Finite Group Databasebeta

← C2×#20275910 #27411547

Subgroups — C2×#20275910 #27411547

Order 13568.

1662 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133193C211yesmaximal, characteristic
2C22×#13133149C211yesmaximal, characteristic, complemented
3C22×#13133141C211yesmaximal, characteristic, complemented
4C22×#13133129C211yesmaximal, characteristic
5C2×#16810480C211yesmaximal, characteristic
6#20275910C218yesmaximal, direct product factor
7C2×#16810385C211yesmaximal, characteristic, complemented
8C22×#103×C53C211yesmaximal, characteristic, complemented
9C23×C4×D53C2211yescharacteristic
10C23×#88322C2211yescharacteristic
11C2×#13133193—22no—
12C2×#13133193—24no—
13C2×#13133193—24no—
14C2×#13133193C2214yes—
15C22×C4⋊Dic53C2211yescharacteristic
16C22×C4⋊Dic53C2211yescharacteristic
17C22×#11719919—21no—
18C22×#11719921—21no—
19C22×#11719921—21no—
20C2×#13133149—24no—
21C2×#13133149C2214yescomplemented
22C2×#13133141—24no—
23C2×#13133141C2214yescomplemented
24C2×#13133129—24no—
25C2×#13133129C2214yes—
26#16810480—22no—
27#16810480C2214yes—
28C2×#13133124—21no—
29#16810385C2214yescomplemented
30C23×Q8×C53C2211yescharacteristic
31C23×C8×C53C2211yescharacteristic
32C22×C4⋊C4×C53C2211yescharacteristic
33C2×#103×C53—24no—
34C2×#103×C53C2214yescomplemented
35C24×D53D411yescharacteristic
36C23×Dic53—21no—
37C23×Dic53—21no—
38C23×Dic53D411yescharacteristic
39C22×C4×D53—22no—
40C22×C4×D53—22no—
41C22×C4×D53—22no—
42C22×C4×D53—22no—
43C22×C4×D53—22no—
44C22×C4×D53—22no—
45C22×C4×D53D412yes—
46C22×C4×D53C2312yes—
47C22×#88322—22no—
48C22×#88322—22no—
49C22×#88322—22no—
50C22×#88322C2312yes—
51#13133193—216no—
52#13133193—28no—
53#13133193—28no—
54C2×C4⋊Dic53—22no—
55C2×C4⋊Dic53—21no—
56C2×C4⋊Dic53—21no—
57C2×C4⋊Dic53—22no—
58C2×C4⋊Dic53—22no—
59C2×C4⋊Dic53D414yes—
60C2×C4⋊Dic53C2312yes—
61C2×C4⋊Dic53C2312yes—
62C2×#11719919—28no—
63C2×#11719919—24no—
64C2×#11719921—28no—
65C2×#11719921—24no—
66C2×#11719921—24no—
67C2×#11719921—44no—
68#13133149—28no—
69#13133141—28no—
70#13133129—28no—
71#13133124—24no—
72C22×Q8×C53—22no—
73C22×Q8×C53—21no—
74C22×Q8×C53—22no—
75C22×Q8×C53—22no—
76C22×Q8×C53—22no—
77C22×Q8×C53—22no—
78C22×Q8×C53D414yes—
79C22×Q8×C53C2312yes—
80C23×C4×C53—21no—
81C23×C4×C53—21no—
82C23×C4×C53C2311yescharacteristic
83C22×C8×C53—22no—
84C22×C8×C53—22no—
85C22×C8×C53D414yes—
86C22×C8×C53C2312yes—
87C2×C4⋊C4×C53—21no—
88C2×C4⋊C4×C53—22no—
89C2×C4⋊C4×C53—22no—
90C2×C4⋊C4×C53C2312yes—
91#103×C53—28no—
92C24×C53C2×D411yescharacteristic
93C23×D53—22no—
94C23×D53—22no—
95C23×D53—22no—
96C23×D53—22no—
97C23×D53—22no—
98C23×D53—22no—
