Finite Group Databasebeta

← C31×#18973 #21475732

Subgroups — C31×#18973 #21475732

Order 7936.

818 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3555C211yesmaximal, characteristic
2C31×#3230C211yesmaximal, characteristic, complemented
3C31×#3229C211yesmaximal, characteristic, complemented
4C31×#3218C211yesmaximal, characteristic
5C31×#1650C212yesmaximal, complemented
6C31×#1648C212yesmaximal, complemented
7C31×#1645C212yesmaximal, complemented
8C31×#1643C212yesmaximal, complemented
9C31×#1554C211yesmaximal, characteristic, complemented
10C31×#1497C211yesmaximal, characteristic, complemented
11C31×#1482C211yesmaximal, characteristic
12C31×#563—22no—
13Q82×C31C2211yescharacteristic
14C31×#557—21no—
15C31×#557C2211yescharacteristic
16C31×#556—21no—
17C31×#556—21no—
18C31×#556C2211yescharacteristic
19C31×#554C2211yescharacteristic
20C31×#552—21no—
21C31×#544—21no—
22C31×#542—21no—
23C31×#541—21no—
24C31×#541—21no—
25C31×#541—22no—
26C31×#533C2211yescharacteristic
27C31×#533C2211yescharacteristic
28C2×C31×C4⋊Q8C2211yescharacteristic
29C31×#529C2211yescharacteristic
30C31×#500C2212yes—
31C31×#498—21no—
32C31×#497—21no—
33C31×#491—21no—
34C31×#487—21no—
35C31×#487C2212yes—
36C31×#484—22no—
37C31×#483—21no—
38C31×#483—21no—
39C31×#477—22no—
40C31×#475C2212yes—
41C31×#475C2212yes—
42C31×#462C2212yes—
43C31×#461C2212yes—
44C31×#451—21no—
45C31×#448—21no—
46C31×#432C2211yescharacteristic
47C31×#431C2211yescharacteristic
48C31×#431C2211yescharacteristic
49C31×#333C2212yes—
50C31×#332C2212yes—
51C31×Q8⋊C8C2212yes—
52C31×Q8⋊C8C2212yes—
53C31×Q8⋊C8C2212yes—
54C31×D4⋊C8C2212yes—
55C22×Q8×C31—21no—
56C2×Q16×C31—22no—
57C2×SD16×C31—22no—
58C31×C4⋊Q8—22no—
59C31×C4⋊Q8—21no—
60C31×C4⋊Q8—22no—
61C31×C4⋊Q8—21no—
62C31×C4⋊Q8—21no—
63C31×C4⋊Q8—21no—
64C31×C4⋊Q8—21no—
65C31×C4⋊Q8—21no—
66C31×C4⋊Q8D411yescharacteristic
67C31×C4⋊Q8D411yescharacteristic
68C31×C4⋊Q8C2312yes—
69C31×C4⋊Q8C2311yescharacteristic
70C31×C4⋊Q8C2311yescharacteristic
71C31×#126—22no—
72C31×#126—22no—
73C31×#126—22no—
74C31×#126—22no—
75C31×#126—22no—
76C31×#126—22no—
77C31×#126—21no—
78C31×#125—22no—
79C31×#125—22no—
80C31×#125—22no—
81C31×#125—22no—
82C31×#125—21no—
83C31×#125—22no—
84C31×#125—21no—
85C31×#125—21no—
86C31×#125—21no—
87C31×#125D411yescharacteristic
88C31×#124—22no—
89C31×#124—21no—
90C31×#124D411yescharacteristic
91C31×#123—22no—
92C31×#123—21no—
93C31×C22⋊Q8—22no—
94C31×C22⋊Q8—22no—
95C31×C22⋊Q8—21no—
96C31×C22⋊Q8—21no—
97C31×C22⋊Q8—22no—
98C31×C22⋊Q8—21no—
99C31×C22⋊Q8—21no—
100C31×C22⋊Q8D411yescharacteristic
101C31×C22⋊Q8D411yescharacteristic
102C31×C22⋊Q8D411yescharacteristic
103C31×C22⋊Q8D412yes—
104C31×#121D411yescharacteristic
105C4×Q8×C31—22no—
106C4×Q8×C31—21no—
107C4×Q8×C31—21no—
108C4×Q8×C31—21no—
109C4×Q8×C31—21no—
110C4×Q8×C31—21no—
111C4×Q8×C31—21no—
112C4×Q8×C31—21no—
113C4×Q8×C31D412yes—
114C4×Q8×C31C2311yescharacteristic
115C4×Q8×C31C2311yescharacteristic
116C4×Q8×C31C2311yescharacteristic
117C4×D4×C31—21no—
118C4×D4×C31—21no—
119C4×D4×C31C2311yescharacteristic
120C31×#117—22no—
121C31×#117—21no—
122C31×#117—21no—
123C31×#117—21no—
124C31×#117—21no—
125C2×C4⋊C4×C31—21no—
126C2×C4⋊C4×C31—21no—
