Finite Group Databasebeta

← C31×#23950 #21480709

Subgroups — C31×#23950 #21480709

Order 7936.

1242 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3493C211yesmaximal, characteristic, complemented
2C31×#3005C212yesmaximal, complemented
3C31×#2297C211yesmaximal, characteristic, complemented
4C31×#2028C211yesmaximal, characteristic, complemented
5C31×#2027C211yesmaximal, characteristic, complemented
6C31×#2007C211yesmaximal, characteristic, complemented
7C31×#2006C211yesmaximal, characteristic, complemented
8C31×#1896C212yesmaximal, complemented
9C31×#1883C212yesmaximal, complemented
10C31×#1882C212yesmaximal, complemented
11C31×#1822C211yesmaximal, characteristic, complemented
12C22×#41×C31C2211yescharacteristic
13C2×C31×#137C2212yes—
14C2×C31×AGL(1,ℤ/8)C2212yes—
15C31×#556—22no—
16C31×#555—22no—
17C31×#553—22no—
18C31×#552—22no—
19C31×#544C2211yescharacteristic
20C31×#538—22no—
21C31×#537C2211yescharacteristic
22C31×#529—22no—
23C31×#525C2212yes—
24C31×#520C2212yes—
25C2×C31×#117C2211yescharacteristic
26C22×C42×C31C2211yescharacteristic
27C31×#452—22no—
28C31×#452—22no—
29C31×#449—22no—
30C31×#449—22no—
31C2×C31×#104C2212yes—
32C2×C31×#104C2212yes—
33C31×#419C2212yes—
34C31×#418C2212yes—
35C31×#407C2212yes—
36C31×#391C2211yescharacteristic
37C31×#391C2211yescharacteristic
38C31×#390C2211yescharacteristic
39C31×#390C2211yescharacteristic
40C31×#386—22no—
41C4×C22⋊C4×C31—22no—
42C31×#343C2212yes—
43C31×#343C2211yescharacteristic
44C31×#343C2211yescharacteristic
45C31×#339C2212yes—
46C31×#339C2212yes—
47C2×#41×C31—22no—
48C2×#41×C31—22no—
49C2×#41×C31—22no—
50C2×#41×C31—22no—
51C2×#41×C31—42no—
52C2×#41×C31C2312yes—
53C22×Q8×C31C2311yescharacteristic
54C22×D4×C31—21no—
55C22×D4×C31C2311yescharacteristic
56C23×C4×C31—21no—
57C23×C4×C31—21no—
58C23×C4×C31C2311yescharacteristic
59C31×#137—24no—
60C31×#137—42no—
61C31×AGL(1,ℤ/8)—24no—
62C31×AGL(1,ℤ/8)—42no—
63C2×Q16×C31—22no—
64C2×SD16×C31—22no—
65C2×SD16×C31—22no—
66C2×D8×C31—22no—
67C2×M16×C31C2312yes—
68C2×M16×C31C2312yes—
69C31×C4⋊Q8—22no—
70C31×#126—24no—
71C31×#126—24no—
72C31×#125—22no—
73C31×#125—22no—
74C31×#124—22no—
75C31×#124—24no—
76C31×#123—24no—
77C31×#123—22no—
78C31×C22⋊Q8—22no—
79C31×C22⋊Q8—22no—
80C31×C22⋊Q8D412yes—
81C31×C22⋊Q8D412yes—
82C31×#121—24no—
83C31×#121—22no—
84C31×#121—22no—
85C31×#121D412yes—
86C31×#121D412yes—
87C31×#120—22no—
88C4×Q8×C31—22no—
89C4×Q8×C31—22no—
90C4×D4×C31—22no—
91C4×D4×C31—22no—
92C4×D4×C31—24no—
93C4×D4×C31—22no—
94C4×D4×C31—22no—
95C31×#117—24no—
96C31×#117—22no—
97C31×#117—22no—
98C31×#117—22no—
99C31×#117C2312yes—
100C2×C4⋊C4×C31—22no—
101C2×C4⋊C4×C31C2311yescharacteristic
102C2×C4⋊C4×C31C2311yescharacteristic
103C2×C22⋊C4×C31—22no—
104C2×C22⋊C4×C31—22no—
105C2×C22⋊C4×C31D412yes—
106C2×C42×C31—21no—
107C2×C42×C31—42no—
108C2×C42×C31D412yes—
109C2×C42×C31C2311yescharacteristic
110C2×C42×C31C2311yescharacteristic
111C31×#104—42no—
112C31×#104—42no—
113C31×#103—22no—
114C31×#103—22no—
115C31×#102—22no—
116C31×#102—22no—
117C31×C22⋊C8—22no—
118C31×C22⋊C8—22no—
119C31×#95—22no—
120C31×#95—22no—
121C24×C31—21no—
122C24×C31C2×D411yescharacteristic
123#41×C31—22no—
124#41×C31—22no—
125#41×C31—22no—
