Finite Group Databasebeta

← C2×#20277698 #27543585

Subgroups — C2×#20277698 #27543585

Order 13568.

5942 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133185C211yesmaximal, characteristic
2C22×C4×C22⋊D53C212yesmaximal, complemented
3C22×C22⋊C4×D53C212yesmaximal, complemented
4#20277698C2116yesmaximal, direct product factor
5C2×#16810526C211yesmaximal, characteristic, complemented
6C2×#16810517C212yesmaximal, complemented
7C2×#16810499C212yesmaximal, complemented
8C2×#16810497C212yesmaximal, complemented
9C2×#16810470C212yesmaximal, complemented
10C2×C53×#386C211yesmaximal, characteristic, complemented
11C25×D53C2211yescharacteristic
12C23×C22⋊D53C2212yes—
13C23×C4×D53C2212yes—
14C23×C4×D53C2212yes—
15C2×#13133185C4116yescomplemented
16C2×#13133185C2218yes—
17C2×C4×C22⋊D53—216no—
18C2×C4×C22⋊D53C22116yescomplemented
19C22×C4⋊Dic53C2211yescharacteristic, complemented
20C22×#11719919C2212yescomplemented
21C22×C22⋊Dic53C2212yes—
22C22×C22⋊Dic53C2211yescharacteristic
23C22×C22⋊Dic53C2212yescomplemented
24C22×#11719921C2212yes—
25C22×#11719921C2212yes—
26C22×#11719921C2212yes—
27C2×C22⋊C4×D53—216no—
28C2×C22⋊C4×D53C22116yescomplemented
29C22×C4×Dic53C2212yescomplemented
30C22×C4×Dic53C2211yescharacteristic, complemented
31#16810526C2218yescomplemented
32#16810517C22116yescomplemented
33#16810499C22116yescomplemented
34#16810497C22116yescomplemented
35#16810470C22116yescomplemented
36C2×#13133124C2212yescomplemented
37C2×#13133124C2212yescomplemented
38C23×D4×C53C2211yescharacteristic
39C24×C4×C53C2211yescharacteristic, complemented
40C22×C22⋊C4×C53C2212yes—
41C22×C22⋊C4×C53C2211yescharacteristic, complemented
42C53×#386C2218yescomplemented
43C2×#95×C53C2212yescomplemented
44C24×D53—28no—
45C24×D53—28no—
46C24×D53—42no—
47C24×D53—42no—
48C24×D53C2×C418yescomplemented
49C24×D53C2314yes—
50C24×D53C2312yes—
51C22×C22⋊D53—28no—
52C22×C22⋊D53—28no—
53C22×C22⋊D53C2×C4116yescomplemented
54C22×C22⋊D53C2318yes—
55C23×Dic53C2312yes—
56C23×Dic53C2312yes—
57C23×Dic53C2312yes—
58C22×C4×D53—28no—
59C22×C4×D53—28no—
60C22×C4×D53—28no—
61C22×C4×D53—28no—
62C22×C4×D53—44no—
63C22×C4×D53—44no—
64C22×C4×D53C2318yes—
65C22×C4×D53C2318yes—
66#13133185C2×C4164yescomplemented
67C4×C22⋊D53—264no—
68C2×C4⋊Dic53—42no—
69C2×C4⋊Dic53C2314yescomplemented
70C2×#11719919—28no—
71C2×#11719919C2318yescomplemented
72C2×C22⋊Dic53—28no—
73C2×C22⋊Dic53—28no—
74C2×C22⋊Dic53C2×C418yescomplemented
75C2×C22⋊Dic53C2318yes—
76C2×C22⋊Dic53C2314yes—
77C2×C22⋊Dic53C2318yescomplemented
78C2×#11719921—28no—
79C2×#11719921C2×C4116yescomplemented
80C2×#11719921D4116yescomplemented
81C2×#11719921C2318yes—
82C2×#11719921C2318yes—
