Finite Group Databasebeta

← C2×#20289980 #28314120

Subgroups — C2×#20289980 #28314120

Order 13568.

1490 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811593C211yesmaximal, characteristic, complemented
2C22×C22⋊#88324C211yesmaximal, characteristic, complemented
3#20289980C218yesmaximal, direct product factor
4C2×#16811469C212yesmaximal, complemented
5C2×#16811466C212yesmaximal, complemented
6C22×#13133198C211yesmaximal, characteristic
7C22×#11719940—21no—
8C23×#88324—21no—
9#16811593—22no—
10#16811593C2214yescomplemented
11C2×C22⋊#88324—22no—
12C2×C22⋊#88324—21no—
13C2×C22⋊#88324—21no—
14C2×C22⋊#88324C2212yes—
15C2×C22⋊#88324C2212yes—
16C2×C22⋊#88324C2214yescomplemented
17#16811469C2218yescomplemented
18#16811466C2218yescomplemented
19C24×Dic53C2211yescharacteristic
20C2×#13133198—21no—
21C2×#13133198—21no—
22C2×#13133198—21no—
23C2×#13133198—21no—
24C2×#13133198—22no—
25C2×#13133198—22no—
26C2×#13133198C414yes—
27C2×#13133198C2214yes—
28C22×#11719919C411yescharacteristic
29C22×#11719919C2211yescharacteristic
30C22×C22⋊Dic53C411yescharacteristic
31C22×C22⋊Dic53C411yescharacteristic
32C22×C22⋊Dic53C2211yescharacteristic
33C23×D4×C53C411yescharacteristic
34C2×#11719940—24no—
35C2×#11719940—44no—
36C22×#88324—22no—
37C22×#88324—22no—
38C22×#88324—24no—
39C22×#88324—44no—
40C22⋊#88324—22no—
41C22⋊#88324—24no—
42C22⋊#88324—24no—
43C22⋊#88324—22no—
44C22⋊#88324C2314yes—
45C22⋊#88324C2314yes—
46C23×Dic53—44no—
47C23×Dic53C2×C411yescharacteristic
48C23×Dic53C2×C411yescharacteristic
49C23×Dic53C2×C411yescharacteristic
50C23×Dic53C2×C412yes—
51C23×Dic53D411yescharacteristic
52C23×Dic53D411yescharacteristic
53C23×Dic53D411yescharacteristic
54C23×Dic53D411yescharacteristic
55C23×Dic53D412yes—
56C23×Dic53D412yes—
57C23×Dic53C2311yescharacteristic
58C23×Dic53C2312yes—
59#13133198—24no—
60#13133198—24no—
61#13133198—24no—
62#13133198—24no—
63#13133198—24no—
64#13133198—24no—
65#13133198—24no—
66#13133198—24no—
67C2×#11719919—21no—
68C2×#11719919—21no—
69C2×#11719919—22no—
70C2×#11719919—21no—
71C2×#11719919—21no—
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919C2×C414yes—
75C2×#11719919C2×C412yes—
76C2×#11719919C2312yes—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—22no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53—22no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53C2×C414yes—
88C2×C22⋊Dic53C2×C414yes—
89C2×C22⋊Dic53C2×C412yes—
90C2×C22⋊Dic53C2312yes—
91C25×C53C2×C411yescharacteristic
92C25×C53C2×C411yescharacteristic
93C22×D4×C53—21no—
94C22×D4×C53—22no—
95C22×D4×C53—21no—
96C22×D4×C53—44no—
97C22×D4×C53C2×C414yes—
98C23×C4×C53C2×C411yescharacteristic
99C24×C53—24no—
100C24×C53—24no—
101C24×C53—21no—
102C24×C53—21no—
103C24×C53—24no—
104C24×C53—42no—
105C24×C53—42no—
106C24×C53—42no—
107C24×C53C22⋊C411yescharacteristic
108C24×C53C22⋊C412yes—
109C24×C53C22⋊C411yescharacteristic
110C24×C53C22×C412yes—
111C24×C53C22×C411yescharacteristic
112C24×C53C22×C412yes—
113C2×D4×C53—24no—
114C2×D4×C53—24no—
115C2×D4×C53—48no—
116C2×D4×C53—44no—
117C2×D4×C53—44no—
118C2×D4×C53—48no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—42no—
122C22×C4×C53C22×C412yes—
123C22×Dic53—24no—
124C22×Dic53—24no—
