Finite Group Databasebeta

← C2×#20279010 #27553608

Subgroups — C2×#20279010 #27553608

Order 13568.

3070 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D212C211yesmaximal, characteristic
2C22×#13133175C211yesmaximal, characteristic, complemented
3#20279010C2116yesmaximal, direct product factor
4C2×#16810552C212yesmaximal, complemented
5C2×#16810540C212yesmaximal
6C2×#16810521C211yesmaximal, characteristic, complemented
7C2×#16810512C212yesmaximal, complemented
8C2×#16810509C212yesmaximal, 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
16C2×C4×D212—28no—
17C2×C4×D212C2218yes—
18C2×#13133175—28no—
19C2×#13133175C2218yescomplemented
20C22×C4⋊Dic53C2211yescharacteristic
21C22×C4⋊Dic53C2211yescharacteristic
22C22×C4⋊Dic53C2211yescharacteristic
23C22×#11719919C2212yes—
24C22×#11719921C2212yes—
25C22×#11719921C2212yes—
26C22×#11719921C2211yescharacteristic
27C22×#11719921C2211yescharacteristic
28C22×#11719921C2212yes—
29C22×#11719921C2212yes—
30C22×C4×Dic53C2212yes—
31#16810552C22116yescomplemented
32#16810540C22116yes—
33#16810521C2218yescomplemented
34#16810512C22116yescomplemented
35#16810509C22116yescomplemented
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×D212—24no—
54C22×D212C2314yes—
55C23×Dic53C2312yes—
56C23×Dic53C2312yes—
57C23×Dic53C2311yescharacteristic
58C23×Dic53C2311yescharacteristic
59C22×C4×D53—24no—
60C22×C4×D53—24no—
61C22×C4×D53—24no—
62C22×C4×D53—24no—
63C22×C4×D53—24no—
64C22×C4×D53D418yes—
65C22×C4×D53C2314yes—
66C22×C4×D53C2314yes—
67C4×D212—232no—
68#13133175—232no—
69C2×C4⋊Dic53—24no—
70C2×C4⋊Dic53—24no—
71C2×C4⋊Dic53—24no—
72C2×C4⋊Dic53C2314yes—
73C2×C4⋊Dic53C2314yes—
74C2×C4⋊Dic53C2314yes—
75C2×#11719919—44no—
76C2×#11719919C2318yes—
77C2×#11719921—28no—
78C2×#11719921—24no—
79C2×#11719921—28no—
80C2×#11719921—24no—
81C2×#11719921—44no—
82C2×#11719921—44no—
83C2×#11719921C2314yes—
84C2×#11719921C2318yes—
85C2×#11719921C2318yes—
86C2×#11719921C2314yes—
87C2×#11719921C2318yes—
88C2×#11719921C2318yes—
89C2×C4×Dic53—28no—
90C2×C4×Dic53C2318yes—
91#13133124C2318yes—
92#13133124C2318yes—
93#13133124C2318yes—
94#13133124C2318yes—
95C23×C4×C53C2311yescharacteristic
96C23×C4×C53C2312yes—
97C23×C4×C53C2312yes—
98C23×C4×C53C2311yescharacteristic
99C23×C4×C53C2311yescharacteristic
100C2×C4⋊C4×C53—24no—
101C2×C4⋊C4×C53—28no—
102C2×C4⋊C4×C53—24no—
103C2×C4⋊C4×C53C2318yes—
104C2×C4⋊C4×C53C2314yes—
105C2×C4⋊C4×C53C2314yes—
106C2×C42×C53—24no—
107C2×C42×C53C2314yes—
108#95×C53C2318yes—
109C24×C53C2411yescharacteristic
110C23×D53—22no—
111C23×D53—22no—
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×D53#4114yes—
124C23×D53C2412yes—
125C23×D53C2412yes—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—24no—
