Finite Group Databasebeta

← C2×#13115059 #15863770

Subgroups — C2×#13115059 #15863770

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×#11719228C211yesmaximal, characteristic
2C22×#88065C211yesmaximal, characteristic
3C2×#11718627C211yesmaximal, characteristic
4C2×#11718637C211yesmaximal, characteristic, complemented
5#13115059C218yesmaximal, direct product factor
6C2×#11718146C211yesmaximal, characteristic, complemented
7C2×#11718124C211yesmaximal, characteristic, complemented
8C2×C13×#428C211yesmaximal, characteristic, complemented
9C22×#6127C2211yescharacteristic
10C22×C13×M16C2211yescharacteristic
11C2×C13×#117C2211yescharacteristic
12C22×C13×C4⋊C4C2211yescharacteristic
13C22×C4⋊Dic13C2211yescharacteristic
14C22×#6101—22no—
15C22×C4×Dic13C2211yescharacteristic
16C22×#6099C2211yescharacteristic
17#11719228—22no—
18#11719228—22no—
19#11719228—22no—
20#11719228C2214yes—
21C2×#88146—21no—
22C2×C4⋊C4×D13—21no—
23C2×#88135—22no—
24C2×C4×D52—21no—
25C2×C4×Dic26—21no—
26C13×#428—22no—
27C13×#428C2214yescomplemented
28C2×C13×#107—21no—
29C2×C13×#106—21no—
30C2×#88065C418yescomplemented
31C2×#88065C2214yes—
32#11718637—22no—
33#11718637C2214yescomplemented
34C2×#88050—21no—
35#11718627—22no—
36#11718627C2214yes—
37C2×#88049—21no—
38#11718146C2214yescomplemented
39#11718124C2214yescomplemented
40C23×C4×C13—21no—
41C23×C4×C13C2311yescharacteristic
42C22×C22⋊D13—21no—
43C23×Dic13—21no—
44C23×Dic13D412yes—
45C2×#6127—21no—
46C2×#6127—21no—
47C2×#6127—22no—
48C2×#6127—22no—
49C2×#6127C2×C414yes—
50C2×#6127D412yes—
51C2×#6127Q814yes—
52C2×#6127C2312yes—
53C22×D52D411yescharacteristic
54C22×C4×D13—21no—
55C22×C4×D13—22no—
56C22×Dic26D411yescharacteristic
57C2×C13×M16—21no—
58C2×C13×M16—22no—
59C2×C13×M16C2×C414yescomplemented
60C2×C13×M16C2312yes—
61C22×C8×C13—21no—
62C13×#117—21no—
63C13×#117—21no—
64C13×#117D412yes—
65C13×#117C2312yes—
66C2×C13×C4⋊C4—21no—
67C2×C13×C4⋊C4—21no—
68C2×C13×C4⋊C4—21no—
69C2×C13×C4⋊C4—21no—
70C2×C13×C4⋊C4—21no—
71C2×C13×C4⋊C4D411yescharacteristic
72C2×C13×C4⋊C4D411yescharacteristic
73C2×C13×C4⋊C4C2312yes—
74C2×C13×C22⋊C4—21no—
75C2×C42×C13—21no—
76C2×#6103—22no—
77C2×C4⋊Dic13—21no—
78C2×C4⋊Dic13—21no—
79C2×C4⋊Dic13—21no—
80C2×C4⋊Dic13—21no—
81C2×C4⋊Dic13D412yes—
82C2×C4⋊Dic13C2312yes—
83C2×#6101—22no—
84C2×#6101—28no—
85C2×#6101—48no—
86C2×C4×Dic13—22no—
87C2×C4×Dic13—21no—
88C2×C4×Dic13—22no—
89C2×C4×Dic13C2×C414yescomplemented
90C2×C4×Dic13C2312yes—
91C2×#6099—21no—
92C2×#6099—22no—
93C2×#6099—22no—
94C2×#6099C2312yes—
95C22×#1055—21no—
96#88146—24no—
97#88146—24no—
98C4⋊C4×D13—24no—
99C4⋊C4×D13—24no—
100#88135—28no—
101#88135—28no—
102C4×D52—24no—
103C4×D52—24no—
104C4×Dic26—24no—
105C4×Dic26—24no—
