Finite Group Databasebeta

← C31×#18161 #21474920

Subgroups — C31×#18161 #21474920

Order 7936.

908 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3461C211yesmaximal, characteristic
2C31×#3117C212yesmaximal, complemented
3C31×#2581C211yesmaximal, characteristic, complemented
4C31×#1617C211yesmaximal, characteristic, complemented
5C31×#1616C211yesmaximal, characteristic
6C31×#1615C212yesmaximal, complemented
7C31×#1605C211yesmaximal, characteristic, complemented
8C31×#1604C211yesmaximal, characteristic
9C31×#1603C212yesmaximal, complemented
10C31×#1512C212yesmaximal, complemented
11C31×#1467C211yesmaximal, characteristic, complemented
12C31×#544C2211yescharacteristic
13C31×#542—22no—
14C31×#541—22no—
15C31×#539—21no—
16C31×#539C2212yes—
17C31×#537C2211yescharacteristic
18C31×#536—21no—
19C31×#533—22no—
20C31×#529—22no—
21C31×#529—21no—
22C31×#529C2212yes—
23C4×#41×C31—21no—
24C2×C4×Q8×C31C2211yescharacteristic
25C2×C4×Q8×C31C2211yescharacteristic
26C31×#484C2212yes—
27C31×#481C2212yes—
28C31×#479—22no—
29C31×#464—22no—
30C31×#451C2212yes—
31C31×#448C2212yes—
32C31×#441—22no—
33C31×#440—22no—
34C31×#432C2212yes—
35C31×#400C2211yescharacteristic
36C31×#400C2211yescharacteristic
37C31×#395C2211yescharacteristic
38C31×#395C2211yescharacteristic
39C31×#391—21no—
40C31×#389—21no—
41C31×#387—21no—
42C31×#387—21no—
43C31×#382C2211yescharacteristic
44C31×#328C2212yes—
45C31×#328C2212yes—
46C31×#327C2212yes—
47C31×#327C2212yes—
48C31×#324C2212yes—
49C31×#324C2211yescharacteristic
50C31×#324C2211yescharacteristic
51C2×#41×C31—21no—
52C22×Q8×C31D411yescharacteristic
53C22×Q8×C31C2311yescharacteristic
54C2×Q16×C31—22no—
55C2×SD16×C31—22no—
56C31×C4⋊Q8—22no—
57C31×C4⋊Q8—22no—
58C31×#126—22no—
59C31×#126—22no—
60C31×#126—22no—
61C31×#126—22no—
62C31×#125—21no—
63C31×#125—22no—
64C31×#125—22no—
65C31×#125—22no—
66C31×#125D412yes—
67C31×#124—22no—
68C31×#124—21no—
69C31×#124D412yes—
70C31×#123—22no—
71C31×#123—22no—
72C31×#123—22no—
73C31×#123—22no—
74C31×C22⋊Q8—22no—
75C31×C22⋊Q8—22no—
76C31×C22⋊Q8—22no—
77C31×C22⋊Q8—22no—
78C31×C22⋊Q8—21no—
79C31×C22⋊Q8—22no—
80C31×C22⋊Q8C2312yes—
81C31×#121—22no—
82C31×#121—21no—
83C31×#121C2312yes—
84C4×Q8×C31—21no—
85C4×Q8×C31—21no—
86C4×Q8×C31—21no—
87C4×Q8×C31—22no—
88C4×Q8×C31—22no—
89C4×Q8×C31—22no—
90C4×Q8×C31—22no—
91C4×Q8×C31D412yes—
92C4×D4×C31—21no—
93C4×D4×C31—21no—
94C4×D4×C31—21no—
95C4×D4×C31—22no—
96C4×D4×C31—22no—
97C4×D4×C31D412yes—
98C31×#117—22no—
99C31×#117—22no—
100C31×#117—21no—
101C31×#117—22no—
102C2×C4⋊C4×C31—21no—
103C2×C4⋊C4×C31—21no—
104C2×C4⋊C4×C31—21no—
105C2×C4⋊C4×C31—21no—
106C2×C4⋊C4×C31D411yescharacteristic
107C2×C4⋊C4×C31C2311yescharacteristic
108C2×C4⋊C4×C31C2311yescharacteristic
109C2×C4⋊C4×C31C2311yescharacteristic
110C2×C42×C31—22no—
111C2×C42×C31—21no—
112C2×C42×C31D411yescharacteristic
113C2×C42×C31D411yescharacteristic
114C2×C42×C31C2311yescharacteristic
115C31×#107—22no—
116C31×#106—22no—
117C31×C4⋊C8—22no—
118C31×#103—22no—