99C23×D53SD1614yes—
100C23×D53C2×D412yes—
101C2×Q8×C53—24no—
102C2×Q8×C53—22no—
103C2×Q8×C53—28no—
104C2×Q8×C53—28no—
105C2×Q8×C53—22no—
106C2×Q8×C53—22no—
107C2×Q8×C53—22no—
108C2×Q8×C53—22no—
109C2×Q8×C53—42no—
110C2×Q8×C53—44no—
111C2×Q8×C53—42no—
112C2×Q8×C53—44no—
113C2×Q8×C53C2×D414yes—
114C22×C4×C53—24no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—42no—
118C22×C4×C53—42no—
119C22×C4×C53—42no—
120C22×C4×C53—42no—
121C22×C4×C53—42no—
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D412yes—
124C22×C4×C53C2×D412yes—
125C22×C4×C53C2×D412yes—
126C22×C4×C53#4112yes—
127C22×C4×C53#4112yes—
128C22×C4×C53#4112yes—
129C22×C4×C53C2411yescharacteristic
130C22×Dic53—24no—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—42no—
137C22×Dic53—42no—
138C22×Dic53—42no—
139C22×Dic53—42no—
140C22×Dic53—42no—
141C22×Dic53C2×D412yes—
142C2×C4×D53—24no—
143C2×C4×D53—24no—
144C2×C4×D53—24no—
145C2×C4×D53—22no—
146C2×C4×D53—22no—
147C2×C4×D53—22no—
148C2×C4×D53—22no—
149C2×C4×D53—22no—
150C2×C4×D53—22no—
151C2×C4×D53—44no—
152C2×C4×D53—44no—
153C2×C4×D53—44no—
154C2×C4×D53—44no—
155C2×C8×C53—28no—
156C2×C8×C53—22no—
157C2×C8×C53—22no—
158C2×C8×C53C2×D414yes—
159C4⋊C4×C53—22no—
160C4⋊C4×C53—22no—
161C4⋊C4×C53—42no—
162#11719921—216no—
163#11719921—48no—
164C4⋊Dic53—24no—
165C4⋊Dic53—22no—
166C4⋊Dic53—22no—
167C4⋊Dic53—22no—
168C4⋊Dic53—42no—
169C4⋊Dic53C2×D414yes—
170#11719919—216no—
171C2×#88322—22no—
172C2×#88322—22no—
173C2×#88322—22no—
174C2×#88322—44no—
175C23×C53#12012yes—
176C23×C53#12112yes—
177C23×C53#12312yes—
178C23×C53C2×SD1612yes—
179C23×C53#13512yes—
180C23×C53#13712yes—
181C23×C53#13712yes—
182C23×C53C22×D411yescharacteristic
183C22×D53—22no—
184C22×D53—22no—
185C22×D53—22no—
186C22×D53—22no—
187C22×D53—22no—
188C22×D53—22no—
189C22×D53—28no—
190C22×D53—28no—
191C22×D53—28no—
192C22×D53—44no—
193C22×D53—44no—
194C22×D53—44no—
195C22×D53—44no—
196C22×D53C2×SD1614yes—
197Q8×C53—24no—
198Q8×C53—24no—
199Q8×C53—42no—
200Q8×C53—42no—
201Q8×C53—48no—
202C2×C4×C53—24no—
203C2×C4×C53—42no—
204C2×C4×C53—42no—
205C2×C4×C53—48no—
206C2×C4×C53—42no—
207C2×C4×C53—42no—
208C2×C4×C53—42no—
209C2×C4×C53—84no—
210C2×C4×C53#12012yes—
211C2×C4×C53#12112yes—
212C2×C4×C53#12114yes—
213C2×C4×C53#12114yes—
214C2×C4×C53#12312yes—
215C2×C4×C53#12314yes—
216C2×C4×C53#12414yes—
217C2×C4×C53C22×D411yescharacteristic
218C2×C4×C53C22×D411yescharacteristic
219C2×C4×C53C22×D411yescharacteristic
220C2×C4×C53C2×#4111yescharacteristic
221C2×C4×C53C2×#4111yescharacteristic
222C2×C4×C53C2×#4111yescharacteristic
223C2×Dic53—22no—
224C2×Dic53—24no—
225C2×Dic53—22no—