125C22×Dic53—21no—
126C22×Dic53—21no—
127C22×Dic53—21no—
128C22×Dic53—21no—
129C22×Dic53—24no—
130C22×Dic53—42no—
131C22×Dic53—44no—
132C22×Dic53—44no—
133C22×Dic53—42no—
134C22×Dic53—42no—
135C22×Dic53—44no—
136C22×Dic53—42no—
137C22×Dic53—44no—
138C22×Dic53—42no—
139C22×Dic53C22⋊C411yescharacteristic
140C22×Dic53C22⋊C411yescharacteristic
141C22×Dic53C22⋊C412yes—
142C22×Dic53C22⋊C412yes—
143C22×Dic53C22⋊C412yes—
144C22×Dic53C22⋊C412yes—
145C22×Dic53C22⋊C412yes—
146C22×Dic53C22×C412yes—
147C22×Dic53C22×C412yes—
148C22×Dic53C22×C412yes—
149C22×Dic53C22×C411yescharacteristic
150C22×Dic53C2×D411yescharacteristic
151C22×Dic53C2×D411yescharacteristic
152C22×Dic53C2×D412yes—
153C22×Dic53C2×D412yes—
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53C2×D411yescharacteristic
156C22×Dic53C2×D412yes—
157C22×Dic53C2×D412yes—
158C22×Dic53C2411yescharacteristic
159#11719940—48no—
160C2×#88324—24no—
161C2×#88324—24no—
162C2×#88324—44no—
163C2×#88324—48no—
164C2×#88324—44no—
165C22⋊Dic53—24no—
166C22⋊Dic53—24no—
167C22⋊Dic53—24no—
168C22⋊Dic53—24no—
169C22⋊Dic53—22no—
170C22⋊Dic53—22no—
171C22⋊Dic53—22no—
172C22⋊Dic53—22no—
173#11719919—24no—
174#11719919—24no—
175#11719919—22no—
176#11719919—22no—
177#11719919—22no—
178#11719919—22no—
179C23×C53—22no—
180C23×C53—22no—
181C23×C53—22no—
182C23×C53—22no—
183C23×C53—24no—
184C23×C53—24no—
185C23×C53—22no—
186C23×C53—24no—
187C23×C53—22no—
188C23×C53—24no—
189C23×C53—22no—
190C23×C53—22no—
191C23×C53—24no—
192C23×C53—42no—
193C23×C53—42no—
194C23×C53—416no—
195C23×C53—42no—
196C23×C53—42no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53—42no—
200C23×C53—42no—
201C23×C53—42no—
202C23×C53—42no—
203C23×C53—88no—
204C23×C53#9914yes—
205C23×C53C2×C22⋊C412yes—
206C23×C53C2×C22⋊C411yescharacteristic
207C23×C53C2×C22⋊C412yes—
208C23×C53C2×C22⋊C411yescharacteristic
209C23×C53#13114yes—
210C23×C53C23×C411yescharacteristic
211D4×C53—48no—
212D4×C53—48no—
213C2×C4×C53—22no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×Dic53—24no—
218C2×Dic53—22no—
219C2×Dic53—22no—
220C2×Dic53—24no—
221C2×Dic53—24no—
222C2×Dic53—24no—
223C2×Dic53—22no—
224C2×Dic53—22no—
225C2×Dic53—22no—
226C2×Dic53—22no—
227C2×Dic53—24no—
228C2×Dic53—42no—
229C2×Dic53—42no—
230C2×Dic53—42no—
231C2×Dic53—41no—
232C2×Dic53—42no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—42no—
236C2×Dic53—42no—
237C2×Dic53—41no—
238C2×Dic53—42no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53C2×C22⋊C412yes—
244C2×Dic53C2×C22⋊C412yes—
245C2×Dic53C2×C22⋊C411yescharacteristic
246C2×Dic53C2×C22⋊C411yescharacteristic
247C2×Dic53C22×D411yescharacteristic
248C2×Dic53C22×D411yescharacteristic
249#88324—44no—
250#88324—44no—
251C2×#1468—531nomaximal, Sylow
252C22×C53—22no—
253C22×C53—21no—
254C22×C53—21no—
255C22×C53—21no—
256C22×C53—21no—
257C22×C53—22no—
258C22×C53—28no—
259C22×C53—28no—
260C22×C53—22no—
261C22×C53—22no—
262C22×C53—22no—
263C22×C53—48no—
264C22×C53—48no—
265C22×C53—44no—
266C22×C53—42no—
267C22×C53—42no—
268C22×C53—42no—
269C22×C53—84no—
270C22×C53—84no—
271C22×C53—84no—
272C22×C53#40812yes—
273C22×C53#40812yes—
274C22×C53C2×#9914yes—
275C22×C53C2×#9912yes—
276C22×C53#41312yes—
277C22×C53C22×C22⋊C411yescharacteristic
