Finite Group Databasebeta

← C2×#20278051 #27552784

Subgroups — C2×#20278051 #27552784

Order 13568.

5718 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11719949C211yesmaximal, characteristic
2C22×#13133202C212yesmaximal, complemented
3C22×C4⋊C4×D53C211yesmaximal, characteristic, complemented
4#20278051C2116yesmaximal, direct product factor
5C2×#16810542C211yesmaximal, characteristic, complemented
6C2×#16810541C211yesmaximal, characteristic, complemented
7C2×#16810540C211yesmaximal, characteristic, complemented
8C2×#16810501C212yesmaximal, complemented
9C2×#16810497C212yesmaximal, complemented
10C2×#16810494C212yesmaximal, complemented
11C2×#16810485C211yesmaximal, characteristic, complemented
12C2×C53×#395C211yesmaximal, characteristic, complemented
13C22×#11719949—216no—
14C22×#11719949—28no—
15C22×#11719949C2218yes—
16C23×D212C2211yescharacteristic
17C23×D212C2212yes—
18C23×C4×D53C2211yescharacteristic
19C23×C4×D53C2211yescharacteristic
20C23×C4×D53C2212yes—
21C23×C4×D53C2211yescharacteristic
22C2×#13133202—216no—
23C2×#13133202C22116yescomplemented
24C22×C4⋊Dic53C2211yescharacteristic
25C22×#11719919C2211yescharacteristic
26C22×#11719919C2211yescharacteristic
27C22×#11719919C2212yes—
28C2×C4⋊C4×D53—28no—
29C2×C4⋊C4×D53C2218yescomplemented
30C22×#11719921C2212yes—
31C22×#11719921C2212yes—
32C22×#11719921C2212yes—
33C22×#11719921C2212yes—
34C22×#11719921C2212yes—
35#16810542C2218yescomplemented
36#16810541C2218yescomplemented
37#16810540C2218yescomplemented
38#16810501C22116yescomplemented
39#16810497C22116yescomplemented
40#16810494C22116yescomplemented
41#16810485C2218yescomplemented
42C2×#13133124C2212yes—
43C2×#13133124C2211yescharacteristic
44C2×#13133124C2212yes—
45C23×Q8×C53C2211yescharacteristic
46C22×C4⋊C4×C53C2211yescharacteristic
47C22×C4⋊C4×C53C2212yes—
48C53×#395C2218yescomplemented
49C2×#95×C53C2211yescharacteristic
50C2×#95×C53C2212yes—
51C24×D53C2312yes—
52C24×D53C2311yescharacteristic
53C2×#11719949—2128no—
54C2×#11719949—264no—
55C2×#11719949—232no—
56C22×D212—28no—
57C22×D212—28no—
58C22×D212—28no—
59C22×D212—24no—
60C22×D212D4116yes—
61C22×D212C2314yes—
62C22×D212C2318yes—
63C23×Dic53C2312yes—
64C23×Dic53C2311yescharacteristic
65C23×Dic53C2311yescharacteristic
66C23×Dic53C2311yescharacteristic
67C22×C4×D53—24no—
68C22×C4×D53—28no—
69C22×C4×D53—28no—
70C22×C4×D53—28no—
71C22×C4×D53—24no—
72C22×C4×D53—28no—
73C22×C4×D53—24no—
74C22×C4×D53—24no—
75C22×C4×D53—28no—
76C22×C4×D53D418yes—
77C22×C4×D53C2318yes—
78C22×C4×D53C2314yes—
79C22×C4×D53C2314yes—
80C22×C4×D53C2314yes—
81#13133202—264no—
82C2×C4⋊Dic53—24no—
83C2×C4⋊Dic53C2314yes—
84C2×#11719919—24no—
85C2×#11719919—28no—
86C2×#11719919—24no—
87C2×#11719919C2318yes—
88C2×#11719919C2314yes—
89C2×#11719919C2314yes—
90C4⋊C4×D53—232no—
91C2×#11719921—28no—
92C2×#11719921—28no—
93C2×#11719921—28no—
94C2×#11719921—28no—
95C2×#11719921—28no—
