Finite Group Databasebeta

← C2×#20274950 #27384956

Subgroups — C2×#20274950 #27384956

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×#16811106C211yesmaximal, characteristic
2C2×#16810665C211yesmaximal, characteristic, complemented
3C2×#16810624C211yesmaximal, characteristic
4C2×#16810489C211yesmaximal, characteristic, complemented
5#20274950C218yesmaximal, direct product factor
6C2×#16810391C211yesmaximal, characteristic, complemented
7C2×#16810383C211yesmaximal, characteristic, complemented
8C22×#104×C53C211yesmaximal, characteristic
9C22×#11719917C2211yescharacteristic
10#16811106—22no—
11#16811106—22no—
12#16811106—22no—
13#16811106C2214yes—
14C2×Q8×Dic53—21no—
15C2×D4×Dic53—22no—
16C2×D4×Dic53—21no—
17C2×#13133176—21no—
18C2×#13133176C2211yescharacteristic
19C22×C4⋊Dic53—21no—
20C22×C4⋊Dic53C2211yescharacteristic
21C22×C4⋊Dic53C2211yescharacteristic
22#16810665—22no—
23#16810665C2214yescomplemented
24C2×#13133151—21no—
25C2×#13133149—21no—
26#16810624—22no—
27#16810624C2214yes—
28C2×#13133141—21no—
29C2×#13133138—21no—
30#16810489—22no—
31#16810489C2214yescomplemented
32C2×#13133124—22no—
33#16810391C2214yescomplemented
34#16810383C2214yescomplemented
35C22×#41×C53C2211yescharacteristic
36C22×M16×C53C2211yescharacteristic
37C22×C42×C53C2211yescharacteristic
38C2×#104×C53C418yescomplemented
39C2×#104×C53C2214yes—
40C23×Dic53—22no—
41C23×Dic53—21no—
42C23×Dic53D412yes—
43C2×#11719917—21no—
44C2×#11719917—22no—
45C2×#11719917—22no—
46C2×#11719917C2312yes—
47C22×#88322—21no—
48Q8×Dic53—24no—
49Q8×Dic53—24no—
50D4×Dic53—24no—
51D4×Dic53—28no—
52D4×Dic53—24no—
53D4×Dic53—28no—
54#13133176—24no—
55#13133176—24no—
56#13133176—21no—
57#13133176—21no—
58#13133176D412yes—
59#13133176C2312yes—
60C2×C4⋊Dic53—22no—
61C2×C4⋊Dic53—24no—
62C2×C4⋊Dic53—24no—
63C2×C4⋊Dic53—21no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—21no—
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⋊Dic53D412yes—
75C2×C4⋊Dic53C2312yes—
76C2×C4⋊Dic53C2312yes—
77C2×C22⋊Dic53—22no—
78C2×C22⋊Dic53—21no—
79C2×C4×Dic53—22no—
80C2×C4×Dic53—21no—
81#13133151—24no—
82#13133149—24no—
83#13133141—24no—
84#13133138—24no—
85#13133124—28no—
86C2×#41×C53—21no—
87C2×#41×C53—21no—
88C2×#41×C53—22no—
89C2×#41×C53—22no—
90C2×#41×C53C2×C414yes—
91C2×#41×C53D412yes—
92C2×#41×C53Q814yes—
93C2×#41×C53C2312yes—
94C22×Q8×C53D411yescharacteristic
95C22×D4×C53—21no—
96C22×D4×C53D411yescharacteristic
97C23×C4×C53—21no—
98C23×C4×C53—21no—
99C23×C4×C53C2311yescharacteristic
100C2×M16×C53—21no—
101C2×M16×C53—22no—
102C2×M16×C53C2×C414yescomplemented
103C2×M16×C53C2312yes—
104C22×C8×C53—21no—
105C2×C42×C53—21no—
106C2×C42×C53—22no—
107C2×C42×C53C2×C414yescomplemented
108C2×C42×C53C2312yes—
109#104×C53C2×C4116yescomplemented
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—24no—
140C22×C4×C53—22no—
141C22×C4×C53—42no—
142C22×C4×C53—42no—
143C22×C4×C53C22×C412yes—
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×Q812yes—
