Finite Group Databasebeta

← C2×#20279184 #27553149

Subgroups — C2×#20279184 #27553149

Order 13568.

4222 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133201C211yesmaximal, characteristic, complemented
2C22×#13133200C211yesmaximal, characteristic, complemented
3C22×#13133198C211yesmaximal, characteristic, complemented
4C22×#13133189C211yesmaximal, characteristic
5C23×#11719919C211yesmaximal, characteristic, complemented
6#20279184C2116yesmaximal, direct product factor
7C2×#16810559C212yesmaximal, complemented
8C2×#16810558C212yesmaximal, complemented
9C2×#16810523C212yesmaximal, complemented
10C2×#16810492C212yesmaximal, complemented
11C2×C22⋊C4×Dic53C211yesmaximal, characteristic, complemented
12C22×#123×C53C211yesmaximal, characteristic, complemented
13C23×C22⋊D53C2211yescharacteristic
14C23×C22⋊D53C2211yescharacteristic
15C24×Dic53C2211yescharacteristic
16C24×Dic53C2211yescharacteristic
17C2×#13133201—28no—
18C2×#13133201C2218yescomplemented
19C2×#13133200—28no—
20C2×#13133200C2218yescomplemented
21C2×#13133198—28no—
22C2×#13133198C2218yescomplemented
23C2×#13133189—28no—
24C2×#13133189C2218yes—
25C22×C4⋊Dic53C2212yes—
26C22×#11719919—28no—
27C22×#11719919C2211yescharacteristic
28C22×#11719919C2212yes—
29C22×#11719919C2212yes—
30C22×#11719919C2212yes—
31C22×#11719919C2212yes—
32C22×#11719919C2218yescomplemented
33C22×C22⋊Dic53C2212yes—
34C22×C22⋊Dic53C2211yescharacteristic
35C22×C22⋊Dic53C2211yescharacteristic
36C22×#11719921C2212yes—
37C22×#11719921C2212yes—
38C22×#11719921C2211yescharacteristic
39C22×C4×Dic53C2212yes—
40#16810559C22116yescomplemented
41#16810558C22116yescomplemented
42#16810523C22116yescomplemented
43#16810492C22116yescomplemented
44C22⋊C4×Dic53C2218yescomplemented
45C2×#13133124C2212yes—
46C23×D4×C53C2211yescharacteristic
47C24×C4×C53C2211yescharacteristic
48C2×#123×C53—28no—
49C2×#123×C53C2218yescomplemented
50C22×C4⋊C4×C53C2212yes—
51C22×C22⋊C4×C53C2212yes—
52C22×C22⋊C4×C53C2211yescharacteristic
53C24×D53C2311yescharacteristic
54C22×C22⋊D53—24no—
55C22×C22⋊D53—24no—
56C22×C22⋊D53—24no—
57C22×C22⋊D53—24no—
58C22×C22⋊D53—28no—
59C22×C22⋊D53C2314yes—
60C22×C22⋊D53C2314yes—
61C23×Dic53—28no—
62C23×Dic53—24no—
63C23×Dic53—24no—
64C23×Dic53—24no—
65C23×Dic53D418yes—
66C23×Dic53C2312yes—
67C23×Dic53C2311yescharacteristic
68C23×Dic53C2314yes—
69C23×Dic53C2312yes—
70C23×Dic53C2311yescharacteristic
71C23×Dic53C2314yes—
72C23×Dic53C2311yescharacteristic
73#13133201—232no—
74#13133200—232no—
75#13133198—232no—
76#13133189—232no—
77C2×C4⋊Dic53—28no—
78C2×C4⋊Dic53C2318yes—
79C2×#11719919—28no—
