Finite Group Databasebeta

← C13×#18351 #15704690

Subgroups — C13×#18351 #15704690

Order 3328.

1226 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C13×#3496C211yesmaximal, characteristic
2C13×#3193C212yesmaximal, complemented
3C13×#2638C211yesmaximal, characteristic, complemented
4C13×#1610C212yesmaximal, complemented
5C13×#1607C212yesmaximal, complemented
6C13×#1604C212yesmaximal, complemented
7C13×#1601C212yesmaximal, complemented
8C13×#1510C212yesmaximal, complemented
9C13×#1465C211yesmaximal, characteristic, complemented
10C2×C13×H(5,𝔽2)—21no—
11C13×#547—22no—
12C13×#546—24no—
13D42×C13—22no—
14C13×#537C2212yes—
15C13×#534—21no—
16C13×#534C2212yes—
17C13×#532—21no—
18C2×C13×C22⋊Q8C2211yescharacteristic, complemented
19C2×C13×#121—21no—
20C2×C13×#121C2212yes—
21C2×C4×D4×C13C2211yescharacteristic
22C13×#484—22no—
23C13×#480C2214yescomplemented
24C13×#468—24no—
25C13×#450—22no—
26C13×#448C2214yescomplemented
27C13×#407C2212yescomplemented
28C13×#398C2211yescharacteristic, complemented
29C13×#397—21no—
30C13×#397C2211yescharacteristic, complemented
31C13×#395C2211yescharacteristic, complemented
32C13×#394—21no—
33C13×#393C2211yescharacteristic, complemented
34C13×#387—21no—
35C13×#385—21no—
36C13×#381C2211yescharacteristic, complemented
37C13×#327C2214yescomplemented
38C13×#327C2214yescomplemented
39C13×#324C2212yescomplemented
40C13×#323C2212yescomplemented
41C13×H(5,𝔽2)—44no—
42C13×H(5,𝔽2)—44no—
43C2×C13×#41—24no—
44C2×C13×#41—22no—
45C22×Q8×C13C2311yescharacteristic
46C22×D4×C13—24no—
47C22×D4×C13—22no—
48C22×D4×C13—22no—
49C22×D4×C13—21no—
50C22×D4×C13D411yescharacteristic
51C22×D4×C13D412yes—
52C23×C4×C13D411yescharacteristic
53C23×C4×C13C2311yescharacteristic
54C2×C13×SD16—24no—
55C2×C13×M16—22no—
56C13×C4⋊Q8—21no—
57C13×#127—21no—
58C13×#124—22no—
59C13×#123—24no—
60C13×#123—24no—
61C13×C22⋊Q8—24no—
62C13×C22⋊Q8—22no—
63C13×C22⋊Q8—22no—
64C13×#121—22no—
65C13×#121—22no—
66C13×#121—24no—
67C13×#121—22no—
68C13×#121—22no—
69C13×#121—22no—
70C13×#121—22no—
71C13×#121D414yes—
72C13×#121C2314yes—
73C13×#120—24no—
74C13×#120—24no—
75C4×D4×C13—22no—
76C4×D4×C13—22no—
77C4×D4×C13—22no—
78C4×D4×C13—22no—
79C2×C13×C4⋊C4—21no—
80C2×C13×C4⋊C4—21no—
81C2×C13×C4⋊C4—21no—
82C2×C13×C4⋊C4—21no—
83C2×C13×C4⋊C4D412yes—
84C2×C13×C4⋊C4C2311yescharacteristic
85C2×C13×C4⋊C4C2311yescharacteristic
86C2×C13×C22⋊C4—22no—
87C2×C13×C22⋊C4—22no—
88C2×C13×C22⋊C4—21no—
89C2×C13×C22⋊C4—21no—
90C2×C13×C22⋊C4—21no—
91C2×C13×C22⋊C4—21no—
92C2×C13×C22⋊C4D411yescharacteristic
93C2×C13×C22⋊C4C2311yescharacteristic
94C2×C42×C13D411yescharacteristic
95C13×#107—24no—
96C13×#103—24no—
97C13×#102—24no—
98C13×C22⋊C8—22no—
99C13×C22⋊C8C2314yes—
100C13×#95—21no—
101C13×#95—21no—
102C13×#95—21no—
103C13×#95—21no—
104C13×#95—21no—
105C13×#95C2311yescharacteristic
106C13×#95C2311yescharacteristic
107#18351C1311yesmaximal, characteristic, direct product factor, Sylow
108C24×C13—22no—
109C24×C13—22no—
110C24×C13C2×D411yescharacteristic
111C24×C13C2×D411yescharacteristic
112C13×#41—44no—
113C13×#41—416no—
114C13×#41—44no—
115C2×Q8×C13—22no—
116C2×Q8×C13—22no—
117C2×Q8×C13—81no—
118C2×D4×C13—22no—
119C2×D4×C13—22no—
120C2×D4×C13—24no—
121C2×D4×C13—22no—
122C2×D4×C13—24no—
123C2×D4×C13—24no—
124C2×D4×C13—22no—
125C2×D4×C13—22no—
126C2×D4×C13—22no—
127C2×D4×C13—22no—
128C2×D4×C13—416no—
129C2×D4×C13—44no—
130C2×D4×C13—44no—
131C2×D4×C13—44no—
132C2×D4×C13—44no—
133C2×D4×C13—82no—
134C2×D4×C13—82no—
135C2×D4×C13C2×D414yes—
136C22×C4×C13—22no—
137C22×C4×C13—24no—
138C22×C4×C13—22no—
139C22×C4×C13—21no—
140C22×C4×C13—21no—
141C22×C4×C13—21no—
142C22×C4×C13—21no—
143C22×C4×C13—21no—
144C22×C4×C13—21no—
145C22×C4×C13—81no—
146C22×C4×C13—81no—
147C22×C4×C13C2×D411yescharacteristic
148C22×C4×C13C2×D411yescharacteristic
149C22×C4×C13C2×D411yescharacteristic
