Finite Group Databasebeta

← C2×C13×#3509 #15749143

Subgroups — C2×C13×#3509 #15749143

Order 3328.

7022 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C13×#3509C2132yesmaximal, direct product factor
2C2×C13×#557C214yesmaximal, complemented
3C2×C13×#556C214yesmaximal, complemented
4C2×C13×#552C218yesmaximal, complemented
5C2×C13×#541C214yesmaximal, complemented
6C2×C13×#536C212yesmaximal
7C2×C13×#533C211yesmaximal, characteristic, complemented
8C2×C13×#533C214yesmaximal, complemented
9C22×C13×C4⋊Q8C211yesmaximal, characteristic, complemented
10C2×C13×#519C212yesmaximal, complemented
11C2×C4×C13×#41C211yesmaximal, characteristic
12C22×C13×#41C2211yescharacteristic
13C23×Q8×C13C2212yes—
14C13×#557C22164yescomplemented
15C13×#556C22164yescomplemented
16C13×#552C221128yescomplemented
17C13×#541C22164yescomplemented
18C13×#536C22132yes—
19C13×#533C22116yescomplemented
20C13×#533C22164yescomplemented
21C2×C13×C4⋊Q8C2212yes—
22C2×C13×C4⋊Q8C2212yes—
23C2×C13×C4⋊Q8C2214yes—
24C2×C13×C4⋊Q8C2212yes—
25C2×C13×C4⋊Q8C2214yes—
26C2×C13×C4⋊Q8C22116yescomplemented
27C2×C13×#126C22116yes—
28C2×C13×#125C2212yes—
29C2×C13×#125C2218yes—
30C2×C13×#125C22116yes—
31C2×C13×#124C2214yes—
32C2×C13×#123C2218yes—
33C2×C13×C22⋊Q8C2218yes—
34C2×C13×C22⋊Q8C2214yes—
35C2×C13×C22⋊Q8C2218yes—
36C13×#519C22132yescomplemented
37C4×C13×#41C22116yes—
38C2×C4×Q8×C13C2212yes—
39C2×C4×Q8×C13C2218yes—
40C2×C4×Q8×C13C2218yes—
41C2×C4×Q8×C13C2218yes—
42C2×C4×Q8×C13C2214yes—
43C2×C4×D4×C13C2214yes—
44C2×C4×D4×C13C2212yes—
45C2×C13×#117C2211yescharacteristic
46C2×C13×#117C2212yes—
47C2×C13×#117C2212yes—
48C2×C13×#117C2218yes—
49C2×C13×#117C2218yes—
50C22×C13×C4⋊C4C2214yes—
51C22×C42×C13C2211yescharacteristic
52C22×C42×C13C2212yes—
53C2×C13×#41—28no—
54C2×C13×#41C2318yes—
55C22×Q8×C13—216no—
56C22×Q8×C13C2314yes—
57C22×Q8×C13C2314yes—
58C22×Q8×C13C2314yes—
59C22×Q8×C13C2311yescharacteristic
60C22×Q8×C13C23116yes—
61C22×D4×C13C2312yes—
62C22×D4×C13C2311yescharacteristic
63C23×C4×C13C2312yes—
64C23×C4×C13C2314yes—
65C23×C4×C13C2312yes—
66C23×C4×C13C2314yes—
67C23×C4×C13C2311yescharacteristic
68C13×C4⋊Q8C23116yes—
69C13×C4⋊Q8C23116yes—
70C13×C4⋊Q8C23132yes—
71C13×C4⋊Q8C23116yes—
72C13×C4⋊Q8C23132yes—
73C13×#126C231128yes—
74C13×#125C23116yes—
75C13×#125C23164yes—
76C13×#125C231128yes—
77C13×#124C23132yes—
78C13×#123C23164yes—
79C13×C22⋊Q8C23164yes—
80C13×C22⋊Q8C23132yes—
81C13×C22⋊Q8C23164yes—
82C4×Q8×C13C23116yes—
83C4×Q8×C13C23164yes—
84C4×Q8×C13C23164yes—
85C4×Q8×C13C23164yes—
86C4×Q8×C13C23132yes—
87C4×D4×C13C23132yes—
88C4×D4×C13C23116yes—
89C13×#117C2318yes—
90C13×#117C23116yes—
91C13×#117C23116yes—
92C13×#117C23164yes—
93C13×#117C23164yes—
94C2×C13×C4⋊C4C2318yes—
95C2×C13×C4⋊C4C2318yes—
96C2×C13×C4⋊C4C23132yes—
97C2×C13×C4⋊C4C2314yes—
98C2×C13×C4⋊C4C2312yes—
99C2×C13×C4⋊C4C2314yes—
100C2×C13×C4⋊C4C2314yes—
