—1062no—
543C22×C4—1062no—
544C22×C4—1062no—
545C22×C4—1062no—
546C22×C4—1062no—
547C22×C4—2124no—
548C22×C4—2124no—
549C22×C4—2128no—
550C22×C4—2128no—
551C22×C4C2×C4×D5312yes—
552C22×C4#1171994714yes—
553C4⋊C4—1068no—
554C4⋊C4—1068no—
555C4⋊C4—2128no—
556C22⋊C4—28no—
557C22⋊C4—48no—
558C22⋊C4—44no—
559C22⋊C4—1068no—
560C22⋊C4—1068no—
561C22⋊C4—1068no—
562C22⋊C4—1068no—
563C22⋊C4—2128no—
564C22⋊C4—2124no—
565C42—2128no—
566C23—24no—
567C23—22no—
568C23—22no—
569C23—24no—
570C23—22no—
571C23—22no—
572C23—24no—
573C23—24no—
574C23—24no—
575C23—22no—
576C23—22no—
577C23—24no—
578C23—22no—
579C23—24no—
580C23—22no—
581C23—22no—
582C23—24no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—41no—
592C23—41no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—41no—
597C23—41no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23C22⋊C4×D5314yes—
608C23C22×C4×D5311yescharacteristic
609C23C2×D4×D5311yescharacteristic
610C23C2×D4×D5311yescharacteristic
611C23#41×D5312yes—
612C23C2×#1171994712yes—
613C23#1313326212yes—
614C23#1313326212yes—
615C23#1313326212yes—
616C23#1313326412yes—
617D4—416no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—534no—
638C2×C4—534no—
639C2×C4—1062no—
640C2×C4—1062no—
641C2×C4—1062no—
642C2×C4—1062no—
643C2×C4—1064no—
644C2×C4—1064no—
645C2×C4—1064no—
646C2×C4—1064no—
647C2×C4—1062no—
648C2×C4—1062no—
649C2×C4—1064no—
650C2×C4—1064no—
651C2×C4—1064no—
652C2×C4—1064no—
653C2×C4—1064no—
654C2×C4—21216no—
655C2×C4—2122no—
656C2×C4—2122no—
657C2×C4—2122no—
658C2×C4—2122no—
659C2×C4—2122no—
660C2×C4—2122no—
661C2×C4—2122no—
662C2×C4—2121no—
663C2×C4—2121no—
664C2×C4—2122no—
665C2×C4—2122no—
666C2×C4—2121no—
667C2×C4—2121no—
668C2×C4—2122no—
669C2×C4—2122no—
670C2×C4—2122no—
671C2×C4—2122no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—44no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—44no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22C2×C22⋊C4×D5311yescharacteristic
691C22C4×#1171994712yes—
692C22#1681109611yescharacteristic
693C22#1681109611yescharacteristic
694C22#1681110212yes—
695C22#1681112911yescharacteristic
696C22#1681113111yescharacteristic
697C22#1681116812yes—
698C22#1681118312yes—
699C22#1681118312yes—
700C22#1681118811yescharacteristic
701C22#1681119511yescharacteristic
702C22#1681120012yes—
703C22#1681120512yes—
704C22#1681120812yes—
705C22#1681121612yes—
706C22#1681121812yes—
707C22#1681122312yes—
708C22#1681122712yes—
709C22#1681123712yes—
710C22#1681124012yes—
711C4—42no—
712C4—44no—
713C4—82no—
714C4—1064no—
715C4—1064no—
716C4—4242no—
717C4—2122no—
718C4—2122no—
719C4—2124no—
720C2—44no—
721C2—82no—
722C2#2027720011yescharacteristic
723C2#2027729111yescharacteristic
724C2#2027730211yescharacteristic
725C2#2027750212yes—
726C2#2027750612yes—
727C2#2028055112yes—
728C2#2028055512yes—
729C2#2028084012yes—
730C2#2028086212yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1438C2
socle, cosocleC24×C53C24
prime radicalC23×D4×C53C22
hypercenter#1438D53
hypocenterC53#14287
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1438

Complements

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

NH ≅ G/Ntype
2. C2×#16810481C2semidirect
3. C2×C22⋊C4×Dic53C2semidirect
4. #20272966C2semidirect
5. #20272962C2semidirect
6. #20272962C2semidirect
7. #20272958C2semidirect
9. #20272952C2semidirect
10. #20272946C2semidirect
15. C2×#13133198C4semidirect
177. C22×Dic53C22⋊C4semidirect
405. C53#14287semidirect

Maximal subgroups

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

subgroupindex
C22×#131331982
C2×#168104812
C2×C22⋊C4×Dic532
#202729662
#202729622
#202729582
#202729562
#202729522
#202729462
C53×#14382
#1428753