Finite Group Databasebeta

← C31×#9524 #21466283

Subgroups — C31×#9524 #21466283

Order 7936.

686 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3007C211yesmaximal, characteristic
2C31×#2332C211yesmaximal, characteristic, complemented
3C31×#1789C211yesmaximal, characteristic
4C31×#1782C211yesmaximal, characteristic
5C31×#1511C211yesmaximal, characteristic, complemented
6C31×#1306C211yesmaximal, characteristic, complemented
7C31×#1284C211yesmaximal, characteristic, complemented
8C22×#41×C31C2211yescharacteristic
9C2×C31×#137—21no—
10C2×C31×C22⋊Q8C2211yescharacteristic
11C31×#452—22no—
12C31×#451C412yescomplemented
13C31×#450C412yescomplemented
14C2×C31×#104—21no—
15C2×C31×#104—21no—
16C31×#417—21no—
17C2×C31×#99—21no—
18C2×C31×#99—21no—
19C31×#407C2211yescharacteristic
20C31×#407C2211yescharacteristic
21C31×#391—21no—
22C31×#388—21no—
23C31×#388—21no—
24C31×#387—21no—
25C31×#387—21no—
26C31×#381C2211yescharacteristic
27C31×#381C2211yescharacteristic
28C4×C4⋊C4×C31—21no—
29C4×C22⋊C4×C31C2211yescharacteristic
30C31×#328—22no—
31C2×#41×C31—21no—
32C2×#41×C31—21no—
33C2×#41×C31—21no—
34C2×#41×C31—21no—
35C2×#41×C31—42no—
36C2×#41×C31—42no—
37C22×Q8×C31C2×C411yescharacteristic
38C22×Q8×C31D411yescharacteristic
39C22×D4×C31—21no—
40C22×D4×C31C2×C411yescharacteristic
41C23×C4×C31—21no—
42C23×C4×C31C2311yescharacteristic
43C31×#137—42no—
44C31×#137—42no—
45C2×Q16×C31—22no—
46C2×SD16×C31—22no—
47C2×M16×C31—21no—
48C2×M16×C31—21no—
49C31×C22⋊Q8—21no—
50C31×C22⋊Q8—21no—
51C31×C22⋊Q8C2×C412yes—
52C31×C22⋊Q8D412yes—
53C2×C4⋊C4×C31—21no—
54C2×C4⋊C4×C31—21no—
55C2×C4⋊C4×C31—21no—
56C2×C4⋊C4×C31D411yescharacteristic
57C2×C22⋊C4×C31—21no—
58C2×C22⋊C4×C31—21no—
59C2×C22⋊C4×C31—21no—
60C2×C22⋊C4×C31D411yescharacteristic
61C2×C22⋊C4×C31D411yescharacteristic
62C2×C42×C31—21no—
63C2×C42×C31—21no—
64C2×C42×C31D411yescharacteristic
65C2×C42×C31D411yescharacteristic
66C31×#104—42no—
67C31×#104—42no—
68C31×#103—22no—
69C31×#102—22no—
70C31×#99—42no—
71C31×#99—42no—
72C31×C22⋊C8—21no—
73C31×C22⋊C8—21no—
74C31×C22⋊C8—21no—
75C31×C22⋊C8C2×C412yes—
76C31×#95—21no—
77C31×#95—21no—
78C31×#95—21no—
79C31×#95—21no—
80C31×#95—21no—
81C31×#95—21no—
82C31×#95—21no—
83C31×#95—21no—
84C31×#95—21no—
85C24×C31—21no—
86C24×C31C2×D411yescharacteristic
87#41×C31—22no—
88#41×C31—22no—
89#41×C31—22no—
90#41×C31—22no—
91#41×C31—41no—
92#41×C31—41no—
93#41×C31—41no—
94#41×C31—41no—
95#41×C31—42no—
96#41×C31—42no—
97#41×C31—42no—
98#41×C31—42no—
99C2×Q8×C31—21no—
100C2×Q8×C31—21no—
101C2×Q8×C31—21no—
102C2×Q8×C31—42no—
103C2×Q8×C31—41no—
104C2×Q8×C31—41no—
105C2×Q8×C31C22⋊C412yes—
106C2×D4×C31—22no—
107C2×D4×C31—22no—
108C2×D4×C31—21no—
109C2×D4×C31—21no—
