Finite Group Databasebeta

← C2×#20280456 #27550450

Subgroups — C2×#20280456 #27550450

Order 13568.

3718 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133199C212yesmaximal, complemented
2C22×#13133188C211yesmaximal, characteristic
3C22×C4⋊C4×D53C211yesmaximal, characteristic, complemented
4#20280456C2116yesmaximal, direct product factor
5C2×#16810555C211yesmaximal, characteristic, complemented
6C2×#16810553C211yesmaximal, characteristic, complemented
7C2×#16810546C212yesmaximal, complemented
8C2×#16810546C212yesmaximal, complemented
9C2×#16810537C211yesmaximal, characteristic, complemented
10C2×#16810499C212yesmaximal, complemented
11C2×#16810482C211yesmaximal, characteristic, complemented
12C2×C53×#394C211yesmaximal, characteristic, complemented
13C23×C22⋊D53C2212yes—
14C23×C4×D53C2211yescharacteristic
15C23×C4×D53C2211yescharacteristic
16C23×C4×D53C2211yescharacteristic
17C2×#13133199—216no—
18C2×#13133199C22116yescomplemented
19C2×#13133188—28no—
20C2×#13133188C2218yes—
21C22×C4⋊Dic53C2211yescharacteristic
22C22×C4⋊Dic53C2212yes—
23C22×C4⋊Dic53C2211yescharacteristic
24C22×#11719919C2211yescharacteristic
25C22×#11719919C2211yescharacteristic
26C22×#11719919C2212yes—
27C22×#11719919C2212yes—
28C2×C4⋊C4×D53—28no—
29C2×C4⋊C4×D53C2218yescomplemented
30C22×C22⋊Dic53C2212yes—
31C22×C22⋊Dic53C2212yes—
32C22×C22⋊Dic53C2212yes—
33C22×#11719921C2212yes—
34#16810555C2218yescomplemented
35#16810553C2218yescomplemented
36#16810546C22116yescomplemented
37#16810546C22116yescomplemented
38#16810537C2218yescomplemented
39#16810499C22116yescomplemented
40#16810482C2218yescomplemented
41C2×#13133124C2211yescharacteristic
42C2×#13133124C2212yes—
43C2×#13133124C2212yes—
44C23×D4×C53C2211yescharacteristic
45C22×C4⋊C4×C53C2211yescharacteristic
46C22×C22⋊C4×C53C2212yes—
47C22×C22⋊C4×C53C2212yes—
48C53×#394C2218yescomplemented
49C2×#95×C53C2211yescharacteristic
50C24×D53C2311yescharacteristic
51C22×C22⋊D53—28no—
52C22×C22⋊D53—28no—
53C22×C22⋊D53—28no—
54C22×C22⋊D53C2318yes—
55C23×Dic53C2312yes—
56C23×Dic53C2311yescharacteristic
57C23×Dic53C2312yes—
58C23×Dic53C2311yescharacteristic
59C23×Dic53C2311yescharacteristic
60C22×C4×D53—24no—
61C22×C4×D53—24no—
62C22×C4×D53—24no—
63C22×C4×D53—44no—
64C22×C4×D53—44no—
65C22×C4×D53D418yes—
66C22×C4×D53D418yescomplemented
67C22×C4×D53C2314yes—
68C22×C4×D53C2314yes—
69C22×C4×D53C2314yes—
70#13133199—264no—
71#13133188—232no—
72C2×C4⋊Dic53—24no—
73C2×C4⋊Dic53—24no—
74C2×C4⋊Dic53—28no—
75C2×C4⋊Dic53C2314yes—
76C2×C4⋊Dic53C2318yes—
77C2×C4⋊Dic53C2314yes—
78C2×#11719919—24no—
79C2×#11719919—28no—
80C2×#11719919—24no—
81C2×#11719919—44no—
82C2×#11719919C2314yes—
83C2×#11719919C2314yes—
84C2×#11719919C2318yes—
85C2×#11719919C2318yes—
86C4⋊C4×D53—232no—
