Finite Group Databasebeta

← C13×#33568 #15719907

Subgroups — C13×#33568 #15719907

Order 3328.

1110 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C13×#3545C211yesmaximal, characteristic, complemented
2C13×#3020C212yesmaximal, complemented
3C2×C13×#454C211yesmaximal, characteristic, complemented
4C13×#2204C214yesmaximal, complemented
5C13×#2203C212yesmaximal, complemented
6C13×#2202C212yesmaximal, complemented
7C13×#2133C212yesmaximal, complemented
8C2×C13×#355C211yesmaximal, characteristic, complemented
9C2×C13×2−1+4C2211yescharacteristic, complemented
10C13×#575—22no—
11C2×C13×AGL(1,ℤ/8)—21no—
12C13×#563C2211yescharacteristic, complemented
13C13×#562—21no—
14C13×H(3,𝔽4)C2211yescharacteristic, complemented
15C13×#555C2212yescomplemented
16C13×#551—21no—
17C13×#551—22no—
18C13×#546—22no—
19C13×#542—22no—
20C13×#538—22no—
21C13×#536C2212yes—
22C13×#534—21no—
23C13×#529—21no—
24C2×C13×#124C2211yescharacteristic
25C13×#456—22no—
26C13×#456C2214yescomplemented
27C13×#454—21no—
28C13×#454—21no—
29C13×#454—22no—
30C13×#454C2212yes—
31C13×#454C2212yescomplemented
32C13×#454C2214yescomplemented
33C13×#421—22no—
34C2×C13×#104—21no—
35C2×C13×#101C2211yescharacteristic, complemented
36C2×C13×#101C2212yescomplemented
37C13×#356C2214yescomplemented
38C13×#355C2212yescomplemented
39C13×#355C2212yescomplemented
40C13×#355C2214yescomplemented
41C13×2−1+4—21no—
42C13×2−1+4—22no—
43C13×2−1+4—42no—
44C13×2−1+4C2312yes—
45C2×C13×#41—21no—
46C2×C13×#41—22no—
47C2×C13×#41—41no—
48C2×C13×#41D412yes—
49C22×Q8×C13—41no—
50C22×Q8×C13C2311yescharacteristic
51C22×D4×C13—21no—
52C22×D4×C13D411yescharacteristic
53C13×#137—24no—
54C13×#137—42no—
55C13×AGL(1,ℤ/8)—22no—
56C13×AGL(1,ℤ/8)—22no—
57C13×AGL(1,ℤ/8)—24no—
58C13×AGL(1,ℤ/8)—42no—
59C13×AGL(1,ℤ/8)—42no—
60C13×#135—44no—
61C2×C13×SD16—41no—
62C2×C13×D8—41no—
63C2×C13×M16—21no—
64C2×C13×M16—22no—
65C13×C4⋊Q8D412yes—
66C13×C4⋊Q8C2312yes—
67C13×#126—22no—
68C13×#126—22no—
69C13×#126—22no—
70C13×#125—21no—
71C13×#124—22no—
72C13×#124—22no—
73C13×#124—21no—
74C13×#124—22no—
75C13×#124D411yescharacteristic
76C13×#124D411yescharacteristic
77C13×#124D412yes—
78C13×#124C2311yescharacteristic
79C13×#124C2311yescharacteristic
80C13×#124C2312yes—
81C13×#123—21no—
82C13×#123—22no—
83C13×#123—22no—
84C13×#123—22no—
85C13×#123—21no—
86C13×#123—22no—
87C13×#123—22no—
88C13×C22⋊Q8—21no—
89C13×C22⋊Q8—22no—
90C13×C22⋊Q8—22no—
91C13×C22⋊Q8—22no—
92C13×C22⋊Q8—22no—
93C13×#121—21no—
94C13×#121—22no—
95C13×#121—22no—
96C13×#121—22no—
97C13×#121—22no—
98C13×#120—21no—
99C13×#120—22no—
100C13×#120—21no—
101C13×#120—22no—
102C4×Q8×C13—21no—
103C4×D4×C13—21no—
104C4×D4×C13—22no—
105C4×D4×C13—22no—
106C4×D4×C13—22no—
107C13×#117—22no—
108C13×#117—22no—
109C13×#117D412yes—
110C2×C13×C22⋊C4—21no—
111C2×C13×C22⋊C4—22no—
112C2×C13×C22⋊C4—22no—
113C2×C42×C13D411yescharacteristic
114C13×#104—22no—
115C13×#104—22no—
116C13×#104—24no—
117C13×#104—24no—
118C13×#101—21no—
119C13×#101—22no—
120C13×#101C2314yes—
121C13×#101C2312yes—
