Finite Group Databasebeta

← C2×#20279021 #27553144

Subgroups — C2×#20279021 #27553144

Order 13568.

3678 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×#13133202C212yesmaximal
2C22×C4×D212C211yesmaximal, characteristic
3C22×C4⋊C4×D53C211yesmaximal, characteristic, complemented
4#20279021C2116yesmaximal, direct product factor
5C2×#16810556C212yesmaximal, complemented
6C2×#16810552C212yesmaximal, complemented
7C2×#16810521C211yesmaximal, characteristic, complemented
8C2×#16810485C212yesmaximal, complemented
9C2×#16810480C211yesmaximal, characteristic, complemented
10C2×#16810480C211yesmaximal, characteristic, complemented
11C2×#16810480C211yesmaximal, characteristic, complemented
12C2×C53×#389C211yesmaximal, characteristic, complemented
13C23×D212C2211yescharacteristic
14C23×C4×D53C2211yescharacteristic
15C23×C4×D53C2212yes—
16C23×C4×D53C2211yescharacteristic
17C2×#13133202—216no—
18C2×#13133202C22116yes—
19C2×C4×D212—28no—
20C2×C4×D212C2218yes—
21C22×C4⋊Dic53C2211yescharacteristic
22C22×C4⋊Dic53C2211yescharacteristic
23C22×C4⋊Dic53C2211yescharacteristic
24C22×#11719919C2212yes—
25C22×#11719919C2212yes—
26C2×C4⋊C4×D53—28no—
27C2×C4⋊C4×D53C2218yescomplemented
28C22×#11719921C2212yes—
29C22×#11719921C2212yes—
30C22×#11719921C2212yes—
31C22×#11719921C2211yescharacteristic
32C22×#11719921C2211yescharacteristic
33#16810556C22116yescomplemented
34#16810552C22116yescomplemented
35#16810521C2218yescomplemented
36#16810485C22116yescomplemented
37#16810480C2218yescomplemented
38#16810480C2218yescomplemented
39#16810480C2218yescomplemented
40C2×#13133124C2212yes—
41C2×#13133124C2212yes—
42C2×#13133124C2212yes—
43C2×#13133124C2212yes—
44C22×C4⋊C4×C53C2211yescharacteristic
45C22×C4⋊C4×C53C2211yescharacteristic
46C22×C4⋊C4×C53C2212yes—
47C22×C42×C53C2211yescharacteristic
48C53×#389C2218yescomplemented
49C2×#95×C53C2212yes—
50C24×D53C2311yescharacteristic
51C24×D53C2311yescharacteristic
52C22×D212—24no—
53C22×D212—24no—
54C22×D212D418yes—
55C22×D212C2314yes—
56C23×Dic53C2312yes—
57C23×Dic53C2312yes—
58C23×Dic53C2311yescharacteristic
59C23×Dic53C2311yescharacteristic
60C22×C4×D53—24no—
61C22×C4×D53—24no—
62C22×C4×D53—28no—
63C22×C4×D53—28no—
64C22×C4×D53—28no—
65C22×C4×D53—24no—
66C22×C4×D53—24no—
67C22×C4×D53—24no—
68C22×C4×D53D418yes—
69C22×C4×D53C2318yes—
70C22×C4×D53C2314yes—
71C22×C4×D53C2314yes—
72#13133202—264no—
73C4×D212—232no—
74C2×C4⋊Dic53—24no—
75C2×C4⋊Dic53—24no—
76C2×C4⋊Dic53—24no—
77C2×C4⋊Dic53C2314yes—
78C2×C4⋊Dic53C2314yes—
79C2×C4⋊Dic53C2314yes—
80C2×#11719919—28no—
81C2×#11719919—28no—
82C2×#11719919C2318yes—
83C2×#11719919C2318yes—
84C4⋊C4×D53—232no—
85C2×#11719921—24no—
86C2×#11719921—28no—
87C2×#11719921—28no—
88C2×#11719921—28no—
89C2×#11719921—24no—
90C2×#11719921C2314yes—
91C2×#11719921C2318yes—
92C2×#11719921C2318yes—
93C2×#11719921C2314yes—
94C2×#11719921C2318yes—
95#13133124C2318yes—
96#13133124C2318yes—
97#13133124C2318yes—
98#13133124C2318yes—
99C23×C4×C53C2311yescharacteristic
