Finite Group Databasebeta

← #27320640

Subgroups — #27320640

Order 13568.

2038 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133215C211yesmaximal, characteristic, complemented
2C2×#16810535C211yesmaximal, characteristic
3C2×#16810481C212yesmaximal
4C2×C22⋊C4×Dic53C211yesmaximal, characteristic
5#20272986C211yesmaximal, characteristic, complemented
6#20272985C211yesmaximal, characteristic, complemented
7#20272970C212yesmaximal, complemented
8#20272958C212yesmaximal
9#20272943C212yesmaximal, complemented
10#20272943C211yesmaximal, characteristic, complemented
11C2×C53×#393C211yesmaximal, characteristic
12C23×Dic106C2211yescharacteristic
13C24×Dic53C2211yescharacteristic
14C2×#13133215—28no—
15C2×#13133215—24no—
16C22×#11719919C2212yes—
17C22×#11719919C2212yes—
18C22×#11719919C2212yes—
19C22×C22⋊Dic53C2212yes—
20C22×C22⋊Dic53C2211yescharacteristic
21C22×C4×Dic53C2211yescharacteristic
22C22×C4×Dic53C2211yescharacteristic
23#16810535—24no—
24#16810481—28no—
25C22⋊C4×Dic53—24no—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2211yescharacteristic
29C2×#13133124C2211yescharacteristic
30C2×#13133124C2211yescharacteristic
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2212yes—
35C23×Q8×C53C2211yescharacteristic
36C22×C22⋊C4×C53C2212yes—
37C22×C22⋊C4×C53C2211yescharacteristic
38C53×#393—24no—
39C2×#95×C53C2212yes—
40C2×#95×C53C2211yescharacteristic
41C22×Dic106—22no—
42C22×Dic106—24no—
43C22×Dic106—22no—
44C22×Dic106—22no—
45C22×Dic106—42no—
46C23×Dic53—22no—
47C23×Dic53—24no—
48C23×Dic53—24no—
49C23×Dic53—22no—
50C23×Dic53Q814yescomplemented
51C23×Dic53C2312yes—
52C23×Dic53C2312yes—
53C23×Dic53C2311yescharacteristic
54C23×Dic53C2311yescharacteristic
55C23×Dic53C2311yescharacteristic
56C23×Dic53C2311yescharacteristic
57#13133215—232no—
58C2×#11719919—24no—
59C2×#11719919—24no—
60C2×#11719919—24no—
61C2×#11719919—24no—
62C2×#11719919—24no—
63C2×#11719919—24no—
64C2×#11719919—24no—
65C2×#11719919—24no—
66C2×#11719919D418yes—
67C2×C22⋊Dic53—24no—
68C2×C22⋊Dic53—22no—
69C2×C22⋊Dic53—24no—
70C2×C22⋊Dic53—22no—
71C2×C22⋊Dic53—42no—
72C2×C22⋊Dic53Q814yescomplemented
73C2×C4×Dic53—24no—
74C2×C4×Dic53—22no—
75C2×C4×Dic53—24no—
76C2×C4×Dic53D414yes—
77#13133124—24no—
78#13133124—22no—
79#13133124—22no—
80#13133124—24no—
81#13133124—22no—
82#13133124—24no—
83#13133124—24no—
84#13133124—42no—
85#13133124—42no—
86C25×C53C2311yescharacteristic
87C22×Q8×C53—24no—
88C22×Q8×C53—22no—
89C22×Q8×C53—22no—
90C22×Q8×C53—22no—
91C22×Q8×C53—42no—
92C23×C4×C53C2312yes—
93C23×C4×C53C2311yescharacteristic
94C23×C4×C53C2312yes—
95C23×C4×C53C2311yescharacteristic
96C2×C22⋊C4×C53—24no—
97C2×C22⋊C4×C53—22no—
98C2×C22⋊C4×C53—24no—
99C2×C22⋊C4×C53—22no—
100C2×C22⋊C4×C53—42no—
101C2×C22⋊C4×C53D414yes—
102#95×C53—24no—
103#95×C53—22no—
104C24×C53—21no—
105C24×C53—22no—
106C24×C53—21no—
107C24×C53—21no—
108C24×C53—21no—
109C24×C53—22no—
110C24×C53—22no—
