Finite Group Databasebeta

← C2×#20279880 #27555377

Subgroups — C2×#20279880 #27555377

Order 13568.

3006 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×#13133208C211yesmaximal, characteristic, complemented
2C22×C4×C22⋊D53C211yesmaximal, characteristic
3#20279880C2116yesmaximal, direct product factor
4C2×#16810565C212yesmaximal, complemented
5C2×#16810539C212yesmaximal
6C2×#16810512C212yesmaximal, complemented
7C2×#16810511C212yesmaximal, complemented
8C2×#16810483C211yesmaximal, characteristic, complemented
9C2×#16810480C211yesmaximal, characteristic, complemented
10C2×#16810480C211yesmaximal, characteristic, complemented
11C2×C4⋊C4×Dic53C211yesmaximal, characteristic, complemented
12C2×C53×#380C211yesmaximal, characteristic, complemented
13C23×C22⋊D53C2211yescharacteristic
14C23×C4×D53C2211yescharacteristic
15C2×#13133208—28no—
16C2×#13133208C2218yescomplemented
17C2×C4×C22⋊D53—28no—
18C2×C4×C22⋊D53C2218yes—
19C22×C4⋊Dic53C2211yescharacteristic
20C22×C4⋊Dic53C2211yescharacteristic
21C22×C4⋊Dic53C2212yes—
22C22×#11719919C2211yescharacteristic
23C22×#11719919C2212yes—
24C22×C22⋊Dic53C2212yes—
25C22×C22⋊Dic53C2211yescharacteristic
26C22×#11719921C2212yes—
27C22×#11719921C2211yescharacteristic
28C22×#11719921C2212yes—
29C22×C4×Dic53C2212yes—
30C22×C4×Dic53C2211yescharacteristic
31#16810565C22116yescomplemented
32#16810539C22116yes—
33#16810512C22116yescomplemented
34#16810511C22116yescomplemented
35#16810483C2218yescomplemented
36#16810480C2218yescomplemented
37#16810480C2218yescomplemented
38C4⋊C4×Dic53C2218yescomplemented
39C2×#13133124C2212yes—
40C2×#13133124C2212yes—
41C2×#13133124C2212yes—
42C2×#13133124C2212yes—
43C24×C4×C53C2211yescharacteristic
44C22×C4⋊C4×C53C2211yescharacteristic
45C22×C4⋊C4×C53C2211yescharacteristic
46C22×C22⋊C4×C53C2212yes—
47C53×#380C2218yescomplemented
48C2×#95×C53C2212yes—
49C24×D53C2311yescharacteristic
50C22×C22⋊D53—24no—
51C22×C22⋊D53—24no—
52C22×C22⋊D53—24no—
53C22×C22⋊D53C2314yes—
54C23×Dic53C2312yes—
55C23×Dic53C2312yes—
56C23×Dic53C2311yescharacteristic
57C23×Dic53C2311yescharacteristic
58C23×Dic53C2311yescharacteristic
59C22×C4×D53—24no—
60C22×C4×D53—24no—
61C22×C4×D53—24no—
62C22×C4×D53C2314yes—
63#13133208—232no—
64C4×C22⋊D53—232no—
65C2×C4⋊Dic53—24no—
66C2×C4⋊Dic53—24no—
67C2×C4⋊Dic53—28no—
68C2×C4⋊Dic53C2318yes—
69C2×C4⋊Dic53C2314yes—
70C2×C4⋊Dic53C2314yes—
71C2×#11719919—24no—
72C2×#11719919—28no—
73C2×#11719919C2318yes—
74C2×#11719919C2314yes—
75C2×C22⋊Dic53—28no—
76C2×C22⋊Dic53—24no—
77C2×C22⋊Dic53C2314yes—
78C2×C22⋊Dic53C2318yes—
79C2×#11719921—28no—
80C2×#11719921—24no—
81C2×#11719921—44no—
82C2×#11719921C2314yes—
83C2×#11719921C2318yes—
84C2×#11719921C2318yes—
85C2×C4×Dic53—44no—
86C2×C4×Dic53D418yes—
87C2×C4×Dic53C2318yes—
88C2×C4×Dic53C2314yes—
89#13133124C2318yes—
90#13133124C2318yes—
