Finite Group Databasebeta

← C31×#24058 #21480817

Subgroups — C31×#24058 #21480817

Order 7936.

1556 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×#3484C211yesmaximal, characteristic
2C31×#2622C212yesmaximal, complemented
3C31×#2308C211yesmaximal, characteristic, complemented
4C31×#2044C214yesmaximal, complemented
5C31×#1898C212yesmaximal, complemented
6C31×#1867C212yesmaximal, complemented
7C31×#1790C212yesmaximal, complemented
8C31×#1781C211yesmaximal, characteristic, complemented
9C2×C31×H(5,𝔽2)C2211yescharacteristic
10C23×D4×C31C2211yescharacteristic
11C31×#546—24no—
12D42×C31—24no—
13C31×#535—21no—
14C31×#534—21no—
15C31×#534C2212yes—
16C31×#525—21no—
17C2×C31×#120—21no—
18C2×C31×#120C2212yes—
19C31×#518C2211yescharacteristic
20C22×C22⋊C4×C31C2211yescharacteristic, complemented
21C22×C22⋊C4×C31C2211yescharacteristic, complemented
22C2×C31×#99C2212yes—
23C2×C31×#99C2214yescomplemented
24C2×C31×#99C2212yescomplemented
25C31×#397—24no—
26C31×#394—24no—
27C31×#386—21no—
28C31×#386—21no—
29C31×#386C2212yes—
30C31×#386C2212yescomplemented
31C31×#386C2214yescomplemented
32C31×#385C2214yescomplemented
33C31×#381—21no—
34C31×#379—21no—
35C31×#379C2212yescomplemented
36C31×#342C2214yescomplemented
37C25×C31D411yescharacteristic
38C25×C31C2311yescharacteristic
39C31×H(5,𝔽2)—24no—
40C31×H(5,𝔽2)—42no—
41C2×#41×C31—21no—
42C2×#41×C31—22no—
43C22×D4×C31—21no—
44C22×D4×C31—22no—
45C22×D4×C31—24no—
46C22×D4×C31—22no—
47C22×D4×C31—24no—
48C22×D4×C31—42no—
49C22×D4×C31D411yescharacteristic
50C22×D4×C31D411yescharacteristic
51C22×D4×C31D414yes—
52C22×D4×C31C2312yes—
53C22×D4×C31C2311yescharacteristic
54C22×D4×C31C2311yescharacteristic
55C23×C4×C31—21no—
56C23×C4×C31D411yescharacteristic
57C23×C4×C31C2312yes—
58C31×#127—22no—
59C31×#124—22no—
60C31×#123—22no—
61C31×#123—22no—
62C31×#123—22no—
63C31×C22⋊Q8—22no—
64C31×#121—22no—
65C31×#121—24no—
66C31×#121—24no—
67C31×#121—24no—
68C31×#120—24no—
69C31×#120—22no—
70C31×#120—22no—
71C31×#120—22no—
72C31×#120—22no—
73C31×#120—24no—
74C31×#120—22no—
75C4×D4×C31—24no—
76C4×D4×C31—24no—
77C31×#117—21no—
78C2×C4⋊C4×C31—24no—
79C2×C22⋊C4×C31—22no—
80C2×C22⋊C4×C31—21no—
81C2×C22⋊C4×C31—24no—
82C2×C22⋊C4×C31—21no—
83C2×C22⋊C4×C31—21no—
84C2×C22⋊C4×C31—21no—
85C2×C22⋊C4×C31—24no—
86C2×C22⋊C4×C31—22no—
87C2×C22⋊C4×C31—22no—
88C2×C22⋊C4×C31—24no—
89C2×C22⋊C4×C31—42no—
90C2×C22⋊C4×C31D414yes—
91C2×C22⋊C4×C31C2314yes—
92C2×C22⋊C4×C31C2312yes—
93C2×C22⋊C4×C31C2312yes—
94C31×#99—24no—
95C31×#99—42no—
96C31×#99—44no—
97C31×#95—24no—
98C31×#95—24no—
99C31×#95—22no—
100C31×#95—22no—
101C24×C31—22no—
102C24×C31—22no—
103C24×C31—21no—
104C24×C31—22no—
105C24×C31—22no—
106C24×C31—41no—
107C24×C31—82no—
108C24×C31—82no—
109C24×C31C2×D411yescharacteristic
110C24×C31C2×D412yes—
111C24×C31C2×D414yes—
112C24×C31C2×D411yescharacteristic
113C24×C31C2×D411yescharacteristic
114C24×C31C2×D411yescharacteristic
115C24×C31C2411yescharacteristic
116#41×C31—48no—
117#41×C31—42no—
118#41×C31—42no—
119C2×Q8×C31—21no—
120C2×D4×C31—22no—
121C2×D4×C31—22no—
122C2×D4×C31—22no—
123C2×D4×C31—21no—
124C2×D4×C31—24no—
125C2×D4×C31—22no—
126C2×D4×C31—22no—
127C2×D4×C31—22no—
128C2×D4×C31—22no—
