Finite Group Databasebeta

← C13×#13361 #15699700

Subgroups — C13×#13361 #15699700

Order 3328.

686 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C13×#3200C211yesmaximal, characteristic, complemented
2C13×#3022C211yesmaximal, characteristic, complemented
3C13×#3021C211yesmaximal, characteristic
4C13×#2123C211yesmaximal, characteristic, complemented
5C13×#2122C211yesmaximal, characteristic, complemented
6C13×#1794C211yesmaximal, characteristic, complemented
7C13×#1392C211yesmaximal, characteristic, complemented
8C13×#585C2211yescharacteristic
9C2×C13×#137—21no—
10C2×C13×AGL(1,ℤ/8)—21no—
11C13×#536C2211yescharacteristic
12C13×#535C2211yescharacteristic
13C13×#485—21no—
14C13×#482—21no—
15C13×#468—21no—
16C13×#465—21no—
17C13×#456—22no—
18C13×#455—22no—
19C13×#454—22no—
20C13×#453—22no—
21C13×#450—21no—
22C13×#448—21no—
23C13×#421—21no—
24C13×#421—21no—
25C13×#421—21no—
26C13×#416—21no—
27C13×#413C2211yescharacteristic
28C13×#413C2211yescharacteristic
29C13×#410C2211yescharacteristic
30C13×#408C2211yescharacteristic
31C13×#352—22no—
32C13×AΣL(1,𝔽16)—22no—
33C13×2−1+4—41no—
34C13×2−1+4D412yes—
35C13×H(5,𝔽2)—21no—
36C13×H(5,𝔽2)—21no—
37C13×H(5,𝔽2)—21no—
38C13×H(5,𝔽2)—41no—
39C2×C13×#41—21no—
40C2×C13×#41—21no—
41C2×C13×#41—21no—
42C2×C13×#41—42no—
43C2×C13×#41C2311yescharacteristic
44C22×Q8×C13D411yescharacteristic
45C22×D4×C13D411yescharacteristic
46C13×#137—24no—
47C13×#137—42no—
48C13×AGL(1,ℤ/8)—24no—
49C13×AGL(1,ℤ/8)—42no—
50C2×C13×Q16—41no—
51C2×C13×SD16—21no—
52C2×C13×SD16—21no—
53C2×C13×SD16—41no—
54C2×C13×SD16—41no—
55C2×C13×D8—41no—
56C2×C13×M16—21no—
57C2×C13×M16—21no—
58C2×C13×M16D411yescharacteristic
59C22×C8×C13D411yescharacteristic
60C13×C4⋊Q8—21no—
61C13×#127—21no—
62C13×#124—21no—
63C13×#124—21no—
64C13×#123—21no—
65C13×#123—21no—
66C13×C22⋊Q8—21no—
67C13×C22⋊Q8—21no—
68C13×#121—21no—
69C13×#121—21no—
70C13×#120—21no—
71C13×#120—21no—
72C13×#117—21no—
73C13×#117D411yescharacteristic
74C13×#117D411yescharacteristic
75C13×#107—21no—
76C13×#106—21no—
77C13×#104—24no—
78C13×#104—24no—
79C13×#104—42no—
80C13×#103—21no—
81C13×#103—21no—
82C13×#103—41no—
83C13×#102—21no—
84C13×#102—21no—
85C13×#102—41no—
86C13×#101—21no—
87C13×#101—21no—
88C13×#100D412yes—
89C13×#100D412yes—
90C13×#99—21no—
91C13×#99—21no—
92C13×C22⋊C8—21no—
93C13×C22⋊C8—21no—
94#13361C1311yesmaximal, characteristic, direct product factor, Sylow
95C24×C13—21no—
96C13×#41—22no—
97C13×#41—22no—
98C13×#41—22no—
99C13×#41—22no—
100C13×#41—22no—
101C13×#41—22no—
102C13×#41—41no—
103C13×#41—41no—
104C13×#41—41no—
105C13×#41—41no—
106C13×#41—41no—
107C13×#41—42no—
108C13×#41—42no—
109C13×#41—42no—
110C13×#41—42no—