101C2×C13×C4⋊C4C23116yes—
102C2×C13×C4⋊C4C23116yes—
103C2×C13×C4⋊C4C2318yes—
104C2×C13×C22⋊C4C2312yes—
105C2×C13×C22⋊C4C2314yes—
106C2×C13×C22⋊C4C2314yes—
107C2×C13×C22⋊C4C23116yes—
108C2×C42×C13C2311yescharacteristic
109C2×C42×C13C2312yes—
110C2×C42×C13C2311yescharacteristic
111C2×C42×C13C2314yes—
112C2×C42×C13C2312yes—
113C2×C42×C13C2318yes—
114C2×C42×C13C23116yes—
115C2×C42×C13C23116yes—
116C2×C42×C13C2314yes—
117C2×#3509C1311yesmaximal, characteristic, direct product factor, Sylow
118C24×C13C2412yes—
119C24×C13C2411yescharacteristic
120C13×#41—232no—
121C2×Q8×C13—216no—
122C2×Q8×C13—216no—
123C2×Q8×C13—264no—
124C2×Q8×C13—24no—
125C2×Q8×C13—216no—
126C2×Q8×C13C24116yes—
127C2×Q8×C13C24116yes—
128C2×Q8×C13C24116yes—
129C2×Q8×C13C2414yes—
130C2×D4×C13—28no—
131C2×D4×C13—24no—
132C2×D4×C13C2418yes—
133C2×D4×C13C2414yes—
134C22×C4×C13—216no—
135C22×C4×C13—28no—
136C22×C4×C13—28no—
137C22×C4×C13—216no—
138C22×C4×C13#4118yes—
139C22×C4×C13C2418yes—
140C22×C4×C13C24116yes—
141C22×C4×C13C2418yes—
142C22×C4×C13C24116yes—
143C22×C4×C13C2414yes—
144C22×C4×C13C2418yes—
145C22×C4×C13C2412yes—
146C22×C4×C13C2418yes—
147C22×C4×C13C2412yes—
148C22×C4×C13C2414yes—
149C22×C4×C13C2411yescharacteristic
150C22×C4×C13C2411yescharacteristic
151C22×C4×C13C2412yes—
152C13×C4⋊C4C24132yes—
153C13×C4⋊C4C24132yes—
154C13×C4⋊C4C24116yes—
155C13×C4⋊C4C2418yes—
156C13×C4⋊C4C24116yes—
157C13×C4⋊C4C24116yes—
158C13×C4⋊C4C24164yes—
159C13×C4⋊C4C24164yes—
160C13×C4⋊C4C24132yes—
161C13×C22⋊C4C2418yes—
162C13×C22⋊C4C24116yes—
163C13×C22⋊C4C24116yes—
164C13×C22⋊C4C24164yes—
165C42×C13C2414yes—
166C42×C13C2418yes—
167C42×C13C2414yes—
168C42×C13C24116yes—
169C42×C13C2418yes—
170C42×C13C24164yes—
171C42×C13C24116yes—
172#3509C2×C13132yesdirect product factor
173C2×#557C2×C1314yescomplemented
174C2×#556C2×C1314yescomplemented
175C2×#552C2×C1318yescomplemented
176C2×#541C2×C1314yescomplemented
177C2×#536C2×C1312yes—
178C2×#533C2×C1311yescharacteristic, complemented
179C2×#533C2×C1314yescomplemented
180C22×C4⋊Q8C2×C1311yescharacteristic, complemented
181C2×#519C2×C1312yescomplemented
182C2×C4×#41C2×C1311yescharacteristic
183C23×C13—24no—
184C23×C13—24no—
185C23×C13—28no—
186C23×C13C2×#4114yes—
187C23×C13C2514yes—
188C23×C13C2512yes—
189C23×C13C2511yescharacteristic
190Q8×C13—232no—
191Q8×C13—232no—
192Q8×C13—28no—
193Q8×C13—232no—
194D4×C13—216no—
195D4×C13—28no—
196C2×C4×C13—24no—
197C2×C4×C13—232no—
198C2×C4×C13—216no—
199C2×C4×C13—28no—
200C2×C4×C13—24no—
201C2×C4×C13—216no—
202C2×C4×C13—216no—
203C2×C4×C13—216no—
204C2×C4×C13—232no—
205C2×C4×C13C2×#4114yes—
206C2×C4×C13C2×#4114yes—
207C2×C4×C13C2×#41116yes—
208C2×C4×C132−1+418yes—
209C2×C4×C13C2518yes—
210C2×C4×C13C25116yes—
211C2×C4×C13C2514yes—
212C2×C4×C13C25116yes—
213C2×C4×C13C2514yes—
214C2×C4×C13C2512yes—
215C2×C4×C13C2512yes—