83C2×#11719921C2318yes—
84C22⋊C4×D53—264no—
85C2×C4×Dic53—28no—
86C2×C4×Dic53—42no—
87C2×C4×Dic53C2318yescomplemented
88C2×C4×Dic53C2314yescomplemented
89#13133124C2318yescomplemented
90#13133124C2318yescomplemented
91C25×C53C2311yescharacteristic
92C25×C53C2311yescharacteristic
93C22×D4×C53—42no—
94C22×D4×C53—42no—
95C22×D4×C53C2×C418yescomplemented
96C22×D4×C53C2314yes—
97C23×C4×C53—28no—
98C23×C4×C53—42no—
99C23×C4×C53C2312yes—
100C23×C4×C53C2311yescharacteristic
101C23×C4×C53C2312yes—
102C23×C4×C53C2314yescomplemented
103C2×C22⋊C4×C53—28no—
104C2×C22⋊C4×C53—42no—
105C2×C22⋊C4×C53C2318yes—
106C2×C22⋊C4×C53C2314yescomplemented
107#95×C53C2318yescomplemented
108C24×C53—24no—
109C24×C53—24no—
110C24×C53—24no—
111C24×C53—24no—
112C24×C53—22no—
113C24×C53—22no—
114C24×C53—22no—
115C24×C53—22no—
116C24×C53C22×C414yes—
117C24×C53C22×C414yes—
118C24×C53C2412yes—
119C24×C53C2412yes—
120C24×C53C2411yescharacteristic
121C23×D53—24no—
122C23×D53—24no—
123C23×D53—24no—
124C23×D53—24no—
125C23×D53—264no—
126C23×D53—216no—
127C23×D53—216no—
128C23×D53—416no—
129C23×D53—44no—
130C23×D53—44no—
131C23×D53—416no—
132C23×D53—432no—
133C23×D53—416no—
134C23×D53C22×C418yes—
135C23×D53C22×C4116yescomplemented
136C23×D53C2×D418yes—
137C23×D53#4118yes—
138C23×D53C2414yes—
139C2×D4×C53—416no—
140C2×D4×C53—44no—
141C2×D4×C53—44no—
142C2×D4×C53C22×C4116yescomplemented
143C22×C4×C53—24no—
144C22×C4×C53—24no—
145C22×C4×C53—22no—
146C22×C4×C53—22no—
147C22×C4×C53—24no—
148C22×C4×C53—24no—
149C22×C4×C53—24no—
150C22×C4×C53—216no—
151C22×C4×C53—44no—
152C22×C4×C53—416no—
153C22×C4×C53C22×C414yes—
154C22×C4×C53C2×D418yes—
155C22×C4×C53C2414yes—
156C22×C4×C53C2412yes—
157C22×C4×C53C2414yes—
158C2×C22⋊D53—264no—
159C2×C22⋊D53—216no—
160C2×C22⋊D53—216no—
161C2×C22⋊D53C22×C4132yescomplemented
162C22×Dic53—24no—
163C22×Dic53—24no—
164C22×Dic53—24no—
165C22×Dic53—24no—
166C22×Dic53—24no—
167C22×Dic53—24no—
168C22×Dic53—24no—
169C22×Dic53C22×C418yes—
170C22×Dic53C2×D418yes—
171C22×Dic53C2414yes—
172C22×Dic53C2414yes—
173C22×Dic53C2414yes—
174C2×C4×D53—216no—
175C2×C4×D53—216no—
176C2×C4×D53—216no—
177C2×C4×D53—216no—
178C2×C4×D53—48no—
179C2×C4×D53—48no—
180C2×C4×D53—432no—
181C2×C4×D53—432no—
182C22⋊C4×C53—216no—
183C22⋊C4×C53—44no—
184C22⋊Dic53—216no—
185C22⋊Dic53—216no—
186C22⋊Dic53C22×C4116yescomplemented
187#11719921—216no—
188#11719921C22×C4132yescomplemented
189#11719921C2×D4132yescomplemented
190C4⋊Dic53—44no—
191#11719919—216no—
192C4×Dic53—216no—
193C4×Dic53—44no—
194C23×C53—216no—
195C23×C53—216no—
196C23×C53—28no—
197C23×C53—28no—
198C23×C53—24no—
199C23×C53—24no—
200C23×C53—24no—