100C23×C4×C53C2312yes—
101C23×C4×C53C2312yes—
102C23×C4×C53C2311yescharacteristic
103C23×C4×C53C2311yescharacteristic
104C2×C4⋊C4×C53—24no—
105C2×C4⋊C4×C53—28no—
106C2×C4⋊C4×C53—24no—
107C2×C4⋊C4×C53C2318yes—
108C2×C4⋊C4×C53C2314yes—
109C2×C4⋊C4×C53C2314yes—
110C2×C42×C53—24no—
111C2×C42×C53C2314yes—
112#95×C53C2318yes—
113C24×C53C2411yescharacteristic
114C23×D53—22no—
115C23×D53—22no—
116C23×D53—22no—
117C23×D53—22no—
118C23×D53—22no—
119C23×D53—22no—
120C23×D53—22no—
121C23×D53—22no—
122C23×D53—22no—
123C23×D53—22no—
124C23×D53C2×D414yes—
125C23×D53C2×D414yes—
126C23×D53C2×D414yes—
127C23×D53#4114yes—
128C23×D53C2412yes—
129C23×D53C2412yes—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—24no—
134C22×C4×C53—24no—
135C22×C4×C53—24no—
136C22×C4×C53—24no—
137C22×C4×C53—24no—
138C22×C4×C53—22no—
139C22×C4×C53—22no—
140C22×C4×C53—24no—
141C22×C4×C53C2×D414yes—
142C22×C4×C53C2×D414yes—
143C22×C4×C53#4114yes—
144C22×C4×C53#4114yes—
145C22×C4×C53C2412yes—
146C22×C4×C53C2414yes—
147C22×C4×C53C2414yes—
148C22×C4×C53C2412yes—
149C22×C4×C53C2412yes—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—24no—
153C22×Dic53—24no—
154C22×Dic53—22no—
155C22×Dic53—24no—
156C22×Dic53—24no—
157C22×Dic53—24no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—24no—
161C22×Dic53C2×D414yes—
162C22×Dic53C2412yes—
163C22×Dic53C2414yes—
164C22×Dic53C2414yes—
165C22×Dic53C2412yes—
166C2×D212—232no—
167C2×D212—28no—
168C2×D212—28no—
169C2×D212C2×D4116yes—
170C2×C4×D53—232no—
171C2×C4×D53—28no—
172C2×C4×D53—28no—
173C2×C4×D53—216no—
174C2×C4×D53—216no—
175C2×C4×D53—216no—
176C2×C4×D53—28no—
177C2×C4×D53—28no—
178C2×C4×D53—28no—
179C2×C4×D53—432no—
180C2×C4×D53—416no—
181C2×C4×D53C2×D4116yes—
182C4⋊C4×C53—28no—
183C4⋊C4×C53—216no—
184C4⋊C4×C53—28no—
185C42×C53—28no—
186#11719921—28no—
187#11719921—216no—
188#11719921—216no—
189#11719921—216no—
190#11719921—28no—
191C4⋊Dic53—28no—
192C4⋊Dic53—28no—
193C4⋊Dic53—28no—
194#11719919—216no—
195#11719919—216no—
196C23×C53C22×D412yes—
197C23×C53C22×D412yes—
198C23×C53C2×#4112yes—
199C23×C53C2×#4112yes—
200C23×C53C2×#4112yes—
201C23×C532−1+412yes—
202C23×C532−1+412yes—
203C23×C53C2511yescharacteristic
204C22×D53—28no—
205C22×D53—28no—
206C22×D53—28no—
207C22×D53—28no—
208C22×D53—28no—
209C22×D53—28no—
210C22×D53—22no—
211C22×D53—22no—
212C22×D53—22no—
213C22×D53—22no—
214C22×D53—28no—
215C22×D53—28no—
216C22×D53—28no—
217C22×D53—22no—
218C22×D53—22no—
219C22×D53—22no—
220C22×D53—22no—
221C22×D53—22no—
222C22×D53—22no—
223C22×D53—44no—
224C22×D53—44no—
225C22×D53—44no—
226C22×D53—44no—
227C22×D53#121116yes—
228C22×D53C22×D414yes—
229C22×D53C22×D414yes—
230C22×D53C22×D414yes—
231C22×D53C2×#4114yes—
232C2×C4×C53—28no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—24no—
236C2×C4×C53—24no—
237C2×C4×C53—22no—
238C2×C4×C53—24no—
239C2×C4×C53—24no—
240C2×C4×C53—24no—
241C2×C4×C53—22no—
242C2×C4×C53—24no—
243C2×C4×C53—22no—
244C2×C4×C53—48no—