87C2×C22⋊Dic53—28no—
88C2×C22⋊Dic53—28no—
89C2×C22⋊Dic53—44no—
90C2×C22⋊Dic53C2318yes—
91C2×C22⋊Dic53C2318yes—
92C2×C22⋊Dic53C2318yes—
93C2×#11719921—28no—
94C2×#11719921C2318yes—
95#13133124C2318yes—
96#13133124C2318yes—
97#13133124C2314yes—
98C25×C53C2312yes—
99C22×D4×C53—24no—
100C22×D4×C53—44no—
101C22×D4×C53C2314yes—
102C23×C4×C53C2311yescharacteristic
103C23×C4×C53C2312yes—
104C23×C4×C53C2311yescharacteristic
105C23×C4×C53C2311yescharacteristic
106C2×C4⋊C4×C53—24no—
107C2×C4⋊C4×C53C2314yes—
108C2×C22⋊C4×C53—28no—
109C2×C22⋊C4×C53—44no—
110C2×C22⋊C4×C53C2318yes—
111C2×C22⋊C4×C53C2318yes—
112#95×C53C2314yes—
113C24×C53—24no—
114C24×C53—24no—
115C24×C53—24no—
116C24×C53—24no—
117C24×C53—24no—
118C24×C53—24no—
119C24×C53—24no—
120C24×C53C2414yes—
121C24×C53C2411yescharacteristic
122C23×D53—24no—
123C23×D53—24no—
124C23×D53C2×D414yes—
125C23×D53C2×D414yes—
126C23×D53#4114yes—
127C23×D53C2412yes—
128C2×D4×C53—28no—
129C2×D4×C53—48no—
130C2×D4×C53—416no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—24no—
134C22×C4×C53—24no—
135C22×C4×C53—22no—
136C22×C4×C53—24no—
137C22×C4×C53—24no—
138C22×C4×C53—24no—
139C22×C4×C53C2×D414yes—
140C22×C4×C53C2×D414yes—
141C22×C4×C53C2414yes—
142C22×C4×C53C2412yes—
143C22×C4×C53C2412yes—
144C22×C4×C53C2412yes—
145C2×C22⋊D53—216no—
146C2×C22⋊D53—216no—
147C2×C22⋊D53—216no—
148C2×C22⋊D53—432no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—24no—
152C22×Dic53—24no—
153C22×Dic53—24no—
154C22×Dic53—22no—
155C22×Dic53—24no—
156C22×Dic53—24no—
157C22×Dic53—24no—
158C22×Dic53—24no—
159C22×Dic53—24no—
160C22×Dic53C2×D414yes—
161C22×Dic53C2×D414yes—
162C22×Dic53C2414yes—
163C22×Dic53C2412yes—
164C22×Dic53C2414yes—
165C22×Dic53C2412yes—
166C22×Dic53C2412yes—
167C2×C4×D53—232no—
168C2×C4×D53—28no—
169C2×C4×D53—28no—
170C2×C4×D53—28no—
171C2×C4×D53—416no—
172C2×C4×D53—48no—
173C2×C4×D53—48no—
174C2×C4×D53—416no—
175C2×C4×D53C2×D4116yes—
176C2×C4×D53C2×D4116yescomplemented
177C4⋊C4×C53—28no—
178C22⋊C4×C53—216no—
179C22⋊C4×C53—48no—
180C22⋊Dic53—216no—
181C22⋊Dic53—216no—
182C22⋊Dic53—48no—
183#11719921—216no—
184C4⋊Dic53—28no—
185C4⋊Dic53—28no—
186C4⋊Dic53—216no—
187#11719919—28no—
188#11719919—216no—
189#11719919—28no—
190#11719919—48no—
191C23×C53—24no—
192C23×C53—24no—
193C23×C53—24no—
194C23×C53—24no—
195C23×C53—24no—
196C23×C53—24no—
197C23×C53—24no—
198C23×C53—48no—
199C23×C53—48no—
200C23×C53—48no—
201C23×C53—48no—
202C23×C53—48no—
203C23×C53—48no—
204C23×C53—48no—
205C23×C53C22×D412yes—
206C23×C53C22×D412yes—
207C23×C53C2×#4112yes—
208C23×C53H(5,𝔽2)14yes—
209C23×C532−1+414yes—
210C23×C53C2511yescharacteristic
211C22×D53—28no—
212C22×D53—28no—
213C22×D53—216no—
214C22×D53—216no—
215C22×D53—24no—
216C22×D53—24no—
217C22×D53#121116yes—
218C22×D53C22×D414yes—