150C22×C4×C53#4112yes—
151C22×C4×C53#4112yes—
152C22×C4×C53C2411yescharacteristic
153C22×Dic53—21no—
154C22×Dic53—22no—
155C22×Dic53—21no—
156C22×Dic53—28no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—44no—
160C22×Dic53—44no—
161C22×Dic53—41no—
162C22×Dic53—41no—
163C22×Dic53—84no—
164C22×Dic53C2×D414yes—
165M16×C53—24no—
166M16×C53—22no—
167M16×C53C22×C414yescomplemented
168C2×C8×C53—24no—
169C2×C8×C53—22no—
170C42×C53—24no—
171C42×C53—22no—
172C42×C53C22×C414yescomplemented
173C22⋊Dic53—28no—
174C22⋊Dic53—22no—
175C22⋊Dic53—22no—
176C4⋊Dic53—28no—
177C4⋊Dic53—21no—
178C4⋊Dic53—22no—
179C4⋊Dic53—22no—
180C4⋊Dic53—21no—
181C4⋊Dic53—22no—
182C4⋊Dic53—22no—
183C4⋊Dic53—22no—
184C4⋊Dic53—22no—
185C4⋊Dic53—41no—
186C4⋊Dic53—41no—
187C4⋊Dic53C2×D412yes—
188C4⋊Dic53C2×D412yes—
189C4×Dic53—28no—
190C4×Dic53—22no—
191C4×Dic53—22no—
192#11719917—22no—
193#11719917—22no—
194#11719917—42no—
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—24no—
223C2×C4×C53—48no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53C2×C4⋊C412yes—
227C2×C4×C53C2×C4⋊C412yes—
228C2×C4×C53C4×D412yes—
229C2×C4×C53C4×D412yes—
230C2×C4×C53C4×D412yes—
231C2×C4×C53#12011yescharacteristic
232C2×C4×C53#12011yescharacteristic
233C2×C4×C53C22⋊Q812yes—
234C2×C4×C53C22⋊Q812yes—
235C2×C4×C53C22⋊Q812yes—
236C2×C4×C53#12312yes—
237C2×C4×C53#12312yes—
238C2×C4×C53C23×C411yescharacteristic
239C2×C4×C53C22×D411yescharacteristic
240C2×C4×C53C22×D411yescharacteristic
241C2×C4×C53C22×Q811yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×Dic53—22no—
245C2×Dic53—24no—
246C2×Dic53—22no—
247C2×Dic53—42no—
248C2×Dic53—44no—
249C2×Dic53—42no—
250C2×Dic53—44no—
251C2×Dic53—44no—
252C2×Dic53—42no—
253C2×Dic53—82no—
254C2×Dic53—88no—
255C2×Dic53—82no—
256C2×Dic53—82no—
257C8×C53—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#38512yes—
280C4×C53#38512yes—
281C4×C53C22×C4⋊C411yescharacteristic
282C4×C53C2×C4×D411yescharacteristic
283C4×C53C2×C22⋊Q811yescharacteristic
284C4×C53C2×#12311yescharacteristic
285Dic53—44no—
286Dic53—42no—
287Dic53—42no—
288Dic53—82no—
289C2×#520—531no—
290C2×#428—531no—
291C22×#104D5311yescharacteristic
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×M16C2×D5311yescharacteristic
308#520—534no—
309#520—1062no—
310#520—1062no—
311#520—1062no—
312C2×C4×Q8—1061no—
313C2×C4×D4—1062no—
314C2×C4×D4—1061no—
315C2×#117—531no—
316C2×#117—1061no—
317C22×C4⋊C4—531no—
318C22×C4⋊C4—531no—
319C22×C4⋊C4—1061no—
320C22×C42C2×D5311yescharacteristic
321#428—534no—
322#428—1062no—
323C2×#107—1061no—
324C2×#106—1061no—
325C2×#104Dic5318yescomplemented
326C2×#104C2×D5314yes—
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—21no—
338C2×#41—21no—
339C2×#41—22no—
340C2×#41—22no—
341C2×#41Dic10614yes—
342C2×#41D21212yes—
343C2×#41C2×Dic5314yes—
344C2×#41C22×D5312yes—
345C22×Q8D21211yescharacteristic
346C22×D4—21no—
347C22×D4D21211yescharacteristic
348C23×C4—21no—
349C23×C4—21no—
350C23×C4—532no—
