Finite Group Databasebeta

← C13×#9996 #15696335

Subgroups — C13×#9996 #15696335

Order 3328.

686 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C13×#3100C211yesmaximal, characteristic, complemented
2C13×#3036C211yesmaximal, characteristic, complemented
3C13×#2535C211yesmaximal, characteristic, complemented
4C13×#1517C211yesmaximal, characteristic, complemented
5C13×#1486C211yesmaximal, characteristic, complemented
6C13×#1478C211yesmaximal, characteristic, complemented
7C13×#1286C211yesmaximal, characteristic, complemented
8C2×C13×AGL(1,ℤ/8)—21no—
9C13×#532C2211yescharacteristic
10C2×C13×#121—21no—
11C2×C4×D4×C13C2211yescharacteristic
12C2×C4×D4×C13C2211yescharacteristic
13C13×#480—21no—
14C13×#474—21no—
15C13×#463—22no—
16C13×#461—21no—
17C13×#459—21no—
18C13×#459—21no—
19C13×#447—21no—
20C4×C13×D8—22no—
21C13×#431C2211yescharacteristic
22C13×#431C2211yescharacteristic
23C13×#423C2211yescharacteristic
24C13×#417—21no—
25C13×#397—21no—
26C13×#392—21no—
27C13×#386—22no—
28C13×#382C2211yescharacteristic
29C13×#380—21no—
30C13×#327—22no—
31C13×D4⋊C8—22no—
32C13×D4⋊C8—22no—
33C2×C13×#41—21no—
34C22×D4×C13—22no—
35C22×D4×C13D411yescharacteristic
36C22×D4×C13D411yescharacteristic
37C23×C4×C13—21no—
38C23×C4×C13—21no—
39C13×AGL(1,ℤ/8)—42no—
40C13×AGL(1,ℤ/8)—42no—
41C2×C13×SD16—22no—
42C2×C13×D8—22no—
43C2×C13×D8—22no—
44C2×C13×M16—21no—
45C13×C4⋊Q8D411yescharacteristic, complemented
46C13×#127D411yescharacteristic, complemented
47C13×#124—21no—
48C13×#121—21no—
49C13×#121—21no—
50C13×#121—42no—
51C13×#121—42no—
52C4×D4×C13—21no—
53C4×D4×C13—21no—
54C4×D4×C13—21no—
55C4×D4×C13—21no—
56C4×D4×C13—21no—
57C4×D4×C13—21no—
58C4×D4×C13—21no—
59C4×D4×C13—21no—
60C4×D4×C13—21no—
61C4×D4×C13—21no—
62C2×C13×C4⋊C4—21no—
63C2×C13×C4⋊C4D411yescharacteristic
64C2×C13×C4⋊C4D411yescharacteristic
65C2×C13×C22⋊C4—22no—
66C2×C13×C22⋊C4—21no—
67C2×C13×C22⋊C4—21no—
68C2×C13×C22⋊C4—22no—
69C2×C42×C13C2311yescharacteristic
70C13×#107—22no—
71C13×C4⋊C8—21no—
72C13×C4⋊C8—21no—
73C13×C4⋊C8—21no—
74C13×C4⋊C8D412yescomplemented
75C13×#103—22no—
76C13×#102—22no—
77C13×#102—22no—
78C13×#102—22no—
79C13×C22⋊C8—21no—
80C13×C22⋊C8—21no—
81C4×C8×C13—21no—
82C4×C8×C13D412yescomplemented
83C13×#95—22no—
84C13×#95—21no—
85C13×#95D412yes—
86#9996C1311yesmaximal, characteristic, direct product factor, Sylow
87C24×C13—21no—
88C24×C13—21no—
89C13×#41—41no—
90C13×#41—41no—
91C13×#41—41no—
92C13×#41—41no—
93C2×Q8×C13—21no—
94C2×D4×C13—21no—
95C2×D4×C13—21no—
96C2×D4×C13—21no—
97C2×D4×C13—21no—
98C2×D4×C13—21no—
99C2×D4×C13—21no—
100C2×D4×C13—21no—
101C2×D4×C13—21no—
102C2×D4×C13—21no—
103C2×D4×C13—21no—
104C2×D4×C13—42no—
105C2×D4×C13—44no—
106C2×D4×C13—44no—
107C2×D4×C13—41no—
108C2×D4×C13—82no—
109C2×D4×C13D812yescomplemented
110C22×C4×C13—22no—
111C22×C4×C13—21no—
112C22×C4×C13—21no—
