Finite Group Databasebeta

← C131×#25503 #45711387

Subgroups — C131×#25503 #45711387

Order 33536.

1170 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#533×C131C211yesmaximal, characteristic, complemented
2C2×#468×C131C212yesmaximal
3#2854×C131C211yesmaximal, characteristic
4#2057×C131C211yesmaximal, characteristic, complemented
5#2043×C131C211yesmaximal, characteristic, complemented
6#2023×C131C211yesmaximal, characteristic, complemented
7#2013×C131C211yesmaximal, characteristic, complemented
8#1894×C131C212yesmaximal, complemented
9#1880×C131C212yesmaximal, complemented
10#1863×C131C212yesmaximal, complemented
11#1728×C131C211yesmaximal, characteristic, complemented
12C22×SD16×C131C2212yes—
13C22×M16×C131C2212yes—
14#533×C131—28no—
15C2×C4⋊Q8×C131C2211yescharacteristic
16C2×C4⋊Q8×C131C2211yescharacteristic
17C2×#125×C131—21no—
18C2×#123×C131—21no—
19C2×C22⋊Q8×C131—21no—
20C2×C22⋊Q8×C131C2212yes—
21C2×#121×C131C2211yescharacteristic
22C2×C4×Q8×C131—21no—
23C2×C4×Q8×C131C2212yes—
24C2×#117×C131—21no—
25C22×C42×C131C2211yescharacteristic
26#468×C131—28no—
27C2×#107×C131C2212yes—
28C2×#103×C131C2212yes—
29C2×#103×C131C2212yes—
30C2×#103×C131C2212yes—
31C2×#102×C131C2212yes—
32#390×C131C2211yescharacteristic
33#390×C131C2211yescharacteristic
34#389×C131C2211yescharacteristic
35#389×C131C2211yescharacteristic
36#388×C131—21no—
37#388×C131—21no—
38#386×C131—21no—
39#380×C131C2211yescharacteristic
40#340×C131C2212yes—
41#340×C131C2212yes—
42#336×C131C2211yescharacteristic
43#336×C131C2211yescharacteristic
44#336×C131C2212yes—
45C22×Q8×C131—21no—
46C22×Q8×C131C2312yes—
47C22×D4×C131—21no—
48C22×D4×C131C2311yescharacteristic
49C23×C4×C131D411yescharacteristic
50C23×C4×C131D411yescharacteristic
51C23×C4×C131C2311yescharacteristic
52C2×SD16×C131—24no—
53C2×SD16×C131—24no—
54C2×SD16×C131—42no—
55C2×M16×C131—24no—
56C2×M16×C131—24no—
57C2×M16×C131—42no—
58C22×C8×C131C2312yes—
59C22×C8×C131C2312yes—
60C4⋊Q8×C131—24no—
61C4⋊Q8×C131—24no—
62#125×C131—28no—
63#123×C131—44no—
64C22⋊Q8×C131—28no—
65C22⋊Q8×C131—24no—
66C22⋊Q8×C131D418yes—
67#121×C131—24no—
68C4×Q8×C131—28no—
69C4×Q8×C131—24no—
70C4×Q8×C131—24no—
71#117×C131—28no—
72C2×C4⋊C4×C131—21no—
73C2×C4⋊C4×C131—21no—
74C2×C4⋊C4×C131—21no—
75C2×C4⋊C4×C131—22no—
76C2×C4⋊C4×C131—22no—
77C2×C4⋊C4×C131—21no—
78C2×C4⋊C4×C131—21no—
79C2×C4⋊C4×C131C2311yescharacteristic
80C2×C4⋊C4×C131C2311yescharacteristic
81C2×C4⋊C4×C131C2311yescharacteristic
82C2×C4⋊C4×C131C2312yes—
83C2×C22⋊C4×C131—22no—
84C2×C22⋊C4×C131—21no—
85C2×C22⋊C4×C131—21no—
86C2×C22⋊C4×C131—21no—
87C2×C22⋊C4×C131—21no—
88C2×C22⋊C4×C131—21no—
89C2×C22⋊C4×C131—21no—
90C2×C42×C131—24no—
91C2×C42×C131—22no—
92C2×C42×C131D411yescharacteristic
93C2×C42×C131D411yescharacteristic
94C2×C42×C131C2311yescharacteristic
95C2×C42×C131C2311yescharacteristic
96#107×C131—24no—
97#103×C131—24no—
98#103×C131—24no—
99#103×C131—42no—
100#102×C131—24no—
101#95×C131—21no—
102#95×C131—21no—
103#95×C131—21no—
104C24×C131—21no—
105C24×C131C2×D411yescharacteristic
106C2×Q8×C131—24no—
107C2×Q8×C131—22no—
108C2×Q8×C131—22no—
109C2×Q8×C131—22no—
110C2×Q8×C131—22no—
