Finite Group Databasebeta

← C131×#23481 #45709365

Subgroups — C131×#23481 #45709365

Order 33536.

1558 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊Q8×C131C211yesmaximal, characteristic
2C2×#428×C131C211yesmaximal, characteristic, complemented
3C2×#419×C131C211yesmaximal, characteristic, complemented
4C2×#419×C131C211yesmaximal, characteristic, complemented
5C2×#407×C131C211yesmaximal, characteristic
6#2280×C131C211yesmaximal, characteristic
7#2036×C131C214yesmaximal, complemented
8#2035×C131C214yesmaximal, complemented
9#1736×C131C211yesmaximal, characteristic, complemented
10C23×Q8×C131C2211yescharacteristic
11C24×C4×C131C2211yescharacteristic
12C22×M16×C131C2211yescharacteristic
13C22×M16×C131C2211yescharacteristic
14C2×C22⋊Q8×C131—24no—
15C2×C22⋊Q8×C131—22no—
16C2×C22⋊Q8×C131—24no—
17C2×C22⋊Q8×C131C2214yes—
18C2×#117×C131C2211yescharacteristic
19C2×#117×C131C2211yescharacteristic
20C22×C4⋊C4×C131—21no—
21C22×C4⋊C4×C131C2211yescharacteristic
22C22×C22⋊C4×C131—21no—
23#428×C131—24no—
24C2×#107×C131C2212yescomplemented
25C2×#106×C131C2212yescomplemented
26#419×C131—24no—
27#419×C131—24no—
28C2×#103×C131C2212yescomplemented
29C2×#103×C131C2214yescomplemented
30C2×#103×C131C2212yescomplemented
31#407×C131—24no—
32C2×C22⋊C8×C131C2214yes—
33#381×C131—21no—
34#380×C131C2212yes—
35#380×C131C2212yes—
36#379×C131—21no—
37C4×C22⋊C4×C131—22no—
38#340×C131C2214yescomplemented
39C25×C131D411yescharacteristic
40C22×Q8×C131—22no—
41C22×Q8×C131—21no—
42C22×Q8×C131—22no—
43C22×Q8×C131—22no—
44C22×Q8×C131—22no—
45C22×Q8×C131—22no—
46C22×Q8×C131D414yes—
47C22×Q8×C131C2312yes—
48C23×C4×C131—21no—
49C23×C4×C131—21no—
50C23×C4×C131—22no—
51C23×C4×C131—22no—
52C23×C4×C131—22no—
53C23×C4×C131—22no—
54C23×C4×C131—22no—
55C23×C4×C131—22no—
56C23×C4×C131D411yescharacteristic
57C23×C4×C131D412yes—
58C23×C4×C131C2312yes—
59C23×C4×C131C2311yescharacteristic
60C2×M16×C131—22no—
61C2×M16×C131—22no—
62C2×M16×C131—22no—
63C2×M16×C131—22no—
64C2×M16×C131—22no—
65C2×M16×C131D414yes—
66C22×C8×C131C2312yes—
67C22×C8×C131C2312yes—
68C22⋊Q8×C131—216no—
69C22⋊Q8×C131—28no—
70C22⋊Q8×C131—28no—
71#117×C131—22no—
72#117×C131—22no—
73#117×C131—22no—
74#117×C131—22no—
75C2×C4⋊C4×C131—28no—
76C2×C4⋊C4×C131—24no—
77C2×C4⋊C4×C131—21no—
78C2×C4⋊C4×C131—22no—
79C2×C4⋊C4×C131D414yes—
80C2×C4⋊C4×C131C2311yescharacteristic
81C2×C4⋊C4×C131C2311yescharacteristic
82C2×C4⋊C4×C131C2312yes—
83C2×C4⋊C4×C131C2312yes—
84C2×C22⋊C4×C131—28no—
85C2×C22⋊C4×C131—24no—
86C2×C22⋊C4×C131—21no—
87C2×C22⋊C4×C131—21no—
88C2×C22⋊C4×C131—22no—
89C2×C22⋊C4×C131—22no—
90C2×C42×C131—21no—
91C2×C42×C131—21no—
92#107×C131—24no—
93#106×C131—24no—
94#103×C131—24no—
95#103×C131—24no—
96#103×C131—28no—
97C22⋊C8×C131—28no—
98#95×C131—22no—
99#95×C131—22no—
100C24×C131—22no—
101C24×C131—24no—
102C24×C131—22no—
103C24×C131—22no—
104C24×C131—42no—
105C24×C131C2×D412yes—
106C24×C131C2×D411yescharacteristic
107C2×Q8×C131—24no—
108C2×Q8×C131—22no—
109C2×Q8×C131—28no—
110C2×Q8×C131—28no—
111C2×Q8×C131—22no—
112C2×Q8×C131—22no—
113C2×Q8×C131—22no—
114C2×Q8×C131—22no—
115C2×Q8×C131—42no—
116C2×Q8×C131—44no—
117C2×Q8×C131—42no—
118C2×Q8×C131—44no—
119C2×Q8×C131C2×D414yes—
120C22×C4×C131—24no—
121C22×C4×C131—24no—