96C2×#11719921C2318yes—
97C2×#11719921C2318yes—
98C2×#11719921C2318yes—
99C2×#11719921C2318yes—
100C2×#11719921C2318yes—
101#13133124C2318yes—
102#13133124C2318yes—
103#13133124C2314yes—
104C22×Q8×C53—28no—
105C22×Q8×C53—24no—
106C22×Q8×C53C2314yes—
107C23×C4×C53C2311yescharacteristic
108C23×C4×C53C2312yes—
109C23×C4×C53C2312yes—
110C23×C4×C53C2311yescharacteristic
111C23×C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53—24no—
113C2×C4⋊C4×C53—28no—
114C2×C4⋊C4×C53C2314yes—
115C2×C4⋊C4×C53C2318yes—
116#95×C53C2318yes—
117#95×C53C2314yes—
118C24×C53C2411yescharacteristic
119C23×D53—22no—
120C23×D53—22no—
121C23×D53—24no—
122C23×D53—24no—
123C23×D53—24no—
124C23×D53—24no—
125C23×D53—24no—
126C23×D53—24no—
127C23×D53—24no—
128C23×D53C2×D418yes—
129C23×D53C2×D418yes—
130C23×D53C2×D414yes—
131C23×D53C2414yes—
132C23×D53C2412yes—
133C2×Q8×C53—232no—
134C2×Q8×C53—216no—
135C2×Q8×C53—28no—
136C22×C4×C53—22no—
137C22×C4×C53—24no—
138C22×C4×C53—24no—
139C22×C4×C53—24no—
140C22×C4×C53—22no—
141C22×C4×C53—24no—
142C22×C4×C53—24no—
143C22×C4×C53—22no—
144C22×C4×C53—24no—
145C22×C4×C53—24no—
146C22×C4×C53C2×D418yes—
147C22×C4×C53C2×D414yes—
148C22×C4×C53#4114yescomplemented
149C22×C4×C53C2414yes—
150C22×C4×C53C2412yes—
151C22×C4×C53C2414yes—
152C22×C4×C53C2412yes—
153C22×C4×C53C2412yes—
154C22×Dic53—22no—
155C22×Dic53—24no—
156C22×Dic53—22no—
157C22×Dic53—24no—
158C22×Dic53—24no—
159C22×Dic53—22no—
160C22×Dic53—24no—
161C22×Dic53—24no—
162C22×Dic53C2×D414yes—
163C22×Dic53#4114yescomplemented
164C22×Dic53C2414yes—
165C22×Dic53C2412yes—
166C22×Dic53C2412yes—
167C22×Dic53C2412yes—
168#11719949—2256no—
169C2×D212—232no—
170C2×D212—264no—
171C2×D212—216no—
172C2×D212—216no—
173C2×D212—216no—
174C2×D212—28no—
175C2×D212C2×D4132yes—
176C2×C4×D53—232no—
177C2×C4×D53—264no—
178C2×C4×D53—232no—
179C2×C4×D53—28no—
180C2×C4×D53—216no—
181C2×C4×D53—216no—
182C2×C4×D53—216no—
183C2×C4×D53—28no—
184C2×C4×D53—216no—
185C2×C4×D53—28no—
186C2×C4×D53—28no—
187C2×C4×D53—216no—
188C2×C4×D53—416no—
189C2×C4×D53C2×D4116yes—
190C4⋊C4×C53—28no—
191C4⋊C4×C53—216no—
192#11719921—216no—
193#11719921—216no—
194#11719921—216no—
195#11719921—216no—
196#11719921—216no—
197C4⋊Dic53—28no—
198#11719919—28no—
199#11719919—216no—
200#11719919—28no—
201C23×C53C22×D412yes—
202C23×C53C22×D414yes—
203C23×C53C2×#4112yes—
204C23×C53C2×#4112yes—
205C23×C532−1+414yes—
206C23×C53C2511yescharacteristic
207C22×D53—28no—
208C22×D53—28no—
209C22×D53—216no—
210C22×D53—216no—
211C22×D53—22no—
212C22×D53—22no—
213C22×D53—24no—
214C22×D53—216no—
215C22×D53—216no—
216C22×D53—216no—
217C22×D53—24no—
218C22×D53—24no—
219C22×D53—24no—
220C22×D53—24no—
221C22×D53—24no—
222C22×D53—24no—
223C22×D53—48no—
224C22×D53—48no—
225C22×D53—48no—
226C22×D53—48no—
227C22×D53#120116yes—
228C22×D53C22×D418yes—
229C22×D53C22×D414yes—
230C22×D53C22×D418yes—
231Q8×C53—232no—
232C2×C4×C53—216no—