127C2×C4⋊C4×C31—21no—
128C2×C42×C31C2311yescharacteristic
129C31×#107—24no—
130C31×#106—22no—
131C31×#106—22no—
132C31×C4⋊C8—21no—
133C31×C4⋊C8C2312yes—
134C31×C4⋊C8C2312yes—
135C31×#103—22no—
136C31×#103—22no—
137C31×#103—24no—
138C31×#103—22no—
139C31×#103—22no—
140C31×C22⋊C8—21no—
141C31×C22⋊C8—21no—
142C31×C22⋊C8—21no—
143C31×#97—21no—
144C4×C8×C31—21no—
145C4×C8×C31C2312yes—
146C2×Q8×C31—22no—
147C2×Q8×C31—21no—
148C2×Q8×C31—21no—
149C2×Q8×C31—22no—
150C2×Q8×C31—41no—
151C2×Q8×C31—82no—
152C2×Q8×C31C2×D411yescharacteristic
153C2×Q8×C31C2×D411yescharacteristic
154C2×Q8×C31C2×D411yescharacteristic
155C2×Q8×C31C2×D412yes—
156C2×D4×C31—21no—
157C2×D4×C31C2×D411yescharacteristic
158C22×C4×C31—22no—
159C22×C4×C31—21no—
160C22×C4×C31—41no—
161C22×C4×C31C2×D411yescharacteristic
162C22×C4×C31C2×D411yescharacteristic
163C22×C4×C31C2×D411yescharacteristic
164Q16×C31—82no—
165SD16×C31—82no—
166C2×C8×C31—22no—
167C2×C8×C31—22no—
168C2×C8×C31—22no—
169C4⋊C4×C31—22no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—22no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—22no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—21no—
178C4⋊C4×C31—21no—
179C4⋊C4×C31—21no—
180C4⋊C4×C31—21no—
181C4⋊C4×C31—21no—
182C4⋊C4×C31—21no—
183C4⋊C4×C31—21no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—21no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—21no—
188C4⋊C4×C31—21no—
189C4⋊C4×C31—21no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—21no—
192C4⋊C4×C31—21no—
193C4⋊C4×C31—22no—
194C4⋊C4×C31—22no—
195C4⋊C4×C31—41no—
196C4⋊C4×C31—41no—
197C4⋊C4×C31C2×D411yescharacteristic
198C4⋊C4×C31C2×D411yescharacteristic
199C4⋊C4×C31C2×D411yescharacteristic
200C4⋊C4×C31C2×D411yescharacteristic
201C4⋊C4×C31C2×D412yes—
202C22⋊C4×C31—22no—
203C22⋊C4×C31—22no—
204C22⋊C4×C31—22no—
205C22⋊C4×C31—21no—
206C22⋊C4×C31—22no—
207C22⋊C4×C31—21no—
208C22⋊C4×C31—21no—
209C22⋊C4×C31—21no—
210C22⋊C4×C31—21no—
211C22⋊C4×C31—21no—
212C22⋊C4×C31—21no—
213C42×C31—22no—
214C42×C31—22no—
215C42×C31—22no—
216C42×C31—22no—
217C42×C31—21no—
218C42×C31—21no—
219C42×C31—21no—
220C42×C31C2×D411yescharacteristic
221C42×C31C2×D411yescharacteristic
222C42×C31C2×D411yescharacteristic
223C42×C31C2411yescharacteristic
224#18973C3111yesmaximal, characteristic, direct product factor, Sylow
225C23×C31—21no—
226C23×C31#12011yescharacteristic
227Q8×C31—22no—
228Q8×C31—22no—
229Q8×C31—22no—
230Q8×C31—42no—
231Q8×C31—42no—
232Q8×C31—42no—
233Q8×C31—42no—
234Q8×C31—81no—
235Q8×C31—82no—
236Q8×C31—82no—
237D4×C31—22no—
238D4×C31—81no—
239C2×C4×C31—22no—
240C2×C4×C31—22no—
241C2×C4×C31—21no—
242C2×C4×C31—21no—
243C2×C4×C31—21no—
244C2×C4×C31—22no—
245C2×C4×C31—21no—
246C2×C4×C31—21no—
247C2×C4×C31—21no—
248C2×C4×C31—22no—
249C2×C4×C31—21no—
250C2×C4×C31—41no—
251C2×C4×C31—42no—
252C2×C4×C31—42no—
253C2×C4×C31—82no—
254C2×C4×C31—82no—
255C2×C4×C31#12011yescharacteristic
256C2×C4×C31#12011yescharacteristic
257C2×C4×C31#12011yescharacteristic
258C2×C4×C31C22×D411yescharacteristic
259C2×C4×C31C22×D411yescharacteristic
260C2×C4×C31C22×D411yescharacteristic
261C8×C31—42no—
262C8×C31—42no—
263C8×C31—42no—
264#3555C2×C3111yescharacteristic