122C22×C4×C131—22no—
123C22×C4×C131—22no—
124C22×C4×C131—22no—
125C22×C4×C131—22no—
126C22×C4×C131—22no—
127C22×C4×C131—24no—
128C22×C4×C131—24no—
129C22×C4×C131—24no—
130C22×C4×C131—22no—
131C22×C4×C131—22no—
132C22×C4×C131—22no—
133C22×C4×C131—22no—
134C22×C4×C131—22no—
135C22×C4×C131—22no—
136C22×C4×C131—21no—
137C22×C4×C131—21no—
138C22×C4×C131—22no—
139C22×C4×C131—44no—
140C22×C4×C131—42no—
141C22×C4×C131—42no—
142C22×C4×C131—44no—
143C22×C4×C131—44no—
144C22×C4×C131—44no—
145C22×C4×C131—44no—
146C22×C4×C131C2×D412yes—
147C22×C4×C131C2×D411yescharacteristic
148C22×C4×C131C2×D412yes—
149C22×C4×C131C2×D412yes—
150C22×C4×C131C2×D412yes—
151C22×C4×C131#4112yes—
152C22×C4×C131#4112yes—
153C22×C4×C131#4112yes—
154C22×C4×C131C2411yescharacteristic
155M16×C131—28no—
156M16×C131—44no—
157C2×C8×C131—22no—
158C2×C8×C131—22no—
159C2×C8×C131—22no—
160C2×C8×C131—22no—
161C2×C8×C131—22no—
162C2×C8×C131C2×D414yes—
163C4⋊C4×C131—216no—
164C4⋊C4×C131—24no—
165C4⋊C4×C131—22no—
166C4⋊C4×C131—21no—
167C4⋊C4×C131—21no—
168C4⋊C4×C131—22no—
169C4⋊C4×C131—21no—
170C4⋊C4×C131—21no—
171C4⋊C4×C131—22no—
172C4⋊C4×C131C2×D414yes—
173C22⋊C4×C131—216no—
174C22⋊C4×C131—44no—
175C22⋊C4×C131—42no—
176C22⋊C4×C131—42no—
177C22⋊C4×C131—44no—
178C42×C131—42no—
179C42×C131—42no—
180C23×C131—22no—
181C23×C131—22no—
182C23×C131—28no—
183C23×C131—22no—
184C23×C131—22no—
185C23×C131—22no—
186C23×C131—22no—
187C23×C131—22no—
188C23×C131—22no—
189C23×C131—48no—
190C23×C131—48no—
191C23×C131—44no—
192C23×C131—44no—
193C23×C131#12012yes—
194C23×C131#12112yes—
195C23×C131#12312yes—
196C23×C131#13714yes—
197C23×C131C22×D411yescharacteristic
198Q8×C131—24no—
199Q8×C131—24no—
200Q8×C131—42no—
201Q8×C131—42no—
202Q8×C131—48no—
203C2×C4×C131—24no—
204C2×C4×C131—24no—
205C2×C4×C131—22no—
206C2×C4×C131—22no—
207C2×C4×C131—22no—
208C2×C4×C131—48no—
209C2×C4×C131—42no—
210C2×C4×C131—42no—
211C2×C4×C131—44no—
212C2×C4×C131—44no—
213C2×C4×C131—48no—
214C2×C4×C131—42no—
215C2×C4×C131—42no—
216C2×C4×C131—42no—
217C2×C4×C131—42no—
218C2×C4×C131—48no—
219C2×C4×C131—41no—
220C2×C4×C131—41no—
221C2×C4×C131—42no—
222C2×C4×C131—42no—
223C2×C4×C131—42no—
224C2×C4×C131—42no—
225C2×C4×C131—42no—
226C2×C4×C131#12012yes—
227C2×C4×C131#12114yes—
228C2×C4×C131#12114yes—
229C2×C4×C131#12112yes—
230C2×C4×C131#12314yes—
231C2×C4×C131#12312yes—
232C2×C4×C131#12414yes—
233C2×C4×C131C22×D411yescharacteristic
234C2×C4×C131C22×D411yescharacteristic
235C2×C4×C131C22×D411yescharacteristic
236C2×C4×C131C2×#4111yescharacteristic
237C2×C4×C131C2×#4111yescharacteristic
238C2×C4×C131C2×#4111yescharacteristic
239C8×C131—24no—
240C8×C131—42no—
241C22×C131—24no—
242C22×C131—22no—
243C22×C131—22no—
244C22×C131—24no—
245C22×C131—24no—
246C22×C131—24no—
247C22×C131—24no—
248C22×C131—416no—
249C22×C131—42no—
250C22×C131—42no—
251C22×C131—42no—
252C22×C131—42no—
253C22×C131#39414yes—
254C22×C131C2×#12011yescharacteristic
255C22×C131C2×#12111yescharacteristic
256C22×C131C2×#12311yescharacteristic
257C22×C131C2×#13712yes—
258C22×C131#57512yes—
259C4×C131—42no—
260C4×C131—44no—
261C4×C131—44no—
262C4×C131—81no—
263C4×C131—81no—
264C4×C131—82no—
265C4×C131#39414yes—
266C4×C131C2×#12111yescharacteristic
267C4×C131C2×#12111yescharacteristic