130C22×C4×C53—24no—
131C22×C4×C53—24no—
132C22×C4×C53—24no—
133C22×C4×C53—24no—
134C22×C4×C53—24no—
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53C2×D414yes—
138C22×C4×C53#4114yes—
139C22×C4×C53#4114yes—
140C22×C4×C53#4114yes—
141C22×C4×C53C2412yes—
142C22×C4×C53C2414yes—
143C22×C4×C53C2414yes—
144C22×C4×C53C2412yes—
145C22×C4×C53C2412yes—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—24no—
149C22×Dic53—22no—
150C22×Dic53—24no—
151C22×Dic53—24no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—24no—
155C22×Dic53—24no—
156C22×Dic53—24no—
157C22×Dic53C2×D414yes—
158C22×Dic53C2412yes—
159C22×Dic53C2414yes—
160C22×Dic53C2414yes—
161C22×Dic53C2412yes—
162C2×D212—232no—
163C2×D212—28no—
164C2×D212—28no—
165C2×D212—28no—
166C2×C4×D53—232no—
167C2×C4×D53—28no—
168C2×C4×D53—28no—
169C2×C4×D53—28no—
170C2×C4×D53—28no—
171C2×C4×D53—28no—
172C2×C4×D53—416no—
173C2×C4×D53C2×D4116yes—
174C4⋊C4×C53—28no—
175C4⋊C4×C53—216no—
176C4⋊C4×C53—28no—
177C42×C53—28no—
178#11719921—216no—
179#11719921—28no—
180#11719921—216no—
181#11719921—28no—
182#11719921—48no—
183#11719921—48no—
184C4⋊Dic53—28no—
185C4⋊Dic53—28no—
186C4⋊Dic53—28no—
187#11719919—48no—
188C4×Dic53—216no—
189C23×C53C22×D412yes—
190C23×C53C2×#4112yes—
191C23×C53C2×#4112yes—
192C23×C53C2×#4112yes—
193C23×C53C2×#4112yes—
194C23×C53H(5,𝔽2)12yes—
195C23×C532−1+412yes—
196C23×C53C2511yescharacteristic
197C22×D53—28no—
198C22×D53—28no—
199C22×D53—28no—
200C22×D53—28no—
201C22×D53—28no—
202C22×D53—28no—
203C22×D53—22no—
204C22×D53—22no—
205C22×D53—22no—
206C22×D53—22no—
207C22×D53—22no—
208C22×D53—22no—
209C22×D53—22no—
210C22×D53—22no—
211C22×D53—22no—
212C22×D53—22no—
213C22×D53—22no—
214C22×D53—22no—
215C22×D53—44no—
216C22×D53—44no—
217C22×D53—44no—
218C22×D53—44no—
219C22×D53—44no—
220C22×D53—44no—
221C22×D53—44no—
222C22×D53—44no—
223C22×D53C22×D414yes—
224C22×D53C2×#4114yes—
225C2×C4×C53—28no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—24no—
229C2×C4×C53—22no—
230C2×C4×C53—24no—
231C2×C4×C53—24no—
232C2×C4×C53—24no—
233C2×C4×C53—24no—
234C2×C4×C53—22no—
235C2×C4×C53—24no—
236C2×C4×C53—22no—
237C2×C4×C53—48no—
238C2×C4×C53—44no—
239C2×C4×C53—48no—
240C2×C4×C53#123116yes—
241C2×C4×C53C22×D414yes—
242C2×C4×C53C2×#4114yes—
243C2×C4×C53C2×#4114yes—
244C2×C4×C53C2×#4114yes—
245C2×Dic53—28no—
246C2×Dic53—22no—
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—24no—
250C2×Dic53—22no—
251C2×Dic53—22no—
252C2×Dic53—24no—
253C2×Dic53—24no—
254C2×Dic53—24no—
255C2×Dic53—24no—
256C2×Dic53—24no—
257C2×Dic53—44no—
258C2×Dic53—48no—
259C2×Dic53—48no—
260C2×Dic53C22×D414yes—
261D212—232no—
262C4×D53—232no—
263C4×D53—416no—
264C2×#2485—531nomaximal, Sylow
