Finite Group Databasebeta

← #27413695

Subgroups — #27413695

Order 13568.

958 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810723C211yesmaximal, characteristic, complemented
2C2×#16810680C211yesmaximal, characteristic, complemented
3C4×#13133196C211yesmaximal, characteristic, complemented
4#20276206C211yesmaximal, characteristic, complemented
5#20276146C211yesmaximal, characteristic, complemented
6#20276103C211yesmaximal, characteristic, complemented
7#20276093C211yesmaximal, characteristic, complemented
8#20276032C211yesmaximal, characteristic, complemented
9#20275918C211yesmaximal, characteristic, complemented
10#20275901C211yesmaximal, characteristic
11#20275718C211yesmaximal, characteristic
12#20274613C211yesmaximal, characteristic, complemented
13#20274548C211yesmaximal, characteristic, complemented
14#20274540C211yesmaximal, characteristic, complemented
15C4×#121×C53C211yesmaximal, characteristic
16C22×#11719923C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×C4×Dic106C2211yescharacteristic
19#16810723—24no—
20#16810680—24no—
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25C2×C4⋊#88322C2211yescharacteristic
26C2×#13133140C2211yescharacteristic
27C2×#13133140C2211yescharacteristic
28C2×#13133140C2211yescharacteristic
29C2×#13133138C2211yescharacteristic
30C2×#13133138C2211yescharacteristic
31C2×#13133138C2211yescharacteristic
32C2×C22⋊#88322C2211yescharacteristic
33C2×C22⋊#88322C2211yescharacteristic
34C2×#13133128C2211yescharacteristic
35C2×#13133128C2211yescharacteristic
36#16810529C2211yescharacteristic
37#16810521C2211yescharacteristic
38#16810507—21no—
39#16810496—21no—
40#16810483C2211yescharacteristic
41C4×#11719919—21no—
42C4×C4⋊Dic53C2211yescharacteristic
43C4×C22⋊Dic53—21no—
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50C2×#121×C53C2211yescharacteristic
51C2×C4×D4×C53—21no—
52C2×C4×D4×C53C2211yescharacteristic
53C22×C42×C53C2211yescharacteristic
54C53×#390C2211yescharacteristic
55C53×#389C2211yescharacteristic
56C53×#388—21no—
57C53×#386—21no—
58C53×#380C2211yescharacteristic
59C4×C4⋊C4×C53C2211yescharacteristic
60C4×C22⋊C4×C53—21no—
61C22×Dic106C2311yescharacteristic
62C2×#11719923—22no—
63C2×#11719923—22no—
64C2×#11719923—22no—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69#13133196—22no—
70#13133196—22no—
71C4×Dic106—22no—
72C4×Dic106—22no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83#13133150—22no—
84#13133150—22no—
85#13133148—22no—
86C4⋊#88322—22no—
87C4⋊#88322—22no—
88#13133140—22no—
89#13133140—22no—
90#13133140—22no—
91#13133138—22no—
92#13133138—22no—
93#13133138—22no—
94C22⋊#88322—22no—
95C22⋊#88322—22no—
96#13133128—22no—
97#13133128D414yescomplemented
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103C22×D4×C53—21no—
104C22×D4×C53C2311yescharacteristic
105C23×C4×C53—21no—
106C23×C4×C53D411yescharacteristic
107C23×C4×C53D411yescharacteristic
