Finite Group Databasebeta

← C13×#61376 #15747715

Subgroups — C13×#61376 #15747715

Order 3328.

1706 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C13×#3504C212yesmaximal, complemented
2C13×#3388C218yesmaximal, complemented
3C13×#3383C214yesmaximal, complemented
4C13×#3380C214yesmaximal, complemented
5C13×#3164C212yesmaximal, complemented
6C2×C13×#499C211yesmaximal, characteristic, complemented
7C13×#3084C214yesmaximal, complemented
8C13×#3080C212yesmaximal, complemented
9C13×#2960C211yesmaximal, characteristic, complemented
10C13×#2951C212yesmaximal
11C4×C13×#135C211yesmaximal, characteristic
12C2×C13×#135C2211yescharacteristic
13C13×#557C2218yes—
14C13×#554—28no—
15C13×#554C2218yes—
16C13×#544—24no—
17C13×#544C2214yes—
18C13×#533—22no—
19C13×#533C2211yescharacteristic
20C2×C13×#125—22no—
21C2×C13×#125C2212yes—
22C13×#520C2214yes—
23C4×C13×#41C2212yes—
24C13×#501C2218yes—
25C13×#500C2214yes—
26C13×#499C2214yes—
27C13×#499C2214yes—
28C13×#499C2214yes—
29C13×#479C2218yes—
30C13×#479C2218yes—
31C13×#478C2218yes—
32C13×#478C2218yes—
33C13×#476C2218yes—
34C13×#475C2218yes—
35C13×#446C2214yes—
36C8×Q8×C13C2214yes—
37C13×#440C2218yes—
38C4×C13×Q16C2212yes—
39C4×C13×Q16C2214yes—
40C4×C13×SD16C2214yes—
41C4×C13×D8C2212yes—
42C4×C13×D8C2214yes—
43C13×#428C2214yes—
44C13×#427C2211yescharacteristic
45C2×C13×#107C2212yes—
46C2×C13×#106C2212yes—
47C13×#424C2212yes—
48C13×#423C2211yescharacteristic
49C13×#417C2214yes—
50C13×#416C2212yes—
51C13×#405C2212yes—
52C2×C4×C8×C13C2211yescharacteristic
53C2×C13×#41C2312yes—
54C13×#135Q818yescomplemented
55C2×C13×Q16C2311yescharacteristic
56C2×C13×SD16C2312yes—
57C2×C13×D8C2311yescharacteristic
58C2×C13×M16C2312yes—
59C22×C8×C13C2311yescharacteristic
60C22×C8×C13C2311yescharacteristic
61C13×C4⋊Q8—28no—
62C13×C4⋊Q8—24no—
63C13×C4⋊Q8D412yes—
64C13×C4⋊Q8C2318yes—
65C13×#125—24no—
66C13×#125—28no—
67C13×#125—28no—
68C13×#125—22no—
69C13×#125—22no—
70C13×#125D411yescharacteristic
71C13×#125D411yescharacteristic
72C13×#125C2318yes—
73C13×#125C2318yes—
74C13×C22⋊Q8—28no—
75C13×C22⋊Q8—28no—
76C13×C22⋊Q8—28no—
77C13×C22⋊Q8—28no—
78C13×C22⋊Q8D414yes—
79C4×Q8×C13—28no—
80C4×Q8×C13C2314yes—
81C4×Q8×C13C2318yes—
82C4×Q8×C13C2314yes—
83C4×D4×C13—24no—
84C4×D4×C13—28no—
85C4×D4×C13C2314yes—
86C4×D4×C13C2318yes—
87C13×#117—24no—
88C13×#117—22no—
89C13×#117C2312yes—
90C13×#117C2312yes—
91C2×C13×C4⋊C4—24no—
92C2×C13×C4⋊C4—24no—
93C2×C13×C4⋊C4—28no—
94C2×C13×C4⋊C4C2314yes—
95C2×C42×C13—22no—
96C2×C42×C13C2311yescharacteristic
97C13×#107C2314yes—
98C13×#107C2314yes—
99C13×#107C2312yes—
100C13×#107C2318yes—
101C13×#106C2318yes—
102C13×#106C2312yes—
103C13×#106C2318yes—
104C13×C4⋊C8C2314yes—
105C13×C4⋊C8C2314yes—
