Finite Group Databasebeta

← C2×#20274948 #27384954

Subgroups — C2×#20274948 #27384954

Order 13568.

1478 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810666C211yesmaximal, characteristic, complemented
2C22×#13133144C211yesmaximal, characteristic
3C2×#16810624C211yesmaximal, characteristic
4C2×#16810488C211yesmaximal, characteristic, complemented
5#20274948C218yesmaximal, direct product factor
6C2×#16810392C211yesmaximal, characteristic, complemented
7C2×#16810382C211yesmaximal, characteristic, complemented
8C2×C53×#520C211yesmaximal, characteristic
9C22×#11719917C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11C22×C4⋊Dic53C2211yescharacteristic
12C22×C4×Dic53C2211yescharacteristic
13#16810666—22no—
14#16810666C2214yescomplemented
15C2×#13133150—21no—
16C2×#13133148—21no—
17C2×#13133144C418yescomplemented
18C2×#13133144C2214yes—
19#16810624—22no—
20#16810624C2214yes—
21C2×#13133141—21no—
22C2×#13133138—21no—
23#16810488—22no—
24#16810488C2214yescomplemented
25C2×#13133124—22no—
26#16810392C2214yescomplemented
27#16810382C2214yescomplemented
28C22×#41×C53C2211yescharacteristic
29C53×#520—22no—
30C53×#520—22no—
31C53×#520—22no—
32C53×#520C2214yes—
33C2×C4×Q8×C53—21no—
34C2×C4×D4×C53—21no—
35C2×C4×D4×C53—22no—
36C2×#117×C53—21no—
37C2×#117×C53C2211yescharacteristic
38C22×C4⋊C4×C53—21no—
39C22×C4⋊C4×C53C2211yescharacteristic
40C23×Dic53—21no—
41C23×Dic53—21no—
42C2×#11719917—21no—
43C2×#11719917—21no—
44C2×#11719917—22no—
45C2×#11719917—22no—
46C2×#11719917—22no—
47C2×#11719917C2×C414yescomplemented
48C2×#11719917C2312yes—
49C2×#11719917C2312yes—
50C22×#88322—21no—
51C22×#88322—21no—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53—21no—
55C2×C4⋊Dic53—21no—
56C2×C4⋊Dic53—21no—
57C2×C4⋊Dic53C2312yes—
58C2×C4×Dic53—21no—
59C2×C4×Dic53—22no—
60C2×C4×Dic53C2×C414yescomplemented
61C2×C4×Dic53C2312yes—
62#13133150—24no—
63#13133148—24no—
64#13133144C2×C4116yescomplemented
65#13133141—24no—
66#13133138—24no—
67#13133124—28no—
68C2×#41×C53—21no—
69C2×#41×C53—21no—
70C2×#41×C53—22no—
71C2×#41×C53—22no—
72C2×#41×C53C2×C414yes—
73C2×#41×C53D412yes—
74C2×#41×C53Q814yes—
75C2×#41×C53C2312yes—
76C22×Q8×C53D411yescharacteristic
77C22×D4×C53—21no—
78C22×D4×C53D411yescharacteristic
79C23×C4×C53—21no—
80C23×C4×C53—22no—
81C23×C4×C53D412yes—
82C23×C4×C53C2311yescharacteristic
83C4×Q8×C53—24no—
84C4×Q8×C53—24no—
85C4×D4×C53—24no—
86C4×D4×C53—28no—
87C4×D4×C53—24no—
88C4×D4×C53—28no—
89#117×C53—24no—
90#117×C53—24no—
91#117×C53—21no—
92#117×C53—21no—
93#117×C53D412yes—
94#117×C53C2312yes—
95C2×C4⋊C4×C53—22no—
96C2×C4⋊C4×C53—24no—
97C2×C4⋊C4×C53—24no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53D411yescharacteristic