111C13×#41—41no—
112C2×Q8×C13—21no—
113C2×Q8×C13—21no—
114C2×Q8×C13—21no—
115C2×Q8×C13—21no—
116C2×Q8×C13—21no—
117C2×Q8×C13—41no—
118C2×Q8×C13—41no—
119C2×Q8×C13—42no—
120C2×Q8×C13C2×D411yescharacteristic
121C2×D4×C13—21no—
122C2×D4×C13—21no—
123C2×D4×C13—21no—
124C2×D4×C13—21no—
125C2×D4×C13—21no—
126C2×D4×C13—41no—
127C2×D4×C13—41no—
128C2×D4×C13—42no—
129C2×D4×C13—42no—
130C2×D4×C13—42no—
131C2×D4×C13—41no—
132C2×D4×C13—41no—
133C2×D4×C13—41no—
134C2×D4×C13—41no—
135C2×D4×C13—41no—
136C2×D4×C13C2×D411yescharacteristic
137C2×D4×C13C2×D411yescharacteristic
138C2×D4×C13#4111yescharacteristic
139C22×C4×C13—21no—
140C22×C4×C13—41no—
141C22×C4×C13—41no—
142C22×C4×C13—41no—
143C22×C4×C13C2×D411yescharacteristic
144C22×C4×C13C2×D411yescharacteristic
145C22×C4×C13C2×D411yescharacteristic
146C13×Q16—44no—
147C13×SD16—44no—
148C13×SD16—44no—
149C13×SD16—81no—
150C13×SD16—81no—
151C13×D8—44no—
152C13×M16—22no—
153C13×M16—22no—
154C13×M16—41no—
155C13×M16—41no—
156C13×M16—41no—
157C2×C8×C13—21no—
158C2×C8×C13—21no—
159C2×C8×C13—41no—
160C2×C8×C13—41no—
161C2×C8×C13—41no—
162C13×C4⋊C4—21no—
163C13×C4⋊C4—21no—
164C13×C4⋊C4—41no—
165C13×C4⋊C4—41no—
166C13×C4⋊C4—41no—
167C13×C22⋊C4—21no—
168C13×C22⋊C4—21no—
169C13×C22⋊C4—41no—
170C13×C22⋊C4—41no—
171C13×C22⋊C4—41no—
172C13×C22⋊C4—41no—
173C13×C22⋊C4—41no—
174C42×C13—21no—
175C42×C13—21no—
176C42×C13—41no—
177#3200C2×C1311yescharacteristic, complemented
178#3022C2×C1311yescharacteristic, complemented
179#3021C2×C1311yescharacteristic
180#2123C2×C1311yescharacteristic, complemented
181#2122C2×C1311yescharacteristic, complemented
182#1794C2×C1311yescharacteristic, complemented
183#1392C2×C1311yescharacteristic, complemented
184C23×C13—41no—
185C23×C13—41no—
186C23×C13—41no—
187C23×C13—41no—
188C23×C13—81no—
189C23×C13—161no—
190C23×C13#12111yescharacteristic
191C23×C13#12111yescharacteristic
192C23×C13#12711yescharacteristic
193Q8×C13—22no—
194Q8×C13—22no—
195Q8×C13—22no—
196Q8×C13—41no—
197Q8×C13—42no—
198Q8×C13—41no—
199Q8×C13—41no—
200Q8×C13—41no—
201D4×C13—22no—
202D4×C13—22no—
203D4×C13—22no—
204D4×C13—41no—
205D4×C13—41no—
206D4×C13—41no—
207D4×C13—42no—
208D4×C13—42no—
209D4×C13—42no—
210D4×C13—42no—
211D4×C13—42no—
212D4×C13—41no—
213D4×C13—41no—
214D4×C13—41no—
215D4×C13—161no—
216D4×C13—161no—
217C2×C4×C13—21no—
218C2×C4×C13—21no—
219C2×C4×C13—21no—
220C2×C4×C13—21no—
221C2×C4×C13—21no—
222C2×C4×C13—21no—
223C2×C4×C13—42no—
224C2×C4×C13—42no—
225C2×C4×C13—42no—
226C2×C4×C13—41no—
227C2×C4×C13—41no—
228C2×C4×C13—41no—
229C2×C4×C13—42no—
230C2×C4×C13—41no—
231C2×C4×C13—41no—
232C2×C4×C13—41no—
233C2×C4×C13—81no—
234C2×C4×C13—81no—
235C2×C4×C13#12011yescharacteristic
