Finite Group Databasebeta

← C2×#20279709 #27555902

Subgroups — C2×#20279709 #27555902

Order 13568.

3526 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133216C211yesmaximal, characteristic, complemented
2C22×#13133203C212yesmaximal
3C22×C4×C22⋊D53C211yesmaximal, characteristic
4#20279709C2116yesmaximal, direct product factor
5C2×#16810552C212yesmaximal, complemented
6C2×#16810551C212yesmaximal, complemented
7C2×#16810527C212yesmaximal, complemented
8C2×#16810483C211yesmaximal, characteristic, complemented
9C2×#16810480C211yesmaximal, characteristic, complemented
10C2×#16810480C211yesmaximal, characteristic, complemented
11C2×C4⋊C4×Dic53C211yesmaximal, characteristic, complemented
12C2×C53×#380C211yesmaximal, characteristic, complemented
13C23×Dic106C2212yes—
14C23×C22⋊D53C2211yescharacteristic
15C23×C4×D53C2211yescharacteristic
16C2×#13133216—28no—
17C2×#13133216C2218yescomplemented
18C2×#13133203—216no—
19C2×#13133203C22116yes—
20C2×C4×C22⋊D53—28no—
21C2×C4×C22⋊D53C2218yes—
22C22×C4⋊Dic53C2211yescharacteristic
23C22×C4⋊Dic53C2211yescharacteristic
24C22×#11719919C2211yescharacteristic
25C22×#11719919C2212yes—
26C22×C22⋊Dic53C2212yes—
27C22×C22⋊Dic53C2211yescharacteristic
28C22×#11719921C2212yes—
29C22×#11719921C2212yes—
30C22×#11719921C2211yescharacteristic
31C22×C4×Dic53C2212yes—
32C22×C4×Dic53C2211yescharacteristic
33#16810552C22116yescomplemented
34#16810551C22116yescomplemented
35#16810527C22116yescomplemented
36#16810483C2218yescomplemented
37#16810480C2218yescomplemented
38#16810480C2218yescomplemented
39C4⋊C4×Dic53C2218yescomplemented
40C2×#13133124C2212yes—
41C2×#13133124C2212yes—
42C2×#13133124C2212yes—
43C2×#13133124C2212yes—
44C24×C4×C53C2211yescharacteristic
45C22×C4⋊C4×C53C2211yescharacteristic
46C22×C4⋊C4×C53C2211yescharacteristic
47C22×C22⋊C4×C53C2212yes—
48C53×#380C2218yescomplemented
49C2×#95×C53C2212yes—
50C24×D53C2311yescharacteristic
51C22×Dic106—28no—
52C22×Dic106—28no—
53C22×Dic106—28no—
54C22×Dic106C2318yes—
55C22×C22⋊D53—24no—
56C22×C22⋊D53—24no—
57C22×C22⋊D53D418yes—
58C22×C22⋊D53C2314yes—
59C23×Dic53C2312yes—
60C23×Dic53C2312yes—
61C23×Dic53C2311yescharacteristic
62C23×Dic53C2311yescharacteristic
63C23×Dic53C2311yescharacteristic
64C22×C4×D53—24no—
65C22×C4×D53—24no—
66C22×C4×D53—24no—
67C22×C4×D53C2314yes—
68#13133216—232no—
69#13133203—264no—
70C4×C22⋊D53—232no—
71C2×C4⋊Dic53—24no—
72C2×C4⋊Dic53—24no—
73C2×C4⋊Dic53C2314yes—
74C2×C4⋊Dic53C2314yes—
75C2×#11719919—24no—
76C2×#11719919—28no—
77C2×#11719919C2314yes—
78C2×#11719919C2318yes—
79C2×C22⋊Dic53—28no—
80C2×C22⋊Dic53—24no—
81C2×C22⋊Dic53C2314yes—
82C2×C22⋊Dic53C2318yes—
83C2×#11719921—28no—
84C2×#11719921—28no—
85C2×#11719921—24no—
86C2×#11719921C2314yes—
87C2×#11719921C2318yes—
88C2×#11719921C2318yes—
89C2×C4×Dic53—28no—
90C2×C4×Dic53D418yes—
91C2×C4×Dic53C2318yes—
92C2×C4×Dic53C2314yes—
93#13133124C2318yes—
94#13133124C2318yes—
95#13133124C2318yes—
96#13133124C2318yes—
97C25×C53C2311yescharacteristic
