Finite Group Databasebeta

← #27309396

Subgroups — #27309396

Order 13568.

2062 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
1C22×Q8×Dic53C211yesmaximal, characteristic, complemented
2C2×#16810534C211yesmaximal, characteristic
3C2×#16810497C212yesmaximal
4C2×#16810470C211yesmaximal, characteristic
5#20272979C211yesmaximal, characteristic, complemented
6#20272961C211yesmaximal, characteristic
7#20272951C212yesmaximal
8#20272949C212yesmaximal, complemented
9#20272946C212yesmaximal, complemented
10#20272940C211yesmaximal, characteristic, complemented
11C53×#1446C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C2×Q8×Dic53—28no—
14C2×Q8×Dic53—24no—
15C22×C4⋊Dic53C2211yescharacteristic
16C22×C4⋊Dic53C2212yes—
17C22×#11719919C2212yes—
18C22×#11719921C2212yes—
19C22×#11719921C2211yescharacteristic
20C22×#11719921C2212yes—
21C22×#11719921C2211yescharacteristic
22C22×C4×Dic53C2212yes—
23C22×C4×Dic53C2211yescharacteristic
24C22×C4×Dic53C2211yescharacteristic
25#16810534C418yescomplemented
26#16810497—28no—
27#16810470—24no—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2211yescharacteristic
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2211yescharacteristic
34C2×#13133124C2212yes—
35C23×Q8×C53C2211yescharacteristic
36C2×#95×C53C2211yescharacteristic
37C2×#95×C53C2212yes—
38C2×#95×C53C2211yescharacteristic
39C2×#95×C53C2212yes—
40C24×D53C2311yescharacteristic
41C23×Dic53C2311yescharacteristic
42C23×Dic53C2312yes—
43C23×Dic53C2312yes—
44C23×Dic53C2311yescharacteristic
45C23×Dic53C2311yescharacteristic
46C22×C4×D53—22no—
47C22×C4×D53—22no—
48C22×C4×D53—22no—
49C22×C4×D53—24no—
50C22×C4×D53—42no—
51Q8×Dic53—232no—
52C2×C4⋊Dic53—22no—
53C2×C4⋊Dic53—24no—
54C2×C4⋊Dic53—24no—
55C2×C4⋊Dic53—24no—
56C2×C4⋊Dic53D418yes—
57C2×#11719919—24no—
58C2×#11719919—24no—
59C2×#11719919—42no—
60C2×#11719921—24no—
61C2×#11719921—24no—
62C2×#11719921—22no—
63C2×#11719921—22no—
64C2×#11719921—24no—
65C2×#11719921—22no—
66C2×#11719921—24no—
67C2×#11719921—22no—
68C2×#11719921—42no—
69C2×#11719921C2×C414yes—
70C2×#11719921C2×C418yes—
71C2×#11719921Q814yescomplemented
72C2×C4×Dic53—24no—
73C2×C4×Dic53—24no—
74C2×C4×Dic53—24no—
75C2×C4×Dic53—24no—
76C2×C4×Dic53—22no—
77C2×C4×Dic53—42no—
78C2×C4×Dic53D414yes—
79#13133124—24no—
80#13133124—22no—
81#13133124—24no—
82#13133124—42no—
83#13133124—42no—
84#13133124C2×C414yescomplemented
85#13133124C2×C418yescomplemented
86C22×Q8×C53—24no—
87C22×Q8×C53—22no—
88C22×Q8×C53—22no—
89C22×Q8×C53—24no—
90C22×Q8×C53C2×C414yescomplemented
91C23×C4×C53C2312yes—
92C23×C4×C53C2311yescharacteristic
93C23×C4×C53C2311yescharacteristic
94C23×C4×C53C2312yes—
95C23×C4×C53C2311yescharacteristic
96#95×C53—24no—
97#95×C53—22no—
98#95×C53—24no—
99#95×C53—22no—
