Finite Group Databasebeta

← #27320735

Subgroups — #27320735

Order 13568.

1814 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810559C211yesmaximal, characteristic
2C2×#16810518C212yesmaximal, complemented
3C2×#16810492C211yesmaximal, characteristic, complemented
4C2×#16810481C212yesmaximal
5C2×C22⋊C4×Dic53C211yesmaximal, characteristic
6#20272983C211yesmaximal, characteristic, complemented
7#20272982C211yesmaximal, characteristic, complemented
8#20272978C212yesmaximal, complemented
9#20272958C212yesmaximal
10#20272943C211yesmaximal, characteristic, complemented
11C2×C53×#399C211yesmaximal, characteristic
12C24×Dic53C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14C22×#11719919C2212yes—
15C22×#11719919C2212yes—
16C22×#11719919C2212yes—
17C22×#11719919C2212yes—
18C22×C22⋊Dic53C2212yes—
19C22×C22⋊Dic53C2211yescharacteristic
20C22×C4×Dic53C2211yescharacteristic
21C22×C4×Dic53C2211yescharacteristic
22#16810559—24no—
23#16810518—28no—
24#16810492—24no—
25#16810481—28no—
26C22⋊C4×Dic53—24no—
27C2×#13133124C2212yes—
28C2×#13133124C2211yescharacteristic
29C2×#13133124C2211yescharacteristic
30C2×#13133124C2211yescharacteristic
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2212yes—
35C22×C4⋊C4×C53C2211yescharacteristic
36C22×C4⋊C4×C53C2212yes—
37C22×C22⋊C4×C53C2212yes—
38C22×C22⋊C4×C53C2211yescharacteristic
39C53×#399—24no—
40C2×#95×C53C2211yescharacteristic
41C23×Dic53—22no—
42C23×Dic53—24no—
43C23×Dic53—24no—
44C23×Dic53—22no—
45C23×Dic53Q814yes—
46C23×Dic53C2312yes—
47C23×Dic53C2312yes—
48C23×Dic53C2311yescharacteristic
49C23×Dic53C2311yescharacteristic
50C23×Dic53C2311yescharacteristic
51C23×Dic53C2311yescharacteristic
52C2×C4⋊Dic53—22no—
53C2×C4⋊Dic53—22no—
54C2×C4⋊Dic53—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—42no—
64C2×#11719919—42no—
65C2×#11719919—42no—
66C2×#11719919D418yes—
67C2×C22⋊Dic53—24no—
68C2×C22⋊Dic53—22no—
69C2×C22⋊Dic53—24no—
70C2×C22⋊Dic53—22no—
71C2×C22⋊Dic53—42no—
72C2×C22⋊Dic53Q814yes—
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—24no—
82#13133124—22no—
83#13133124—24no—
84#13133124—24no—
85C25×C53C2311yescharacteristic
86C23×C4×C53C2312yes—
87C23×C4×C53C2311yescharacteristic
88C23×C4×C53C2312yes—
89C23×C4×C53C2311yescharacteristic
90C2×C4⋊C4×C53—24no—
91C2×C4⋊C4×C53—24no—
92C2×C4⋊C4×C53—22no—
93C2×C4⋊C4×C53—22no—
94C2×C4⋊C4×C53—22no—
95C2×C4⋊C4×C53—42no—
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—22no—
103C24×C53—21no—
104C24×C53—22no—
105C24×C53—21no—
106C24×C53—21no—
107C24×C53—21no—
108C24×C53—22no—
109C24×C53—22no—
110C24×C53—22no—
111C24×C53C2×D412yes—
112C24×C53C2×Q812yes—
113C24×C53C2×Q812yes—
114C24×C53C2411yescharacteristic
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—21no—
119C22×C4×C53—22no—
