Finite Group Databasebeta

← C2×#13115064 #15863775

Subgroups — C2×#13115064 #15863775

Order 3328.

1566 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719532C211yesmaximal, characteristic
2C2×#11718782C211yesmaximal, characteristic, complemented
3C2×#11718893C211yesmaximal, characteristic
4C2×#11718596C211yesmaximal, characteristic, complemented
5#13115064C218yesmaximal, direct product factor
6C2×#11718146C211yesmaximal, characteristic, complemented
7C2×#11718115C211yesmaximal, characteristic, complemented
8C22×C13×#104C211yesmaximal, characteristic
9C22×C13×#41C2211yescharacteristic
10C22×C13×M16C2211yescharacteristic
11C22×C42×C13C2211yescharacteristic
12#11719532—22no—
13#11719532—22no—
14#11719532—22no—
15#11719532C2214yes—
16C2×Q8×Dic13—21no—
17C2×D4×Dic13—21no—
18C2×D4×Dic13—22no—
19C2×#88180—21no—
20C2×#88180C2211yescharacteristic
21C22×C4⋊Dic13—21no—
22C22×C4⋊Dic13—22no—
23C22×C4⋊Dic13C2211yescharacteristic
24C22×C4⋊Dic13C2211yescharacteristic
25C22×#6099C2211yescharacteristic
26C2×C13×#104C418yescomplemented
27C2×C13×#104C2214yes—
28#11718893—22no—
29#11718893C2214yes—
30C2×#88075—21no—
31C2×#88072—21no—
32#11718782—22no—
33#11718782C2214yescomplemented
34C2×#88058—21no—
35C2×#88057—21no—
36#11718596—22no—
37#11718596C2214yescomplemented
38#11718146C2214yescomplemented
39#11718115C2214yescomplemented
40C2×C13×#41—21no—
41C2×C13×#41—21no—
42C2×C13×#41—22no—
43C2×C13×#41—22no—
44C2×C13×#41C2×C414yes—
45C2×C13×#41D412yes—
46C2×C13×#41Q814yes—
47C2×C13×#41C2312yes—
48C22×Q8×C13D411yescharacteristic
49C22×D4×C13—21no—
50C22×D4×C13D411yescharacteristic
51C23×C4×C13—21no—
52C23×C4×C13—21no—
53C23×C4×C13C2311yescharacteristic
54C23×Dic13—22no—
55C23×Dic13—21no—
56C23×Dic13D412yes—
57C2×C13×M16—21no—
58C2×C13×M16—22no—
59C2×C13×M16C2×C414yescomplemented
60C2×C13×M16C2312yes—
61C22×C8×C13—21no—
62C2×C42×C13—21no—
63C2×C42×C13—22no—
64C2×C42×C13C2×C414yescomplemented
65C2×C42×C13C2312yes—
66C2×C22⋊Dic13—22no—
67C2×C22⋊Dic13—21no—
68Q8×Dic13—24no—
69Q8×Dic13—24no—
70D4×Dic13—28no—
71D4×Dic13—24no—
72D4×Dic13—24no—
73D4×Dic13—28no—
74#88180—24no—
75#88180—21no—
76#88180—21no—
77#88180—24no—
78#88180D412yes—
79#88180C2312yes—
80C2×C4⋊Dic13—22no—
81C2×C4⋊Dic13—24no—
82C2×C4⋊Dic13—21no—
83C2×C4⋊Dic13—21no—
84C2×C4⋊Dic13—21no—
85C2×C4⋊Dic13—21no—
86C2×C4⋊Dic13—24no—
87C2×C4⋊Dic13—21no—
88C2×C4⋊Dic13—21no—
89C2×C4⋊Dic13—21no—
90C2×C4⋊Dic13—21no—
91C2×C4⋊Dic13—21no—
92C2×C4⋊Dic13—28no—
93C2×C4⋊Dic13—48no—
94C2×C4⋊Dic13D412yes—
95C2×C4⋊Dic13D411yescharacteristic
96C2×C4⋊Dic13D411yescharacteristic
97C2×C4⋊Dic13C2312yes—
98C2×C4⋊Dic13C2312yes—
99C2×C4×Dic13—22no—
100C2×C4×Dic13—21no—
101C2×#6099—21no—
102C2×#6099—22no—
103C2×#6099—22no—
104C2×#6099C2312yes—
105C22×#1055—21no—
106C13×#104C2×C4116yescomplemented
107#88075—24no—
108#88072—24no—
109#88058—24no—
110#88057—24no—
111C2×#1859—131nomaximal, Sylow