126#41×C31—22no—
127#41×C31—44no—
128#41×C31—44no—
129#41×C31—44no—
130#41×C31—44no—
131C2×Q8×C31—22no—
132C2×Q8×C31—21no—
133C2×Q8×C31—22no—
134C2×Q8×C31—22no—
135C2×Q8×C31C2×D412yes—
136C2×Q8×C31#4112yes—
137C2×D4×C31—24no—
138C2×D4×C31—22no—
139C2×D4×C31—22no—
140C2×D4×C31—22no—
141C2×D4×C31—22no—
142C2×D4×C31—22no—
143C2×D4×C31—21no—
144C2×D4×C31—22no—
145C2×D4×C31—42no—
146C2×D4×C31—42no—
147C2×D4×C31C2×D412yes—
148C2×D4×C31#4112yes—
149C22×C4×C31—21no—
150C22×C4×C31—21no—
151C22×C4×C31—22no—
152C22×C4×C31—24no—
153C22×C4×C31—21no—
154C22×C4×C31—21no—
155C22×C4×C31—21no—
156C22×C4×C31—24no—
157C22×C4×C31—22no—
158C22×C4×C31—42no—
159C22×C4×C31—42no—
160C22×C4×C31—42no—
161C22×C4×C31—42no—
162C22×C4×C31—41no—
163C22×C4×C31—41no—
164C22×C4×C31C2×D411yescharacteristic
165C22×C4×C31C2×D412yes—
166C22×C4×C31C2×D412yes—
167C22×C4×C31C2×D412yes—
168C22×C4×C31C2×D412yes—
169C22×C4×C31C2411yescharacteristic
170Q16×C31—44no—
171SD16×C31—44no—
172SD16×C31—44no—
173D8×C31—44no—
174M16×C31—22no—
175M16×C31—22no—
176M16×C31—22no—
177M16×C31—22no—
178C2×C8×C31—22no—
179C2×C8×C31—22no—
180C4⋊C4×C31—24no—
181C4⋊C4×C31—24no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31C2×D412yes—
189C4⋊C4×C31C2×D412yes—
190C22⋊C4×C31—24no—
191C22⋊C4×C31—24no—
192C22⋊C4×C31—24no—
193C22⋊C4×C31—22no—
194C22⋊C4×C31—22no—
195C22⋊C4×C31—22no—
196C22⋊C4×C31—22no—
197C22⋊C4×C31—22no—
198C22⋊C4×C31—44no—
199C22⋊C4×C31—44no—
200C42×C31—24no—
201C42×C31—22no—
202C42×C31—22no—
203C42×C31—22no—
204C42×C31—22no—
205C42×C31—22no—
206C42×C31—42no—
207#23950C3111yesmaximal, characteristic, direct product factor, Sylow
208C23×C31—21no—
209C23×C31—21no—
210C23×C31—22no—
211C23×C31—22no—
212C23×C31—41no—
213C23×C31—41no—
214C23×C31—42no—
215C23×C31—42no—
216C23×C31—41no—
217C23×C31—41no—
218C23×C31#12012yes—
219C23×C31C22×D411yescharacteristic
220Q8×C31—22no—
221Q8×C31—24no—
222Q8×C31—22no—
223Q8×C31—42no—
224Q8×C31—82no—
225D4×C31—22no—
226D4×C31—24no—
227D4×C31—22no—
228D4×C31—48no—
229D4×C31—42no—
230D4×C31—42no—
231D4×C31—42no—
232D4×C31—82no—
233D4×C31—82no—
234D4×C31—82no—
235C2×C4×C31—21no—
236C2×C4×C31—21no—
237C2×C4×C31—22no—
238C2×C4×C31—22no—
239C2×C4×C31—22no—
240C2×C4×C31—22no—
241C2×C4×C31—22no—
242C2×C4×C31—22no—
243C2×C4×C31—22no—
244C2×C4×C31—22no—
245C2×C4×C31—22no—
246C2×C4×C31—22no—
247C2×C4×C31—22no—
248C2×C4×C31—22no—
249C2×C4×C31—44no—
250C2×C4×C31—44no—
251C2×C4×C31—41no—
252C2×C4×C31—41no—
253C2×C4×C31—48no—
254C2×C4×C31—41no—
255C2×C4×C31—41no—
256C2×C4×C31—41no—
257C2×C4×C31—41no—
258C2×C4×C31—42no—
259C2×C4×C31—42no—
260C2×C4×C31—42no—
261C2×C4×C31—42no—
262C2×C4×C31—84no—
263C2×C4×C31#12011yescharacteristic
264C2×C4×C31#12011yescharacteristic
265C2×C4×C31C22×D412yes—
266C2×C4×C31C2×#4111yescharacteristic
267C2×C4×C31C2×#4111yescharacteristic
268C2×C4×C31H(5,𝔽2)12yes—
269C8×C31—42no—
270C8×C31—42no—
271#3493C2×C3111yescharacteristic, complemented
272#3005C2×C3112yescomplemented
273#2297C2×C3111yescharacteristic, complemented
274#2028C2×C3111yescharacteristic, complemented
275#2027C2×C3111yescharacteristic, complemented
276#2007C2×C3111yescharacteristic, complemented
277#2006C2×C3111yescharacteristic, complemented