103C2×C4⋊C4×C53D411yescharacteristic
104C2×C4⋊C4×C53D412yes—
105C2×C4⋊C4×C53C2312yes—
106C2×C22⋊C4×C53—22no—
107C2×C22⋊C4×C53—21no—
108C2×C42×C53—22no—
109C2×C42×C53—21no—
110C24×C53—21no—
111C24×C53C2×D411yescharacteristic
112#41×C53—24no—
113#41×C53—24no—
114#41×C53—24no—
115#41×C53—24no—
116#41×C53—24no—
117#41×C53—22no—
118#41×C53—22no—
119#41×C53C4⋊C418yes—
120#41×C53C22×C414yes—
121#41×C53C2×Q814yes—
122C2×Q8×C53—21no—
123C2×Q8×C53—22no—
124C2×Q8×C53C4⋊C414yes—
125C2×Q8×C53C2×D412yes—
126C2×D4×C53—24no—
127C2×D4×C53—22no—
128C2×D4×C53—24no—
129C2×D4×C53—22no—
130C2×D4×C53—21no—
131C2×D4×C53—22no—
132C2×D4×C53C4⋊C414yes—
133C2×D4×C53C2×D412yes—
134C22×C4×C53—21no—
135C22×C4×C53—24no—
136C22×C4×C53—22no—
137C22×C4×C53—24no—
138C22×C4×C53—22no—
139C22×C4×C53—21no—
140C22×C4×C53—22no—
141C22×C4×C53—21no—
142C22×C4×C53—28no—
143C22×C4×C53—22no—
144C22×C4×C53—44no—
145C22×C4×C53—84no—
146C22×C4×C53C22×C412yes—
147C22×C4×C53C2×D414yes—
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×Q812yes—
154C22×C4×C53#4112yes—
155C22×C4×C53#4112yes—
156C22×C4×C53C2411yescharacteristic
157C22×Dic53—24no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—42no—
161C22×Dic53—42no—
162C22×Dic53—44no—
163C22×Dic53—41no—
164C22×Dic53—41no—
165C4⋊C4×C53—28no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—22no—
168C4⋊C4×C53—22no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—22no—
171C4⋊C4×C53—22no—
172C4⋊C4×C53C2×D412yes—
173C4⋊C4×C53C2×D412yes—
174C22⋊C4×C53—28no—
175C22⋊C4×C53—22no—
176C22⋊C4×C53—22no—
177C42×C53—28no—
178C42×C53—22no—
179C42×C53—22no—
180C4⋊Dic53—22no—
181C4⋊Dic53—22no—
182C4⋊Dic53—41no—
183C4⋊Dic53—41no—
184C4×Dic53—24no—
185C4×Dic53—22no—
186C4×Dic53C22×C414yescomplemented
187#11719917—24no—
188#11719917—22no—
189#11719917—22no—
190#11719917—22no—
191#11719917—42no—
192#11719917C22×C414yescomplemented
193C2×#88322—24no—
194C2×#88322—22no—
195C2×#88322—22no—
196C2×#88322—42no—
197C23×C53—22no—
198C23×C53—22no—
199C23×C53—22no—
200C23×C53—24no—
201C23×C53—42no—
202C23×C53—42no—
203C23×C53C4×D412yes—
204C23×C53#12011yescharacteristic
205C23×C53#12011yescharacteristic
206C23×C53C22⋊Q812yes—
207C23×C53C22×D411yescharacteristic
208Q8×C53—22no—
209Q8×C53—24no—
210Q8×C53C2×C4⋊C414yes—
211D4×C53—24no—
212D4×C53—24no—
213D4×C53—28no—
214D4×C53—22no—
215D4×C53—24no—
216D4×C53C2×C4⋊C414yes—
217C2×C4×C53—22no—
218C2×C4×C53—24no—
219C2×C4×C53—24no—
220C2×C4×C53—24no—
221C2×C4×C53—28no—
222C2×C4×C53—22no—
223C2×C4×C53—24no—
224C2×C4×C53—22no—
225C2×C4×C53—42no—
226C2×C4×C53—44no—
227C2×C4×C53—42no—
228C2×C4×C53—44no—
229C2×C4×C53—82no—
230C2×C4×C53—88no—
231C2×C4×C53C2×C4⋊C412yes—
232C2×C4×C53C2×C4⋊C412yes—