91#13133124C2318yes—
92#13133124C2318yes—
93C25×C53C2311yescharacteristic
94C23×C4×C53—24no—
95C23×C4×C53—24no—
96C23×C4×C53—24no—
97C23×C4×C53C2312yes—
98C23×C4×C53C2312yes—
99C23×C4×C53C2311yescharacteristic
100C23×C4×C53C2314yes—
101C23×C4×C53C2311yescharacteristic
102C2×C4⋊C4×C53—24no—
103C2×C4⋊C4×C53—24no—
104C2×C4⋊C4×C53C2314yes—
105C2×C4⋊C4×C53C2314yes—
106C2×C22⋊C4×C53—44no—
107C2×C22⋊C4×C53C2318yes—
108#95×C53C2318yes—
109C24×C53—22no—
110C24×C53—22no—
111C24×C53—22no—
112C24×C53—22no—
113C24×C53—22no—
114C24×C53—22no—
115C24×C53#4114yes—
116C24×C53C2412yes—
117C24×C53C2411yescharacteristic
118C23×D53—22no—
119C23×D53—22no—
120C23×D53—22no—
121C23×D53—22no—
122C23×D53—22no—
123C23×D53—22no—
124C23×D53#4114yes—
125C23×D53C2412yes—
126C22×C4×C53—24no—
127C22×C4×C53—22no—
128C22×C4×C53—24no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—24no—
132C22×C4×C53—24no—
133C22×C4×C53—24no—
134C22×C4×C53—24no—
135C22×C4×C53—28no—
136C22×C4×C53—28no—
137C22×C4×C53—28no—
138C22×C4×C53—416no—
139C22×C4×C53C2×D414yes—
140C22×C4×C53#4114yes—
141C22×C4×C53#4114yes—
142C22×C4×C53C2412yes—
143C22×C4×C53C2414yes—
144C22×C4×C53C2414yes—
145C22×C4×C53C2412yes—
146C2×C22⋊D53—232no—
147C2×C22⋊D53—28no—
148C2×C22⋊D53—28no—
149C2×C22⋊D53—28no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—24no—
156C22×Dic53—24no—
157C22×Dic53—24no—
158C22×Dic53—24no—
159C22×Dic53—24no—
160C22×Dic53—22no—
161C22×Dic53—22no—
162C22×Dic53—24no—
163C22×Dic53C2×D414yes—
164C22×Dic53C2×D414yes—
165C22×Dic53C2412yes—
166C22×Dic53C2414yes—
167C22×Dic53C2414yes—
168C22×Dic53C2412yes—
169C22×Dic53C2412yes—
170C2×C4×D53—28no—
171C2×C4×D53—28no—
172C2×C4×D53—28no—
173C2×C4×D53—416no—
174C4⋊C4×C53—28no—
175C4⋊C4×C53—28no—
176C22⋊C4×C53—48no—
177C22⋊Dic53—216no—
178C22⋊Dic53—28no—
179#11719921—216no—
180#11719921—28no—
181#11719921—48no—
182C4⋊Dic53—28no—
183C4⋊Dic53—28no—
184C4⋊Dic53—216no—
185#11719919—28no—
186#11719919—216no—
187C4×Dic53—48no—
188C4×Dic53C2×D4116yes—
189C23×C53—28no—
190C23×C53—28no—
191C23×C53—28no—
192C23×C53—22no—
193C23×C53—22no—
194C23×C53—22no—
195C23×C53—22no—
196C23×C53—22no—
197C23×C53—22no—
198C23×C53—44no—
199C23×C53—44no—
200C23×C53—44no—
201C23×C53—44no—
202C23×C53C22×D412yes—
203C23×C53C2×#4112yes—
204C23×C53C2×#4112yes—
205C23×C53C2×#4112yes—
206C23×C53C2×#4114yes—
207C23×C53C2×#4112yes—
208C23×C53H(5,𝔽2)12yes—
209C23×C532−1+412yes—
210C23×C53C2511yescharacteristic
211C22×D53—28no—
212C22×D53—28no—
213C22×D53—28no—
214C22×D53—22no—
215C22×D53—22no—
216C22×D53—22no—
217C22×D53—22no—
218C22×D53—22no—
219C22×D53—22no—
220C22×D53—44no—
221C22×D53—44no—
222C22×D53—44no—
223C22×D53—44no—
224C22×D53C2×#4114yes—
225C2×C4×C53—22no—
226C2×C4×C53—24no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—24no—
230C2×C4×C53—24no—
231C2×C4×C53—24no—
232C2×C4×C53—24no—
233C2×C4×C53—24no—