216C2×C4×C13C2514yes—
217C22×#41C22×C1311yescharacteristic
218C23×Q8C22×C1312yes—
219#557C22×C13164yescomplemented
220#556C22×C13164yescomplemented
221#552C22×C131128yescomplemented
222#541C22×C13164yescomplemented
223#536C22×C13132yes—
224#533C22×C13116yescomplemented
225#533C22×C13164yescomplemented
226C2×C4⋊Q8C22×C1312yes—
227C2×C4⋊Q8C22×C1312yes—
228C2×C4⋊Q8C22×C1314yes—
229C2×C4⋊Q8C22×C1312yes—
230C2×C4⋊Q8C22×C1314yes—
231C2×C4⋊Q8C22×C13116yescomplemented
232C2×#126C22×C13116yes—
233C2×#125C22×C1312yes—
234C2×#125C22×C1318yes—
235C2×#125C22×C13116yes—
236C2×#124C22×C1314yes—
237C2×#123C22×C1318yes—
238C2×C22⋊Q8C22×C1318yes—
239C2×C22⋊Q8C22×C1314yes—
240C2×C22⋊Q8C22×C1318yes—
241#519C22×C13132yescomplemented
242C4×#41C22×C13116yes—
243C2×C4×Q8C22×C1312yes—
244C2×C4×Q8C22×C1318yes—
245C2×C4×Q8C22×C1318yes—
246C2×C4×Q8C22×C1318yes—
247C2×C4×Q8C22×C1314yes—
248C2×C4×D4C22×C1314yes—
249C2×C4×D4C22×C1312yes—
250C2×#117C22×C1311yescharacteristic
251C2×#117C22×C1312yes—
252C2×#117C22×C1312yes—
253C2×#117C22×C1318yes—
254C2×#117C22×C1318yes—
255C22×C4⋊C4C22×C1314yes—
256C22×C42C22×C1311yescharacteristic
257C22×C42C22×C1312yes—
258C22×C13—24no—
259C22×C13—28no—
260C22×C13—24no—
261C22×C13—28no—
262C22×C13—48no—
263C22×C13C22×#4114yes—
264C22×C13C22×#4112yes—
265C22×C13C2×2−1+414yes—
266C22×C13C2611yescharacteristic
267C4×C13—24no—
268C4×C13—216no—
269C4×C13—28no—
270C4×C13—24no—
271C4×C13—216no—
272C4×C13C22×#4114yes—
273C4×C13C22×#4114yes—
274C4×C13C2×2−1+418yes—
275C2×#41—28no—
276C2×#41C23×C1318yes—
277C22×Q8—216no—
278C22×Q8C23×C1314yes—
279C22×Q8C23×C1314yes—
280C22×Q8C23×C1314yes—
281C22×Q8C23×C1311yescharacteristic
282C22×Q8C23×C13116yes—
283C22×D4C23×C1312yes—
284C22×D4C23×C1311yescharacteristic
285C23×C4C23×C1314yes—
286C23×C4C23×C1312yes—
287C23×C4C23×C1314yes—
288C23×C4C23×C1312yes—
289C23×C4C23×C1311yescharacteristic
290C4⋊Q8C23×C13116yes—
291C4⋊Q8C23×C13116yes—
292C4⋊Q8C23×C13132yes—
293C4⋊Q8C23×C13116yes—
294C4⋊Q8C23×C13132yes—
295#126C23×C131128yes—
296#125C23×C13116yes—
297#125C23×C13164yes—
298#125C23×C131128yes—
299#124C23×C13132yes—
300#123C23×C13164yes—
301C22⋊Q8C23×C13164yes—
302C22⋊Q8C23×C13132yes—
303C22⋊Q8C23×C13164yes—
304C4×Q8C23×C13116yes—
305C4×Q8C23×C13164yes—
306C4×Q8C23×C13164yes—
307C4×Q8C23×C13164yes—
308C4×Q8C23×C13132yes—
309C4×D4C23×C13132yes—
310C4×D4C23×C13116yes—
311#117C23×C1318yes—
312#117C23×C13116yes—
313#117C23×C13116yes—
314#117C23×C13164yes—
315#117C23×C13164yes—
316C2×C4⋊C4C23×C1318yes—
317C2×C4⋊C4C23×C1318yes—
318C2×C4⋊C4C23×C13132yes—
319C2×C4⋊C4C23×C1314yes—
320C2×C4⋊C4C23×C1312yes—
321C2×C4⋊C4C23×C1314yes—
322C2×C4⋊C4C23×C1314yes—
323C2×C4⋊C4C23×C13116yes—
324C2×C4⋊C4C23×C13116yes—
325C2×C4⋊C4C23×C1318yes—
326C2×C22⋊C4C23×C1312yes—
327C2×C22⋊C4C23×C1314yes—
328C2×C22⋊C4C23×C1314yes—
329C2×C22⋊C4C23×C13116yes—
330C2×C42C23×C1312yes—
