Finite Group Databasebeta

← C131×#23494 #45709378

Subgroups — C131×#23494 #45709378

Order 33536.

1362 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#536×C131C211yesmaximal, characteristic
2#2911×C131C211yesmaximal, characteristic, complemented
3C2×#421×C131C211yesmaximal, characteristic
4#2896×C131C211yesmaximal, characteristic
5C22×#101×C131C211yesmaximal, characteristic, complemented
6#2310×C131C211yesmaximal, characteristic, complemented
7#2038×C131C218yesmaximal, complemented
8#1733×C131C211yesmaximal, characteristic, complemented
9C22×#41×C131C2211yescharacteristic
10C23×Q8×C131C2211yescharacteristic
11C22×M16×C131C2211yescharacteristic
12C22×M16×C131C2211yescharacteristic
13#536×C131—22no—
14#536×C131—22no—
15#536×C131—22no—
16#536×C131C2214yes—
17C2×C4⋊Q8×C131—21no—
18C2×#124×C131—21no—
19C2×#123×C131—22no—
20C2×C22⋊Q8×C131—22no—
21C2×#117×C131C2211yescharacteristic
22C2×#117×C131C2211yescharacteristic
23C2×#117×C131C2211yescharacteristic
24#428×C131—22no—
25#428×C131C2214yescomplemented
26#427×C131—22no—
27#421×C131—24no—
28C2×#104×C131C2214yes—
29#419×C131—21no—
30#418×C131—21no—
31#417×C131C2214yes—
32#416×C131—22no—
33C2×#101×C131—24no—
34C2×#101×C131C2214yescomplemented
35#391×C131—22no—
36#391×C131C2214yescomplemented
37#381×C131—22no—
38#339×C131C2214yescomplemented
39C2×#41×C131—21no—
40C2×#41×C131—21no—
41C2×#41×C131—22no—
42C2×#41×C131—22no—
43C2×#41×C131—22no—
44C2×#41×C131—22no—
45C2×#41×C131D412yes—
46C2×#41×C131C2312yes—
47C22×Q8×C131—22no—
48C22×Q8×C131—21no—
49C22×Q8×C131—21no—
50C22×Q8×C131—21no—
51C22×Q8×C131—21no—
52C22×Q8×C131—22no—
53C22×Q8×C131—22no—
54C22×Q8×C131—22no—
55C22×Q8×C131D411yescharacteristic
56C22×Q8×C131D412yes—
57C22×Q8×C131C2312yes—
58C22×D4×C131—21no—
59C22×D4×C131D411yescharacteristic
60C23×C4×C131—21no—
61C23×C4×C131D412yes—
62C23×C4×C131C2311yescharacteristic
63C2×M16×C131—21no—
64C2×M16×C131—22no—
65C2×M16×C131—21no—
66C2×M16×C131—22no—
67C2×M16×C131—22no—
68C2×M16×C131D414yes—
69C2×M16×C131C2312yes—
70C2×M16×C131C2312yes—
71C22×C8×C131—21no—
72C22×C8×C131—21no—
73C4⋊Q8×C131—24no—
74C4⋊Q8×C131—24no—
75#124×C131—24no—
76#124×C131—24no—
77#123×C131—28no—
78#123×C131—28no—
79C22⋊Q8×C131—28no—
80C22⋊Q8×C131—28no—
81#117×C131—21no—
82#117×C131—21no—
83#117×C131—22no—
84#117×C131—22no—
85#117×C131—21no—
86#117×C131—21no—
87#117×C131—21no—
88#117×C131—21no—
89#117×C131D412yes—
90#117×C131C2312yes—
91C2×C4⋊C4×C131—24no—
92C2×C4⋊C4×C131—21no—
93C2×C4⋊C4×C131D412yes—
94C2×C4⋊C4×C131C2312yes—
95C2×C22⋊C4×C131—24no—
96C2×C22⋊C4×C131—21no—
97C2×C22⋊C4×C131—21no—
98C2×C22⋊C4×C131—21no—
99C2×C42×C131—21no—
100C2×C42×C131—21no—
101C2×C42×C131C2312yes—
102#107×C131—24no—
103#106×C131—24no—
104#104×C131—28no—
105#103×C131—24no—
106#102×C131—24no—
107#101×C131—28no—
108#95×C131—28no—
109C24×C131—21no—
110C24×C131C2×D411yescharacteristic
111#41×C131—24no—
112#41×C131—24no—
113#41×C131—22no—
114#41×C131—22no—
115#41×C131—22no—
116#41×C131—22no—
117#41×C131—42no—
118#41×C131—42no—
119#41×C131—42no—
120#41×C131—42no—
121C2×Q8×C131—22no—
122C2×Q8×C131—22no—
123C2×Q8×C131—21no—
124C2×Q8×C131—24no—
125C2×Q8×C131—24no—
126C2×Q8×C131—22no—
127C2×Q8×C131—22no—
128C2×Q8×C131—22no—
129C2×Q8×C131—22no—
130C2×Q8×C131—22no—
131C2×Q8×C131—22no—
132C2×Q8×C131—22no—
133C2×Q8×C131—22no—
134C2×Q8×C131—42no—
135C2×Q8×C131—42no—