234C2×C4×C53—44no—
235C2×C4×C53—48no—
236C2×C4×C53—48no—
237C2×C4×C53—416no—
238C2×C4×C53#124116yes—
239C2×C4×C53C22×D414yes—
240C2×C4×C53C2×#4114yes—
241C2×C4×C53C2×#4114yes—
242C22⋊D53—232no—
243C2×Dic53—28no—
244C2×Dic53—28no—
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—22no—
250C2×Dic53—24no—
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—24no—
254C2×Dic53—24no—
255C2×Dic53—24no—
256C2×Dic53—24no—
257C2×Dic53—24no—
258C2×Dic53—48no—
259C2×Dic53—44no—
260C2×Dic53—48no—
261C2×Dic53C22×D414yes—
262C2×Dic53C22×D414yes—
263C4×D53—416no—
264C2×#2562—531nomaximal, Sylow
265C22×C53—24no—
266C22×C53—24no—
267C22×C53—24no—
268C22×C53—416no—
269C22×C53—42no—
270C22×C53—42no—
271C22×C53—42no—
272C22×C53—42no—
273C22×C53#52514yes—
274C22×C53C2×#12414yes—
275C22×C53#53714yes—
276C22×C53#55214yes—
277C22×C53#55214yes—
278C22×C53#55314yes—
279C22×C53#55314yes—
280C22×C53C23×D411yescharacteristic
281C22×C53C22×#4111yescharacteristic
282C22×C53C22×#4111yescharacteristic
283C22×C53C22×#4111yescharacteristic
284C22×C53C22×#4111yescharacteristic
285C22×C53C2×H(5,𝔽2)11yescharacteristic
286C22×C53C2×2−1+411yescharacteristic
287C2×D53—24no—
288C2×D53—24no—
289C2×D53—24no—
290C2×D53—416no—
291C2×D53—42no—
292C2×D53—42no—
293C2×D53—42no—
294C2×D53—42no—
295C4×C53—42no—
296C4×C53—44no—
297C4×C53—44no—
298C4×C53C2×#12418yes—
299Dic53—24no—
300Dic53—24no—
301Dic53—44no—
302Dic53—42no—
303Dic53—44no—
304C22×#125—531no—
305C22×C4×D4—531no—
306#2562—5316no—
307C2×#400—532no—
308C2×#394—532no—
309C2×#388—532no—
310C2×#388—532no—
311C2×#380—531no—
312C2×#380—531no—
313C2×#380—531no—
314C2×#380D5311yescharacteristic, complemented
315C2×C4×C4⋊C4—531no—
316C2×C53—44no—
317C2×C53#256218yescomplemented
318C2×C53C2×#52511yescharacteristic
319C2×C53C22×#12411yescharacteristic
320C2×C53C2×#53711yescharacteristic
321C2×C53C2×#55211yescharacteristic
322C2×C53C2×#55211yescharacteristic
323C2×C53C2×#55311yescharacteristic
324C2×C53C2×#55311yescharacteristic
325D53—44no—
326C23×D4—531no—
327C24×C4—531no—
328C24×C4C2×D5311yescharacteristic
329C2×#125—538no—
330C2×#125—1068no—
331C2×C4×D4—538no—
332C2×C4×D4—1068no—
333C22×C4⋊C4—531no—
334C22×C4⋊C4—532no—
335C22×C4⋊C4—532no—
336C22×C4⋊C4—531no—
337C22×C4⋊C4—531no—
338C22×C4⋊C4C2×D5311yescharacteristic
339C22×C4⋊C4C2×D5311yescharacteristic
340C22×C22⋊C4—532no—
341C22×C22⋊C4—531no—
342C22×C22⋊C4—532no—
343C22×C22⋊C4—531no—
344C22×C22⋊C4—532no—
345C22×C22⋊C4C2×D5312yes—
346C22×C42—531no—
347C22×C42—532no—
348#400—5316no—
349#394—5316no—
350#388—5316no—
351#388—5316no—
352#380—538no—
353#380—538no—
354#380—538no—
355#380C2×D5318yescomplemented
356C4×C4⋊C4—538no—
357C2×#95—532no—
358C2×#95—532no—
359C2×#95—532no—
360C2×#95—532no—
361C2×#95C2×D5312yes—
362C53C2×#256211yescharacteristic, complemented, Sylow
363C25—531no—
