Finite Group Databasebeta

← #27420299

Subgroups — #27420299

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×#16810932C211yesmaximal, characteristic, complemented
3C2×#16810916C211yesmaximal, characteristic, complemented
4#20277399C211yesmaximal, characteristic
5#20276056C211yesmaximal, characteristic, complemented
6#20276042C211yesmaximal, characteristic, complemented
7#20275864C211yesmaximal, characteristic, complemented
8#20275854C211yesmaximal, characteristic, complemented
9#20275671C211yesmaximal, characteristic, complemented
10#20275665C211yesmaximal, characteristic, complemented
11#20275023C211yesmaximal, characteristic
12#20274914C211yesmaximal, characteristic
13#20274905C211yesmaximal, characteristic
14#20274697C211yesmaximal, characteristic
15C53×#1873C211yesmaximal, characteristic
16#16811251—24no—
17#16811251—24no—
18C2×#13133217—21no—
19C2×#13133216C2211yescharacteristic
20C2×#13133216C2211yescharacteristic
21C2×#13133209—21no—
22C2×#13133208C2211yescharacteristic
23C2×#13133208C2211yescharacteristic
24C2×#13133194—21no—
25C2×#13133194—21no—
26C2×Q8×Dic53C2211yescharacteristic
27C2×#13133176C2211yescharacteristic
28C22×C4⋊Dic53C2211yescharacteristic
29#16810932—24no—
30#16810916—24no—
31C2×#13133151C2211yescharacteristic
32C2×#13133151C2211yescharacteristic
33C2×#13133150C2211yescharacteristic
34C2×#13133148C2211yescharacteristic
35C2×#13133146C2211yescharacteristic
36C2×#13133146C2211yescharacteristic
37C2×#13133141C2211yescharacteristic
38C2×#13133141C2211yescharacteristic
39C2×C22⋊#88322C2211yescharacteristic
40#16810530C2211yescharacteristic
41#16810522—21no—
42#16810522C2211yescharacteristic
43#16810516—21no—
44#16810516—21no—
45#16810509—21no—
46#16810483C2211yescharacteristic
47C4⋊C4×Dic53C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53#16810385C2211yescharacteristic
54#16810377C2211yescharacteristic
55#16810377C2211yescharacteristic
56C2×C22⋊Q8×C53C2211yescharacteristic
57C2×#117×C53C2211yescharacteristic
58C2×#103×C53C2211yescharacteristic
59C2×#103×C53C2211yescharacteristic
60C2×C22⋊C8×C53C2211yescharacteristic
61C53×#340C2211yescharacteristic
62C53×#340C2211yescharacteristic
63C22×Dic106—21no—
64C22×Dic106—21no—
65C23×Dic53—21no—
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68#13133217—28no—
69#13133216—22no—
70#13133216—22no—
71#13133216—22no—
72#13133216D414yes—
73#13133209—28no—
74#13133208—22no—
75#13133208—22no—
76#13133208—22no—
77#13133208D414yes—
78#13133194—28no—
79#13133194—28no—
80Q8×Dic53—22no—
81Q8×Dic53—22no—
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⋊Dic53—21no—
90C2×C4⋊Dic53—21no—
91C2×C4⋊Dic53D411yescharacteristic
92C2×C4⋊Dic53D411yescharacteristic
93C2×C4⋊Dic53C2311yescharacteristic
94C2×C4⋊Dic53C2311yescharacteristic
95C2×C4⋊Dic53C2311yescharacteristic
96C2×C4⋊Dic53C2311yescharacteristic
97C2×#11719919—21no—
98C2×#11719919—21no—
99C2×#11719919—21no—
100C2×#11719919—21no—
101C2×C22⋊Dic53—21no—
102C2×C22⋊Dic53D411yescharacteristic