150C22×C4×C13C2×D411yescharacteristic
151C22×C4×C13C2×D412yes—
152C22×C4×C13C2×D412yes—
153C22×C4×C13C2411yescharacteristic
154C13×SD16—84no—
155C13×M16—82no—
156C2×C8×C13—24no—
157C13×C4⋊C4—22no—
158C13×C4⋊C4—22no—
159C13×C4⋊C4—22no—
160C13×C4⋊C4—22no—
161C13×C4⋊C4—22no—
162C13×C4⋊C4—41no—
163C13×C4⋊C4—41no—
164C13×C4⋊C4—42no—
165C13×C4⋊C4—81no—
166C13×C4⋊C4C2×D414yes—
167C13×C22⋊C4—24no—
168C13×C22⋊C4—24no—
169C13×C22⋊C4—22no—
170C13×C22⋊C4—24no—
171C13×C22⋊C4—24no—
172C13×C22⋊C4—22no—
173C13×C22⋊C4—42no—
174C13×C22⋊C4—42no—
175C13×C22⋊C4—81no—
176C13×C22⋊C4—81no—
177C42×C13—21no—
178C42×C13—21no—
179#3496C2×C1311yescharacteristic
180#3193C2×C1312yescomplemented
181#2638C2×C1311yescharacteristic, complemented
182#1610C2×C1312yescomplemented
183#1607C2×C1312yescomplemented
184#1604C2×C1312yescomplemented
185#1601C2×C1312yescomplemented
186#1510C2×C1312yescomplemented
187#1465C2×C1311yescharacteristic, complemented
188C23×C13—21no—
189C23×C13—21no—
190C23×C13—21no—
191C23×C13—21no—
192C23×C13—24no—
193C23×C13—41no—
194C23×C13—41no—
195C23×C13—41no—
196C23×C13—41no—
197C23×C13—44no—
198C23×C13—44no—
199C23×C13—82no—
200C23×C13—82no—
201C23×C13—82no—
202C23×C13—82no—
203C23×C13#12012yes—
204C23×C13AGL(1,ℤ/8)12yes—
205C23×C13C22×D411yescharacteristic
206Q8×C13—44no—
207Q8×C13—44no—
208D4×C13—28no—
209D4×C13—44no—
210D4×C13—44no—
211D4×C13—44no—
212D4×C13—48no—
213D4×C13—48no—
214D4×C13—44no—
215D4×C13—44no—
216D4×C13—82no—
217D4×C13—82no—
218C2×C4×C13—21no—
219C2×C4×C13—21no—
220C2×C4×C13—21no—
221C2×C4×C13—21no—
222C2×C4×C13—24no—
223C2×C4×C13—42no—
224C2×C4×C13—41no—
225C2×C4×C13—42no—
226C2×C4×C13—41no—
227C2×C4×C13—42no—
228C2×C4×C13—48no—
229C2×C4×C13—44no—
230C2×C4×C13—44no—
231C2×C4×C13—41no—
232C2×C4×C13—41no—
233C2×C4×C13—41no—
234C2×C4×C13—41no—
235C2×C4×C13—41no—
236C2×C4×C13—41no—
237C2×C4×C13—41no—
238C2×C4×C13—41no—
239C2×C4×C13—41no—
240C2×C4×C13—41no—
241C2×C4×C13—41no—
242C2×C4×C13—41no—
243C2×C4×C13—41no—
244C2×C4×C13—41no—
245C2×C4×C13—81no—
246C2×C4×C13#12011yescharacteristic
247C2×C4×C13#12011yescharacteristic
248C2×C4×C13AGL(1,ℤ/8)12yes—
249C2×C4×C13C22×D412yes—
250C8×C13—44no—
251C2×H(5,𝔽2)—21no—
252#547—22no—
253#546—24no—
254D42—22no—
255#537C22×C1312yes—
256#534—21no—
257#534C22×C1312yes—
258#532—21no—
259C2×C22⋊Q8C22×C1311yescharacteristic, complemented
260C2×#121—21no—
261C2×#121C22×C1312yes—
262C2×C4×D4C22×C1311yescharacteristic
263#484—22no—
264#480C22×C1314yescomplemented
265#468—24no—
266#450—22no—
267#448C22×C1314yescomplemented
268#407C22×C1312yescomplemented
269#398C22×C1311yescharacteristic, complemented
270#397—21no—
271#397C22×C1311yescharacteristic, complemented
272#395C22×C1311yescharacteristic, complemented
273#394—21no—
274#393C22×C1311yescharacteristic, complemented
275#387—21no—
276#385—21no—
277#381C22×C1311yescharacteristic, complemented
278#327C22×C1314yescomplemented
279#327C22×C1314yescomplemented
280#324C22×C1312yescomplemented
281#323C22×C1312yescomplemented
282C22×C13—21no—
283C22×C13—21no—
284C22×C13—22no—
285C22×C13—22no—
286C22×C13—22no—
287C22×C13—22no—
288C22×C13—42no—
289C22×C13—41no—
290C22×C13—41no—
291C22×C13—44no—
292C22×C13—44no—
293C22×C13—44no—
294C22×C13—81no—
295C22×C13—81no—
296C22×C13—81no—
297C22×C13—164no—
298C22×C13—164no—
299C22×C13C2×#12011yescharacteristic
300C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
301C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
302C4×C13—22no—
303C4×C13—22no—
304C4×C13—21no—
305C4×C13—21no—
306C4×C13—44no—
307C4×C13—41no—
308C4×C13—41no—
309C4×C13—81no—
310C4×C13—81no—
311C4×C13—81no—
312C4×C13—81no—
313H(5,𝔽2)—44no—
314H(5,𝔽2)—44no—
315C2×#41—24no—
316C2×#41—22no—
317C22×Q8C23×C1311yescharacteristic
318C22×D4—24no—
319C22×D4—22no—
320C22×D4—22no—
321C22×D4—21no—
322C22×D4D4×C1311yescharacteristic