351C23×C4—1062no—
352C23×C4—1061no—
353C23×C4C22×D5311yescharacteristic
354C2×M16—21no—
355C2×M16—22no—
356C2×M16—532no—
357C2×M16—1061no—
358C2×M16—1062no—
359C2×M16—1062no—
360C2×M16C2×Dic5314yescomplemented
361C2×M16C22×D5312yes—
362C22×C8—21no—
363C22×C8—1061no—
364C4×Q8—1064no—
365C4×Q8—1064no—
366C4×D4—1068no—
367C4×D4—1068no—
368C4×D4—1064no—
369C4×D4—1064no—
370#117—532no—
371#117—532no—
372#117—1064no—
373#117—1064no—
374#117—1061no—
375#117—1061no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—532no—
379C2×C4⋊C4—532no—
380C2×C4⋊C4—532no—
381C2×C4⋊C4—1064no—
382C2×C4⋊C4—1062no—
383C2×C4⋊C4—1064no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4—1061no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4—1061no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—1061no—
393C2×C22⋊C4—1062no—
394C2×C22⋊C4—1061no—
395C2×C42—21no—
396C2×C42—22no—
397C2×C42—1062no—
398C2×C42—1061no—
399C2×C42C2×Dic5314yescomplemented
400C2×C42C22×D5312yes—
401#107—1064no—
402#106—1064no—
403#104C2×Dic53116yescomplemented
404#103—1064no—
405#102—1064no—
406#95—1068no—
407C24—21no—
408C24D4×D5311yescharacteristic
409#41—24no—
410#41—24no—
411#41—24no—
412#41—24no—
413#41—24no—
414#41—22no—
415#41—22no—
416#41C4⋊Dic5318yes—
417#41C2×Dic10614yes—
418#41C22×Dic5314yes—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8C4⋊Dic5314yes—
422C2×Q8C2×D21212yes—
423C2×D4—24no—
424C2×D4—22no—
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4C4⋊Dic5314yes—
430C2×D4C2×D21212yes—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—534no—
441C22×C4—1068no—
442C22×C4—1062no—
443C22×C4—1061no—
444C22×C4—1061no—
445C22×C4—1062no—
446C22×C4—1062no—
447C22×C4—4244no—
448C22×C4—2124no—
449C22×C4—2121no—
450C22×C4—2121no—
451C22×C4—2124no—
452C22×C4C2×Dic10612yes—
453C22×C4C2×D21211yescharacteristic
454C22×C4C2×D21211yescharacteristic
455C22×C4D4×D5311yescharacteristic
456C22×C4D4×D5311yescharacteristic
457C22×C4D4×D5311yescharacteristic
458C22×C4#1171994712yes—
459C22×C4#1171994712yes—
460C22×C4C22×Dic5312yes—
461C22×C4C23×D5311yescharacteristic
462M16—24no—
463M16—22no—
464M16—1062no—
465M16—1062no—
466M16—2122no—
467M16C22×Dic5314yescomplemented
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—1062no—
471C2×C8—2122no—
472C4⋊C4—532no—
473C4⋊C4—532no—
474C4⋊C4—1068no—
475C4⋊C4—1061no—
476C4⋊C4—1062no—
477C4⋊C4—1062no—
478C4⋊C4—1061no—
479C4⋊C4—1062no—
480C4⋊C4—1062no—
481C4⋊C4—1062no—
482C4⋊C4—1062no—
483C4⋊C4—2121no—
484C4⋊C4—2121no—
485C22⋊C4—1068no—
486C22⋊C4—1062no—
487C22⋊C4—1062no—
488C42—24no—
489C42—22no—
490C42—1068no—
491C42—1062no—
492C42—1062no—
493C42C22×Dic5314yescomplemented
494C23—22no—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—42no—
499C23—42no—
500C23D4×Dic5312yes—
501C23#1313318711yescharacteristic
502C23#1313318711yescharacteristic
503C23#1313319412yes—
504C23C2×D4×D5311yescharacteristic
505Q8—24no—
506Q8—22no—
507Q8C2×C4⋊Dic5314yes—
508D4—28no—
509D4—24no—