119C31×#103—22no—
120C31×#103—22no—
121C31×#102—22no—
122C31×C22⋊C8C2312yes—
123C31×C22⋊C8C2312yes—
124C31×#97—22no—
125C31×#95—21no—
126C31×#95—21no—
127C31×#95—21no—
128C31×#95—21no—
129C31×#95—21no—
130C31×#95—21no—
131C31×#95—21no—
132C31×#95C2311yescharacteristic
133C31×#95C2311yescharacteristic
134#41×C31—42no—
135#41×C31—42no—
136C2×Q8×C31—21no—
137C2×Q8×C31—21no—
138C2×Q8×C31—22no—
139C2×Q8×C31—22no—
140C2×Q8×C31—22no—
141C2×Q8×C31—41no—
142C2×Q8×C31—42no—
143C2×Q8×C31C2×D412yes—
144C2×D4×C31—21no—
145C2×D4×C31—22no—
146C2×D4×C31—21no—
147C2×D4×C31C2×D412yes—
148C22×C4×C31—22no—
149C22×C4×C31—21no—
150C22×C4×C31—22no—
151C22×C4×C31—22no—
152C22×C4×C31—21no—
153C22×C4×C31—21no—
154C22×C4×C31—21no—
155C22×C4×C31C2×D411yescharacteristic
156C22×C4×C31C2×D411yescharacteristic
157C22×C4×C31C2×D411yescharacteristic
158C22×C4×C31C2×D411yescharacteristic
159C22×C4×C31C2×D411yescharacteristic
160C22×C4×C31C2×D411yescharacteristic
161C22×C4×C31C2411yescharacteristic
162Q16×C31—44no—
163SD16×C31—44no—
164C2×C8×C31—22no—
165C2×C8×C31—22no—
166C4⋊C4×C31—21no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—22no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—21no—
172C4⋊C4×C31—21no—
173C4⋊C4×C31—22no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—22no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—42no—
183C4⋊C4×C31—42no—
184C4⋊C4×C31—42no—
185C4⋊C4×C31—41no—
186C4⋊C4×C31—41no—
187C4⋊C4×C31C2×D412yes—
188C4⋊C4×C31C2×D412yes—
189C22⋊C4×C31—22no—
190C22⋊C4×C31—22no—
191C22⋊C4×C31—22no—
192C22⋊C4×C31—22no—
193C22⋊C4×C31—22no—
194C22⋊C4×C31—22no—
195C22⋊C4×C31—22no—
196C22⋊C4×C31—22no—
197C22⋊C4×C31—22no—
198C42×C31—22no—
199C42×C31—22no—
200C42×C31—21no—
201C42×C31—22no—
202C42×C31—21no—
203C42×C31—21no—
204C42×C31—21no—
205C42×C31—42no—
206C42×C31C2×D412yes—
207C42×C31C2×D412yes—
208#18161C3111yesmaximal, characteristic, direct product factor, Sylow
209C23×C31—22no—
210C23×C31C22×D411yescharacteristic
211Q8×C31—24no—
212Q8×C31—44no—
213Q8×C31—42no—
214Q8×C31—42no—
215Q8×C31—42no—
216Q8×C31—81no—
217D4×C31—42no—
218D4×C31—42no—
219D4×C31—42no—
220D4×C31—42no—
221D4×C31—81no—
222C2×C4×C31—21no—
223C2×C4×C31—21no—
224C2×C4×C31—21no—
225C2×C4×C31—21no—
226C2×C4×C31—22no—
227C2×C4×C31—21no—
228C2×C4×C31—22no—
229C2×C4×C31—22no—
230C2×C4×C31—21no—
231C2×C4×C31—21no—
232C2×C4×C31—22no—
233C2×C4×C31—21no—
234C2×C4×C31—21no—
235C2×C4×C31—41no—
236C2×C4×C31—41no—
237C2×C4×C31—44no—
238C2×C4×C31—44no—
239C2×C4×C31—41no—
240C2×C4×C31—41no—
241C2×C4×C31—41no—
242C2×C4×C31—41no—
243C2×C4×C31—41no—
244C2×C4×C31—41no—
245C2×C4×C31—41no—
246C2×C4×C31—41no—
247C2×C4×C31—42no—
248C2×C4×C31—41no—
249C2×C4×C31—41no—
250C2×C4×C31—41no—
251C2×C4×C31—82no—
252C2×C4×C31#12011yescharacteristic
253C2×C4×C31#12011yescharacteristic
254C2×C4×C31#12012yes—
255C2×C4×C31#13712yes—
256C2×C4×C31C22×D412yes—
257C8×C31—42no—
258C8×C31—42no—
259#3461C2×C3111yescharacteristic
260#3117C2×C3112yescomplemented
