Finite Group Databasebeta

← C131×#16364 #45702248

Subgroups — C131×#16364 #45702248

Order 33536.

1706 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊Q8×C131C211yesmaximal, characteristic
2C2×#397×C131C212yesmaximal
3C2×#381×C131C211yesmaximal, characteristic
4#1446×C131C211yesmaximal, characteristic, complemented
5#1446×C131C211yesmaximal, characteristic, complemented
6#1444×C131C211yesmaximal, characteristic, complemented
7#1442×C131C212yesmaximal, complemented
8#1442×C131C211yesmaximal, characteristic, complemented
9#1441×C131C212yesmaximal, complemented
10#1440×C131C212yesmaximal
11C22⋊#95×C131C211yesmaximal, characteristic
12C23×Q8×C131C2211yescharacteristic
13C24×C4×C131C2211yescharacteristic
14C2×C22⋊Q8×C131—24no—
15C2×C22⋊Q8×C131—28no—
16C22×C4⋊C4×C131C2212yes—
17C22×C4⋊C4×C131C2211yescharacteristic
18C22×C4⋊C4×C131C2212yes—
19C22×C22⋊C4×C131C2211yescharacteristic
20C22×C22⋊C4×C131C2212yes—
21C22×C22⋊C4×C131C2212yes—
22C22×C22⋊C4×C131C2211yescharacteristic
23#397×C131—28no—
24#381×C131—24no—
25C2×#95×C131C2211yescharacteristic
26C2×#95×C131C2211yescharacteristic
27C2×#95×C131C2211yescharacteristic
28C2×#95×C131C2211yescharacteristic
29C2×#95×C131C2212yes—
30C2×#95×C131C2212yes—
31C2×#95×C131C2212yes—
32C2×#95×C131C2211yescharacteristic
33C2×#95×C131C2211yescharacteristic
34C2×#95×C131C2212yes—
35C2×#95×C131C2212yes—
36C2×#95×C131C2212yes—
37C2×#95×C131C2212yes—
38C2×#95×C131C2212yes—
39C25×C131C2311yescharacteristic
40C22×Q8×C131—24no—
41C22×Q8×C131—22no—
42C22×Q8×C131—22no—
43C22×Q8×C131—22no—
44C22×Q8×C131—42no—
45C23×C4×C131—24no—
46C23×C4×C131—22no—
47C23×C4×C131—22no—
48C23×C4×C131—42no—
49C23×C4×C131D414yes—
50C23×C4×C131C2311yescharacteristic
51C23×C4×C131C2311yescharacteristic
52C23×C4×C131C2311yescharacteristic
53C23×C4×C131C2312yes—
54C23×C4×C131C2311yescharacteristic
55C23×C4×C131C2312yes—
56C23×C4×C131C2312yes—
57C23×C4×C131C2311yescharacteristic
58C23×C4×C131C2312yes—
59C23×C4×C131C2311yescharacteristic
60C22⋊Q8×C131—232no—
61C2×C4⋊C4×C131—24no—
62C2×C4⋊C4×C131—22no—
63C2×C4⋊C4×C131—24no—
64C2×C4⋊C4×C131—24no—
65C2×C4⋊C4×C131—24no—
66C2×C4⋊C4×C131—24no—
67C2×C4⋊C4×C131—24no—
68C2×C4⋊C4×C131—42no—
69C2×C22⋊C4×C131—24no—
70C2×C22⋊C4×C131—24no—
71C2×C22⋊C4×C131—24no—
72C2×C22⋊C4×C131—24no—
73C2×C22⋊C4×C131—22no—
74C2×C22⋊C4×C131—22no—
75C2×C22⋊C4×C131—24no—
76C2×C22⋊C4×C131—42no—
77C2×C22⋊C4×C131—42no—
78C2×C22⋊C4×C131D418yes—
79#95×C131—24no—
80#95×C131—24no—
81#95×C131—24no—
82#95×C131—22no—
83#95×C131—22no—
84#95×C131—22no—
85#95×C131—22no—
86#95×C131—22no—
87#95×C131—24no—
88#95×C131—22no—
89#95×C131—42no—
90#95×C131—42no—
91#95×C131—42no—
92#95×C131—42no—
93C24×C131—22no—
94C24×C131—21no—
95C24×C131—21no—
96C24×C131—21no—