201C23×C53—24no—
202C23×C53—22no—
203C23×C53—22no—
204C23×C53—22no—
205C23×C53—22no—
206C23×C53—44no—
207C23×C53—44no—
208C23×C53—48no—
209C23×C53—48no—
210C23×C53—44no—
211C23×C53—44no—
212C23×C53C23×C414yes—
213C23×C53C23×C414yes—
214C23×C53C23×C412yes—
215C23×C53C22×D414yes—
216C23×C53C2×#4114yes—
217C23×C53H(5,𝔽2)12yes—
218C23×C53H(5,𝔽2)12yes—
219C23×C53C2511yescharacteristic
220C22×D53—216no—
221C22×D53—216no—
222C22×D53—216no—
223C22×D53—216no—
224C22×D53—216no—
225C22×D53—216no—
226C22×D53—24no—
227C22×D53—24no—
228C22×D53—24no—
229C22×D53—24no—
230C22×D53—264no—
231C22×D53—416no—
232C22×D53—4128no—
233C22×D53—416no—
234C22×D53—432no—
235C22×D53—416no—
236C22×D53C4×D4132yescomplemented
237C22×D53C23×C418yes—
238C22×D53C22×D418yes—
239C22×D53C2×#4118yes—
240D4×C53—416no—
241C2×C4×C53—28no—
242C2×C4×C53—216no—
243C2×C4×C53—24no—
244C2×C4×C53—24no—
245C2×C4×C53—22no—
246C2×C4×C53—22no—
247C2×C4×C53—24no—
248C2×C4×C53—24no—
249C2×C4×C53—24no—
250C2×C4×C53—416no—
251C2×C4×C53—48no—
252C2×C4×C53C23×C414yes—
253C2×C4×C53C22×D418yes—
254C22⋊D53—264no—
255C2×Dic53—216no—
256C2×Dic53—216no—
257C2×Dic53—24no—
258C2×Dic53—24no—
259C2×Dic53—24no—
260C2×Dic53—24no—
261C2×Dic53—24no—
262C2×Dic53—24no—
263C2×Dic53—24no—
264C2×Dic53—48no—
265C2×Dic53C23×C418yes—
266C2×Dic53C22×D418yes—
267C4×D53—432no—
268C4×D53—432no—
269C2×#2358—531nomaximal, Sylow
270C22×C53—28no—
271C22×C53—28no—
272C22×C53—24no—
273C22×C53—24no—
274C22×C53—416no—
275C22×C53—416no—
276C22×C53—42no—
277C22×C53—42no—
278C22×C53—44no—
279C22×C53—44no—
280C22×C53—42no—
281C22×C53—42no—
282C22×C53C2×C4×D418yes—
283C22×C53#51814yes—
284C22×C53D4214yes—
285C22×C53#54618yes—
286C22×C53#55114yes—
287C22×C53C24×C411yescharacteristic
288C22×C53C23×D412yes—
289C22×C53C22×#4112yes—
290C22×C53C2×H(5,𝔽2)11yescharacteristic
291C22×C53C2×H(5,𝔽2)11yescharacteristic
292C2×D53—264no—
293C2×D53—28no—
294C2×D53—28no—
295C2×D53—28no—
296C2×D53—28no—
297C2×D53—28no—
298C2×D53—28no—
299C2×D53—464no—
300C2×D53C2×C4×D4116yescomplemented
301C4×C53—24no—
302C4×C53—28no—
303C4×C53—44no—
304Dic53—28no—
305Dic53—28no—
306Dic53—44no—
307C22×#120—531no—
308C22×C4×D4—532no—
309C23×C22⋊C4—532no—
310#2358—5316no—
311C2×#390—531no—
312C2×#388—532no—
313C2×#386—532no—
314C2×#386D5311yescharacteristic, complemented
315C2×#385—532no—
316C2×C4×C22⋊C4—532no—
317C2×C53—44no—
318C2×C53—44no—
319C2×C53#235818yescomplemented
320C2×C53C22×C4×D412yes—
321C2×C53C2×#51811yescharacteristic
322C2×C53C2×D4211yescharacteristic
323C2×C53C2×#54612yes—
324C2×C53C2×#55111yescharacteristic
325D53—216no—
326C26—531no—
327C23×D4—532no—
328C23×D4C2×D5311yescharacteristic
329C24×C4—532no—
