Finite Group Databasebeta

← #27429772

Subgroups — #27429772

Order 13568.

886 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810850C211yesmaximal, characteristic
2C2×C4×#11719923C211yesmaximal, characteristic, complemented
3#20278240C211yesmaximal, characteristic, complemented
4#20276124C211yesmaximal, characteristic, complemented
5#20276118C211yesmaximal, characteristic, complemented
6#20275911C211yesmaximal, characteristic, complemented
7#20275901C211yesmaximal, characteristic
8#20275568C211yesmaximal, characteristic, complemented
9#20275440C211yesmaximal, characteristic, complemented
10#20274982C211yesmaximal, characteristic, complemented
11#20274976C211yesmaximal, characteristic, complemented
12#20274871C211yesmaximal, characteristic, complemented
13#20274738C211yesmaximal, characteristic, complemented
14C22⋊C4×#88322C211yesmaximal, characteristic, complemented
15C53×#2422C211yesmaximal, characteristic
16C22×#11719923C2211yescharacteristic
17C23×#88322C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×C4×Dic106C2211yescharacteristic
20#16810850—24no—
21C4×#11719923—24no—
22C2×#13133148C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×#13133140C2211yescharacteristic
28C2×#13133140C2211yescharacteristic
29C2×#13133140C2211yescharacteristic
30C2×#13133138C2211yescharacteristic
31C2×#13133138C2211yescharacteristic
32C2×#13133138C2211yescharacteristic
33C2×C22⋊#88322C2211yescharacteristic
34C2×C4×#88322C2211yescharacteristic
35C2×C4×#88322C2211yescharacteristic
36#16810559—21no—
37#16810557—21no—
38#16810529C2211yescharacteristic
39#16810521C2211yescharacteristic
40#16810516—21no—
41#16810506—21no—
42#16810493C2211yescharacteristic
43#16810483C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×#123×C53—21no—
51C2×#121×C53C2211yescharacteristic
52C2×C4×D4×C53C2211yescharacteristic
53C53×#397—21no—
54C53×#394—21no—
55C53×#390C2211yescharacteristic
56C53×#389C2211yescharacteristic
57C53×#385—21no—
58C53×#384C2211yescharacteristic
59C53×#380C2211yescharacteristic
60C4×C22⋊C4×C53C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62C2×#11719923—22no—
63C2×#11719923—22no—
64C2×#11719923D414yes—
65C22×#88322—22no—
66C22×#88322—22no—
67C22×#88322—22no—
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71C22×#88322C2311yescharacteristic
72#13133196—22no—
73#13133196—41no—
74C4×Dic106—22no—
75C4×Dic106—41no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×C4×Dic53—21no—
87#13133148—22no—
88#13133148—22no—
89#13133148—22no—
90#13133148—41no—
91#13133148—41no—
92#13133140—22no—
93#13133140—22no—
94#13133140—22no—
95#13133138—22no—
96#13133138—22no—
97#13133138—22no—
98C22⋊#88322—22no—
99C4×#88322—22no—
100C4×#88322—22no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105C22×D4×C53—21no—
106C22×D4×C53C2311yescharacteristic
107C23×C4×C53—21no—
108C23×C4×C53C2311yescharacteristic
109#123×C53—44no—
110#121×C53—22no—
111#121×C53—41no—
112C4×D4×C53—22no—
113C4×D4×C53—41no—
114C2×C4⋊C4×C53—21no—
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53—21no—
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C22⋊C4×C53—21no—
121C2×C22⋊C4×C53—21no—
122C2×C22⋊C4×C53—21no—
123C2×C22⋊C4×C53—21no—
124C2×C22⋊C4×C53—21no—
125C2×C22⋊C4×C53D411yescharacteristic
126C2×C22⋊C4×C53D411yescharacteristic
127C2×C42×C53C2311yescharacteristic
128C2×C42×C53C2311yescharacteristic
129#95×C53—21no—
130#95×C53—21no—