120C22×C4×C53—21no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
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×C53C2×D414yes—
135C22×C4×C53C2×D412yes—
136C22×C4×C53C2×D412yes—
137C22×C4×C53C2×D412yes—
138C22×C4×C53#4112yes—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—21no—
143C22×Dic53—22no—
144C22×Dic53—21no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—21no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—21no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—28no—
156C22×Dic53—216no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53—22no—
162C22×Dic53—22no—
163C22×Dic53—22no—
164C22×Dic53—44no—
165C22×Dic53—48no—
166C22×Dic53—44no—
167C22×Dic53C2×D414yes—
168C22×Dic53C2×D412yes—
169C22×Dic53C2×D412yes—
170C22×Dic53C2×D414yes—
171C22×Dic53C2×Q812yes—
172C22×Dic53C2×Q812yes—
173C22×Dic53C2×Q812yes—
174C22×Dic53C2×Q812yes—
175C22×Dic53#4112yes—
176C22×Dic53#4114yes—
177C22×Dic53#4112yes—
178C4⋊C4×C53—48no—
179C4⋊C4×C53—44no—
180C22⋊C4×C53—28no—
181C22⋊C4×C53—48no—
182C22⋊Dic53—28no—
183C22⋊Dic53—48no—
184C4⋊Dic53—44no—
185#11719919—216no—
186#11719919—48no—
187#11719919—48no—
188#11719919—48no—
189C4×Dic53—28no—
190C4×Dic53—44no—
191C23×C53—22no—
192C23×C53—22no—
193C23×C53—24no—
194C23×C53—22no—
195C23×C53—22no—
196C23×C53—22no—
197C23×C53—22no—
198C23×C53—24no—
199C23×C53—24no—
200C23×C53—24no—
201C23×C53—24no—
202C23×C53—24no—
203C23×C53—42no—
204C23×C53—41no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53—41no—
208C23×C53—42no—
209C23×C53—42no—
210C23×C53—42no—
211C23×C53—42no—
212C23×C53—42no—
213C23×C53C4⋊Q814yes—
214C23×C53C22×D411yescharacteristic
215C23×C53C22×D412yes—
216C23×C53C22×D411yescharacteristic
217C23×C53C22×Q811yescharacteristic
218C23×C53C22×Q811yescharacteristic
219C23×C53C2×#4111yescharacteristic
220C23×C53C2×#4112yes—
221C23×C53C2×#4111yescharacteristic
222C23×C53C2×#4111yescharacteristic
223C23×C53H(5,𝔽2)12yes—
224C23×C532−1+412yes—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—24no—
228C2×C4×C53—24no—
229C2×C4×C53—22no—
230C2×C4×C53—24no—
231C2×C4×C53—24no—
232C2×C4×C53—24no—
233C2×C4×C53—24no—
234C2×C4×C53—41no—
235C2×C4×C53—42no—
236C2×C4×C53—41no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53—42no—
241C2×C4×C53—42no—
242C2×C4×C53—42no—
243C2×C4×C53—42no—
244C2×C4×C53—42no—
245C2×C4×C53—42no—
246C2×C4×C53—42no—
247C2×C4×C53—42no—
248C2×C4×C53#12114yes—
249C2×Dic53—22no—
250C2×Dic53—22no—
251C2×Dic53—24no—
252C2×Dic53—24no—
253C2×Dic53—22no—
254C2×Dic53—24no—
255C2×Dic53—22no—
256C2×Dic53—24no—
257C2×Dic53—24no—
258C2×Dic53—24no—
259C2×Dic53—24no—
260C2×Dic53—42no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—42no—
265C2×Dic53—41no—
266C2×Dic53—42no—
267C2×Dic53—42no—
