Finite Group Databasebeta

← C2×#20279030 #27553265

Subgroups — C2×#20279030 #27553265

Order 13568.

3374 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D212C211yesmaximal, characteristic
2C22×C4⋊C4×D53C211yesmaximal, characteristic, complemented
3#20279030C2116yesmaximal, direct product factor
4C2×#16810565C212yesmaximal, complemented
5C2×#16810558C212yesmaximal, complemented
6C2×#16810521C211yesmaximal, characteristic, complemented
7C2×#16810501C212yesmaximal
8C2×#16810494C212yesmaximal, 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×D53C2211yescharacteristic
16C23×C4×D53C2212yes—
17C2×C4×D212—28no—
18C2×C4×D212C2218yes—
19C22×C4⋊Dic53C2211yescharacteristic
20C22×C4⋊Dic53C2211yescharacteristic
21C22×C4⋊Dic53C2211yescharacteristic
22C22×#11719919C2212yes—
23C22×#11719919C2212yes—
24C2×C4⋊C4×D53—28no—
25C2×C4⋊C4×D53C2218yescomplemented
26C22×#11719921C2212yes—
27C22×#11719921C2211yescharacteristic
28C22×#11719921C2211yescharacteristic
29C22×#11719921C2212yes—
30C22×#11719921C2212yes—
31#16810565C22116yescomplemented
32#16810558C22116yescomplemented
33#16810521C2218yescomplemented
34#16810501C22116yes—
35#16810494C22116yescomplemented
36#16810480C2218yescomplemented
37#16810480C2218yescomplemented
38#16810480C2218yescomplemented
39C2×#13133124C2212yes—
40C2×#13133124C2212yes—
41C2×#13133124C2212yes—
42C2×#13133124C2212yes—
43C22×C4⋊C4×C53C2211yescharacteristic
44C22×C4⋊C4×C53C2211yescharacteristic
45C22×C4⋊C4×C53C2212yes—
46C22×C42×C53C2211yescharacteristic
47C53×#389C2218yescomplemented
48C2×#95×C53C2212yes—
49C24×D53C2311yescharacteristic
50C24×D53C2311yescharacteristic
51C22×D212—24no—
52C22×D212—24no—
53C22×D212Q818yes—
54C22×D212C2314yes—
55C23×Dic53C2312yes—
56C23×Dic53C2312yes—
57C23×Dic53C2311yescharacteristic
58C23×Dic53C2311yescharacteristic
59C22×C4×D53—24no—
60C22×C4×D53—24no—
61C22×C4×D53—28no—
62C22×C4×D53—24no—
63C22×C4×D53—24no—
64C22×C4×D53—24no—
65C22×C4×D53—44no—
66C22×C4×D53—44no—
67C22×C4×D53D418yes—
68C22×C4×D53C2314yes—
69C22×C4×D53C2314yes—
70C22×C4×D53C2318yes—
71C4×D212—232no—
72C2×C4⋊Dic53—24no—
73C2×C4⋊Dic53—24no—
74C2×C4⋊Dic53—24no—
75C2×C4⋊Dic53C2314yes—
76C2×C4⋊Dic53C2314yes—
77C2×C4⋊Dic53C2314yes—
78C2×#11719919—28no—
79C2×#11719919—44no—
80C2×#11719919C2318yes—
81C2×#11719919C2318yes—
82C4⋊C4×D53—232no—
83C2×#11719921—24no—
84C2×#11719921—28no—
85C2×#11719921—24no—
86C2×#11719921—44no—
87C2×#11719921—44no—
88C2×#11719921C2318yes—
89C2×#11719921C2314yes—
90C2×#11719921C2314yes—
91C2×#11719921C2318yes—
92C2×#11719921C2318yes—
93#13133124C2318yes—
94#13133124C2318yes—
95#13133124C2318yes—
96#13133124C2318yes—
97C23×C4×C53C2311yescharacteristic
98C23×C4×C53C2312yes—
99C23×C4×C53C2312yes—
100C23×C4×C53C2311yescharacteristic
101C23×C4×C53C2311yescharacteristic
102C2×C4⋊C4×C53—24no—
103C2×C4⋊C4×C53—24no—
104C2×C4⋊C4×C53—44no—