323C22×D4D4×C1312yes—
324C23×C4D4×C1311yescharacteristic
325C23×C4C23×C1311yescharacteristic
326C2×SD16—24no—
327C2×M16—22no—
328C4⋊Q8—21no—
329#127—21no—
330#124—22no—
331#123—24no—
332#123—24no—
333C22⋊Q8—24no—
334C22⋊Q8—22no—
335C22⋊Q8—22no—
336#121—22no—
337#121—22no—
338#121—24no—
339#121—22no—
340#121—22no—
341#121—22no—
342#121—22no—
343#121D4×C1314yes—
344#121C23×C1314yes—
345#120—24no—
346#120—24no—
347C4×D4—22no—
348C4×D4—22no—
349C4×D4—22no—
350C4×D4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4D4×C1312yes—
356C2×C4⋊C4C23×C1311yescharacteristic
357C2×C4⋊C4C23×C1311yescharacteristic
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4D4×C1311yescharacteristic
365C2×C22⋊C4C23×C1311yescharacteristic
366C2×C42D4×C1311yescharacteristic
367#107—24no—
368#103—24no—
369#102—24no—
370C22⋊C8—22no—
371C22⋊C8C23×C1314yes—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95C23×C1311yescharacteristic
378#95C23×C1311yescharacteristic
379C2×C13—21no—
380C2×C13—21no—
381C2×C13—42no—
382C2×C13—42no—
383C2×C13—84no—
384C2×C13#300311yescharacteristic
385C2×C13#302211yescharacteristic
386C2×C13#302611yescharacteristic
387C24—22no—
388C24—22no—
389C24C2×D4×C1311yescharacteristic
390C24C2×D4×C1311yescharacteristic
391#41—44no—
392#41—416no—
393#41—44no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—81no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—24no—
400C2×D4—22no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—416no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—82no—
413C2×D4—82no—
414C2×D4C2×D4×C1314yes—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—81no—
425C22×C4—81no—
426C22×C4C2×D4×C1311yescharacteristic
427C22×C4C2×D4×C1311yescharacteristic
428C22×C4C2×D4×C1311yescharacteristic
429C22×C4C2×D4×C1311yescharacteristic
430C22×C4C2×D4×C1312yes—
431C22×C4C2×D4×C1312yes—
432C22×C4C24×C1311yescharacteristic
433SD16—84no—
434M16—82no—
435C2×C8—24no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—42no—
444C4⋊C4—81no—
445C4⋊C4C2×D4×C1314yes—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—22no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—22no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—81no—
455C22⋊C4—81no—
456C42—21no—
457C42—21no—
458C13#1835111yescharacteristic, direct product factor, Sylow
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—24no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—44no—
469C23—44no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23C13×#12012yes—
475C23C13×AGL(1,ℤ/8)12yes—
476C23C22×D4×C1311yescharacteristic
477Q8—44no—
478Q8—44no—
479D4—28no—
480D4—44no—
481D4—44no—
482D4—44no—
483D4—48no—
484D4—48no—
485D4—44no—
486D4—44no—
487D4—82no—
488D4—82no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—48no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—81no—
517C2×C4C13×#12011yescharacteristic
518C2×C4C13×#12011yescharacteristic
519C2×C4C13×AGL(1,ℤ/8)12yes—
520C2×C4C22×D4×C1312yes—
521C8—44no—
522C22—21no—
523C22—21no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—42no—
529C22—41no—
530C22—41no—
531C22—44no—
532C22—44no—
533C22—44no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—164no—
538C22—164no—
539C22C2×C13×#12011yescharacteristic
540C22C2×C13×AGL(1,ℤ/8)11yescharacteristic
541C22C2×C13×AGL(1,ℤ/8)11yescharacteristic
542C4—22no—
543C4—21no—
544C4—21no—
545C4—22no—
546C4—41no—
547C4—41no—
548C4—44no—
549C4—81no—
550C4—81no—
551C4—81no—
552C4—81no—
553C2—21no—
554C2—21no—
555C2—42no—
556C2—42no—
557C2—84no—
558C2C13×#300311yescharacteristic
559C2C13×#302211yescharacteristic
560C2C13×#302611yescharacteristic
subgroupG/H
center, socleC22×C13C2×#120
commutator, Frattini, cosocleC22×C4C24×C13
prime radicalC13×#3496C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C13 ◁ C23×C4×C13 ◁ G