265#3230C2×C3111yescharacteristic, complemented
266#3229C2×C3111yescharacteristic, complemented
267#3218C2×C3111yescharacteristic
268#1650C2×C3112yescomplemented
269#1648C2×C3112yescomplemented
270#1645C2×C3112yescomplemented
271#1643C2×C3112yescomplemented
272#1554C2×C3111yescharacteristic, complemented
273#1497C2×C3111yescharacteristic, complemented
274#1482C2×C3111yescharacteristic
275C22×C31—21no—
276C22×C31—21no—
277C22×C31—21no—
278C22×C31—41no—
279C22×C31—41no—
280C22×C31—41no—
281C22×C31C2×#12011yescharacteristic
282C4×C31—21no—
283C4×C31—21no—
284C4×C31—21no—
285C4×C31—21no—
286C4×C31—41no—
287C4×C31—42no—
288C4×C31—41no—
289C4×C31—41no—
290C4×C31—41no—
291C4×C31—41no—
292C4×C31—42no—
293C4×C31—41no—
294C4×C31—41no—
295C4×C31—82no—
296C4×C31#57712yes—
297C4×C31#57812yes—
298#563—22no—
299Q82C22×C3111yescharacteristic
300#557—21no—
301#557C22×C3111yescharacteristic
302#556—21no—
303#556—21no—
304#556C22×C3111yescharacteristic
305#554C22×C3111yescharacteristic
306#552—21no—
307#544—21no—
308#542—21no—
309#541—21no—
310#541—21no—
311#541—22no—
312#533C22×C3111yescharacteristic
313#533C22×C3111yescharacteristic
314C2×C4⋊Q8C22×C3111yescharacteristic
315#529C22×C3111yescharacteristic
316#500C22×C3112yes—
317#498—21no—
318#497—21no—
319#491—21no—
320#487—21no—
321#487C22×C3112yes—
322#484—22no—
323#483—21no—
324#483—21no—
325#477—22no—
326#475C22×C3112yes—
327#475C22×C3112yes—
328#462C22×C3112yes—
329#461C22×C3112yes—
330#451—21no—
331#448—21no—
332#432C22×C3111yescharacteristic
333#431C22×C3111yescharacteristic
334#431C22×C3111yescharacteristic
335#333C22×C3112yes—
336#332C22×C3112yes—
337Q8⋊C8C22×C3112yes—
338Q8⋊C8C22×C3112yes—
339Q8⋊C8C22×C3112yes—
340D4⋊C8C22×C3112yes—
341C2×C31—41no—
342C2×C31—81no—
343C2×C31#301011yescharacteristic
344C2×C31#301111yescharacteristic
345C2×C31#302211yescharacteristic
346C22×Q8—21no—
347C2×Q16—22no—
348C2×SD16—22no—
349C4⋊Q8—22no—
350C4⋊Q8—21no—
351C4⋊Q8—22no—
352C4⋊Q8—21no—
353C4⋊Q8—21no—
354C4⋊Q8—21no—
355C4⋊Q8—21no—
356C4⋊Q8—21no—
357C4⋊Q8D4×C3111yescharacteristic
358C4⋊Q8D4×C3111yescharacteristic
359C4⋊Q8C23×C3112yes—
360C4⋊Q8C23×C3111yescharacteristic
361C4⋊Q8C23×C3111yescharacteristic
362#126—22no—
363#126—22no—
364#126—22no—
365#126—22no—
366#126—22no—
367#126—22no—
368#126—21no—
369#125—22no—
370#125—22no—
371#125—22no—
372#125—22no—
373#125—21no—
374#125—22no—
375#125—21no—
376#125—21no—
377#125—21no—
378#125D4×C3111yescharacteristic
379#124—22no—
380#124—21no—
381#124D4×C3111yescharacteristic
382#123—22no—
383#123—21no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8—21no—
387C22⋊Q8—21no—
388C22⋊Q8—22no—
389C22⋊Q8—21no—
390C22⋊Q8—21no—
391C22⋊Q8D4×C3111yescharacteristic
392C22⋊Q8D4×C3111yescharacteristic
393C22⋊Q8D4×C3111yescharacteristic
394C22⋊Q8D4×C3112yes—
395#121D4×C3111yescharacteristic
396C4×Q8—22no—
397C4×Q8—21no—
398C4×Q8—21no—
399C4×Q8—21no—
400C4×Q8—21no—
401C4×Q8—21no—
402C4×Q8—21no—
403C4×Q8—21no—
404C4×Q8D4×C3112yes—
405C4×Q8C23×C3111yescharacteristic
406C4×Q8C23×C3111yescharacteristic
407C4×Q8C23×C3111yescharacteristic
408C4×D4—21no—
409C4×D4—21no—
410C4×D4C23×C3111yescharacteristic
411#117—22no—
412#117—21no—