136C2×Q8×C131—42no—
137C2×Q8×C131—42no—
138C2×Q8×C131—42no—
139C2×Q8×C131—42no—
140C2×Q8×C131—42no—
141C2×Q8×C131—42no—
142C2×Q8×C131C2×D412yes—
143C2×D4×C131—22no—
144C2×D4×C131—22no—
145C2×D4×C131—21no—
146C2×D4×C131—22no—
147C2×D4×C131—22no—
148C2×D4×C131—42no—
149C2×D4×C131—42no—
150C2×D4×C131C2×D412yes—
151C22×C4×C131—22no—
152C22×C4×C131—22no—
153C22×C4×C131—21no—
154C22×C4×C131—22no—
155C22×C4×C131—24no—
156C22×C4×C131—24no—
157C22×C4×C131—22no—
158C22×C4×C131—22no—
159C22×C4×C131—22no—
160C22×C4×C131—22no—
161C22×C4×C131—42no—
162C22×C4×C131—42no—
163C22×C4×C131C2×D414yes—
164C22×C4×C131C2×D411yescharacteristic
165C22×C4×C131C2×D411yescharacteristic
166C22×C4×C131C2×D411yescharacteristic
167C22×C4×C131C2×D411yescharacteristic
168C22×C4×C131C2×D411yescharacteristic
169C22×C4×C131C2×D412yes—
170C22×C4×C131#4112yes—
171C22×C4×C131#4112yes—
172C22×C4×C131#4112yes—
173C22×C4×C131C2411yescharacteristic
174M16×C131—24no—
175M16×C131—22no—
176M16×C131—22no—
177M16×C131—22no—
178M16×C131—42no—
179M16×C131C2×D414yes—
180C2×C8×C131—24no—
181C2×C8×C131—22no—
182C2×C8×C131—22no—
183C2×C8×C131—42no—
184C4⋊C4×C131—216no—
185C4⋊C4×C131—22no—
186C4⋊C4×C131—22no—
187C4⋊C4×C131—22no—
188C4⋊C4×C131—42no—
189C4⋊C4×C131C2×D414yes—
190C22⋊C4×C131—216no—
191C22⋊C4×C131—22no—
192C22⋊C4×C131—22no—
193C22⋊C4×C131—41no—
194C22⋊C4×C131—41no—
195C22⋊C4×C131—42no—
196C42×C131—22no—
197C42×C131—22no—
198C42×C131—22no—
199C42×C131—22no—
200C42×C131—41no—
201C42×C131—41no—
202C23×C131—22no—
203C23×C131—42no—
204C23×C131—44no—
205C23×C131—42no—
206C23×C131#12011yescharacteristic
207C23×C131#12011yescharacteristic
208C23×C131#12112yes—
209C23×C131#12312yes—
210C23×C131C22×D411yescharacteristic
211Q8×C131—22no—
212Q8×C131—22no—
213Q8×C131—42no—
214Q8×C131—42no—
215Q8×C131—42no—
216Q8×C131—42no—
217Q8×C131—44no—
218Q8×C131—44no—
219Q8×C131—42no—
220D4×C131—22no—
221D4×C131—22no—
222D4×C131—42no—
223D4×C131—42no—
224D4×C131—44no—
225D4×C131—42no—
226C2×C4×C131—24no—
227C2×C4×C131—24no—
228C2×C4×C131—22no—
229C2×C4×C131—24no—
230C2×C4×C131—24no—
231C2×C4×C131—44no—
232C2×C4×C131—44no—
233C2×C4×C131—44no—
234C2×C4×C131—42no—
235C2×C4×C131—42no—
236C2×C4×C131—42no—
237C2×C4×C131—44no—
238C2×C4×C131—44no—
239C2×C4×C131—44no—
240C2×C4×C131—42no—
241C2×C4×C131—42no—
242C2×C4×C131#12011yescharacteristic
243C2×C4×C131#12011yescharacteristic
244C2×C4×C131#12112yes—
245C2×C4×C131#12112yes—
246C2×C4×C131#12112yes—
247C2×C4×C131#12112yes—
248C2×C4×C131#12112yes—
249C2×C4×C131#12312yes—
250C2×C4×C131#12312yes—
251C2×C4×C131#12312yes—
252C2×C4×C131#12412yes—
253C2×C4×C131#12412yes—
254C2×C4×C131C22×D411yescharacteristic
255C2×C4×C131C22×D411yescharacteristic
256C2×C4×C131C22×D411yescharacteristic
257C2×C4×C131C2×#4111yescharacteristic
258C2×C4×C131C2×#4111yescharacteristic
259C2×C4×C131C2×#4111yescharacteristic
260C8×C131—24no—
261C8×C131—42no—
262C22×C131—22no—
263C22×C131—22no—
264C22×C131—22no—
265C22×C131—42no—
266C22×C131—42no—
267C22×C131—42no—
268C22×C131—44no—
269C22×C131—84no—
270C22×C131#39412yes—
271C22×C131#39412yes—
272C22×C131C2×#12011yescharacteristic
273C22×C131C2×#12111yescharacteristic
274C22×C131C2×#12311yescharacteristic
275C4×C131—44no—
276C4×C131—42no—
277C4×C131—44no—
278C4×C131—82no—
279C4×C131—82no—
280C4×C131#39412yes—
281C4×C131#39412yes—