268C2×Dic53—42no—
269C2×Dic53—42no—
270C2×Dic53—416no—
271C2×Dic53—42no—
272C2×Dic53—42no—
273C2×Dic53—42no—
274C2×Dic53—42no—
275C2×Dic53—42no—
276C2×Dic53—42no—
277C2×Dic53—42no—
278C2×Dic53C22⋊Q814yes—
279C2×Dic53C22⋊Q814yes—
280C2×Dic53C22⋊Q818yes—
281C2×Dic53#12314yes—
282C2×Dic53#12318yes—
283C2×Dic53#12714yes—
284#15135—531nomaximal, Sylow
285C22×C53—24no—
286C22×C53—28no—
287C22×C53—42no—
288C22×C53—42no—
289C22×C53—44no—
290C22×C53—44no—
291C22×C53—44no—
292C22×C53—42no—
293C22×C53—42no—
294C22×C53—42no—
295C22×C53—42no—
296C22×C53C2×#12111yescharacteristic
297C22×C53C2×C22⋊Q811yescharacteristic
298C22×C53C2×C22⋊Q811yescharacteristic
299C22×C53C2×C22⋊Q812yes—
300C22×C53C2×#12312yes—
301C22×C53C2×#12311yescharacteristic
302C22×C53C2×#12711yescharacteristic
303C22×C53C2×C4⋊Q811yescharacteristic
304C22×C53#53211yescharacteristic
305C22×C53#53311yescharacteristic
306C22×C53#53311yescharacteristic
307C22×C53#53511yescharacteristic
308C22×C53#53611yescharacteristic
309C22×C53#54212yes—
310C22×C53#54412yes—
311C22×C53D4212yes—
312C22×C53#54712yes—
313C22×C53#54712yes—
314C22×C53D4×Q812yes—
315C22×C53#55012yes—
316C22×C53#55212yes—
317C22×C53#55312yes—
318C22×C53#55412yes—
319C4×C53—24no—
320C4×C53—44no—
321C4×C53—42no—
322C4×C53—82no—
323Dic53—24no—
324Dic53—42no—
325Dic53—44no—
326Dic53—42no—
327Dic53—82no—
328Dic53#39918yescomplemented
329C2×#399—531no—
330C2×#399D5311yescharacteristic
331C2×#389—532no—
332C2×#384—531no—
333C2×#380—532no—
334C2×C4×C22⋊C4—531no—
335#1445—531no—
336#1445—531no—
337#1444—532no—
338#1439—532no—
339C4⋊#95—531no—
340C2×C53—44no—
341C2×C53C2×#39911yescharacteristic
342C2×C53#242212yes—
343C2×C53#242312yes—
344C2×C53#244012yes—
345C2×C53#244112yes—
346C2×C53#248611yescharacteristic
347C2×C53#248711yescharacteristic
348C2×C53#254111yescharacteristic
349C2×C53#254211yescharacteristic
350C2×C53#262811yescharacteristic
351C2×C53#285711yescharacteristic
352C24×C4—531no—
353C22×C4⋊C4—532no—
354C22×C4⋊C4—532no—
355C22×C4⋊C4—531no—
356C22×C4⋊C4—532no—
357C22×C4⋊C4—532no—
358C22×C4⋊C4C2×D5311yescharacteristic
359C22×C4⋊C4C2×D5312yes—
360C22×C22⋊C4—531no—
361C22×C22⋊C4—532no—
362C22×C22⋊C4C2×D5312yes—
363C22×C22⋊C4C2×D5311yescharacteristic
364C22×C42—531no—
365C22×C42—531no—
366#399—24no—
367#399—1064no—
368#389—1068no—
369#384—1064no—
370#380—1068no—
371C4×C22⋊C4—1064no—
372C2×#95—531no—
373C2×#95—532no—
374C2×#95—532no—
375C2×#95—532no—
376C2×#95—531no—
377C2×#95—532no—
378C2×#95—532no—
379C2×#95—531no—
380C2×#95C2×D5311yescharacteristic
381C53#1513511yescharacteristic, complemented, Sylow
382C25C22×D5311yescharacteristic
383C23×C4—531no—
384C23×C4—531no—
385C23×C4—534no—
386C23×C4—532no—
387C23×C4—531no—
388C23×C4—531no—
389C23×C4—532no—
390C23×C4—1062no—