Complements

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

NH ≅ G/Ntype
2. C13×#3193C2semidirect
3. C13×#2638C2semidirect
4. C13×#1610C2semidirect
5. C13×#1607C2semidirect
6. C13×#1604C2semidirect
7. C13×#1601C2semidirect
8. C13×#1510C2semidirect
9. C13×#1465C2semidirect
18. C2×C13×C22⋊Q8C22semidirect
23. C13×#480C22semidirect
26. C13×#448C22semidirect
27. C13×#407C22semidirect
28. C13×#398C22semidirect
30. C13×#397C22semidirect
31. C13×#395C22semidirect
33. C13×#393C22semidirect
36. C13×#381C22semidirect
37. C13×#327C22semidirect
38. C13×#327C22semidirect
39. C13×#324C22semidirect
40. C13×#323C22semidirect
107. #18351C13direct (N × H)
180. #3193C2×C13semidirect
181. #2638C2×C13semidirect
182. #1610C2×C13semidirect
183. #1607C2×C13semidirect
184. #1604C2×C13semidirect
185. #1601C2×C13semidirect
186. #1510C2×C13semidirect
187. #1465C2×C13semidirect
259. C2×C22⋊Q8C22×C13semidirect
264. #480C22×C13semidirect
267. #448C22×C13semidirect
268. #407C22×C13semidirect
269. #398C22×C13semidirect
271. #397C22×C13semidirect
272. #395C22×C13semidirect
274. #393C22×C13semidirect
277. #381C22×C13semidirect
278. #327C22×C13semidirect
279. #327C22×C13semidirect
280. #324C22×C13semidirect
281. #323C22×C13semidirect
458. C13#18351direct (N × H)

Maximal subgroups

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

subgroupindex
C13×#34962
C13×#31932
C13×#26382
C13×#16102
C13×#16072
C13×#16042
C13×#16012
C13×#15102
C13×#14652
#1835113