106C13×#107—24no—
107C13×#106—24no—
108#88065C2×C4116yescomplemented
109#88050—24no—
110#88049—24no—
111C2×#1859—131nomaximal, Sylow
112C24×C13C2×D411yescharacteristic
113C23×D13—21no—
114C22×C4×C13—21no—
115C22×C4×C13—21no—
116C22×C4×C13—22no—
117C22×C4×C13—44no—
118C22×C4×C13—41no—
119C22×C4×C13—41no—
120C22×C4×C13C22×C412yes—
121C22×C4×C13C2×D411yescharacteristic
122C22×C4×C13C2×D411yescharacteristic
123C22×C4×C13C2×D411yescharacteristic
124C22×C4×C13C2×D411yescharacteristic
125C22×C4×C13C2×D411yescharacteristic
126C22×C4×C13C2×Q812yes—
127C22×C4×C13#4112yes—
128C22×C4×C13#4112yes—
129C22×C4×C13C2411yescharacteristic
130C2×C22⋊D13—24no—
131C2×C22⋊D13—22no—
132C2×C22⋊D13—24no—
133C2×C22⋊D13—22no—
134C22×Dic13—24no—
135C22×Dic13—24no—
136C22×Dic13—24no—
137C22×Dic13—21no—
138C22×Dic13—22no—
139C22×Dic13—22no—
140C22×Dic13—22no—
141C22×Dic13—44no—
142C22×Dic13C2×D414yes—
143#6127—24no—
144#6127—24no—
145#6127—24no—
146#6127—24no—
147#6127—24no—
148#6127—22no—
149#6127—22no—
150#6127C4⋊C418yescomplemented
151#6127C22×C414yes—
152#6127C2×Q814yes—
153C2×D52—21no—
154C2×D52—22no—
155C2×D52C4⋊C414yescomplemented
156C2×D52C2×D412yes—
157C2×C4×D13—24no—
158C2×C4×D13—22no—
159C2×C4×D13—24no—
160C2×C4×D13—22no—
161C2×C4×D13—28no—
162C2×C4×D13—84no—
163C2×Dic26—21no—
164C2×Dic26—22no—
165C2×Dic26C4⋊C414yescomplemented
166C2×Dic26C2×D412yes—
167C13×M16—24no—
168C13×M16—22no—
169C13×M16C22×C414yescomplemented
170C2×C8×C13—24no—
171C2×C8×C13—22no—
172C13×C4⋊C4—21no—
173C13×C4⋊C4—21no—
174C13×C4⋊C4—22no—
175C13×C4⋊C4—22no—
176C13×C4⋊C4—41no—
177C13×C4⋊C4—41no—
178C13×C4⋊C4C2×D412yes—
179C13×C4⋊C4C2×D412yes—
180C13×C22⋊C4—22no—
181C13×C22⋊C4—22no—
182C42×C13—22no—
183C42×C13—22no—
184#6103—28no—
185C4⋊Dic13—22no—
186C4⋊Dic13—22no—
187C4⋊Dic13—22no—
188C4⋊Dic13—22no—
189#6101—28no—
190#6101—416no—
191C4×Dic13—28no—
192C4×Dic13—24no—
193C4×Dic13—22no—
194C4×Dic13C22×C414yescomplemented
195#6099—22no—
196#6099—22no—
197#6099—42no—
198C2×#1055—22no—
199C2×#1055—42no—
200C2×#520—131no—
201C2×#428D1311yescharacteristic, complemented
202C22×#104—131no—
203C2×#417—131no—
204C2×#384—131no—
205#1859—138no—
206C2×#343—131no—
207C2×#339—131no—
208C23×C13—22no—
209C23×C13—22no—
210C23×C13C4×D412yes—
211C23×C13#12011yescharacteristic
212C23×C13#12011yescharacteristic
213C23×C13C22⋊Q812yes—
214C23×C13C22×D411yescharacteristic
215C22×D13—22no—
216C22×D13—24no—
217C22×D13—42no—
218C22×D13—42no—
219C2×C4×C13—22no—
220C2×C4×C13—22no—
221C2×C4×C13—42no—
222C2×C4×C13—42no—
223C2×C4×C13—44no—
224C2×C4×C13—42no—
225C2×C4×C13—82no—
226C2×C4×C13—82no—
227C2×C4×C13C2×C4⋊C412yes—
228C2×C4×C13C2×C4⋊C412yes—
229C2×C4×C13C4×D412yes—
230C2×C4×C13C4×D412yes—
231C2×C4×C13C4×D412yes—