233C2×C4×C53—28no—
234C2×C4×C53—28no—
235C2×C4×C53—24no—
236C2×C4×C53—24no—
237C2×C4×C53—24no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—24no—
242C2×C4×C53—24no—
243C2×C4×C53—24no—
244C2×C4×C53—24no—
245C2×C4×C53—44no—
246C2×C4×C53—48no—
247C2×C4×C53C22×D418yes—
248C2×C4×C53C22×D414yes—
249C2×C4×C53C2×#4114yescomplemented
250C2×Dic53—28no—
251C2×Dic53—28no—
252C2×Dic53—24no—
253C2×Dic53—22no—
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—24no—
257C2×Dic53—24no—
258C2×Dic53—24no—
259C2×Dic53—24no—
260C2×Dic53—44no—
261C2×Dic53—48no—
262C2×Dic53C22×D414yes—
263C2×Dic53C2×#4114yescomplemented
264D212—232no—
265D212—264no—
266C4×D53—232no—
267C4×D53—264no—
268C4×D53—232no—
269C4×D53—416no—
270C2×#2406—531nomaximal, Sylow
271C22×C53C2×#12014yes—
272C22×C53#52514yes—
273C22×C53#53614yes—
274C22×C53#54718yes—
275C22×C53#54718yes—
276C22×C53C23×D411yescharacteristic
277C22×C53C23×D412yes—
278C22×C53C22×#4111yescharacteristic
279C22×C53C22×#4111yescharacteristic
280C22×C53C2×2−1+412yes—
281C2×D53—232no—
282C2×D53—24no—
283C2×D53—24no—
284C2×D53—28no—
285C2×D53—28no—
286C2×D53—28no—
287C2×D53—28no—
288C2×D53—28no—
289C2×D53—432no—
290C2×D53—44no—
291C2×D53—44no—
292C2×D53—44no—
293C2×D53—44no—
294C2×D53C2×#12018yes—
295C4×C53—28no—
296C4×C53—24no—
297C4×C53—24no—
298C4×C53—42no—
299C4×C53—44no—
300Dic53—24no—
301Dic53—24no—
302Dic53—42no—
303Dic53—44no—
304C23×#41—531no—
305C22×#123—532no—
306C23×C4⋊C4—531no—
307#2406—5316no—
308C2×#395—531no—
309C2×#395D5311yescharacteristic, complemented
310C2×#394—531no—
311C2×#394—531no—
312C2×#386—532no—
313C2×#385—532no—
314C2×#385—532no—
315C2×#381—531no—
316C2×C53#240618yescomplemented
317C2×C53C22×#12011yescharacteristic
318C2×C53C2×#52511yescharacteristic
319C2×C53C2×#53611yescharacteristic
320C2×C53C2×#54712yes—
321C2×C53C2×#54712yes—
322D53—28no—
323D53—48no—
324C22×#41—538no—
325C22×#41—1068no—
326C22×#41—10616no—
327C23×Q8C2×D5311yescharacteristic
328C23×D4—532no—
329C23×D4—531no—
330C24×C4—531no—
331C24×C4—532no—
332C24×C4—531no—
333C24×C4—531no—
334C2×#123—5316no—
335C2×#123—10616no—
336C22×C4⋊C4—531no—
337C22×C4⋊C4—532no—
338C22×C4⋊C4—531no—
339C22×C4⋊C4—531no—
340C22×C4⋊C4—538no—
341C22×C4⋊C4—1068no—
342C22×C4⋊C4C2×D5311yescharacteristic
343C22×C4⋊C4C2×D5312yes—
344C22×C22⋊C4—532no—
345C22×C22⋊C4—532no—
346C22×C22⋊C4—532no—
347C22×C22⋊C4—532no—
348C22×C22⋊C4—532no—
349#395—538no—
350#395C2×D5318yescomplemented
351#394—538no—
352#394—538no—
353#386—5316no—
354#385—5316no—
355#385—5316no—
356#381—538no—
357C2×#95—531no—
358C2×#95—532no—
359C2×#95—532no—
360C2×#95C2×D5311yescharacteristic
361C2×#95C2×D5312yes—
362C53C2×#240611yescharacteristic, complemented, Sylow
363C25—531no—
364C25—532no—
365C2×#41—10632no—
366C2×#41—10664no—
367C2×#41—106128no—
368C22×Q8—24no—
369C22×Q8—28no—
370C22×Q8C22×D5314yes—
371C22×D4—5316no—
372C22×D4—538no—
373C22×D4—534no—