391C23×C4—1064no—
392C23×C4—1064no—
393C23×C4—1062no—
394C23×C4C22×D5312yes—
395C23×C4C22×D5312yes—
396C23×C4C22×D5311yescharacteristic
397C23×C4C22×D5311yescharacteristic
398C2×C4⋊C4—22no—
399C2×C4⋊C4—24no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—24no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—42no—
404C2×C4⋊C4—538no—
405C2×C4⋊C4—1064no—
406C2×C4⋊C4—1064no—
407C2×C4⋊C4—1064no—
408C2×C4⋊C4—1064no—
409C2×C4⋊C4—1064no—
410C2×C4⋊C4—1064no—
411C2×C4⋊C4—1064no—
412C2×C4⋊C4—1062no—
413C2×C4⋊C4—1062no—
414C2×C4⋊C4—1064no—
415C2×C4⋊C4—1062no—
416C2×C4⋊C4—2122no—
417C2×C4⋊C4—2122no—
418C2×C4⋊C4—2122no—
419C2×C22⋊C4—22no—
420C2×C22⋊C4—22no—
421C2×C22⋊C4—24no—
422C2×C22⋊C4—24no—
423C2×C22⋊C4—42no—
424C2×C22⋊C4—534no—
425C2×C22⋊C4—1062no—
426C2×C22⋊C4—1062no—
427C2×C22⋊C4—1064no—
428C2×C22⋊C4—1064no—
429C2×C22⋊C4—2122no—
430C2×C22⋊C4C22⋊D5314yes—
431C2×C42—534no—
432C2×C42—1064no—
433C2×C42—1062no—
434C2×C42—1064no—
435#95—22no—
436#95—1064no—
437#95—1062no—
438#95—1064no—
439#95—1064no—
440#95—1062no—
441#95—1064no—
442#95—1064no—
443#95—1062no—
444C24—22no—
445C24—21no—
446C24—21no—
447C24—22no—
448C24—21no—
449C24—21no—
450C24—22no—
451C24—22no—
452C24Q8×D5312yes—
453C24Q8×D5312yes—
454C24C2×C22⋊D5312yes—
455C24C23×D5311yescharacteristic
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
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—22no—
474C22×C4—22no—
475C22×C4—532no—
476C22×C4—534no—
477C22×C4—532no—
478C22×C4—534no—
479C22×C4—532no—
480C22×C4—532no—
481C22×C4—532no—
482C22×C4—532no—
483C22×C4—534no—
484C22×C4—532no—
485C22×C4—532no—
486C22×C4—1068no—
487C22×C4—10616no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1061no—
493C22×C4—1061no—
494C22×C4—1062no—
495C22×C4—1062no—
496C22×C4—1061no—
497C22×C4—1062no—
498C22×C4—1062no—
499C22×C4—1062no—
500C22×C4—1061no—
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—1062no—
510C22×C4—1062no—
511C22×C4—2124no—
512C22×C4—2128no—
513C22×C4—2124no—
514C22×C4D4×D5314yes—
515C22×C4D4×D5312yes—
516C22×C4#1171994712yes—
517C22×C4C2×C22⋊D5312yes—
518C22×C4C2×C22⋊D5312yes—
519C4⋊C4—48no—
520C4⋊C4—44no—
521C4⋊C4—10616no—
522C4⋊C4—2128no—
523C4⋊C4—2128no—
524C4⋊C4—2128no—
525C4⋊C4—2124no—
526C22⋊C4—28no—
527C22⋊C4—48no—
528C22⋊C4—1068no—
529C22⋊C4—2128no—
530C42—1068no—
531C42—2124no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—22no—
536C23—22no—
537C23—24no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—24no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—42no—
554C23#1313321514yes—
555C23C2×D4×D5312yes—
556C23C2×D4×D5311yescharacteristic
557C23C22×C22⋊D5311yescharacteristic
558C23C2×Q8×D5311yescharacteristic
559C23C2×Q8×D5311yescharacteristic
560C23#41×D5311yescharacteristic