265C22×C53C2×#12314yes—
266C22×C53#52514yes—
267C22×C53#53714yes—
268C22×C53#55214yes—
269C22×C53#55314yes—
270C22×C53#55514yes—
271C22×C53#55614yes—
272C22×C53C23×D411yescharacteristic
273C22×C53C22×#4111yescharacteristic
274C22×C53C22×#4111yescharacteristic
275C22×C53C22×#4111yescharacteristic
276C22×C53C22×#4111yescharacteristic
277C22×C53C2×H(5,𝔽2)11yescharacteristic
278C22×C53C2×2−1+411yescharacteristic
279C2×D53—24no—
280C2×D53—24no—
281C2×D53—24no—
282C2×D53—24no—
283C2×D53—24no—
284C2×D53—24no—
285C2×D53—42no—
286C2×D53—42no—
287C2×D53—42no—
288C2×D53—42no—
289C2×D53—42no—
290C2×D53—42no—
291C2×D53—42no—
292C2×D53—42no—
293C2×D53—416no—
294C2×D53—416no—
295C4×C53—24no—
296C4×C53—44no—
297C4×C53—42no—
298C4×C53—44no—
299C4×C53C2×#12318yes—
300Dic53—24no—
301Dic53—42no—
302Dic53—44no—
303Dic53—44no—
304C22×C4×D4—531no—
305C22×#117—531no—
306#2485—5316no—
307C2×#397—532no—
308C2×#394—532no—
309C2×#389—531no—
310C2×#389D5311yescharacteristic, complemented
311C2×#388—532no—
312C2×#387—532no—
313C2×#380—531no—
314C2×#380—531no—
315C2×#380—531no—
316C2×C53#248518yescomplemented
317C2×C53C22×#12311yescharacteristic
318C2×C53C2×#52511yescharacteristic
319C2×C53C2×#53711yescharacteristic
320C2×C53C2×#55211yescharacteristic
321C2×C53C2×#55311yescharacteristic
322C2×C53C2×#55511yescharacteristic
323C2×C53C2×#55611yescharacteristic
324D53—44no—
325D53—44no—
326C23×D4—531no—
327C24×C4—531no—
328C24×C4—531no—
329C2×C4×D4—538no—
330C2×C4×D4—1068no—
331C2×#117—538no—
332C2×#117—1068no—
333C22×C4⋊C4—531no—
334C22×C4⋊C4—531no—
335C22×C4⋊C4—531no—
336C22×C4⋊C4—532no—
337C22×C4⋊C4C2×D5312yes—
338C22×C4⋊C4C2×D5311yescharacteristic
339C22×C4⋊C4C2×D5311yescharacteristic
340C22×C22⋊C4—532no—
341C22×C22⋊C4—532no—
342C22×C22⋊C4—531no—
343C22×C22⋊C4—531no—
344C22×C22⋊C4—532no—
345C22×C22⋊C4—532no—
346C22×C42—532no—
347C22×C42C2×D5311yescharacteristic
348#397—5316no—
349#394—5316no—
350#389—538no—
351#389C2×D5318yescomplemented
352#388—5316no—
353#387—5316no—
354#380—538no—
355#380—538no—
356#380—538no—
357C2×#95—532no—
358C2×#95—532no—
359C2×#95—532no—
360C2×#95—532no—
361C2×#95C2×D5312yes—
362C53C2×#248511yescharacteristic, complemented, Sylow
363C25—531no—
364C25—531no—
365C22×D4—534no—
366C22×D4—1064no—
367C22×D4—1064no—
368C22×D4—1064no—
369C23×C4—531no—
370C23×C4—538no—
371C23×C4—534no—
372C23×C4—532no—
373C23×C4—532no—
374C23×C4—531no—
375C23×C4—534no—
376C23×C4—1064no—
377C23×C4—1064no—
378C23×C4—1064no—
379C23×C4—1064no—
380C23×C4—1064no—
381C23×C4C22×D5311yescharacteristic
382C23×C4C22×D5312yes—
383C23×C4C22×D5312yes—
384C23×C4C22×D5311yescharacteristic
385C23×C4C22×D5311yescharacteristic
386C4×D4—10632no—