108C23×C4×C53C2311yescharacteristic
109#121×C53—22no—
110#121×C53—22no—
111C4×D4×C53—22no—
112C4×D4×C53—22no—
113C4×D4×C53—44no—
114C2×C4⋊C4×C53—21no—
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C22⋊C4×C53—21no—
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121C2×C22⋊C4×C53—21no—
122C2×C22⋊C4×C53—21no—
123C2×C42×C53—22no—
124C2×C42×C53—22no—
125C2×C42×C53—21no—
126C2×C42×C53D411yescharacteristic
127C2×C42×C53D411yescharacteristic
128C2×C42×C53C2311yescharacteristic
129C2×C42×C53C2311yescharacteristic
130#95×C53—21no—
131#95×C53—21no—
132#95×C53—21no—
133C24×C53—21no—
134C24×C53C2×D411yescharacteristic
135C2×D4×C53—24no—
136C2×D4×C53—21no—
137C2×D4×C53—21no—
138C2×D4×C53—21no—
139C2×D4×C53—21no—
140C2×D4×C53—21no—
141C2×D4×C53—42no—
142C2×D4×C53—42no—
143C2×D4×C53#4112yes—
144C22×C4×C53—22no—
145C22×C4×C53—21no—
146C22×C4×C53—22no—
147C22×C4×C53—22no—
148C22×C4×C53—22no—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53—21no—
156C22×C4×C53—21no—
157C22×C4×C53—41no—
158C22×C4×C53—41no—
159C22×C4×C53—42no—
160C22×C4×C53—42no—
161C22×C4×C53—42no—
162C22×C4×C53C2×D411yescharacteristic
163C22×C4×C53C2×D411yescharacteristic
164C22×C4×C53C2×D411yescharacteristic
165C22×C4×C53C2×D411yescharacteristic
166C22×C4×C53C2×D411yescharacteristic
167C22×C4×C53C2411yescharacteristic
168C22×Dic53—21no—
169C22×Dic53—21no—
170C22×Dic53—21no—
171C22×Dic53—21no—
172C2×Dic106—21no—
173C2×Dic106—21no—
174C2×Dic106—21no—
175C2×Dic106—21no—
176C2×Dic106—21no—
177C2×Dic106#4112yes—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53—42no—
184C4⋊C4×C53#4112yes—
185C22⋊C4×C53—24no—
186C22⋊C4×C53—42no—
187C22⋊C4×C53—42no—
188C22⋊C4×C53—42no—
189C22⋊C4×C53—42no—
190C42×C53—22no—
191C42×C53—22no—
192C42×C53—21no—
193C42×C53—21no—
194C42×C53—42no—
195C42×C53C2×D412yes—
196C42×C53#4112yes—
197C22⋊Dic53—24no—
198C22⋊Dic53—42no—
199#11719923—28no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53—21no—
204C4⋊Dic53—21no—
205C4⋊Dic53#4112yes—
206#11719919—24no—
207#11719919—42no—
208#11719919—42no—
209C4×Dic53—42no—
210C4×Dic53—42no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C2×#88322—21no—
216C2×#88322—21no—
217C2×#88322—21no—
218C2×#88322—21no—
219C2×#88322—21no—
220C2×#88322—21no—
221C2×#88322C2×D412yes—
222C2×#88322C2×D412yes—
223C23×C53—21no—
224C23×C53—21no—
225C23×C53—21no—
226C23×C53—21no—
227C23×C53—21no—
228C23×C53—21no—
229C23×C53—22no—
230C23×C53—41no—
231C23×C53—41no—
232C23×C53—41no—
233C23×C53—41no—
234C23×C53—41no—
235C23×C53—41no—
236C23×C53#13712yes—
237C23×C53C22×D411yescharacteristic
238D4×C53—22no—
239D4×C53—22no—
240D4×C53—48no—
241D4×C53—41no—
242D4×C53—41no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—21no—
251C2×C4×C53—21no—
252C2×C4×C53—21no—
253C2×C4×C53—22no—
254C2×C4×C53—21no—
255C2×C4×C53—21no—
256C2×C4×C53—44no—
257C2×C4×C53—44no—
258C2×C4×C53—41no—
259C2×C4×C53—41no—
260C2×C4×C53—41no—
