Finite Group Databasebeta

← C131×#12768 #45698652

Subgroups — C131×#12768 #45698652

Order 33536.

778 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3545×C131C211yesmaximal, characteristic
2C2×#354×C131C211yesmaximal, characteristic, complemented
3#2121×C131C212yesmaximal, complemented
4#1913×C131C212yesmaximal, complemented
5#1885×C131C211yesmaximal, characteristic, complemented
6#563×C131C411yescharacteristic, complemented
7#562×C131—21no—
8H(3,𝔽4)×C131C411yescharacteristic, complemented
9#555×C131—21no—
10#551×C131—22no—
11#551×C131C412yescomplemented
12#546×C131—22no—
13#542×C131—22no—
14#538×C131—22no—
15#536×C131—21no—
16#534×C131C2212yes—
17#529×C131—21no—
18C2×#124×C131C2211yescharacteristic
19#518×C131C2211yescharacteristic
20#419×C131—21no—
21C2×#100×C131C2211yescharacteristic
22C2×#99×C131—22no—
23C2×#99×C131C2212yes—
24#354×C131—21no—
25#354×C131—21no—
26#352×C131—22no—
27AΣL(1,𝔽16)×C131—22no—
28#342×C131—22no—
29#339×C131—21no—
30C2×#41×C131—21no—
31C22×Q8×C131D411yescharacteristic
32C22×D4×C131D412yes—
33C22×D4×C131C2311yescharacteristic
34C2×M16×C131—21no—
35C4⋊Q8×C131—21no—
36C4⋊Q8×C131—21no—
37#126×C131—22no—
38#126×C131—22no—
39#126×C131—22no—
40#125×C131—21no—
41#124×C131—21no—
42#124×C131—22no—
43#124×C131—21no—
44#124×C131—22no—
45#124×C131—22no—
46#124×C131—21no—
47#124×C131C2×C411yescharacteristic
48#124×C131C2×C411yescharacteristic
49#124×C131D411yescharacteristic
50#124×C131D411yescharacteristic
51#123×C131—22no—
52#123×C131—22no—
53#123×C131—22no—
54#123×C131—22no—
55#123×C131—22no—
56#123×C131C2×C412yes—
57#123×C131D412yes—
58C22⋊Q8×C131—21no—
59C22⋊Q8×C131—22no—
60C22⋊Q8×C131—22no—
61C22⋊Q8×C131—22no—
62C22⋊Q8×C131—22no—
63#121×C131—21no—
64#121×C131—22no—
65#121×C131—22no—
66#121×C131—22no—
67#121×C131—22no—
68#120×C131—22no—
69#120×C131—22no—
70#120×C131C2×C412yes—
71#120×C131D412yes—
72C4×Q8×C131—21no—
73C4×D4×C131—21no—
74C4×D4×C131—22no—
75C4×D4×C131—22no—
76C4×D4×C131—22no—
77C4×D4×C131—42no—
78#117×C131—21no—
79#117×C131—22no—
80#117×C131—22no—
81#117×C131—21no—
82C2×C22⋊C4×C131—22no—
83C2×C22⋊C4×C131—22no—
84C2×C22⋊C4×C131—22no—
85C2×C22⋊C4×C131—22no—
86C2×C22⋊C4×C131D412yes—
87C2×C42×C131D411yescharacteristic
88#103×C131—42no—
89#100×C131—22no—
90#99×C131—22no—
91#99×C131—22no—
92#99×C131—44no—
93C24×C131C22⋊C412yes—
94C24×C131C2×D412yes—
95#41×C131—81no—
96#41×C131—81no—
97C2×Q8×C131—21no—
98C2×Q8×C131—21no—
99C2×Q8×C131—21no—
100C2×Q8×C131—41no—
101C2×Q8×C131—41no—
102C2×Q8×C131C22⋊C411yescharacteristic
103C2×Q8×C131C22⋊C411yescharacteristic
104C2×D4×C131—22no—
105C2×D4×C131—21no—
106C2×D4×C131—21no—
107C2×D4×C131—22no—
108C2×D4×C131—22no—
109C2×D4×C131—22no—
110C2×D4×C131—41no—
111C2×D4×C131—41no—
112C2×D4×C131—42no—
113C2×D4×C131—42no—
114C2×D4×C131C22×C411yescharacteristic
115C2×D4×C131C2×D411yescharacteristic
116C2×D4×C131C2×D412yes—
117C22×C4×C131—21no—
118C22×C4×C131—22no—
119C22×C4×C131—22no—
120C22×C4×C131—22no—
121C22×C4×C131—21no—
122C22×C4×C131—42no—
123C22×C4×C131—42no—
124C22×C4×C131C22⋊C412yes—
125C22×C4×C131C2×D411yescharacteristic
126M16×C131—41no—
127M16×C131—41no—
128C2×C8×C131—41no—
129C4⋊C4×C131—22no—
130C4⋊C4×C131—22no—
131C4⋊C4×C131—22no—
132C4⋊C4×C131—22no—
133C4⋊C4×C131—22no—
134C4⋊C4×C131—21no—
135C4⋊C4×C131—21no—
136C4⋊C4×C131—22no—
137C4⋊C4×C131—22no—
138C4⋊C4×C131—22no—
139C22⋊C4×C131—22no—
140C22⋊C4×C131—22no—
141C22⋊C4×C131—22no—
142C22⋊C4×C131—22no—