111C2×Q8×C131—22no—
112C2×Q8×C131—42no—
113C2×Q8×C131—42no—
114C2×Q8×C131C2×D414yes—
115C2×D4×C131—22no—
116C2×D4×C131—22no—
117C2×D4×C131—21no—
118C2×D4×C131—21no—
119C2×D4×C131—42no—
120C2×D4×C131—42no—
121C2×D4×C131—42no—
122C22×C4×C131—22no—
123C22×C4×C131—22no—
124C22×C4×C131—22no—
125C22×C4×C131—22no—
126C22×C4×C131—21no—
127C22×C4×C131—21no—
128C22×C4×C131—22no—
129C22×C4×C131—22no—
130C22×C4×C131—22no—
131C22×C4×C131—22no—
132C22×C4×C131—21no—
133C22×C4×C131—21no—
134C22×C4×C131—21no—
135C22×C4×C131—41no—
136C22×C4×C131C2×D411yescharacteristic
137C22×C4×C131C2×D411yescharacteristic
138C22×C4×C131C2×D414yes—
139C22×C4×C131C2×D411yescharacteristic
140C22×C4×C131C2×D411yescharacteristic
141C22×C4×C131C2×D411yescharacteristic
142C22×C4×C131C2411yescharacteristic
143SD16×C131—48no—
144M16×C131—48no—
145C2×C8×C131—22no—
146C2×C8×C131—22no—
147C2×C8×C131—22no—
148C2×C8×C131—22no—
149C2×C8×C131—22no—
150C2×C8×C131#4114yes—
151C4⋊C4×C131—24no—
152C4⋊C4×C131—24no—
153C4⋊C4×C131—28no—
154C4⋊C4×C131—24no—
155C4⋊C4×C131—28no—
156C4⋊C4×C131—22no—
157C4⋊C4×C131—22no—
158C4⋊C4×C131—22no—
159C4⋊C4×C131—21no—
160C4⋊C4×C131—21no—
161C4⋊C4×C131—42no—
162C4⋊C4×C131—42no—
163C4⋊C4×C131C2×D414yes—
164C22⋊C4×C131—28no—
165C22⋊C4×C131—42no—
166C22⋊C4×C131—42no—
167C22⋊C4×C131—42no—
168C22⋊C4×C131—42no—
169C22⋊C4×C131—42no—
170C22⋊C4×C131—42no—
171C42×C131—28no—
172C42×C131—22no—
173C42×C131—22no—
174C42×C131—22no—
175C42×C131—22no—
176C23×C131—21no—
177C23×C131—21no—
178C23×C131—21no—
179C23×C131—21no—
180C23×C131—21no—
181C23×C131—21no—
182C23×C131—41no—
183C23×C131—41no—
184C23×C131—41no—
185C23×C131—41no—
186C23×C131—41no—
187C23×C131—41no—
188C23×C131—41no—
189C23×C131#12012yes—
190C23×C131C22×D411yescharacteristic
191Q8×C131—24no—
192Q8×C131—24no—
193Q8×C131—48no—
194Q8×C131—42no—
195Q8×C131—42no—
196D4×C131—42no—
197D4×C131—42no—
198D4×C131—84no—
199C2×C4×C131—28no—
200C2×C4×C131—21no—
201C2×C4×C131—21no—
202C2×C4×C131—21no—
203C2×C4×C131—21no—
204C2×C4×C131—21no—
205C2×C4×C131—21no—
206C2×C4×C131—21no—
207C2×C4×C131—21no—
208C2×C4×C131—24no—
209C2×C4×C131—24no—
210C2×C4×C131—21no—
211C2×C4×C131—24no—
212C2×C4×C131—24no—
213C2×C4×C131—21no—
214C2×C4×C131—21no—
215C2×C4×C131—44no—
216C2×C4×C131—42no—
217C2×C4×C131—42no—
218C2×C4×C131—42no—
219C2×C4×C131—42no—
220C2×C4×C131—41no—
221C2×C4×C131—41no—
222C2×C4×C131—41no—
223C2×C4×C131—41no—
224C2×C4×C131—41no—
225C2×C4×C131—41no—
226C2×C4×C131—41no—
227C2×C4×C131—41no—
228C2×C4×C131—41no—
229C2×C4×C131#12012yes—
230C2×C4×C131AGL(1,ℤ/8)12yes—
231C2×C4×C131#13712yes—
232C2×C4×C131#13712yes—
233C2×C4×C131C22×D412yes—
234C2×C4×C131C2×#4112yes—
235C2×C4×C131H(5,𝔽2)11yescharacteristic
236C2×C4×C1312−1+411yescharacteristic
237C8×C131—24no—
238C8×C131—42no—
239C22×C131—22no—
240C22×C131—22no—
241C22×C131—22no—
242C22×C131—41no—
243C22×C131—41no—
244C22×C131—41no—
245C22×C131—41no—
246C22×C131—81no—
247C22×C131—81no—
248C22×C131—81no—
249C22×C131—81no—
250C22×C131—81no—
251C22×C131—81no—
252C22×C131—81no—
253C22×C131C2×#12011yescharacteristic
254C22×C131#53211yescharacteristic
255C22×C131#53711yescharacteristic