100C24×C53C2411yescharacteristic
101C23×D53—21no—
102C23×D53—22no—
103C23×D53—21no—
104C23×D53—22no—
105C23×D53—21no—
106C23×D53—21no—
107C23×D53—22no—
108C23×D53—22no—
109C23×D53C22×C412yes—
110C23×D53C2×Q812yes—
111C23×D53#4112yes—
112C2×Q8×C53—28no—
113C2×Q8×C53—216no—
114C2×Q8×C53—28no—
115C2×Q8×C53—28no—
116C2×Q8×C53—216no—
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×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
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×C53C22×C414yes—
139C22×C4×C53C22×C412yes—
140C22×C4×C53C2×D414yes—
141C22×C4×C53C2×D412yes—
142C22×C4×C53C2×Q812yes—
143C22×C4×C53#4112yes—
144C22×C4×C53#4112yes—
145C22×C4×C53#4114yes—
146C22×Dic53—22no—
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—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×Dic53C22×C414yes—
168C22×Dic53C22×C412yes—
169C22×Dic53C2×D414yes—
170C22×Dic53C2×D414yes—
171C22×Dic53C2×D412yes—
172C22×Dic53C2×D412yes—
173C22×Dic53C2×Q812yes—
174C22×Dic53#4112yes—
175C2×C4×D53—28no—
176C2×C4×D53—48no—
177C2×C4×D53—48no—
178C2×C4×D53—44no—
179C2×C4×D53—44no—
180#11719921—216no—
181#11719921—28no—
182#11719921—28no—
183#11719921—48no—
184C4⋊Dic53—28no—
185C4⋊Dic53—216no—
186#11719919—48no—
187C4×Dic53—216no—
188C4×Dic53—28no—
189C4×Dic53—44no—
190C23×C53C23×C411yescharacteristic
191C23×C53C22×D411yescharacteristic
192C23×C53C22×D412yes—
193C23×C53C22×Q811yescharacteristic
194C23×C53C2×#4111yescharacteristic
195C23×C53C2×#4111yescharacteristic
196C23×C53C2×#4111yescharacteristic
197C23×C53C2×#4112yes—
198C23×C53C2×#4111yescharacteristic
199C23×C53H(5,𝔽2)12yes—
200C23×C532−1+412yes—
201C22×D53—24no—
202C22×D53—22no—
203C22×D53—22no—
204C22×D53—24no—
205C22×D53—22no—
206C22×D53—22no—
207C22×D53—22no—
208C22×D53—22no—
209C22×D53—24no—
210C22×D53—24no—
211C22×D53—24no—
212C22×D53—24no—
213C22×D53—41no—
214C22×D53—41no—
215C22×D53—42no—
216C22×D53—41no—
217C22×D53—41no—
218C22×D53—42no—
219C22×D53—42no—
220C22×D53—42no—
221C22×D53—42no—
222C22×D53—42no—
223C22×D53C4×Q814yescomplemented
224Q8×C53—232no—
225C2×C4×C53—24no—
226C2×C4×C53—24no—
227C2×C4×C53—24no—
228C2×C4×C53—24no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—24no—
232C2×C4×C53—24no—
233C2×C4×C53—24no—
234C2×C4×C53—24no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—24no—
238C2×C4×C53—24no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—42no—
244C2×C4×C53—42no—
245C2×C4×C53—42no—
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×C53C4×D414yes—
253C2×C4×C53C4×D418yes—
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—24no—
259C2×Dic53—24no—
260C2×Dic53—24no—
261C2×Dic53—24no—
262C2×Dic53—22no—
263C2×Dic53—24no—
264C2×Dic53—24no—
265C2×Dic53—24no—
266C2×Dic53—24no—
267C2×Dic53—24no—
268C2×Dic53—24no—