226C2×Dic53—22no—
227C2×Dic53—42no—
228C2×Dic53—42no—
229C2×Dic53—44no—
230C2×Dic53—48no—
231C2×Dic53—42no—
232C2×Dic53—42no—
233C2×Dic53—42no—
234C2×Dic53—84no—
235C4×D53—42no—
236C4×D53—42no—
237C4×D53—42no—
238C4×D53—42no—
239C4×D53—48no—
240C8×C53—24no—
241#88322—42no—
242C2×#2035—531nomaximal, Sylow
243C22×C53#39414yes—
244C22×C53#44814yes—
245C22×C53#45214yes—
246C22×C53#46514yes—
247C22×C53#47014yes—
248C22×C53#48314yes—
249C22×C53#48514yes—
250C22×C53C2×#12011yescharacteristic
251C22×C53C2×#12111yescharacteristic
252C22×C53C2×#12311yescharacteristic
253C22×C53C22×SD1611yescharacteristic
254C22×C53C2×#13511yescharacteristic
255C22×C53C2×#13711yescharacteristic
256C22×C53C2×#13711yescharacteristic
257C2×D53—24no—
258C2×D53—24no—
259C2×D53—24no—
260C2×D53—42no—
261C2×D53—42no—
262C2×D53—42no—
263C2×D53—42no—
264C2×D53—416no—
265C4×C53—44no—
266C4×C53—82no—
267C4×C53#39414yes—
268C4×C53C2×#12111yescharacteristic
269C4×C53C2×#12111yescharacteristic
270C4×C53C2×#12311yescharacteristic
271C4×C53C2×#12411yescharacteristic
272Dic53—42no—
273Dic53—44no—
274Dic53—82no—
275C22×C22⋊Q8—531no—
276C22×#106—531no—
277C22×#103—531no—
278C22×#103D5311yescharacteristic, complemented
279C22×C22⋊C8—531no—
280C2×#380—531no—
281#2035—538no—
282C2×#340—531no—
283C2×C53#203518yescomplemented
284C2×C53C2×#39411yescharacteristic
285C2×C53C2×#44811yescharacteristic
286C2×C53C2×#45211yescharacteristic
287C2×C53C2×#46511yescharacteristic
288C2×C53C2×#47011yescharacteristic
289C2×C53C2×#48311yescharacteristic
290C2×C53C2×#48511yescharacteristic
291D53—44no—
292C23×Q8C2×D5311yescharacteristic
293C24×C4—531no—
294C23×C8—531no—
295C23×C8C2×D5311yescharacteristic
296C2×C22⋊Q8—534no—
297C2×C22⋊Q8—1064no—
298C2×C22⋊Q8—1062no—
299C2×C22⋊Q8—1064no—
300C22×C4⋊C4—531no—
301C22×C4⋊C4—531no—
302C22×C4⋊C4—1061no—
303C22×C4⋊C4C2×D5311yescharacteristic
304C22×C22⋊C4—1061no—
305C22×C22⋊C4—1061no—
306C2×#106—534no—
307C2×#106—1064no—
308C2×#103—24no—
309C2×#103—534no—
310C2×#103—1064no—
311C2×#103C2×D5314yescomplemented
312C2×C22⋊C8—534no—
313C2×C22⋊C8—1064no—
314#380—534no—
315#380—1062no—
316C2×#95—1061no—
317#340—534no—
318C53C2×#203511yescharacteristic, complemented, Sylow
319C25—531no—
320C22×Q8—22no—
321C22×Q8—21no—
322C22×Q8—22no—
323C22×Q8—22no—
324C22×Q8—22no—
325C22×Q8—22no—
326C22×Q8D21214yes—
327C22×Q8C22×D5312yes—
328C23×C4—21no—
329C23×C4—21no—
330C23×C4—531no—
331C23×C4—532no—
332C23×C4—532no—
333C23×C4—1061no—
334C23×C4—1062no—
335C23×C4—1062no—
336C23×C4—1062no—
337C23×C4—1062no—
338C23×C4—1062no—
339C23×C4—1062no—
340C23×C4—1061no—