232C2×C4×C13#12011yescharacteristic
233C2×C4×C13#12011yescharacteristic
234C2×C4×C13C22⋊Q812yes—
235C2×C4×C13C22⋊Q812yes—
236C2×C4×C13C22⋊Q812yes—
237C2×C4×C13#12312yes—
238C2×C4×C13#12312yes—
239C2×C4×C13C23×C411yescharacteristic
240C2×C4×C13C22×D411yescharacteristic
241C2×C4×C13C22×D411yescharacteristic
242C2×C4×C13C22×Q811yescharacteristic
243C2×C4×C13C2×#4111yescharacteristic
244C2×C4×C13C2×#4111yescharacteristic
245C22⋊D13—24no—
246C22⋊D13—24no—
247C22⋊D13—28no—
248C2×Dic13—24no—
249C2×Dic13—216no—
250C2×Dic13—24no—
251C2×Dic13—24no—
252C2×Dic13—22no—
253C2×Dic13—24no—
254C2×Dic13—24no—
255C2×Dic13—44no—
256C2×Dic13—44no—
257C2×Dic13—82no—
258D52—22no—
259D52—24no—
260D52C2×C4⋊C414yescomplemented
261C4×D13—24no—
262C4×D13—24no—
263C4×D13—28no—
264C4×D13—88no—
265Dic26—22no—
266Dic26—24no—
267Dic26C2×C4⋊C414yescomplemented
268C8×C13—24no—
269#1055—42no—
270C22×#41—131no—
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—261no—
279C2×C4×D4—262no—
280C2×#117—261no—
281C2×#117C2×D1311yescharacteristic
282C22×C4⋊C4—131no—
283C22×C4⋊C4—261no—
284C22×C4⋊C4—262no—
285C22×C4⋊C4C2×D1311yescharacteristic
286C22×C42—131no—
287#428—22no—
288#428C2×D1314yescomplemented
289C2×#107—21no—
290C2×#106—21no—
291C2×#104—138no—
292C2×#104—134no—
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#38512yes—
308C22×C13#38512yes—
309C22×C13#45312yes—
310C22×C13#45612yes—
311C22×C13C2×C4×D411yescharacteristic
312C22×C13C2×#12011yescharacteristic
313C22×C13C2×C22⋊Q811yescharacteristic
314C2×D13—24no—
315C2×D13—48no—
316C2×D13—42no—
317C2×D13—42no—
318C4×C13—42no—
319C4×C13—42no—
320C4×C13—82no—
321C4×C13#38512yes—
322C4×C13#38512yes—
323C4×C13C22×C4⋊C411yescharacteristic
324C4×C13C2×C4×D411yescharacteristic
325C4×C13C2×C22⋊Q811yescharacteristic
326C4×C13C2×#12311yescharacteristic
327Dic13—28no—
328Dic13—24no—
329Dic13—44no—
330C2×#41—134no—
331C2×#41—132no—
332C2×#41—134no—
333C2×#41—132no—
334C2×#41—262no—
335C2×#41—261no—
336C2×#41—262no—
337C2×#41—261no—
338C22×Q8—131no—
339C22×D4—131no—
340C22×D4—261no—
341C23×C4—21no—
342C23×C4—132no—
343C23×C4—262no—
344C23×C4—261no—
345C23×C4—261no—
346C23×C4C22×D1311yescharacteristic
347C2×M16—22no—
348C2×M16—21no—
349C2×M16—132no—
350C2×M16—261no—
351C2×M16—262no—
352C2×M16—262no—
353C2×M16C4×D1314yescomplemented
354C2×M16C22×D1312yes—
355C22×C8—21no—
356C22×C8—261no—
357C4×Q8—264no—
358C4×Q8—264no—
359C4×D4—264no—
360C4×D4—264no—
361C4×D4—268no—
362C4×D4—268no—
363#117—21no—
364#117—21no—
365#117—264no—
366#117—264no—
367#117C22⋊D1312yes—
368#117C22×D1312yes—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—132no—
375C2×C4⋊C4—132no—
376C2×C4⋊C4—264no—
377C2×C4⋊C4—262no—
378C2×C4⋊C4—261no—