261C2×C4×C53—41no—
262C2×C4×C53—41no—
263C2×C4×C53—41no—
264C2×C4×C53—41no—
265C2×C4×C53—41no—
266C2×C4×C53—41no—
267C2×C4×C53—41no—
268C2×C4×C53—41no—
269C2×C4×C53—41no—
270C2×C4×C53—84no—
271C2×C4×C53#12112yes—
272C2×C4×C53#12112yes—
273C2×C4×C53AGL(1,ℤ/8)12yes—
274C2×C4×C53C22×D411yescharacteristic
275C2×C4×C53C2×#4111yescharacteristic
276C2×C4×C53C2×#4111yescharacteristic
277C2×C4×C53C2×#4111yescharacteristic
278C2×C4×C53C2×#4111yescharacteristic
279C2×C4×C53C2×#4111yescharacteristic
280C2×Dic53—22no—
281C2×Dic53—22no—
282C2×Dic53—41no—
283C2×Dic53—41no—
284C2×Dic53—41no—
285C2×Dic53—41no—
286C2×Dic53—41no—
287C2×Dic53—41no—
288C2×Dic53—41no—
289C2×Dic53—41no—
290C2×Dic53—41no—
291C2×Dic53—41no—
292Dic106—22no—
293Dic106—22no—
294Dic106—41no—
295Dic106—41no—
296#88322—22no—
297#88322—22no—
298#88322—41no—
299#88322—41no—
300#23711—531nomaximal, Sylow
301C22×C53—22no—
302C22×C53—22no—
303C22×C53—22no—
304C22×C53—41no—
305C22×C53—41no—
306C22×C53—41no—
307C22×C53—41no—
308C22×C53—42no—
309C22×C53—42no—
310C22×C53—42no—
311C22×C53—81no—
312C22×C53—81no—
313C22×C53—81no—
314C22×C53—81no—
315C22×C53C2×#12111yescharacteristic
316C22×C53#52511yescharacteristic
317C22×C53#52511yescharacteristic
318C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
319C22×C53C2×#13711yescharacteristic
320C22×C53#57511yescharacteristic
321C22×C53#57511yescharacteristic
322C4×C53—22no—
323C4×C53—22no—
324C4×C53—22no—
325C4×C53—42no—
326C4×C53—41no—
327C4×C53—41no—
328C4×C53—81no—
329C4×C53—81no—
330C4×C53#52511yescharacteristic
331C4×C53#52911yescharacteristic
332C4×C53#52911yescharacteristic
333C4×C53#53211yescharacteristic
334Dic53—42no—
335Dic53—42no—
336Dic53—81no—
337Dic53—81no—
338C2×#440—531no—
339C2×#432—531no—
340C4×C22⋊Q8—531no—
341C4×#121D5311yescharacteristic
342#2086—531no—
343#2076—531no—
344#2067—531no—
345#2066—531no—
346#2055—531no—
347#2036—531no—
348#2034—531no—
349#2000—531no—
350#1776—531no—
351#1763—531no—
352#1762—531no—
353C2×C53—42no—
354C2×C53—82no—
355C2×C53#239211yescharacteristic
356C2×C53#305511yescharacteristic
357C2×C53#305611yescharacteristic
358C2×C53#311011yescharacteristic
359C2×C53#311111yescharacteristic
360C2×C53#311511yescharacteristic
361C2×C53#311711yescharacteristic
362C22×SD16—531no—
363C2×C22⋊Q8—531no—
364C2×#121C2×D5311yescharacteristic
365C2×C4×Q8—531no—
366C2×C4×D4—21no—
367C2×C4×D4C2×D5311yescharacteristic
368C22×C42C2×D5311yescharacteristic
369#440—1064no—
370#432—1064no—
371C2×#107—531no—
372C2×#107—531no—
373C2×#106—531no—
374C2×C4⋊C8—531no—
375C2×C4⋊C8—531no—
376C2×#103—531no—
377C2×#103—531no—
378C2×#103—531no—
379C2×#102—531no—
380C2×#102—531no—
381C2×#102—531no—
382C2×C22⋊C8—531no—
383C2×C22⋊C8—531no—
384C2×#97—531no—
385C2×#97—531no—
386#391—531no—
387#390C2×D5311yescharacteristic
388#389—531no—
389#389C2×D5311yescharacteristic
390#388—21no—
391#387—1061no—
392#386—21no—
393#385—1061no—