122#33568C1311yesmaximal, characteristic, direct product factor, Sylow
123C24×C13—21no—
124C24×C13—21no—
125C13×#41—24no—
126C13×#41—22no—
127C13×#41—22no—
128C13×#41—24no—
129C13×#41—42no—
130C13×#41—41no—
131C13×#41—42no—
132C13×#41—42no—
133C13×#41—42no—
134C13×#41—42no—
135C13×#41—44no—
136C13×#41—41no—
137C13×#41—42no—
138C2×Q8×C13—21no—
139C2×Q8×C13—21no—
140C2×Q8×C13—22no—
141C2×Q8×C13—41no—
142C2×Q8×C13—41no—
143C2×Q8×C13—42no—
144C2×Q8×C13—42no—
145C2×Q8×C13—44no—
146C2×Q8×C13C2×D412yes—
147C2×Q8×C13C2×D411yescharacteristic
148C2×Q8×C13C2×D412yes—
149C2×Q8×C13C2411yescharacteristic
150C2×D4×C13—21no—
151C2×D4×C13—22no—
152C2×D4×C13—22no—
153C2×D4×C13—22no—
154C2×D4×C13—22no—
155C2×D4×C13—21no—
156C2×D4×C13—22no—
157C2×D4×C13—21no—
158C2×D4×C13—41no—
159C2×D4×C13—41no—
160C2×D4×C13—41no—
161C2×D4×C13—81no—
162C2×D4×C13—81no—
163C2×D4×C13C2×D412yes—
164C2×D4×C13C2×D411yescharacteristic
165C2×D4×C13C2×D411yescharacteristic
166C22×C4×C13—22no—
167C22×C4×C13—22no—
168C22×C4×C13—21no—
169C22×C4×C13—21no—
170C22×C4×C13—22no—
171C22×C4×C13—41no—
172C22×C4×C13—42no—
173C22×C4×C13C2×D411yescharacteristic
174C22×C4×C13C2×D412yes—
175C13×Q16—44no—
176C13×SD16—44no—
177C13×SD16—42no—
178C13×SD16—42no—
179C13×SD16—44no—
180C13×D8—42no—
181C13×D8—42no—
182C13×D8—44no—
183C13×M16—22no—
184C13×M16—24no—
185C13×M16—41no—
186C13×M16—42no—
187C2×C8×C13—41no—
188C2×C8×C13—42no—
189C13×C4⋊C4—22no—
190C13×C4⋊C4—22no—
191C13×C4⋊C4—21no—
192C13×C4⋊C4—21no—
193C13×C4⋊C4—22no—
194C13×C4⋊C4—22no—
195C13×C4⋊C4—22no—
196C13×C4⋊C4—22no—
197C13×C4⋊C4—22no—
198C13×C22⋊C4—22no—
199C13×C22⋊C4—22no—
200C13×C22⋊C4—21no—
201C13×C22⋊C4—21no—
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—22no—
209C13×C22⋊C4—22no—
210C13×C22⋊C4—22no—
211C13×C22⋊C4—22no—
212C13×C22⋊C4—22no—
213C42×C13—22no—
214C42×C13—21no—
215C42×C13C2×D411yescharacteristic
216C42×C13C2×D411yescharacteristic
217C42×C13C2×D412yes—
218C42×C13C2×D412yes—
219#3545C2×C1311yescharacteristic, complemented
220#3020C2×C1312yescomplemented
221C2×#454C2×C1311yescharacteristic, complemented
222#2204C2×C1314yescomplemented
223#2203C2×C1312yescomplemented
224#2202C2×C1312yescomplemented
225#2133C2×C1312yescomplemented
226C2×#355C2×C1311yescharacteristic, complemented
227C23×C13—21no—
228C23×C13—21no—
229C23×C13—22no—
230C23×C13—41no—
231C23×C13—41no—
232C23×C13—41no—
233C23×C13—81no—
234C23×C13—82no—
235C23×C13—82no—
236C23×C13#12011yescharacteristic
237Q8×C13—22no—
238Q8×C13—24no—
239Q8×C13—41no—
240Q8×C13—42no—
241Q8×C13—41no—
242Q8×C13—42no—
243Q8×C13—41no—
244Q8×C13—42no—
245Q8×C13—81no—
246D4×C13—22no—
247D4×C13—24no—
248D4×C13—42no—
249D4×C13—41no—
250D4×C13—41no—
251D4×C13—41no—
252D4×C13—42no—
253D4×C13—42no—
254D4×C13—42no—
255D4×C13—42no—
256D4×C13—42no—
257D4×C13—81no—
258D4×C13—82no—
259D4×C13—82no—
260D4×C13—81no—
261D4×C13—82no—
262C2×C4×C13—21no—
263C2×C4×C13—22no—
264C2×C4×C13—22no—
265C2×C4×C13—21no—
266C2×C4×C13—21no—
267C2×C4×C13—22no—
268C2×C4×C13—21no—
269C2×C4×C13—22no—
270C2×C4×C13—22no—
271C2×C4×C13—42no—