330C24×C4—532no—
331C24×C4C2×D5311yescharacteristic, complemented
332C2×#120—5316no—
333C2×#120—538no—
334C2×C4×D4—5316no—
335C2×C4×D4—10616no—
336C22×C4⋊C4—532no—
337C22×C4⋊C4—531no—
338C22×C22⋊C4—531no—
339C22×C22⋊C4—532no—
340C22×C22⋊C4—532no—
341C22×C22⋊C4—532no—
342C22×C22⋊C4—532no—
343C22×C22⋊C4—532no—
344C22×C22⋊C4—5316no—
345C22×C22⋊C4—10616no—
346C22×C22⋊C4C2×D5312yes—
347C22×C22⋊C4C2×D5311yescharacteristic, complemented
348C22×C42—532no—
349C22×C42—531no—
350#390—538no—
351#388—5316no—
352#386—5316no—
353#386C2×D5318yescomplemented
354#385—5316no—
355C4×C22⋊C4—5316no—
356C2×#95—532no—
357C2×#95—532no—
358C2×#95C2×D5312yescomplemented
359C53C2×#235811yescharacteristic, complemented, Sylow
360C25—532no—
361C25—538no—
362C25—534no—
363C25—1068no—
364C25—1068no—
365C25—2122no—
366C25—2122no—
367C25C22×D5311yescharacteristic
368C25C22×D5311yescharacteristic
369C22×D4—42no—
370C22×D4—42no—
371C22×D4—5316no—
372C22×D4—538no—
373C22×D4—1068no—
374C22×D4—1068no—
375C22×D4C4×D5318yescomplemented
376C22×D4C22×D5314yes—
377C23×C4—28no—
378C23×C4—42no—
379C23×C4—532no—
380C23×C4—532no—
381C23×C4—532no—
382C23×C4—538no—
383C23×C4—538no—
384C23×C4—1068no—
385C23×C4—1068no—
386C23×C4—1068no—
387C23×C4—1068no—
388C23×C4—2124no—
389C23×C4—2124no—
390C23×C4C22×D5312yes—
391C23×C4C22×D5311yescharacteristic
392C23×C4C22×D5312yes—
393C23×C4C22×D5314yescomplemented
394#120—5364no—
395C4×D4—10664no—
396C2×C4⋊C4—538no—
397C2×C4⋊C4—534no—
398C2×C4⋊C4—1068no—
399C2×C4⋊C4—2122no—
400C2×C22⋊C4—28no—
401C2×C22⋊C4—42no—
402C2×C22⋊C4—538no—
403C2×C22⋊C4—534no—
404C2×C22⋊C4—5316no—
405C2×C22⋊C4—5316no—
406C2×C22⋊C4—538no—
407C2×C22⋊C4—538no—
408C2×C22⋊C4—538no—
409C2×C22⋊C4—538no—
410C2×C22⋊C4—538no—
411C2×C22⋊C4—10664no—
412C2×C22⋊C4—1068no—
413C2×C22⋊C4—1068no—
414C2×C22⋊C4—1068no—
415C2×C22⋊C4C22×D5318yes—
416C2×C22⋊C4C22×D5314yescomplemented
417C2×C42—538no—
418C2×C42—534no—
419C2×C42—1068no—
420C2×C42—2122no—
421#95—538no—
422#95—538no—
423#95C22×D5318yescomplemented
424C24—24no—
425C24—24no—
426C24—24no—
427C24—24no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—538no—
433C24—538no—
434C24—538no—
435C24—534no—
436C24—5316no—
437C24—10616no—
438C24—10664no—
439C24—10616no—
440C24—1064no—
441C24—1064no—
442C24—1064no—
443C24—1064no—
444C24—21216no—
445C24—2124no—
446C24—2124no—
447C24—21232no—
448C24—21216no—
449C24—21216no—
450C24C2×C4×D5314yes—
451C24C2×C4×D5314yes—
452C24C23×D5312yes—
453C24C23×D5312yes—
454C24C23×D5311yescharacteristic
455C2×D4—416no—
456C2×D4—44no—
457C2×D4—44no—
458C2×D4—5332no—
459C2×D4—10616no—
460C2×D4—10664no—
461C2×D4—10616no—
462C2×D4C2×C4×D53116yescomplemented