131#95×C53D411yescharacteristic
132#95×C53D411yescharacteristic
133C24×C53—21no—
134C24×C53C2×D411yescharacteristic
135C2×D4×C53—21no—
136C2×D4×C53—21no—
137C2×D4×C53—21no—
138C2×D4×C53—21no—
139C2×D4×C53—21no—
140C2×D4×C53—42no—
141C2×D4×C53—42no—
142C2×D4×C53—81no—
143C2×D4×C53C2×D412yes—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—42no—
154C22×C4×C53—42no—
155C22×C4×C53—81no—
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53C2×D411yescharacteristic
160C22×C4×C53C2×D411yescharacteristic
161C22×C4×C53C2411yescharacteristic
162C22×Dic53—21no—
163C22×Dic53—21no—
164C22×Dic53—21no—
165C22×Dic53—21no—
166C2×Dic106—21no—
167C2×Dic106—21no—
168C2×Dic106—21no—
169C2×Dic106—21no—
170C2×Dic106—21no—
171C2×Dic106C2×D412yes—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—42no—
179C4⋊C4×C53—42no—
180C4⋊C4×C53—81no—
181C22⋊C4×C53—22no—
182C22⋊C4×C53—42no—
183C22⋊C4×C53—42no—
184C22⋊C4×C53—42no—
185C22⋊C4×C53—42no—
186C22⋊C4×C53—42no—
187C22⋊C4×C53SD1614yescomplemented
188C42×C53—21no—
189C42×C53—21no—
190C42×C53—21no—
191C42×C53—21no—
192C22⋊Dic53—42no—
193C22⋊Dic53—81no—
194#11719923—28no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—42no—
202C4⋊Dic53—81no—
203#11719919—42no—
204#11719919—42no—
205#11719919—81no—
206C4×Dic53—42no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C2×#88322—44no—
216C2×#88322C2×D412yes—
217C2×#88322#4112yes—
218C2×#88322#4112yes—
219C2×#88322#4112yes—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53—21no—
223C23×C53—21no—
224C23×C53—21no—
225C23×C53—21no—
226C23×C53—22no—
227C23×C53—41no—
228C23×C53—41no—
229C23×C53—41no—
230C23×C53—41no—
231C23×C53—41no—
232C23×C53—41no—
233C23×C53C2×SD1612yes—
234C23×C53C22×D411yescharacteristic
235D4×C53—22no—
236D4×C53—22no—
237D4×C53—41no—
238D4×C53—41no—
239D4×C53—84no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
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—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
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—84no—
266C2×C4×C53C2×SD1612yes—
267C2×C4×C53C2×SD1612yes—
268C2×C4×C53#13512yes—
269C2×C4×C53C22×D411yescharacteristic
270C2×C4×C53C2×#4111yescharacteristic
271C2×C4×C53C2×#4111yescharacteristic
272C2×C4×C53C2×#4111yescharacteristic
273C2×C4×C53H(5,𝔽2)11yescharacteristic
274C2×C4×C532−1+411yescharacteristic
275C2×Dic53—22no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—41no—
284C2×Dic53—41no—
285C2×Dic53—41no—
286C2×Dic53—41no—
287Dic106—22no—
288Dic106—22no—
289Dic106—41no—
290Dic106—41no—
291#88322—22no—
292#88322—41no—
293#88322—41no—
294#88322#12314yescomplemented
295#25030—531nomaximal, Sylow
296C22×C53—22no—
297C22×C53—22no—
298C22×C53—22no—
299C22×C53—41no—
300C22×C53—41no—
301C22×C53—41no—
302C22×C53—41no—
303C22×C53—42no—
304C22×C53—42no—
305C22×C53—42no—
306C22×C53—81no—
307C22×C53—81no—
308C22×C53—81no—
309C22×C53—81no—
310C22×C53#53211yescharacteristic
311C22×C53D4211yescharacteristic
312C22×C53#54711yescharacteristic
313C22×C53C22×SD1611yescharacteristic
314C22×C53C2×#13511yescharacteristic
315C22×C53#57711yescharacteristic
316C22×C53#57711yescharacteristic
317C4×C53—22no—
318C4×C53—22no—
319C4×C53—22no—
320C4×C53—41no—
321C4×C53—41no—
322C4×C53—81no—
323C4×C53—81no—
324C4×C53—81no—
325C4×C53C2×#12311yescharacteristic
326C4×C53#54211yescharacteristic
327C4×C53#54711yescharacteristic
328C4×C53#55311yescharacteristic
