Finite Group Databasebeta

← C13×#24059 #15710398

Subgroups — C13×#24059 #15710398

Order 3328.

1348 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C13×#3485C211yesmaximal, characteristic
2C13×#2623C212yesmaximal, complemented
3C13×#2311C211yesmaximal, characteristic, complemented
4C13×#2044C214yesmaximal, complemented
5C13×#1899C212yesmaximal, complemented
6C13×#1867C212yesmaximal, complemented
7C13×#1790C212yesmaximal, complemented
8C13×#1782C211yesmaximal, characteristic, complemented
9C2×C13×H(5,𝔽2)C2211yescharacteristic
10C22×C13×#41C2211yescharacteristic
11C13×#547—22no—
12C13×#547—22no—
13C13×#546—22no—
14C13×#546—22no—
15C13×#536—21no—
16C13×#534—21no—
17C13×#534C2212yes—
18C13×#525—21no—
19C2×C13×#123—21no—
20C2×C13×#123C2212yes—
21C13×#518C2211yescharacteristic
22C2×C13×#117C2211yescharacteristic, complemented
23C22×C13×C4⋊C4C2211yescharacteristic, complemented
24C2×C13×#99C2212yes—
25C2×C13×#99C2214yes—
26C2×C13×#99C2212yescomplemented
27C13×#400—22no—
28C13×#399—22no—
29C13×#398—22no—
30C13×#397—22no—
31C13×#391—21no—
32C13×#390—21no—
33C13×#386C2214yes—
34C13×#386C2212yes—
35C13×#386C2212yescomplemented
36C13×#385C2214yescomplemented
37C13×#381C2212yescomplemented
38C13×#380—21no—
39C13×#380—21no—
40C13×#342C2214yescomplemented
41C13×H(5,𝔽2)—24no—
42C13×H(5,𝔽2)—42no—
43C2×C13×#41—24no—
44C2×C13×#41—21no—
45C2×C13×#41—22no—
46C2×C13×#41—22no—
47C2×C13×#41—22no—
48C2×C13×#41—42no—
49C22×Q8×C13D411yescharacteristic
50C22×D4×C13—21no—
51C22×D4×C13—22no—
52C22×D4×C13D411yescharacteristic
53C22×D4×C13C2312yes—
54C22×D4×C13C2311yescharacteristic
55C22×D4×C13C2311yescharacteristic
56C23×C4×C13D411yescharacteristic
57C23×C4×C13D411yescharacteristic
58C23×C4×C13C2312yes—
59C23×C4×C13C2311yescharacteristic
60C13×C4⋊Q8—22no—
61C13×#124—22no—
62C13×#123—22no—
63C13×#123—22no—
64C13×#123—22no—
65C13×#123—22no—
66C13×#123—22no—
67C13×#123—22no—
68C13×#123—22no—
69C13×#123—22no—
70C13×#123—22no—
71C13×C22⋊Q8—21no—
72C13×C22⋊Q8—21no—
73C13×C22⋊Q8—22no—
74C13×C22⋊Q8—22no—
75C13×#121—21no—
76C13×#121—21no—
77C13×#121—22no—
78C13×#121—22no—
79C13×#121—22no—
80C13×#121—22no—
81C13×#120—22no—
82C13×#120—22no—
83C13×#120—22no—
84C4×D4×C13—22no—
85C4×D4×C13—22no—
86C4×D4×C13—22no—
87C4×D4×C13—22no—
88C13×#117—21no—
89C13×#117—42no—
90C2×C13×C4⋊C4—21no—
91C2×C13×C4⋊C4—21no—
92C2×C13×C4⋊C4—21no—
93C2×C13×C4⋊C4—21no—
94C2×C13×C4⋊C4—21no—
95C2×C13×C4⋊C4—22no—
96C2×C13×C4⋊C4—22no—
97C2×C13×C4⋊C4—24no—
98C2×C13×C4⋊C4D412yes—
99C2×C13×C4⋊C4D412yes—
100C2×C13×C22⋊C4—21no—
101C2×C13×C22⋊C4—21no—
102C2×C13×C22⋊C4—21no—
103C2×C13×C22⋊C4—22no—
104C2×C13×C22⋊C4—22no—
105C2×C13×C22⋊C4—24no—
106C2×C13×C22⋊C4D412yes—
107C2×C13×C22⋊C4D412yes—
108C2×C13×C22⋊C4C2314yes—
109C2×C13×C22⋊C4C2312yes—
110C2×C13×C22⋊C4C2312yes—
111C2×C42×C13—21no—
112C13×#99—24no—
113C13×#99—44no—
114C13×#99—42no—
115C13×#95—22no—
116C13×#95—22no—
117C13×#95—22no—
118C13×#95—22no—
119C13×#95—24no—
120C13×#95—24no—
121#24059C1311yesmaximal, characteristic, direct product factor, Sylow
122C24×C13—21no—
123C24×C13C2×D412yes—
124C24×C13C2×D411yescharacteristic
125C24×C13C2×D411yescharacteristic
126C24×C13C2411yescharacteristic
127C13×#41—48no—