97C24×C131—22no—
98C24×C131—21no—
99C24×C131—21no—
100C24×C131—22no—
101C24×C131—21no—
102C24×C131C2×D412yes—
103C24×C131C2×D414yes—
104C24×C131C2411yescharacteristic
105C2×Q8×C131—28no—
106C2×Q8×C131—48no—
107C2×Q8×C131—48no—
108C2×Q8×C131—44no—
109C2×Q8×C131—44no—
110C22×C4×C131—22no—
111C22×C4×C131—22no—
112C22×C4×C131—28no—
113C22×C4×C131—22no—
114C22×C4×C131—22no—
115C22×C4×C131—22no—
116C22×C4×C131—22no—
117C22×C4×C131—22no—
118C22×C4×C131—22no—
119C22×C4×C131—22no—
120C22×C4×C131—22no—
121C22×C4×C131—22no—
122C22×C4×C131—21no—
123C22×C4×C131—21no—
124C22×C4×C131—22no—
125C22×C4×C131—22no—
126C22×C4×C131—22no—
127C22×C4×C131—22no—
128C22×C4×C131—22no—
129C22×C4×C131—22no—
130C22×C4×C131—22no—
131C22×C4×C131—22no—
132C22×C4×C131—22no—
133C22×C4×C131—22no—
134C22×C4×C131—22no—
135C22×C4×C131—28no—
136C22×C4×C131—21no—
137C22×C4×C131—21no—
138C22×C4×C131—22no—
139C22×C4×C131—21no—
140C22×C4×C131—21no—
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—21no—
148C22×C4×C131—22no—
149C22×C4×C131—22no—
150C22×C4×C131—22no—
151C22×C4×C131—22no—
152C22×C4×C131—21no—
153C22×C4×C131—21no—
154C22×C4×C131—21no—
155C22×C4×C131—22no—
156C22×C4×C131—21no—
157C22×C4×C131—21no—
158C22×C4×C131—22no—
159C22×C4×C131—22no—
160C22×C4×C131—22no—
161C22×C4×C131—21no—
162C22×C4×C131—21no—
163C22×C4×C131—44no—
164C22×C4×C131—48no—
165C22×C4×C131—48no—
166C22×C4×C131C2×D414yes—
167C22×C4×C131C2×D412yes—
168C22×C4×C131C2×D414yes—
169C22×C4×C131C2×D412yes—
170C22×C4×C131#4114yes—
171C22×C4×C131#4112yes—
172C22×C4×C131#4112yes—
173C4⋊C4×C131—28no—
174C4⋊C4×C131—216no—
175C4⋊C4×C131—48no—
176C22⋊C4×C131—216no—
177C22⋊C4×C131—48no—
178C22⋊C4×C131—44no—
179C22⋊C4×C131—44no—
180C23×C131—22no—
181C23×C131—22no—
182C23×C131—22no—
183C23×C131—22no—
184C23×C131—24no—
185C23×C131—24no—
186C23×C131—22no—
187C23×C131—22no—
188C23×C131—24no—
189C23×C131—24no—
190C23×C131—24no—
191C23×C131—24no—
192C23×C131—42no—
193C23×C131—41no—
194C23×C131—41no—
195C23×C131—41no—
196C23×C131—41no—
197C23×C131—42no—
198C23×C131—42no—
199C23×C131—42no—
200C23×C131—42no—
201C23×C131—42no—
202C23×C131#12014yes—
203C23×C131C22×D411yescharacteristic
204C23×C131C22×D412yes—
205C23×C131C2×#4111yescharacteristic
206C23×C131C2×#4111yescharacteristic
207C23×C131C2×#4112yes—
208C23×C131H(5,𝔽2)12yes—
209C23×C131H(5,𝔽2)11yescharacteristic
210C23×C1312−1+411yescharacteristic
211C23×C1312−1+411yescharacteristic
212C23×C1312−1+412yes—
213C23×C1312−1+411yescharacteristic
214Q8×C131—416no—
215C2×C4×C131—22no—
216C2×C4×C131—22no—
217C2×C4×C131—24no—
218C2×C4×C131—24no—
219C2×C4×C131—24no—
220C2×C4×C131—24no—
221C2×C4×C131—22no—
222C2×C4×C131—22no—
223C2×C4×C131—22no—
224C2×C4×C131—24no—
225C2×C4×C131—24no—