113C22×C4×C13—41no—
114C22×C4×C13—41no—
115C22×C4×C13—41no—
116C22×C4×C13—41no—
117C22×C4×C13—41no—
118C22×C4×C13—41no—
119C22×C4×C13—41no—
120C22×C4×C13—41no—
121C22×C4×C13—41no—
122C22×C4×C13—82no—
123C22×C4×C13C2×D411yescharacteristic
124C22×C4×C13C2×D411yescharacteristic
125C22×C4×C13C2×D411yescharacteristic
126C13×SD16—44no—
127C13×D8—44no—
128C13×D8—44no—
129C13×M16—41no—
130C13×M16—41no—
131C2×C8×C13—22no—
132C2×C8×C13—22no—
133C2×C8×C13—41no—
134C13×C4⋊C4—21no—
135C13×C4⋊C4—21no—
136C13×C4⋊C4—21no—
137C13×C4⋊C4—21no—
138C13×C4⋊C4—21no—
139C13×C4⋊C4—21no—
140C13×C4⋊C4—41no—
141C13×C4⋊C4—41no—
142C13×C22⋊C4—44no—
143C13×C22⋊C4—41no—
144C13×C22⋊C4—41no—
145C13×C22⋊C4—41no—
146C13×C22⋊C4—41no—
147C13×C22⋊C4—41no—
148C13×C22⋊C4—41no—
149C13×C22⋊C4—82no—
150C42×C13C2×D411yescharacteristic
151C42×C13C2×D411yescharacteristic
152C42×C13C2×D411yescharacteristic
153C42×C13#4111yescharacteristic
154#3100C2×C1311yescharacteristic, complemented
155#3036C2×C1311yescharacteristic, complemented
156#2535C2×C1311yescharacteristic, complemented
157#1517C2×C1311yescharacteristic, complemented
158#1486C2×C1311yescharacteristic, complemented
159#1478C2×C1311yescharacteristic, complemented
160#1286C2×C1311yescharacteristic, complemented
161C23×C13—22no—
162C23×C13—22no—
163C23×C13—41no—
164C23×C13—42no—
165C23×C13—42no—
166C23×C13—41no—
167C23×C13—41no—
168C23×C13—41no—
169C23×C13—41no—
170C23×C13—41no—
171C23×C13—41no—
172C23×C13—41no—
173C23×C13—41no—
174C23×C13#12011yescharacteristic
175Q8×C13—41no—
176Q8×C13—41no—
177D4×C13—22no—
178D4×C13—22no—
179D4×C13—22no—
180D4×C13—22no—
181D4×C13—41no—
182D4×C13—41no—
183D4×C13—42no—
184D4×C13—41no—
185D4×C13—41no—
186D4×C13—41no—
187D4×C13—41no—
188D4×C13—81no—
189D4×C13—88no—
190D4×C13—81no—
191D4×C13—81no—
192C2×C4×C13—41no—
193C2×C4×C13—41no—
194C2×C4×C13—42no—
195C2×C4×C13—42no—
196C2×C4×C13—42no—
197C2×C4×C13—41no—
198C2×C4×C13—41no—
199C2×C4×C13—41no—
200C2×C4×C13—41no—
201C2×C4×C13—42no—
202C2×C4×C13—41no—
203C2×C4×C13—82no—
204C2×C4×C13—82no—
205C2×C4×C13—81no—
206C2×C4×C13—81no—
207C2×C4×C13—81no—
208C2×C4×C13—81no—
209C2×C4×C13—81no—
210C2×C4×C13—81no—
211C2×C4×C13#12011yescharacteristic
212C2×C4×C13#12011yescharacteristic
213C2×C4×C13#12111yescharacteristic
214C2×C4×C13#12111yescharacteristic
215C2×C4×C13#12111yescharacteristic
216C2×C4×C13#12711yescharacteristic
217C2×C4×C13C2×D811yescharacteristic
218C2×C4×C13#13511yescharacteristic
219C2×C4×C13AGL(1,ℤ/8)11yescharacteristic
220C2×C4×C13AGL(1,ℤ/8)11yescharacteristic
221C8×C13—42no—
222C8×C13—42no—
223C8×C13—81no—
224C2×AGL(1,ℤ/8)—21no—
225#532C22×C1311yescharacteristic
226C2×#121—21no—
227C2×C4×D4C22×C1311yescharacteristic
228C2×C4×D4C22×C1311yescharacteristic
229#480—21no—
230#474—21no—
231#463—22no—
232#461—21no—
233#459—21no—
234#459—21no—
235#447—21no—