413#117—21no—
414#117—21no—
415#117—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C42C23×C3111yescharacteristic
420#107—24no—
421#106—22no—
422#106—22no—
423C4⋊C8—21no—
424C4⋊C8C23×C3112yes—
425C4⋊C8C23×C3112yes—
426#103—22no—
427#103—22no—
428#103—24no—
429#103—22no—
430#103—22no—
431C22⋊C8—21no—
432C22⋊C8—21no—
433C22⋊C8—21no—
434#97—21no—
435C4×C8—21no—
436C4×C8C23×C3112yes—
437C31#1897311yescharacteristic, direct product factor, Sylow
438C2×Q8—22no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8—41no—
443C2×Q8—82no—
444C2×Q8C2×D4×C3111yescharacteristic
445C2×Q8C2×D4×C3111yescharacteristic
446C2×Q8C2×D4×C3111yescharacteristic
447C2×Q8C2×D4×C3112yes—
448C2×D4—21no—
449C2×D4C2×D4×C3111yescharacteristic
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—41no—
453C22×C4C2×D4×C3111yescharacteristic
454C22×C4C2×D4×C3111yescharacteristic
455C22×C4C2×D4×C3111yescharacteristic
456Q16—82no—
457SD16—82no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4C2×D4×C3111yescharacteristic
490C4⋊C4C2×D4×C3111yescharacteristic
491C4⋊C4C2×D4×C3111yescharacteristic
492C4⋊C4C2×D4×C3111yescharacteristic
493C4⋊C4C2×D4×C3112yes—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—21no—
498C22⋊C4—22no—
499C22⋊C4—21no—
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—21no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42C2×D4×C3111yescharacteristic
513C42C2×D4×C3111yescharacteristic
514C42C2×D4×C3111yescharacteristic
515C42C24×C3111yescharacteristic
516C23—21no—
517C23C31×#12011yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—22no—
521Q8—42no—
522Q8—42no—
523Q8—42no—
524Q8—42no—
525Q8—81no—
526Q8—82no—
527Q8—82no—
528D4—22no—
529D4—81no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4C31×#12011yescharacteristic
547C2×C4C31×#12011yescharacteristic
548C2×C4C31×#12011yescharacteristic
549C2×C4C22×D4×C3111yescharacteristic
550C2×C4C22×D4×C3111yescharacteristic
551C2×C4C22×D4×C3111yescharacteristic
552C8—42no—
553C8—42no—
554C8—42no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22C2×C31×#12011yescharacteristic
562C4—21no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—41no—
567C4—42no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—42no—
573C4—41no—
574C4—41no—
575C4—82no—
576C4C31×#57712yes—
577C4C31×#57812yes—
578C2—41no—
579C2—81no—
580C2C31×#301011yescharacteristic
581C2C31×#301111yescharacteristic
582C2C31×#302211yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#121D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3230C2semidirect
3. C31×#3229C2semidirect
5. C31×#1650C2semidirect
6. C31×#1648C2semidirect
7. C31×#1645C2semidirect
8. C31×#1643C2semidirect
9. C31×#1554C2semidirect
10. C31×#1497C2semidirect
224. #18973C31direct (N × H)
265. #3230C2×C31semidirect
266. #3229C2×C31semidirect
268. #1650C2×C31semidirect
269. #1648C2×C31semidirect
270. #1645C2×C31semidirect
271. #1643C2×C31semidirect
272. #1554C2×C31semidirect
273. #1497C2×C31semidirect
437. C31#18973direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35552
C31×#32302
C31×#32292
C31×#32182
C31×#16502
C31×#16482
C31×#16452
C31×#16432
C31×#15542
C31×#14972
C31×#14822
#1897331