Finite Group Databasebeta

← C2×#20274951 #27384957

Subgroups — C2×#20274951 #27384957

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×#16811104C211yesmaximal, characteristic
2C2×#16810665C211yesmaximal, characteristic, complemented
3C22×#13133143C211yesmaximal, characteristic
4C2×#16810625C211yesmaximal, characteristic
5C2×#16810488C211yesmaximal, characteristic, complemented
6#20274951C218yesmaximal, direct product factor
7C2×#16810384C211yesmaximal, characteristic, complemented
8C2×C53×#343C211yesmaximal, characteristic, complemented
9C22×#11719945C2211yescharacteristic
10#16811104—22no—
11#16811104—22no—
12#16811104—22no—
13#16811104C2214yes—
14C2×#13133182—21no—
15C2×#13133179—21no—
16C2×C4×C22⋊D53—22no—
17C2×#13133176C2211yescharacteristic
18C2×#13133175—21no—
19C22×C4⋊Dic53C2211yescharacteristic
20C2×C4⋊C4×D53—21no—
21C22×C4×Dic53C2211yescharacteristic
22#16810665—22no—
23#16810665C2214yescomplemented
24C2×#13133151—21no—
25C2×#13133149—21no—
26C2×#13133143C418yescomplemented
27C2×#13133143C2214yes—
28#16810625—22no—
29#16810625C2214yes—
30C2×#13133142—21no—
31C2×#13133139—21no—
32#16810488—22no—
33#16810488C2214yescomplemented
34C2×#13133124—22no—
35#16810384C2214yescomplemented
36C22×M16×C53C2211yescharacteristic
37C22×M16×C53C2211yescharacteristic
38C22×C4⋊C4×C53C2211yescharacteristic
39C53×#343C2214yescomplemented
40C2×#11719945—21no—
41C2×#11719945—21no—
42C2×#11719945—22no—
43C2×#11719945—22no—
44C2×#11719945C2×C414yes—
45C2×#11719945D412yes—
46C2×#11719945Q814yes—
47C2×#11719945C2312yes—
48C22×Dic106D411yescharacteristic
49C22×D212D411yescharacteristic
50C22×C22⋊D53—21no—
51C23×Dic53—21no—
52C23×Dic53—21no—
53C22×C4×D53—21no—
54C22×C4×D53—22no—
55#13133182—24no—
56#13133182—24no—
57#13133179—24no—
58#13133179—24no—
59C4×C22⋊D53—28no—
60C4×C22⋊D53—28no—
61#13133176—21no—
62#13133176—21no—
63#13133176D412yes—
64#13133176C2312yes—
65#13133175—24no—
66#13133175—24no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53D411yescharacteristic
73C2×C4⋊Dic53D411yescharacteristic
74C2×C4⋊Dic53C2312yes—
75C2×#11719919—22no—
76C4⋊C4×D53—24no—
77C4⋊C4×D53—24no—
78C2×C22⋊Dic53—21no—
79C2×#11719921—22no—
80C2×C4×Dic53—22no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—22no—
84C2×C4×Dic53C2×C414yescomplemented
85C2×C4×Dic53C2312yes—
86#13133151—24no—
87#13133149—24no—
88#13133143C2×C4116yescomplemented
89#13133142—24no—
90#13133139—24no—
91#13133124—28no—
92C23×C4×C53D412yes—
93C23×C4×C53C2311yescharacteristic
94C2×M16×C53—21no—
95C2×M16×C53—21no—
96C2×M16×C53—22no—
97C2×M16×C53—22no—
98C2×M16×C53—22no—
99C2×M16×C53C2×C414yescomplemented
100C2×M16×C53C2312yes—
101C2×M16×C53C2312yes—
102C22×C8×C53—21no—
103C22×C8×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53D412yes—
109C2×C4⋊C4×C53C2312yes—
110C24×C53C2×D411yescharacteristic
111C23×D53—21no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—44no—
115C22×C4×C53C22×C412yes—
116C22×C4×C53C2×D414yes—
117C22×C4×C53C2×D411yescharacteristic
118C22×C4×C53C2×D411yescharacteristic
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×Q812yes—
123C22×C4×C53#4112yes—
124C22×C4×C53#4112yes—
125C22×C4×C53C2411yescharacteristic
126C2×C22⋊D53—24no—
127C2×C22⋊D53—22no—
128C2×C22⋊D53—24no—
129C2×C22⋊D53—22no—
130C22×Dic53—21no—