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—216no—
471C22×C4—44no—
472C22×C4—416no—
473C22×C4—538no—
474C22×C4—538no—
475C22×C4—534no—
476C22×C4—534no—
477C22×C4—534no—
478C22×C4—10616no—
479C22×C4—10616no—
480C22×C4—10616no—
481C22×C4—1064no—
482C22×C4—1064no—
483C22×C4—1064no—
484C22×C4—1064no—
485C22×C4—1064no—
486C22×C4—1064no—
487C22×C4—1064no—
488C22×C4—10616no—
489C22×C4—2128no—
490C22×C4—2128no—
491C22×C4—21232no—
492C22×C4—21232no—
493C22×C4C2×C4×D5314yes—
494C22×C4D4×D5318yes—
495C22×C4C23×D5314yes—
496C22×C4C23×D5312yes—
497C22×C4C23×D5314yes—
498C4⋊C4—10616no—
499C4⋊C4—2124no—
500C22⋊C4—216no—
501C22⋊C4—44no—
502C22⋊C4—5316no—
503C22⋊C4—5332no—
504C22⋊C4—5332no—
505C22⋊C4—10616no—
506C22⋊C4—10616no—
507C22⋊C4—10616no—
508C42—10616no—
509C42—2124no—
510C23—216no—
511C23—216no—
512C23—28no—
513C23—28no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—48no—
523C23—48no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—5332no—
529C23—538no—
530C23—538no—
531C23—538no—
532C23—10664no—
533C23—10616no—
534C23—10616no—
535C23—10616no—
536C23—10616no—
537C23—10616no—
538C23—1064no—
539C23—1064no—
540C23—1064no—
541C23—1064no—
542C23—10616no—
543C23—21216no—
544C23—21216no—
545C23—21232no—
546C23—212128no—
547C23—21216no—
548C23C22×C4×D5314yes—
549C23C22×C4×D5314yes—
550C23C22×C4×D5312yes—
551C23C2×D4×D5314yes—
552C23#41×D5314yes—
553C23#1313326212yes—
554C23#1313326212yes—
555C23C24×D5311yescharacteristic
556D4—416no—
557D4—10664no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—216no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—416no—
568C2×C4—48no—
569C2×C4—538no—
570C2×C4—538no—
571C2×C4—10616no—
572C2×C4—10616no—
573C2×C4—1064no—
574C2×C4—1064no—
575C2×C4—1064no—
576C2×C4—1064no—
577C2×C4—1064no—
578C2×C4—1064no—
579C2×C4—1064no—
580C2×C4—21232no—
581C2×C4—21232no—
582C2×C4—2128no—
583C2×C4C22×C4×D5314yes—
584C2×C4C2×D4×D5318yes—
585C22—28no—
586C22—28no—
587C22—24no—
588C22—24no—
589C22—42no—
590C22—42no—
591C22—44no—
592C22—44no—
593C22—416no—
594C22—416no—
595C22—42no—
596C22—42no—
597C22—5316no—
598C22—1068no—
599C22—1068no—
600C22—1068no—
601C22—1068no—
602C22—1068no—
603C22—10664no—
604C22—1068no—
605C22—21264no—
606C22C4×D4×D5318yes—
607C22#1681109614yes—
608C22#1681125514yes—
609C22#1681127018yes—
610C22#1681130114yes—
611C22C23×C4×D5311yescharacteristic
612C22C22×D4×D5312yes—
613C22C2×#41×D5312yes—
614C22C2×#1313326211yescharacteristic
615C22C2×#1313326211yescharacteristic
616C4—28no—
617C4—24no—
618C4—44no—
619C4—1068no—
620C4—1068no—
621C4—2124no—
622C2—44no—
623C2—44no—
624C2—10616no—
625C2#2027769818yesdirect product factor