106C13×C4⋊C8C2314yes—
107C13×#103C2314yes—
108C13×#103C2318yes—
109C13×#102C2314yes—
110C13×#102C2318yes—
111C13×#97C2314yes—
112C4×C8×C13C2312yes—
113C4×C8×C13C2312yes—
114C4×C8×C13C2314yes—
115#61376C1311yesmaximal, characteristic, direct product factor, Sylow
116C13×#41—24no—
117C13×#41C2×Q818yes—
118C2×Q8×C13—28no—
119C2×Q8×C13C2×D414yes—
120C2×Q8×C13C2412yes—
121C2×D4×C13—22no—
122C2×D4×C13C2412yes—
123C22×C4×C13—22no—
124C22×C4×C13—24no—
125C22×C4×C13—24no—
126C22×C4×C13—28no—
127C22×C4×C13C2×D412yes—
128C22×C4×C13C2411yescharacteristic
129C13×Q16C2×Q814yes—
130C13×SD16C2×Q818yes—
131C13×D8C2×Q814yes—
132C13×M16—24no—
133C2×C8×C13—22no—
134C2×C8×C13C2×Q814yes—
135C2×C8×C13C2412yes—
136C2×C8×C13C2414yes—
137C2×C8×C13C2411yescharacteristic
138C2×C8×C13C2411yescharacteristic
139C13×C4⋊C4—28no—
140C13×C4⋊C4—24no—
141C13×C4⋊C4—28no—
142C13×C4⋊C4—28no—
143C13×C4⋊C4—28no—
144C13×C4⋊C4—28no—
145C13×C4⋊C4—24no—
146C13×C4⋊C4—28no—
147C13×C4⋊C4C2×D412yes—
148C13×C4⋊C4C2×D414yes—
149C13×C4⋊C4C2×D412yes—
150C13×C4⋊C4C2×D414yes—
151C13×C4⋊C4C2×D414yes—
152C13×C4⋊C4C2418yes—
153C13×C4⋊C4C2414yes—
154C13×C22⋊C4—22no—
155C13×C22⋊C4—24no—
156C13×C22⋊C4—28no—
157C13×C22⋊C4C2×D414yes—
158C42×C13—22no—
159C42×C13—24no—
160C42×C13—28no—
161C42×C13C2×D411yescharacteristic
162C42×C13C2×D411yescharacteristic
163C42×C13C2412yes—
164C42×C13C2414yes—
165#3504C2×C1312yescomplemented
166#3388C2×C1318yescomplemented
167#3383C2×C1314yescomplemented
168#3380C2×C1314yescomplemented
169#3164C2×C1312yescomplemented
170C2×#499C2×C1311yescharacteristic, complemented
171#3084C2×C1314yescomplemented
172#3080C2×C1312yescomplemented
173#2960C2×C1311yescharacteristic, complemented
174#2951C2×C1312yes—
175C4×#135C2×C1311yescharacteristic
176C23×C13—22no—
177C23×C13C22×D411yescharacteristic
178Q8×C13—22no—
179Q8×C13—44no—
180Q8×C13—88no—
181Q8×C13C22×Q814yes—
182D4×C13—28no—
183D4×C13—22no—
184D4×C13C22×Q814yes—
185C2×C4×C13—22no—
186C2×C4×C13—28no—
187C2×C4×C13—24no—
188C2×C4×C13—28no—
189C2×C4×C13—42no—
190C2×C4×C13—84no—
191C2×C4×C13—88no—
192C2×C4×C13—84no—
193C2×C4×C13C22×D412yes—
194C2×C4×C13C22×D414yes—
195C2×C4×C13C22×D412yes—
196C2×C4×C13C22×D414yes—
197C2×C4×C13C22×D411yescharacteristic
198C2×C4×C13C22×Q812yes—
199C2×C4×C132−1+412yes—
200C2×C4×C13C2511yescharacteristic
201C8×C13—22no—
202C8×C13—24no—
203C8×C13C22×Q812yes—
204C8×C13C22×Q812yes—
205C2×#135C22×C1311yescharacteristic
206#557C22×C1318yes—
207#554—28no—
208#554C22×C1318yes—
209#544—24no—
210#544C22×C1314yes—
211#533—22no—
212#533C22×C1311yescharacteristic
213C2×#125—22no—
214C2×#125C22×C1312yes—
215#520C22×C1314yes—
216C4×#41C22×C1312yes—
217#501C22×C1318yes—
218#500C22×C1314yes—
219#499C22×C1314yes—