226C2×C4×C131—24no—
227C2×C4×C131—24no—
228C2×C4×C131—24no—
229C2×C4×C131—24no—
230C2×C4×C131—24no—
231C2×C4×C131—416no—
232C2×C4×C131—42no—
233C2×C4×C131—42no—
234C2×C4×C131—42no—
235C2×C4×C131—42no—
236C2×C4×C131—42no—
237C2×C4×C131—42no—
238C2×C4×C131—42no—
239C2×C4×C131—42no—
240C2×C4×C131—42no—
241C2×C4×C131—42no—
242C2×C4×C131—42no—
243C2×C4×C131—42no—
244C2×C4×C131—42no—
245C2×C4×C131—42no—
246C2×C4×C131—41no—
247C2×C4×C131—41no—
248C2×C4×C131—41no—
249C2×C4×C131—41no—
250C2×C4×C131—41no—
251C2×C4×C131—41no—
252C2×C4×C131—41no—
253C2×C4×C131—42no—
254C2×C4×C131—42no—
255C2×C4×C131—41no—
256C2×C4×C131—41no—
257C2×C4×C131—42no—
258C2×C4×C131—41no—
259C2×C4×C131—41no—
260C2×C4×C131—41no—
261C2×C4×C131—42no—
262C2×C4×C131—42no—
263C2×C4×C131—42no—
264C2×C4×C131—42no—
265C2×C4×C131—42no—
266C2×C4×C131—42no—
267C2×C4×C131—42no—
268C2×C4×C131—42no—
269C2×C4×C131—42no—
270C2×C4×C131—42no—
271C2×C4×C131—42no—
272C2×C4×C131—42no—
273C2×C4×C131#12314yes—
274C22×C131—24no—
275C22×C131—24no—
276C22×C131—24no—
277C22×C131—42no—
278C22×C131—42no—
279C22×C131—44no—
280C22×C131—44no—
281C22×C131—44no—
282C22×C131—44no—
283C22×C131—42no—
284C22×C131—42no—
285C22×C131C2×#12011yescharacteristic
286C22×C131C2×#12311yescharacteristic
287C22×C131#53211yescharacteristic
288C22×C131#53411yescharacteristic
289C22×C131#53611yescharacteristic
290C22×C131#53612yes—
291C22×C131#53612yes—
292C22×C131#53611yescharacteristic
293C22×C131#53711yescharacteristic
294C22×C131#53912yes—
295C22×C131#54111yescharacteristic
296C22×C131#54111yescharacteristic
297C22×C131#54211yescharacteristic
298C22×C131#54211yescharacteristic
299C22×C131#54212yes—
300C22×C131#54612yes—
301C22×C131#54612yes—
302C22×C131#54612yes—
303C22×C131#54712yes—
304C22×C131#55212yes—
305C22×C131#55612yes—
306C22×C131#56212yes—
307C22×C131#56312yes—
308C4×C131—24no—
309C4×C131—42no—
310C4×C131—44no—
311C4×C131—44no—
312C4×C131—44no—
313C4×C131—81no—
314C4×C131—81no—
315C4×C131—81no—
316C4×C131—82no—
317C4×C131—81no—
318C2×C131—44no—
319C2×C131#240511yescharacteristic
320C2×C131#243011yescharacteristic
321C2×C131#249911yescharacteristic
322C2×C131#261411yescharacteristic
323C2×C131#262511yescharacteristic
324C2×C131#266311yescharacteristic
325C2×C131#268712yes—
326C2×C131#268912yes—
327C2×C131#269812yes—
328C2×C131#270012yes—
329C2×C131#281411yescharacteristic
330#16364C13111yesmaximal, characteristic, direct product factor, Sylow
331C131#1636411yescharacteristic, direct product factor, Sylow
332C22×C22⋊Q8C2×C13111yescharacteristic
333C2×#397C2×C13112yes—
334C2×#381C2×C13111yescharacteristic
335#1446C2×C13111yescharacteristic, complemented
336#1446C2×C13111yescharacteristic, complemented
337#1444C2×C13111yescharacteristic, complemented