103C2×C22⋊Dic53D411yescharacteristic
104C2×C4×Dic53—21no—
105C2×C4×Dic53—21no—
106C2×C4×Dic53C2311yescharacteristic
107C2×C4×Dic53C2311yescharacteristic
108#13133151—22no—
109#13133151—22no—
110#13133150—22no—
111#13133148—22no—
112#13133146—22no—
113#13133146—22no—
114#13133141—22no—
115#13133141—22no—
116C22⋊#88322—22no—
117#13133124—21no—
118#13133124—21no—
119#13133124—21no—
120#13133124—21no—
121#13133124—21no—
122C22×Q8×C53C2311yescharacteristic
123C23×C4×C53C2311yescharacteristic
124C22×C8×C53C2311yescharacteristic
125C22×C8×C53C2311yescharacteristic
126C22⋊Q8×C53—22no—
127C22⋊Q8×C53Q814yescomplemented
128#117×C53—22no—
129#117×C53—22no—
130C2×C4⋊C4×C53—21no—
131C2×C4⋊C4×C53C2311yescharacteristic
132C2×C4⋊C4×C53C2311yescharacteristic
133C2×C4⋊C4×C53C2311yescharacteristic
134C2×C22⋊C4×C53—21no—
135C2×C22⋊C4×C53—21no—
136C2×C42×C53—21no—
137#103×C53—22no—
138#103×C53—22no—
139C22⋊C8×C53—22no—
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×Q812yes—
157C22×C4×C53#4112yes—
158C22×C4×C53C2411yescharacteristic
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53—24no—
162C22×Dic53—21no—
163C22×Dic53—21no—
164C22×Dic53—81no—
165C22×Dic53—81no—
166C22×Dic53C2×D411yescharacteristic
167C22×Dic53C2×D411yescharacteristic
168C22×Dic53C2×D411yescharacteristic
169C22×Dic53C2×D411yescharacteristic
170C2×Dic106—24no—
171C2×Dic106—24no—
172C2×Dic106—81no—
173C2×Dic106—81no—
174C2×Dic106—81no—
175C2×Dic106—81no—
176C2×C8×C53—21no—
177C2×C8×C53—21no—
178C2×C8×C53—21no—
179C2×C8×C53—21no—
180C2×C8×C53—21no—
181C2×C8×C53—21no—
182C4⋊C4×C53—24no—
183C4⋊C4×C53—21no—
184C4⋊C4×C53—21no—
185C4⋊C4×C53—21no—
186C4⋊C4×C53C2×D412yes—
187C4⋊C4×C53C2×D412yes—
188C4⋊C4×C53C2×Q812yes—
189C22⋊C4×C53—24no—
190C22⋊C4×C53—24no—
191C42×C53—24no—
192C22⋊Dic53—22no—
193C22⋊Dic53—22no—
194C22⋊Dic53—81no—
195C4⋊Dic53—24no—
196C4⋊Dic53—24no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—24no—
200C4⋊Dic53—22no—
201C4⋊Dic53—22no—
202C4⋊Dic53—21no—
203C4⋊Dic53—21no—
204C4⋊Dic53—21no—
205C4⋊Dic53—21no—
206C4⋊Dic53—42no—
207C4⋊Dic53—81no—
208C4⋊Dic53C2×D412yes—
209C4⋊Dic53C2×D412yes—
210#11719919—24no—
211#11719919—24no—
212#11719919—24no—
213#11719919—24no—
214C4×Dic53—21no—
215C4×Dic53—21no—
216C4×Dic53—42no—
217C4×Dic53—81no—
218C4×Dic53C2×D412yes—
219C4×Dic53C2×D412yes—
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—22no—
241C2×C4×C53—22no—
242C2×C4×C53—22no—
243C2×C4×C53—22no—
244C2×C4×C53—22no—
245C2×C4×C53—22no—
246C2×C4×C53—22no—
247C2×C4×C53—22no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53C22⋊Q812yes—
257C2×C4×C53C22×D411yescharacteristic
258C2×C4×C53C22×D411yescharacteristic
259C2×C4×C53C22×Q811yescharacteristic
260C2×C4×C53C2×#4111yescharacteristic
261C2×C4×C532−1+411yescharacteristic
262C2×C4×C532−1+411yescharacteristic
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—21no—
268C2×Dic53—22no—
269C2×Dic53—22no—
270C2×Dic53—21no—
271C2×Dic53—21no—
272C2×Dic53—21no—
273C2×Dic53—21no—
274C2×Dic53—21no—
275C2×Dic53—41no—