98C23×C4×C53—24no—
99C23×C4×C53—24no—
100C23×C4×C53—24no—
101C23×C4×C53C2312yes—
102C23×C4×C53C2312yes—
103C23×C4×C53C2311yescharacteristic
104C23×C4×C53C2314yes—
105C23×C4×C53C2311yescharacteristic
106C2×C4⋊C4×C53—24no—
107C2×C4⋊C4×C53—24no—
108C2×C4⋊C4×C53C2314yes—
109C2×C4⋊C4×C53C2314yes—
110C2×C22⋊C4×C53—28no—
111C2×C22⋊C4×C53C2318yes—
112#95×C53C2318yes—
113C24×C53—22no—
114C24×C53—22no—
115C24×C53—22no—
116C24×C53—22no—
117C24×C53—22no—
118C24×C53—22no—
119C24×C53C2×D414yes—
120C24×C53C2412yes—
121C24×C53C2411yescharacteristic
122C23×D53—22no—
123C23×D53—22no—
124C23×D53—22no—
125C23×D53—22no—
126C23×D53—22no—
127C23×D53—22no—
128C23×D53C2×D414yes—
129C23×D53C2412yes—
130C22×C4×C53—24no—
131C22×C4×C53—24no—
132C22×C4×C53—24no—
133C22×C4×C53—22no—
134C22×C4×C53—24no—
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53—24no—
138C22×C4×C53—24no—
139C22×C4×C53—28no—
140C22×C4×C53—28no—
141C22×C4×C53—28no—
142C22×C4×C53—416no—
143C22×C4×C53C2×D414yes—
144C22×C4×C53#4114yes—
145C22×C4×C53#4114yes—
146C22×C4×C53C2412yes—
147C22×C4×C53C2414yes—
148C22×C4×C53C2414yes—
149C22×C4×C53C2412yes—
150C2×C22⋊D53—232no—
151C2×C22⋊D53—28no—
152C2×C22⋊D53—28no—
153C2×C22⋊D53C2×D4116yes—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—24no—
157C22×Dic53—24no—
158C22×Dic53—22no—
159C22×Dic53—24no—
160C22×Dic53—24no—
161C22×Dic53—22no—
162C22×Dic53—22no—
163C22×Dic53—24no—
164C22×Dic53—24no—
165C22×Dic53C2×D414yes—
166C22×Dic53C2×D414yes—
167C22×Dic53C2×D414yes—
168C22×Dic53#4114yes—
169C22×Dic53C2412yes—
170C22×Dic53C2414yes—
171C22×Dic53C2414yes—
172C22×Dic53C2412yes—
173C22×Dic53C2412yes—
174C2×C4×D53—28no—
175C2×C4×D53—28no—
176C2×C4×D53—28no—
177C2×C4×D53—416no—
178C2×Dic106—216no—
179C2×Dic106—216no—
180C2×Dic106—216no—
181C2×Dic106—432no—
182C4⋊C4×C53—28no—
183C4⋊C4×C53—28no—
184C22⋊C4×C53—216no—
185C22⋊Dic53—216no—
186C22⋊Dic53—28no—
187#11719921—216no—
188#11719921—216no—
189#11719921—28no—
190C4⋊Dic53—28no—
191C4⋊Dic53—28no—
192#11719919—28no—
193#11719919—216no—
194C4×Dic53—216no—
195C4×Dic53C2×D4116yes—
196C23×C53—28no—
197C23×C53—28no—
198C23×C53—28no—
199C23×C53—22no—
200C23×C53—22no—
201C23×C53—22no—
202C23×C53—22no—
203C23×C53—22no—
204C23×C53—22no—
205C23×C53—44no—
206C23×C53—44no—
207C23×C53—44no—
208C23×C53—44no—
209C23×C53C22×D414yes—
210C23×C53C22×D412yes—
211C23×C53C22×D412yes—
212C23×C53C2×#4112yes—
213C23×C53C2×#4112yes—
214C23×C53C2×#4112yes—
215C23×C532−1+412yes—
216C23×C532−1+412yes—
217C23×C53C2511yescharacteristic
218C22×D53—28no—
219C22×D53—28no—
220C22×D53—28no—
221C22×D53—22no—
222C22×D53—22no—
223C22×D53—22no—
224C22×D53—22no—
225C22×D53—22no—
226C22×D53—22no—
227C22×D53—44no—
228C22×D53—44no—
229C22×D53—44no—
230C22×D53—44no—
231C22×D53C22×D414yes—
232C2×C4×C53—24no—
233C2×C4×C53—24no—
234C2×C4×C53—22no—
235C2×C4×C53—24no—
236C2×C4×C53—24no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—24no—
240C2×C4×C53—24no—
241C2×C4×C53—48no—
242C2×C4×C53—44no—
243C2×C4×C53—48no—