111C24×C53—22no—
112C24×C53C2×D412yes—
113C24×C53C2×Q812yes—
114C24×C53C2×Q812yes—
115C24×C53C2411yescharacteristic
116C2×Q8×C53—28no—
117C2×Q8×C53—48no—
118C2×Q8×C53—48no—
119C2×Q8×C53—44no—
120C2×Q8×C53—44no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—21no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—21no—
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×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53—22no—
138C22×C4×C53—22no—
139C22×C4×C53—22no—
140C22×C4×C53C2×D414yes—
141C22×C4×C53C2×D412yes—
142C22×C4×C53C2×D412yes—
143C22×C4×C53C2×D412yes—
144C22×C4×C53#4112yes—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—21no—
148C22×Dic53—22no—
149C22×Dic53—21no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53—28no—
162C22×Dic53—216no—
163C22×Dic53—22no—
164C22×Dic53—22no—
165C22×Dic53—22no—
166C22×Dic53—22no—
167C22×Dic53—22no—
168C22×Dic53—22no—
169C22×Dic53—22no—
170C22×Dic53—44no—
171C22×Dic53—44no—
172C22×Dic53—48no—
173C22×Dic53C2×D414yes—
174C22×Dic53C2×D412yes—
175C22×Dic53C2×D412yes—
176C22×Dic53C2×D414yes—
177C22×Dic53C2×Q812yes—
178C22×Dic53C2×Q812yes—
179C22×Dic53C2×Q812yes—
180C22×Dic53C2×Q812yes—
181C22×Dic53#4112yes—
182C22×Dic53#4114yes—
183C22×Dic53#4112yes—
184C2×Dic106—28no—
185C2×Dic106—48no—
186C2×Dic106—48no—
187C2×Dic106—44no—
188C2×Dic106—44no—
189C22⋊C4×C53—28no—
190C22⋊C4×C53—48no—
191C22⋊Dic53—28no—
192C22⋊Dic53—48no—
193#11719919—216no—
194#11719919—216no—
195#11719919—48no—
196C4×Dic53—28no—
197C4×Dic53—44no—
198C23×C53—22no—
199C23×C53—22no—
200C23×C53—24no—
201C23×C53—22no—
202C23×C53—22no—
203C23×C53—22no—
204C23×C53—22no—
205C23×C53—24no—
206C23×C53—24no—
207C23×C53—24no—
208C23×C53—24no—
209C23×C53—24no—
210C23×C53—41no—
211C23×C53—41no—
212C23×C53—42no—
213C23×C53—41no—
214C23×C53—41no—
215C23×C53—42no—
216C23×C53—42no—
217C23×C53—42no—
218C23×C53—42no—
219C23×C53—42no—
220C23×C53C4⋊Q814yescomplemented
221C23×C53C22×D411yescharacteristic
222C23×C53C22×D412yes—
223C23×C53C22×D411yescharacteristic
224C23×C53C22×Q811yescharacteristic
225C23×C53C22×Q811yescharacteristic
226C23×C53C2×#4111yescharacteristic
227C23×C53C2×#4112yes—
228C23×C53C2×#4111yescharacteristic
229C23×C53C2×#4111yescharacteristic
230C23×C53H(5,𝔽2)12yes—
231C23×C532−1+412yes—
232Q8×C53—416no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—24no—
236C2×C4×C53—24no—
237C2×C4×C53—24no—
238C2×C4×C53—22no—
239C2×C4×C53—24no—
240C2×C4×C53—24no—
241C2×C4×C53—24no—
242C2×C4×C53—42no—
243C2×C4×C53—41no—
244C2×C4×C53—42no—
245C2×C4×C53—41no—
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×C53—42no—
254C2×C4×C53—42no—
255C2×C4×C53—42no—
256C2×C4×C53#12114yes—
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—24no—
260C2×Dic53—24no—
261C2×Dic53—24no—
262C2×Dic53—22no—
263C2×Dic53—24no—
264C2×Dic53—22no—
265C2×Dic53—24no—
266C2×Dic53—24no—
267C2×Dic53—24no—
268C2×Dic53—41no—