331C2×C42C23×C13116yes—
332C2×C42C23×C1311yescharacteristic
333C2×C42C23×C1314yes—
334C2×C42C23×C1314yes—
335C2×C42C23×C1312yes—
336C2×C42C23×C1318yes—
337C2×C42C23×C13116yes—
338C2×C42C23×C1311yescharacteristic
339C2×C13—24no—
340C2×C13—44no—
341C2×C13#350914yesdirect product factor
342C2×C13C23×#4111yescharacteristic
343C2×C13C22×2−1+412yes—
344C24C24×C1312yes—
345C24C24×C1311yescharacteristic
346#41—232no—
347C2×Q8—216no—
348C2×Q8—216no—
349C2×Q8—264no—
350C2×Q8—24no—
351C2×Q8—216no—
352C2×Q8C24×C13116yes—
353C2×Q8C24×C13116yes—
354C2×Q8C24×C13116yes—
355C2×Q8C24×C1314yes—
356C2×D4—28no—
357C2×D4—24no—
358C2×D4C24×C1318yes—
359C2×D4C24×C1314yes—
360C22×C4—216no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—216no—
364C22×C4C13×#4118yes—
365C22×C4C24×C1314yes—
366C22×C4C24×C1318yes—
367C22×C4C24×C1312yes—
368C22×C4C24×C1318yes—
369C22×C4C24×C1312yes—
370C22×C4C24×C1318yes—
371C22×C4C24×C13116yes—
372C22×C4C24×C1318yes—
373C22×C4C24×C13116yes—
374C22×C4C24×C1311yescharacteristic
375C22×C4C24×C1311yescharacteristic
376C22×C4C24×C1314yes—
377C22×C4C24×C1312yes—
378C4⋊C4C24×C13132yes—
379C4⋊C4C24×C13132yes—
380C4⋊C4C24×C13116yes—
381C4⋊C4C24×C1318yes—
382C4⋊C4C24×C13116yes—
383C4⋊C4C24×C13116yes—
384C4⋊C4C24×C13164yes—
385C4⋊C4C24×C13164yes—
386C4⋊C4C24×C13132yes—
387C22⋊C4C24×C1318yes—
388C22⋊C4C24×C13116yes—
389C22⋊C4C24×C13116yes—
390C22⋊C4C24×C13164yes—
391C42C24×C1314yes—
392C42C24×C1318yes—
393C42C24×C1314yes—
394C42C24×C13116yes—
395C42C24×C1318yes—
396C42C24×C13164yes—
397C42C24×C13116yes—
398C13C2×#350911yescharacteristic, direct product factor, Sylow
399C23—24no—
400C23—24no—
401C23—28no—
402C23C2×C13×#4114yes—
403C23C25×C1314yes—
404C23C25×C1312yes—
405C23C25×C1311yescharacteristic
406Q8—232no—
407Q8—232no—
408Q8—28no—
409Q8—232no—
410D4—216no—
411D4—28no—
412C2×C4—24no—
413C2×C4—232no—
414C2×C4—216no—
415C2×C4—28no—
416C2×C4—24no—
417C2×C4—216no—
418C2×C4—216no—
419C2×C4—216no—
420C2×C4—232no—
421C2×C4C2×C13×#4114yes—
422C2×C4C2×C13×#4114yes—
423C2×C4C2×C13×#41116yes—
424C2×C4C13×2−1+418yes—
425C2×C4C25×C1318yes—
426C2×C4C25×C13116yes—
427C2×C4C25×C1314yes—
428C2×C4C25×C13116yes—
429C2×C4C25×C1314yes—
430C2×C4C25×C1312yes—
431C2×C4C25×C1312yes—
432C2×C4C25×C1314yes—
433C22—24no—
434C22—28no—
435C22—24no—
436C22—28no—
437C22—48no—
438C22C22×C13×#4114yes—
439C22C22×C13×#4112yes—
440C22C2×C13×2−1+414yes—
441C22C26×C1311yescharacteristic
442C4—24no—
443C4—216no—
444C4—28no—
445C4—24no—
446C4—216no—
447C4C22×C13×#4114yes—
448C4C22×C13×#4114yes—
449C4C2×C13×2−1+418yes—
450C2—24no—
451C2—44no—
452C2C13×#350914yesdirect product factor
453C2C23×C13×#4111yescharacteristic
454C2C22×C13×2−1+412yes—
subgroupG/H
center, socleC23×C13C25
commutator, Frattini, cosocleC22C26×C13
prime radicalC22×C13×#41C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C13 ◁ G