245C2×C4×C53—44no—
246C2×C4×C53—48no—
247C2×C4×C53#123116yescomplemented
248C2×C4×C53C22×D414yes—
249C2×C4×C53C22×D414yes—
250C2×C4×C53C2×#4114yes—
251C2×C4×C53C2×#4114yes—
252C2×Dic53—28no—
253C2×Dic53—22no—
254C2×Dic53—22no—
255C2×Dic53—24no—
256C2×Dic53—24no—
257C2×Dic53—22no—
258C2×Dic53—24no—
259C2×Dic53—24no—
260C2×Dic53—22no—
261C2×Dic53—22no—
262C2×Dic53—24no—
263C2×Dic53—24no—
264C2×Dic53—48no—
265C2×Dic53—44no—
266C2×Dic53—48no—
267C2×Dic53C22×D414yes—
268D212—232no—
269C4×D53—232no—
270C4×D53—432no—
271C4×D53—416no—
272C2×#2486—531nomaximal, Sylow
273C22×C53C2×#12114yes—
274C22×C53C2×#12314yes—
275C22×C53#53614yes—
276C22×C53#54714yes—
277C22×C53#54714yes—
278C22×C53#55214yes—
279C22×C53#55214yes—
280C22×C53C23×D411yescharacteristic
281C22×C53C23×D411yescharacteristic
282C22×C53C22×#4111yescharacteristic
283C22×C53C22×#4111yescharacteristic
284C22×C53C22×#4111yescharacteristic
285C22×C53C2×2−1+411yescharacteristic
286C22×C53C2×2−1+411yescharacteristic
287C2×D53—24no—
288C2×D53—24no—
289C2×D53—24no—
290C2×D53—24no—
291C2×D53—24no—
292C2×D53—24no—
293C2×D53—232no—
294C2×D53—24no—
295C2×D53—24no—
296C2×D53—24no—
297C2×D53—42no—
298C2×D53—42no—
299C2×D53—42no—
300C2×D53—42no—
301C2×D53—416no—
302C2×D53C2×#12118yes—
303C4×C53—24no—
304C4×C53—44no—
305C4×C53—42no—
306C4×C53—44no—
307C4×C53C2×#12318yescomplemented
308Dic53—24no—
309Dic53—44no—
310Dic53—42no—
311Dic53—44no—
312C22×#123—532no—
313C22×C4×D4—531no—
314C23×C4⋊C4—531no—
315#2486—5316no—
316C2×#398—532no—
317C2×#397—532no—
318C2×#389—531no—
319C2×#389D5311yescharacteristic, complemented
320C2×#381—532no—
321C2×#380—531no—
322C2×#380—531no—
323C2×#380—531no—
324C2×C53#248618yescomplemented
325C2×C53C22×#12111yescharacteristic
326C2×C53C22×#12311yescharacteristic
327C2×C53C2×#53611yescharacteristic
328C2×C53C2×#54711yescharacteristic
329C2×C53C2×#54711yescharacteristic
330C2×C53C2×#55211yescharacteristic
331C2×C53C2×#55211yescharacteristic
332D53—28no—
333D53—44no—
334C23×D4—531no—
335C24×C4—531no—
336C24×C4—532no—
337C24×C4—531no—
338C2×#123—5316no—
339C2×#123—10616no—
340C2×C4×D4—538no—
341C2×C4×D4—1068no—
342C22×C4⋊C4—532no—
343C22×C4⋊C4—532no—
344C22×C4⋊C4—531no—
345C22×C4⋊C4—531no—
346C22×C4⋊C4—531no—
347C22×C4⋊C4—538no—
348C22×C4⋊C4—1068no—
349C22×C4⋊C4C2×D5312yes—
350C22×C4⋊C4C2×D5311yescharacteristic
351C22×C4⋊C4C2×D5311yescharacteristic
352C22×C22⋊C4—531no—
353C22×C22⋊C4—532no—
354C22×C22⋊C4—532no—
355C22×C22⋊C4—531no—
356C22×C22⋊C4—532no—
357C22×C42C2×D5311yescharacteristic
358#398—5316no—
359#397—5316no—
360#389—538no—
361#389C2×D5318yescomplemented
362#381—5316no—
363#380—538no—
364#380—538no—
365#380—538no—
366C2×#95—532no—
367C2×#95—532no—
368C2×#95—532no—
369C2×#95—532no—
370C2×#95C2×D5312yes—
371C53C2×#248611yescharacteristic, complemented, Sylow
372C25—531no—
373C25—531no—
374C22×D4—538no—
375C22×D4—534no—
376C22×D4—1064no—
377C22×D4—1064no—
378C23×C4—531no—
379C23×C4—538no—
380C23×C4—534no—
381C23×C4—532no—