219C22×D53C22×D414yes—
220C22×D53C2×#4114yes—
221D4×C53—416no—
222C2×C4×C53—28no—
223C2×C4×C53—28no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—24no—
227C2×C4×C53—22no—
228C2×C4×C53—24no—
229C2×C4×C53—24no—
230C2×C4×C53—24no—
231C2×C4×C53—24no—
232C2×C4×C53—44no—
233C2×C4×C53—48no—
234C2×C4×C53C22×D414yes—
235C2×C4×C53C22×D414yes—
236C22⋊D53—432no—
237C2×Dic53—28no—
238C2×Dic53—28no—
239C2×Dic53—22no—
240C2×Dic53—22no—
241C2×Dic53—24no—
242C2×Dic53—22no—
243C2×Dic53—24no—
244C2×Dic53—24no—
245C2×Dic53—24no—
246C2×Dic53—24no—
247C2×Dic53—24no—
248C2×Dic53—24no—
249C2×Dic53—24no—
250C2×Dic53—48no—
251C2×Dic53—44no—
252C2×Dic53—48no—
253C2×Dic53C22×D414yes—
254C2×Dic53C22×D414yes—
255C4×D53—232no—
256C4×D53—416no—
257C4×D53—416no—
258C2×#2618—531nomaximal, Sylow
259C22×C53—44no—
260C22×C53—44no—
261C22×C53—44no—
262C22×C53—44no—
263C22×C53—44no—
264C22×C53—44no—
265C22×C53—44no—
266C22×C53—816no—
267C22×C53C2×#12114yes—
268C22×C53#53414yes—
269C22×C53#53614yes—
270C22×C53#53718yes—
271C22×C53#53918yes—
272C22×C53C23×D411yescharacteristic
273C22×C53C23×D411yescharacteristic
274C22×C53C22×#4111yescharacteristic
275C22×C53C2×H(5,𝔽2)12yes—
276C22×C53C2×2−1+412yes—
277C2×D53—232no—
278C2×D53—24no—
279C2×D53—24no—
280C2×D53—28no—
281C2×D53—28no—
282C2×D53C2×#12118yes—
283C4×C53—24no—
284C4×C53—24no—
285C4×C53—42no—
286C4×C53—44no—
287Dic53—24no—
288Dic53—24no—
289Dic53—44no—
290Dic53—42no—
291Dic53—44no—
292C22×#123—532no—
293C22×#121—531no—
294C23×C4⋊C4—531no—
295#2618—5316no—
296C2×#398—531no—
297C2×#398—531no—
298C2×#396—532no—
299C2×#396—532no—
300C2×#394—531no—
301C2×#394D5311yescharacteristic, complemented
302C2×#386—532no—
303C2×#380—531no—
304C2×C53—84no—
305C2×C53#261818yescomplemented
306C2×C53C22×#12111yescharacteristic
307C2×C53C2×#53411yescharacteristic
308C2×C53C2×#53611yescharacteristic
309C2×C53C2×#53712yes—
310C2×C53C2×#53912yes—
311D53—28no—
312C23×D4—532no—
313C23×D4C2×D5311yescharacteristic
314C24×C4—531no—
315C24×C4—531no—
316C24×C4—531no—
317C2×#123—5316no—
318C2×#123—10616no—
319C2×#121—538no—
320C2×#121—1068no—
321C22×C4⋊C4—531no—
322C22×C4⋊C4—531no—
323C22×C4⋊C4—532no—
324C22×C4⋊C4—531no—
325C22×C4⋊C4—532no—
326C22×C4⋊C4—531no—
327C22×C4⋊C4—532no—
328C22×C4⋊C4—538no—
329C22×C4⋊C4—1068no—
330C22×C4⋊C4C2×D5311yescharacteristic
331C22×C22⋊C4—532no—
332C22×C22⋊C4—532no—
333C22×C22⋊C4—532no—
334C22×C22⋊C4—532no—
335C22×C22⋊C4C2×D5312yes—
336C22×C22⋊C4C2×D5312yes—
337#398—538no—
338#398—538no—
339#396—5316no—
340#396—5316no—
341#394—538no—
342#394C2×D5318yescomplemented
343#386—5316no—
344#380—538no—
345C2×#95—531no—
346C2×#95—532no—
347C2×#95—532no—
348C2×#95C2×D5311yescharacteristic
349C53C2×#261811yescharacteristic, complemented, Sylow
350C25—531no—