233C2×C4×C53C4×D412yes—
234C2×C4×C53C4×D412yes—
235C2×C4×C53C4×D412yes—
236C2×C4×C53#12011yescharacteristic
237C2×C4×C53#12011yescharacteristic
238C2×C4×C53C22⋊Q812yes—
239C2×C4×C53C22⋊Q812yes—
240C2×C4×C53C22⋊Q812yes—
241C2×C4×C53#12312yes—
242C2×C4×C53#12312yes—
243C2×C4×C53C23×C411yescharacteristic
244C2×C4×C53C22×D411yescharacteristic
245C2×C4×C53C22×D411yescharacteristic
246C2×C4×C53C22×Q811yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×Dic53—24no—
250C2×Dic53—48no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—44no—
254C2×Dic53—42no—
255C2×Dic53—82no—
256C2×Dic53—82no—
257#88322—24no—
258#88322—42no—
259C2×#1860—531nomaximal, Sylow
260C22×C53—24no—
261C22×C53—24no—
262C22×C53—21no—
263C22×C53—21no—
264C22×C53—21no—
265C22×C53—21no—
266C22×C53—24no—
267C22×C53—48no—
268C22×C53—42no—
269C22×C53—42no—
270C22×C53#38512yes—
271C22×C53#38512yes—
272C22×C53#45412yes—
273C22×C53#45512yes—
274C22×C53C2×C4×D411yescharacteristic
275C22×C53C2×#12011yescharacteristic
276C22×C53C2×C22⋊Q811yescharacteristic
277C4×C53—24no—
278C4×C53—44no—
279C4×C53—42no—
280C4×C53—42no—
281C4×C53#38512yes—
282C4×C53#38512yes—
283C4×C53C22×C4⋊C411yescharacteristic
284C4×C53C2×C4×D411yescharacteristic
285C4×C53C2×C22⋊Q811yescharacteristic
286C4×C53C2×#12311yescharacteristic
287Dic53—44no—
288Dic53—82no—
289C2×#520D5311yescharacteristic
290C2×#428—531no—
291C22×#104—531no—
292C2×#417—531no—
293C2×#382—531no—
294#1860—538no—
295C2×#343—531no—
296C2×#339—531no—
297C2×C53—21no—
298C2×C53—21no—
299C2×C53—22no—
300C2×C53—44no—
301C2×C53#186014yescomplemented
302C2×C53C2×#38511yescharacteristic
303C2×C53C2×#45511yescharacteristic
304C2×C53C2×#45411yescharacteristic
305C22×#41C2×D5311yescharacteristic
306C22×M16—531no—
307C22×M16—531no—
308#520—22no—
309#520—22no—
310#520—22no—
311#520C2×D5314yes—
312C2×C4×Q8—21no—
313C2×C4×D4—21no—
314C2×C4×D4—22no—
315C2×#117—21no—
316C2×#117C2×D5311yescharacteristic
317C22×C4⋊C4—21no—
318C22×C4⋊C4—531no—
319C22×C4⋊C4C2×D5311yescharacteristic
320C22×C42—531no—
321#428—534no—
322#428—1062no—
323C2×#107—1061no—
324C2×#106—1061no—
325C2×#104—538no—
326C2×#104—534no—
327#417—534no—
328#417—1062no—
329C2×#103—1061no—
330C2×#102—1061no—
331#382—534no—
332#382—1062no—
333C2×#95—1062no—
334#343—534no—
335#339—534no—
336C53C2×#186011yescharacteristic, complemented, Sylow
337C2×#41—22no—
338C2×#41—22no—
339C2×#41—21no—
340C2×#41—21no—
341C2×#41Dic10614yes—
342C2×#41C4×D5314yes—
343C2×#41C22⋊D5312yes—
344C2×#41C22×D5312yes—
345C22×Q8C22⋊D5311yescharacteristic
346C22×D4—21no—
347C22×D4C22⋊D5311yescharacteristic
348C23×C4—21no—
349C23×C4—22no—
350C23×C4—1061no—
351C23×C4—1061no—
352C23×C4C22⋊D5312yes—
353C23×C4C22×D5311yescharacteristic