105C2×C4⋊C4×C53C2314yes—
106C2×C4⋊C4×C53C2314yes—
107C2×C4⋊C4×C53C2318yes—
108C2×C42×C53—24no—
109C2×C42×C53C2314yes—
110#95×C53C2318yes—
111C24×C53C2411yescharacteristic
112C23×D53—22no—
113C23×D53—22no—
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×D53C2×D414yes—
123C23×D53C2×Q814yes—
124C23×D53C2×Q814yes—
125C23×D53#4114yes—
126C23×D53C2412yes—
127C23×D53C2412yes—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—24no—
132C22×C4×C53—24no—
133C22×C4×C53—24no—
134C22×C4×C53—22no—
135C22×C4×C53—24no—
136C22×C4×C53—24no—
137C22×C4×C53—24no—
138C22×C4×C53—22no—
139C22×C4×C53C2×D414yes—
140C22×C4×C53C2×Q814yes—
141C22×C4×C53#4114yes—
142C22×C4×C53#4114yes—
143C22×C4×C53C2414yes—
144C22×C4×C53C2414yes—
145C22×C4×C53C2412yes—
146C22×C4×C53C2412yes—
147C22×C4×C53C2412yes—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—24no—
152C22×Dic53—24no—
153C22×Dic53—24no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—24no—
157C22×Dic53—24no—
158C22×Dic53—24no—
159C22×Dic53C2×D414yes—
160C22×Dic53C2412yes—
161C22×Dic53C2414yes—
162C22×Dic53C2414yes—
163C22×Dic53C2412yes—
164C2×D212—232no—
165C2×D212—28no—
166C2×D212—28no—
167C2×D212C2×Q8116yes—
168C2×C4×D53—232no—
169C2×C4×D53—28no—
170C2×C4×D53—28no—
171C2×C4×D53—216no—
172C2×C4×D53—28no—
173C2×C4×D53—28no—
174C2×C4×D53—28no—
175C2×C4×D53—48no—
176C2×C4×D53—48no—
177C2×C4×D53—432no—
178C2×C4×D53—416no—
179C2×C4×D53C2×D4116yes—
180C4⋊C4×C53—28no—
181C4⋊C4×C53—28no—
182C4⋊C4×C53—48no—
183C42×C53—28no—
184#11719921—28no—
185#11719921—216no—
186#11719921—28no—
187#11719921—48no—
188#11719921—48no—
189C4⋊Dic53—28no—
190C4⋊Dic53—28no—
191C4⋊Dic53—28no—
192#11719919—216no—
193#11719919—48no—
194C23×C53C22×D412yes—
195C23×C53C22×Q812yes—
196C23×C53C2×#4112yes—
197C23×C53C2×#4112yes—
198C23×C53C2×#4112yes—
199C23×C53H(5,𝔽2)12yes—
200C23×C53H(5,𝔽2)12yes—
201C23×C53C2511yescharacteristic
202C22×D53—28no—
203C22×D53—28no—
204C22×D53—28no—
205C22×D53—28no—
206C22×D53—28no—
207C22×D53—28no—
208C22×D53—22no—
209C22×D53—22no—
210C22×D53—22no—
211C22×D53—22no—
212C22×D53—28no—
213C22×D53—28no—
214C22×D53—28no—
215C22×D53—22no—
216C22×D53—22no—
217C22×D53—22no—
218C22×D53—22no—
219C22×D53—22no—
220C22×D53—22no—
221C22×D53—44no—
222C22×D53—44no—
223C22×D53—44no—
224C22×D53—44no—
225C22×D53C22⋊Q8116yes—
226C22×D53C22×D414yes—
227C22×D53C22×Q814yes—
228C22×D53C22×Q814yes—
229C22×D53C2×#4114yes—
230C2×C4×C53—28no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—24no—
234C2×C4×C53—22no—
235C2×C4×C53—24no—
236C2×C4×C53—24no—
237C2×C4×C53—24no—
238C2×C4×C53—24no—
239C2×C4×C53—22no—
240C2×C4×C53—24no—
241C2×C4×C53—22no—
242C2×C4×C53—48no—
243C2×C4×C53—44no—
244C2×C4×C53—48no—
245C2×C4×C53#123116yes—
246C2×C4×C53C22×D414yes—
247C2×C4×C53C22×Q814yes—
248C2×C4×C53C2×#4114yes—
249C2×C4×C53C2×#4114yes—
250C2×Dic53—28no—