236C4×D8—22no—
237#431C22×C1311yescharacteristic
238#431C22×C1311yescharacteristic
239#423C22×C1311yescharacteristic
240#417—21no—
241#397—21no—
242#392—21no—
243#386—22no—
244#382C22×C1311yescharacteristic
245#380—21no—
246#327—22no—
247D4⋊C8—22no—
248D4⋊C8—22no—
249C22×C13—21no—
250C22×C13—21no—
251C22×C13—42no—
252C22×C13—44no—
253C22×C13—42no—
254C22×C13—42no—
255C22×C13—42no—
256C22×C13—81no—
257C22×C13—82no—
258C22×C13—81no—
259C22×C13—81no—
260C22×C13—81no—
261C22×C13—81no—
262C22×C13—81no—
263C22×C13—81no—
264C22×C13—81no—
265C22×C13—81no—
266C22×C13#39211yescharacteristic
267C22×C13#44711yescharacteristic
268C22×C13#44911yescharacteristic
269C4×C13—21no—
270C4×C13—21no—
271C4×C13—21no—
272C4×C13—21no—
273C4×C13—82no—
274C4×C13—81no—
275C4×C13—81no—
276C4×C13—81no—
277C4×C13#44711yescharacteristic
278C4×C13#44911yescharacteristic
279C4×C13#46611yescharacteristic
280C4×C13#46911yescharacteristic
281C2×#41—21no—
282C22×D4—22no—
283C22×D4D4×C1311yescharacteristic
284C22×D4D4×C1311yescharacteristic
285C23×C4—21no—
286C23×C4—21no—
287AGL(1,ℤ/8)—42no—
288AGL(1,ℤ/8)—42no—
289C2×SD16—22no—
290C2×D8—22no—
291C2×D8—22no—
292C2×M16—21no—
293C4⋊Q8D4×C1311yescharacteristic, complemented
294#127D4×C1311yescharacteristic, complemented
295#124—21no—
296#121—21no—
297#121—21no—
298#121—42no—
299#121—42no—
300C4×D4—21no—
301C4×D4—21no—
302C4×D4—21no—
303C4×D4—21no—
304C4×D4—21no—
305C4×D4—21no—
306C4×D4—21no—
307C4×D4—21no—
308C4×D4—21no—
309C4×D4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4D4×C1311yescharacteristic
312C2×C4⋊C4D4×C1311yescharacteristic
313C2×C22⋊C4—22no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—22no—
317C2×C42C23×C1311yescharacteristic
318#107—22no—
319C4⋊C8—21no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8D4×C1312yescomplemented
323#103—22no—
324#102—22no—
325#102—22no—
326#102—22no—
327C22⋊C8—21no—
328C22⋊C8—21no—
329C4×C8—21no—
330C4×C8D4×C1312yescomplemented
331#95—22no—
332#95—21no—
333#95D4×C1312yes—
334C2×C13—21no—
335C2×C13—21no—
336C2×C13—82no—
337C2×C13—82no—
338C2×C13—161no—
339C2×C13#199911yescharacteristic
340C2×C13#200811yescharacteristic
341C2×C13#201011yescharacteristic
342C24—21no—
343C24—21no—
344#41—41no—
345#41—41no—
346#41—41no—
347#41—41no—
348C2×Q8—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—42no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—41no—
363C2×D4—82no—
364C2×D4C13×D812yescomplemented
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—82no—
378C22×C4C2×D4×C1311yescharacteristic
379C22×C4C2×D4×C1311yescharacteristic
380C22×C4C2×D4×C1311yescharacteristic
381SD16—44no—
382D8—44no—
383D8—44no—
384M16—41no—
385M16—41no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—41no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C22⋊C4—44no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—82no—