278#1896C2×C3112yescomplemented
279#1883C2×C3112yescomplemented
280#1882C2×C3112yescomplemented
281#1822C2×C3111yescharacteristic, complemented
282C22×C31—21no—
283C22×C31—21no—
284C22×C31—22no—
285C22×C31—22no—
286C22×C31—22no—
287C22×C31—42no—
288C22×C31—42no—
289C22×C31—44no—
290C22×C31—42no—
291C22×C31—42no—
292C22×C31—42no—
293C22×C31—42no—
294C22×C31—82no—
295C22×C31—81no—
296C22×C31—81no—
297C22×C31C2×#12011yescharacteristic
298C22×C31#52511yescharacteristic
299C22×C31#53511yescharacteristic
300C4×C31—21no—
301C4×C31—21no—
302C4×C31—42no—
303C4×C31—42no—
304C4×C31—42no—
305C4×C31—42no—
306C4×C31—42no—
307C4×C31—42no—
308C4×C31—42no—
309C4×C31—42no—
310C4×C31#54612yes—
311C4×C31#54612yes—
312C22×#41C22×C3111yescharacteristic
313C2×#137C22×C3112yes—
314C2×AGL(1,ℤ/8)C22×C3112yes—
315#556—22no—
316#555—22no—
317#553—22no—
318#552—22no—
319#544C22×C3111yescharacteristic
320#538—22no—
321#537C22×C3111yescharacteristic
322#529—22no—
323#525C22×C3112yes—
324#520C22×C3112yes—
325C2×#117C22×C3111yescharacteristic
326C22×C42C22×C3111yescharacteristic
327#452—22no—
328#452—22no—
329#449—22no—
330#449—22no—
331C2×#104C22×C3112yes—
332C2×#104C22×C3112yes—
333#419C22×C3112yes—
334#418C22×C3112yes—
335#407C22×C3112yes—
336#391C22×C3111yescharacteristic
337#391C22×C3111yescharacteristic
338#390C22×C3111yescharacteristic
339#390C22×C3111yescharacteristic
340#386—22no—
341C4×C22⋊C4—22no—
342#343C22×C3112yes—
343#343C22×C3111yescharacteristic
344#343C22×C3111yescharacteristic
345#339C22×C3112yes—
346#339C22×C3112yes—
347C2×C31—21no—
348C2×C31—21no—
349C2×C31—42no—
350C2×C31—82no—
351C2×C31—82no—
352C2×C31#240411yescharacteristic
353C2×C31#301811yescharacteristic
354C2×C31#301811yescharacteristic
355C2×#41—22no—
356C2×#41—22no—
357C2×#41—22no—
358C2×#41—22no—
359C2×#41—42no—
360C2×#41C23×C3112yes—
361C22×Q8C23×C3111yescharacteristic
362C22×D4—21no—
363C22×D4C23×C3111yescharacteristic
364C23×C4—21no—
365C23×C4—21no—
366C23×C4C23×C3111yescharacteristic
367#137—24no—
368#137—42no—
369AGL(1,ℤ/8)—24no—
370AGL(1,ℤ/8)—42no—
371C2×Q16—22no—
372C2×SD16—22no—
373C2×SD16—22no—
374C2×D8—22no—
375C2×M16C23×C3112yes—
376C2×M16C23×C3112yes—
377C4⋊Q8—22no—
378#126—24no—
379#126—24no—
380#125—22no—
381#125—22no—
382#124—22no—
383#124—24no—
384#123—24no—
385#123—22no—
386C22⋊Q8—22no—
387C22⋊Q8—22no—
388C22⋊Q8D4×C3112yes—
389C22⋊Q8D4×C3112yes—
390#121—24no—
391#121—22no—
392#121—22no—
393#121D4×C3112yes—
394#121D4×C3112yes—
395#120—22no—
396C4×Q8—22no—
397C4×Q8—22no—
398C4×D4—22no—
399C4×D4—22no—
400C4×D4—24no—
401C4×D4—22no—
402C4×D4—22no—
403#117—24no—
404#117—22no—
405#117—22no—
406#117—22no—
407#117C23×C3112yes—
408C2×C4⋊C4—22no—
409C2×C4⋊C4C23×C3111yescharacteristic
410C2×C4⋊C4C23×C3111yescharacteristic
411C2×C22⋊C4—22no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4D4×C3112yes—
414C2×C42—21no—
415C2×C42—42no—
416C2×C42D4×C3112yes—
417C2×C42C23×C3111yescharacteristic
418C2×C42C23×C3111yescharacteristic
419#104—42no—
420#104—42no—
421#103—22no—
422#103—22no—
423#102—22no—
424#102—22no—
425C22⋊C8—22no—
426C22⋊C8—22no—
427#95—22no—
428#95—22no—
429C31#2395011yescharacteristic, direct product factor, Sylow