394#380—531no—
395#380C2×D5311yescharacteristic
396C4×C4⋊C4—531no—
397C4×C4⋊C4—1061no—
398C4×C4⋊C4C2×D5311yescharacteristic
399C4×C22⋊C4—21no—
400C4×C22⋊C4—1061no—
401#340—531no—
402#340—531no—
403#340—531no—
404#340—531no—
405#340—531no—
406#340—531no—
407C53#2371111yescharacteristic, complemented, Sylow
408C22×Q8—531no—
409C22×D4—21no—
410C22×D4C22×D5311yescharacteristic
411C23×C4—21no—
412C23×C4C22⋊D5311yescharacteristic
413C23×C4C22⋊D5311yescharacteristic
414C23×C4C22×D5311yescharacteristic
415C2×SD16—1062no—
416C2×SD16—1062no—
417C2×SD16—1062no—
418C22×C8—531no—
419C22×C8—531no—
420C22×C8—531no—
421C22×C8—531no—
422C22⋊Q8—1062no—
423C22⋊Q8—1062no—
424#121—22no—
425#121—22no—
426C4×Q8—1062no—
427C4×Q8—1062no—
428C4×D4—22no—
429C4×D4—22no—
430C4×D4—44no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—531no—
433C2×C4⋊C4—531no—
434C2×C4⋊C4—531no—
435C2×C4⋊C4—1061no—
436C2×C4⋊C4—1061no—
437C2×C4⋊C4—1061no—
438C2×C4⋊C4C22×D5311yescharacteristic
439C2×C4⋊C4C22×D5311yescharacteristic
440C2×C4⋊C4C22×D5311yescharacteristic
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—21no—
444C2×C22⋊C4—21no—
445C2×C22⋊C4—21no—
446C2×C22⋊C4—1061no—
447C2×C22⋊C4—1061no—
448C2×C42—22no—
449C2×C42—22no—
450C2×C42—21no—
451C2×C42—1061no—
452C2×C42—1061no—
453C2×C42C22⋊D5311yescharacteristic
454C2×C42C22⋊D5311yescharacteristic
455C2×C42C22×D5311yescharacteristic
456C2×C42C22×D5311yescharacteristic
457#107—1062no—
458#107—1062no—
459#106—1062no—
460C4⋊C8—1062no—
461C4⋊C8—1062no—
462#103—1062no—
463#103—1062no—
464#103—1062no—
465#102—1062no—
466#102—1062no—
467#102—1062no—
468C22⋊C8—1062no—
469C22⋊C8—1062no—
470#97—534no—
471#97—1062no—
472#95—21no—
473#95—21no—
474#95—21no—
475#95—1061no—
476#95—1061no—
477#95—1061no—
478#95—1061no—
479#95—1061no—
480C24—21no—
481C24C2×C22⋊D5311yescharacteristic
482C2×Q8—532no—
483C2×Q8—1061no—
484C2×Q8—1061no—
485C2×Q8—1061no—
486C2×Q8—1061no—
487C2×Q8—1061no—
488C2×D4—24no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—42no—
495C2×D4—42no—
496C2×D4#1171994512yes—
497C22×C4—22no—
498C22×C4—21no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—41no—
511C22×C4—41no—
512C22×C4—42no—
513C22×C4—42no—
514C22×C4—42no—
515C22×C4—1061no—
516C22×C4—1061no—
517C22×C4—1061no—
518C22×C4—1061no—
519C22×C4C2×C22⋊D5311yescharacteristic
520C22×C4C2×C22⋊D5311yescharacteristic
521C22×C4C2×C22⋊D5311yescharacteristic
522C22×C4C2×C22⋊D5311yescharacteristic
523C22×C4C2×C22⋊D5311yescharacteristic
524C22×C4C23×D5311yescharacteristic
525SD16—1068no—
526C2×C8—532no—
527C2×C8—532no—
528C2×C8—1061no—
529C2×C8—1061no—
530C2×C8—1061no—
531C2×C8—1061no—
532C2×C8—1061no—
533C2×C8—1061no—
534C2×C8—1061no—
535C2×C8—1061no—
536C2×C8—1061no—
537C2×C8—1061no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—42no—
544C4⋊C4—532no—
545C4⋊C4—1064no—
546C4⋊C4—1061no—
547C4⋊C4—1061no—
548C4⋊C4—1061no—
549C4⋊C4—1061no—
550C4⋊C4—1061no—