128C13×#41—48no—
129C13×#41—48no—
130C13×#41—42no—
131C13×#41—42no—
132C2×Q8×C13—21no—
133C2×Q8×C13—22no—
134C2×Q8×C13—22no—
135C2×Q8×C13—41no—
136C2×D4×C13—22no—
137C2×D4×C13—22no—
138C2×D4×C13—22no—
139C2×D4×C13—21no—
140C2×D4×C13—22no—
141C2×D4×C13—22no—
142C2×D4×C13—22no—
143C2×D4×C13—22no—
144C2×D4×C13—22no—
145C2×D4×C13—24no—
146C2×D4×C13—22no—
147C2×D4×C13—24no—
148C2×D4×C13—22no—
149C2×D4×C13—22no—
150C2×D4×C13—22no—
151C2×D4×C13—41no—
152C2×D4×C13—48no—
153C2×D4×C13—42no—
154C2×D4×C13—42no—
155C2×D4×C13#4114yes—
156C22×C4×C13—22no—
157C22×C4×C13—22no—
158C22×C4×C13—21no—
159C22×C4×C13—22no—
160C22×C4×C13—22no—
161C22×C4×C13—21no—
162C22×C4×C13—21no—
163C22×C4×C13—21no—
164C22×C4×C13—21no—
165C22×C4×C13—21no—
166C22×C4×C13—21no—
167C22×C4×C13—22no—
168C22×C4×C13—24no—
169C22×C4×C13—22no—
170C22×C4×C13—22no—
171C22×C4×C13—24no—
172C22×C4×C13—41no—
173C22×C4×C13—41no—
174C22×C4×C13—82no—
175C22×C4×C13C2×D412yes—
176C22×C4×C13C2×D411yescharacteristic
177C22×C4×C13C2×D412yes—
178C22×C4×C13C2×D412yes—
179C22×C4×C13C2×D411yescharacteristic
180C22×C4×C13C2×D411yescharacteristic
181C22×C4×C13C2×D411yescharacteristic
182C22×C4×C13C2×D412yes—
183C22×C4×C13C2×D412yes—
184C13×C4⋊C4—22no—
185C13×C4⋊C4—22no—
186C13×C4⋊C4—21no—
187C13×C4⋊C4—21no—
188C13×C4⋊C4—22no—
189C13×C4⋊C4—22no—
190C13×C4⋊C4—22no—
191C13×C4⋊C4—22no—
192C13×C4⋊C4—42no—
193C13×C4⋊C4—41no—
194C13×C4⋊C4—41no—
195C13×C4⋊C4—42no—
196C13×C4⋊C4—84no—
197C13×C22⋊C4—22no—
198C13×C22⋊C4—22no—
199C13×C22⋊C4—22no—
200C13×C22⋊C4—22no—
201C13×C22⋊C4—22no—
202C13×C22⋊C4—22no—
203C13×C22⋊C4—22no—
204C13×C22⋊C4—22no—
205C13×C22⋊C4—22no—
206C13×C22⋊C4—22no—
207C13×C22⋊C4—22no—
208C13×C22⋊C4—24no—
209C13×C22⋊C4—24no—
210C13×C22⋊C4—24no—
211C13×C22⋊C4—82no—
212C13×C22⋊C4—82no—
213C13×C22⋊C4—84no—
214C42×C13—22no—
215C42×C13—42no—
216#3485C2×C1311yescharacteristic
217#2623C2×C1312yescomplemented
218#2311C2×C1311yescharacteristic, complemented
219#2044C2×C1314yescomplemented
220#1899C2×C1312yescomplemented
221#1867C2×C1312yescomplemented
222#1790C2×C1312yescomplemented
223#1782C2×C1311yescharacteristic, complemented
224C23×C13—21no—
225C23×C13—21no—
226C23×C13—22no—
227C23×C13—21no—
228C23×C13—21no—
229C23×C13—21no—
230C23×C13—24no—
231C23×C13—41no—
232C23×C13—41no—
233C23×C13—41no—
234C23×C13—41no—
235C23×C13—41no—
236C23×C13—41no—
237C23×C13—42no—
238C23×C13—42no—
239C23×C13—42no—
240C23×C13—81no—
241C23×C13—81no—
242C23×C13#12312yes—
243C23×C13C22×D411yescharacteristic
244C23×C13C22×D412yes—
245C23×C13C2×#4112yes—
246C23×C13H(5,𝔽2)12yes—
247Q8×C13—44no—
248Q8×C13—42no—
249D4×C13—28no—
250D4×C13—44no—
251D4×C13—42no—
252D4×C13—44no—
253D4×C13—42no—
254D4×C13—42no—
255D4×C13—44no—
256D4×C13—44no—
257D4×C13—82no—
258D4×C13—82no—
259C2×C4×C13—21no—
260C2×C4×C13—21no—
261C2×C4×C13—21no—
262C2×C4×C13—21no—
263C2×C4×C13—21no—
264C2×C4×C13—21no—
265C2×C4×C13—24no—
266C2×C4×C13—24no—
267C2×C4×C13—22no—
268C2×C4×C13—22no—
269C2×C4×C13—21no—
270C2×C4×C13—22no—
271C2×C4×C13—21no—
272C2×C4×C13—21no—
273C2×C4×C13—24no—
274C2×C4×C13—42no—
275C2×C4×C13—42no—
276C2×C4×C13—42no—
277C2×C4×C13—42no—
278C2×C4×C13—44no—
279C2×C4×C13—42no—
280C2×C4×C13—42no—
281C2×C4×C13—42no—