244C2×C4×C53—416no—
245C2×C4×C53#124116yescomplemented
246C2×C4×C53C22×D414yes—
247C2×C4×C53C2×#4114yes—
248C2×C4×C53C2×#4114yes—
249C22⋊D53—232no—
250C2×Dic53—28no—
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—24no—
254C2×Dic53—24no—
255C2×Dic53—22no—
256C2×Dic53—24no—
257C2×Dic53—24no—
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—24no—
261C2×Dic53—24no—
262C2×Dic53—48no—
263C2×Dic53—44no—
264C2×Dic53—48no—
265C2×Dic53#121116yes—
266C2×Dic53C22×D414yes—
267C2×Dic53C22×D414yes—
268C2×Dic53C22×D414yes—
269C2×Dic53C2×#4114yes—
270C4×D53—416no—
271Dic106—432no—
272C2×#2547—531nomaximal, Sylow
273C22×C53—24no—
274C22×C53—24no—
275C22×C53—24no—
276C22×C53—42no—
277C22×C53—42no—
278C22×C53—42no—
279C22×C53—42no—
280C22×C53—416no—
281C22×C53C2×#12114yes—
282C22×C53C2×#12414yes—
283C22×C53#53614yes—
284C22×C53#54714yes—
285C22×C53#54714yes—
286C22×C53#55614yes—
287C22×C53#55614yes—
288C22×C53C23×D411yescharacteristic
289C22×C53C23×D411yescharacteristic
290C22×C53C22×#4111yescharacteristic
291C22×C53C22×#4111yescharacteristic
292C22×C53C22×#4111yescharacteristic
293C22×C53C2×2−1+411yescharacteristic
294C22×C53C2×2−1+411yescharacteristic
295C2×D53—24no—
296C2×D53—24no—
297C2×D53—24no—
298C2×D53—42no—
299C2×D53—42no—
300C2×D53—42no—
301C2×D53—42no—
302C2×D53—416no—
303C4×C53—44no—
304C4×C53—42no—
305C4×C53—44no—
306C4×C53C2×#12418yescomplemented
307Dic53—24no—
308Dic53—44no—
309Dic53—42no—
310Dic53—44no—
311Dic53C2×#12118yes—
312C22×C4⋊Q8—531no—
313C22×#124—532no—
314C22×C4×D4—531no—
315#2547—5316no—
316C2×#397—532no—
317C2×#397—532no—
318C2×#391—532no—
319C2×#380—531no—
320C2×#380—531no—
321C2×#380—531no—
322C2×#380D5311yescharacteristic, complemented
323C2×C4×C4⋊C4—531no—
324C2×C53—44no—
325C2×C53#254718yescomplemented
326C2×C53C22×#12111yescharacteristic
327C2×C53C22×#12411yescharacteristic
328C2×C53C2×#53611yescharacteristic
329C2×C53C2×#54711yescharacteristic
330C2×C53C2×#54711yescharacteristic
331C2×C53C2×#55611yescharacteristic
332C2×C53C2×#55611yescharacteristic
333D53—44no—
334C23×Q8—532no—
335C23×D4—531no—
336C24×C4—531no—
337C24×C4C2×D5311yescharacteristic
338C2×C4⋊Q8—538no—
339C2×C4⋊Q8—1068no—
340C2×#124—5316no—
341C2×#124—10616no—
342C2×C4×D4—538no—
343C2×C4×D4—1068no—
344C22×C4⋊C4—532no—
345C22×C4⋊C4—531no—
346C22×C4⋊C4—531no—
347C22×C4⋊C4—531no—
348C22×C4⋊C4C2×D5311yescharacteristic
349C22×C4⋊C4C2×D5311yescharacteristic
350C22×C22⋊C4—531no—
351C22×C22⋊C4—532no—
352C22×C22⋊C4—531no—
353C22×C22⋊C4—532no—
354C22×C22⋊C4—532no—
355C22×C22⋊C4C2×D5312yes—
356C22×C42—531no—
357C22×C42—532no—
358#397—5316no—
359#397—5316no—
360#391—5316no—
361#380—538no—
362#380—538no—
363#380—538no—
364#380C2×D5318yescomplemented
365C4×C4⋊C4—538no—
366C2×#95—532no—
367C2×#95—532no—
368C2×#95—532no—
369C2×#95—532no—
370C2×#95C2×D5312yes—
371C53C2×#254711yescharacteristic, complemented, Sylow
372C25—531no—
373C25C22×D5311yescharacteristic
374C22×Q8—538no—
375C22×Q8—1068no—
376C22×Q8—1068no—