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—24no—
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—24no—
258C2×Dic53—24no—
259C2×Dic53—24no—
260C2×Dic53—24no—
261C2×Dic53—24no—
262C2×Dic53—44no—
263C2×Dic53—48no—
264C2×Dic53—48no—
265C2×Dic53C22×D414yes—
266D212—232no—
267C4×D53—232no—
268C4×D53—432no—
269C4×D53—416no—
270C2×#2487—531nomaximal, Sylow
271C22×C53C2×C22⋊Q814yes—
272C22×C53C2×#12314yes—
273C22×C53#53514yes—
274C22×C53#55314yes—
275C22×C53#55314yes—
276C22×C53#55414yes—
277C22×C53#55414yes—
278C22×C53C23×D411yescharacteristic
279C22×C53C23×Q811yescharacteristic
280C22×C53C22×#4111yescharacteristic
281C22×C53C22×#4111yescharacteristic
282C22×C53C22×#4111yescharacteristic
283C22×C53C2×H(5,𝔽2)11yescharacteristic
284C22×C53C2×H(5,𝔽2)11yescharacteristic
285C2×D53—24no—
286C2×D53—24no—
287C2×D53—24no—
288C2×D53—24no—
289C2×D53—24no—
290C2×D53—24no—
291C2×D53—232no—
292C2×D53—24no—
293C2×D53—24no—
294C2×D53—24no—
295C2×D53—42no—
296C2×D53—42no—
297C2×D53—42no—
298C2×D53—42no—
299C2×D53—416no—
300C2×D53C2×C22⋊Q818yes—
301C4×C53—24no—
302C4×C53—44no—
303C4×C53—42no—
304C4×C53—44no—
305C4×C53C2×#12318yes—
306Dic53—24no—
307Dic53—42no—
308Dic53—44no—
309Dic53—44no—
310C22×C4×D4—531no—
311C23×C4⋊C4—531no—
312#2487—5316no—
313C2×#400—532no—
314C2×#399—532no—
315C2×#389—531no—
316C2×#389D5311yescharacteristic, complemented
317C2×#386—532no—
318C2×#385—532no—
319C2×#380—531no—
320C2×#380—531no—
321C2×#380—531no—
322C2×C53#248718yescomplemented
323C2×C53C22×C22⋊Q811yescharacteristic
324C2×C53C22×#12311yescharacteristic
325C2×C53C2×#53511yescharacteristic
326C2×C53C2×#55311yescharacteristic
327C2×C53C2×#55311yescharacteristic
328C2×C53C2×#55411yescharacteristic
329C2×C53C2×#55411yescharacteristic
330D53—28no—
331D53—44no—
332C23×D4—531no—
333C24×C4—531no—
334C24×C4—531no—
335C24×C4—532no—
336C2×C4×D4—538no—
337C2×C4×D4—1068no—
338C22×C4⋊C4—532no—
339C22×C4⋊C4—531no—
340C22×C4⋊C4—531no—
341C22×C4⋊C4—531no—
342C22×C4⋊C4—532no—
343C22×C4⋊C4—538no—
344C22×C4⋊C4—1068no—
345C22×C4⋊C4C2×D5311yescharacteristic
346C22×C4⋊C4C2×D5311yescharacteristic
347C22×C4⋊C4C2×D5312yes—
348C22×C22⋊C4—531no—
349C22×C22⋊C4—531no—
350C22×C22⋊C4—532no—
351C22×C22⋊C4—532no—
352C22×C22⋊C4—532no—
353C22×C42C2×D5311yescharacteristic
354#400—5316no—
355#399—5316no—
356#389—538no—
357#389C2×D5318yescomplemented
358#386—5316no—
359#385—5316no—
360#380—538no—
361#380—538no—
362#380—538no—
363C2×#95—532no—
364C2×#95—532no—
365C2×#95—532no—
366C2×#95—532no—
367C2×#95C2×D5312yes—
368C53C2×#248711yescharacteristic, complemented, Sylow
369C25—531no—
370C25—531no—
371C22×D4—538no—
372C22×D4—534no—
373C22×D4—1064no—
374C22×D4—1064no—
375C23×C4—531no—
376C23×C4—538no—
377C23×C4—534no—
378C23×C4—532no—
379C23×C4—532no—
380C23×C4—531no—
381C23×C4—534no—
382C23×C4—538no—
383C23×C4—1068no—
384C23×C4—1064no—
385C23×C4—1064no—