341C23×C4C22×D5311yescharacteristic
342C22×C8—22no—
343C22×C8—22no—
344C22×C8—532no—
345C22×C8—1062no—
346C22×C8—1062no—
347C22×C8—1062no—
348C22×C8C22⋊D5314yes—
349C22×C8C22×D5312yes—
350C22⋊Q8—1068no—
351C22⋊Q8—10616no—
352C22⋊Q8—1068no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—534no—
357C2×C4⋊C4—532no—
358C2×C4⋊C4—532no—
359C2×C4⋊C4—1064no—
360C2×C4⋊C4—1068no—
361C2×C4⋊C4—1062no—
362C2×C4⋊C4—1061no—
363C2×C4⋊C4—1062no—
364C2×C4⋊C4—1062no—
365C2×C4⋊C4—1061no—
366C2×C4⋊C4C22×D5312yes—
367C2×C22⋊C4—1064no—
368C2×C22⋊C4—1068no—
369C2×C22⋊C4—1064no—
370C2×C22⋊C4—2124no—
371#106—1068no—
372#103—28no—
373#103—1068no—
374C22⋊C8—1068no—
375#95—1064no—
376C24—534no—
377C24—532no—
378C24—1062no—
379C24—1062no—
380C24—1062no—
381C24—1062no—
382C24—1062no—
383C24—1062no—
384C24D4×D5311yescharacteristic
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—28no—
388C2×Q8—28no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8C2×D21214yes—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—532no—
407C22×C4—1064no—
408C22×C4—1062no—
409C22×C4—1062no—
410C22×C4—1062no—
411C22×C4—1062no—
412C22×C4—1064no—
413C22×C4—1064no—
414C22×C4—1064no—
415C22×C4—1062no—
416C22×C4—1062no—
417C22×C4—1062no—
418C22×C4—1062no—
419C22×C4—1062no—
420C22×C4—1062no—
421C22×C4—1062no—
422C22×C4—2122no—
423C22×C4—2122no—
424C22×C4—2122no—
425C22×C4—2122no—
426C22×C4—2122no—
427C22×C4—2124no—
428C22×C4—2124no—
429C22×C4—2124no—
430C22×C4—2124no—
431C22×C4C2×D21212yes—
432C22×C4#1171994512yes—
433C22×C4D4×D5311yescharacteristic
434C22×C4D4×D5312yes—
435C22×C4#1171994712yes—
436C22×C4#1171994712yes—
437C22×C4C2×C22⋊D5312yes—
438C22×C4C23×D5311yescharacteristic
439C2×C8—28no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—1062no—
443C2×C8—1062no—
444C2×C8—1062no—
445C2×C8—2124no—
446C2×C8C2×C22⋊D5314yes—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—42no—
450C4⋊C4—534no—
451C4⋊C4—10616no—
452C4⋊C4—1064no—
453C4⋊C4—1062no—
454C4⋊C4—1062no—
455C4⋊C4—1062no—
456C4⋊C4—2122no—
457C22⋊C4—10616no—
458C22⋊C4—2128no—
459C23—534no—
460C23—1068no—
461C23—1068no—
462C23—1062no—
463C23—1062no—
464C23—1062no—
465C23—1062no—
466C23—1062no—
467C23—1062no—
468C23—1068no—
469C23—2124no—
470C23—2124no—
471C23—2124no—
472C23—2124no—
473C23#1313318712yes—
474C23#1313318812yes—
475C23#1313319912yes—
476C23SD16×D5312yes—
477C23#1313323812yes—
478C23#1313324912yes—
479C23#1313324912yes—
480C23C2×D4×D5311yescharacteristic
481Q8—24no—
482Q8—24no—
483Q8—48no—
484Q8—42no—
485Q8—42no—
486C2×C4—24no—
487C2×C4—48no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—84no—
494C2×C4—1064no—