329Dic53—42no—
330Dic53—81no—
331Dic53—81no—
332Dic53—81no—
333C2×#465—531no—
334C2×C4×SD16—531no—
335#2423—531no—
336#2422D5311yescharacteristic
337#2072—531no—
338#2071—531no—
339#2035—531no—
340#2034—531no—
341#1968—531no—
342#1941—531no—
343#1865—531no—
344#1864—531no—
345#1844—531no—
346#1807—531no—
347C8×C22⋊C4—531no—
348C2×C53—42no—
349C2×C53—82no—
350C2×C53#242211yescharacteristic
351C2×C53#316711yescharacteristic
352C2×C53#317311yescharacteristic
353C2×C53D4×SD1611yescharacteristic
354C2×C53#328411yescharacteristic
355C2×C53#333511yescharacteristic
356C2×C53#333711yescharacteristic
357C22×SD16—531no—
358C23×C8—531no—
359C2×#123—21no—
360C2×C22⋊Q8—531no—
361C2×#121C2×D5311yescharacteristic
362C2×C4×Q8—531no—
363C2×C4×D4C2×D5311yescharacteristic
364#465—1064no—
365C4×SD16—1064no—
366C2×#106—531no—
367C2×#106—531no—
368C2×#106—531no—
369C2×#106—531no—
370C2×#106—531no—
371C2×#103—531no—
372C2×#103—531no—
373C2×#103—531no—
374C2×#102—531no—
375C2×#102—531no—
376C2×#102—531no—
377C2×C22⋊C8—531no—
378C2×C4×C8—531no—
379C2×C4×C8—531no—
380#399—1061no—
381#398—1061no—
382#397—21no—
383#394—21no—
384#391—531no—
385#390C2×D5311yescharacteristic
386#389—531no—
387#389C2×D5311yescharacteristic
388#388—1061no—
389#387—1061no—
390#385—21no—
391#384—531no—
392#384C2×D5311yescharacteristic
393#380—531no—
394#380C2×D5311yescharacteristic
395C4×C22⋊C4C2×D5311yescharacteristic
396#340—531no—
397#340—531no—
398#340—531no—
399#340—531no—
400#336—531no—
401#336—531no—
402C53#2503011yescharacteristic, complemented, Sylow
403C22×Q8—531no—
404C22×D4—21no—
405C22×D4C22×D5311yescharacteristic
406C23×C4—21no—
407C23×C4C22×D5311yescharacteristic
408C2×SD16—534no—
409C2×SD16—1062no—
410C2×SD16—1062no—
411C22×C8—531no—
412C22×C8—531no—
413C22×C8—531no—
414C22×C8—531no—
415C22×C8—1062no—
416C22×C8—1062no—
417C22×C8—1062no—
418#123—44no—
419C22⋊Q8—1062no—
420C22⋊Q8—2121no—
421#121—22no—
422#121—41no—
423C4×Q8—1062no—
424C4×Q8—2121no—
425C4×D4—22no—
426C4×D4—41no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—531no—
431C2×C4⋊C4—531no—
432C2×C4⋊C4—531no—
433C2×C4⋊C4—1061no—
434C2×C4⋊C4—1061no—
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×C22⋊C4C22⋊D5311yescharacteristic
449C2×C22⋊C4C22⋊D5311yescharacteristic
450C2×C42—1061no—
451C2×C42C22×D5311yescharacteristic
452C2×C42C22×D5311yescharacteristic
453#106—1062no—
454#106—1062no—
455#106—1062no—
456#106—2121no—
457#106—2121no—
458#103—1062no—
459#103—1062no—
460#103—1062no—
461#102—1062no—
462#102—1062no—
463#102—1062no—
464C22⋊C8—1062no—
465C4×C8—1062no—
466C4×C8—1062no—
467#95—21no—
468#95—21no—
469#95—1061no—
470#95—1061no—
471#95—1061no—
472#95—1061no—
473#95C22⋊D5311yescharacteristic
474#95C22⋊D5311yescharacteristic
475C24—21no—
476C24C2×C22⋊D5311yescharacteristic
477C2×Q8—532no—
478C2×Q8—1061no—
479C2×Q8—1061no—
480C2×Q8—1061no—
481C2×Q8—1061no—
482C2×Q8—1061no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—42no—
489C2×D4—42no—
490C2×D4—81no—
491C2×D4D4×D5312yes—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—81no—
504C22×C4—1061no—
505C22×C4—1061no—
506C22×C4—1061no—
507C22×C4—1061no—
508C22×C4C2×C22⋊D5311yescharacteristic
509C22×C4C2×C22⋊D5311yescharacteristic
510C22×C4C2×C22⋊D5311yescharacteristic
511C22×C4C2×C22⋊D5311yescharacteristic
512C22×C4C2×C22⋊D5311yescharacteristic
513C22×C4C23×D5311yescharacteristic
514SD16—1068no—
515C2×C8—532no—