112C24×C13—21no—
113C24×C13C2×D411yescharacteristic
114C13×#41—24no—
115C13×#41—24no—
116C13×#41—24no—
117C13×#41—24no—
118C13×#41—24no—
119C13×#41—22no—
120C13×#41—22no—
121C13×#41C4⋊C418yescomplemented
122C13×#41C22×C414yes—
123C13×#41C2×Q814yes—
124C2×Q8×C13—21no—
125C2×Q8×C13—22no—
126C2×Q8×C13C4⋊C414yescomplemented
127C2×Q8×C13C2×D412yes—
128C2×D4×C13—24no—
129C2×D4×C13—22no—
130C2×D4×C13—24no—
131C2×D4×C13—22no—
132C2×D4×C13—21no—
133C2×D4×C13—22no—
134C2×D4×C13C4⋊C414yescomplemented
135C2×D4×C13C2×D412yes—
136C22×C4×C13—24no—
137C22×C4×C13—24no—
138C22×C4×C13—24no—
139C22×C4×C13—21no—
140C22×C4×C13—22no—
141C22×C4×C13—24no—
142C22×C4×C13—22no—
143C22×C4×C13—24no—
144C22×C4×C13—22no—
145C22×C4×C13C22×C412yes—
146C22×C4×C13C2×D411yescharacteristic
147C22×C4×C13C2×D411yescharacteristic
148C22×C4×C13C2×D411yescharacteristic
149C22×C4×C13C2×D411yescharacteristic
150C22×C4×C13C2×D411yescharacteristic
151C22×C4×C13C2×Q812yes—
152C22×C4×C13#4112yes—
153C22×C4×C13#4112yes—
154C22×C4×C13C2411yescharacteristic
155C22×Dic13—21no—
156C22×Dic13—22no—
157C22×Dic13—21no—
158C22×Dic13—28no—
159C22×Dic13—22no—
160C22×Dic13—22no—
161C22×Dic13—44no—
162C22×Dic13—44no—
163C22×Dic13—41no—
164C22×Dic13—41no—
165C22×Dic13—84no—
166C22×Dic13C2×D414yes—
167C13×M16—24no—
168C13×M16—22no—
169C13×M16C22×C414yescomplemented
170C2×C8×C13—24no—
171C2×C8×C13—22no—
172C42×C13—24no—
173C42×C13—22no—
174C42×C13C22×C414yescomplemented
175C22⋊Dic13—28no—
176C22⋊Dic13—22no—
177C22⋊Dic13—22no—
178C4⋊Dic13—28no—
179C4⋊Dic13—22no—
180C4⋊Dic13—21no—
181C4⋊Dic13—22no—
182C4⋊Dic13—21no—
183C4⋊Dic13—22no—
184C4⋊Dic13—22no—
185C4⋊Dic13—22no—
186C4⋊Dic13—22no—
187C4⋊Dic13—416no—
188C4⋊Dic13—41no—
189C4⋊Dic13—41no—
190C4⋊Dic13C2×D412yes—
191C4⋊Dic13C2×D412yes—
192C4×Dic13—28no—
193C4×Dic13—22no—
194C4×Dic13—22no—
195#6099—22no—
196#6099—22no—
197#6099—42no—
198C2×#1055—22no—
199C2×#1055—42no—
200C2×#520—131no—
201C2×#428—131no—
202C22×#104D1311yescharacteristic
203C2×#417—131no—
204C2×#384—131no—
205#1859—138no—
206C2×#343—131no—
207C2×#339—131no—
208C23×C13—22no—
209C23×C13—22no—
210C23×C13—22no—
211C23×C13—24no—
212C23×C13—42no—
213C23×C13—42no—
214C23×C13C4×D412yes—
215C23×C13#12011yescharacteristic
216C23×C13#12011yescharacteristic
217C23×C13C22⋊Q812yes—
218C23×C13C22×D411yescharacteristic
219Q8×C13—22no—
220Q8×C13—24no—
221Q8×C13C2×C4⋊C414yescomplemented
222D4×C13—24no—
223D4×C13—24no—
224D4×C13—28no—
225D4×C13—22no—
226D4×C13—24no—
227D4×C13C2×C4⋊C414yescomplemented
228C2×C4×C13—24no—
229C2×C4×C13—216no—
230C2×C4×C13—24no—
231C2×C4×C13—24no—
232C2×C4×C13—22no—
233C2×C4×C13—24no—
234C2×C4×C13—24no—
235C2×C4×C13—24no—
236C2×C4×C13—28no—
237C2×C4×C13C2×C4⋊C412yes—
238C2×C4×C13C2×C4⋊C412yes—
239C2×C4×C13C4×D412yes—
240C2×C4×C13C4×D412yes—
241C2×C4×C13C4×D412yes—
242C2×C4×C13#12011yescharacteristic
243C2×C4×C13#12011yescharacteristic