495C2×C4—1062no—
496C2×C4—1062no—
497C2×C4—1062no—
498C2×C4—4244no—
499C2×C4—2122no—
500C2×C4—2122no—
501C2×C4—2122no—
502C2×C4—2128no—
503C2×C4—2124no—
504C2×C4—2122no—
505C2×C4—2122no—
506C2×C4—2128no—
507C2×C4—2122no—
508C2×C4—2122no—
509C2×C4—2122no—
510C2×C4—2122no—
511C2×C4#1313318712yes—
512C2×C4#1313318812yes—
513C2×C4#1313318914yes—
514C2×C4#1313319114yes—
515C2×C4#1313319912yes—
516C2×C4#1313320114yes—
517C2×C4#1313320314yes—
518C2×C4C2×D4×D5311yescharacteristic
519C2×C4C22×C22⋊D5311yescharacteristic
520C2×C4C22×D21211yescharacteristic
521C2×C4C2×#1171994711yescharacteristic
522C2×C4C2×#1171994711yescharacteristic
523C2×C4C2×#1171994511yescharacteristic
524C8—24no—
525C8—2122no—
526C22—1064no—
527C22—1064no—
528C22—1064no—
529C22—2122no—
530C22—2122no—
531C22—2122no—
532C22—2122no—
533C22—21216no—
534C22#1681053914yes—
535C22#1681076314yes—
536C22#1681078814yes—
537C22#1681084914yes—
538C22#1681087814yes—
539C22#1681095214yes—
540C22#1681096214yes—
541C22C2×#1313318711yescharacteristic
542C22C2×#1313318811yescharacteristic
543C22C2×#1313319911yescharacteristic
544C22C2×SD16×D5311yescharacteristic
545C22C2×#1313323811yescharacteristic
546C22C2×#1313324911yescharacteristic
547C22C2×#1313324911yescharacteristic
548C4—44no—
549C4—82no—
550C4—4242no—
551C4—2124no—
552C4—2122no—
553C4#1681053914yes—
554C4C2×#1313318911yescharacteristic
555C4C2×#1313319111yescharacteristic
556C4C2×#1313320111yescharacteristic
557C4C2×#1313320311yescharacteristic
558C2—2124no—
559C2#2027591018yesdirect product factor
560C2C2×#1681053911yescharacteristic
561C2C2×#1681076311yescharacteristic
562C2C2×#1681078811yescharacteristic
563C2C2×#1681084911yescharacteristic
564C2C2×#1681087811yescharacteristic
565C2C2×#1681095211yescharacteristic
566C2C2×#1681096211yescharacteristic
subgroupG/H
centerC24D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC22×#103×C53C2
socleC24×C53C2×D4
prime radicalC24×D53D4
hypercenterC22×#103D53
hypocenterC53C2×#2035
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
2. C22×#13133149C2semidirect
3. C22×#13133141C2semidirect
6. #20275910C2direct (N × H)
7. C2×#16810385C2semidirect
8. C22×#103×C53C2semidirect
21. C2×#13133149C22semidirect
23. C2×#13133141C22semidirect
29. #16810385C22semidirect
34. C2×#103×C53C22semidirect
278. C22×#103D53semidirect
283. C2×C53#2035semidirect
311. C2×#103C2×D53semidirect
318. C53C2×#2035semidirect
559. C2#20275910direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131331932
C22×#131331492
C22×#131331412
C22×#131331292
C2×#168104802
#202759102
C2×#168103852
C22×#103×C532
C2×#203553