Finite Group Databasebeta

← #27420302

Subgroups — #27420302

Order 13568.

1012 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811251C211yesmaximal, characteristic
2C2×#16810930C211yesmaximal, characteristic, complemented
3C2×#16810918C211yesmaximal, characteristic, complemented
4#20276053C211yesmaximal, characteristic, complemented
5#20276043C211yesmaximal, characteristic, complemented
6#20275866C211yesmaximal, characteristic, complemented
7#20275856C211yesmaximal, characteristic, complemented
8#20275672C211yesmaximal, characteristic, complemented
9#20275667C211yesmaximal, characteristic, complemented
10#20275028C211yesmaximal, characteristic
11#20275023C211yesmaximal, characteristic
12#20274912C211yesmaximal, characteristic
13#20274902C211yesmaximal, characteristic
14#20274699C211yesmaximal, characteristic
15C53×#2323C211yesmaximal, characteristic
16#16811251—24no—
17#16811251—24no—
18C2×#13133217C2211yescharacteristic
19C2×#13133217C2211yescharacteristic
20C2×#13133216—21no—
21C2×#13133209C2211yescharacteristic
22C2×#13133209C2211yescharacteristic
23C2×#13133208—21no—
24C2×#13133194—21no—
25C2×#13133194—21no—
26C2×#13133176C2211yescharacteristic
27C22×C4⋊Dic53C2211yescharacteristic
28#16810930—24no—
29#16810918—24no—
30C2×#13133150C2211yescharacteristic
31C2×#13133150C2211yescharacteristic
32C2×#13133150C2211yescharacteristic
33C2×#13133148C2211yescharacteristic
34C2×C4⋊#88322C2211yescharacteristic
35C2×C4⋊#88322C2211yescharacteristic
36C2×#13133141C2211yescharacteristic
37C2×#13133141C2211yescharacteristic
38C2×#13133141C2211yescharacteristic
39C2×#13133141C2211yescharacteristic
40C2×C22⋊#88322C2211yescharacteristic
41C2×C22⋊#88322C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810378C2211yescharacteristic
51#16810378C2211yescharacteristic
52C2×C22⋊Q8×C53C2211yescharacteristic
53C2×C4×Q8×C53C2211yescharacteristic
54C2×#117×C53C2211yescharacteristic
55C53×#391C2211yescharacteristic
56C53×#389—21no—
57C53×#389C2211yescharacteristic
58C53×#388—21no—
59C53×#388—21no—
60C53×#387—21no—
61C53×#380C2211yescharacteristic
62C4×C4⋊C4×C53C2211yescharacteristic
63C22×Dic106—21no—
64C22×Dic106—21no—
65C23×Dic53—21no—
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70#13133217—22no—
71#13133217—22no—
72#13133217—22no—
73#13133217D414yes—
74#13133216—28no—
75#13133209—22no—
76#13133209—22no—
77#13133209—22no—
78#13133209D414yes—
79#13133208—28no—
80#13133194—28no—
81#13133194—28no—
82#13133176—22no—
83#13133176—22no—
84C2×C4⋊Dic53—22no—
85C2×C4⋊Dic53—22no—
86C2×C4⋊Dic53—21no—
87C2×C4⋊Dic53—21no—
88C2×C4⋊Dic53—21no—
89C2×C4⋊Dic53C2311yescharacteristic
90C2×C4⋊Dic53C2311yescharacteristic
91C2×C4⋊Dic53C2311yescharacteristic
92C2×C4⋊Dic53C2311yescharacteristic
93C2×#11719919—21no—
94C2×#11719919—21no—
95C2×#11719919—21no—
96C2×#11719919—21no—
97C2×C22⋊Dic53—21no—
98C2×C4×Dic53—21no—