110C2×D4×C31—42no—
111C2×D4×C31—41no—
112C2×D4×C31—41no—
113C2×D4×C31—41no—
114C2×D4×C31—41no—
115C22×C4×C31—21no—
116C22×C4×C31—21no—
117C22×C4×C31—21no—
118C22×C4×C31—21no—
119C22×C4×C31—21no—
120C22×C4×C31—21no—
121C22×C4×C31—21no—
122C22×C4×C31—41no—
123C22×C4×C31—41no—
124C22×C4×C31—42no—
125C22×C4×C31—42no—
126C22×C4×C31—41no—
127C22×C4×C31—41no—
128C22×C4×C31—41no—
129C22×C4×C31C22×C411yescharacteristic
130C22×C4×C31C2×D411yescharacteristic
131C22×C4×C31C2×D411yescharacteristic
132C22×C4×C31C2×D411yescharacteristic
133C22×C4×C31#4111yescharacteristic
134C22×C4×C31#4111yescharacteristic
135Q16×C31—44no—
136SD16×C31—44no—
137M16×C31—41no—
138M16×C31—41no—
139M16×C31—41no—
140M16×C31—41no—
141C2×C8×C31—22no—
142C2×C8×C31—41no—
143C4⋊C4×C31—22no—
144C4⋊C4×C31—22no—
145C4⋊C4×C31—21no—
146C4⋊C4×C31—42no—
147C4⋊C4×C31—81no—
148C4⋊C4×C31C22⋊C412yes—
149C22⋊C4×C31—22no—
150C22⋊C4×C31—22no—
151C22⋊C4×C31—42no—
152C22⋊C4×C31—41no—
153C22⋊C4×C31—41no—
154C22⋊C4×C31—41no—
155C22⋊C4×C31—41no—
156C42×C31—42no—
157C42×C31—41no—
158C42×C31—41no—
159C42×C31—81no—
160#9524C3111yesmaximal, characteristic, direct product factor, Sylow
161C23×C31—21no—
162C23×C31—21no—
163C23×C31—22no—
164C23×C31—41no—
165C23×C31—41no—
166C23×C31—41no—
167C23×C31—41no—
168C23×C31—81no—
169C23×C31—81no—
170C23×C31C4×D411yescharacteristic
171C23×C31#12011yescharacteristic
172C23×C31#12111yescharacteristic
173Q8×C31—22no—
174Q8×C31—22no—
175Q8×C31—42no—
176Q8×C31—41no—
177Q8×C31—41no—
178Q8×C31—81no—
179D4×C31—22no—
180D4×C31—22no—
181D4×C31—41no—
182D4×C31—41no—
183D4×C31—42no—
184D4×C31—42no—
185D4×C31—42no—
186D4×C31—42no—
187C2×C4×C31—21no—
188C2×C4×C31—21no—
189C2×C4×C31—21no—
190C2×C4×C31—21no—
191C2×C4×C31—22no—
192C2×C4×C31—22no—
193C2×C4×C31—22no—
194C2×C4×C31—22no—
195C2×C4×C31—41no—
196C2×C4×C31—41no—
197C2×C4×C31—41no—
198C2×C4×C31—41no—
199C2×C4×C31—42no—
200C2×C4×C31—42no—
201C2×C4×C31—42no—
202C2×C4×C31—42no—
203C2×C4×C31—41no—
204C2×C4×C31—41no—
205C2×C4×C31—41no—
206C2×C4×C31—41no—
207C2×C4×C31—41no—
208C2×C4×C31—41no—
209C2×C4×C31—41no—
210C2×C4×C31—41no—
211C2×C4×C31—41no—
212C2×C4×C31—41no—
213C2×C4×C31—41no—
214C2×C4×C31—41no—
215C2×C4×C31—41no—
216C2×C4×C31—41no—
217C2×C4×C31—41no—
218C2×C4×C31—41no—
219C2×C4×C31—41no—
220C2×C4×C31—81no—
221C2×C4×C31—81no—
222C2×C4×C31—81no—
223C2×C4×C31C2×C22⋊C411yescharacteristic
224C2×C4×C31C4×D411yescharacteristic
225C2×C4×C31#12111yescharacteristic
226C2×C4×C31#12311yescharacteristic
227C8×C31—42no—
228C8×C31—81no—
229#3007C2×C3111yescharacteristic
230#2332C2×C3111yescharacteristic, complemented
231#1789C2×C3111yescharacteristic
232#1782C2×C3111yescharacteristic