282C4×C131C2×#12111yescharacteristic
283C4×C131C2×#12111yescharacteristic
284C4×C131C2×#12311yescharacteristic
285C4×C131C2×#12411yescharacteristic
286C2×C131—21no—
287C2×C131—22no—
288C2×C131—21no—
289C2×C131—22no—
290C2×C131—82no—
291C2×C131C2×#39411yescharacteristic
292C2×C131#302012yes—
293#23494C13111yesmaximal, characteristic, direct product factor, Sylow
294C131#2349411yescharacteristic, direct product factor, Sylow
295C2×#536C2×C13111yescharacteristic
296#2911C2×C13111yescharacteristic, complemented
297C2×#421C2×C13111yescharacteristic
298#2896C2×C13111yescharacteristic
299C22×#101C2×C13111yescharacteristic, complemented
300#2310C2×C13111yescharacteristic, complemented
301#2038C2×C13118yescomplemented
302#1733C2×C13111yescharacteristic, complemented
303C22×#41C22×C13111yescharacteristic
304C23×Q8C22×C13111yescharacteristic
305C22×M16C22×C13111yescharacteristic
306C22×M16C22×C13111yescharacteristic
307#536—22no—
308#536—22no—
309#536—22no—
310#536C22×C13114yes—
311C2×C4⋊Q8—21no—
312C2×#124—21no—
313C2×#123—22no—
314C2×C22⋊Q8—22no—
315C2×#117C22×C13111yescharacteristic
316C2×#117C22×C13111yescharacteristic
317C2×#117C22×C13111yescharacteristic
318#428—22no—
319#428C22×C13114yescomplemented
320#427—22no—
321#421—24no—
322C2×#104C22×C13114yes—
323#419—21no—
324#418—21no—
325#417C22×C13114yes—
326#416—22no—
327C2×#101—24no—
328C2×#101C22×C13114yescomplemented
329#391—22no—
330#391C22×C13114yescomplemented
331#381—22no—
332#339C22×C13114yescomplemented
333C2×#41—21no—
334C2×#41—21no—
335C2×#41—22no—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41—22no—
339C2×#41D4×C13112yes—
340C2×#41C23×C13112yes—
341C22×Q8—21no—
342C22×Q8—21no—
343C22×Q8—21no—
344C22×Q8—21no—
345C22×Q8—22no—
346C22×Q8—22no—
347C22×Q8—22no—
348C22×Q8—22no—
349C22×Q8D4×C13111yescharacteristic
350C22×Q8D4×C13112yes—
351C22×Q8C23×C13112yes—
352C22×D4—21no—
353C22×D4D4×C13111yescharacteristic
354C23×C4—21no—
355C23×C4D4×C13112yes—
356C23×C4C23×C13111yescharacteristic
357C2×M16—21no—
358C2×M16—22no—
359C2×M16—21no—
360C2×M16—22no—
361C2×M16—22no—
362C2×M16D4×C13114yes—
363C2×M16C23×C13112yes—
364C2×M16C23×C13112yes—
365C22×C8—21no—
366C22×C8—21no—
367C4⋊Q8—24no—
368C4⋊Q8—24no—
369#124—24no—
370#124—24no—
371#123—28no—
372#123—28no—
373C22⋊Q8—28no—
374C22⋊Q8—28no—
375#117—21no—
376#117—21no—
377#117—22no—
378#117—22no—
379#117—21no—
380#117—21no—
381#117—21no—
382#117—21no—
383#117D4×C13112yes—
384#117C23×C13112yes—
385C2×C4⋊C4—24no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4D4×C13112yes—
388C2×C4⋊C4C23×C13112yes—
389C2×C22⋊C4—24no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42C23×C13112yes—
396#107—24no—
397#106—24no—
398#104—28no—
399#103—24no—
400#102—24no—
401#101—28no—
402#95—28no—
403C24—21no—
404C24C2×D4×C13111yescharacteristic
405#41—24no—
406#41—24no—
407#41—22no—
408#41—22no—
409#41—22no—
410#41—22no—
411#41—42no—
412#41—42no—
413#41—42no—
414#41—42no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—21no—
418C2×Q8—24no—
419C2×Q8—24no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8C2×D4×C13112yes—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—21no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4C2×D4×C13112yes—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4C2×D4×C13114yes—
458C22×C4C2×D4×C13111yescharacteristic