129C2×D4×C31—22no—
130C2×D4×C31—22no—
131C2×D4×C31—22no—
132C2×D4×C31—24no—
133C2×D4×C31—22no—
134C2×D4×C31—24no—
135C2×D4×C31—24no—
136C2×D4×C31—24no—
137C2×D4×C31—22no—
138C2×D4×C31—41no—
139C2×D4×C31—41no—
140C2×D4×C31—416no—
141C2×D4×C31—48no—
142C2×D4×C31—42no—
143C2×D4×C31—42no—
144C2×D4×C31—84no—
145C2×D4×C31—84no—
146C2×D4×C31#4114yes—
147C22×C4×C31—21no—
148C22×C4×C31—22no—
149C22×C4×C31—22no—
150C22×C4×C31—22no—
151C22×C4×C31—22no—
152C22×C4×C31—22no—
153C22×C4×C31—24no—
154C22×C4×C31—22no—
155C22×C4×C31—24no—
156C22×C4×C31—41no—
157C22×C4×C31—42no—
158C22×C4×C31—82no—
159C22×C4×C31—82no—
160C22×C4×C31C2×D412yes—
161C22×C4×C31C2×D411yescharacteristic
162C22×C4×C31C2×D411yescharacteristic
163C22×C4×C31C2×D414yes—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—24no—
166C4⋊C4×C31—84no—
167C22⋊C4×C31—22no—
168C22⋊C4×C31—24no—
169C22⋊C4×C31—24no—
170C22⋊C4×C31—24no—
171C22⋊C4×C31—24no—
172C22⋊C4×C31—22no—
173C22⋊C4×C31—22no—
174C22⋊C4×C31—24no—
175C22⋊C4×C31—24no—
176C22⋊C4×C31—24no—
177C22⋊C4×C31—42no—
178C22⋊C4×C31—42no—
179C22⋊C4×C31—42no—
180C22⋊C4×C31—84no—
181C22⋊C4×C31—84no—
182C22⋊C4×C31—82no—
183C22⋊C4×C31—82no—
184C42×C31—22no—
185#24058C3111yesmaximal, characteristic, direct product factor, Sylow
186C23×C31—21no—
187C23×C31—21no—
188C23×C31—21no—
189C23×C31—21no—
190C23×C31—21no—
191C23×C31—21no—
192C23×C31—28no—
193C23×C31—21no—
194C23×C31—22no—
195C23×C31—21no—
196C23×C31—21no—
197C23×C31—24no—
198C23×C31—42no—
199C23×C31—42no—
200C23×C31—42no—
201C23×C31—44no—
202C23×C31—44no—
203C23×C31—44no—
204C23×C31—42no—
205C23×C31—42no—
206C23×C31—44no—
207C23×C31—42no—
208C23×C31—44no—
209C23×C31—82no—
210C23×C31—84no—
211C23×C31—84no—
212C23×C31—84no—
213C23×C31—82no—
214C23×C31—84no—
215C23×C31#12012yes—
216C23×C31#12012yes—
217C23×C31#12312yes—
218C23×C31C22×D411yescharacteristic
219C23×C31C22×D412yes—
220C23×C31C2×#4112yes—
221C23×C31H(5,𝔽2)12yes—
222Q8×C31—42no—
223D4×C31—28no—
224D4×C31—44no—
225D4×C31—42no—
226D4×C31—44no—
227D4×C31—44no—
228D4×C31—48no—
229D4×C31—42no—
230D4×C31—42no—
231D4×C31—44no—
232D4×C31—44no—
233D4×C31—84no—
234D4×C31—84no—
235C2×C4×C31—21no—
236C2×C4×C31—21no—
237C2×C4×C31—24no—
238C2×C4×C31—22no—
239C2×C4×C31—21no—
240C2×C4×C31—21no—
241C2×C4×C31—21no—
242C2×C4×C31—24no—
243C2×C4×C31—44no—
244C2×C4×C31—44no—
245C2×C4×C31—42no—
246C2×C4×C31—44no—
247C2×C4×C31—44no—
248C2×C4×C31—44no—
249C2×C4×C31—42no—
250C2×C4×C31—42no—
251C2×C4×C31—44no—
252C2×C4×C31—42no—
253C2×C4×C31—42no—
254C2×C4×C31—44no—
255C2×C4×C31—42no—
256C2×C4×C31—42no—
257C2×C4×C31—44no—
258C2×C4×C31—82no—
259C2×C4×C31—84no—
260C2×C4×C31#12312yes—
261#3484C2×C3111yescharacteristic
262#2622C2×C3112yescomplemented
263#2308C2×C3111yescharacteristic, complemented
264#2044C2×C3114yescomplemented
265#1898C2×C3112yescomplemented
266#1867C2×C3112yescomplemented
267#1790C2×C3112yescomplemented
268#1781C2×C3111yescharacteristic, complemented
269C22×C31—24no—
270C22×C31—22no—
271C22×C31—22no—
272C22×C31—22no—
273C22×C31—22no—
274C22×C31—22no—
275C22×C31—24no—
276C22×C31—22no—
277C22×C31—42no—
278C22×C31—42no—
279C22×C31—42no—
280C22×C31—42no—
281C22×C31—42no—
282C22×C31—42no—