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—84no—
286C2×Dic53#12012yes—
287C2×Dic53#12012yes—
288Dic106—84no—
289Dic106—84no—
290C8×C53—41no—
291C8×C53—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—42no—
311C4×C53—42no—
312C4×C53—42no—
313C4×C53—42no—
314C4×C53#54811yescharacteristic
315C4×C53#54811yescharacteristic
316C4×C53D4×Q811yescharacteristic
317C4×C53D4×Q811yescharacteristic
318Dic53—22no—
319Dic53—22no—
320Dic53—41no—
321Dic53—41no—
322Dic53—42no—
323Dic53—42no—
324Dic53—81no—
325Dic53—81no—
326C2×#544—531no—
327C2×#479—531no—
328C2×#477—531no—
329#2323—531no—
330#2059—531no—
331#2057—531no—
332#2026—531no—
333#2024—531no—
334#1989—531no—
335#1988—531no—
336#1873D5311yescharacteristic
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×Q8—531no—
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×#103C2×D5311yescharacteristic
375C2×#103C2×D5311yescharacteristic
376C2×C22⋊C8—531no—
377C2×C22⋊C8C2×D5311yescharacteristic
378#391—531no—
379#389—531no—
380#389—1061no—
381#388—1061no—
382#388—1061no—
383#387—1061no—
384#380—531no—
385C4×C4⋊C4—531no—
386#340—531no—
387#340—531no—
388#340—531no—
389#340—531no—
390#340—531no—
391#340—531no—
392#340C2×D5311yescharacteristic
393#340C2×D5311yescharacteristic
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×C8C22×D5311yescharacteristic
405C22×C8C22×D5311yescharacteristic
406C4⋊Q8—534no—
407C4⋊Q8—1068no—
408C4⋊Q8—1062no—
409C4⋊Q8—1062no—
410C4⋊Q8—1062no—
411#125—534no—
412#125—1068no—
413#125—1062no—
414#125—1062no—
415#125—1062no—
416C22⋊Q8—22no—
417C22⋊Q8—1068no—
418C22⋊Q8—1068no—
419C22⋊Q8Dic10614yescomplemented
420C4×Q8—1062no—
421C4×Q8—1062no—
422#117—22no—
423#117—22no—
424#117—1062no—
425#117—1062no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—531no—
428C2×C4⋊C4—531no—
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⋊C4—1061no—
443C2×C4⋊C4—1061no—
444C2×C4⋊C4C22×D5311yescharacteristic
445C2×C4⋊C4C22×D5311yescharacteristic
446C2×C4⋊C4C22×D5311yescharacteristic
447C2×C22⋊C4—21no—
448C2×C22⋊C4—21no—
449C2×C22⋊C4—531no—
450C2×C22⋊C4—531no—
451C2×C22⋊C4—1061no—
452C2×C42—21no—
453C2×C42—531no—
454C2×C42—531no—
455C2×C42—1061no—
456C2×C42—1061no—
457#107—1062no—
458#107—1062no—
459#107—1062no—
460#106—1062no—
461C4⋊C8—1062no—
462C4⋊C8—1062no—
463#103—22no—
464#103—22no—
465#103—1062no—
466#103—1062no—
467C22⋊C8—22no—
468C22⋊C8—1062no—
469#95—1061no—
470#95—1061no—
471#95—1061no—
472#95—1061no—
473#95—1061no—
474C24D4×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—531no—
496C22×C4—531no—
497C22×C4—531no—
498C22×C4—531no—
499C22×C4—1064no—
500C22×C4—1061no—
501C22×C4—1061no—
502C22×C4—1061no—
503C22×C4—1061no—
504C22×C4—4241no—
505C22×C4—4241no—
506C22×C4C2×Dic10612yes—
507C22×C4D4×D5311yescharacteristic
508C22×C4#1171994712yes—
509C22×C4C23×D5311yescharacteristic
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—1061no—
517C2×C8—1061no—
518C2×C8—1061no—
519C2×C8—1061no—
520C2×C8—1061no—
521C2×C8—1061no—