405C42C2×D4×C1311yescharacteristic
406C42C2×D4×C1311yescharacteristic
407C42C2×D4×C1311yescharacteristic
408C42C13×#4111yescharacteristic
409C13#999611yescharacteristic, direct product factor, Sylow
410C23—22no—
411C23—22no—
412C23—41no—
413C23—42no—
414C23—42no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23C13×#12011yescharacteristic
424Q8—41no—
425Q8—41no—
426D4—22no—
427D4—22no—
428D4—22no—
429D4—22no—
430D4—41no—
431D4—41no—
432D4—42no—
433D4—41no—
434D4—41no—
435D4—41no—
436D4—41no—
437D4—81no—
438D4—88no—
439D4—81no—
440D4—81no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4C13×#12011yescharacteristic
461C2×C4C13×#12011yescharacteristic
462C2×C4C13×#12111yescharacteristic
463C2×C4C13×#12111yescharacteristic
464C2×C4C13×#12111yescharacteristic
465C2×C4C13×#12711yescharacteristic
466C2×C4C2×C13×D811yescharacteristic
467C2×C4C13×#13511yescharacteristic
468C2×C4C13×AGL(1,ℤ/8)11yescharacteristic
469C2×C4C13×AGL(1,ℤ/8)11yescharacteristic
470C8—42no—
471C8—42no—
472C8—81no—
473C22—21no—
474C22—21no—
475C22—42no—
476C22—44no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—81no—
481C22—82no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22C13×#39211yescharacteristic
491C22C13×#44711yescharacteristic
492C22C13×#44911yescharacteristic
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—82no—
498C4—81no—
499C4—81no—
500C4—81no—
501C4C13×#44711yescharacteristic
502C4C13×#44911yescharacteristic
503C4C13×#46611yescharacteristic
504C4C13×#46911yescharacteristic
505C2—21no—
506C2—21no—
507C2—82no—
508C2—82no—
509C2—161no—
510C2C13×#199911yescharacteristic
511C2C13×#200811yescharacteristic
512C2C13×#201011yescharacteristic
subgroupG/H
center, socleC22×C13#392
commutator, Frattini, cosocleC2×C42C23×C13
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C13 ◁ C2×C42×C13 ◁ G

Complements

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

NH ≅ G/Ntype
1. C13×#3100C2semidirect
2. C13×#3036C2semidirect
3. C13×#2535C2semidirect
4. C13×#1517C2semidirect
5. C13×#1486C2semidirect
6. C13×#1478C2semidirect
7. C13×#1286C2semidirect
45. C13×C4⋊Q8D4semidirect
46. C13×#127D4semidirect
74. C13×C4⋊C8D4semidirect
82. C4×C8×C13D4semidirect
86. #9996C13direct (N × H)
109. C2×D4×C13D8semidirect
154. #3100C2×C13semidirect
155. #3036C2×C13semidirect
156. #2535C2×C13semidirect
157. #1517C2×C13semidirect
158. #1486C2×C13semidirect
159. #1478C2×C13semidirect
160. #1286C2×C13semidirect
293. C4⋊Q8D4×C13semidirect
294. #127D4×C13semidirect
322. C4⋊C8D4×C13semidirect
330. C4×C8D4×C13semidirect
364. C2×D4C13×D8semidirect
409. C13#9996direct (N × H)

Maximal subgroups

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

subgroupindex
C13×#31002
C13×#30362
C13×#25352
C13×#15172
C13×#14862
C13×#14782
C13×#12862
#999613