386C23×C4—1064no—
387C23×C4—1064no—
388C23×C4—1064no—
389C23×C4—2124no—
390C23×C4—2124no—
391C23×C4C22×D5311yescharacteristic
392C23×C4C22×D5312yes—
393C23×C4C22×D5312yes—
394C23×C4C22×D5311yescharacteristic
395C23×C4C22×D5311yescharacteristic
396C4×D4—10632no—
397C2×C4⋊C4—24no—
398C2×C4⋊C4—24no—
399C2×C4⋊C4—44no—
400C2×C4⋊C4—538no—
401C2×C4⋊C4—534no—
402C2×C4⋊C4—534no—
403C2×C4⋊C4—534no—
404C2×C4⋊C4—538no—
405C2×C4⋊C4—10632no—
406C2×C4⋊C4—1068no—
407C2×C4⋊C4—1064no—
408C2×C4⋊C4—1064no—
409C2×C4⋊C4—1064no—
410C2×C4⋊C4—2124no—
411C2×C4⋊C4C22×D5314yes—
412C2×C4⋊C4C22×D5314yes—
413C2×C4⋊C4C22×D5318yes—
414C2×C22⋊C4—534no—
415C2×C22⋊C4—534no—
416C2×C22⋊C4—538no—
417C2×C22⋊C4—538no—
418C2×C22⋊C4—538no—
419C2×C22⋊C4—1064no—
420C2×C22⋊C4—1064no—
421C2×C22⋊C4—1068no—
422C2×C22⋊C4—2124no—
423C2×C22⋊C4—2124no—
424C2×C42—24no—
425C2×C42C22×D5314yes—
426#95—538no—
427#95—538no—
428#95—538no—
429#95—538no—
430#95C22×D5318yes—
431C24—534no—
432C24—534no—
433C24—534no—
434C24—532no—
435C24—534no—
436C24—532no—
437C24—1062no—
438C24—1062no—
439C24—1062no—
440C24—1062no—
441C24—1062no—
442C24—1062no—
443C24—1062no—
444C24—1062no—
445C24—1062no—
446C24—1062no—
447C24C23×D5311yescharacteristic
448C2×D4—5316no—
449C2×D4—1068no—
450C2×D4—10632no—
451C2×D4—1068no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—534no—
464C22×C4—532no—
465C22×C4—5316no—
466C22×C4—532no—
467C22×C4—534no—
468C22×C4—534no—
469C22×C4—10616no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—10632no—
473C22×C4—1068no—
474C22×C4—1068no—
475C22×C4—1062no—
476C22×C4—1064no—
477C22×C4—1064no—
478C22×C4—1062no—
479C22×C4—1064no—
480C22×C4—1062no—
481C22×C4—1064no—
482C22×C4—1064no—
483C22×C4—1064no—
484C22×C4—1068no—
485C22×C4—1068no—
486C22×C4—1068no—
487C22×C4—2128no—
488C22×C4—2128no—
489C22×C4—21232no—
490C22×C4—21216no—
491C22×C4#1171994514yes—
492C22×C4D4×D5314yes—
493C22×C4Q8×D5314yes—
494C22×C4#1171994914yes—
495C22×C4C23×D5314yes—
496C22×C4C23×D5314yes—
497C22×C4C23×D5312yes—
498C22×C4C23×D5312yes—
499C22×C4C23×D5312yes—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—48no—
503C4⋊C4—10616no—
504C4⋊C4—1068no—
505C4⋊C4—1068no—
506C4⋊C4—1068no—
507C4⋊C4—2128no—
508C22⋊C4—1068no—
509C22⋊C4—1068no—
510C22⋊C4—10616no—
511C22⋊C4—2128no—
512C22⋊C4—2128no—
513C42—28no—
514C23—5316no—
515C23—534no—
516C23—534no—
517C23—534no—
518C23—534no—
519C23—1068no—
520C23—1068no—
521C23—1068no—
522C23—1068no—
523C23—1068no—
524C23—1068no—
525C23—1062no—
526C23—1062no—
527C23—1062no—
528C23—1062no—
529C23—1068no—
530C23—1068no—
531C23—1068no—
532C23—1062no—
533C23—1062no—
534C23—1062no—
535C23—1062no—
536C23—1062no—
537C23—1062no—
538C23—2124no—
539C23—2124no—
540C23—2124no—
541C23—2124no—
542C23C2×D4×D5312yes—
543C23C2×Q8×D5312yes—
544C23#41×D5312yes—
545C23C2×#1171994512yes—
546C23C2×#1171994912yes—
547C23#1313326212yes—