364C25C22×D5311yescharacteristic
365C22×D4—534no—
366C22×D4—1064no—
367C22×D4—1064no—
368C22×D4—1064no—
369C23×C4—24no—
370C23×C4—24no—
371C23×C4—24no—
372C23×C4—531no—
373C23×C4—531no—
374C23×C4—532no—
375C23×C4—532no—
376C23×C4—531no—
377C23×C4—534no—
378C23×C4—1064no—
379C23×C4—1064no—
380C23×C4—1064no—
381C23×C4C22×D5314yes—
382C23×C4C22×D5312yes—
383C23×C4C22×D5312yes—
384C23×C4C22×D5311yescharacteristic
385C23×C4C22×D5311yescharacteristic
386#125—10632no—
387C4×D4—10632no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—534no—
391C2×C4⋊C4—538no—
392C2×C4⋊C4—534no—
393C2×C4⋊C4—538no—
394C2×C4⋊C4—534no—
395C2×C4⋊C4—1064no—
396C2×C4⋊C4—1068no—
397C2×C4⋊C4—1068no—
398C2×C4⋊C4—1064no—
399C2×C4⋊C4—1064no—
400C2×C4⋊C4C22×D5314yes—
401C2×C4⋊C4C22×D5314yes—
402C2×C22⋊C4—44no—
403C2×C22⋊C4—534no—
404C2×C22⋊C4—538no—
405C2×C22⋊C4—534no—
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⋊C4C22×D5318yes—
414C2×C42—538no—
415C2×C42—534no—
416C2×C42—538no—
417C2×C42—2124no—
418#95—538no—
419#95—538no—
420#95—538no—
421#95—538no—
422#95C22×D5318yes—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—534no—
430C24—532no—
431C24—1062no—
432C24—1062no—
433C24—1062no—
434C24—1062no—
435C24—1062no—
436C24—1062no—
437C24#1171994914yes—
438C24C23×D5312yes—
439C24C23×D5311yescharacteristic
440C2×D4—1068no—
441C2×D4—1068no—
442C2×D4—10632no—
443C2×D4—1068no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—28no—
454C22×C4—28no—
455C22×C4—28no—
456C22×C4—416no—
457C22×C4—534no—
458C22×C4—534no—
459C22×C4—532no—
460C22×C4—532no—
461C22×C4—532no—
462C22×C4—534no—
463C22×C4—534no—
464C22×C4—1062no—
465C22×C4—1062no—
466C22×C4—1062no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1064no—
470C22×C4—1064no—
471C22×C4—1062no—
472C22×C4—1064no—
473C22×C4—1062no—
474C22×C4—1064no—
475C22×C4—1064no—
476C22×C4—1064no—
477C22×C4—1068no—
478C22×C4—1068no—
479C22×C4—1068no—
480C22×C4—21216no—
481C22×C4#1171994514yes—
482C22×C4D4×D5314yes—
483C22×C4#1171994714yes—
484C22×C4C23×D5312yes—
485C22×C4C23×D5314yes—
486C22×C4C23×D5314yes—
487C22×C4C23×D5312yes—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—1068no—
491C4⋊C4—10616no—
492C4⋊C4—10616no—
493C4⋊C4—1068no—
494C4⋊C4—1068no—
495C22⋊C4—48no—
496C22⋊C4—1068no—
497C22⋊C4—10616no—
498C22⋊C4—1068no—
499C22⋊C4—10616no—
500C22⋊C4—2128no—
501C42—5316no—
502C42—2128no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—534no—
517C23—1068no—
518C23—1068no—
519C23—1068no—
520C23—1062no—
521C23—1062no—
522C23—1062no—
523C23—1062no—
524C23—1062no—
525C23—1062no—
526C23—2124no—
527C23—2124no—
528C23—2124no—
529C23—2124no—
530C23C2×D4×D5312yes—
531C23#41×D5312yes—
532C23C2×#1171994712yes—
533C23C2×#1171994512yes—
534C23C2×#1171994914yes—
535C23C2×#1171994912yes—
536C23#1313326212yes—
537C23#1313326412yes—
538C23C24×D5311yescharacteristic