233#1511C2×C3111yescharacteristic, complemented
234#1306C2×C3111yescharacteristic, complemented
235#1284C2×C3111yescharacteristic, complemented
236C22×C31—21no—
237C22×C31—21no—
238C22×C31—21no—
239C22×C31—21no—
240C22×C31—21no—
241C22×C31—21no—
242C22×C31—21no—
243C22×C31—21no—
244C22×C31—42no—
245C22×C31—81no—
246C22×C31—81no—
247C22×C31—82no—
248C22×C31—82no—
249C22×C31—81no—
250C22×C31—81no—
251C22×C31#38611yescharacteristic
252C22×C31B(4,𝔽2)11yescharacteristic
253C22×C31B(4,𝔽2)11yescharacteristic
254C4×C31—21no—
255C4×C31—21no—
256C4×C31—21no—
257C4×C31—21no—
258C4×C31—42no—
259C4×C31—42no—
260C4×C31—42no—
261C4×C31—42no—
262C4×C31—81no—
263C4×C31—81no—
264C4×C31—81no—
265C4×C31—161no—
266C22×#41C22×C3111yescharacteristic
267C2×#137—21no—
268C2×C22⋊Q8C22×C3111yescharacteristic
269#452—22no—
270#451C4×C3112yescomplemented
271#450C4×C3112yescomplemented
272C2×#104—21no—
273C2×#104—21no—
274#417—21no—
275C2×#99—21no—
276C2×#99—21no—
277#407C22×C3111yescharacteristic
278#407C22×C3111yescharacteristic
279#391—21no—
280#388—21no—
281#388—21no—
282#387—21no—
283#387—21no—
284#381C22×C3111yescharacteristic
285#381C22×C3111yescharacteristic
286C4×C4⋊C4—21no—
287C4×C22⋊C4C22×C3111yescharacteristic
288#328—22no—
289C2×C31—21no—
290C2×C31—21no—
291C2×C31—41no—
292C2×C31—41no—
293C2×C31—82no—
294C2×C31#188911yescharacteristic
295C2×C31#190511yescharacteristic
296C2×C31#190611yescharacteristic
297C2×#41—21no—
298C2×#41—21no—
299C2×#41—21no—
300C2×#41—21no—
301C2×#41—42no—
302C2×#41—42no—
303C22×Q8C2×C4×C3111yescharacteristic
304C22×Q8D4×C3111yescharacteristic
305C22×D4—21no—
306C22×D4C2×C4×C3111yescharacteristic
307C23×C4—21no—
308C23×C4C23×C3111yescharacteristic
309#137—42no—
310#137—42no—
311C2×Q16—22no—
312C2×SD16—22no—
313C2×M16—21no—
314C2×M16—21no—
315C22⋊Q8—21no—
316C22⋊Q8—21no—
317C22⋊Q8C2×C4×C3112yes—
318C22⋊Q8D4×C3112yes—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4D4×C3111yescharacteristic
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4D4×C3111yescharacteristic
327C2×C22⋊C4D4×C3111yescharacteristic
328C2×C42—21no—
329C2×C42—21no—
330C2×C42D4×C3111yescharacteristic
331C2×C42D4×C3111yescharacteristic
332#104—42no—
333#104—42no—
334#103—22no—
335#102—22no—
336#99—42no—
337#99—42no—
338C22⋊C8—21no—
339C22⋊C8—21no—
340C22⋊C8—21no—
341C22⋊C8C2×C4×C3112yes—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—21no—
351C31#952411yescharacteristic, direct product factor, Sylow
352C24—21no—
353C24C2×D4×C3111yescharacteristic
354#41—22no—
355#41—22no—
356#41—22no—
357#41—22no—
358#41—41no—
359#41—41no—
360#41—41no—
361#41—41no—
362#41—42no—
363#41—42no—
364#41—42no—
365#41—42no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—42no—