377C22×Q8—1068no—
378C22×D4—538no—
379C22×D4—534no—
380C22×D4—1064no—
381C22×D4—1064no—
382C23×C4—24no—
383C23×C4—24no—
384C23×C4—24no—
385C23×C4—531no—
386C23×C4—531no—
387C23×C4—532no—
388C23×C4—532no—
389C23×C4—531no—
390C23×C4—534no—
391C23×C4—1064no—
392C23×C4—1064no—
393C23×C4—1064no—
394C23×C4C22×D5314yes—
395C23×C4C22×D5312yes—
396C23×C4C22×D5312yes—
397C23×C4C22×D5311yescharacteristic
398C23×C4C22×D5311yescharacteristic
399C4⋊Q8—10632no—
400#124—10664no—
401C4×D4—10632no—
402C2×C4⋊C4—24no—
403C2×C4⋊C4—24no—
404C2×C4⋊C4—534no—
405C2×C4⋊C4—538no—
406C2×C4⋊C4—534no—
407C2×C4⋊C4—534no—
408C2×C4⋊C4—1064no—
409C2×C4⋊C4—1068no—
410C2×C4⋊C4—1064no—
411C2×C4⋊C4—1064no—
412C2×C4⋊C4C22×D5314yes—
413C2×C4⋊C4C22×D5314yes—
414C2×C22⋊C4—28no—
415C2×C22⋊C4—534no—
416C2×C22⋊C4—538no—
417C2×C22⋊C4—534no—
418C2×C22⋊C4—538no—
419C2×C22⋊C4—538no—
420C2×C22⋊C4—1064no—
421C2×C22⋊C4—1068no—
422C2×C22⋊C4—1068no—
423C2×C22⋊C4—1064no—
424C2×C22⋊C4—1068no—
425C2×C22⋊C4C22×D5318yes—
426C2×C42—538no—
427C2×C42—534no—
428C2×C42—538no—
429C2×C42—1068no—
430#95—538no—
431#95—538no—
432#95—538no—
433#95—538no—
434#95C22×D5318yes—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—22no—
440C24—22no—
441C24—534no—
442C24—532no—
443C24—1062no—
444C24—1062no—
445C24—1062no—
446C24—1062no—
447C24—1062no—
448C24—1062no—
449C24D4×D5314yes—
450C24C23×D5312yes—
451C24C23×D5311yescharacteristic
452C2×Q8—10616no—
453C2×Q8—10616no—
454C2×Q8—10616no—
455C2×Q8—21232no—
456C2×D4—5316no—
457C2×D4—1068no—
458C2×D4—10632no—
459C2×D4—1068no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—28no—
470C22×C4—28no—
471C22×C4—28no—
472C22×C4—416no—
473C22×C4—534no—
474C22×C4—534no—
475C22×C4—534no—
476C22×C4—532no—
477C22×C4—534no—
478C22×C4—532no—
479C22×C4—532no—
480C22×C4—534no—
481C22×C4—534no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1064no—
485C22×C4—1064no—
486C22×C4—1062no—
487C22×C4—1064no—
488C22×C4—1064no—
489C22×C4—1062no—
490C22×C4—1064no—
491C22×C4—1062no—
492C22×C4—1064no—
493C22×C4—1068no—
494C22×C4—1068no—
495C22×C4—1068no—
496C22×C4—21216no—
497C22×C4#1171994514yes—
498C22×C4D4×D5314yes—
499C22×C4#1171994714yes—
500C22×C4C23×D5312yes—
501C22×C4C23×D5314yes—
502C22×C4C23×D5314yes—
503C22×C4C23×D5312yes—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—1068no—
507C4⋊C4—10616no—
508C4⋊C4—1068no—
509C4⋊C4—1068no—
510C22⋊C4—216no—
511C22⋊C4—1068no—
512C22⋊C4—10616no—
513C22⋊C4—10616no—
514C22⋊C4—1068no—
515C22⋊C4—10616no—
516C42—5316no—
517C42—10616no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—534no—
532C23—1068no—
533C23—1068no—
534C23—1068no—
535C23—1062no—
536C23—1062no—
537C23—1062no—
538C23—1062no—
539C23—1062no—
540C23—1062no—
541C23—2124no—
542C23—2124no—
543C23—2124no—
544C23—2124no—
545C23C2×D4×D5314yes—
546C23C2×D4×D5312yes—
547C23C2×D4×D5312yes—
548C23#41×D5312yes—
549C23C2×#1171994712yes—
550C23C2×#1171994512yes—
551C23#1313326412yes—
552C23#1313326512yes—
553C23C24×D5311yescharacteristic