522C4⋊C4—24no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—532no—
527C4⋊C4—532no—
528C4⋊C4—1062no—
529C4⋊C4—1062no—
530C4⋊C4—1064no—
531C4⋊C4—1064no—
532C4⋊C4—1064no—
533C4⋊C4—1064no—
534C4⋊C4—1064no—
535C4⋊C4—1064no—
536C4⋊C4—1064no—
537C4⋊C4—1061no—
538C4⋊C4—1061no—
539C4⋊C4—1061no—
540C4⋊C4—1061no—
541C4⋊C4—1061no—
542C4⋊C4—1061no—
543C4⋊C4—4241no—
544C4⋊C4—2122no—
545C4⋊C4C2×Dic10612yes—
546C4⋊C4D4×D5312yes—
547C4⋊C4D4×D5312yes—
548C22⋊C4—24no—
549C22⋊C4—24no—
550C22⋊C4—1062no—
551C22⋊C4—1062no—
552C22⋊C4—4241no—
553C42—24no—
554C42—532no—
555C42—532no—
556C42—1061no—
557C42—1061no—
558C42—4241no—
559C42—2122no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23#1313319412yes—
567C23C2×D4×D5311yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—41no—
571Q8—41no—
572Q8—4244no—
573Q8—4244no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—532no—
593C2×C4—532no—
594C2×C4—1061no—
595C2×C4—1061no—
596C2×C4—1061no—
597C2×C4—1061no—
598C2×C4—1061no—
599C2×C4—1061no—
600C2×C4—1062no—
601C2×C4—1062no—
602C2×C4—1061no—
603C2×C4—1061no—
604C2×C4—1061no—
605C2×C4—1061no—
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—2121no—
616C2×C4—2121no—
617C2×C4#1313319412yes—
618C2×C4C2×D4×D5311yescharacteristic
619C2×C4C2×D4×D5311yescharacteristic
620C2×C4C22×Dic10611yescharacteristic
621C2×C4C2×#1171994711yescharacteristic
622C2×C4#1313326411yescharacteristic
623C2×C4#1313326411yescharacteristic
624C8—41no—
625C8—41no—
626C8—2121no—
627C8—2121no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22#120×D5311yescharacteristic
636C22C2×#1313319411yescharacteristic
637C22#1681119511yescharacteristic
638C22#1681142511yescharacteristic
639C22#1681143211yescharacteristic
640C22#1681143711yescharacteristic
641C22#1681143911yescharacteristic
642C4—22no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—1062no—
648C4—1062no—
649C4—4241no—
650C4—4241no—
651C4—2121no—
652C4—2121no—
653C4—2122no—
654C4—2122no—
655C4#1681127911yescharacteristic
656C4#1681127911yescharacteristic
657C4D4×Dic10611yescharacteristic
658C4D4×Dic10611yescharacteristic
659C2—42no—
660C2#2027807711yescharacteristic
661C2#2028400711yescharacteristic
662C2#2028402311yescharacteristic
663C2#2028455911yescharacteristic
664C2#2028456411yescharacteristic
665C2#2028598111yescharacteristic
666C2#2028600411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1873C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1873D53
hypocenterC53#24311
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

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

NH ≅ G/Ntype
2. C2×#16810932C2semidirect
3. C2×#16810916C2semidirect
5. #20276056C2semidirect
6. #20276042C2semidirect
7. #20275864C2semidirect
8. #20275854C2semidirect
9. #20275671C2semidirect
10. #20275665C2semidirect
127. 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×#168109322
C2×#168109162
#202773992
#202760562
#202760422
#202758642
#202758542
#202756712
#202756652
#202750232
#202749142
#202749052
#202746972
C53×#18732
#2431153