626C2C2×C4×D4×D5312yes—
627C2C2×#1681109611yescharacteristic
628C2C2×#1681125511yescharacteristic
629C2C2×#1681127012yes—
630C2C2×#1681130111yescharacteristic
subgroupG/H
centerC24C23×D53
commutatorC22×C53C24×C4
FrattiniC23C24×D53
FittingC2×C53×#386C2
socleC24×C53C24
prime radicalC22×#13133185C2
cosocleC23×C53C25
hypercenterC2×#386D53
hypocenterC53C2×#2358
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#386

Complements

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

NH ≅ G/Ntype
2. C22×C4×C22⋊D53C2semidirect
3. C22×C22⋊C4×D53C2semidirect
4. #20277698C2direct (N × H)
5. C2×#16810526C2semidirect
6. C2×#16810517C2semidirect
7. C2×#16810499C2semidirect
8. C2×#16810497C2semidirect
9. C2×#16810470C2semidirect
10. C2×C53×#386C2semidirect
15. C2×#13133185C4semidirect
18. C2×C4×C22⋊D53C22semidirect
19. C22×C4⋊Dic53C22semidirect
20. C22×#11719919C22semidirect
23. C22×C22⋊Dic53C22semidirect
28. C2×C22⋊C4×D53C22semidirect
29. C22×C4×Dic53C22semidirect
30. C22×C4×Dic53C22semidirect
31. #16810526C22semidirect
32. #16810517C22semidirect
33. #16810499C22semidirect
34. #16810497C22semidirect
35. #16810470C22semidirect
36. C2×#13133124C22semidirect
37. C2×#13133124C22semidirect
39. C24×C4×C53C22semidirect
41. C22×C22⋊C4×C53C22semidirect
42. C53×#386C22semidirect
43. C2×#95×C53C22semidirect
48. C24×D53C2×C4semidirect
53. C22×C22⋊D53C2×C4semidirect
66. #13133185C2×C4semidirect
69. C2×C4⋊Dic53C23semidirect
71. C2×#11719919C23semidirect
74. C2×C22⋊Dic53C2×C4semidirect
77. C2×C22⋊Dic53C23semidirect
79. C2×#11719921C2×C4semidirect
80. C2×#11719921D4semidirect
87. C2×C4×Dic53C23semidirect
88. C2×C4×Dic53C23semidirect
89. #13133124C23semidirect
90. #13133124C23semidirect
95. C22×D4×C53C2×C4semidirect
102. C23×C4×C53C23semidirect
106. C2×C22⋊C4×C53C23semidirect
107. #95×C53C23semidirect
135. C23×D53C22×C4semidirect
142. C2×D4×C53C22×C4semidirect
161. C2×C22⋊D53C22×C4semidirect
186. C22⋊Dic53C22×C4semidirect
188. #11719921C22×C4semidirect
189. #11719921C2×D4semidirect
236. C22×D53C4×D4semidirect
300. C2×D53C2×C4×D4semidirect
314. C2×#386D53semidirect
319. C2×C53#2358semidirect
331. C24×C4C2×D53semidirect
347. C22×C22⋊C4C2×D53semidirect
353. #386C2×D53semidirect
358. C2×#95C2×D53semidirect
359. C53C2×#2358semidirect
375. C22×D4C4×D53semidirect
393. C23×C4C22×D53semidirect
416. C2×C22⋊C4C22×D53semidirect
423. #95C22×D53semidirect
462. C2×D4C2×C4×D53semidirect
625. C2#20277698direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131331852
C22×C4×C22⋊D532
C22×C22⋊C4×D532
#202776982
C2×#168105262
C2×#168105172
C2×#168104992
C2×#168104972
C2×#168104702
C2×C53×#3862
C2×#235853