459C22×C4C2×D4×C13111yescharacteristic
460C22×C4C2×D4×C13111yescharacteristic
461C22×C4C2×D4×C13111yescharacteristic
462C22×C4C2×D4×C13111yescharacteristic
463C22×C4C2×D4×C13112yes—
464C22×C4#41×C13112yes—
465C22×C4#41×C13112yes—
466C22×C4#41×C13112yes—
467C22×C4C24×C13111yescharacteristic
468M16—24no—
469M16—22no—
470M16—22no—
471M16—22no—
472M16—42no—
473M16C2×D4×C13114yes—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—42no—
478C4⋊C4—216no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—42no—
483C4⋊C4C2×D4×C13114yes—
484C22⋊C4—216no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—42no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—41no—
495C42—41no—
496C23—22no—
497C23—42no—
498C23—44no—
499C23—42no—
500C23#120×C13111yescharacteristic
501C23#120×C13111yescharacteristic
502C23#121×C13112yes—
503C23#123×C13112yes—
504C23C22×D4×C13111yescharacteristic
505Q8—22no—
506Q8—22no—
507Q8—44no—
508Q8—44no—
509Q8—42no—
510Q8—42no—
511Q8—42no—
512Q8—42no—
513Q8—42no—
514D4—22no—
515D4—22no—
516D4—44no—
517D4—42no—
518D4—42no—
519D4—42no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4#120×C13111yescharacteristic
537C2×C4#120×C13111yescharacteristic
538C2×C4#121×C13112yes—
539C2×C4#121×C13112yes—
540C2×C4#121×C13112yes—
541C2×C4#121×C13112yes—
542C2×C4#121×C13112yes—
543C2×C4#123×C13112yes—
544C2×C4#123×C13112yes—
545C2×C4#123×C13112yes—
546C2×C4#124×C13112yes—
547C2×C4#124×C13112yes—
548C2×C4C22×D4×C13111yescharacteristic
549C2×C4C22×D4×C13111yescharacteristic
550C2×C4C22×D4×C13111yescharacteristic
551C2×C4C2×#41×C13111yescharacteristic
552C2×C4C2×#41×C13111yescharacteristic
553C2×C4C2×#41×C13111yescharacteristic
554C8—24no—
555C8—42no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—42no—
560C22—42no—
561C22—44no—
562C22—42no—
563C22—84no—
564C22#394×C13112yes—
565C22#394×C13112yes—
566C22C2×#120×C13111yescharacteristic
567C22C2×#121×C13111yescharacteristic
568C22C2×#123×C13111yescharacteristic
569C4—44no—
570C4—44no—
571C4—42no—
572C4—82no—
573C4—82no—
574C4#394×C13112yes—
575C4#394×C13112yes—
576C4C2×#121×C13111yescharacteristic
577C4C2×#121×C13111yescharacteristic
578C4C2×#123×C13111yescharacteristic
579C4C2×#124×C13111yescharacteristic
580C2—21no—
581C2—22no—
582C2—22no—
583C2—21no—
584C2—82no—
585C2C2×#394×C13111yescharacteristic
586C2#3020×C13112yes—
subgroupG/H
center, socleC22×C131C2×#120
commutator, Frattini, cosocleC22×C4C24×C131
prime radicalC22×#41×C131C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C131 ◁ C23×C4×C131 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2911×C131C2semidirect
5. C22×#101×C131C2semidirect
6. #2310×C131C2semidirect
7. #2038×C131C2semidirect
8. #1733×C131C2semidirect
25. #428×C131C22semidirect
34. C2×#101×C131C22semidirect
36. #391×C131C22semidirect
38. #339×C131C22semidirect
293. #23494C131direct (N × H)
294. C131#23494direct (N × H)
296. #2911C2×C131semidirect
299. C22×#101C2×C131semidirect
300. #2310C2×C131semidirect
301. #2038C2×C131semidirect
302. #1733C2×C131semidirect
319. #428C22×C131semidirect
328. C2×#101C22×C131semidirect
330. #391C22×C131semidirect
332. #339C22×C131semidirect

Maximal subgroups

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

subgroupindex
C2×#536×C1312
#2911×C1312
C2×#421×C1312
#2896×C1312
C22×#101×C1312
#2310×C1312
#2038×C1312
#1733×C1312
#23494131