220#499C22×C1314yes—
221#499C22×C1314yes—
222#479C22×C1318yes—
223#479C22×C1318yes—
224#478C22×C1318yes—
225#478C22×C1318yes—
226#476C22×C1318yes—
227#475C22×C1318yes—
228#446C22×C1314yes—
229C8×Q8C22×C1314yes—
230#440C22×C1318yes—
231C4×Q16C22×C1312yes—
232C4×Q16C22×C1314yes—
233C4×SD16C22×C1314yes—
234C4×D8C22×C1312yes—
235C4×D8C22×C1314yes—
236#428C22×C1314yes—
237#427C22×C1311yescharacteristic
238C2×#107C22×C1312yes—
239C2×#106C22×C1312yes—
240#424C22×C1312yes—
241#423C22×C1311yescharacteristic
242#417C22×C1314yes—
243#416C22×C1312yes—
244#405C22×C1312yes—
245C2×C4×C8C22×C1311yescharacteristic
246C22×C13—22no—
247C22×C13—24no—
248C22×C13—44no—
249C22×C13D4×Q812yes—
250C22×C13C23×D411yescharacteristic
251C4×C13—22no—
252C4×C13—22no—
253C4×C13—24no—
254C4×C13—24no—
255C4×C13—24no—
256C4×C13—48no—
257C4×C13—44no—
258C4×C13D4×Q812yes—
259C4×C13C23×Q811yescharacteristic
260C4×C13C2×2−1+411yescharacteristic
261C2×#41C23×C1312yes—
262#135Q8×C1318yescomplemented
263C2×Q16C23×C1311yescharacteristic
264C2×SD16C23×C1312yes—
265C2×D8C23×C1311yescharacteristic
266C2×M16C23×C1312yes—
267C22×C8C23×C1311yescharacteristic
268C22×C8C23×C1311yescharacteristic
269C4⋊Q8—28no—
270C4⋊Q8—24no—
271C4⋊Q8D4×C1312yes—
272C4⋊Q8C23×C1318yes—
273#125—24no—
274#125—28no—
275#125—28no—
276#125—22no—
277#125—22no—
278#125D4×C1311yescharacteristic
279#125D4×C1311yescharacteristic
280#125C23×C1318yes—
281#125C23×C1318yes—
282C22⋊Q8—28no—
283C22⋊Q8—28no—
284C22⋊Q8—28no—
285C22⋊Q8—28no—
286C22⋊Q8D4×C1314yes—
287C4×Q8—28no—
288C4×Q8C23×C1314yes—
289C4×Q8C23×C1318yes—
290C4×Q8C23×C1314yes—
291C4×D4—24no—
292C4×D4—28no—
293C4×D4C23×C1314yes—
294C4×D4C23×C1318yes—
295#117—24no—
296#117—22no—
297#117C23×C1312yes—
298#117C23×C1312yes—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—28no—
302C2×C4⋊C4C23×C1314yes—
303C2×C42—22no—
304C2×C42C23×C1311yescharacteristic
305#107C23×C1314yes—
306#107C23×C1314yes—
307#107C23×C1312yes—
308#107C23×C1318yes—
309#106C23×C1318yes—
310#106C23×C1312yes—
311#106C23×C1318yes—
312C4⋊C8C23×C1314yes—
313C4⋊C8C23×C1314yes—
314C4⋊C8C23×C1314yes—
315#103C23×C1314yes—
316#103C23×C1318yes—
317#102C23×C1314yes—
318#102C23×C1318yes—
319#97C23×C1314yes—
320C4×C8C23×C1312yes—
321C4×C8C23×C1312yes—
322C4×C8C23×C1314yes—
323C2×C13—22no—
324C2×C13—44no—
325C2×C13C2×D4×Q811yescharacteristic
326C2×C13#358412yes—
327#41—24no—
328#41C2×Q8×C1318yes—
329C2×Q8—28no—
330C2×Q8C2×D4×C1314yes—
331C2×Q8C24×C1312yes—
332C2×D4—22no—
333C2×D4C24×C1312yes—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—28no—
338C22×C4C2×D4×C1312yes—
339C22×C4C24×C1311yescharacteristic
340Q16C2×Q8×C1314yes—
341SD16C2×Q8×C1318yes—
342D8C2×Q8×C1314yes—
343M16—24no—
344C2×C8—22no—