272C2×C4×C13—44no—
273C2×C4×C13—44no—
274C2×C4×C13—42no—
275C2×C4×C13—41no—
276C2×C4×C13—82no—
277C2×C4×C13—82no—
278C2×C4×C13—81no—
279C2×C4×C13—82no—
280C2×C4×C13—81no—
281C2×C4×C13#12011yescharacteristic
282C2×C4×C13#12012yes—
283C2×C4×C13C22×D412yes—
284C2×C4×C13C22×D411yescharacteristic
285C8×C13—42no—
286C8×C13—44no—
287C2×2−1+4C22×C1311yescharacteristic, complemented
288#575—22no—
289C2×AGL(1,ℤ/8)—21no—
290#563C22×C1311yescharacteristic, complemented
291#562—21no—
292H(3,𝔽4)C22×C1311yescharacteristic, complemented
293#555C22×C1312yescomplemented
294#551—21no—
295#551—22no—
296#546—22no—
297#542—22no—
298#538—22no—
299#536C22×C1312yes—
300#534—21no—
301#529—21no—
302C2×#124C22×C1311yescharacteristic
303#456—22no—
304#456C22×C1314yescomplemented
305#454—21no—
306#454—21no—
307#454—22no—
308#454C22×C1312yes—
309#454C22×C1314yescomplemented
310#454C22×C1312yescomplemented
311#421—22no—
312C2×#104—21no—
313C2×#101C22×C1312yescomplemented
314C2×#101C22×C1311yescharacteristic, complemented
315#356C22×C1314yescomplemented
316#355C22×C1312yescomplemented
317#355C22×C1312yescomplemented
318#355C22×C1314yescomplemented
319C22×C13—41no—
320C22×C13—42no—
321C22×C13—42no—
322C22×C13—42no—
323C22×C13—42no—
324C22×C13—42no—
325C22×C13—42no—
326C22×C13—41no—
327C22×C13—41no—
328C22×C13—82no—
329C22×C13—161no—
330C22×C13—161no—
331C22×C13—162no—
332C22×C13B(4,𝔽2)11yescharacteristic
333C22×C13B(4,𝔽2)11yescharacteristic
334C22×C13C2×#12011yescharacteristic
335C4×C13—21no—
336C4×C13—22no—
337C4×C13—22no—
338C4×C13—21no—
339C4×C13—41no—
340C4×C13—42no—
341C4×C13—42no—
342C4×C13—41no—
343C4×C13—42no—
344C4×C13—42no—
3452−1+4—21no—
3462−1+4—22no—
3472−1+4—42no—
3482−1+4C23×C1312yes—
349C2×#41—21no—
350C2×#41—22no—
351C2×#41—41no—
352C2×#41D4×C1312yes—
353C22×Q8—41no—
354C22×Q8C23×C1311yescharacteristic
355C22×D4—21no—
356C22×D4D4×C1311yescharacteristic
357#137—24no—
358#137—42no—
359AGL(1,ℤ/8)—22no—
360AGL(1,ℤ/8)—22no—
361AGL(1,ℤ/8)—24no—
362AGL(1,ℤ/8)—42no—
363AGL(1,ℤ/8)—42no—
364#135—44no—
365C2×SD16—41no—
366C2×D8—41no—
367C2×M16—21no—
368C2×M16—22no—
369C4⋊Q8D4×C1312yes—
370C4⋊Q8C23×C1312yes—
371#126—22no—
372#126—22no—
373#126—22no—
374#125—21no—
375#124—22no—
376#124—22no—
377#124—21no—
378#124—22no—
379#124D4×C1311yescharacteristic
380#124D4×C1311yescharacteristic
381#124D4×C1312yes—
382#124C23×C1311yescharacteristic
383#124C23×C1311yescharacteristic
384#124C23×C1312yes—
385#123—21no—
386#123—22no—
387#123—22no—
388#123—22no—
389#123—21no—
390#123—22no—
391#123—22no—
392C22⋊Q8—21no—
393C22⋊Q8—22no—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8—22no—
397#121—21no—
398#121—22no—
399#121—22no—
400#121—22no—
401#121—22no—
402#120—21no—
403#120—22no—
404#120—21no—
405#120—22no—
406C4×Q8—21no—
407C4×D4—21no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—22no—
411#117—22no—
412#117—22no—
413#117D4×C1312yes—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—22no—
416C2×C22⋊C4—22no—
417C2×C42D4×C1311yescharacteristic
418#104—22no—
419#104—22no—
420#104—24no—