261#2581C2×C3111yescharacteristic, complemented
262#1617C2×C3111yescharacteristic, complemented
263#1616C2×C3111yescharacteristic
264#1615C2×C3112yescomplemented
265#1605C2×C3111yescharacteristic, complemented
266#1604C2×C3111yescharacteristic
267#1603C2×C3112yescomplemented
268#1512C2×C3112yescomplemented
269#1467C2×C3111yescharacteristic, complemented
270C22×C31—21no—
271C22×C31—21no—
272C22×C31—42no—
273C22×C31—42no—
274C22×C31—42no—
275C22×C31C2×#12011yescharacteristic
276C22×C31C2×#13711yescharacteristic
277C22×C31#57511yescharacteristic
278C4×C31—22no—
279C4×C31—22no—
280C4×C31—21no—
281C4×C31—21no—
282C4×C31—22no—
283C4×C31—41no—
284C4×C31—42no—
285C4×C31—42no—
286C4×C31—42no—
287C4×C31—41no—
288C4×C31—41no—
289C4×C31—42no—
290C4×C31—81no—
291C4×C31—81no—
292C4×C31—81no—
293#544C22×C3111yescharacteristic
294#542—22no—
295#541—22no—
296#539—21no—
297#539C22×C3112yes—
298#537C22×C3111yescharacteristic
299#536—21no—
300#533—22no—
301#529—22no—
302#529—21no—
303#529C22×C3112yes—
304C4×#41—21no—
305C2×C4×Q8C22×C3111yescharacteristic
306C2×C4×Q8C22×C3111yescharacteristic
307#484C22×C3112yes—
308#481C22×C3112yes—
309#479—22no—
310#464—22no—
311#451C22×C3112yes—
312#448C22×C3112yes—
313#441—22no—
314#440—22no—
315#432C22×C3112yes—
316#400C22×C3111yescharacteristic
317#400C22×C3111yescharacteristic
318#395C22×C3111yescharacteristic
319#395C22×C3111yescharacteristic
320#391—21no—
321#389—21no—
322#387—21no—
323#387—21no—
324#382C22×C3111yescharacteristic
325#328C22×C3112yes—
326#328C22×C3112yes—
327#327C22×C3112yes—
328#327C22×C3112yes—
329#324C22×C3112yes—
330#324C22×C3111yescharacteristic
331#324C22×C3111yescharacteristic
332C2×C31—21no—
333C2×C31—21no—
334C2×C31—82no—
335C2×C31#300711yescharacteristic
336C2×C31#302011yescharacteristic
337C2×C31#302511yescharacteristic
338C2×#41—21no—
339C22×Q8D4×C3111yescharacteristic
340C22×Q8C23×C3111yescharacteristic
341C2×Q16—22no—
342C2×SD16—22no—
343C4⋊Q8—22no—
344C4⋊Q8—22no—
345#126—22no—
346#126—22no—
347#126—22no—
348#126—22no—
349#125—21no—
350#125—22no—
351#125—22no—
352#125—22no—
353#125D4×C3112yes—
354#124—22no—
355#124—21no—
356#124D4×C3112yes—
357#123—22no—
358#123—22no—
359#123—22no—
360#123—22no—
361C22⋊Q8—22no—
362C22⋊Q8—22no—
363C22⋊Q8—22no—
364C22⋊Q8—22no—
365C22⋊Q8—21no—
366C22⋊Q8—22no—
367C22⋊Q8C23×C3112yes—
368#121—22no—
369#121—21no—
370#121C23×C3112yes—
371C4×Q8—21no—
372C4×Q8—21no—
373C4×Q8—21no—
374C4×Q8—22no—
375C4×Q8—22no—
376C4×Q8—22no—
377C4×Q8—22no—
378C4×Q8D4×C3112yes—
379C4×D4—21no—
380C4×D4—21no—
381C4×D4—21no—
382C4×D4—22no—
383C4×D4—22no—
384C4×D4D4×C3112yes—
385#117—22no—
386#117—22no—
387#117—21no—
388#117—22no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4D4×C3111yescharacteristic
394C2×C4⋊C4C23×C3111yescharacteristic
395C2×C4⋊C4C23×C3111yescharacteristic
396C2×C4⋊C4C23×C3111yescharacteristic
397C2×C42—22no—
398C2×C42—21no—
399C2×C42D4×C3111yescharacteristic
400C2×C42D4×C3111yescharacteristic
401C2×C42C23×C3111yescharacteristic