351C25C22×D5312yes—
352C22×D4—24no—
353C22×D4—44no—
354C22×D4—538no—
355C22×D4—1068no—
356C22×D4—1068no—
357C22×D4—1068no—
358C22×D4C22×D5314yes—
359C23×C4—531no—
360C23×C4—531no—
361C23×C4—538no—
362C23×C4—534no—
363C23×C4—538no—
364C23×C4—534no—
365C23×C4—532no—
366C23×C4—531no—
367C23×C4—532no—
368C23×C4—534no—
369C23×C4—1064no—
370C23×C4—1064no—
371C23×C4—1064no—
372C23×C4—2124no—
373C23×C4—2124no—
374C23×C4C22×D5312yes—
375C23×C4C22×D5311yescharacteristic
376C23×C4C22×D5311yescharacteristic
377C23×C4C22×D5311yescharacteristic
378#123—10664no—
379#121—10632no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—534no—
382C2×C4⋊C4—534no—
383C2×C4⋊C4—534no—
384C2×C4⋊C4—534no—
385C2×C4⋊C4—538no—
386C2×C4⋊C4—538no—
387C2×C4⋊C4—538no—
388C2×C4⋊C4—10632no—
389C2×C4⋊C4—1064no—
390C2×C4⋊C4—1064no—
391C2×C4⋊C4—1068no—
392C2×C4⋊C4—1064no—
393C2×C4⋊C4—1064no—
394C2×C4⋊C4—1068no—
395C2×C4⋊C4—2124no—
396C2×C4⋊C4C22×D5314yes—
397C2×C22⋊C4—28no—
398C2×C22⋊C4—44no—
399C2×C22⋊C4—538no—
400C2×C22⋊C4—538no—
401C2×C22⋊C4—538no—
402C2×C22⋊C4—538no—
403C2×C22⋊C4—1068no—
404C2×C22⋊C4—1068no—
405C2×C22⋊C4—1068no—
406C2×C22⋊C4—2124no—
407C2×C22⋊C4C22×D5318yes—
408C2×C22⋊C4C22×D5318yes—
409#95—534no—
410#95—538no—
411#95—538no—
412#95C22×D5314yes—
413C24—24no—
414C24—24no—
415C24—24no—
416C24—24no—
417C24—24no—
418C24—24no—
419C24—24no—
420C24—534no—
421C24—534no—
422C24—534no—
423C24—532no—
424C24—1064no—
425C24—1064no—
426C24C23×D5314yes—
427C24C23×D5311yescharacteristic
428C2×D4—28no—
429C2×D4—48no—
430C2×D4—416no—
431C2×D4—10616no—
432C2×D4—10616no—
433C2×D4—10616no—
434C2×D4—21232no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—534no—
444C22×C4—534no—
445C22×C4—532no—
446C22×C4—532no—
447C22×C4—5316no—
448C22×C4—5316no—
449C22×C4—534no—
450C22×C4—532no—
451C22×C4—534no—
452C22×C4—1068no—
453C22×C4—10632no—
454C22×C4—1068no—
455C22×C4—1068no—
456C22×C4—1062no—
457C22×C4—1062no—
458C22×C4—1064no—
459C22×C4—1064no—
460C22×C4—1064no—
461C22×C4—1062no—
462C22×C4—1064no—
463C22×C4—1064no—
464C22×C4—1064no—
465C22×C4—1064no—
466C22×C4—1064no—
467C22×C4—21216no—
468C22×C4—2128no—
469C22×C4—2128no—
470C22×C4—21216no—
471C22×C4D4×D5314yes—
472C22×C4D4×D5314yes—
473C22×C4C23×D5312yes—
474C22×C4C23×D5314yes—
475C22×C4C23×D5312yes—
476C22×C4C23×D5312yes—
477C4⋊C4—28no—
478C4⋊C4—1068no—
479C4⋊C4—1068no—
480C4⋊C4—10616no—
481C4⋊C4—1068no—
482C4⋊C4—1068no—
483C4⋊C4—10616no—
484C4⋊C4—2128no—
485C22⋊C4—216no—
486C22⋊C4—48no—
487C22⋊C4—10616no—
488C22⋊C4—10616no—
489C22⋊C4—10616no—
490C22⋊C4—2128no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—48no—
499C23—48no—
500C23—48no—
501C23—48no—
502C23—48no—
503C23—48no—
504C23—48no—
505C23—5316no—
506C23—534no—
507C23—534no—
508C23—534no—
509C23—1068no—