387#117—10632no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—28no—
391C2×C4⋊C4—534no—
392C2×C4⋊C4—534no—
393C2×C4⋊C4—534no—
394C2×C4⋊C4—538no—
395C2×C4⋊C4—1064no—
396C2×C4⋊C4—1064no—
397C2×C4⋊C4—1064no—
398C2×C4⋊C4—2124no—
399C2×C4⋊C4C22×D5314yes—
400C2×C4⋊C4C22×D5314yes—
401C2×C4⋊C4C22×D5318yes—
402C2×C22⋊C4—538no—
403C2×C22⋊C4—534no—
404C2×C22⋊C4—534no—
405C2×C22⋊C4—538no—
406C2×C22⋊C4—538no—
407C2×C22⋊C4—538no—
408C2×C22⋊C4—1064no—
409C2×C22⋊C4—1068no—
410C2×C22⋊C4—1064no—
411C2×C22⋊C4—1068no—
412C2×C22⋊C4—2124no—
413C2×C22⋊C4—2124no—
414C2×C42—24no—
415C2×C42—538no—
416C2×C42—1068no—
417C2×C42C22×D5314yes—
418#95—538no—
419#95—538no—
420#95—538no—
421#95—538no—
422#95C22×D5318yes—
423C24—534no—
424C24—534no—
425C24—532no—
426C24—532no—
427C24—1062no—
428C24—1062no—
429C24—1062no—
430C24—1062no—
431C24—1062no—
432C24—1062no—
433C24—1062no—
434C24—1062no—
435C24—1062no—
436C24—1062no—
437C24—1062no—
438C24—1062no—
439C24C23×D5311yescharacteristic
440C2×D4—1068no—
441C2×D4—1068no—
442C2×D4—10632no—
443C2×D4—1068no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—534no—
456C22×C4—532no—
457C22×C4—5316no—
458C22×C4—532no—
459C22×C4—534no—
460C22×C4—534no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—10632no—
464C22×C4—1068no—
465C22×C4—1068no—
466C22×C4—1064no—
467C22×C4—1062no—
468C22×C4—1064no—
469C22×C4—1062no—
470C22×C4—1064no—
471C22×C4—1062no—
472C22×C4—1064no—
473C22×C4—1064no—
474C22×C4—1064no—
475C22×C4—1068no—
476C22×C4—1068no—
477C22×C4—1068no—
478C22×C4—21216no—
479C22×C4#1171994514yes—
480C22×C4D4×D5314yes—
481C22×C4#1171994714yes—
482C22×C4#1171994914yes—
483C22×C4C23×D5314yes—
484C22×C4C23×D5314yes—
485C22×C4C23×D5312yes—
486C22×C4C23×D5312yes—
487C22×C4C23×D5312yes—
488C4⋊C4—28no—
489C4⋊C4—216no—
490C4⋊C4—28no—
491C4⋊C4—1068no—
492C4⋊C4—1068no—
493C4⋊C4—1068no—
494C4⋊C4—2128no—
495C22⋊C4—1068no—
496C22⋊C4—10616no—
497C22⋊C4—1068no—
498C22⋊C4—10616no—
499C22⋊C4—2128no—
500C22⋊C4—2128no—
501C42—28no—
502C42—10616no—
503C23—534no—
504C23—534no—
505C23—1068no—
506C23—1068no—
507C23—1068no—
508C23—1068no—
509C23—1068no—
510C23—1062no—
511C23—1062no—
512C23—1062no—
513C23—1062no—
514C23—1062no—
515C23—1062no—
516C23—1062no—
517C23—1062no—
518C23—1062no—
519C23—1062no—
520C23—1062no—
521C23—1062no—
522C23—1068no—
523C23—2124no—
524C23—2124no—
525C23—2124no—
526C23—2124no—
527C23—2124no—
528C23—2124no—
529C23—2124no—
530C23—2124no—
531C23C2×D4×D5312yes—
532C23#41×D5312yes—
533C23C2×#1171994712yes—
534C23C2×#1171994512yes—
535C23C2×#1171994912yes—
536C23#1313326312yes—
537C23#1313326512yes—
538C23C24×D5311yescharacteristic
539D4—10632no—
540C2×C4—28no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—48no—