402#107—22no—
403#106—22no—
404C4⋊C8—22no—
405#103—22no—
406#103—22no—
407#103—22no—
408#102—22no—
409C22⋊C8C23×C3112yes—
410C22⋊C8C23×C3112yes—
411#97—22no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95C23×C3111yescharacteristic
420#95C23×C3111yescharacteristic
421C31#1816111yescharacteristic, direct product factor, Sylow
422#41—42no—
423#41—42no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—41no—
430C2×Q8—42no—
431C2×Q8C2×D4×C3112yes—
432C2×D4—21no—
433C2×D4—22no—
434C2×D4—21no—
435C2×D4C2×D4×C3112yes—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4C2×D4×C3111yescharacteristic
444C22×C4C2×D4×C3111yescharacteristic
445C22×C4C2×D4×C3111yescharacteristic
446C22×C4C2×D4×C3111yescharacteristic
447C22×C4C2×D4×C3111yescharacteristic
448C22×C4C2×D4×C3111yescharacteristic
449C22×C4C24×C3111yescharacteristic
450Q16—44no—
451SD16—44no—
452C2×C8—22no—
453C2×C8—22no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
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—22no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4C2×D4×C3112yes—
476C4⋊C4C2×D4×C3112yes—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C42—22no—
487C42—22no—
488C42—21no—
489C42—22no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—42no—
494C42C2×D4×C3112yes—
495C42C2×D4×C3112yes—
496C23—22no—
497C23C22×D4×C3111yescharacteristic
498Q8—24no—
499Q8—44no—
500Q8—42no—
501Q8—42no—
502Q8—42no—
503Q8—81no—
504D4—42no—
505D4—42no—
506D4—42no—
507D4—42no—
508D4—81no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—82no—
539C2×C4C31×#12011yescharacteristic
540C2×C4C31×#12011yescharacteristic
541C2×C4C31×#12012yes—
542C2×C4C31×#13712yes—
543C2×C4C22×D4×C3112yes—
544C8—42no—
545C8—42no—
546C22—21no—
547C22—21no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22C2×C31×#12011yescharacteristic
552C22C2×C31×#13711yescharacteristic
553C22C31×#57511yescharacteristic
554C4—22no—
555C4—22no—
556C4—21no—
557C4—21no—
558C4—22no—
559C4—41no—
560C4—42no—
561C4—42no—
562C4—42no—
563C4—41no—
564C4—41no—
565C4—42no—
566C4—81no—
567C4—81no—
568C4—81no—
569C2—21no—
570C2—21no—
571C2—82no—
572C2C31×#300711yescharacteristic
573C2C31×#302011yescharacteristic
574C2C31×#302511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#537C22
LCSG ▷ C22×C4 ▷ 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×#3117C2semidirect
3. C31×#2581C2semidirect
4. C31×#1617C2semidirect
6. C31×#1615C2semidirect
7. C31×#1605C2semidirect
9. C31×#1603C2semidirect
10. C31×#1512C2semidirect
11. C31×#1467C2semidirect
208. #18161C31direct (N × H)
260. #3117C2×C31semidirect
261. #2581C2×C31semidirect
262. #1617C2×C31semidirect
264. #1615C2×C31semidirect
265. #1605C2×C31semidirect
267. #1603C2×C31semidirect
268. #1512C2×C31semidirect
269. #1467C2×C31semidirect
421. C31#18161direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34612
C31×#31172
C31×#25812
C31×#16172
C31×#16162
C31×#16152
C31×#16052
C31×#16042
C31×#16032
C31×#15122
C31×#14672
#1816131