379C2×C4⋊C4—261no—
380C2×C4⋊C4—261no—
381C2×C4⋊C4—261no—
382C2×C4⋊C4—264no—
383C2×C4⋊C4—268no—
384C2×C4⋊C4—528no—
385C2×C4⋊C4C22⋊D1311yescharacteristic
386C2×C4⋊C4C22⋊D1311yescharacteristic
387C2×C4⋊C4C22×D1312yes—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—262no—
390C2×C42—21no—
391C2×C42—134no—
392C2×C42—132no—
393C2×C42—262no—
394C2×C42—261no—
395C2×C42—262no—
396#107—24no—
397#106—24no—
398#104—1316no—
399#103—264no—
400#102—264no—
401C2×C13—21no—
402C2×C13—21no—
403C2×C13—22no—
404C2×C13#185914yescomplemented
405C2×C13C2×#38511yescharacteristic
406C2×C13C2×#45311yescharacteristic
407C2×C13C2×#45611yescharacteristic
408D13—44no—
409C24—261no—
410C24D4×D1311yescharacteristic
411#41—138no—
412#41—134no—
413#41—134no—
414#41—264no—
415#41—264no—
416#41—264no—
417#41—264no—
418#41—264no—
419#41—262no—
420#41—262no—
421C2×Q8—134no—
422C2×Q8—132no—
423C2×Q8—262no—
424C2×Q8—261no—
425C2×D4—134no—
426C2×D4—132no—
427C2×D4—264no—
428C2×D4—264no—
429C2×D4—262no—
430C2×D4—262no—
431C2×D4—262no—
432C2×D4—261no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—44no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—134no—
440C22×C4—268no—
441C22×C4—264no—
442C22×C4—264no—
443C22×C4—264no—
444C22×C4—261no—
445C22×C4—262no—
446C22×C4—264no—
447C22×C4—262no—
448C22×C4—264no—
449C22×C4—262no—
450C22×C4—262no—
451C22×C4—262no—
452C22×C4—524no—
453C22×C4—1044no—
454C22×C4C2×C4×D1312yes—
455C22×C4#612712yes—
456C22×C4D4×D1311yescharacteristic
457C22×C4D4×D1311yescharacteristic
458C22×C4D4×D1311yescharacteristic
459C22×C4Q8×D1312yes—
460C22×C4#613112yes—
461C22×C4C2×C22⋊D1311yescharacteristic
462C22×C4C2×C22⋊D1311yescharacteristic
463C22×C4C23×D1311yescharacteristic
464M16—24no—
465M16—22no—
466M16—262no—
467M16—262no—
468M16—522no—
469M16C2×C4×D1314yescomplemented
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—262no—
473C2×C8—522no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—268no—
481C4⋊C4—262no—
482C4⋊C4—262no—
483C4⋊C4—262no—
484C4⋊C4—262no—
485C4⋊C4—5216no—
486C4⋊C4C2×C22⋊D1312yes—
487C4⋊C4C2×C22⋊D1312yes—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—268no—
491C42—22no—
492C42—22no—
493C42—134no—
494C42—268no—
495C42—264no—
496C42—262no—
497C13C2×#185911yescharacteristic, complemented, Sylow
498C23—22no—
499C23—22no—
500C23—262no—
501C23—264no—
502C23—522no—
503C23—522no—
504C23#8814712yes—
505C23#8815012yes—
506C23#8819111yescharacteristic
507C23#8819111yescharacteristic
508C23C2×D4×D1311yescharacteristic
509Q8—134no—
510Q8—262no—
511Q8—264no—
512D4—134no—
513D4—264no—
514D4—264no—
515D4—268no—
516D4—262no—
517D4—264no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—264no—
527C2×C4—264no—