510C23—1068no—
511C23—1064no—
512C23—1064no—
513C23—10616no—
514C23—10616no—
515C23C2×D4×D5312yes—
516C23C2×D4×D5312yes—
517C23#41×D5312yes—
518C23#1313326214yes—
519C23#1313326414yes—
520C23C24×D5311yescharacteristic
521D4—416no—
522D4—21232no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—44no—
534C2×C4—48no—
535C2×C4—534no—
536C2×C4—534no—
537C2×C4—10632no—
538C2×C4—1068no—
539C2×C4—1068no—
540C2×C4—1062no—
541C2×C4—1062no—
542C2×C4—1064no—
543C2×C4—1064no—
544C2×C4—1064no—
545C2×C4—1062no—
546C2×C4—1064no—
547C2×C4—1064no—
548C2×C4—1064no—
549C2×C4—1064no—
550C2×C4—1064no—
551C2×C4—21216no—
552C2×C4—21216no—
553C2×C4—2128no—
554C2×C4—2124no—
555C2×C4—2128no—
556C2×C4C2×D4×D5314yes—
557C2×C4C2×D4×D5314yes—
558C22—44no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—816no—
566C22—538no—
567C22—1064no—
568C22—1064no—
569C22—1068no—
570C22—1068no—
571C22—10632no—
572C22#121×D5314yes—
573C22#1681118314yes—
574C22#1681119314yes—
575C22#1681120218yes—
576C22#1681121618yes—
577C22C22×D4×D5311yescharacteristic
578C22C22×D4×D5311yescharacteristic
579C22C2×#41×D5311yescharacteristic
580C22C2×#1313326212yes—
581C22C2×#1313326412yes—
582C4—24no—
583C4—24no—
584C4—42no—
585C4—44no—
586C4—1064no—
587C4—1064no—
588C4—2122no—
589C4—2124no—
590C4—2124no—
591C2—84no—
592C2—1068no—
593C2#2028045618yesdirect product factor
594C2C2×#121×D5311yescharacteristic
595C2C2×#1681118311yescharacteristic
596C2C2×#1681119311yescharacteristic
597C2C2×#1681120212yes—
598C2C2×#1681121612yes—
subgroupG/H
centerC24C23×D53
commutator, cosocleC23×C53C25
FrattiniC23C24×D53
FittingC2×C53×#394C2
socleC24×C53C24
prime radicalC22×#13133188C2
hypercenterC2×#394D53
hypocenterC53C2×#2618
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
1. C22×#13133199C2semidirect
3. C22×C4⋊C4×D53C2semidirect
4. #20280456C2direct (N × H)
5. C2×#16810555C2semidirect
6. C2×#16810553C2semidirect
7. C2×#16810546C2semidirect
8. C2×#16810546C2semidirect
9. C2×#16810537C2semidirect
10. C2×#16810499C2semidirect
11. C2×#16810482C2semidirect
12. C2×C53×#394C2semidirect
18. C2×#13133199C22semidirect
29. C2×C4⋊C4×D53C22semidirect
34. #16810555C22semidirect
35. #16810553C22semidirect
36. #16810546C22semidirect
37. #16810546C22semidirect
38. #16810537C22semidirect
39. #16810499C22semidirect
40. #16810482C22semidirect
48. C53×#394C22semidirect
66. C22×C4×D53D4semidirect
176. C2×C4×D53C2×D4semidirect
301. C2×#394D53semidirect
305. C2×C53#2618semidirect
342. #394C2×D53semidirect
349. C53C2×#2618semidirect
593. C2#20280456direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131331992
C22×#131331882
C22×C4⋊C4×D532
#202804562
C2×#168105552
C2×#168105532
C2×#168105462
C2×#168105372
C2×#168104992
C2×#168104822
C2×C53×#3942
C2×#261853