131C22×Dic53—24no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—42no—
137C22×Dic53—42no—
138C22×Dic53—44no—
139C22×Dic53—41no—
140C22×Dic53—41no—
141#11719945—24no—
142#11719945—24no—
143#11719945—24no—
144#11719945—24no—
145#11719945—24no—
146#11719945—22no—
147#11719945—22no—
148#11719945C4⋊C418yes—
149#11719945C22×C414yes—
150#11719945C2×Q814yes—
151C2×D212—21no—
152C2×D212—22no—
153C2×D212C4⋊C414yes—
154C2×D212C2×D412yes—
155C2×C4×D53—24no—
156C2×C4×D53—22no—
157C2×C4×D53—24no—
158C2×C4×D53—22no—
159C2×C4×D53—28no—
160C2×C4×D53—84no—
161C2×Dic106—21no—
162C2×Dic106—22no—
163C2×Dic106C4⋊C414yes—
164C2×Dic106C2×D412yes—
165M16×C53—24no—
166M16×C53—22no—
167M16×C53—22no—
168M16×C53—22no—
169M16×C53—42no—
170M16×C53C22×C414yescomplemented
171C2×C8×C53—24no—
172C2×C8×C53—22no—
173C2×C8×C53—22no—
174C2×C8×C53—42no—
175C4⋊C4×C53—22no—
176C4⋊C4×C53—22no—
177C4⋊C4×C53—22no—
178C4⋊C4×C53—22no—
179C22⋊Dic53—22no—
180C22⋊Dic53—22no—
181#11719921—28no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—22no—
185C4⋊Dic53—22no—
186C4⋊Dic53—41no—
187C4⋊Dic53—41no—
188C4⋊Dic53C2×D412yes—
189C4⋊Dic53C2×D412yes—
190#11719919—28no—
191C4×Dic53—28no—
192C4×Dic53—24no—
193C4×Dic53—22no—
194C4×Dic53—22no—
195C4×Dic53—22no—
196C4×Dic53C22×C414yescomplemented
197C23×C53—22no—
198C23×C53—22no—
199C23×C53C4×D412yes—
200C23×C53#12011yescharacteristic
201C23×C53#12011yescharacteristic
202C23×C53C22⋊Q812yes—
203C23×C53C22×D411yescharacteristic
204C22×D53—22no—
205C22×D53—24no—
206C22×D53—42no—
207C22×D53—42no—
208C2×C4×C53—24no—
209C2×C4×C53—44no—
210C2×C4×C53—44no—
211C2×C4×C53—82no—
212C2×C4×C53C2×C4⋊C412yes—
213C2×C4×C53C2×C4⋊C412yes—
214C2×C4×C53C4×D412yes—
215C2×C4×C53C4×D412yes—
216C2×C4×C53C4×D412yes—
217C2×C4×C53#12011yescharacteristic
218C2×C4×C53#12011yescharacteristic
219C2×C4×C53C22⋊Q812yes—
220C2×C4×C53C22⋊Q812yes—
221C2×C4×C53C22⋊Q812yes—
222C2×C4×C53#12312yes—
223C2×C4×C53#12312yes—
224C2×C4×C53C23×C411yescharacteristic
225C2×C4×C53C22×D411yescharacteristic
226C2×C4×C53C22×D411yescharacteristic
227C2×C4×C53C22×Q811yescharacteristic
228C2×C4×C53C2×#4111yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C22⋊D53—24no—
231C22⋊D53—24no—
232C22⋊D53—28no—
233C2×Dic53—22no—
234C2×Dic53—24no—
235C2×Dic53—24no—
236C2×Dic53—22no—
237C2×Dic53—22no—
238C2×Dic53—48no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—44no—
244C2×Dic53—42no—
245C2×Dic53—82no—
246C2×Dic53—82no—
247D212—22no—
248D212—24no—
249D212C2×C4⋊C414yes—
250C4×D53—24no—
251C4×D53—24no—
252C4×D53—28no—
253C4×D53—88no—
254Dic106—22no—
255Dic106—24no—
256Dic106C2×C4⋊C414yes—
257C8×C53—24no—
258C8×C53—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#38512yes—
267C22×C53#38512yes—
268C22×C53#45412yes—
269C22×C53#45512yes—
270C22×C53C2×C4×D411yescharacteristic
271C22×C53C2×#12011yescharacteristic
272C22×C53C2×C22⋊Q811yescharacteristic
273C2×D53—24no—
274C2×D53—48no—
275C2×D53—42no—
276C2×D53—42no—
277C4×C53—44no—
278C4×C53#38512yes—
279C4×C53#38512yes—
280C4×C53C22×C4⋊C411yescharacteristic
281C4×C53C2×C4×D411yescharacteristic
282C4×C53C2×C22⋊Q811yescharacteristic
283C4×C53C2×#12311yescharacteristic
284Dic53—24no—
285Dic53—44no—
286Dic53—42no—
287Dic53—42no—
288Dic53—82no—
289C2×#520—531no—
290C2×#428—531no—
291C22×#104—531no—
292C2×#417—531no—
293C2×#382—531no—
294#1860—538no—