80C2×#11719919—28no—
81C2×#11719919—232no—
82C2×#11719919—24no—
83C2×#11719919—28no—
84C2×#11719919—28no—
85C2×#11719919C2318yes—
86C2×#11719919C2318yes—
87C2×#11719919C2318yes—
88C2×#11719919C2318yes—
89C2×#11719919C2314yes—
90C2×C22⋊Dic53—24no—
91C2×C22⋊Dic53—28no—
92C2×C22⋊Dic53D418yescomplemented
93C2×C22⋊Dic53C2318yes—
94C2×C22⋊Dic53C2314yes—
95C2×C22⋊Dic53C2314yes—
96C2×#11719921—28no—
97C2×#11719921—24no—
98C2×#11719921—28no—
99C2×#11719921C2318yes—
100C2×#11719921C2318yes—
101C2×#11719921C2314yes—
102C2×C4×Dic53—28no—
103C2×C4×Dic53C2318yes—
104#13133124C2318yes—
105C25×C53C2311yescharacteristic
106C25×C53C2311yescharacteristic
107C22×D4×C53—24no—
108C22×D4×C53—24no—
109C22×D4×C53—24no—
110C22×D4×C53C2314yes—
111C23×C4×C53—24no—
112C23×C4×C53—24no—
113C23×C4×C53—24no—
114C23×C4×C53C2312yes—
115C23×C4×C53C2311yescharacteristic
116C23×C4×C53C2314yes—
117C23×C4×C53C2312yes—
118#123×C53—232no—
119C2×C4⋊C4×C53—28no—
120C2×C4⋊C4×C53C2318yes—
121C2×C22⋊C4×C53—28no—
122C2×C22⋊C4×C53D418yescomplemented
123C2×C22⋊C4×C53C2318yes—
124C2×C22⋊C4×C53C2314yes—
125C24×C53—24no—
126C24×C53—24no—
127C24×C53—22no—
128C24×C53—22no—
129C24×C53—22no—
130C24×C53—24no—
131C24×C53—24no—
132C24×C53C2×D414yes—
133C24×C53C2×D414yes—
134C24×C53C2×D414yes—
135C24×C53C2412yes—
136C24×C53C2412yes—
137C24×C53C2411yescharacteristic
138C23×D53—22no—
139C23×D53—22no—
140C23×D53—22no—
141C23×D53—24no—
142C23×D53—24no—
143C23×D53C2412yes—
144C2×D4×C53—28no—
145C2×D4×C53—28no—
146C2×D4×C53—28no—
147C2×D4×C53—416no—
148C22×C4×C53—24no—
149C22×C4×C53—24no—
150C22×C4×C53—24no—
151C22×C4×C53—22no—
152C22×C4×C53—28no—
153C22×C4×C53—24no—
154C22×C4×C53—28no—
155C22×C4×C53—24no—
156C22×C4×C53—28no—
157C22×C4×C53—22no—
158C22×C4×C53—22no—
159C22×C4×C53—416no—
160C22×C4×C53C2×D418yes—
161C22×C4×C53C2414yes—
162C22×C4×C53C2412yes—
163C22×C4×C53C2414yes—
164C2×C22⋊D53—28no—
165C2×C22⋊D53—28no—
166C2×C22⋊D53—28no—
167C2×C22⋊D53—28no—
168C2×C22⋊D53—216no—
169C2×C22⋊D53—416no—
170C2×C22⋊D53—416no—
171C22×Dic53—24no—
172C22×Dic53—24no—
173C22×Dic53—24no—
174C22×Dic53—232no—
175C22×Dic53—216no—
176C22×Dic53—24no—
177C22×Dic53—24no—
178C22×Dic53—22no—
179C22×Dic53—28no—
180C22×Dic53—24no—
181C22×Dic53—22no—
182C22×Dic53—22no—
183C22×Dic53—28no—
184C22×Dic53—28no—
185C22×Dic53—416no—
186C22×Dic53C2×D418yes—
187C22×Dic53C2×D414yes—
188C22×Dic53C2×D4116yes—
189C22×Dic53C2×D414yes—
190C22×Dic53#4118yes—
191C22×Dic53C2414yes—
192C22×Dic53C2412yes—
193C22×Dic53C2412yes—
194C22×Dic53C2414yes—
195C22×Dic53C2412yes—
196C4⋊C4×C53—216no—
197C22⋊C4×C53—216no—