283C22×C31—42no—
284C22×C31—42no—
285C22×C31—42no—
286C22×C31—42no—
287C22×C31—41no—
288C22×C31—44no—
289C22×C31—42no—
290C22×C31—81no—
291C22×C31—88no—
292C22×C31—84no—
293C22×C31—81no—
294C22×C31—88no—
295C22×C31—162no—
296C22×C31C2×#12011yescharacteristic
297C22×C31C2×#12311yescharacteristic
298C22×C31#53411yescharacteristic
299C22×C31#54614yes—
300C4×C31—22no—
301C4×C31—41no—
302C4×C31—42no—
303C4×C31—44no—
304C4×C31—84no—
305C4×C31—82no—
306C4×C31—82no—
307C2×H(5,𝔽2)C22×C3111yescharacteristic
308C23×D4C22×C3111yescharacteristic
309#546—24no—
310D42—24no—
311#535—21no—
312#534—21no—
313#534C22×C3112yes—
314#525—21no—
315C2×#120—21no—
316C2×#120C22×C3112yes—
317#518C22×C3111yescharacteristic
318C22×C22⋊C4C22×C3111yescharacteristic, complemented
319C22×C22⋊C4C22×C3111yescharacteristic, complemented
320C2×#99C22×C3112yes—
321C2×#99C22×C3114yescomplemented
322C2×#99C22×C3112yescomplemented
323#397—24no—
324#394—24no—
325#386—21no—
326#386—21no—
327#386C22×C3112yes—
328#386C22×C3112yescomplemented
329#386C22×C3114yescomplemented
330#385C22×C3114yescomplemented
331#381—21no—
332#379—21no—
333#379C22×C3112yescomplemented
334#342C22×C3114yescomplemented
335C2×C31—21no—
336C2×C31—24no—
337C2×C31—21no—
338C2×C31—42no—
339C2×C31—42no—
340C2×C31—42no—
341C2×C31—81no—
342C2×C31—82no—
343C2×C31#240511yescharacteristic
344C2×C31#302612yes—
345C25D4×C3111yescharacteristic
346C25C23×C3111yescharacteristic
347H(5,𝔽2)—24no—
348H(5,𝔽2)—42no—
349C2×#41—21no—
350C2×#41—22no—
351C22×D4—21no—
352C22×D4—22no—
353C22×D4—24no—
354C22×D4—22no—
355C22×D4—24no—
356C22×D4—42no—
357C22×D4D4×C3111yescharacteristic
358C22×D4D4×C3111yescharacteristic
359C22×D4D4×C3114yes—
360C22×D4C23×C3112yes—
361C22×D4C23×C3111yescharacteristic
362C22×D4C23×C3111yescharacteristic
363C23×C4—21no—
364C23×C4D4×C3111yescharacteristic
365C23×C4C23×C3112yes—
366#127—22no—
367#124—22no—
368#123—22no—
369#123—22no—
370#123—22no—
371C22⋊Q8—22no—
372#121—22no—
373#121—24no—
374#121—24no—
375#121—24no—
376#120—24no—
377#120—22no—
378#120—22no—
379#120—22no—
380#120—22no—
381#120—24no—
382#120—22no—
383C4×D4—24no—
384C4×D4—24no—
385#117—21no—
386C2×C4⋊C4—24no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—42no—
398C2×C22⋊C4D4×C3114yes—
399C2×C22⋊C4C23×C3112yes—
400C2×C22⋊C4C23×C3114yes—
401C2×C22⋊C4C23×C3112yes—
402#99—24no—
403#99—42no—
404#99—44no—
405#95—22no—
406#95—22no—
407#95—24no—
408#95—24no—
409C31#2405811yescharacteristic, direct product factor, Sylow
410C24—22no—
411C24—22no—
412C24—21no—
413C24—22no—
414C24—22no—
415C24—41no—
416C24—82no—
417C24—82no—
418C24C2×D4×C3111yescharacteristic
419C24C2×D4×C3112yes—
420C24C2×D4×C3114yes—
421C24C2×D4×C3111yescharacteristic
422C24C2×D4×C3111yescharacteristic
423C24C2×D4×C3111yescharacteristic
424C24C24×C3111yescharacteristic
425#41—48no—
426#41—42no—
427#41—42no—
428C2×Q8—21no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—21no—
433C2×D4—24no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—24no—
442C2×D4—22no—
443C2×D4—24no—
444C2×D4—24no—
445C2×D4—24no—
446C2×D4—22no—
447C2×D4—41no—
448C2×D4—41no—
449C2×D4—416no—
450C2×D4—48no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—84no—