338#1442C2×C13112yescomplemented
339#1442C2×C13111yescharacteristic, complemented
340#1441C2×C13112yescomplemented
341#1440C2×C13112yes—
342C22⋊#95C2×C13111yescharacteristic
343C23×Q8C22×C13111yescharacteristic
344C24×C4C22×C13111yescharacteristic
345C2×C22⋊Q8—24no—
346C2×C22⋊Q8—28no—
347C22×C4⋊C4C22×C13112yes—
348C22×C4⋊C4C22×C13111yescharacteristic
349C22×C4⋊C4C22×C13112yes—
350C22×C22⋊C4C22×C13111yescharacteristic
351C22×C22⋊C4C22×C13112yes—
352C22×C22⋊C4C22×C13112yes—
353C22×C22⋊C4C22×C13111yescharacteristic
354#397—28no—
355#381—24no—
356C2×#95C22×C13112yes—
357C2×#95C22×C13112yes—
358C2×#95C22×C13112yes—
359C2×#95C22×C13111yescharacteristic
360C2×#95C22×C13111yescharacteristic
361C2×#95C22×C13112yes—
362C2×#95C22×C13111yescharacteristic
363C2×#95C22×C13111yescharacteristic
364C2×#95C22×C13111yescharacteristic
365C2×#95C22×C13111yescharacteristic
366C2×#95C22×C13112yes—
367C2×#95C22×C13112yes—
368C2×#95C22×C13112yes—
369C2×#95C22×C13112yes—
370C25C23×C13111yescharacteristic
371C22×Q8—24no—
372C22×Q8—22no—
373C22×Q8—22no—
374C22×Q8—22no—
375C22×Q8—42no—
376C23×C4—24no—
377C23×C4—22no—
378C23×C4—22no—
379C23×C4—42no—
380C23×C4D4×C13114yes—
381C23×C4C23×C13111yescharacteristic
382C23×C4C23×C13111yescharacteristic
383C23×C4C23×C13111yescharacteristic
384C23×C4C23×C13112yes—
385C23×C4C23×C13111yescharacteristic
386C23×C4C23×C13112yes—
387C23×C4C23×C13112yes—
388C23×C4C23×C13111yescharacteristic
389C23×C4C23×C13112yes—
390C23×C4C23×C13111yescharacteristic
391C22⋊Q8—232no—
392C2×C4⋊C4—24no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—24no—
395C2×C4⋊C4—24no—
396C2×C4⋊C4—24no—
397C2×C4⋊C4—24no—
398C2×C4⋊C4—24no—
399C2×C4⋊C4—42no—
400C2×C22⋊C4—24no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—24no—
406C2×C22⋊C4—24no—
407C2×C22⋊C4—42no—
408C2×C22⋊C4—42no—
409C2×C22⋊C4D4×C13118yes—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—24no—
415#95—24no—
416#95—24no—
417#95—22no—
418#95—24no—
419#95—22no—
420#95—42no—
421#95—42no—
422#95—42no—
423#95—42no—
424C24—22no—
425C24—21no—
426C24—21no—
427C24—21no—
428C24—22no—
429C24—21no—
430C24—21no—
431C24—22no—
432C24—21no—
433C24C2×D4×C13112yes—
434C24C2×D4×C13114yes—
435C24C24×C13111yescharacteristic
436C2×Q8—28no—
437C2×Q8—48no—
438C2×Q8—48no—
439C2×Q8—44no—
440C2×Q8—44no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—28no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—28no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—44no—
495C22×C4—48no—
496C22×C4—48no—
497C22×C4C2×D4×C13114yes—
498C22×C4C2×D4×C13112yes—
499C22×C4C2×D4×C13114yes—
500C22×C4C2×D4×C13112yes—
501C22×C4#41×C13114yes—
502C22×C4#41×C13112yes—
503C22×C4#41×C13112yes—
504C4⋊C4—28no—
505C4⋊C4—216no—