143C22⋊C4×C131—22no—
144C22⋊C4×C131—22no—
145C22⋊C4×C131—22no—
146C22⋊C4×C131—22no—
147C22⋊C4×C131—22no—
148C22⋊C4×C131—22no—
149C22⋊C4×C131—22no—
150C22⋊C4×C131—22no—
151C22⋊C4×C131—22no—
152C22⋊C4×C131—24no—
153C22⋊C4×C131—41no—
154C22⋊C4×C131—42no—
155C22⋊C4×C131—42no—
156C22⋊C4×C131C22⋊C412yes—
157C22⋊C4×C131C22⋊C412yes—
158C42×C131—21no—
159C42×C131—22no—
160C42×C131—21no—
161C42×C131—21no—
162C42×C131—41no—
163C42×C131C22⋊C411yescharacteristic
164C42×C131C22⋊C411yescharacteristic
165C23×C131—22no—
166C23×C131—22no—
167C23×C131—22no—
168C23×C131—42no—
169C23×C131—82no—
170C23×C131—82no—
171C23×C131C2×C22⋊C412yes—
172C23×C131#12012yes—
173C23×C131#12011yescharacteristic
174Q8×C131—41no—
175Q8×C131—41no—
176Q8×C131—81no—
177Q8×C131—81no—
178Q8×C131—81no—
179D4×C131—41no—
180D4×C131—42no—
181D4×C131—41no—
182D4×C131—42no—
183D4×C131—42no—
184D4×C131—82no—
185D4×C131—82no—
186D4×C131—82no—
187D4×C131—81no—
188D4×C131—81no—
189C2×C4×C131—21no—
190C2×C4×C131—21no—
191C2×C4×C131—21no—
192C2×C4×C131—21no—
193C2×C4×C131—22no—
194C2×C4×C131—21no—
195C2×C4×C131—22no—
196C2×C4×C131—22no—
197C2×C4×C131—21no—
198C2×C4×C131—41no—
199C2×C4×C131—41no—
200C2×C4×C131—42no—
201C2×C4×C131—42no—
202C2×C4×C131—82no—
203C2×C4×C131—81no—
204C2×C4×C131—82no—
205C2×C4×C131—82no—
206C2×C4×C131—82no—
207C2×C4×C131—162no—
208C2×C4×C131—162no—
209C2×C4×C131C2×C22⋊C411yescharacteristic
210C2×C4×C131#12011yescharacteristic
211C8×C131—81no—
212C22×C131—22no—
213C22×C131—22no—
214C22×C131—41no—
215C22×C131—42no—
216C22×C131—42no—
217C22×C131—42no—
218C22×C131—42no—
219C22×C131—42no—
220C22×C131—82no—
221C22×C131—82no—
222C22×C131—162no—
223C22×C131#37911yescharacteristic
224C22×C131B(4,𝔽2)11yescharacteristic
225C22×C131#45811yescharacteristic
226C4×C131—21no—
227C4×C131—21no—
228C4×C131—41no—
229C4×C131—42no—
230C4×C131—42no—
231C4×C131—41no—
232C4×C131—41no—
233C4×C131—41no—
234C4×C131—42no—
235C4×C131—82no—
236C4×C131—162no—
237C2×C131—21no—
238C2×C131—21no—
239C2×C131—21no—
240C2×C131—42no—
241C2×C131—42no—
242C2×C131—82no—
243C2×C131#179011yescharacteristic
244#12768C13111yesmaximal, characteristic, direct product factor, Sylow
245C131#1276811yescharacteristic, direct product factor, Sylow
246#3545C2×C13111yescharacteristic
247C2×#354C2×C13111yescharacteristic, complemented
248#2121C2×C13112yescomplemented
249#1913C2×C13112yescomplemented
250#1885C2×C13111yescharacteristic, complemented
251#563C4×C13111yescharacteristic, complemented
252#562—21no—
253H(3,𝔽4)C4×C13111yescharacteristic, complemented
254#555—21no—
255#551—22no—
256#551C4×C13112yescomplemented
257#546—22no—
258#542—22no—
259#538—22no—
260#536—21no—
261#534C22×C13112yes—
262#529—21no—
263C2×#124C22×C13111yescharacteristic
264#518C22×C13111yescharacteristic
265#419—21no—
266C2×#100C22×C13111yescharacteristic
267C2×#99—22no—
268C2×#99C22×C13112yes—
269#354—21no—
270#354—21no—
271#352—22no—
272AΣL(1,𝔽16)—22no—
273#342—22no—
274#339—21no—
275C2×#41—21no—
276C22×Q8D4×C13111yescharacteristic
277C22×D4D4×C13112yes—
278C22×D4C23×C13111yescharacteristic
279C2×M16—21no—
280C4⋊Q8—21no—
281C4⋊Q8—21no—
282#126—22no—
283#126—22no—
284#126—22no—
285#125—21no—
286#124—21no—
287#124—22no—
288#124—21no—
289#124—22no—
290#124—22no—
291#124—21no—
292#124C2×C4×C13111yescharacteristic
293#124C2×C4×C13111yescharacteristic
294#124D4×C13111yescharacteristic
295#124D4×C13111yescharacteristic
296#123—22no—