382C23×C4—532no—
383C23×C4—531no—
384C23×C4—538no—
385C23×C4—534no—
386C23×C4—1068no—
387C23×C4—1068no—
388C23×C4—1068no—
389C23×C4—1064no—
390C23×C4—1064no—
391C23×C4—1064no—
392C23×C4—1064no—
393C23×C4—1064no—
394C23×C4C22×D5312yes—
395C23×C4C22×D5311yescharacteristic
396C23×C4C22×D5312yes—
397C23×C4C22×D5311yescharacteristic
398C23×C4C22×D5311yescharacteristic
399#123—10664no—
400C4×D4—10632no—
401C2×C4⋊C4—24no—
402C2×C4⋊C4—24no—
403C2×C4⋊C4—28no—
404C2×C4⋊C4—538no—
405C2×C4⋊C4—534no—
406C2×C4⋊C4—534no—
407C2×C4⋊C4—538no—
408C2×C4⋊C4—534no—
409C2×C4⋊C4—10632no—
410C2×C4⋊C4—1068no—
411C2×C4⋊C4—1064no—
412C2×C4⋊C4—1064no—
413C2×C4⋊C4—1068no—
414C2×C4⋊C4—1064no—
415C2×C4⋊C4C22×D5314yes—
416C2×C4⋊C4C22×D5318yes—
417C2×C4⋊C4C22×D5314yes—
418C2×C22⋊C4—534no—
419C2×C22⋊C4—534no—
420C2×C22⋊C4—538no—
421C2×C22⋊C4—538no—
422C2×C22⋊C4—538no—
423C2×C22⋊C4—1064no—
424C2×C22⋊C4—1064no—
425C2×C22⋊C4—1068no—
426C2×C22⋊C4—1068no—
427C2×C22⋊C4—1068no—
428C2×C42—24no—
429C2×C42C22×D5314yes—
430#95—538no—
431#95—538no—
432#95—538no—
433#95—538no—
434#95C22×D5318yes—
435C24—534no—
436C24—534no—
437C24—534no—
438C24—532no—
439C24—534no—
440C24—532no—
441C24—1062no—
442C24—1062no—
443C24—1062no—
444C24—1062no—
445C24—1062no—
446C24—1062no—
447C24—1062no—
448C24—1062no—
449C24—1062no—
450C24—1062no—
451C24C23×D5311yescharacteristic
452C2×D4—5316no—
453C2×D4—1068no—
454C2×D4—10632no—
455C2×D4—1068no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—534no—
468C22×C4—532no—
469C22×C4—5316no—
470C22×C4—532no—
471C22×C4—534no—
472C22×C4—534no—
473C22×C4—10616no—
474C22×C4—10616no—
475C22×C4—10616no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—10632no—
479C22×C4—1068no—
480C22×C4—1068no—
481C22×C4—1064no—
482C22×C4—1064no—
483C22×C4—1062no—
484C22×C4—1064no—
485C22×C4—1064no—
486C22×C4—1062no—
487C22×C4—1064no—
488C22×C4—1062no—
489C22×C4—1064no—
490C22×C4—1068no—
491C22×C4—1068no—
492C22×C4—1068no—
493C22×C4—21232no—
494C22×C4—21216no—
495C22×C4#1171994514yes—
496C22×C4D4×D5314yes—
497C22×C4D4×D5314yes—
498C22×C4#1171994914yes—
499C22×C4C23×D5312yes—
500C22×C4C23×D5314yes—
501C22×C4C23×D5314yes—
502C22×C4C23×D5312yes—
503C22×C4C23×D5312yes—
504C4⋊C4—28no—
505C4⋊C4—216no—
506C4⋊C4—28no—
507C4⋊C4—10616no—
508C4⋊C4—1068no—
509C4⋊C4—1068no—
510C4⋊C4—10616no—
511C4⋊C4—1068no—
512C22⋊C4—1068no—
513C22⋊C4—1068no—
514C22⋊C4—10616no—
515C22⋊C4—10616no—
516C22⋊C4—10616no—
517C42—28no—
518C23—5316no—
519C23—534no—
520C23—534no—
521C23—534no—
522C23—534no—
523C23—1068no—
524C23—1068no—
525C23—1068no—
526C23—1068no—
527C23—1068no—
528C23—1062no—
529C23—1062no—
530C23—1068no—
531C23—1062no—
532C23—1062no—
533C23—1068no—
534C23—1068no—
535C23—1068no—
536C23—1062no—
537C23—1062no—
538C23—1062no—
539C23—1062no—
540C23—1062no—
541C23—1062no—
542C23—2124no—
543C23—2124no—
544C23—2124no—
545C23—2124no—
546C23C2×D4×D5312yes—
547C23C2×D4×D5312yes—
548C23#41×D5312yes—
549C23C2×#1171994512yes—
550C23C2×#1171994912yes—