370C2×Q8—41no—
371C2×Q8—41no—
372C2×Q8C22⋊C4×C3112yes—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—42no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—41no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4C22×C4×C3111yescharacteristic
397C22×C4C2×D4×C3111yescharacteristic
398C22×C4C2×D4×C3111yescharacteristic
399C22×C4C2×D4×C3111yescharacteristic
400C22×C4#41×C3111yescharacteristic
401C22×C4#41×C3111yescharacteristic
402Q16—44no—
403SD16—44no—
404M16—41no—
405M16—41no—
406M16—41no—
407M16—41no—
408C2×C8—22no—
409C2×C8—41no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—42no—
414C4⋊C4—81no—
415C4⋊C4C22⋊C4×C3112yes—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—42no—
419C22⋊C4—41no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—41no—
423C42—41no—
424C42—41no—
425C42—42no—
426C42—81no—
427C23—21no—
428C23—21no—
429C23—22no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—81no—
435C23—81no—
436C23C4×D4×C3111yescharacteristic
437C23C31×#12011yescharacteristic
438C23C31×#12111yescharacteristic
439Q8—22no—
440Q8—22no—
441Q8—42no—
442Q8—41no—
443Q8—41no—
444Q8—81no—
445D4—22no—
446D4—22no—
447D4—41no—
448D4—41no—
449D4—42no—
450D4—42no—
451D4—42no—
452D4—42no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4C2×C22⋊C4×C3111yescharacteristic
490C2×C4C4×D4×C3111yescharacteristic
491C2×C4C31×#12111yescharacteristic
492C2×C4C31×#12311yescharacteristic
493C8—42no—
494C8—81no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—42no—
504C22—81no—
505C22—81no—
506C22—82no—
507C22—82no—
508C22—81no—
509C22—81no—
510C22C31×#38611yescharacteristic
511C22C31×B(4,𝔽2)11yescharacteristic
512C22C31×B(4,𝔽2)11yescharacteristic
513C4—21no—
514C4—21no—
515C4—21no—
516C4—21no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—81no—
522C4—81no—
523C4—81no—
524C4—161no—
525C2—21no—
526C2—21no—
527C2—41no—
528C2—41no—
529C2—82no—
530C2C31×#188911yescharacteristic
531C2C31×#190511yescharacteristic
532C2C31×#190611yescharacteristic
subgroupG/H
center, socleC22×C31#386
commutatorC22×C4C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC22×#41×C31C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2332C2semidirect
5. C31×#1511C2semidirect
6. C31×#1306C2semidirect
7. C31×#1284C2semidirect
12. C31×#451C4semidirect
13. C31×#450C4semidirect
160. #9524C31direct (N × H)
230. #2332C2×C31semidirect
233. #1511C2×C31semidirect
234. #1306C2×C31semidirect
235. #1284C2×C31semidirect
270. #451C4×C31semidirect
271. #450C4×C31semidirect
351. C31#9524direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#30072
C31×#23322
C31×#17892
C31×#17822
C31×#15112
C31×#13062
C31×#12842
#952431