278C22×C53C2×#13112yes—
279C4×C53—42no—
280Dic53—21no—
281Dic53—21no—
282Dic53—22no—
283Dic53—21no—
284Dic53—21no—
285Dic53—22no—
286Dic53—42no—
287Dic53—42no—
288C22×#123—531no—
289C2×#407—531no—
290C22×C22⋊C8—531no—
291#1468—538no—
292C2×#324—532no—
293C2×#323—532no—
294C2×C53—22no—
295C2×C53—22no—
296C2×C53—24no—
297C2×C53—42no—
298C2×C53—82no—
299C2×C53#146814yescomplemented
300C2×C53C2×#40811yescharacteristic
301C2×C53C22×#9911yescharacteristic
302C2×C53C2×#41311yescharacteristic
303C23×D4#615711yescharacteristic
304C24×C4—531no—
305C22×M16—1061no—
306C23×C8—1061no—
307C2×#123—534no—
308C2×#123—534no—
309C2×#123—1061no—
310C2×#123—1061no—
311C2×#123—1061no—
312C2×#123—1061no—
313C2×#123—1062no—
314C2×#123—1062no—
315C22×C4⋊C4—531no—
316C22×C4⋊C4—531no—
317C22×C22⋊C4—531no—
318C22×C22⋊C4—531no—
319C22×C22⋊C4—531no—
320#407—534no—
321#407—1062no—
322C2×C22⋊C8—532no—
323C2×C22⋊C8—534no—
324C2×C22⋊C8—532no—
325C2×C22⋊C8—1062no—
326C2×C22⋊C8—1061no—
327C2×C22⋊C8—1061no—
328#324—538no—
329#323—538no—
330C53C2×#146811yescharacteristic, complemented, Sylow
331C25C2×#615711yescharacteristic
332C25C2×#615711yescharacteristic
333C22×D4—22no—
334C22×D4—21no—
335C22×D4—21no—
336C22×D4—44no—
337C22×D4C2×#615714yes—
338C23×C4—531no—
339C23×C4—531no—
340C23×C4—532no—
341C23×C4—531no—
342C23×C4—531no—
343C23×C4—532no—
344C23×C4—531no—
345C23×C4—531no—
346C23×C4—532no—
347C23×C4—532no—
348C23×C4—531no—
349C23×C4—531no—
350C23×C4—2124no—
351C23×C4C2×#615711yescharacteristic
352C2×M16—1064no—
353C2×M16—2124no—
354C22×C8—1064no—
355C22×C8—1062no—
356C22×C8—1062no—
357C22×C8—2124no—
358#123—1064no—
359#123—1064no—
360#123—1064no—
361#123—1064no—
362#123—1064no—
363#123—1064no—
364#123—1064no—
365#123—1064no—
366C2×C4⋊C4—532no—
367C2×C4⋊C4—532no—
368C2×C4⋊C4—534no—
369C2×C4⋊C4—1062no—
370C2×C4⋊C4—1061no—
371C2×C4⋊C4—1061no—
372C2×C4⋊C4—1061no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1061no—
376C2×C22⋊C4—532no—
377C2×C22⋊C4—534no—
378C2×C22⋊C4—532no—
379C2×C22⋊C4—534no—
380C2×C22⋊C4—1061no—
381C2×C22⋊C4—1061no—
382C2×C22⋊C4—1061no—
383C2×C22⋊C4—1061no—
384C2×C22⋊C4—1062no—
385C2×C22⋊C4—1061no—
386C2×C22⋊C4—1061no—
387C2×C22⋊C4—1062no—
388C2×C22⋊C4—1061no—
389C2×C22⋊C4—1061no—
390C22⋊C8—534no—
391C22⋊C8—534no—
392C22⋊C8—1062no—
393C22⋊C8—1064no—
394C22⋊C8—1062no—
395C22⋊C8—1064no—
396C24—24no—
397C24—24no—
398C24—21no—
399C24—21no—
400C24—24no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24C22⋊#615711yescharacteristic
405C24C22⋊#615712yes—
406C24C22⋊#615711yescharacteristic
407C24C22×#615712yes—
408C24C22×#615711yescharacteristic
409C24C22×#615712yes—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—48no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—48no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—42no—
419C22×C4—532no—
420C22×C4—532no—
421C22×C4—532no—
422C22×C4—532no—
423C22×C4—531no—
424C22×C4—531no—
425C22×C4—531no—
426C22×C4—531no—
427C22×C4—532no—
428C22×C4—532no—
429C22×C4—532no—
430C22×C4—531no—
431C22×C4—531no—
432C22×C4—532no—
433C22×C4—532no—
434C22×C4—531no—
435C22×C4—532no—
436C22×C4—531no—
437C22×C4—532no—
438C22×C4—532no—
439C22×C4—1064no—
440C22×C4—1064no—
441C22×C4—1061no—