269C2×Dic53—24no—
270C2×Dic53—42no—
271C2×Dic53—42no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—42no—
275C2×Dic53—42no—
276C2×Dic53—42no—
277C2×Dic53—42no—
278C2×Dic53—42no—
279C2×Dic53—42no—
280C2×Dic53—42no—
281C2×Dic53#12014yes—
282C2×Dic53C22⋊Q814yes—
283C4×D53—416no—
284#14172—531nomaximal, Sylow
285C22×C53C2×C4×D411yescharacteristic
286C22×C53C2×C4×D412yes—
287C22×C53C2×C4×Q811yescharacteristic
288C22×C53C4×#4111yescharacteristic
289C22×C53#52012yes—
290C22×C53C2×#12011yescharacteristic
291C22×C53C2×C22⋊Q811yescharacteristic
292C22×C53#52511yescharacteristic
293C22×C53#52512yes—
294C22×C53#52511yescharacteristic
295C22×C53#52911yescharacteristic
296C22×C53#53311yescharacteristic
297C22×C53#53511yescharacteristic
298C22×C53#53611yescharacteristic
299C22×C53#54212yes—
300C22×C53#54612yes—
301C22×C53#54612yes—
302C22×C53#54812yes—
303C22×C53D4×Q812yes—
304C22×C53#55212yes—
305C22×C53#55412yes—
306C22×C53#55612yes—
307C22×C53#55912yes—
308C2×D53—28no—
309C2×D53—24no—
310C2×D53—42no—
311C2×D53—42no—
312C2×D53—42no—
313C2×D53—44no—
314C2×D53—42no—
315C2×D53—42no—
316C2×D53—44no—
317C2×D53—44no—
318C2×D53—42no—
319C4×C53—28no—
320C4×C53—24no—
321C4×C53—42no—
322C4×C53—42no—
323C4×C53—82no—
324Dic53—24no—
325Dic53—24no—
326Dic53—42no—
327Dic53—44no—
328Dic53—44no—
329C22×C4×Q8—531no—
330C2×#393—531no—
331C2×#385—532no—
332C2×C4×C22⋊C4—531no—
333#1446D5311yescharacteristic, complemented
334#1444—531no—
335#1439—531no—
336#1437—532no—
337#1436—532no—
338#1436—532no—
339#1434—531no—
340C2×C53C4×#12011yescharacteristic
341C2×C53C4×C22⋊Q811yescharacteristic
342C2×C53#232611yescharacteristic
343C2×C53#236012yes—
344C2×C53#236712yes—
345C2×C53#240411yescharacteristic
346C2×C53#240911yescharacteristic
347C2×C53#245812yes—
348C2×C53#246512yes—
349C2×C53#258011yescharacteristic
350C2×C53#259911yescharacteristic
351D53—44no—
352C23×Q8C2×D5311yescharacteristic
353C24×C4—531no—
354C2×C4×Q8—1064no—
355C2×C4×Q8—1068no—
356C22×C4⋊C4—532no—
357C22×C4⋊C4—531no—
358C22×C4⋊C4—532no—
359C22×C22⋊C4—532no—
360C22×C22⋊C4—531no—
361C22×C22⋊C4—531no—
362C22×C22⋊C4—532no—
363C22×C42—531no—
364C22×C42—532no—
365C22×C42—531no—
366#393—538no—
367#385—1068no—
368C4×C22⋊C4—1064no—
369C2×#95—531no—
370C2×#95—532no—
371C2×#95—532no—
372C2×#95—532no—
373C2×#95—531no—
374C2×#95—532no—
375C2×#95—532no—
376C2×#95C2×D5311yescharacteristic
377C2×#95C2×D5312yes—
378C2×#95C2×D5312yes—
379C2×#95C2×D5311yescharacteristic
380C53#1417211yescharacteristic, complemented, Sylow
381C25—531no—
382C22×Q8—22no—
383C22×Q8—24no—
384C22×Q8—24no—
385C22×Q8—22no—
386C22×Q8C4×D5314yescomplemented
387C23×C4—531no—
388C23×C4—531no—
389C23×C4—532no—
390C23×C4—532no—
391C23×C4—531no—
392C23×C4—1062no—
393C23×C4—1064no—