528C2×C4—2616no—
529C2×C4—264no—
530C2×C4—262no—
531C2×C4—264no—
532C2×C4—264no—
533C2×C4—264no—
534C2×C4—268no—
535C2×C4—264no—
536C2×C4—524no—
537C2×C4—524no—
538C2×C4—1048no—
539C2×C4—1042no—
540C2×C4C4⋊C4×D1312yes—
541C2×C4C4⋊C4×D1312yes—
542C2×C4#8814712yes—
543C2×C4#8814812yes—
544C2×C4#8814812yes—
545C2×C4#8815012yes—
546C2×C4C4×C22⋊D1312yes—
547C2×C4C4×C22⋊D1312yes—
548C2×C4#8819111yescharacteristic
549C2×C4#8819111yescharacteristic
550C2×C4#8820012yes—
551C2×C4#8820012yes—
552C2×C4C22×C4×D1311yescharacteristic
553C2×C4C2×#612711yescharacteristic
554C2×C4C2×D4×D1311yescharacteristic
555C2×C4C2×Q8×D1311yescharacteristic
556C2×C4C2×#613111yescharacteristic
557C2×C4C22×C22⋊D1311yescharacteristic
558C8—24no—
559C8—522no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—24no—
565C22—24no—
566C22—264no—
567C22—522no—
568C22—522no—
569C22—528no—
570C22#1171865012yes—
571C22#1171865012yes—
572C22#1171885912yes—
573C22#1171886612yes—
574C22C2×#8814711yescharacteristic
575C22C2×#8815011yescharacteristic
576C22C2×#8819111yescharacteristic
577C4—42no—
578C4—42no—
579C4—82no—
580C4—268no—
581C4—264no—
582C4—524no—
583C4#1171865012yes—
584C4#1171865012yes—
585C4C2×C4⋊C4×D1311yescharacteristic
586C4C2×#8814811yescharacteristic
587C4C2×C4×C22⋊D1311yescharacteristic
588C4C2×#8820011yescharacteristic
589C2—21no—
590C2—21no—
591C2—22no—
592C2—524no—
593C2#1311505914yesdirect product factor
594C2C2×#1171865011yescharacteristic
595C2C2×#1171885911yescharacteristic
596C2C2×#1171886611yescharacteristic
subgroupG/H
centerC23#88191
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C13×#428C2
socleC23×C13#120
prime radicalC22×#6127C22
cosocleC22×C4×C13C24
hypercenterC2×#428D13
hypocenterC13C2×#1859
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
4. C2×#11718637C2semidirect
5. #13115059C2direct (N × H)
6. C2×#11718146C2semidirect
7. C2×#11718124C2semidirect
8. C2×C13×#428C2semidirect
27. C13×#428C22semidirect
30. C2×#88065C4semidirect
33. #11718637C22semidirect
38. #11718146C22semidirect
39. #11718124C22semidirect
59. C2×C13×M16C2×C4semidirect
89. C2×C4×Dic13C2×C4semidirect
108. #88065C2×C4semidirect
150. #6127C4⋊C4semidirect
155. C2×D52C4⋊C4semidirect
165. C2×Dic26C4⋊C4semidirect
169. C13×M16C22×C4semidirect
194. C4×Dic13C22×C4semidirect
201. C2×#428D13semidirect
260. D52C2×C4⋊C4semidirect
267. Dic26C2×C4⋊C4semidirect
288. #428C2×D13semidirect
353. C2×M16C4×D13semidirect
404. C2×C13#1859semidirect
469. M16C2×C4×D13semidirect
497. C13C2×#1859semidirect
593. C2#13115059direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117192282
C22×#880652
C2×#117186272
C2×#117186372
#131150592
C2×#117181462
C2×#117181242
C2×C13×#4282
C2×#185913