510D4—24no—
511D4—24no—
512D4—22no—
513D4C2×C4⋊Dic5314yes—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—48no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—1062no—
524C2×C4—1062no—
525C2×C4—1064no—
526C2×C4—4242no—
527C2×C4—4242no—
528C2×C4—4242no—
529C2×C4—4248no—
530C2×C4—2124no—
531C2×C4—2122no—
532C2×C4—2122no—
533C2×C4—2122no—
534C2×C4—2124no—
535C2×C4—2124no—
536C2×C4C2×C4⋊Dic5312yes—
537C2×C4C2×C4⋊Dic5312yes—
538C2×C4D4×Dic5312yes—
539C2×C4D4×Dic5312yes—
540C2×C4D4×Dic5312yes—
541C2×C4#1313318711yescharacteristic
542C2×C4#1313318711yescharacteristic
543C2×C4#1313319412yes—
544C2×C4#1313319412yes—
545C2×C4#1313319412yes—
546C2×C4#1313320112yes—
547C2×C4#1313320112yes—
548C2×C4C23×Dic5311yescharacteristic
549C2×C4C2×D4×D5311yescharacteristic
550C2×C4C22×D21211yescharacteristic
551C2×C4C22×Dic10611yescharacteristic
552C2×C4C2×#1171994711yescharacteristic
553C2×C4C2×#1171994711yescharacteristic
554C8—24no—
555C8—2122no—
556C22—24no—
557C22—24no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—24no—
563C22—48no—
564C22—42no—
565C22—42no—
566C22#1681049812yes—
567C22#1681049812yes—
568C22#1681079712yes—
569C22#1681080212yes—
570C22C2×D4×Dic5311yescharacteristic
571C22C2×#1313318711yescharacteristic
572C22C2×#1313319411yescharacteristic
573C4—24no—
574C4—44no—
575C4—4242no—
576C4—2122no—
577C4—2124no—
578C4—2122no—
579C4#1681049812yes—
580C4#1681049812yes—
581C4C22×C4⋊Dic5311yescharacteristic
582C4C2×D4×Dic5311yescharacteristic
583C4C2×#1313319411yescharacteristic
584C4C2×#1313320111yescharacteristic
585C2—21no—
586C2—22no—
587C2—21no—
588C2—44no—
589C2#2027495014yesdirect product factor
590C2C2×#1681049811yescharacteristic
591C2C2×#1681080211yescharacteristic
592C2C2×#1681079711yescharacteristic
subgroupG/H
centerC23#13133187
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC22×#104×C53C2
socleC23×C53#120
prime radicalC22×#41×C53C22
cosocleC22×C4×C53C24
hypercenterC22×#104D53
hypocenterC53C2×#1860
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#104

Complements

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

NH ≅ G/Ntype
2. C2×#16810665C2semidirect
4. C2×#16810489C2semidirect
5. #20274950C2direct (N × H)
6. C2×#16810391C2semidirect
7. C2×#16810383C2semidirect
23. #16810665C22semidirect
31. #16810489C22semidirect
33. #16810391C22semidirect
34. #16810383C22semidirect
38. C2×#104×C53C4semidirect
102. C2×M16×C53C2×C4semidirect
107. C2×C42×C53C2×C4semidirect
109. #104×C53C2×C4semidirect
167. M16×C53C22×C4semidirect
172. C42×C53C22×C4semidirect
301. C2×C53#1860semidirect
325. C2×#104Dic53semidirect
336. C53C2×#1860semidirect
360. C2×M16C2×Dic53semidirect
399. C2×C42C2×Dic53semidirect
403. #104C2×Dic53semidirect
467. M16C22×Dic53semidirect
493. C42C22×Dic53semidirect
589. C2#20274950direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111062
C2×#168106652
C2×#168106242
C2×#168104892
#202749502
C2×#168103912
C2×#168103832
C22×#104×C532
C2×#186053