282C2×C4×C13—42no—
283C2×C4×C13—44no—
284C2×C4×C13—44no—
285C2×C4×C13—42no—
286C2×C4×C13—42no—
287C2×C4×C13—44no—
288C2×C4×C13—44no—
289C2×C4×C13—42no—
290C2×C4×C13—42no—
291C2×C4×C13—42no—
292C2×C4×C13—44no—
293C2×C4×C13—81no—
294C2×C4×C13—81no—
295C2×C4×C13#12011yescharacteristic
296C2×C4×C13#12011yescharacteristic
297C2×C4×C13#12012yes—
298C2×C4×C13#12312yes—
299C2×H(5,𝔽2)C22×C1311yescharacteristic
300C22×#41C22×C1311yescharacteristic
301#547—22no—
302#547—22no—
303#546—22no—
304#546—22no—
305#536—21no—
306#534—21no—
307#534C22×C1312yes—
308#525—21no—
309C2×#123—21no—
310C2×#123C22×C1312yes—
311#518C22×C1311yescharacteristic
312C2×#117C22×C1311yescharacteristic, complemented
313C22×C4⋊C4C22×C1311yescharacteristic, complemented
314C2×#99C22×C1312yes—
315C2×#99C22×C1314yes—
316C2×#99C22×C1312yescomplemented
317#400—22no—
318#399—22no—
319#398—22no—
320#397—22no—
321#391—21no—
322#390—21no—
323#386C22×C1312yes—
324#386C22×C1314yes—
325#386C22×C1312yescomplemented
326#385C22×C1314yescomplemented
327#381C22×C1312yescomplemented
328#380—21no—
329#380—21no—
330#342C22×C1314yescomplemented
331C22×C13—22no—
332C22×C13—21no—
333C22×C13—22no—
334C22×C13—21no—
335C22×C13—22no—
336C22×C13—42no—
337C22×C13—42no—
338C22×C13—42no—
339C22×C13—42no—
340C22×C13—41no—
341C22×C13—41no—
342C22×C13—41no—
343C22×C13—42no—
344C22×C13—41no—
345C22×C13—42no—
346C22×C13—41no—
347C22×C13—42no—
348C22×C13—81no—
349C22×C13—81no—
350C22×C13—84no—
351C22×C13—162no—
352C22×C13C2×#12011yescharacteristic
353C22×C13C2×#12311yescharacteristic
354C22×C13#53411yescharacteristic
355C22×C13#54614yes—
356C4×C13—21no—
357C4×C13—21no—
358C4×C13—22no—
359C4×C13—22no—
360C4×C13—42no—
361C4×C13—41no—
362C4×C13—42no—
363C4×C13—42no—
364C4×C13—82no—
365C4×C13—84no—
366C4×C13—82no—
367H(5,𝔽2)—24no—
368H(5,𝔽2)—42no—
369C2×#41—24no—
370C2×#41—21no—
371C2×#41—22no—
372C2×#41—22no—
373C2×#41—22no—
374C2×#41—42no—
375C22×Q8D4×C1311yescharacteristic
376C22×D4—21no—
377C22×D4—22no—
378C22×D4D4×C1311yescharacteristic
379C22×D4C23×C1312yes—
380C22×D4C23×C1311yescharacteristic
381C22×D4C23×C1311yescharacteristic
382C23×C4D4×C1311yescharacteristic
383C23×C4D4×C1311yescharacteristic
384C23×C4C23×C1311yescharacteristic
385C23×C4C23×C1312yes—
386C4⋊Q8—22no—
387#124—22no—
388#123—22no—
389#123—22no—
390#123—22no—
391#123—22no—
392#123—22no—
393#123—22no—
394#123—22no—
395#123—22no—
396#123—22no—
397C22⋊Q8—21no—
398C22⋊Q8—21no—
399C22⋊Q8—22no—
400C22⋊Q8—22no—
401#121—21no—
402#121—21no—
403#121—22no—
404#121—22no—
405#121—22no—
406#121—22no—
407#120—22no—
408#120—22no—
409#120—22no—
410C4×D4—22no—
411C4×D4—22no—
412C4×D4—22no—
413C4×D4—22no—
414#117—21no—
415#117—42no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—22no—
421C2×C4⋊C4—22no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—24no—
424C2×C4⋊C4D4×C1312yes—
425C2×C4⋊C4D4×C1312yes—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—22no—
430C2×C22⋊C4—22no—
431C2×C22⋊C4—24no—
432C2×C22⋊C4D4×C1312yes—
433C2×C22⋊C4D4×C1312yes—
434C2×C22⋊C4C23×C1312yes—
435C2×C22⋊C4C23×C1314yes—
436C2×C22⋊C4C23×C1312yes—
437C2×C42—21no—
438#99—24no—
439#99—44no—
440#99—42no—
441#95—22no—
442#95—22no—
443#95—22no—