99#13133150—22no—
100#13133150—22no—
101#13133150—22no—
102#13133148—22no—
103C4⋊#88322—22no—
104C4⋊#88322—22no—
105#13133141—22no—
106#13133141—22no—
107#13133141—22no—
108#13133141—22no—
109C22⋊#88322—22no—
110C22⋊#88322—22no—
111C22×Q8×C53C2311yescharacteristic
112C23×C4×C53C2311yescharacteristic
113C22⋊Q8×C53—22no—
114C22⋊Q8×C53Q814yescomplemented
115C4×Q8×C53—22no—
116C4×Q8×C53—22no—
117#117×C53—22no—
118#117×C53—22no—
119C2×C4⋊C4×C53—21no—
120C2×C4⋊C4×C53—21no—
121C2×C4⋊C4×C53—21no—
122C2×C4⋊C4×C53D411yescharacteristic
123C2×C4⋊C4×C53D411yescharacteristic
124C2×C4⋊C4×C53C2311yescharacteristic
125C2×C4⋊C4×C53C2311yescharacteristic
126C2×C4⋊C4×C53C2311yescharacteristic
127C2×C22⋊C4×C53—21no—
128C2×C22⋊C4×C53—21no—
129C2×C22⋊C4×C53D411yescharacteristic
130C2×C22⋊C4×C53D411yescharacteristic
131C2×C42×C53—21no—
132C2×C42×C53—21no—
133C2×C42×C53C2311yescharacteristic
134C2×C42×C53C2311yescharacteristic
135#95×C53—21no—
136#95×C53—21no—
137#95×C53—21no—
138#95×C53—21no—
139#95×C53—21no—
140C24×C53C2×D411yescharacteristic
141C2×Q8×C53—21no—
142C2×Q8×C53—21no—
143C2×Q8×C53—21no—
144C2×Q8×C53—21no—
145C2×Q8×C53—21no—
146C2×Q8×C53C2×Q812yes—
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—21no—
154C22×C4×C53—21no—
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53C2×D411yescharacteristic
160C22×C4×C53C2×Q812yes—
161C22×C4×C53#4112yes—
162C22×C4×C53C2411yescharacteristic
163C22×Dic53—21no—
164C22×Dic53—21no—
165C22×Dic53—21no—
166C22×Dic53—21no—
167C22×Dic53—24no—
168C22×Dic53—81no—
169C22×Dic53—81no—
170C2×Dic106—24no—
171C2×Dic106—24no—
172C2×Dic106—81no—
173C2×Dic106—81no—
174C2×Dic106—81no—
175C2×Dic106—81no—
176C4⋊C4×C53—22no—
177C4⋊C4×C53—22no—
178C4⋊C4×C53—24no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53—21no—
184C4⋊C4×C53—42no—
185C4⋊C4×C53—81no—
186C4⋊C4×C53C2×Q812yes—
187C22⋊C4×C53—24no—
188C22⋊C4×C53—22no—
189C22⋊C4×C53—22no—
190C22⋊C4×C53—81no—
191C42×C53—21no—
192C42×C53—21no—
193C42×C53—42no—
194C42×C53—81no—
195C42×C53C2×D412yes—
196C42×C53C2×D412yes—
197C22⋊Dic53—24no—
198C4⋊Dic53—24no—
199C4⋊Dic53—24no—
200C4⋊Dic53—24no—
201C4⋊Dic53—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53—21no—
204C4⋊Dic53—21no—
205C4⋊Dic53C2×D412yes—
206C4⋊Dic53C2×D412yes—
207C4⋊Dic53C2×D412yes—
208C4⋊Dic53C2×D412yes—
209#11719919—24no—
210#11719919—24no—
211#11719919—24no—
212#11719919—24no—
213C4×Dic53—24no—
214C2×#88322—21no—
215C2×#88322—21no—
216C2×#88322—21no—
217C2×#88322—21no—
218C2×#88322—21no—
219C2×#88322—21no—
220C2×#88322—21no—
221C2×#88322—21no—
222C2×#88322—21no—
223C2×#88322—21no—
224C2×#88322—21no—
225C2×#88322—21no—
226C23×C53—21no—
227C23×C53—21no—
228C23×C53—21no—
229C23×C53—21no—
230C23×C53—21no—
231C23×C53—21no—
232C23×C53C22⋊Q812yes—
233C23×C53C22×D411yescharacteristic
234Q8×C53—22no—
235Q8×C53—22no—
236Q8×C53—41no—