548C23#1313326312yes—
549C23C24×D5311yescharacteristic
550D4—10632no—
551C2×C4—28no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—48no—
564C2×C4—44no—
565C2×C4—48no—
566C2×C4—534no—
567C2×C4—1068no—
568C2×C4—1062no—
569C2×C4—1062no—
570C2×C4—10632no—
571C2×C4—1062no—
572C2×C4—1064no—
573C2×C4—1064no—
574C2×C4—1062no—
575C2×C4—1064no—
576C2×C4—1062no—
577C2×C4—1064no—
578C2×C4—1064no—
579C2×C4—1064no—
580C2×C4—21232no—
581C2×C4—2128no—
582C2×C4—2124no—
583C2×C4—2128no—
584C2×C4—21216no—
585C2×C4#13133202116yes—
586C2×C4C2×D4×D5314yes—
587C2×C4C2×Q8×D5314yes—
588C2×C4C2×#1171994514yes—
589C2×C4C2×#1171994914yes—
590C22—538no—
591C22—1064no—
592C22—1064no—
593C22—1064no—
594C22—1064no—
595C22—10632no—
596C22—1064no—
597C22—1064no—
598C22—1064no—
599C22—1064no—
600C22—1064no—
601C22—2122no—
602C22—2122no—
603C22—2122no—
604C22—2122no—
605C22—21216no—
606C22C22⋊Q8×D5314yes—
607C22C2×#1313320214yes—
608C22#1681118914yes—
609C22#1681132314yes—
610C22#1681132614yes—
611C22#1681133314yes—
612C22#1681133414yes—
613C22C22×D4×D5311yescharacteristic
614C22C22×Q8×D5311yescharacteristic
615C22C2×#41×D5311yescharacteristic
616C22C22×#1171994511yescharacteristic
617C22C22×#1171994911yescharacteristic
618C22C2×#1313326211yescharacteristic
619C22C2×#1313326311yescharacteristic
620C4—24no—
621C4—44no—
622C4—42no—
623C4—44no—
624C4—1064no—
625C4—2124no—
626C4—2122no—
627C4—2124no—
628C4C2×#1313320218yes—
629C2—1068no—
630C2—2124no—
631C2#2027903018yesdirect product factor
632C2C2×C22⋊Q8×D5311yescharacteristic
633C2C22×#1313320211yescharacteristic
634C2C2×#1681118911yescharacteristic
635C2C2×#1681132311yescharacteristic
636C2C2×#1681132611yescharacteristic
637C2C2×#1681133311yescharacteristic
638C2C2×#1681133411yescharacteristic
subgroupG/H
centerC24C23×D53
commutator, cosocleC23×C53C25
FrattiniC23C24×D53
FittingC2×C53×#389C2
socleC24×C53C24
prime radicalC23×D212C22
hypercenterC2×#389D53
hypocenterC53C2×#2487
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C4×D53C2semidirect
3. #20279030C2direct (N × H)
4. C2×#16810565C2semidirect
5. C2×#16810558C2semidirect
6. C2×#16810521C2semidirect
8. C2×#16810494C2semidirect
9. C2×#16810480C2semidirect
10. C2×#16810480C2semidirect
11. C2×#16810480C2semidirect
12. C2×C53×#389C2semidirect
25. C2×C4⋊C4×D53C22semidirect
31. #16810565C22semidirect
32. #16810558C22semidirect
33. #16810521C22semidirect
35. #16810494C22semidirect
36. #16810480C22semidirect
37. #16810480C22semidirect
38. #16810480C22semidirect
47. C53×#389C22semidirect
316. C2×#389D53semidirect
322. C2×C53#2487semidirect
357. #389C2×D53semidirect
368. C53C2×#2487semidirect
631. C2#20279030direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×D2122
C22×C4⋊C4×D532
#202790302
C2×#168105652
C2×#168105582
C2×#168105212
C2×#168105012
C2×#168104942
C2×#168104802
C2×C53×#3892
C2×#248753