198C22⋊C4×C53C2×D4116yescomplemented
199C22⋊Dic53—28no—
200C22⋊Dic53—216no—
201C22⋊Dic53C2×D4116yescomplemented
202#11719921—216no—
203#11719921—28no—
204#11719921—216no—
205C4⋊Dic53—216no—
206#11719919—216no—
207#11719919—216no—
208#11719919—28no—
209#11719919—216no—
210#11719919—216no—
211C4×Dic53—216no—
212C23×C53—28no—
213C23×C53—28no—
214C23×C53—216no—
215C23×C53—216no—
216C23×C53—24no—
217C23×C53—24no—
218C23×C53—22no—
219C23×C53—22no—
220C23×C53—22no—
221C23×C53—24no—
222C23×C53—24no—
223C23×C53—44no—
224C23×C53—44no—
225C23×C53—44no—
226C23×C53—48no—
227C23×C53—48no—
228C23×C53#127116yes—
229C23×C53C22×D414yes—
230C23×C53C22×D414yes—
231C23×C53C22×D412yes—
232C23×C53C22×D412yes—
233C23×C53C22×D414yes—
234C23×C53C22×D412yes—
235C23×C53C2×#4114yes—
236C23×C532−1+414yes—
237C23×C53C2511yescharacteristic
238C22×D53—22no—
239C22×D53—22no—
240C22×D53—22no—
241C22×D53—24no—
242C22×D53—24no—
243C22×D53—44no—
244C22×D53—44no—
245C22×D53—44no—
246C22×D53—48no—
247C22×D53—48no—
248D4×C53—416no—
249C2×C4×C53—216no—
250C2×C4×C53—24no—
251C2×C4×C53—24no—
252C2×C4×C53—24no—
253C2×C4×C53—24no—
254C2×C4×C53—24no—
255C2×C4×C53—22no—
256C2×C4×C53—22no—
257C2×C4×C53—22no—
258C2×C4×C53—416no—
259C2×C4×C53—48no—
260C2×C4×C53—44no—
261C2×C4×C53C22×D418yes—
262C22⋊D53—416no—
263C22⋊D53—416no—
264C2×Dic53—28no—
265C2×Dic53—216no—
266C2×Dic53—24no—
267C2×Dic53—24no—
268C2×Dic53—24no—
269C2×Dic53—232no—
270C2×Dic53—24no—
271C2×Dic53—24no—
272C2×Dic53—24no—
273C2×Dic53—22no—
274C2×Dic53—22no—
275C2×Dic53—22no—
276C2×Dic53—416no—
277C2×Dic53—48no—
278C2×Dic53—44no—
279C2×Dic53#123116yescomplemented
280C2×Dic53C22×D414yes—
281C2×Dic53C22×D418yes—
282C2×Dic53C22×D414yes—
283C2×Dic53C2×#4118yes—
284C2×#2501—531nomaximal, Sylow
285C22×C53—24no—
286C22×C53—24no—
287C22×C53—28no—
288C22×C53—28no—
289C22×C53—232no—
290C22×C53—42no—
291C22×C53—42no—
292C22×C53—42no—
293C22×C53—44no—
294C22×C53—44no—
295C22×C53—88no—
296C22×C53C2×#12314yes—
297C22×C53C2×#12718yes—
298C22×C53C2×#12714yes—
299C22×C53#53614yes—
300C22×C53#54718yes—
301C22×C53#54718yes—
302C22×C53C23×D411yescharacteristic
303C22×C53C23×D411yescharacteristic
304C22×C53C23×D411yescharacteristic
305C22×C53C22×#4112yes—
306C22×C53C2×2−1+412yes—
307C2×D53—42no—
308C2×D53—42no—
309C2×D53—42no—
310C2×D53—44no—
311C2×D53—44no—
312C2×D53—88no—
313C4×C53—28no—
314C4×C53—44no—
315C4×C53—42no—
316Dic53—24no—
317Dic53—28no—
318Dic53—44no—
319Dic53—42no—
320Dic53C2×#12318yescomplemented
321C22×#123—531no—
322C22×#123—531no—
323C22×#123—531no—
324C22×#123D5311yescharacteristic, complemented