269C2×Dic53—42no—
270C2×Dic53—41no—
271C2×Dic53—42no—
272C2×Dic53—41no—
273C2×Dic53—42no—
274C2×Dic53—41no—
275C2×Dic53—42no—
276C2×Dic53—42no—
277C2×Dic53—42no—
278C2×Dic53—416no—
279C2×Dic53—42no—
280C2×Dic53—42no—
281C2×Dic53—42no—
282C2×Dic53—42no—
283C2×Dic53—42no—
284C2×Dic53—42no—
285C2×Dic53—42no—
286C2×Dic53#12014yes—
287C2×Dic53C22⋊Q814yes—
288C2×Dic53C22⋊Q814yes—
289C2×Dic53C22⋊Q818yes—
290C2×Dic53#12414yes—
291C2×Dic53#12418yes—
292Dic106—416no—
293#15125—531nomaximal, Sylow
294C22×C53—24no—
295C22×C53—28no—
296C22×C53—42no—
297C22×C53—42no—
298C22×C53—42no—
299C22×C53—42no—
300C22×C53—42no—
301C22×C53—44no—
302C22×C53—44no—
303C22×C53—44no—
304C22×C53—42no—
305C22×C53C2×#12011yescharacteristic
306C22×C53C2×#12111yescharacteristic
307C22×C53C2×C22⋊Q811yescharacteristic
308C22×C53C2×C22⋊Q811yescharacteristic
309C22×C53C2×C22⋊Q812yes—
310C22×C53C2×#12411yescharacteristic
311C22×C53C2×#12412yes—
312C22×C53C2×C4⋊Q811yescharacteristic
313C22×C53#53211yescharacteristic
314C22×C53#53311yescharacteristic
315C22×C53#53311yescharacteristic
316C22×C53#53511yescharacteristic
317C22×C53#53611yescharacteristic
318C22×C53#54212yes—
319C22×C53#54412yes—
320C22×C53D4212yes—
321C22×C53#54612yes—
322C22×C53#54712yes—
323C22×C53#54812yes—
324C22×C53D4×Q812yes—
325C22×C53#55412yes—
326C22×C53#55512yes—
327C22×C53#55612yes—
328C4×C53—24no—
329C4×C53—44no—
330C4×C53—42no—
331C4×C53—82no—
332Dic53—24no—
333Dic53—42no—
334Dic53—42no—
335Dic53—44no—
336Dic53—82no—
337Dic53#39318yescomplemented
338C22×C4⋊Q8—531no—
339C2×#393—531no—
340C2×#393D5311yescharacteristic
341C2×#380—532no—
342C2×C4×C22⋊C4—531no—
343#1446—531no—
344#1446—531no—
345#1442—532no—
346#1439—532no—
347C4⋊#95—532no—
348C4⋊#95—531no—
349C2×C53—44no—
350C2×C53C2×#39311yescharacteristic
351C2×C53#240811yescharacteristic
352C2×C53#240911yescharacteristic
353C2×C53#243912yes—
354C2×C53#244012yes—
355C2×C53#249112yes—
356C2×C53#249212yes—
357C2×C53#254711yescharacteristic
358C2×C53#256311yescharacteristic
359C2×C53#262811yescharacteristic
360C2×C53#285711yescharacteristic
361C23×Q8—531no—
362C23×Q8C2×D5311yescharacteristic
363C24×C4—531no—
364C2×C4⋊Q8—1064no—
365C2×C4⋊Q8—1068no—
366C22×C4⋊C4—532no—
367C22×C4⋊C4—532no—
368C22×C4⋊C4—532no—
369C22×C22⋊C4—531no—
370C22×C22⋊C4—532no—
371C22×C22⋊C4C2×D5312yes—
372C22×C22⋊C4C2×D5311yescharacteristic
373C22×C42—531no—
374C22×C42—531no—
375#393—24no—
376#393—1064no—
377#380—1068no—
378C4×C22⋊C4—1064no—
379C2×#95—532no—
380C2×#95—531no—
381C2×#95—532no—
382C2×#95—531no—
383C2×#95—532no—
384C2×#95—532no—
385C2×#95—531no—
386C2×#95—532no—
387C2×#95—532no—
388C2×#95C2×D5312yes—
389C2×#95C2×D5311yescharacteristic
390C53#1512511yescharacteristic, complemented, Sylow
391C25C22×D5311yescharacteristic
392C22×Q8—24no—
393C22×Q8—22no—
394C22×Q8—22no—
395C22×Q8—22no—
396C22×Q8—42no—
397C22×Q8—1064no—
398C22×Q8—1062no—
399C22×Q8—1062no—
400C22×Q8—1062no—