539D4—10632no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—48no—
550C2×C4—44no—
551C2×C4—48no—
552C2×C4—416no—
553C2×C4—534no—
554C2×C4—534no—
555C2×C4—1068no—
556C2×C4—1062no—
557C2×C4—1062no—
558C2×C4—1062no—
559C2×C4—1062no—
560C2×C4—1068no—
561C2×C4—1062no—
562C2×C4—1064no—
563C2×C4—1064no—
564C2×C4—1062no—
565C2×C4—1064no—
566C2×C4—1062no—
567C2×C4—1064no—
568C2×C4—1064no—
569C2×C4—1064no—
570C2×C4—2128no—
571C2×C4—2128no—
572C2×C4—2124no—
573C2×C4—21216no—
574C2×C4#13133203116yes—
575C2×C4C2×D4×D5314yes—
576C2×C4C2×#1171994714yes—
577C2×C4C2×#1171994514yes—
578C22—24no—
579C22—24no—
580C22—24no—
581C22—42no—
582C22—416no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—1064no—
587C22—1064no—
588C22—1064no—
589C22—2122no—
590C22—21216no—
591C22—2122no—
592C22—2122no—
593C22—2122no—
594C22#1681113614yes—
595C22C2×#1313320314yes—
596C22#1681120214yes—
597C22#1681131214yes—
598C22#1681131514yes—
599C22#1681132014yes—
600C22#1681132114yes—
601C22C22×D4×D5311yescharacteristic
602C22C2×#41×D5311yescharacteristic
603C22C22×#1171994711yescharacteristic
604C22C22×#1171994511yescharacteristic
605C22C22×#1171994911yescharacteristic
606C22C2×#1313326211yescharacteristic
607C22C2×#1313326411yescharacteristic
608C4—44no—
609C4—42no—
610C4—44no—
611C4—1064no—
612C4—1064no—
613C4—2124no—
614C4—2122no—
615C4—2124no—
616C4C2×#1313320318yes—
617C2—44no—
618C2—2124no—
619C2#2027988018yesdirect product factor
620C2C2×#1681113611yescharacteristic
621C2C22×#1313320311yescharacteristic
622C2C2×#1681120211yescharacteristic
623C2C2×#1681131211yescharacteristic
624C2C2×#1681131511yescharacteristic
625C2C2×#1681132011yescharacteristic
626C2C2×#1681132111yescharacteristic
subgroupG/H
centerC24C23×D53
commutator, cosocleC23×C53C25
FrattiniC23C24×D53
FittingC2×C53×#380C2
socleC24×C53C24
prime radicalC23×C22⋊D53C22
hypercenterC2×#380D53
hypocenterC53C2×#2562
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
1. C22×#13133208C2semidirect
3. #20279880C2direct (N × H)
4. C2×#16810565C2semidirect
6. C2×#16810512C2semidirect
7. C2×#16810511C2semidirect
8. C2×#16810483C2semidirect
9. C2×#16810480C2semidirect
10. C2×#16810480C2semidirect
11. C2×C4⋊C4×Dic53C2semidirect
12. C2×C53×#380C2semidirect
16. C2×#13133208C22semidirect
31. #16810565C22semidirect
33. #16810512C22semidirect
34. #16810511C22semidirect
35. #16810483C22semidirect
36. #16810480C22semidirect
37. #16810480C22semidirect
38. C4⋊C4×Dic53C22semidirect
47. C53×#380C22semidirect
314. C2×#380D53semidirect
317. C2×C53#2562semidirect
355. #380C2×D53semidirect
362. C53C2×#2562semidirect
619. C2#20279880direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131332082
C22×C4×C22⋊D532
#202798802
C2×#168105652
C2×#168105392
C2×#168105122
C2×#168105112
C2×#168104832
C2×#168104802
C2×C4⋊C4×Dic532
C2×C53×#3802
C2×#256253