244C2×C4×C13C22⋊Q812yes—
245C2×C4×C13C22⋊Q812yes—
246C2×C4×C13C22⋊Q812yes—
247C2×C4×C13#12312yes—
248C2×C4×C13#12312yes—
249C2×C4×C13C23×C411yescharacteristic
250C2×C4×C13C22×D411yescharacteristic
251C2×C4×C13C22×D411yescharacteristic
252C2×C4×C13C22×Q811yescharacteristic
253C2×C4×C13C2×#4111yescharacteristic
254C2×C4×C13C2×#4111yescharacteristic
255C2×Dic13—22no—
256C2×Dic13—24no—
257C2×Dic13—22no—
258C2×Dic13—44no—
259C2×Dic13—42no—
260C2×Dic13—42no—
261C2×Dic13—44no—
262C2×Dic13—44no—
263C2×Dic13—42no—
264C2×Dic13—82no—
265C2×Dic13—88no—
266C2×Dic13—82no—
267C2×Dic13—82no—
268C8×C13—24no—
269#1055—42no—
270C22×#41C2×D1311yescharacteristic
271C22×M16—131no—
272C22×M16C2×D1311yescharacteristic
273#520—134no—
274#520—262no—
275#520—262no—
276#520—262no—
277C2×C4×Q8—261no—
278C2×C4×D4—262no—
279C2×C4×D4—261no—
280C2×#117—131no—
281C2×#117—261no—
282C22×C4⋊C4—131no—
283C22×C4⋊C4—131no—
284C22×C4⋊C4—261no—
285C22×C4⋊C4—262no—
286C22×C42C2×D1311yescharacteristic
287#428—134no—
288#428—262no—
289C2×#107—261no—
290C2×#106—261no—
291C2×#104Dic1318yescomplemented
292C2×#104C2×D1314yes—
293#417—134no—
294#417—262no—
295C2×#103—261no—
296C2×#102—261no—
297#384—134no—
298#384—262no—
299#343—134no—
300#339—134no—
301C22×C13—24no—
302C22×C13—24no—
303C22×C13—21no—
304C22×C13—21no—
305C22×C13—21no—
306C22×C13—21no—
307C22×C13—24no—
308C22×C13—48no—
309C22×C13—42no—
310C22×C13—42no—
311C22×C13#38512yes—
312C22×C13#38512yes—
313C22×C13#45312yes—
314C22×C13#45612yes—
315C22×C13C2×C4×D411yescharacteristic
316C22×C13C2×#12011yescharacteristic
317C22×C13C2×C22⋊Q811yescharacteristic
318C4×C13—28no—
319C4×C13—24no—
320C4×C13#38512yes—
321C4×C13#38512yes—
322C4×C13C22×C4⋊C411yescharacteristic
323C4×C13C2×C4×D411yescharacteristic
324C4×C13C2×C22⋊Q811yescharacteristic
325C4×C13C2×#12311yescharacteristic
326Dic13—44no—
327Dic13—42no—
328Dic13—42no—
329Dic13—82no—
330C2×#41—21no—
331C2×#41—21no—
332C2×#41—22no—
333C2×#41—22no—
334C2×#41Dic2614yes—
335C2×#41D5212yes—
336C2×#41C2×Dic1314yes—
337C2×#41C22×D1312yes—
338C22×Q8D5211yescharacteristic
339C22×D4—21no—
340C22×D4D5211yescharacteristic
341C23×C4—21no—
342C23×C4—21no—
343C23×C4—132no—
344C23×C4—262no—
345C23×C4—261no—
346C23×C4C22×D1311yescharacteristic
347C2×M16—21no—
348C2×M16—22no—
349C2×M16—132no—
350C2×M16—261no—
351C2×M16—262no—
352C2×M16—262no—
353C2×M16C2×Dic1314yescomplemented
354C2×M16C22×D1312yes—
355C22×C8—21no—
356C22×C8—261no—
357C4×Q8—264no—
358C4×Q8—264no—
359C4×D4—268no—
360C4×D4—268no—
361C4×D4—264no—
362C4×D4—264no—
363#117—132no—
364#117—132no—
365#117—264no—
366#117—264no—
367#117—261no—
368#117—261no—
369C2×C4⋊C4—131no—
370C2×C4⋊C4—131no—
371C2×C4⋊C4—132no—
372C2×C4⋊C4—132no—
373C2×C4⋊C4—132no—
374C2×C4⋊C4—264no—
375C2×C4⋊C4—262no—
376C2×C4⋊C4—264no—
377C2×C4⋊C4—261no—
378C2×C4⋊C4—261no—
379C2×C4⋊C4—261no—
380C2×C4⋊C4—261no—