401C22×Q8—2122no—
402C23×C4—531no—
403C23×C4—531no—
404C23×C4—534no—
405C23×C4—532no—
406C23×C4—531no—
407C23×C4—531no—
408C23×C4—532no—
409C23×C4—1062no—
410C23×C4—1064no—
411C23×C4—1064no—
412C23×C4—1062no—
413C23×C4C22×D5312yes—
414C23×C4C22×D5312yes—
415C23×C4C22×D5311yescharacteristic
416C23×C4C22×D5311yescharacteristic
417C4⋊Q8—10632no—
418C2×C4⋊C4—538no—
419C2×C4⋊C4—1064no—
420C2×C4⋊C4—1064no—
421C2×C4⋊C4—1064no—
422C2×C4⋊C4—1064no—
423C2×C4⋊C4—1064no—
424C2×C4⋊C4—1064no—
425C2×C4⋊C4—1064no—
426C2×C4⋊C4—1064no—
427C2×C22⋊C4—22no—
428C2×C22⋊C4—22no—
429C2×C22⋊C4—24no—
430C2×C22⋊C4—24no—
431C2×C22⋊C4—42no—
432C2×C22⋊C4—534no—
433C2×C22⋊C4—1062no—
434C2×C22⋊C4—1062no—
435C2×C22⋊C4—1064no—
436C2×C22⋊C4—1064no—
437C2×C22⋊C4—2122no—
438C2×C22⋊C4C22⋊D5314yes—
439C2×C42—534no—
440C2×C42—1064no—
441C2×C42—1062no—
442C2×C42—1064no—
443#95—24no—
444#95—22no—
445#95—1064no—
446#95—1062no—
447#95—1064no—
448#95—1062no—
449#95—1064no—
450#95—1064no—
451#95—1062no—
452#95—2122no—
453#95—2122no—
454C24—22no—
455C24—21no—
456C24—21no—
457C24—22no—
458C24—21no—
459C24—21no—
460C24—22no—
461C24—22no—
462C24Q8×D5312yes—
463C24Q8×D5312yes—
464C24C2×C22⋊D5312yes—
465C24C23×D5311yescharacteristic
466C2×Q8—28no—
467C2×Q8—48no—
468C2×Q8—48no—
469C2×Q8—44no—
470C2×Q8—44no—
471C2×Q8—1068no—
472C2×Q8—2128no—
473C2×Q8—2128no—
474C2×Q8—2124no—
475C2×Q8—2124no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—532no—
496C22×C4—534no—
497C22×C4—532no—
498C22×C4—534no—
499C22×C4—532no—
500C22×C4—532no—
501C22×C4—532no—
502C22×C4—532no—
503C22×C4—534no—
504C22×C4—532no—
505C22×C4—532no—
506C22×C4—1068no—
507C22×C4—10616no—
508C22×C4—1062no—
509C22×C4—1062no—
510C22×C4—1062no—
511C22×C4—1061no—
512C22×C4—1061no—
513C22×C4—1062no—
514C22×C4—1062no—
515C22×C4—1062no—
516C22×C4—1062no—
517C22×C4—1061no—
518C22×C4—1062no—
519C22×C4—1061no—
520C22×C4—1062no—
521C22×C4—1062no—
522C22×C4—1062no—
523C22×C4—1062no—
524C22×C4—1062no—
525C22×C4—1062no—
526C22×C4—1062no—
527C22×C4—1062no—
528C22×C4—1062no—
529C22×C4—1062no—
530C22×C4—1062no—
531C22×C4—2124no—
532C22×C4—2124no—
533C22×C4—2128no—
534C22×C4D4×D5314yes—
535C22×C4D4×D5312yes—
536C22×C4#1171994712yes—
537C22×C4C2×C22⋊D5312yes—
538C22×C4C2×C22⋊D5312yes—
539C4⋊C4—10616no—
540C4⋊C4—10616no—
541C4⋊C4—2128no—
542C22⋊C4—28no—
543C22⋊C4—48no—
544C22⋊C4—1068no—
545C22⋊C4—2128no—
546C42—1068no—
547C42—2124no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—22no—
552C23—22no—
553C23—24no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—24no—
559C23—24no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—41no—
564C23—42no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—42no—
569C23—42no—
570C23#1313321514yescomplemented
571C23C2×D4×D5312yes—
572C23C2×D4×D5311yescharacteristic
573C23C22×C22⋊D5311yescharacteristic
574C23C2×Q8×D5311yescharacteristic
575C23C2×Q8×D5311yescharacteristic
576C23#41×D5312yes—