444#95—22no—
445#95—24no—
446#95—24no—
447C2×C13—21no—
448C2×C13—24no—
449C2×C13—21no—
450C2×C13—42no—
451C2×C13—81no—
452C2×C13—82no—
453C2×C13#240511yescharacteristic
454C2×C13#302711yescharacteristic
455C2×C13#302811yescharacteristic
456C24—21no—
457C24C2×D4×C1312yes—
458C24C2×D4×C1311yescharacteristic
459C24C2×D4×C1311yescharacteristic
460C24C24×C1311yescharacteristic
461#41—48no—
462#41—48no—
463#41—48no—
464#41—42no—
465#41—42no—
466C2×Q8—21no—
467C2×Q8—22no—
468C2×Q8—22no—
469C2×Q8—41no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—21no—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—24no—
480C2×D4—22no—
481C2×D4—24no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—41no—
486C2×D4—48no—
487C2×D4—42no—
488C2×D4—42no—
489C2×D4C13×#4114yes—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—21no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—22no—
502C22×C4—24no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—24no—
506C22×C4—41no—
507C22×C4—41no—
508C22×C4—82no—
509C22×C4C2×D4×C1312yes—
510C22×C4C2×D4×C1311yescharacteristic
511C22×C4C2×D4×C1312yes—
512C22×C4C2×D4×C1312yes—
513C22×C4C2×D4×C1311yescharacteristic
514C22×C4C2×D4×C1311yescharacteristic
515C22×C4C2×D4×C1311yescharacteristic
516C22×C4C2×D4×C1312yes—
517C22×C4C2×D4×C1312yes—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—42no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—42no—
530C4⋊C4—84no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—24no—
543C22⋊C4—24no—
544C22⋊C4—24no—
545C22⋊C4—82no—
546C22⋊C4—82no—
547C22⋊C4—84no—
548C42—22no—
549C42—42no—
550C13#2405911yescharacteristic, direct product factor, Sylow
551C23—21no—
552C23—21no—
553C23—22no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—24no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—81no—
568C23—81no—
569C23C13×#12312yes—
570C23C22×D4×C1311yescharacteristic
571C23C22×D4×C1312yes—
572C23C2×C13×#4112yes—
573C23C13×H(5,𝔽2)12yes—
574Q8—44no—
575Q8—42no—
576D4—28no—
577D4—44no—
578D4—42no—
579D4—44no—
580D4—42no—
581D4—42no—
582D4—44no—
583D4—44no—
584D4—82no—
585D4—82no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—24no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4C13×#12011yescharacteristic
623C2×C4C13×#12011yescharacteristic
624C2×C4C13×#12012yes—
625C2×C4C13×#12312yes—
626C22—22no—
627C22—21no—
628C22—22no—
629C22—21no—
630C22—22no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—42no—
639C22—41no—
640C22—42no—
641C22—41no—
642C22—42no—
643C22—81no—
644C22—81no—
645C22—84no—
646C22—162no—
647C22C2×C13×#12011yescharacteristic
648C22C2×C13×#12311yescharacteristic
649C22C13×#53411yescharacteristic
650C22C13×#54614yes—
651C4—21no—
652C4—21no—
653C4—22no—
654C4—22no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—41no—
659C4—82no—
660C4—84no—
661C4—82no—
662C2—21no—
663C2—24no—
664C2—21no—
665C2—42no—
666C2—81no—
667C2—82no—
668C2C13×#240511yescharacteristic
669C2C13×#302711yescharacteristic
670C2C13×#302811yescharacteristic
subgroupG/H
center, socleC22×C13C2×#120
commutator, Frattini, cosocleC24C24×C13
prime radicalC13×#3485C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C13 ◁ C23×C4×C13 ◁ G