237Q8×C53—41no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—22no—
245C2×C4×C53—22no—
246C2×C4×C53—22no—
247C2×C4×C53—22no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—22no—
251C2×C4×C53—22no—
252C2×C4×C53—21no—
253C2×C4×C53—21no—
254C2×C4×C53—21no—
255C2×C4×C53—21no—
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—41no—
266C2×C4×C53#12012yes—
267C2×C4×C53#12012yes—
268C2×C4×C53C22⋊Q812yes—
269C2×C4×C53C22×D411yescharacteristic
270C2×C4×C53C22×D411yescharacteristic
271C2×C4×C53C22×Q811yescharacteristic
272C2×C4×C53C2×#4111yescharacteristic
273C2×C4×C532−1+411yescharacteristic
274C2×C4×C532−1+411yescharacteristic
275C2×Dic53—22no—
276C2×Dic53—22no—
277C2×Dic53—22no—
278C2×Dic53—22no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—41no—
284C2×Dic53—41no—
285C2×Dic53—41no—
286C2×Dic53—41no—
287C2×Dic53—84no—
288Dic106—84no—
289Dic106—84no—
290#88322—41no—
291#88322—41no—
292#88322—41no—
293#88322—41no—
294#24311—531nomaximal, Sylow
295C22×C53—22no—
296C22×C53—22no—
297C22×C53—22no—
298C22×C53—41no—
299C22×C53—41no—
300C22×C53—41no—
301C22×C53—41no—
302C22×C53C2×#12011yescharacteristic
303C22×C53C2×C22⋊Q811yescharacteristic
304C22×C53#53611yescharacteristic
305C22×C53#57611yescharacteristic
306C22×C53#57711yescharacteristic
307C22×C53#57811yescharacteristic
308C22×C53#57811yescharacteristic
309C4×C53—22no—
310C4×C53—22no—
311C4×C53—22no—
312C4×C53—42no—
313C4×C53—42no—
314C4×C53—41no—
315C4×C53—41no—
316C4×C53—81no—
317C4×C53—81no—
318C4×C53#54811yescharacteristic
319C4×C53#54811yescharacteristic
320C4×C53D4×Q811yescharacteristic
321C4×C53D4×Q811yescharacteristic
322Dic53—42no—
323Dic53—42no—
324Dic53—42no—
325Dic53—42no—
326C2×#544—531no—
327C2×#479—531no—
328C2×#477—531no—
329#2323D5311yescharacteristic
330#2059—531no—
331#2057—531no—
332#2026—531no—
333#2024—531no—
334#1989—531no—
335#1988—531no—
336#1873—531no—
337#1872—531no—
338#1854—531no—
339#1852—531no—
340#1798—531no—
341C2×C53—42no—
342C2×C53#240911yescharacteristic
343C2×C53#301111yescharacteristic
344C2×C53#301211yescharacteristic
345C2×C53#308211yescharacteristic
346C2×C53#308311yescharacteristic
347C2×C53#324211yescharacteristic
348C2×C53#324311yescharacteristic
349#544—1064no—
350#544—1064no—
351C2×C4⋊Q8—531no—
352C2×C4⋊Q8—531no—
353C2×C4⋊Q8—1061no—
354C2×#125—531no—
355C2×#125—531no—
356C2×#125—1061no—
357C2×C22⋊Q8—1061no—
358C2×C22⋊Q8—1061no—
359C2×C22⋊Q8C2×D5311yescharacteristic
360C2×C4×Q8C2×D5311yescharacteristic
361C2×#117—531no—
362C2×#117C2×D5311yescharacteristic
363C22×C4⋊C4—531no—
364#479—1064no—
365#477—1064no—
366C2×#107—531no—
367C2×#107—531no—
368C2×#107—531no—
369C2×#106—531no—
370C2×C4⋊C8—531no—
371C2×C4⋊C8—531no—
372C2×#103—531no—
373C2×#103—531no—
374C2×#103—531no—
375C2×#103—531no—
376C2×C22⋊C8—531no—
377C2×C22⋊C8—531no—
378#391C2×D5311yescharacteristic
379#389—21no—
380#389C2×D5311yescharacteristic