381C2×C4⋊C4—261no—
382C2×C4⋊C4—261no—
383C2×C4⋊C4—261no—
384C2×C4⋊C4—261no—
385C2×C4⋊C4—268no—
386C2×C4⋊C4—261no—
387C2×C4⋊C4—528no—
388C2×C22⋊C4—262no—
389C2×C22⋊C4—261no—
390C2×C42—21no—
391C2×C42—22no—
392C2×C42—262no—
393C2×C42—261no—
394C2×C42C2×Dic1314yescomplemented
395C2×C42C22×D1312yes—
396#107—264no—
397#106—264no—
398#104C2×Dic13116yescomplemented
399#103—264no—
400#102—264no—
401C2×C13—21no—
402C2×C13—21no—
403C2×C13—22no—
404C2×C13—44no—
405C2×C13#185914yescomplemented
406C2×C13C2×#38511yescharacteristic
407C2×C13C2×#45311yescharacteristic
408C2×C13C2×#45611yescharacteristic
409C24—21no—
410C24D4×D1311yescharacteristic
411#41—24no—
412#41—24no—
413#41—24no—
414#41—24no—
415#41—24no—
416#41—22no—
417#41—22no—
418#41C4⋊Dic1318yescomplemented
419#41C2×Dic2614yes—
420#41C22×Dic1314yes—
421C2×Q8—21no—
422C2×Q8—22no—
423C2×Q8C4⋊Dic1314yescomplemented
424C2×Q8C2×D5212yes—
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—24no—
428C2×D4—22no—
429C2×D4—21no—
430C2×D4—22no—
431C2×D4C4⋊Dic1314yescomplemented
432C2×D4C2×D5212yes—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—134no—
443C22×C4—261no—
444C22×C4—261no—
445C22×C4—262no—
446C22×C4—268no—
447C22×C4—262no—
448C22×C4—262no—
449C22×C4—524no—
450C22×C4—524no—
451C22×C4—521no—
452C22×C4—521no—
453C22×C4—1044no—
454C22×C4C2×Dic2612yes—
455C22×C4C2×D5211yescharacteristic
456C22×C4C2×D5211yescharacteristic
457C22×C4D4×D1311yescharacteristic
458C22×C4D4×D1311yescharacteristic
459C22×C4D4×D1311yescharacteristic
460C22×C4#612912yes—
461C22×C4#612912yes—
462C22×C4C22×Dic1312yes—
463C22×C4C23×D1311yescharacteristic
464M16—24no—
465M16—22no—
466M16—262no—
467M16—262no—
468M16—522no—
469M16C22×Dic1314yescomplemented
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—262no—
473C2×C8—522no—
474C4⋊C4—132no—
475C4⋊C4—132no—
476C4⋊C4—268no—
477C4⋊C4—261no—
478C4⋊C4—262no—
479C4⋊C4—262no—
480C4⋊C4—261no—
481C4⋊C4—262no—
482C4⋊C4—262no—
483C4⋊C4—262no—
484C4⋊C4—262no—
485C4⋊C4—5216no—
486C4⋊C4—521no—
487C4⋊C4—521no—
488C22⋊C4—268no—
489C22⋊C4—262no—
490C22⋊C4—262no—
491C42—24no—
492C42—22no—
493C42—268no—
494C42—262no—
495C42—262no—
496C42C22×Dic1314yescomplemented
497C13C2×#185911yescharacteristic, complemented, Sylow
498C23—22no—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—42no—
503C23—42no—
504C23#8813212yes—
505C23#8813611yescharacteristic
506C23#8813611yescharacteristic
507C23D4×Dic1312yes—
508C23C2×D4×D1311yescharacteristic
509Q8—22no—
510Q8—24no—
511Q8C2×C4⋊Dic1314yescomplemented
512D4—24no—
513D4—24no—
514D4—28no—
515D4—22no—
516D4—24no—
517D4C2×C4⋊Dic1314yescomplemented
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—216no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—262no—
528C2×C4—262no—
529C2×C4—264no—
530C2×C4—524no—
531C2×C4—522no—
532C2×C4—522no—
533C2×C4—524no—
534C2×C4—524no—
535C2×C4—522no—
536C2×C4—1042no—