454C2×D4—84no—
455C2×D4#41×C3114yes—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—41no—
466C22×C4—42no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4C2×D4×C3112yes—
470C22×C4C2×D4×C3111yescharacteristic
471C22×C4C2×D4×C3111yescharacteristic
472C22×C4C2×D4×C3114yes—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—84no—
476C22⋊C4—22no—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—84no—
490C22⋊C4—84no—
491C22⋊C4—82no—
492C22⋊C4—82no—
493C42—22no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—28no—
501C23—21no—
502C23—22no—
503C23—21no—
504C23—21no—
505C23—24no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—42no—
513C23—42no—
514C23—44no—
515C23—42no—
516C23—44no—
517C23—82no—
518C23—84no—
519C23—84no—
520C23—84no—
521C23—82no—
522C23—84no—
523C23C31×#12012yes—
524C23C31×#12012yes—
525C23C31×#12312yes—
526C23C22×D4×C3111yescharacteristic
527C23C22×D4×C3112yes—
528C23C2×#41×C3112yes—
529C23C31×H(5,𝔽2)12yes—
530Q8—42no—
531D4—28no—
532D4—44no—
533D4—42no—
534D4—44no—
535D4—44no—
536D4—48no—
537D4—42no—
538D4—42no—
539D4—44no—
540D4—44no—
541D4—84no—
542D4—84no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—24no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—82no—
567C2×C4—84no—
568C2×C4C31×#12312yes—
569C22—24no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—24no—
576C22—22no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—41no—
588C22—44no—
589C22—42no—
590C22—81no—
591C22—88no—
592C22—81no—
593C22—84no—
594C22—88no—
595C22—162no—
596C22C2×C31×#12011yescharacteristic
597C22C2×C31×#12311yescharacteristic
598C22C31×#53411yescharacteristic
599C22C31×#54614yes—
600C4—22no—
601C4—41no—
602C4—42no—
603C4—44no—
604C4—82no—
605C4—84no—
606C4—82no—
607C2—21no—
608C2—24no—
609C2—21no—
610C2—42no—
611C2—42no—
612C2—42no—
613C2—81no—
614C2—82no—
615C2C31×#240511yescharacteristic
616C2C31×#302612yes—
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC24C24×C31
prime radicalC31×#3484C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C25×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2622C2semidirect
3. C31×#2308C2semidirect
4. C31×#2044C2semidirect
5. C31×#1898C2semidirect
6. C31×#1867C2semidirect
7. C31×#1790C2semidirect
8. C31×#1781C2semidirect
20. C22×C22⋊C4×C31C22semidirect
21. C22×C22⋊C4×C31C22semidirect
23. C2×C31×#99C22semidirect
24. C2×C31×#99C22semidirect
30. C31×#386C22semidirect
31. C31×#386C22semidirect
32. C31×#385C22semidirect
35. C31×#379C22semidirect
36. C31×#342C22semidirect
185. #24058C31direct (N × H)
262. #2622C2×C31semidirect
263. #2308C2×C31semidirect
264. #2044C2×C31semidirect
265. #1898C2×C31semidirect
266. #1867C2×C31semidirect
267. #1790C2×C31semidirect
268. #1781C2×C31semidirect
318. C22×C22⋊C4C22×C31semidirect
319. C22×C22⋊C4C22×C31semidirect
321. C2×#99C22×C31semidirect
322. C2×#99C22×C31semidirect
328. #386C22×C31semidirect
329. #386C22×C31semidirect
330. #385C22×C31semidirect
333. #379C22×C31semidirect
334. #342C22×C31semidirect
409. C31#24058direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34842
C31×#26222
C31×#23082
C31×#20442
C31×#18982
C31×#18672
C31×#17902
C31×#17812
#2405831