256C22×C131C2×AGL(1,ℤ/8)11yescharacteristic
257C22×C131C2×#13711yescharacteristic
258C22×C131C2×#13711yescharacteristic
259C22×C131#57511yescharacteristic
260C4×C131—22no—
261C4×C131—22no—
262C4×C131—22no—
263C4×C131—22no—
264C4×C131—44no—
265C4×C131—41no—
266C4×C131—44no—
267C4×C131—81no—
268C4×C131—81no—
269C4×C131—81no—
270C4×C131#54612yes—
271C4×C131#54712yes—
272C2×C131—42no—
273C2×C131—161no—
274C2×C131#243011yescharacteristic
275C2×C131#300411yescharacteristic
276C2×C131#300611yescharacteristic
277C2×C131#317211yescharacteristic
278C2×C131#318211yescharacteristic
279C2×C131#325211yescharacteristic
280C2×C131#325411yescharacteristic
281#25503C13111yesmaximal, characteristic, direct product factor, Sylow
282C131#2550311yescharacteristic, direct product factor, Sylow
283C2×#533C2×C13111yescharacteristic, complemented
284C2×#468C2×C13112yes—
285#2854C2×C13111yescharacteristic
286#2057C2×C13111yescharacteristic, complemented
287#2043C2×C13111yescharacteristic, complemented
288#2023C2×C13111yescharacteristic, complemented
289#2013C2×C13111yescharacteristic, complemented
290#1894C2×C13112yescomplemented
291#1880C2×C13112yescomplemented
292#1863C2×C13112yescomplemented
293#1728C2×C13111yescharacteristic, complemented
294C22×SD16C22×C13112yes—
295C22×M16C22×C13112yes—
296#533—28no—
297C2×C4⋊Q8C22×C13111yescharacteristic
298C2×C4⋊Q8C22×C13111yescharacteristic
299C2×#125—21no—
300C2×#123—21no—
301C2×C22⋊Q8—21no—
302C2×C22⋊Q8C22×C13112yes—
303C2×#121C22×C13111yescharacteristic
304C2×C4×Q8—21no—
305C2×C4×Q8C22×C13112yes—
306C2×#117—21no—
307C22×C42C22×C13111yescharacteristic
308#468—28no—
309C2×#107C22×C13112yes—
310C2×#103C22×C13112yes—
311C2×#103C22×C13112yes—
312C2×#103C22×C13112yes—
313C2×#102C22×C13112yes—
314#390C22×C13111yescharacteristic
315#390C22×C13111yescharacteristic
316#389C22×C13111yescharacteristic
317#389C22×C13111yescharacteristic
318#388—21no—
319#388—21no—
320#386—21no—
321#380C22×C13111yescharacteristic
322#340C22×C13112yes—
323#340C22×C13112yes—
324#336C22×C13111yescharacteristic
325#336C22×C13112yes—
326#336C22×C13111yescharacteristic
327C22×Q8—21no—
328C22×Q8C23×C13112yes—
329C22×D4—21no—
330C22×D4C23×C13111yescharacteristic
331C23×C4D4×C13111yescharacteristic
332C23×C4D4×C13111yescharacteristic
333C23×C4C23×C13111yescharacteristic
334C2×SD16—24no—
335C2×SD16—24no—
336C2×SD16—42no—
337C2×M16—24no—
338C2×M16—24no—
339C2×M16—42no—
340C22×C8C23×C13112yes—
341C22×C8C23×C13112yes—
342C4⋊Q8—24no—
343C4⋊Q8—24no—
344#125—28no—
345#123—44no—
346C22⋊Q8—28no—
347C22⋊Q8—24no—
348C22⋊Q8D4×C13118yes—
349#121—24no—
350C4×Q8—28no—
351C4×Q8—24no—
352C4×Q8—24no—
353#117—28no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4C23×C13111yescharacteristic
362C2×C4⋊C4C23×C13111yescharacteristic
363C2×C4⋊C4C23×C13111yescharacteristic
364C2×C4⋊C4C23×C13112yes—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C42—22no—
373C2×C42—24no—
374C2×C42D4×C13111yescharacteristic
375C2×C42D4×C13111yescharacteristic
376C2×C42C23×C13111yescharacteristic
377C2×C42C23×C13111yescharacteristic
378#107—24no—
379#103—24no—
380#103—24no—
381#103—42no—