577C23#41×D5311yescharacteristic
578C23#41×D5311yescharacteristic
579C23C2×#1171994711yescharacteristic
580C23#1313326212yes—
581C23#1313326412yes—
582Q8—416no—
583Q8—21216no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—534no—
608C2×C4—538no—
609C2×C4—534no—
610C2×C4—534no—
611C2×C4—538no—
612C2×C4—534no—
613C2×C4—1064no—
614C2×C4—1064no—
615C2×C4—1062no—
616C2×C4—1062no—
617C2×C4—1064no—
618C2×C4—1064no—
619C2×C4—1062no—
620C2×C4—1064no—
621C2×C4—1062no—
622C2×C4—1064no—
623C2×C4—1064no—
624C2×C4—21216no—
625C2×C4—2122no—
626C2×C4—2122no—
627C2×C4—2122no—
628C2×C4—2122no—
629C2×C4—2122no—
630C2×C4—2122no—
631C2×C4—2122no—
632C2×C4—2122no—
633C2×C4—2121no—
634C2×C4—2122no—
635C2×C4—2121no—
636C2×C4—2122no—
637C2×C4—2121no—
638C2×C4—2121no—
639C2×C4—2122no—
640C2×C4—2122no—
641C2×C4—2122no—
642C2×C4#1313318914yes—
643C22—28no—
644C22—24no—
645C22—44no—
646C22—44no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—44no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22#120×D5311yescharacteristic
655C22C2×#1313318911yescharacteristic
656C22C22⋊Q8×D5311yescharacteristic
657C22C22⋊Q8×D5312yes—
658C22C22⋊Q8×D5311yescharacteristic
659C22#124×D5312yes—
660C22#124×D5311yescharacteristic
661C22C2×#1313321511yescharacteristic
662C22#1681116911yescharacteristic
663C22#1681117511yescharacteristic
664C22#1681117511yescharacteristic
665C22#1681118811yescharacteristic
666C22#1681119311yescharacteristic
667C22#1681123712yes—
668C22#1681125212yes—
669C22D4×C22⋊D5312yes—
670C22#1681126012yes—
671C22#1681127212yes—
672C22#1681128112yes—
673C22#1681128612yes—
674C22#1681132712yes—
675C22#1681133512yes—
676C22#1681134212yes—
677C4—24no—
678C4—44no—
679C4—42no—
680C4—82no—
681C4—538no—
682C4—1064no—
683C4—4242no—
684C4—2124no—
685C4—2122no—
686C4—2122no—
687C2—44no—
688C2#393×D5311yescharacteristic
689C2#2027806811yescharacteristic
690C2#2027807311yescharacteristic
691C2#2027839912yes—
692C2#2027841412yes—
693C2#2027907512yes—
694C2#2027908212yes—
695C2#2027970811yescharacteristic
696C2#2027988311yescharacteristic
697C2#2028053511yescharacteristic
698C2#2028306311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutator, socle, cosocleC24×C53C24
FittingC2×C53×#393C2
prime radicalC25×C53C23
hypercenterC2×#393D53
hypocenterC53#15125
LCSG ▷ C24×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
1. C22×#13133215C2semidirect
5. #20272986C2semidirect
6. #20272985C2semidirect
7. #20272970C2semidirect
9. #20272943C2semidirect
10. #20272943C2semidirect
50. C23×Dic53Q8semidirect
72. C2×C22⋊Dic53Q8semidirect
220. C23×C53C4⋊Q8semidirect
337. Dic53#393semidirect
390. C53#15125semidirect
570. C23#13133215semidirect

Maximal subgroups

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

subgroupindex
C22×#131332152
C2×#168105352
C2×#168104812
C2×C22⋊C4×Dic532
#202729862
#202729852
#202729702
#202729582
#202729432
C2×C53×#3932
#1512553