430C24—21no—
431C24C2×D4×C3111yescharacteristic
432#41—22no—
433#41—22no—
434#41—22no—
435#41—22no—
436#41—44no—
437#41—44no—
438#41—44no—
439#41—44no—
440C2×Q8—22no—
441C2×Q8—21no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8C2×D4×C3112yes—
445C2×Q8#41×C3112yes—
446C2×D4—24no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—22no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4C2×D4×C3112yes—
457C2×D4#41×C3112yes—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4C2×D4×C3111yescharacteristic
474C22×C4C2×D4×C3112yes—
475C22×C4C2×D4×C3112yes—
476C22×C4C2×D4×C3112yes—
477C22×C4C2×D4×C3112yes—
478C22×C4C24×C3111yescharacteristic
479Q16—44no—
480SD16—44no—
481SD16—44no—
482D8—44no—
483M16—22no—
484M16—22no—
485M16—22no—
486M16—22no—
487C2×C8—22no—
488C2×C8—22no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4C2×D4×C3112yes—
498C4⋊C4C2×D4×C3112yes—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C42—24no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—42no—
516C23—21no—
517C23—21no—
518C23—22no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—41no—
525C23—41no—
526C23C31×#12012yes—
527C23C22×D4×C3111yescharacteristic
528Q8—22no—
529Q8—24no—
530Q8—22no—
531Q8—42no—
532Q8—82no—
533D4—22no—
534D4—24no—
535D4—22no—
536D4—48no—
537D4—42no—
538D4—42no—
539D4—42no—
540D4—82no—
541D4—82no—
542D4—82no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—48no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—84no—
571C2×C4C31×#12011yescharacteristic
572C2×C4C31×#12011yescharacteristic
573C2×C4C22×D4×C3112yes—
574C2×C4C2×#41×C3111yescharacteristic
575C2×C4C2×#41×C3111yescharacteristic
576C2×C4C31×H(5,𝔽2)12yes—
577C8—42no—
578C8—42no—
579C22—21no—
580C22—21no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—42no—
585C22—42no—
586C22—44no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—82no—
592C22—81no—
593C22—81no—
594C22C2×C31×#12011yescharacteristic
595C22C31×#52511yescharacteristic
596C22C31×#53511yescharacteristic
597C4—21no—
598C4—21no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—42no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4C31×#54612yes—
608C4C31×#54612yes—
609C2—21no—
610C2—21no—
611C2—42no—
612C2—82no—
613C2—82no—
614C2C31×#240411yescharacteristic
615C2C31×#301811yescharacteristic
616C2C31×#301811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3493C2semidirect
2. C31×#3005C2semidirect
3. C31×#2297C2semidirect
4. C31×#2028C2semidirect
5. C31×#2027C2semidirect
6. C31×#2007C2semidirect
7. C31×#2006C2semidirect
8. C31×#1896C2semidirect
9. C31×#1883C2semidirect
10. C31×#1882C2semidirect
11. C31×#1822C2semidirect
207. #23950C31direct (N × H)
271. #3493C2×C31semidirect
272. #3005C2×C31semidirect
273. #2297C2×C31semidirect
274. #2028C2×C31semidirect
275. #2027C2×C31semidirect
276. #2007C2×C31semidirect
277. #2006C2×C31semidirect
278. #1896C2×C31semidirect
279. #1883C2×C31semidirect
280. #1882C2×C31semidirect
281. #1822C2×C31semidirect
429. C31#23950direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34932
C31×#30052
C31×#22972
C31×#20282
C31×#20272
C31×#20072
C31×#20062
C31×#18962
C31×#18832
C31×#18822
C31×#18222
#2395031