236C2×C4×C13#12011yescharacteristic
237C2×C4×C13#12011yescharacteristic
238C2×C4×C13#12111yescharacteristic
239C8×C13—42no—
240C8×C13—42no—
241C8×C13—41no—
242C8×C13—41no—
243#585C22×C1311yescharacteristic
244C2×#137—21no—
245C2×AGL(1,ℤ/8)—21no—
246#536C22×C1311yescharacteristic
247#535C22×C1311yescharacteristic
248#485—21no—
249#482—21no—
250#468—21no—
251#465—21no—
252#456—22no—
253#455—22no—
254#454—22no—
255#453—22no—
256#450—21no—
257#448—21no—
258#421—21no—
259#421—21no—
260#421—21no—
261#416—21no—
262#413C22×C1311yescharacteristic
263#413C22×C1311yescharacteristic
264#410C22×C1311yescharacteristic
265#408C22×C1311yescharacteristic
266#352—22no—
267AΣL(1,𝔽16)—22no—
268C22×C13—21no—
269C22×C13—21no—
270C22×C13—21no—
271C22×C13—41no—
272C22×C13—41no—
273C22×C13—41no—
274C22×C13—42no—
275C22×C13—81no—
276C22×C13—81no—
277C22×C13—81no—
278C22×C13—161no—
279C22×C13—161no—
280C22×C13—161no—
281C22×C13—162no—
282C22×C13#39211yescharacteristic
283C4×C13—21no—
284C4×C13—21no—
285C4×C13—21no—
286C4×C13—42no—
287C4×C13—81no—
288C4×C13—81no—
289C4×C13—81no—
290C4×C13—161no—
291C4×C13B(4,𝔽2)11yescharacteristic
292C4×C13B(4,𝔽2)11yescharacteristic
2932−1+4—41no—
2942−1+4D4×C1312yes—
295H(5,𝔽2)—21no—
296H(5,𝔽2)—21no—
297H(5,𝔽2)—21no—
298H(5,𝔽2)—41no—
299C2×#41—21no—
300C2×#41—21no—
301C2×#41—21no—
302C2×#41—42no—
303C2×#41C23×C1311yescharacteristic
304C22×Q8D4×C1311yescharacteristic
305C22×D4D4×C1311yescharacteristic
306#137—24no—
307#137—42no—
308AGL(1,ℤ/8)—24no—
309AGL(1,ℤ/8)—42no—
310C2×Q16—41no—
311C2×SD16—21no—
312C2×SD16—21no—
313C2×SD16—41no—
314C2×SD16—41no—
315C2×D8—41no—
316C2×M16—21no—
317C2×M16—21no—
318C2×M16D4×C1311yescharacteristic
319C22×C8D4×C1311yescharacteristic
320C4⋊Q8—21no—
321#127—21no—
322#124—21no—
323#124—21no—
324#123—21no—
325#123—21no—
326C22⋊Q8—21no—
327C22⋊Q8—21no—
328#121—21no—
329#121—21no—
330#120—21no—
331#120—21no—
332#117—21no—
333#117D4×C1311yescharacteristic
334#117D4×C1311yescharacteristic
335#107—21no—
336#106—21no—
337#104—24no—
338#104—24no—
339#104—42no—
340#103—21no—
341#103—21no—
342#103—41no—
343#102—21no—
344#102—21no—
345#102—41no—
346#101—21no—
347#101—21no—
348#100D4×C1312yes—
349#100D4×C1312yes—
350#99—21no—
351#99—21no—
352C22⋊C8—21no—
353C22⋊C8—21no—
354C2×C13—21no—
355C2×C13—41no—
356C2×C13—41no—
357C2×C13—41no—
358C2×C13—82no—
359C2×C13—161no—
360C2×C13#202111yescharacteristic
361C24—21no—
362#41—22no—
363#41—22no—
364#41—22no—
365#41—22no—
366#41—22no—
367#41—22no—
368#41—41no—
369#41—41no—
370#41—41no—
371#41—41no—
372#41—41no—
373#41—42no—
374#41—42no—
375#41—42no—
376#41—42no—
377#41—41no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×Q8—42no—