381#388—21no—
382#388—21no—
383#387—21no—
384#380C2×D5311yescharacteristic
385C4×C4⋊C4C2×D5311yescharacteristic
386#340—531no—
387#340—531no—
388#340—531no—
389#340—531no—
390#340—531no—
391#340—531no—
392#340—531no—
393#340—531no—
394#336—531no—
395#336—531no—
396C53#2431111yescharacteristic, complemented, Sylow
397C22×Q8—1061no—
398C22×Q8—1061no—
399C22×Q8C22×D5311yescharacteristic
400C23×C4—1061no—
401C23×C4C22×D5311yescharacteristic
402C22×C8—531no—
403C22×C8—531no—
404C22×C8—531no—
405C22×C8—531no—
406C4⋊Q8—534no—
407C4⋊Q8—1062no—
408C4⋊Q8—1062no—
409C4⋊Q8—1068no—
410C4⋊Q8—1062no—
411#125—534no—
412#125—1062no—
413#125—1062no—
414#125—1068no—
415#125—1062no—
416C22⋊Q8—22no—
417C22⋊Q8—1068no—
418C22⋊Q8—1068no—
419C22⋊Q8Dic10614yescomplemented
420C4×Q8—22no—
421C4×Q8—22no—
422#117—22no—
423#117—22no—
424#117—1062no—
425#117—1062no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—531no—
430C2×C4⋊C4—531no—
431C2×C4⋊C4—531no—
432C2×C4⋊C4—531no—
433C2×C4⋊C4—1062no—
434C2×C4⋊C4—1062no—
435C2×C4⋊C4—1061no—
436C2×C4⋊C4—1061no—
437C2×C4⋊C4—1061no—
438C2×C4⋊C4—1061no—
439C2×C4⋊C4—1061no—
440C2×C4⋊C4—1061no—
441C2×C4⋊C4—1061no—
442C2×C4⋊C4C22⋊D5311yescharacteristic
443C2×C4⋊C4C22⋊D5311yescharacteristic
444C2×C4⋊C4C22×D5311yescharacteristic
445C2×C4⋊C4C22×D5311yescharacteristic
446C2×C4⋊C4C22×D5311yescharacteristic
447C2×C22⋊C4—21no—
448C2×C22⋊C4—21no—
449C2×C22⋊C4—1061no—
450C2×C22⋊C4C22⋊D5311yescharacteristic
451C2×C22⋊C4C22⋊D5311yescharacteristic
452C2×C42—21no—
453C2×C42—21no—
454C2×C42—1061no—
455C2×C42C22×D5311yescharacteristic
456C2×C42C22×D5311yescharacteristic
457#107—1062no—
458#107—1062no—
459#107—1062no—
460#106—1062no—
461C4⋊C8—1062no—
462C4⋊C8—1062no—
463#103—1062no—
464#103—1062no—
465#103—1062no—
466#103—1062no—
467C22⋊C8—1062no—
468C22⋊C8—1062no—
469#95—21no—
470#95—21no—
471#95—21no—
472#95—21no—
473#95—21no—
474C24C2×C22⋊D5311yescharacteristic
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—1064no—
481C2×Q8—1064no—
482C2×Q8—4241no—
483C2×Q8—4241no—
484C2×Q8—4241no—
485C2×Q8—4241no—
486C2×Q8C2×Dic10612yes—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—1064no—
496C22×C4—1061no—
497C22×C4—1061no—
498C22×C4—1061no—
499C22×C4—1061no—
500C22×C4—4241no—
501C22×C4—4241no—
502C22×C4C2×Dic10612yes—
503C22×C4#1171994512yes—
504C22×C4C2×C22⋊D5311yescharacteristic
505C22×C4C2×C22⋊D5311yescharacteristic
506C22×C4C2×C22⋊D5311yescharacteristic
507C22×C4C2×C22⋊D5311yescharacteristic
508C22×C4C2×C22⋊D5311yescharacteristic
509C22×C4C23×D5311yescharacteristic
510C2×C8—1061no—
511C2×C8—1061no—
512C2×C8—1061no—
513C2×C8—1061no—
514C2×C8—1061no—
515C2×C8—1061no—
516C2×C8—1061no—
517C2×C8—1061no—
518C2×C8—1061no—
519C2×C8—1061no—
520C2×C8—1061no—
521C2×C8—1061no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—24no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—42no—