394C23×C4—1062no—
395C23×C4—1062no—
396C23×C4—2122no—
397C23×C4C22×D5312yes—
398C23×C4C22×D5311yescharacteristic
399C23×C4C22×D5311yescharacteristic
400C23×C4C22×D5311yescharacteristic
401C23×C4C22×D5312yes—
402C4×Q8—10632no—
403C2×C4⋊C4—538no—
404C2×C4⋊C4—1064no—
405C2×C4⋊C4—1064no—
406C2×C4⋊C4—1064no—
407C2×C4⋊C4—1062no—
408C2×C4⋊C4—1064no—
409C2×C4⋊C4—1064no—
410C2×C4⋊C4—2122no—
411C2×C22⋊C4—538no—
412C2×C22⋊C4—534no—
413C2×C22⋊C4—534no—
414C2×C22⋊C4—1062no—
415C2×C22⋊C4—1062no—
416C2×C22⋊C4—1062no—
417C2×C22⋊C4—1064no—
418C2×C22⋊C4—1064no—
419C2×C22⋊C4—1062no—
420C2×C22⋊C4—1064no—
421C2×C22⋊C4—1064no—
422C2×C22⋊C4—2122no—
423C2×C42—534no—
424C2×C42—1064no—
425C2×C42—1064no—
426C2×C42—1064no—
427C2×C42—1064no—
428C2×C42—1062no—
429C2×C42—2122no—
430#95—22no—
431#95—24no—
432#95—24no—
433#95—22no—
434#95—534no—
435#95—538no—
436#95—1064no—
437#95—1062no—
438#95—1064no—
439#95—2122no—
440#95—2122no—
441C24—532no—
442C24—532no—
443C24—532no—
444C24—1062no—
445C24—1061no—
446C24—1061no—
447C24—1062no—
448C24—1061no—
449C24—1061no—
450C24—1062no—
451C24—1062no—
452C24C23×D5311yescharacteristic
453C2×Q8—28no—
454C2×Q8—28no—
455C2×Q8—216no—
456C2×Q8—28no—
457C2×Q8—216no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—532no—
480C22×C4—534no—
481C22×C4—532no—
482C22×C4—532no—
483C22×C4—532no—
484C22×C4—534no—
485C22×C4—532no—
486C22×C4—534no—
487C22×C4—1068no—
488C22×C4—1062no—
489C22×C4—1061no—
490C22×C4—1061no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4—1061no—
494C22×C4—1062no—
495C22×C4—1062no—
496C22×C4—1062no—
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—2128no—
510C22×C4—2124no—
511C22×C4—2128no—
512C22×C4—2124no—
513C22×C4C2×C4×D5314yes—
514C22×C4C2×C4×D5312yes—
515C22×C4D4×D5312yes—
516C22×C4D4×D5314yes—
517C22×C4#1171994714yes—
518C22×C4#1171994712yes—
519C22×C4Q8×D5312yes—
520C22×C4#1171994912yes—
521C4⋊C4—10616no—
522C4⋊C4—1068no—
523C4⋊C4—2128no—
524C22⋊C4—1068no—
525C22⋊C4—1068no—
526C22⋊C4—10616no—
527C22⋊C4—2128no—
528C42—1068no—
529C42—10616no—
530C42—2124no—
531C23—534no—
532C23—1064no—
533C23—1064no—
534C23—1064no—
535C23—1064no—
536C23—1062no—
537C23—1064no—
538C23—1062no—
539C23—1064no—
540C23—1062no—
541C23—1062no—
542C23—1062no—
543C23—1062no—
544C23—2122no—
545C23—2122no—
546C23—2122no—
547C23—2122no—
548C23—2121no—
549C23—2121no—
550C23—2122no—
551C23—2121no—
552C23—2121no—
553C23—2122no—
554C23C22×C4×D5311yescharacteristic
555C23C2×D4×D5312yes—
556C23C2×D4×D5311yescharacteristic
557C23C2×Q8×D5311yescharacteristic
558C23#41×D5311yescharacteristic
559C23#41×D5311yescharacteristic
560C23C2×#1171994712yes—
561C23C2×#1171994711yescharacteristic
562C23C2×#1171994911yescharacteristic
563C23#1313326312yes—
564C23#1313326512yes—