386C2×Q8C2×D4×C1311yescharacteristic
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4C2×D4×C1311yescharacteristic
403C2×D4C2×D4×C1311yescharacteristic
404C2×D4C13×#4111yescharacteristic
405C22×C4—21no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4C2×D4×C1311yescharacteristic
410C22×C4C2×D4×C1311yescharacteristic
411C22×C4C2×D4×C1311yescharacteristic
412Q16—44no—
413SD16—44no—
414SD16—44no—
415SD16—81no—
416SD16—81no—
417D8—44no—
418M16—22no—
419M16—22no—
420M16—41no—
421M16—41no—
422M16—41no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C42—21no—
441C42—21no—
442C42—41no—
443C13#1336111yescharacteristic, direct product factor, Sylow
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—81no—
449C23—161no—
450C23C13×#12111yescharacteristic
451C23C13×#12111yescharacteristic
452C23C13×#12711yescharacteristic
453Q8—22no—
454Q8—22no—
455Q8—22no—
456Q8—41no—
457Q8—42no—
458Q8—41no—
459Q8—41no—
460Q8—41no—
461D4—22no—
462D4—22no—
463D4—22no—
464D4—41no—
465D4—41no—
466D4—41no—
467D4—42no—
468D4—42no—
469D4—42no—
470D4—42no—
471D4—42no—
472D4—41no—
473D4—41no—
474D4—41no—
475D4—161no—
476D4—161no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4C13×#12011yescharacteristic
496C2×C4C13×#12011yescharacteristic
497C2×C4C13×#12011yescharacteristic
498C2×C4C13×#12111yescharacteristic
499C8—42no—
500C8—42no—
501C8—41no—
502C8—41no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—42no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—161no—
514C22—161no—
515C22—161no—
516C22—162no—
517C22C13×#39211yescharacteristic
518C4—21no—
519C4—21no—
520C4—21no—
521C4—42no—
522C4—81no—
523C4—81no—
524C4—81no—
525C4—161no—
526C4C13×B(4,𝔽2)11yescharacteristic
527C4C13×B(4,𝔽2)11yescharacteristic
528C2—21no—
529C2—41no—
530C2—41no—
531C2—41no—
532C2—82no—
533C2—161no—
534C2C13×#202111yescharacteristic
subgroupG/H
center, socleC2×C13#2021
commutator, Frattini, cosocleC2×#41C23×C13
prime radicalC13×#3021C2
LCSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C13 ◁ C2×C4×C13 ◁ C2×C13×#41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C13×#3200C2semidirect
2. C13×#3022C2semidirect
4. C13×#2123C2semidirect
5. C13×#2122C2semidirect
6. C13×#1794C2semidirect
7. C13×#1392C2semidirect
94. #13361C13direct (N × H)
177. #3200C2×C13semidirect
178. #3022C2×C13semidirect
180. #2123C2×C13semidirect
181. #2122C2×C13semidirect
182. #1794C2×C13semidirect
183. #1392C2×C13semidirect
443. C13#13361direct (N × H)

Maximal subgroups

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

subgroupindex
C13×#32002
C13×#30222
C13×#30212
C13×#21232
C13×#21222
C13×#17942
C13×#13922
#1336113