516C2×C8—532no—
517C2×C8—532no—
518C2×C8—532no—
519C2×C8—1061no—
520C2×C8—1061no—
521C2×C8—1061no—
522C2×C8—1061no—
523C2×C8—1061no—
524C2×C8—1061no—
525C2×C8—1061no—
526C2×C8—1061no—
527C2×C8—2124no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—81no—
537C4⋊C4—1061no—
538C4⋊C4—1061no—
539C4⋊C4—1061no—
540C4⋊C4—1061no—
541C4⋊C4—1061no—
542C4⋊C4—1061no—
543C4⋊C4—4241no—
544C4⋊C4—4241no—
545C4⋊C4—2122no—
546C4⋊C4—2122no—
547C4⋊C4—2122no—
548C22⋊C4—22no—
549C22⋊C4—42no—
550C22⋊C4—42no—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C22⋊C4—42no—
554C22⋊C4—4241no—
555C22⋊C4—2122no—
556C22⋊C4#1171992314yescomplemented
557C42—21no—
558C42—21no—
559C42—21no—
560C42—21no—
561C42—2122no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—22no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23C2×#1171992312yes—
576C23C22×C22⋊D5311yescharacteristic
577Q8—1062no—
578Q8—1062no—
579Q8—2121no—
580Q8—2121no—
581D4—22no—
582D4—22no—
583D4—41no—
584D4—41no—
585D4—84no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—84no—
612C2×C4—1062no—
613C2×C4—2121no—
614C2×C4—2121no—
615C2×C4—2121no—
616C2×C4—2121no—
617C2×C4—2121no—
618C2×C4—2121no—
619C2×C4—2121no—
620C2×C4—2121no—
621C2×C4—2121no—
622C2×C4—2121no—
623C2×C4—2121no—
624C2×C4C2×#1171992312yes—
625C2×C4C2×#1171992312yes—
626C2×C4#1313323712yes—
627C2×C4C2×D4×D5311yescharacteristic
628C2×C4#41×D5311yescharacteristic
629C2×C4#41×D5311yescharacteristic
630C2×C4#41×D5311yescharacteristic
631C2×C4#1313326211yescharacteristic
632C2×C4#1313326411yescharacteristic
633C8—534no—
634C8—1062no—
635C8—2121no—
636C8—2121no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22#1681116911yescharacteristic
652C22D4×C22⋊D5311yescharacteristic
653C22#1681127211yescharacteristic
654C22C22×#1171992311yescharacteristic
655C22C2×#1313323711yescharacteristic
656C22#1681143111yescharacteristic
657C22#1681143411yescharacteristic
658C4—22no—
659C4—22no—
660C4—22no—
661C4—41no—
662C4—41no—
663C4—81no—
664C4—81no—
665C4—81no—
666C4—4241no—
667C4—4241no—
668C4—4241no—
669C4—2122no—
670C4#123×D5311yescharacteristic
671C4#1681124111yescharacteristic
672C4#1681127111yescharacteristic
673C4#1681131811yescharacteristic
674C2—42no—
675C2—82no—
676C2#2027822611yescharacteristic
677C2#2028527011yescharacteristic
678C2#2028532611yescharacteristic
679C2D4×#1171992311yescharacteristic
680C2#2028644111yescharacteristic
681C2#2028707211yescharacteristic
682C2#2028708811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2422C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#2422D53
hypocenterC53#25030
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2422

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11719923C2semidirect
3. #20278240C2semidirect
4. #20276124C2semidirect
5. #20276118C2semidirect
6. #20275911C2semidirect
8. #20275568C2semidirect
9. #20275440C2semidirect
10. #20274982C2semidirect
11. #20274976C2semidirect
12. #20274871C2semidirect
13. #20274738C2semidirect
14. C22⋊C4×#88322C2semidirect
187. C22⋊C4×C53SD16semidirect
294. #88322#123semidirect
402. C53#25030semidirect
556. C22⋊C4#11719923semidirect

Maximal subgroups

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

subgroupindex
C2×#168108502
C2×C4×#117199232
#202782402
#202761242
#202761182
#202759112
#202759012
#202755682
#202754402
#202749822
#202749762
#202748712
#202747382
C22⋊C4×#883222
C53×#24222
#2503053