268C4×C131C2×#12311yescharacteristic
269C4×C131C2×#12411yescharacteristic
270C2×C131—24no—
271C2×C131—44no—
272C2×C131C2×#39411yescharacteristic
273C2×C131#300411yescharacteristic
274C2×C131#300511yescharacteristic
275C2×C131#305612yes—
276C2×C131#309512yes—
277#23481C13111yesmaximal, characteristic, direct product factor, Sylow
278C131#2348111yescharacteristic, direct product factor, Sylow
279C22×C22⋊Q8C2×C13111yescharacteristic
280C2×#428C2×C13111yescharacteristic, complemented
281C2×#419C2×C13111yescharacteristic, complemented
282C2×#419C2×C13111yescharacteristic, complemented
283C2×#407C2×C13111yescharacteristic
284#2280C2×C13111yescharacteristic
285#2036C2×C13114yescomplemented
286#2035C2×C13114yescomplemented
287#1736C2×C13111yescharacteristic, complemented
288C23×Q8C22×C13111yescharacteristic
289C24×C4C22×C13111yescharacteristic
290C22×M16C22×C13111yescharacteristic
291C22×M16C22×C13111yescharacteristic
292C2×C22⋊Q8—24no—
293C2×C22⋊Q8—22no—
294C2×C22⋊Q8—24no—
295C2×C22⋊Q8C22×C13114yes—
296C2×#117C22×C13111yescharacteristic
297C2×#117C22×C13111yescharacteristic
298C22×C4⋊C4—21no—
299C22×C4⋊C4C22×C13111yescharacteristic
300C22×C22⋊C4—21no—
301#428—24no—
302C2×#107C22×C13112yescomplemented
303C2×#106C22×C13112yescomplemented
304#419—24no—
305#419—24no—
306C2×#103C22×C13112yescomplemented
307C2×#103C22×C13114yescomplemented
308C2×#103C22×C13112yescomplemented
309#407—24no—
310C2×C22⋊C8C22×C13114yes—
311#381—21no—
312#380C22×C13112yes—
313#380C22×C13112yes—
314#379—21no—
315C4×C22⋊C4—22no—
316#340C22×C13114yescomplemented
317C25D4×C13111yescharacteristic
318C22×Q8—22no—
319C22×Q8—21no—
320C22×Q8—22no—
321C22×Q8—22no—
322C22×Q8—22no—
323C22×Q8—22no—
324C22×Q8D4×C13114yes—
325C22×Q8C23×C13112yes—
326C23×C4—21no—
327C23×C4—21no—
328C23×C4—22no—
329C23×C4—22no—
330C23×C4—22no—
331C23×C4—22no—
332C23×C4—22no—
333C23×C4—22no—
334C23×C4D4×C13111yescharacteristic
335C23×C4D4×C13112yes—
336C23×C4C23×C13112yes—
337C23×C4C23×C13111yescharacteristic
338C2×M16—22no—
339C2×M16—22no—
340C2×M16—22no—
341C2×M16—22no—
342C2×M16—22no—
343C2×M16D4×C13114yes—
344C22×C8C23×C13112yes—
345C22×C8C23×C13112yes—
346C22⋊Q8—216no—
347C22⋊Q8—28no—
348C22⋊Q8—28no—
349#117—22no—
350#117—22no—
351#117—22no—
352#117—22no—
353C2×C4⋊C4—28no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4D4×C13114yes—
358C2×C4⋊C4C23×C13112yes—
359C2×C4⋊C4C23×C13111yescharacteristic
360C2×C4⋊C4C23×C13111yescharacteristic
361C2×C4⋊C4C23×C13112yes—
362C2×C22⋊C4—28no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C42—21no—
369C2×C42—21no—
370#107—24no—
371#106—24no—
372#103—24no—
373#103—24no—
374#103—28no—
375C22⋊C8—28no—
376#95—22no—
377#95—22no—
378C24—22no—
379C24—24no—
380C24—22no—
381C24—22no—
382C24—42no—
383C24C2×D4×C13112yes—
384C24C2×D4×C13111yescharacteristic
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—28no—
388C2×Q8—28no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8C2×D4×C13114yes—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—44no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4C2×D4×C13112yes—
425C22×C4C2×D4×C13111yescharacteristic
426C22×C4C2×D4×C13112yes—
427C22×C4C2×D4×C13112yes—
428C22×C4C2×D4×C13112yes—
429C22×C4#41×C13112yes—