345C2×C8C2×Q8×C1314yes—
346C2×C8C24×C1312yes—
347C2×C8C24×C1314yes—
348C2×C8C24×C1311yescharacteristic
349C2×C8C24×C1311yescharacteristic
350C4⋊C4—28no—
351C4⋊C4—24no—
352C4⋊C4—28no—
353C4⋊C4—28no—
354C4⋊C4—28no—
355C4⋊C4—28no—
356C4⋊C4—24no—
357C4⋊C4—28no—
358C4⋊C4C2×D4×C1312yes—
359C4⋊C4C2×D4×C1314yes—
360C4⋊C4C2×D4×C1312yes—
361C4⋊C4C2×D4×C1314yes—
362C4⋊C4C2×D4×C1314yes—
363C4⋊C4C24×C1318yes—
364C4⋊C4C24×C1314yes—
365C22⋊C4—22no—
366C22⋊C4—24no—
367C22⋊C4—28no—
368C22⋊C4C2×D4×C1314yes—
369C42—22no—
370C42—24no—
371C42—28no—
372C42C2×D4×C1311yescharacteristic
373C42C2×D4×C1311yescharacteristic
374C42C24×C1312yes—
375C42C24×C1314yes—
376C13#6137611yescharacteristic, direct product factor, Sylow
377C23—22no—
378C23C22×D4×C1311yescharacteristic
379Q8—22no—
380Q8—44no—
381Q8—88no—
382Q8C22×Q8×C1314yes—
383D4—28no—
384D4—22no—
385D4C22×Q8×C1314yes—
386C2×C4—22no—
387C2×C4—28no—
388C2×C4—24no—
389C2×C4—28no—
390C2×C4—42no—
391C2×C4—84no—
392C2×C4—88no—
393C2×C4—84no—
394C2×C4C22×D4×C1312yes—
395C2×C4C22×D4×C1314yes—
396C2×C4C22×D4×C1312yes—
397C2×C4C22×D4×C1314yes—
398C2×C4C22×D4×C1311yescharacteristic
399C2×C4C22×Q8×C1312yes—
400C2×C4C13×2−1+412yes—
401C2×C4C25×C1311yescharacteristic
402C8—22no—
403C8—24no—
404C8C22×Q8×C1312yes—
405C8C22×Q8×C1312yes—
406C22—22no—
407C22—24no—
408C22—44no—
409C22D4×Q8×C1312yes—
410C22C23×D4×C1311yescharacteristic
411C4—22no—
412C4—22no—
413C4—24no—
414C4—24no—
415C4—24no—
416C4—48no—
417C4—44no—
418C4D4×Q8×C1312yes—
419C4C23×Q8×C1311yescharacteristic
420C4C2×C13×2−1+411yescharacteristic
421C2—22no—
422C2—44no—
423C2C2×D4×Q8×C1311yescharacteristic
424C2C13×#358412yes—
subgroupG/H
center, socleC22×C13C23×D4
commutator, Frattini, cosocleC2×C4C25×C13
prime radicalC2×C13×#135C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C13 ◁ C13×#533 ◁ G

Complements

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

NH ≅ G/Ntype
1. C13×#3504C2semidirect
2. C13×#3388C2semidirect
3. C13×#3383C2semidirect
4. C13×#3380C2semidirect
5. C13×#3164C2semidirect
6. C2×C13×#499C2semidirect
7. C13×#3084C2semidirect
8. C13×#3080C2semidirect
9. C13×#2960C2semidirect
54. C13×#135Q8semidirect
115. #61376C13direct (N × H)
165. #3504C2×C13semidirect
166. #3388C2×C13semidirect
167. #3383C2×C13semidirect
168. #3380C2×C13semidirect
169. #3164C2×C13semidirect
170. C2×#499C2×C13semidirect
171. #3084C2×C13semidirect
172. #3080C2×C13semidirect
173. #2960C2×C13semidirect
262. #135Q8×C13semidirect
376. C13#61376direct (N × H)

Maximal subgroups

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

subgroupindex
C13×#35042
C13×#33882
C13×#33832
C13×#33802
C13×#31642
C2×C13×#4992
C13×#30842
C13×#30802
C13×#29602
C13×#29512
C4×C13×#1352
#6137613