325C22×#121—531no—
326C23×C4⋊C4—531no—
327#2501—5316no—
328C2×#399—532no—
329C2×#399—532no—
330C2×#390—532no—
331C2×#384—532no—
332C2×C4×C22⋊C4—531no—
333C2×C53—28no—
334C2×C53—82no—
335C2×C53#250118yescomplemented
336C2×C53C22×#12311yescharacteristic
337C2×C53C22×#12711yescharacteristic
338C2×C53C2×#53611yescharacteristic
339C2×C53C2×#54712yes—
340C2×C53C2×#54712yes—
341D53—82no—
342C23×D4—531no—
343C23×D4—531no—
344C23×D4C2×D5311yescharacteristic
345C24×C4—531no—
346C24×C4—531no—
347C24×C4C2×D5311yescharacteristic
348C2×#123—28no—
349C2×#123—538no—
350C2×#123—538no—
351C2×#123—538no—
352C2×#123—1068no—
353C2×#123—1068no—
354C2×#123—1068no—
355C2×#123C2×D5318yescomplemented
356C2×#121—538no—
357C2×#121—1068no—
358C22×C4⋊C4—532no—
359C22×C4⋊C4—532no—
360C22×C4⋊C4—532no—
361C22×C4⋊C4—532no—
362C22×C4⋊C4—531no—
363C22×C4⋊C4—538no—
364C22×C4⋊C4—532no—
365C22×C4⋊C4—1068no—
366C22×C4⋊C4C2×D5312yes—
367C22×C22⋊C4—531no—
368C22×C22⋊C4—531no—
369C22×C22⋊C4—532no—
370C22×C22⋊C4—531no—
371C22×C22⋊C4—532no—
372C22×C22⋊C4—532no—
373C22×C22⋊C4C2×D5312yes—
374C22×C22⋊C4C2×D5311yescharacteristic
375C22×C42—532no—
376#399—5316no—
377#399—5316no—
378#390—5316no—
379#384—5316no—
380C4×C22⋊C4—538no—
381C2×#95—532no—
382C53C2×#250111yescharacteristic, complemented, Sylow
383C25—531no—
384C25C22×D5311yescharacteristic
385C25C22×D5311yescharacteristic
386C22×D4—24no—
387C22×D4—24no—
388C22×D4—24no—
389C22×D4—534no—
390C22×D4—534no—
391C22×D4—1064no—
392C22×D4—1068no—
393C22×D4—1064no—
394C22×D4—1064no—
395C22×D4—1064no—
396C22×D4C22×D5314yes—
397C23×C4—24no—
398C23×C4—24no—
399C23×C4—24no—
400C23×C4—531no—
401C23×C4—538no—
402C23×C4—534no—
403C23×C4—532no—
404C23×C4—531no—
405C23×C4—534no—
406C23×C4—532no—
407C23×C4—531no—
408C23×C4—1068no—
409C23×C4—1064no—
410C23×C4—1064no—
411C23×C4—1064no—
412C23×C4C22×D5314yes—
413C23×C4C22×D5312yes—
414C23×C4C22×D5311yescharacteristic
415C23×C4C22×D5312yes—
416#123—232no—
417#123—10632no—
418#123—10632no—
419#123—10632no—
420#121—10632no—
421C2×C4⋊C4—28no—
422C2×C4⋊C4—538no—
423C2×C4⋊C4—538no—
424C2×C4⋊C4—534no—
425C2×C4⋊C4—538no—
426C2×C4⋊C4—538no—
427C2×C4⋊C4—538no—
428C2×C4⋊C4—1068no—
429C2×C4⋊C4—1068no—
430C2×C4⋊C4—10632no—
431C2×C4⋊C4—1064no—
432C2×C4⋊C4—1068no—
433C2×C4⋊C4—1068no—
434C2×C4⋊C4—1068no—
435C2×C4⋊C4C22×D5318yes—
436C2×C22⋊C4—28no—
437C2×C22⋊C4—538no—
438C2×C22⋊C4—534no—
439C2×C22⋊C4—534no—
440C2×C22⋊C4—538no—
441C2×C22⋊C4—534no—
442C2×C22⋊C4—538no—
443C2×C22⋊C4—538no—
444C2×C22⋊C4—1064no—
445C2×C22⋊C4—1068no—
446C2×C22⋊C4—1064no—
447C2×C22⋊C4—1068no—
448C2×C22⋊C4—1068no—