295C2×#343D5311yescharacteristic, complemented
296C2×#339—531no—
297C2×C53—21no—
298C2×C53—21no—
299C2×C53—22no—
300C2×C53#186014yescomplemented
301C2×C53C2×#38511yescharacteristic
302C2×C53C2×#45511yescharacteristic
303C2×C53C2×#45411yescharacteristic
304D53—44no—
305C22×#41—531no—
306C22×M16C2×D5311yescharacteristic
307C22×M16C2×D5311yescharacteristic
308#520—534no—
309#520—1062no—
310#520—1062no—
311#520—1062no—
312C2×C4×Q8—1061no—
313C2×C4×D4—1061no—
314C2×C4×D4—1062no—
315C2×#117—531no—
316C2×#117—1061no—
317C22×C4⋊C4—531no—
318C22×C4⋊C4—1061no—
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#343C2×D5314yescomplemented
335#339—534no—
336C53C2×#186011yescharacteristic, complemented, Sylow
337C2×#41—534no—
338C2×#41—532no—
339C2×#41—534no—
340C2×#41—532no—
341C2×#41—1061no—
342C2×#41—1062no—
343C2×#41—1061no—
344C2×#41—1062no—
345C22×Q8—531no—
346C22×D4—531no—
347C22×D4—1061no—
348C23×C4—1062no—
349C23×C4—1061no—
350C23×C4—1061no—
351C23×C4—1061no—
352C23×C4D21212yes—
353C23×C4C22×D5311yescharacteristic
354C2×M16—21no—
355C2×M16—22no—
356C2×M16—21no—
357C2×M16—22no—
358C2×M16—22no—
359C2×M16C4×D5314yescomplemented
360C2×M16C22×D5312yes—
361C2×M16C22×D5312yes—
362C22×C8—21no—
363C22×C8—21no—
364C4×Q8—1064no—
365C4×Q8—1064no—
366C4×D4—1064no—
367C4×D4—1064no—
368C4×D4—1068no—
369C4×D4—1068no—
370#117—532no—
371#117—532no—
372#117—1064no—
373#117—1064no—
374#117—1061no—
375#117—1061no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—532no—
383C2×C4⋊C4—1064no—
384C2×C4⋊C4—1064no—
385C2×C4⋊C4—1062no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4—1061no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4—1061no—
391C2×C4⋊C4D21212yes—
392C2×C4⋊C4C22×D5312yes—
393C2×C22⋊C4—1062no—
394C2×C22⋊C4—1061no—
395C2×C42—534no—
396C2×C42—532no—
397C2×C42—1062no—
398C2×C42—1061no—
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—1061no—
408C24C2×D21211yescharacteristic
409#41—538no—
410#41—534no—
411#41—534no—
412#41—1064no—
413#41—1064no—
414#41—1064no—
415#41—1064no—
416#41—1064no—
417#41—1062no—
418#41—1062no—
419C2×Q8—534no—
420C2×Q8—532no—
421C2×Q8—1061no—
422C2×Q8—1062no—
423C2×D4—534no—
424C2×D4—532no—
425C2×D4—1064no—
426C2×D4—1062no—
427C2×D4—1064no—
428C2×D4—1062no—
429C2×D4—1061no—
430C2×D4—1062no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—44no—
434C22×C4—1068no—
435C22×C4—1061no—
436C22×C4—1064no—
437C22×C4—1062no—
438C22×C4—1064no—
439C22×C4—1062no—
440C22×C4—1064no—
441C22×C4—1061no—
442C22×C4—1061no—
443C22×C4—1062no—
444C22×C4—1062no—
445C22×C4—4244no—
446C22×C4—2124no—
447C22×C4—2121no—
448C22×C4—2121no—
449C22×C4—2122no—
450C22×C4—2122no—
451C22×C4C2×C4×D5312yes—
452C22×C4C2×D21214yes—
453C22×C4C2×D21211yescharacteristic
454C22×C4#1171994512yes—
455C22×C4D4×D5311yescharacteristic
456C22×C4D4×D5311yescharacteristic
457C22×C4D4×D5311yescharacteristic
458C22×C4D4×D5311yescharacteristic
459C22×C4#1171994712yes—
460C22×C4Q8×D5312yes—
461C22×C4C23×D5311yescharacteristic
462M16—24no—
463M16—22no—
464M16—22no—
465M16—22no—
466M16—42no—
467M16C2×C4×D5314yescomplemented
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—42no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—532no—
477C4⋊C4—532no—
478C4⋊C4—1068no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1062no—
482C4⋊C4—1062no—
483C4⋊C4—2121no—
484C4⋊C4—2121no—