442C22×C4—1061no—
443C22×C4—1061no—
444C22×C4—1061no—
445C22×C4—1064no—
446C22×C4—2122no—
447C22×C4—2124no—
448C22×C4—2124no—
449C22×C4—2122no—
450C22×C4—2122no—
451C22×C4—2124no—
452C22×C4—2122no—
453C22×C4—2124no—
454C22×C4—2122no—
455C22×C4C22×#615712yes—
456M16—2128no—
457C2×C8—1064no—
458C2×C8—1064no—
459C2×C8—2128no—
460C2×C8—2124no—
461C2×C8—2124no—
462C4⋊C4—1062no—
463C4⋊C4—1062no—
464C4⋊C4—1064no—
465C4⋊C4—1064no—
466C4⋊C4—1062no—
467C4⋊C4—1062no—
468C22⋊C4—1062no—
469C22⋊C4—1062no—
470C22⋊C4—1064no—
471C22⋊C4—1064no—
472C22⋊C4—1062no—
473C22⋊C4—1062no—
474C22⋊C4—1064no—
475C22⋊C4—1064no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—22no—
482C23—24no—
483C23—22no—
484C23—24no—
485C23—22no—
486C23—24no—
487C23—22no—
488C23—24no—
489C23—42no—
490C23—42no—
491C23—416no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—88no—
501C23#1313327514yes—
502C23C2×C22⋊#615712yes—
503C23C2×C22⋊#615711yescharacteristic
504C23C2×C22⋊#615712yes—
505C23C2×C22⋊#615711yescharacteristic
506C23#1313330414yes—
507C23C23×#615711yescharacteristic
508D4—48no—
509D4—48no—
510C2×C4—22no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—531no—
515C2×C4—531no—
516C2×C4—532no—
517C2×C4—532no—
518C2×C4—531no—
519C2×C4—531no—
520C2×C4—1064no—
521C2×C4—1062no—
522C2×C4—1062no—
523C2×C4—1064no—
524C2×C4—1064no—
525C2×C4—1064no—
526C2×C4—1062no—
527C2×C4—1062no—
528C2×C4—1062no—
529C2×C4—1062no—
530C2×C4—1064no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4—2121no—
534C2×C4—2122no—
535C2×C4—2122no—
536C2×C4—2122no—
537C2×C4—2121no—
538C2×C4—2122no—
539C2×C4—2122no—
540C2×C4—2122no—
541C2×C4—2122no—
542C2×C4—2122no—
543C2×C4—2122no—
544C2×C4—2122no—
545C2×C4—2122no—
546C8—2124no—
547C8—2124no—
548C22—28no—
549C22—28no—
550C22—22no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—48no—
560C22—48no—
561C22—44no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—84no—
566C22—84no—
567C22—84no—
568C22#1681159612yes—
569C22#1681159612yes—
570C22C2×#1313327514yes—
571C22C2×#1313327512yes—
572C22#1681161012yes—
573C22C22×C22⋊#615711yescharacteristic
574C22C2×#1313330412yes—
575C4—42no—
576C4—1061no—
577C4—1061no—
578C4—1061no—
579C4—1061no—
580C4—1062no—
581C4—1062no—
582C4—2122no—
583C4—2122no—
584C2—22no—
585C2—22no—
586C2—24no—
587C2—42no—
588C2—82no—
589C2#2028998014yesdirect product factor
590C2C2×#1681159611yescharacteristic
591C2C22×#1313327511yescharacteristic
592C2C2×#1681161011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
Fitting, prime radicalC23×D4×C53C4
socleC23×C53C2×C22⋊C4
cosocleC22×Dic53C24
hypercenterC23×D4#6157
hypocenterC53C2×#1468
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C25 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16811593C2semidirect
2. C22×C22⋊#88324C2semidirect
3. #20289980C2direct (N × H)
4. C2×#16811469C2semidirect
5. C2×#16811466C2semidirect
10. #16811593C22semidirect
16. C2×C22⋊#88324C22semidirect
17. #16811469C22semidirect
18. #16811466C22semidirect
299. C2×C53#1468semidirect
330. C53C2×#1468semidirect
589. C2#20289980direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168115932
C22×C22⋊#883242
#202899802
C2×#168114692
C2×#168114662
C22×#131331982
C2×#146853