537C2×C4—1048no—
538C2×C4—1042no—
539C2×C4—1042no—
540C2×C4#8813212yes—
541C2×C4#8813212yes—
542C2×C4#8813212yes—
543C2×C4#8813611yescharacteristic
544C2×C4#8813611yescharacteristic
545C2×C4#8814012yes—
546C2×C4#8814012yes—
547C2×C4C2×C4⋊Dic1312yes—
548C2×C4C2×C4⋊Dic1312yes—
549C2×C4D4×Dic1312yes—
550C2×C4D4×Dic1312yes—
551C2×C4D4×Dic1312yes—
552C2×C4C22×Dic2611yescharacteristic
553C2×C4C22×D5211yescharacteristic
554C2×C4C2×D4×D1311yescharacteristic
555C2×C4C2×#612911yescharacteristic
556C2×C4C2×#612911yescharacteristic
557C2×C4C23×Dic1311yescharacteristic
558C8—24no—
559C8—522no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—24no—
565C22—24no—
566C22—24no—
567C22—42no—
568C22—42no—
569C22—48no—
570C22#1171848312yes—
571C22#1171848712yes—
572C22#1171861112yes—
573C22#1171861112yes—
574C22C2×#8813211yescharacteristic
575C22C2×#8813611yescharacteristic
576C22C2×D4×Dic1311yescharacteristic
577C4—28no—
578C4—24no—
579C4—524no—
580C4—522no—
581C4—522no—
582C4—1042no—
583C4#1171861112yes—
584C4#1171861112yes—
585C4C2×#8813211yescharacteristic
586C4C2×#8814011yescharacteristic
587C4C22×C4⋊Dic1311yescharacteristic
588C4C2×D4×Dic1311yescharacteristic
589C2—21no—
590C2—21no—
591C2—22no—
592C2—44no—
593C2#1311506414yesdirect product factor
594C2C2×#1171861111yescharacteristic
595C2C2×#1171848311yescharacteristic
596C2C2×#1171848711yescharacteristic
subgroupG/H
centerC23#88136
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC22×C13×#104C2
socleC23×C13#120
prime radicalC22×C13×#41C22
cosocleC22×C4×C13C24
hypercenterC22×#104D13
hypocenterC13C2×#1859
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#104

Complements

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

NH ≅ G/Ntype
2. C2×#11718782C2semidirect
4. C2×#11718596C2semidirect
5. #13115064C2direct (N × H)
6. C2×#11718146C2semidirect
7. C2×#11718115C2semidirect
26. C2×C13×#104C4semidirect
33. #11718782C22semidirect
37. #11718596C22semidirect
38. #11718146C22semidirect
39. #11718115C22semidirect
59. C2×C13×M16C2×C4semidirect
64. C2×C42×C13C2×C4semidirect
106. C13×#104C2×C4semidirect
121. C13×#41C4⋊C4semidirect
126. C2×Q8×C13C4⋊C4semidirect
134. C2×D4×C13C4⋊C4semidirect
169. C13×M16C22×C4semidirect
174. C42×C13C22×C4semidirect
221. Q8×C13C2×C4⋊C4semidirect
227. D4×C13C2×C4⋊C4semidirect
291. C2×#104Dic13semidirect
353. C2×M16C2×Dic13semidirect
394. C2×C42C2×Dic13semidirect
398. #104C2×Dic13semidirect
405. C2×C13#1859semidirect
418. #41C4⋊Dic13semidirect
423. C2×Q8C4⋊Dic13semidirect
431. C2×D4C4⋊Dic13semidirect
469. M16C22×Dic13semidirect
496. C42C22×Dic13semidirect
497. C13C2×#1859semidirect
511. Q8C2×C4⋊Dic13semidirect
517. D4C2×C4⋊Dic13semidirect
593. C2#13115064direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117195322
C2×#117187822
C2×#117188932
C2×#117185962
#131150642
C2×#117181462
C2×#117181152
C22×C13×#1042
C2×#185913