553C2×C4—48no—
554C2×C4—44no—
555C2×C4—534no—
556C2×C4—1068no—
557C2×C4—1062no—
558C2×C4—1062no—
559C2×C4—10632no—
560C2×C4—1064no—
561C2×C4—1062no—
562C2×C4—1064no—
563C2×C4—1062no—
564C2×C4—1064no—
565C2×C4—1062no—
566C2×C4—1064no—
567C2×C4—1064no—
568C2×C4—1064no—
569C2×C4—2128no—
570C2×C4—2124no—
571C2×C4—2128no—
572C2×C4—21216no—
573C2×C4#13133202116yes—
574C2×C4C2×D4×D5314yes—
575C2×C4C2×#1171994714yes—
576C2×C4C2×#1171994514yes—
577C2×C4C2×#1171994914yes—
578C22—1064no—
579C22—1064no—
580C22—1064no—
581C22—1064no—
582C22—1064no—
583C22—1064no—
584C22—2122no—
585C22—2122no—
586C22—2122no—
587C22—2122no—
588C22—21216no—
589C22—21216no—
590C22—2122no—
591C22—2122no—
592C22—2122no—
593C22—2122no—
594C22C2×#1313320214yes—
595C22#1681113314yes—
596C22#1681120314yes—
597C22#1681131414yes—
598C22#1681132614yes—
599C22#1681133814yes—
600C22#1681134714yes—
601C22C22×D4×D5311yescharacteristic
602C22C2×#41×D5311yescharacteristic
603C22C22×#1171994711yescharacteristic
604C22C22×#1171994511yescharacteristic
605C22C22×#1171994911yescharacteristic
606C22C2×#1313326311yescharacteristic
607C22C2×#1313326511yescharacteristic
608C4—24no—
609C4—42no—
610C4—44no—
611C4—44no—
612C4—1064no—
613C4—2124no—
614C4—2122no—
615C4—2124no—
616C4C2×#1313320218yes—
617C2—2124no—
618C2—2124no—
619C2#2027901018yesdirect product factor
620C2C22×#1313320211yescharacteristic
621C2C2×#1681113311yescharacteristic
622C2C2×#1681120311yescharacteristic
623C2C2×#1681131411yescharacteristic
624C2C2×#1681132611yescharacteristic
625C2C2×#1681133811yescharacteristic
626C2C2×#1681134711yescharacteristic
subgroupG/H
centerC24C23×D53
commutator, cosocleC23×C53C25
FrattiniC23C24×D53
FittingC2×C53×#389C2
socleC24×C53C24
prime radicalC23×D212C22
hypercenterC2×#389D53
hypocenterC53C2×#2485
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
2. C22×#13133175C2semidirect
3. #20279010C2direct (N × H)
4. C2×#16810552C2semidirect
6. C2×#16810521C2semidirect
7. C2×#16810512C2semidirect
8. C2×#16810509C2semidirect
9. C2×#16810480C2semidirect
10. C2×#16810480C2semidirect
11. C2×#16810480C2semidirect
12. C2×C53×#389C2semidirect
19. C2×#13133175C22semidirect
31. #16810552C22semidirect
33. #16810521C22semidirect
34. #16810512C22semidirect
35. #16810509C22semidirect
36. #16810480C22semidirect
37. #16810480C22semidirect
38. #16810480C22semidirect
47. C53×#389C22semidirect
310. C2×#389D53semidirect
316. C2×C53#2485semidirect
351. #389C2×D53semidirect
362. C53C2×#2485semidirect
619. C2#20279010direct (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×#131331752
#202790102
C2×#168105522
C2×#168105402
C2×#168105212
C2×#168105122
C2×#168105092
C2×#168104802
C2×C53×#3892
C2×#248553