354C2×M16—534no—
355C2×M16—532no—
356C2×M16—532no—
357C2×M16—1062no—
358C2×M16—1061no—
359C2×M16—1061no—
360C2×M16—1062no—
361C2×M16—1062no—
362C22×C8—1061no—
363C22×C8—1061no—
364C4×Q8—24no—
365C4×Q8—24no—
366C4×D4—28no—
367C4×D4—24no—
368C4×D4—24no—
369C4×D4—28no—
370#117—24no—
371#117—21no—
372#117—21no—
373#117—24no—
374#117C22⋊D5312yes—
375#117C22×D5312yes—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—24no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—24no—
383C2×C4⋊C4—532no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4—1061no—
389C2×C4⋊C4C22⋊D5312yes—
390C2×C4⋊C4C22⋊D5311yescharacteristic
391C2×C4⋊C4C22⋊D5311yescharacteristic
392C2×C4⋊C4C22×D5312yes—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—21no—
395C2×C42—22no—
396C2×C42—21no—
397C2×C42—534no—
398C2×C42—532no—
399C2×C42—1062no—
400C2×C42—1061no—
401#107—1064no—
402#106—1064no—
403#104—5316no—
404#103—1064no—
405#102—1064no—
406#95—1068no—
407C24—21no—
408C24C2×C22⋊D5311yescharacteristic
409#41—24no—
410#41—24no—
411#41—24no—
412#41—24no—
413#41—24no—
414#41—22no—
415#41—22no—
416#41#1171991918yes—
417#41C2×Dic10614yes—
418#41C2×C4×D5314yes—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8#1171991914yes—
422C2×Q8C2×C22⋊D5312yes—
423C2×D4—24no—
424C2×D4—22no—
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4#1171991914yes—
430C2×D4C2×C22⋊D5312yes—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—28no—
440C22×C4—22no—
441C22×C4—44no—
442C22×C4—84no—
443C22×C4—1064no—
444C22×C4—1062no—
445C22×C4—1062no—
446C22×C4—2124no—
447C22×C4—2121no—
448C22×C4—2121no—
449C22×C4—2122no—
450C22×C4—2122no—
451C22×C4C2×Dic10612yes—
452C22×C4C2×C4×D5312yes—
453C22×C4#1171994512yes—
454C22×C4#1171994512yes—
455C22×C4C2×C22⋊D5314yes—
456C22×C4C2×C22⋊D5311yescharacteristic
457C22×C4C2×C22⋊D5311yescharacteristic
458C22×C4C2×C22⋊D5311yescharacteristic
459C22×C4C2×C22⋊D5311yescharacteristic
460C22×C4C2×C22⋊D5311yescharacteristic
461C22×C4C23×D5311yescharacteristic
462M16—534no—
463M16—1064no—
464M16—1062no—
465M16—1062no—
466M16—1062no—
467M16—2122no—
468C2×C8—1064no—
469C2×C8—1062no—
470C2×C8—1062no—
471C2×C8—2122no—
472C4⋊C4—28no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—1062no—
480C4⋊C4—1062no—
481C4⋊C4—2121no—
482C4⋊C4—2121no—
483C4⋊C4C2×C22⋊D5312yes—
484C4⋊C4C2×C22⋊D5312yes—
485C22⋊C4—28no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C42—28no—
489C42—22no—
490C42—22no—
491C42—534no—
492C42—1064no—
493C42—1062no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—42no—
499C23—42no—
500C23C4×C22⋊D5312yes—
501C23#1313318611yescharacteristic
502C23#1313318611yescharacteristic
503C23#1313319612yes—