551C23#1313326412yes—
552C23#1313326512yes—
553C23C24×D5311yescharacteristic
554D4—10632no—
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4—44no—
570C2×C4—534no—
571C2×C4—1068no—
572C2×C4—1062no—
573C2×C4—1062no—
574C2×C4—10632no—
575C2×C4—1064no—
576C2×C4—1064no—
577C2×C4—1062no—
578C2×C4—1064no—
579C2×C4—1064no—
580C2×C4—1062no—
581C2×C4—1064no—
582C2×C4—1062no—
583C2×C4—1064no—
584C2×C4—21232no—
585C2×C4—2128no—
586C2×C4—2128no—
587C2×C4—2124no—
588C2×C4—21216no—
589C2×C4#13133202116yescomplemented
590C2×C4C2×D4×D5314yes—
591C2×C4C2×D4×D5314yes—
592C2×C4C2×#1171994514yes—
593C2×C4C2×#1171994914yes—
594C22—538no—
595C22—1064no—
596C22—1064no—
597C22—1064no—
598C22—1064no—
599C22—1064no—
600C22—1064no—
601C22—10632no—
602C22—1064no—
603C22—1064no—
604C22—1064no—
605C22—2122no—
606C22—2122no—
607C22—2122no—
608C22—2122no—
609C22—21216no—
610C22#121×D5314yes—
611C22C2×#1313320214yes—
612C22#1681119614yes—
613C22#1681127614yes—
614C22#1681127814yes—
615C22#1681131114yes—
616C22#1681131414yes—
617C22C22×D4×D5311yescharacteristic
618C22C22×D4×D5311yescharacteristic
619C22C2×#41×D5311yescharacteristic
620C22C22×#1171994511yescharacteristic
621C22C22×#1171994911yescharacteristic
622C22C2×#1313326411yescharacteristic
623C22C2×#1313326511yescharacteristic
624C4—24no—
625C4—42no—
626C4—44no—
627C4—44no—
628C4—1064no—
629C4—2124no—
630C4—2124no—
631C4—2122no—
632C4C2×#1313320218yescomplemented
633C2—1068no—
634C2—2124no—
635C2#2027902118yesdirect product factor
636C2C2×#121×D5311yescharacteristic
637C2C22×#1313320211yescharacteristic
638C2C2×#1681119611yescharacteristic
639C2C2×#1681127611yescharacteristic
640C2C2×#1681127811yescharacteristic
641C2C2×#1681131111yescharacteristic
642C2C2×#1681131411yescharacteristic
subgroupG/H
centerC24C23×D53
commutator, cosocleC23×C53C25
FrattiniC23C24×D53
FittingC2×C53×#389C2
socleC24×C53C24
prime radicalC23×D212C22
hypercenterC2×#389D53
hypocenterC53C2×#2486
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
3. C22×C4⋊C4×D53C2semidirect
4. #20279021C2direct (N × H)
5. C2×#16810556C2semidirect
6. C2×#16810552C2semidirect
7. C2×#16810521C2semidirect
8. C2×#16810485C2semidirect
9. C2×#16810480C2semidirect
10. C2×#16810480C2semidirect
11. C2×#16810480C2semidirect
12. C2×C53×#389C2semidirect
27. C2×C4⋊C4×D53C22semidirect
33. #16810556C22semidirect
34. #16810552C22semidirect
35. #16810521C22semidirect
36. #16810485C22semidirect
37. #16810480C22semidirect
38. #16810480C22semidirect
39. #16810480C22semidirect
48. C53×#389C22semidirect
247. C2×C4×C53#123semidirect
307. C4×C53C2×#123semidirect
319. C2×#389D53semidirect
324. C2×C53#2486semidirect
361. #389C2×D53semidirect
371. C53C2×#2486semidirect
589. C2×C4#13133202semidirect
632. C4C2×#13133202semidirect
635. C2#20279021direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131332022
C22×C4×D2122
C22×C4⋊C4×D532
#202790212
C2×#168105562
C2×#168105522
C2×#168105212
C2×#168104852
C2×#168104802
C2×C53×#3892
C2×#248653