485C22⋊C4—1068no—
486C22⋊C4—1062no—
487C22⋊C4—1062no—
488C42—534no—
489C42—1068no—
490C42—1064no—
491C42—1062no—
492C42—1062no—
493C42—1062no—
494C23—22no—
495C23—22no—
496C23—1064no—
497C23—1062no—
498C23—2122no—
499C23—2122no—
500C23C4×D21212yes—
501C23#1313318711yescharacteristic
502C23#1313318711yescharacteristic
503C23#1313319712yes—
504C23C22×D21211yescharacteristic
505Q8—534no—
506Q8—1064no—
507Q8—1062no—
508D4—534no—
509D4—1068no—
510D4—1064no—
511D4—1064no—
512D4—1064no—
513D4—1062no—
514C2×C4—24no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—82no—
518C2×C4—1064no—
519C2×C4—1062no—
520C2×C4—1068no—
521C2×C4—1064no—
522C2×C4—1064no—
523C2×C4—1064no—
524C2×C4—1062no—
525C2×C4—1062no—
526C2×C4—4248no—
527C2×C4—4242no—
528C2×C4—4242no—
529C2×C4—2124no—
530C2×C4—2122no—
531C2×C4—2128no—
532C2×C4—2122no—
533C2×C4—2122no—
534C2×C4—2122no—
535C2×C4—2122no—
536C2×C4C4⋊C4×D5312yes—
537C2×C4C4⋊C4×D5312yes—
538C2×C4#1313317712yes—
539C2×C4#1313317712yes—
540C2×C4C4×D21212yes—
541C2×C4#1313318711yescharacteristic
542C2×C4#1313318711yescharacteristic
543C2×C4#1313319512yes—
544C2×C4#1313319512yes—
545C2×C4#1313319712yes—
546C2×C4#1313319912yes—
547C2×C4#1313319912yes—
548C2×C4C22×C4×D5311yescharacteristic
549C2×C4C2×D4×D5311yescharacteristic
550C2×C4C2×D4×D5311yescharacteristic
551C2×C4C2×Q8×D5311yescharacteristic
552C2×C4C2×#1171994711yescharacteristic
553C2×C4C2×#1171994511yescharacteristic
554C8—24no—
555C8—42no—
556C22—24no—
557C22—24no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—1064no—
563C22—2128no—
564C22—2122no—
565C22—2122no—
566C22#1681049712yes—
567C22#1681049712yes—
568C22#1681079812yes—
569C22#1681080312yes—
570C22C2×C4×D21211yescharacteristic
571C22C2×#1313318711yescharacteristic
572C22C2×#1313319711yescharacteristic
573C4—44no—
574C4—1064no—
575C4—4242no—
576C4—2124no—
577C4—2122no—
578C4—2122no—
579C4#1681049712yes—
580C4#1681049712yes—
581C4C2×C4⋊C4×D5311yescharacteristic
582C4C2×#1313317711yescharacteristic
583C4C2×#1313319511yescharacteristic
584C4C2×#1313319911yescharacteristic
585C2—21no—
586C2—22no—
587C2—21no—
588C2—2124no—
589C2#2027495114yesdirect product factor
590C2C2×#1681049711yescharacteristic
591C2C2×#1681080311yescharacteristic
592C2C2×#1681079811yescharacteristic
subgroupG/H
centerC23#13133187
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#343C2
socleC23×C53#120
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenterC2×#343D53
hypocenterC53C2×#1860
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
2. C2×#16810665C2semidirect
5. C2×#16810488C2semidirect
6. #20274951C2direct (N × H)
7. C2×#16810384C2semidirect
8. C2×C53×#343C2semidirect
23. #16810665C22semidirect
26. C2×#13133143C4semidirect
33. #16810488C22semidirect
35. #16810384C22semidirect
39. C53×#343C22semidirect
84. C2×C4×Dic53C2×C4semidirect
88. #13133143C2×C4semidirect
99. C2×M16×C53C2×C4semidirect
170. M16×C53C22×C4semidirect
196. C4×Dic53C22×C4semidirect
295. C2×#343D53semidirect
300. C2×C53#1860semidirect
334. #343C2×D53semidirect
336. C53C2×#1860semidirect
359. C2×M16C4×D53semidirect
467. M16C2×C4×D53semidirect
589. C2#20274951direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111042
C2×#168106652
C22×#131331432
C2×#168106252
C2×#168104882
#202749512
C2×#168103842
C2×C53×#3432
C2×#186053