382#102—24no—
383#95—21no—
384#95—21no—
385#95—21no—
386C24—21no—
387C24C2×D4×C13111yescharacteristic
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8C2×D4×C13114yes—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—41no—
418C22×C4C2×D4×C13111yescharacteristic
419C22×C4C2×D4×C13111yescharacteristic
420C22×C4C2×D4×C13114yes—
421C22×C4C2×D4×C13111yescharacteristic
422C22×C4C2×D4×C13111yescharacteristic
423C22×C4C2×D4×C13111yescharacteristic
424C22×C4C24×C13111yescharacteristic
425SD16—48no—
426M16—48no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8#41×C13114yes—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—28no—
436C4⋊C4—24no—
437C4⋊C4—28no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4C2×D4×C13114yes—
446C22⋊C4—28no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C42—28no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23#120×C13112yes—
472C23C22×D4×C13111yescharacteristic
473Q8—24no—
474Q8—24no—
475Q8—48no—
476Q8—42no—
477Q8—42no—
478D4—42no—
479D4—42no—
480D4—84no—
481C2×C4—28no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4#120×C13112yes—
512C2×C4AGL(1,ℤ/8)×C13112yes—
513C2×C4#137×C13112yes—
514C2×C4#137×C13112yes—
515C2×C4C22×D4×C13112yes—
516C2×C4C2×#41×C13112yes—
517C2×C4H(5,𝔽2)×C13111yescharacteristic
518C2×C42−1+4×C13111yescharacteristic
519C8—24no—
520C8—42no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22C2×#120×C13111yescharacteristic
536C22#532×C13111yescharacteristic
537C22#537×C13111yescharacteristic
538C22C2×AGL(1,ℤ/8)×C13111yescharacteristic
539C22C2×#137×C13111yescharacteristic
540C22C2×#137×C13111yescharacteristic
541C22#575×C13111yescharacteristic
542C4—22no—
543C4—22no—
544C4—22no—
545C4—22no—
546C4—41no—
547C4—44no—
548C4—44no—
549C4—81no—
550C4—81no—
551C4—81no—
552C4#546×C13112yes—
553C4#547×C13112yes—
554C2—42no—
555C2—161no—
556C2#2430×C13111yescharacteristic
557C2#3004×C13111yescharacteristic
558C2#3006×C13111yescharacteristic
559C2#3172×C13111yescharacteristic
560C2#3182×C13111yescharacteristic
561C2#3252×C13111yescharacteristic
562C2#3254×C13111yescharacteristic
subgroupG/H
center, socleC23×C131C22×D4
commutator, Frattini, cosocleC22×C4C24×C131
prime radicalC2×#121×C131C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C131 ◁ C22×C42×C131 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#533×C131C2semidirect
4. #2057×C131C2semidirect
5. #2043×C131C2semidirect
6. #2023×C131C2semidirect
7. #2013×C131C2semidirect
8. #1894×C131C2semidirect
9. #1880×C131C2semidirect
10. #1863×C131C2semidirect
11. #1728×C131C2semidirect
281. #25503C131direct (N × H)
282. C131#25503direct (N × H)
283. C2×#533C2×C131semidirect
286. #2057C2×C131semidirect
287. #2043C2×C131semidirect
288. #2023C2×C131semidirect
289. #2013C2×C131semidirect
290. #1894C2×C131semidirect
291. #1880C2×C131semidirect
292. #1863C2×C131semidirect
293. #1728C2×C131semidirect

Maximal subgroups

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

subgroupindex
C2×#533×C1312
C2×#468×C1312
#2854×C1312
#2057×C1312
#2043×C1312
#2023×C1312
#2013×C1312
#1894×C1312
#1880×C1312
#1863×C1312
#1728×C1312
#25503131