374C22×D4—1068no—
375C22×D4—1068no—
376C22×D4—1068no—
377C22×D4—1064no—
378C23×C4—531no—
379C23×C4—531no—
380C23×C4—538no—
381C23×C4—534no—
382C23×C4—532no—
383C23×C4—531no—
384C23×C4—534no—
385C23×C4—534no—
386C23×C4—538no—
387C23×C4—1068no—
388C23×C4—1064no—
389C23×C4—1064no—
390C23×C4—1068no—
391C23×C4—1068no—
392C23×C4—1064no—
393C23×C4—1068no—
394C23×C4—1068no—
395C23×C4—1064no—
396C23×C4C22×D5312yes—
397C23×C4C22×D5311yescharacteristic
398C23×C4C22×D5311yescharacteristic
399C23×C4C22×D5312yes—
400C23×C4C22×D5311yescharacteristic
401#123—10664no—
402C2×C4⋊C4—24no—
403C2×C4⋊C4—28no—
404C2×C4⋊C4—534no—
405C2×C4⋊C4—534no—
406C2×C4⋊C4—534no—
407C2×C4⋊C4—538no—
408C2×C4⋊C4—10632no—
409C2×C4⋊C4—1064no—
410C2×C4⋊C4—1064no—
411C2×C4⋊C4—1064no—
412C2×C4⋊C4—1068no—
413C2×C4⋊C4C22×D5314yes—
414C2×C4⋊C4C22×D5318yes—
415C2×C22⋊C4—538no—
416C2×C22⋊C4—538no—
417C2×C22⋊C4—538no—
418C2×C22⋊C4—538no—
419C2×C22⋊C4—538no—
420C2×C22⋊C4—1068no—
421C2×C22⋊C4—1068no—
422C2×C22⋊C4—1068no—
423C2×C22⋊C4—1068no—
424C2×C22⋊C4—1068no—
425#95—538no—
426#95—534no—
427#95—538no—
428#95C22×D5314yes—
429#95C22×D5318yes—
430C24—538no—
431C24—534no—
432C24—532no—
433C24—538no—
434C24—534no—
435C24—1062no—
436C24—1062no—
437C24—1064no—
438C24—1064no—
439C24—1064no—
440C24—1064no—
441C24—1064no—
442C24—1064no—
443C24—1064no—
444C24C23×D5311yescharacteristic
445#41—106256no—
446C2×Q8—232no—
447C2×Q8—28no—
448C2×Q8—216no—
449C2×D4—5332no—
450C2×D4—10664no—
451C2×D4—10616no—
452C2×D4—10616no—
453C2×D4—10616no—
454C2×D4—10632no—
455C2×D4—1068no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—534no—
467C22×C4—534no—
468C22×C4—532no—
469C22×C4—532no—
470C22×C4—5316no—
471C22×C4—534no—
472C22×C4—532no—
473C22×C4—1068no—
474C22×C4—10616no—
475C22×C4—1068no—
476C22×C4—10632no—
477C22×C4—10616no—
478C22×C4—10616no—
479C22×C4—10632no—
480C22×C4—10664no—
481C22×C4—1068no—
482C22×C4—10616no—
483C22×C4—1062no—
484C22×C4—1064no—
485C22×C4—1062no—
486C22×C4—10616no—
487C22×C4—1068no—
488C22×C4—1064no—
489C22×C4—1064no—
490C22×C4—1062no—
491C22×C4—1064no—
492C22×C4—1064no—
493C22×C4—21216no—
494C22×C4D4×D5314yes—
495C22×C4D4×D5318yes—
496C22×C4#1171994914yescomplemented
497C22×C4C23×D5314yes—
498C22×C4C23×D5312yes—
499C22×C4C23×D5312yes—
500C22×C4C23×D5314yes—
501C22×C4C23×D5312yes—
502C4⋊C4—28no—
503C4⋊C4—216no—
504C4⋊C4—1068no—
505C4⋊C4—1068no—
506C4⋊C4—1068no—
507C4⋊C4—10616no—
508C22⋊C4—10616no—
509C22⋊C4—10616no—
510C22⋊C4—10616no—
511C22⋊C4—10616no—
512C22⋊C4—10616no—
513C23—5316no—
514C23—538no—
515C23—534no—
516C23—538no—
517C23—10616no—
518C23—10616no—
519C23—1068no—
520C23—1062no—
521C23—1062no—
522C23—1068no—
523C23—1064no—
524C23—10616no—
525C23—10616no—
526C23—10616no—
527C23—1064no—
528C23—1064no—
529C23—1064no—
530C23—1064no—
531C23—1064no—
532C23—1064no—
533C23—2128no—
534C23—2128no—
535C23—2128no—
536C23—2128no—
537C23C2×D4×D5314yes—