551C4⋊C4—2122no—
552C4⋊C4—2122no—
553C4⋊C4#1171994512yes—
554C22⋊C4—24no—
555C22⋊C4—42no—
556C22⋊C4—42no—
557C22⋊C4—42no—
558C22⋊C4—42no—
559C22⋊C4—1064no—
560C22⋊C4—2122no—
561C42—22no—
562C42—22no—
563C42—21no—
564C42—21no—
565C42—42no—
566C42—2122no—
567C42—2122no—
568C42D4×D5312yes—
569C42#1171994712yes—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—22no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23#1313324712yes—
584C23C22×C22⋊D5311yescharacteristic
585Q8—1062no—
586Q8—1062no—
587Q8—2121no—
588Q8—2121no—
589D4—22no—
590D4—22no—
591D4—48no—
592D4—41no—
593D4—41no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—84no—
622C2×C4—1062no—
623C2×C4—1062no—
624C2×C4—2121no—
625C2×C4—2121no—
626C2×C4—2121no—
627C2×C4—2121no—
628C2×C4—2121no—
629C2×C4—2121no—
630C2×C4—2121no—
631C2×C4—2121no—
632C2×C4—2121no—
633C2×C4—2121no—
634C2×C4#1313318912yes—
635C2×C4#1313318912yes—
636C2×C4#1313324312yes—
637C2×C4C2×D4×D5311yescharacteristic
638C2×C4#41×D5311yescharacteristic
639C2×C4#41×D5311yescharacteristic
640C2×C4C2×#1171994711yescharacteristic
641C2×C4C2×#1171994511yescharacteristic
642C2×C4C2×#1171994511yescharacteristic
643C8—1062no—
644C8—1062no—
645C8—2121no—
646C8—2121no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22C2×#1313318911yescharacteristic
662C22#1681113011yescharacteristic
663C22#1681113411yescharacteristic
664C22C2×#1313324311yescharacteristic
665C22C2×#1313324711yescharacteristic
666C22#1681142111yescharacteristic
667C22#1681142111yescharacteristic
668C4—22no—
669C4—22no—
670C4—22no—
671C4—42no—
672C4—41no—
673C4—41no—
674C4—81no—
675C4—81no—
676C4—4241no—
677C4—4241no—
678C4—2122no—
679C4—2122no—
680C4#1681113211yescharacteristic
681C4#1681115311yescharacteristic
682C4#1681115511yescharacteristic
683C4#1681117011yescharacteristic
684C2—42no—
685C2—82no—
686C2#2027795711yescharacteristic
687C2#2028433011yescharacteristic
688C2#2028434711yescharacteristic
689C2#2028481411yescharacteristic
690C2#2028482411yescharacteristic
691C2#2028486011yescharacteristic
692C2#2028488211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC4×#121×C53C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenterC4×#121D53
hypocenterC53#23711
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. C2×#16810723C2semidirect
2. C2×#16810680C2semidirect
3. C4×#13133196C2semidirect
4. #20276206C2semidirect
5. #20276146C2semidirect
6. #20276103C2semidirect
7. #20276093C2semidirect
8. #20276032C2semidirect
9. #20275918C2semidirect
12. #20274613C2semidirect
13. #20274548C2semidirect
14. #20274540C2semidirect
97. #13133128D4semidirect
407. C53#23711semidirect

Maximal subgroups

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

subgroupindex
C2×#168107232
C2×#168106802
C4×#131331962
#202762062
#202761462
#202761032
#202760932
#202760322
#202759182
#202759012
#202757182
#202746132
#202745482
#202745402
C4×#121×C532
#2371153