421#104—24no—
422#101—21no—
423#101—22no—
424#101C23×C1314yes—
425#101C23×C1312yes—
426C2×C13—21no—
427C2×C13—21no—
428C2×C13—21no—
429C2×C13—81no—
430C2×C13—82no—
431C2×C13—81no—
432C2×C13—82no—
433C2×C13C2×B(4,𝔽2)11yescharacteristic
434C24—21no—
435C24—21no—
436#41—24no—
437#41—22no—
438#41—22no—
439#41—24no—
440#41—42no—
441#41—41no—
442#41—42no—
443#41—42no—
444#41—42no—
445#41—42no—
446#41—44no—
447#41—41no—
448#41—42no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—22no—
452C2×Q8—41no—
453C2×Q8—41no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—44no—
457C2×Q8C2×D4×C1312yes—
458C2×Q8C2×D4×C1311yescharacteristic
459C2×Q8C2×D4×C1312yes—
460C2×Q8C24×C1311yescharacteristic
461C2×D4—21no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—21no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—41no—
470C2×D4—41no—
471C2×D4—41no—
472C2×D4—81no—
473C2×D4—81no—
474C2×D4C2×D4×C1312yes—
475C2×D4C2×D4×C1311yescharacteristic
476C2×D4C2×D4×C1311yescharacteristic
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—41no—
483C22×C4—42no—
484C22×C4C2×D4×C1311yescharacteristic
485C22×C4C2×D4×C1312yes—
486Q16—44no—
487SD16—44no—
488SD16—42no—
489SD16—42no—
490SD16—44no—
491D8—42no—
492D8—42no—
493D8—44no—
494M16—22no—
495M16—24no—
496M16—41no—
497M16—42no—
498C2×C8—41no—
499C2×C8—42no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—21no—
512C22⋊C4—21no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C42—22no—
525C42—21no—
526C42C2×D4×C1311yescharacteristic
527C42C2×D4×C1311yescharacteristic
528C42C2×D4×C1312yes—
529C42C2×D4×C1312yes—
530C13#3356811yescharacteristic, direct product factor, Sylow
531C23—21no—
532C23—21no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—81no—
538C23—82no—
539C23—82no—
540C23C13×#12011yescharacteristic
541Q8—22no—
542Q8—24no—
543Q8—41no—
544Q8—42no—
545Q8—41no—
546Q8—42no—
547Q8—42no—
548Q8—41no—
549Q8—81no—
550D4—22no—
551D4—24no—
552D4—41no—
553D4—41no—
554D4—42no—
555D4—42no—
556D4—41no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—81no—
562D4—82no—
563D4—82no—
564D4—82no—
565D4—81no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—82no—
581C2×C4—81no—
582C2×C4—82no—
583C2×C4—81no—
584C2×C4—82no—
585C2×C4C13×#12011yescharacteristic
586C2×C4C13×#12012yes—
587C2×C4C22×D4×C1312yes—
588C2×C4C22×D4×C1311yescharacteristic
589C8—42no—
590C8—44no—
591C22—41no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—41no—
599C22—41no—
600C22—82no—
601C22—161no—
602C22—161no—
603C22—162no—
604C22C13×B(4,𝔽2)11yescharacteristic
605C22C13×B(4,𝔽2)11yescharacteristic
606C22C2×C13×#12011yescharacteristic
607C4—21no—
608C4—22no—
609C4—22no—
610C4—21no—
611C4—41no—
612C4—42no—
613C4—42no—
614C4—41no—
615C4—42no—
616C4—42no—
617C2—21no—
618C2—21no—
619C2—21no—
620C2—81no—
621C2—81no—
622C2—82no—
623C2—82no—
624C2C2×C13×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C13C2×B(4,𝔽2)
commutator, Frattini, cosocleC2×Q8C24×C13
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C13 ◁ C23×C13 ◁ C22×Q8×C13 ◁ G