449C2×C22⋊C4C22⋊D5318yescomplemented
450C2×C22⋊C4C22×D5318yes—
451C2×C22⋊C4C22×D5314yes—
452C2×C42—538no—
453C2×C42—1068no—
454#95—538no—
455C24—24no—
456C24—24no—
457C24—22no—
458C24—22no—
459C24—22no—
460C24—24no—
461C24—24no—
462C24—532no—
463C24—1062no—
464C24—1062no—
465C24—1062no—
466C24—1064no—
467C24—1064no—
468C24D4×D5314yes—
469C24D4×D5314yes—
470C24C2×C22⋊D5314yes—
471C24C23×D5312yes—
472C24C23×D5312yes—
473C24C23×D5311yescharacteristic
474C2×D4—28no—
475C2×D4—28no—
476C2×D4—28no—
477C2×D4—416no—
478C2×D4—1068no—
479C2×D4—10616no—
480C2×D4—1068no—
481C2×D4—1068no—
482C2×D4—1068no—
483C2×D4—21216no—
484C2×D4—21216no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—22no—
489C22×C4—28no—
490C22×C4—24no—
491C22×C4—22no—
492C22×C4—28no—
493C22×C4—24no—
494C22×C4—22no—
495C22×C4—28no—
496C22×C4—416no—
497C22×C4—534no—
498C22×C4—538no—
499C22×C4—532no—
500C22×C4—538no—
501C22×C4—534no—
502C22×C4—5316no—
503C22×C4—532no—
504C22×C4—534no—
505C22×C4—534no—
506C22×C4—532no—
507C22×C4—10632no—
508C22×C4—10616no—
509C22×C4—1064no—
510C22×C4—1064no—
511C22×C4—1064no—
512C22×C4—1064no—
513C22×C4—1062no—
514C22×C4—1068no—
515C22×C4—1064no—
516C22×C4—1062no—
517C22×C4—1068no—
518C22×C4—1064no—
519C22×C4—1062no—
520C22×C4—1068no—
521C22×C4—21216no—
522C22×C4C2×C22⋊D5318yes—
523C22×C4C23×D5314yes—
524C22×C4C23×D5312yes—
525C22×C4C23×D5314yes—
526C4⋊C4—216no—
527C4⋊C4—10616no—
528C4⋊C4—10616no—
529C4⋊C4—1068no—
530C4⋊C4—10616no—
531C4⋊C4—10616no—
532C4⋊C4—10616no—
533C22⋊C4—216no—
534C22⋊C4—5316no—
535C22⋊C4—1068no—
536C22⋊C4—10616no—
537C22⋊C4—1068no—
538C22⋊C4—10616no—
539C22⋊C4—10616no—
540C22⋊C4C2×C22⋊D53116yescomplemented
541C42—10616no—
542C23—216no—
543C23—216no—
544C23—28no—
545C23—24no—
546C23—28no—
547C23—24no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—24no—
552C23—24no—
553C23—48no—
554C23—48no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—1062no—
559C23—1062no—
560C23—1062no—
561C23—1064no—
562C23—1064no—
563C23—2128no—
564C23—2128no—
565C23—2124no—
566C23—2124no—
567C23—2124no—
568C23#13133213116yes—
569C23C2×D4×D5314yes—
570C23C2×D4×D5314yes—
571C23C2×D4×D5312yes—
572C23C2×D4×D5312yes—
573C23C22×C22⋊D5314yes—
574C23C22×C22⋊D5312yes—
575C23#41×D5314yes—
576C23#1313326414yes—
577C23C24×D5311yescharacteristic
578D4—416no—
579D4—21216no—
580D4—21216no—
581C2×C4—216no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—416no—
591C2×C4—48no—
592C2×C4—44no—
593C2×C4—5316no—
594C2×C4—534no—
595C2×C4—538no—
596C2×C4—538no—
597C2×C4—534no—
598C2×C4—10632no—
599C2×C4—10616no—
600C2×C4—1068no—