504C23C22×C22⋊D5311yescharacteristic
505Q8—24no—
506Q8—22no—
507Q8C2×#1171991914yes—
508D4—28no—
509D4—24no—
510D4—24no—
511D4—24no—
512D4—22no—
513D4C2×#1171991914yes—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—82no—
527C2×C4—88no—
528C2×C4—1064no—
529C2×C4—4242no—
530C2×C4—4242no—
531C2×C4—2124no—
532C2×C4—2122no—
533C2×C4—2128no—
534C2×C4—2122no—
535C2×C4—2122no—
536C2×C4C2×#1171991912yes—
537C2×C4C2×#1171991912yes—
538C2×C4C4×C22⋊D5312yes—
539C2×C4C4×C22⋊D5312yes—
540C2×C4C4×C22⋊D5312yes—
541C2×C4#1313318611yescharacteristic
542C2×C4#1313318611yescharacteristic
543C2×C4#1313319612yes—
544C2×C4#1313319612yes—
545C2×C4#1313319612yes—
546C2×C4#1313320012yes—
547C2×C4#1313320012yes—
548C2×C4C22×C4×D5311yescharacteristic
549C2×C4C22×C22⋊D5311yescharacteristic
550C2×C4C22×C22⋊D5311yescharacteristic
551C2×C4C22×Dic10611yescharacteristic
552C2×C4C2×#1171994511yescharacteristic
553C2×C4C2×#1171994511yescharacteristic
554C8—1064no—
555C8—2122no—
556C22—24no—
557C22—24no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—24no—
563C22—48no—
564C22—42no—
565C22—42no—
566C22#1681049612yes—
567C22#1681049612yes—
568C22#1681079612yes—
569C22#1681080112yes—
570C22C2×C4×C22⋊D5311yescharacteristic
571C22C2×#1313318611yescharacteristic
572C22C2×#1313319611yescharacteristic
573C4—24no—
574C4—44no—
575C4—42no—
576C4—42no—
577C4—4242no—
578C4—2124no—
579C4#1681049612yes—
580C4#1681049612yes—
581C4C22×#1171991911yescharacteristic
582C4C2×C4×C22⋊D5311yescharacteristic
583C4C2×#1313319611yescharacteristic
584C4C2×#1313320011yescharacteristic
585C2—21no—
586C2—22no—
587C2—21no—
588C2—44no—
589C2#2027494814yesdirect product factor
590C2C2×#1681049611yescharacteristic
591C2C2×#1681080111yescharacteristic
592C2C2×#1681079611yescharacteristic
subgroupG/H
centerC23#13133186
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#520C2
socleC23×C53#120
prime radicalC22×#41×C53C22
cosocleC22×C4×C53C24
hypercenterC2×#520D53
hypocenterC53C2×#1860
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
1. C2×#16810666C2semidirect
4. C2×#16810488C2semidirect
5. #20274948C2direct (N × H)
6. C2×#16810392C2semidirect
7. C2×#16810382C2semidirect
14. #16810666C22semidirect
17. C2×#13133144C4semidirect
24. #16810488C22semidirect
26. #16810392C22semidirect
27. #16810382C22semidirect
47. C2×#11719917C2×C4semidirect
60. C2×C4×Dic53C2×C4semidirect
64. #13133144C2×C4semidirect
186. C4×Dic53C22×C4semidirect
192. #11719917C22×C4semidirect
301. C2×C53#1860semidirect
336. C53C2×#1860semidirect
589. C2#20274948direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168106662
C22×#131331442
C2×#168106242
C2×#168104882
#202749482
C2×#168103922
C2×#168103822
C2×C53×#5202
C2×#186053