Complements

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

NH ≅ G/Ntype
1. C13×#3545C2semidirect
2. C13×#3020C2semidirect
3. C2×C13×#454C2semidirect
4. C13×#2204C2semidirect
5. C13×#2203C2semidirect
6. C13×#2202C2semidirect
7. C13×#2133C2semidirect
8. C2×C13×#355C2semidirect
9. C2×C13×2−1+4C22semidirect
12. C13×#563C22semidirect
14. C13×H(3,𝔽4)C22semidirect
15. C13×#555C22semidirect
26. C13×#456C22semidirect
31. C13×#454C22semidirect
32. C13×#454C22semidirect
35. C2×C13×#101C22semidirect
36. C2×C13×#101C22semidirect
37. C13×#356C22semidirect
38. C13×#355C22semidirect
39. C13×#355C22semidirect
40. C13×#355C22semidirect
122. #33568C13direct (N × H)
219. #3545C2×C13semidirect
220. #3020C2×C13semidirect
221. C2×#454C2×C13semidirect
222. #2204C2×C13semidirect
223. #2203C2×C13semidirect
224. #2202C2×C13semidirect
225. #2133C2×C13semidirect
226. C2×#355C2×C13semidirect
287. C2×2−1+4C22×C13semidirect
290. #563C22×C13semidirect
292. H(3,𝔽4)C22×C13semidirect
293. #555C22×C13semidirect
304. #456C22×C13semidirect
309. #454C22×C13semidirect
310. #454C22×C13semidirect
313. C2×#101C22×C13semidirect
314. C2×#101C22×C13semidirect
315. #356C22×C13semidirect
316. #355C22×C13semidirect
317. #355C22×C13semidirect
318. #355C22×C13semidirect
530. C13#33568direct (N × H)

Maximal subgroups

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

subgroupindex
C13×#35452
C13×#30202
C2×C13×#4542
C13×#22042
C13×#22032
C13×#22022
C13×#21332
C2×C13×#3552
#3356813