531C4⋊C4—81no—
532C4⋊C4—532no—
533C4⋊C4—532no—
534C4⋊C4—532no—
535C4⋊C4—532no—
536C4⋊C4—1064no—
537C4⋊C4—1064no—
538C4⋊C4—1064no—
539C4⋊C4—1064no—
540C4⋊C4—1064no—
541C4⋊C4—1064no—
542C4⋊C4—1064no—
543C4⋊C4—1061no—
544C4⋊C4—1061no—
545C4⋊C4—1061no—
546C4⋊C4—1061no—
547C4⋊C4C2×Dic10612yes—
548C22⋊C4—24no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—81no—
552C22⋊C4—1064no—
553C42—21no—
554C42—21no—
555C42—42no—
556C42—81no—
557C42—1064no—
558C42D4×D5312yes—
559C42D4×D5312yes—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23#1313319612yes—
567C23C22×C22⋊D5311yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—41no—
571Q8—41no—
572Q8—4244no—
573Q8—4244no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—1062no—
603C2×C4—1062no—
604C2×C4—1062no—
605C2×C4—1062no—
606C2×C4—4244no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4—2121no—
613C2×C4—2121no—
614C2×C4—2121no—
615C2×C4#1313318512yes—
616C2×C4#1313318512yes—
617C2×C4#1313319612yes—
618C2×C4C2×D4×D5311yescharacteristic
619C2×C4C2×D4×D5311yescharacteristic
620C2×C4C22×Dic10611yescharacteristic
621C2×C4C2×#1171994511yescharacteristic
622C2×C4#1313326411yescharacteristic
623C2×C4#1313326411yescharacteristic
624C8—2121no—
625C8—2121no—
626C8—2121no—
627C8—2121no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22C2×#1313318511yescharacteristic
636C22C2×#1313319611yescharacteristic
637C22#1681119411yescharacteristic
638C22#1681142611yescharacteristic
639C22#1681143111yescharacteristic
640C22#1681143811yescharacteristic
641C22#1681143811yescharacteristic
642C4—22no—
643C4—22no—
644C4—22no—
645C4—42no—
646C4—41no—
647C4—41no—
648C4—42no—
649C4—81no—
650C4—81no—
651C4—2122no—
652C4—2122no—
653C4—2122no—
654C4—2122no—
655C4#1681128211yescharacteristic
656C4#1681128211yescharacteristic
657C4D4×Dic10611yescharacteristic
658C4D4×Dic10611yescharacteristic
659C2—42no—
660C2#2027808011yescharacteristic
661C2#2028400811yescharacteristic
662C2#2028401611yescharacteristic
663C2#2028455711yescharacteristic
664C2#2028456611yescharacteristic
665C2#2028597711yescharacteristic
666C2#2028599211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2323C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2323D53
hypocenterC53#24311
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
2. C2×#16810930C2semidirect
3. C2×#16810918C2semidirect
4. #20276053C2semidirect
5. #20276043C2semidirect
6. #20275866C2semidirect
7. #20275856C2semidirect
8. #20275672C2semidirect
9. #20275667C2semidirect
114. C22⋊Q8×C53Q8semidirect
396. C53#24311semidirect
419. C22⋊Q8Dic106semidirect

Maximal subgroups

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

subgroupindex
C2×#168112512
C2×#168109302
C2×#168109182
#202760532
#202760432
#202758662
#202758562
#202756722
#202756672
#202750282
#202750232
#202749122
#202749022
#202746992
C53×#23232
#2431153