Complements

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

NH ≅ G/Ntype
1. C13×#3509C2direct (N × H)
2. C2×C13×#557C2semidirect
3. C2×C13×#556C2semidirect
4. C2×C13×#552C2semidirect
5. C2×C13×#541C2semidirect
7. C2×C13×#533C2semidirect
8. C2×C13×#533C2semidirect
9. C22×C13×C4⋊Q8C2semidirect
10. C2×C13×#519C2semidirect
14. C13×#557C22semidirect
15. C13×#556C22semidirect
16. C13×#552C22semidirect
17. C13×#541C22semidirect
19. C13×#533C22semidirect
20. C13×#533C22semidirect
26. C2×C13×C4⋊Q8C22semidirect
36. C13×#519C22semidirect
117. C2×#3509C13direct (N × H)
172. #3509C2×C13direct (N × H)
173. C2×#557C2×C13semidirect
174. C2×#556C2×C13semidirect
175. C2×#552C2×C13semidirect
176. C2×#541C2×C13semidirect
178. C2×#533C2×C13semidirect
179. C2×#533C2×C13semidirect
180. C22×C4⋊Q8C2×C13semidirect
181. C2×#519C2×C13semidirect
219. #557C22×C13semidirect
220. #556C22×C13semidirect
221. #552C22×C13semidirect
222. #541C22×C13semidirect
224. #533C22×C13semidirect
225. #533C22×C13semidirect
231. C2×C4⋊Q8C22×C13semidirect
241. #519C22×C13semidirect
341. C2×C13#3509direct (N × H)
398. C13C2×#3509direct (N × H)
452. C2C13×#3509direct (N × H)

Maximal subgroups

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

subgroupindex
C13×#35092
C2×C13×#5572
C2×C13×#5562
C2×C13×#5522
C2×C13×#5412
C2×C13×#5362
C2×C13×#5332
C22×C13×C4⋊Q82
C2×C13×#5192
C2×C4×C13×#412
C2×#350913