430C22×C4#41×C13112yes—
431C22×C4#41×C13112yes—
432C22×C4C24×C13111yescharacteristic
433M16—28no—
434M16—44no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8C2×D4×C13114yes—
441C4⋊C4—216no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4C2×D4×C13114yes—
451C22⋊C4—216no—
452C22⋊C4—44no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—44no—
456C42—42no—
457C42—42no—
458C23—22no—
459C23—22no—
460C23—28no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—48no—
468C23—48no—
469C23—44no—
470C23—44no—
471C23#120×C13112yes—
472C23#121×C13112yes—
473C23#123×C13112yes—
474C23#137×C13114yes—
475C23C22×D4×C13111yescharacteristic
476Q8—24no—
477Q8—24no—
478Q8—48no—
479Q8—42no—
480Q8—42no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—48no—
487C2×C4—44no—
488C2×C4—48no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—48no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4#120×C13112yes—
505C2×C4#121×C13112yes—
506C2×C4#121×C13114yes—
507C2×C4#121×C13114yes—
508C2×C4#123×C13112yes—
509C2×C4#123×C13114yes—
510C2×C4#124×C13114yes—
511C2×C4C22×D4×C13111yescharacteristic
512C2×C4C22×D4×C13111yescharacteristic
513C2×C4C22×D4×C13111yescharacteristic
514C2×C4C2×#41×C13111yescharacteristic
515C2×C4C2×#41×C13111yescharacteristic
516C2×C4C2×#41×C13111yescharacteristic
517C8—24no—
518C8—42no—
519C22—22no—
520C22—22no—
521C22—24no—
522C22—24no—
523C22—24no—
524C22—24no—
525C22—24no—
526C22—416no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22#394×C13114yes—
532C22C2×#120×C13111yescharacteristic
533C22C2×#121×C13111yescharacteristic
534C22C2×#123×C13111yescharacteristic
535C22C2×#137×C13112yes—
536C22#575×C13112yes—
537C4—44no—
538C4—42no—
539C4—44no—
540C4—81no—
541C4—81no—
542C4—82no—
543C4#394×C13114yes—
544C4C2×#121×C13111yescharacteristic
545C4C2×#121×C13111yescharacteristic
546C4C2×#123×C13111yescharacteristic
547C4C2×#124×C13111yescharacteristic
548C2—24no—
549C2—44no—
550C2C2×#394×C13111yescharacteristic
551C2#3004×C13111yescharacteristic
552C2#3005×C13111yescharacteristic
553C2#3056×C13112yes—
554C2#3095×C13112yes—
subgroupG/H
center, socleC23×C131C22×D4
commutator, Frattini, cosocleC22×C4C24×C131
prime radicalC25×C131D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C131 ◁ C24×C4×C131 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#428×C131C2semidirect
3. C2×#419×C131C2semidirect
4. C2×#419×C131C2semidirect
7. #2036×C131C2semidirect
8. #2035×C131C2semidirect
9. #1736×C131C2semidirect
24. C2×#107×C131C22semidirect
25. C2×#106×C131C22semidirect
28. C2×#103×C131C22semidirect
29. C2×#103×C131C22semidirect
30. C2×#103×C131C22semidirect
38. #340×C131C22semidirect
277. #23481C131direct (N × H)
278. C131#23481direct (N × H)
280. C2×#428C2×C131semidirect
281. C2×#419C2×C131semidirect
282. C2×#419C2×C131semidirect
285. #2036C2×C131semidirect
286. #2035C2×C131semidirect
287. #1736C2×C131semidirect
302. C2×#107C22×C131semidirect
303. C2×#106C22×C131semidirect
306. C2×#103C22×C131semidirect
307. C2×#103C22×C131semidirect
308. C2×#103C22×C131semidirect
316. #340C22×C131semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Q8×C1312
C2×#428×C1312
C2×#419×C1312
C2×#407×C1312
#2280×C1312
#2036×C1312
#2035×C1312
#1736×C1312
#23481131