Complements

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

NH ≅ G/Ntype
2. C13×#2623C2semidirect
3. C13×#2311C2semidirect
4. C13×#2044C2semidirect
5. C13×#1899C2semidirect
6. C13×#1867C2semidirect
7. C13×#1790C2semidirect
8. C13×#1782C2semidirect
22. C2×C13×#117C22semidirect
23. C22×C13×C4⋊C4C22semidirect
26. C2×C13×#99C22semidirect
35. C13×#386C22semidirect
36. C13×#385C22semidirect
37. C13×#381C22semidirect
40. C13×#342C22semidirect
121. #24059C13direct (N × H)
217. #2623C2×C13semidirect
218. #2311C2×C13semidirect
219. #2044C2×C13semidirect
220. #1899C2×C13semidirect
221. #1867C2×C13semidirect
222. #1790C2×C13semidirect
223. #1782C2×C13semidirect
312. C2×#117C22×C13semidirect
313. C22×C4⋊C4C22×C13semidirect
316. C2×#99C22×C13semidirect
325. #386C22×C13semidirect
326. #385C22×C13semidirect
327. #381C22×C13semidirect
330. #342C22×C13semidirect
550. C13#24059direct (N × H)

Maximal subgroups

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

subgroupindex
C13×#34852
C13×#26232
C13×#23112
C13×#20442
C13×#18992
C13×#18672
C13×#17902
C13×#17822
#2405913