554Q8—21232no—
555D4—10632no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—48no—
566C2×C4—48no—
567C2×C4—44no—
568C2×C4—416no—
569C2×C4—5316no—
570C2×C4—534no—
571C2×C4—534no—
572C2×C4—534no—
573C2×C4—534no—
574C2×C4—1068no—
575C2×C4—1062no—
576C2×C4—1062no—
577C2×C4—1064no—
578C2×C4—1064no—
579C2×C4—1062no—
580C2×C4—1064no—
581C2×C4—1064no—
582C2×C4—1062no—
583C2×C4—1064no—
584C2×C4—1062no—
585C2×C4—1064no—
586C2×C4—2128no—
587C2×C4—2128no—
588C2×C4—2124no—
589C2×C4—21216no—
590C2×C4#13133203116yescomplemented
591C2×C4C2×D4×D5314yes—
592C2×C4C2×#1171994714yes—
593C2×C4C2×#1171994514yes—
594C22—24no—
595C22—24no—
596C22—24no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—416no—
601C22—42no—
602C22—1064no—
603C22—1064no—
604C22—1064no—
605C22—2122no—
606C22—2122no—
607C22—2122no—
608C22—21216no—
609C22—2122no—
610C22#121×D5314yes—
611C22C2×#1313320314yes—
612C22#1681119614yes—
613C22#1681127314yes—
614C22#1681127414yes—
615C22#1681134114yes—
616C22#1681134414yes—
617C22C22×D4×D5311yescharacteristic
618C22C22×D4×D5311yescharacteristic
619C22C2×#41×D5311yescharacteristic
620C22C22×#1171994711yescharacteristic
621C22C22×#1171994511yescharacteristic
622C22C2×#1313326411yescharacteristic
623C22C2×#1313326511yescharacteristic
624C4—44no—
625C4—44no—
626C4—42no—
627C4—538no—
628C4—1064no—
629C4—2124no—
630C4—2124no—
631C4—2122no—
632C4C2×#1313320318yescomplemented
633C2—44no—
634C2—2124no—
635C2#2027970918yesdirect product factor
636C2C2×#121×D5311yescharacteristic
637C2C22×#1313320311yescharacteristic
638C2C2×#1681119611yescharacteristic
639C2C2×#1681127311yescharacteristic
640C2C2×#1681127411yescharacteristic
641C2C2×#1681134111yescharacteristic
642C2C2×#1681134411yescharacteristic
subgroupG/H
centerC24C23×D53
commutator, cosocleC23×C53C25
FrattiniC23C24×D53
FittingC2×C53×#380C2
socleC24×C53C24
prime radicalC23×C22⋊D53C22
hypercenterC2×#380D53
hypocenterC53C2×#2547
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
1. C22×#13133216C2semidirect
4. #20279709C2direct (N × H)
5. C2×#16810552C2semidirect
6. C2×#16810551C2semidirect
7. C2×#16810527C2semidirect
8. C2×#16810483C2semidirect
9. C2×#16810480C2semidirect
10. C2×#16810480C2semidirect
11. C2×C4⋊C4×Dic53C2semidirect
12. C2×C53×#380C2semidirect
17. C2×#13133216C22semidirect
33. #16810552C22semidirect
34. #16810551C22semidirect
35. #16810527C22semidirect
36. #16810483C22semidirect
37. #16810480C22semidirect
38. #16810480C22semidirect
39. C4⋊C4×Dic53C22semidirect
48. C53×#380C22semidirect
245. C2×C4×C53#124semidirect
306. C4×C53C2×#124semidirect
322. C2×#380D53semidirect
325. C2×C53#2547semidirect
364. #380C2×D53semidirect
371. C53C2×#2547semidirect
590. C2×C4#13133203semidirect
632. C4C2×#13133203semidirect
635. C2#20279709direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131332162
C22×#131332032
C22×C4×C22⋊D532
#202797092
C2×#168105522
C2×#168105512
C2×#168105272
C2×#168104832
C2×#168104802
C2×C4⋊C4×Dic532
C2×C53×#3802
C2×#254753