561C23#41×D5312yes—
562C23#41×D5311yescharacteristic
563C23C2×#1171994711yescharacteristic
564C23#1313326212yes—
565C23#1313326412yes—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—538no—
590C2×C4—534no—
591C2×C4—534no—
592C2×C4—538no—
593C2×C4—534no—
594C2×C4—534no—
595C2×C4—1064no—
596C2×C4—1064no—
597C2×C4—1062no—
598C2×C4—1062no—
599C2×C4—1064no—
600C2×C4—1064no—
601C2×C4—1062no—
602C2×C4—1064no—
603C2×C4—1062no—
604C2×C4—1064no—
605C2×C4—1064no—
606C2×C4—21216no—
607C2×C4—2122no—
608C2×C4—2122no—
609C2×C4—2122no—
610C2×C4—2122no—
611C2×C4—2122no—
612C2×C4—2122no—
613C2×C4—2122no—
614C2×C4—2122no—
615C2×C4—2122no—
616C2×C4—2122no—
617C2×C4—2122no—
618C2×C4—2121no—
619C2×C4—2121no—
620C2×C4—2121no—
621C2×C4—2122no—
622C2×C4—2121no—
623C2×C4—2122no—
624C2×C4#1313318914yes—
625C22—28no—
626C22—24no—
627C22—42no—
628C22—42no—
629C22—44no—
630C22—42no—
631C22—44no—
632C22—44no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22C2×#1313318911yescharacteristic
637C22C22⋊Q8×D5311yescharacteristic
638C22C22⋊Q8×D5312yes—
639C22C22⋊Q8×D5311yescharacteristic
640C22#123×D5312yes—
641C22#123×D5311yescharacteristic
642C22#127×D5311yescharacteristic
643C22C2×#1313321511yescharacteristic
644C22#1681116911yescharacteristic
645C22#1681117511yescharacteristic
646C22#1681117511yescharacteristic
647C22#1681118611yescharacteristic
648C22#1681119511yescharacteristic
649C22#1681124112yes—
650C22#1681125212yes—
651C22D4×C22⋊D5312yes—
652C22#1681127112yes—
653C22#1681127212yes—
654C22#1681128612yes—
655C22#1681129312yes—
656C22#1681131012yes—
657C22#1681131812yes—
658C22#1681132712yes—
659C4—24no—
660C4—42no—
661C4—44no—
662C4—82no—
663C4—538no—
664C4—1064no—
665C4—4242no—
666C4—2122no—
667C4—2122no—
668C4—2124no—
669C2—44no—
670C2#399×D5311yescharacteristic
671C2#2027822612yes—
672C2#2027823312yes—
673C2#2027841412yes—
674C2#2027842112yes—
675C2#2027902011yescharacteristic
676C2#2027902511yescharacteristic
677C2#2027965911yescharacteristic
678C2#2027966311yescharacteristic
679C2#2028054111yescharacteristic
680C2#2028306311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutator, socle, cosocleC24×C53C24
FittingC2×C53×#399C2
prime radicalC25×C53C23
hypercenterC2×#399D53
hypocenterC53#15135
LCSG ▷ C24×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
2. C2×#16810518C2semidirect
3. C2×#16810492C2semidirect
6. #20272983C2semidirect
7. #20272982C2semidirect
8. #20272978C2semidirect
10. #20272943C2semidirect
328. Dic53#399semidirect
381. C53#15135semidirect

Maximal subgroups

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

subgroupindex
C2×#168105592
C2×#168105182
C2×#168104922
C2×#168104812
C2×C22⋊C4×Dic532
#202729832
#202729822
#202729782
#202729582
#202729432
C2×C53×#3992
#1513553