506C4⋊C4—48no—
507C22⋊C4—216no—
508C22⋊C4—48no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—24no—
517C23—24no—
518C23—22no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—42no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23#120×C13114yes—
534C23C22×D4×C13111yescharacteristic
535C23C22×D4×C13112yes—
536C23C2×#41×C13111yescharacteristic
537C23C2×#41×C13111yescharacteristic
538C23C2×#41×C13112yes—
539C23H(5,𝔽2)×C13112yes—
540C23H(5,𝔽2)×C13111yescharacteristic
541C232−1+4×C13111yescharacteristic
542C232−1+4×C13111yescharacteristic
543C232−1+4×C13112yes—
544C232−1+4×C13111yescharacteristic
545Q8—416no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—416no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4#123×C13114yes—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22C2×#120×C13111yescharacteristic
617C22C2×#123×C13111yescharacteristic
618C22#532×C13111yescharacteristic
619C22#534×C13111yescharacteristic
620C22#536×C13111yescharacteristic
621C22#536×C13111yescharacteristic
622C22#536×C13112yes—
623C22#536×C13112yes—
624C22#537×C13111yescharacteristic
625C22#539×C13112yes—
626C22#541×C13111yescharacteristic
627C22#541×C13111yescharacteristic
628C22#542×C13111yescharacteristic
629C22#542×C13111yescharacteristic
630C22#542×C13112yes—
631C22#546×C13112yes—
632C22#546×C13112yes—
633C22#546×C13112yes—
634C22#547×C13112yes—
635C22#552×C13112yes—
636C22#556×C13112yes—
637C22#562×C13112yes—
638C22#563×C13112yes—
639C4—24no—
640C4—42no—
641C4—44no—
642C4—44no—
643C4—44no—
644C4—81no—
645C4—81no—
646C4—81no—
647C4—81no—
648C4—82no—
649C2—44no—
650C2#2405×C13111yescharacteristic
651C2#2430×C13111yescharacteristic
652C2#2499×C13111yescharacteristic
653C2#2614×C13111yescharacteristic
654C2#2625×C13111yescharacteristic
655C2#2663×C13111yescharacteristic
656C2#2687×C13112yes—
657C2#2689×C13112yes—
658C2#2698×C13112yes—
659C2#2700×C13112yes—
660C2#2814×C13111yescharacteristic
subgroupG/H
center, socleC24×C131C24
commutator, Frattini, cosocleC24C24×C131
prime radicalC25×C131C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C131 ◁ G

Complements

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

NH ≅ G/Ntype
4. #1446×C131C2semidirect
5. #1446×C131C2semidirect
6. #1444×C131C2semidirect
7. #1442×C131C2semidirect
8. #1442×C131C2semidirect
9. #1441×C131C2semidirect
330. #16364C131direct (N × H)
331. C131#16364direct (N × H)
335. #1446C2×C131semidirect
336. #1446C2×C131semidirect
337. #1444C2×C131semidirect
338. #1442C2×C131semidirect
339. #1442C2×C131semidirect
340. #1441C2×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×#397×C1312
C2×#381×C1312
#1446×C1312
#1444×C1312
#1442×C1312
#1441×C1312
#1440×C1312
C22⋊#95×C1312
#16364131