538C23C2×D4×D5312yes—
539C23#41×D5312yes—
540C23C2×#1171994912yes—
541C23#1313326514yes—
542C23C24×D5311yescharacteristic
543Q8—232no—
544D4—10664no—
545D4—10632no—
546C2×C4—28no—
547C2×C4—216no—
548C2×C4—28no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—48no—
560C2×C4—44no—
561C2×C4—534no—
562C2×C4—534no—
563C2×C4—10632no—
564C2×C4—10632no—
565C2×C4—10664no—
566C2×C4—1068no—
567C2×C4—1068no—
568C2×C4—1062no—
569C2×C4—1064no—
570C2×C4—1062no—
571C2×C4—1064no—
572C2×C4—1064no—
573C2×C4—1062no—
574C2×C4—1064no—
575C2×C4—1064no—
576C2×C4—21216no—
577C2×C4—2128no—
578C2×C4—2124no—
579C2×C4C2×D4×D5314yes—
580C2×C4C2×D4×D5318yes—
581C2×C4C2×#1171994914yescomplemented
582C22—538no—
583C22—1064no—
584C22—10632no—
585C22—1068no—
586C22—1068no—
587C22—1064no—
588C22—1068no—
589C22—1068no—
590C22—1068no—
591C22—2124no—
592C22—21232no—
593C22—2124no—
594C22—2124no—
595C22—2124no—
596C22#120×D5314yes—
597C22#1681113614yes—
598C22#1681119814yes—
599C22#1681127518yes—
600C22#1681127618yes—
601C22C22×D4×D5311yescharacteristic
602C22C22×D4×D5312yes—
603C22C2×#41×D5311yescharacteristic
604C22C22×#1171994911yescharacteristic
605C22C2×#1313326512yes—
606C4—24no—
607C4—28no—
608C4—24no—
609C4—44no—
610C4—42no—
611C4—1064no—
612C4—1064no—
613C4—2124no—
614C4—2122no—
615C2—1068no—
616C2—2128no—
617C2#2027805118yesdirect product factor
618C2C2×#120×D5311yescharacteristic
619C2C2×#1681113611yescharacteristic
620C2C2×#1681119811yescharacteristic
621C2C2×#1681127512yes—
622C2C2×#1681127612yes—
subgroupG/H
centerC24C23×D53
commutator, cosocleC23×C53C25
FrattiniC23C24×D53
FittingC2×C53×#395C2
socleC24×C53C24
prime radicalC23×#11719949C2
hypercenterC2×#395D53
hypocenterC53C2×#2406
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
2. C22×#13133202C2semidirect
3. C22×C4⋊C4×D53C2semidirect
4. #20278051C2direct (N × H)
5. C2×#16810542C2semidirect
6. C2×#16810541C2semidirect
7. C2×#16810540C2semidirect
8. C2×#16810501C2semidirect
9. C2×#16810497C2semidirect
10. C2×#16810494C2semidirect
11. C2×#16810485C2semidirect
12. C2×C53×#395C2semidirect
23. C2×#13133202C22semidirect
29. C2×C4⋊C4×D53C22semidirect
35. #16810542C22semidirect
36. #16810541C22semidirect
37. #16810540C22semidirect
38. #16810501C22semidirect
39. #16810497C22semidirect
40. #16810494C22semidirect
41. #16810485C22semidirect
48. C53×#395C22semidirect
148. C22×C4×C53#41semidirect
163. C22×Dic53#41semidirect
249. C2×C4×C53C2×#41semidirect
263. C2×Dic53C2×#41semidirect
309. C2×#395D53semidirect
316. C2×C53#2406semidirect
350. #395C2×D53semidirect
362. C53C2×#2406semidirect
496. C22×C4#11719949semidirect
581. C2×C4C2×#11719949semidirect
617. C2#20278051direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117199492
C22×#131332022
C22×C4⋊C4×D532
#202780512
C2×#168105422
C2×#168105412
C2×#168105402
C2×#168105012
C2×#168104972
C2×#168104942
C2×#168104852
C2×C53×#3952
C2×#240653