297#123—22no—
298#123—22no—
299#123—22no—
300#123—22no—
301#123C2×C4×C13112yes—
302#123D4×C13112yes—
303C22⋊Q8—21no—
304C22⋊Q8—22no—
305C22⋊Q8—22no—
306C22⋊Q8—22no—
307C22⋊Q8—22no—
308#121—21no—
309#121—22no—
310#121—22no—
311#121—22no—
312#121—22no—
313#120—22no—
314#120—22no—
315#120C2×C4×C13112yes—
316#120D4×C13112yes—
317C4×Q8—21no—
318C4×D4—21no—
319C4×D4—22no—
320C4×D4—22no—
321C4×D4—22no—
322C4×D4—42no—
323#117—21no—
324#117—22no—
325#117—22no—
326#117—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4D4×C13112yes—
332C2×C42D4×C13111yescharacteristic
333#103—42no—
334#100—22no—
335#99—22no—
336#99—22no—
337#99—44no—
338C24C22⋊C4×C13112yes—
339C24C2×D4×C13112yes—
340#41—81no—
341#41—81no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8C22⋊C4×C13111yescharacteristic
348C2×Q8C22⋊C4×C13111yescharacteristic
349C2×D4—22no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4C22×C4×C13111yescharacteristic
360C2×D4C2×D4×C13111yescharacteristic
361C2×D4C2×D4×C13112yes—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4C22⋊C4×C13112yes—
370C22×C4C2×D4×C13111yescharacteristic
371M16—41no—
372M16—41no—
373C2×C8—41no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—22no—
396C22⋊C4—22no—
397C22⋊C4—24no—
398C22⋊C4—41no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4C22⋊C4×C13112yes—
402C22⋊C4C22⋊C4×C13112yes—
403C42—21no—
404C42—22no—
405C42—21no—
406C42—21no—
407C42—41no—
408C42C22⋊C4×C13111yescharacteristic
409C42C22⋊C4×C13111yescharacteristic
410C23—22no—
411C23—22no—
412C23—22no—
413C23—42no—
414C23—82no—
415C23—82no—
416C23C2×C22⋊C4×C13112yes—
417C23#120×C13112yes—
418C23#120×C13111yescharacteristic
419Q8—41no—
420Q8—41no—
421Q8—81no—
422Q8—81no—
423Q8—81no—
424D4—41no—
425D4—42no—
426D4—41no—
427D4—42no—
428D4—42no—
429D4—82no—
430D4—81no—
431D4—82no—
432D4—82no—
433D4—81no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—81no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—162no—
453C2×C4—162no—
454C2×C4C2×C22⋊C4×C13111yescharacteristic
455C2×C4#120×C13111yescharacteristic
456C8—81no—
457C22—22no—
458C22—22no—
459C22—41no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—82no—
466C22—82no—
467C22—162no—
468C22#379×C13111yescharacteristic
469C22B(4,𝔽2)×C13111yescharacteristic
470C22#458×C13111yescharacteristic
471C4—21no—
472C4—21no—
473C4—41no—
474C4—42no—
475C4—42no—
476C4—41no—
477C4—41no—
478C4—41no—
479C4—42no—
480C4—82no—
481C4—162no—
482C2—21no—
483C2—21no—
484C2—21no—
485C2—42no—
486C2—42no—
487C2—82no—
488C2#1790×C13111yescharacteristic
subgroupG/H
center, socleC2×C131#1790
commutatorC2×D4C22×C4×C131
Frattini, cosocleC22×D4C23×C131
prime radical#3545×C131C2
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C131 ◁ C23×C131 ◁ C22×D4×C131 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#354×C131C2semidirect
3. #2121×C131C2semidirect
4. #1913×C131C2semidirect
5. #1885×C131C2semidirect
6. #563×C131C4semidirect
8. H(3,𝔽4)×C131C4semidirect
11. #551×C131C4semidirect
244. #12768C131direct (N × H)
245. C131#12768direct (N × H)
247. C2×#354C2×C131semidirect
248. #2121C2×C131semidirect
249. #1913C2×C131semidirect
250. #1885C2×C131semidirect
251. #563C4×C131semidirect
253. H(3,𝔽4)C4×C131semidirect
256. #551C4×C131semidirect

Maximal subgroups

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

subgroupindex
#3545×C1312
C2×#354×C1312
#2121×C1312
#1913×C1312
#1885×C1312
#12768131