565Q8—232no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—534no—
594C2×C4—534no—
595C2×C4—1064no—
596C2×C4—1064no—
597C2×C4—1064no—
598C2×C4—1064no—
599C2×C4—1062no—
600C2×C4—1062no—
601C2×C4—1062no—
602C2×C4—1062no—
603C2×C4—1064no—
604C2×C4—1064no—
605C2×C4—1064no—
606C2×C4—1062no—
607C2×C4—1064no—
608C2×C4—1064no—
609C2×C4—1064no—
610C2×C4—1064no—
611C2×C4—21216no—
612C2×C4—2122no—
613C2×C4—2122no—
614C2×C4—2122no—
615C2×C4—2122no—
616C2×C4—2122no—
617C2×C4—2122no—
618C2×C4—2122no—
619C2×C4—2121no—
620C2×C4—2122no—
621C2×C4—2121no—
622C2×C4—2122no—
623C2×C4#1313317714yes—
624C2×C4#1313317718yes—
625C22—1064no—
626C22—1068no—
627C22—2122no—
628C22—2122no—
629C22—2124no—
630C22—2122no—
631C22—2124no—
632C22—2122no—
633C22—2124no—
634C22—2122no—
635C22—2122no—
636C22C2×#1313317711yescharacteristic
637C22C2×#1313317712yes—
638C22C4×Q8×D5311yescharacteristic
639C22C4×#1171994911yescharacteristic
640C22#1681110512yes—
641C22#120×D5311yescharacteristic
642C22C22⋊Q8×D5311yescharacteristic
643C22#1681112911yescharacteristic
644C22#1681112912yes—
645C22#1681113611yescharacteristic
646C22#1681115211yescharacteristic
647C22#1681117811yescharacteristic
648C22#1681119111yescharacteristic
649C22#1681119811yescharacteristic
650C22#1681124612yes—
651C22#1681126412yes—
652C22#1681126612yes—
653C22#1681128312yes—
654C22#1681128812yes—
655C22#1681130712yes—
656C22#1681133112yes—
657C22#1681134712yes—
658C22#1681135512yes—
659C4—28no—
660C4—24no—
661C4—42no—
662C4—42no—
663C4—82no—
664C4—1064no—
665C4—1064no—
666C4—2124no—
667C4—2124no—
668C4—2122no—
669C2—2124no—
670C2#2027721411yescharacteristic
671C2#2027725311yescharacteristic
672C2#2027742911yescharacteristic
673C2#2027772512yes—
674C2#2027779112yes—
675C2#2027803511yescharacteristic
676C2#2027807511yescharacteristic
677C2#2027863212yes—
678C2#2027873112yes—
679C2#2028006811yescharacteristic
680C2#2028028711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1446C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenter#1446D53
hypocenterC53#14172
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1446

Complements

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

NH ≅ G/Ntype
1. C22×Q8×Dic53C2semidirect
5. #20272979C2semidirect
8. #20272949C2semidirect
9. #20272946C2semidirect
10. #20272940C2semidirect
11. C53×#1446C2semidirect
25. #16810534C4semidirect
71. C2×#11719921Q8semidirect
84. #13133124C2×C4semidirect
85. #13133124C2×C4semidirect
90. C22×Q8×C53C2×C4semidirect
223. C22×D53C4×Q8semidirect
333. #1446D53semidirect
380. C53#14172semidirect
386. C22×Q8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic532
C2×#168105342
C2×#168104972
C2×#168104702
#202729792
#202729612
#202729512
#202729492
#202729462
#202729402
C53×#14462
#1417253