601C2×C4—1064no—
602C2×C4—1064no—
603C2×C4—1064no—
604C2×C4—1064no—
605C2×C4—1062no—
606C2×C4—1064no—
607C2×C4—1062no—
608C2×C4—1064no—
609C2×C4—1062no—
610C2×C4—21216no—
611C2×C4—2128no—
612C2×C4—2124no—
613C2×C4C22×C22⋊D5318yes—
614C22—24no—
615C22—232no—
616C22—28no—
617C22—28no—
618C22—24no—
619C22—42no—
620C22—44no—
621C22—44no—
622C22—42no—
623C22—42no—
624C22—88no—
625C22—4248no—
626C22—2122no—
627C22—2124no—
628C22—2124no—
629C22—2122no—
630C22—2122no—
631C22#123×D5314yes—
632C22C2×#1313321318yes—
633C22C2×#1313321314yes—
634C22#1681119514yes—
635C22#1681127118yes—
636C22#1681127218yes—
637C22C22×D4×D5311yescharacteristic
638C22C22×D4×D5311yescharacteristic
639C22C23×C22⋊D5311yescharacteristic
640C22C2×#41×D5312yes—
641C22C2×#1313326412yes—
642C4—28no—
643C4—44no—
644C4—42no—
645C4—538no—
646C4—1068no—
647C4—1064no—
648C4—2124no—
649C4—2122no—
650C2—28no—
651C2—82no—
652C2—4242no—
653C2#2027918418yesdirect product factor
654C2C2×#123×D5311yescharacteristic
655C2C22×#1313321311yescharacteristic
656C2C2×#1681119511yescharacteristic
657C2C2×#1681127112yes—
658C2C2×#1681127212yes—
subgroupG/H
centerC24C23×D53
commutator, cosocleC23×C53C25
FrattiniC23C24×D53
FittingC22×#123×C53C2
socleC24×C53C24
prime radicalC22×#13133189C2
hypercenterC22×#123D53
hypocenterC53C2×#2501
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
1. C22×#13133201C2semidirect
2. C22×#13133200C2semidirect
3. C22×#13133198C2semidirect
5. C23×#11719919C2semidirect
6. #20279184C2direct (N × H)
7. C2×#16810559C2semidirect
8. C2×#16810558C2semidirect
9. C2×#16810523C2semidirect
10. C2×#16810492C2semidirect
11. C2×C22⋊C4×Dic53C2semidirect
12. C22×#123×C53C2semidirect
18. C2×#13133201C22semidirect
20. C2×#13133200C22semidirect
22. C2×#13133198C22semidirect
32. C22×#11719919C22semidirect
40. #16810559C22semidirect
41. #16810558C22semidirect
42. #16810523C22semidirect
43. #16810492C22semidirect
44. C22⋊C4×Dic53C22semidirect
49. C2×#123×C53C22semidirect
92. C2×C22⋊Dic53D4semidirect
122. C2×C22⋊C4×C53D4semidirect
198. C22⋊C4×C53C2×D4semidirect
201. C22⋊Dic53C2×D4semidirect
279. C2×Dic53#123semidirect
320. Dic53C2×#123semidirect
324. C22×#123D53semidirect
335. C2×C53#2501semidirect
355. C2×#123C2×D53semidirect
382. C53C2×#2501semidirect
449. C2×C22⋊C4C22⋊D53semidirect
540. C22⋊C4C2×C22⋊D53semidirect
653. C2#20279184direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131332012
C22×#131332002
C22×#131331982
C22×#131331892
C23×#117199192
#202791842
C2×#168105592
C2×#168105582
C2×#168105232
C2×#168104922
C2×C22⋊C4×Dic532
C22×#123×C532
C2×#250153