Finite Group Databasebeta

← C31×#19093 #21475852

Subgroups — C31×#19093 #21475852

Order 7936.

782 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3563C211yesmaximal, characteristic
2C31×#3240C211yesmaximal, characteristic
3C31×#3237C211yesmaximal, characteristic, complemented
4C31×#3236C211yesmaximal, characteristic, complemented
5C31×#1664C212yesmaximal, complemented
6C31×#1664C212yesmaximal, complemented
7C31×#1663C212yesmaximal, complemented
8C31×#1663C212yesmaximal, complemented
9C31×#1537C211yesmaximal, characteristic, complemented
10C31×#1535C211yesmaximal, characteristic
11C31×#1534C211yesmaximal, characteristic, complemented
12C31×#564C2212yes—
13C31×#563C2212yes—
14C31×#557—21no—
15C31×#554—21no—
16C31×#552—21no—
17C31×#552—22no—
18C31×#552—21no—
19C31×#544—21no—
20C31×#544—21no—
21C31×#541—21no—
22C31×#541—22no—
23C31×#539—21no—
24C31×#533C2211yescharacteristic
25C31×#533C2211yescharacteristic
26C31×#529C2211yescharacteristic
27C2×C31×#125C2211yescharacteristic
28C31×#501C2211yescharacteristic
29C31×#501C2211yescharacteristic
30C31×#501C2212yes—
31C31×#501C2212yes—
32C31×#491C2212yes—
33C31×#491C2212yes—
34C31×#489C2211yescharacteristic
35C31×#489C2211yescharacteristic
36C31×#485—21no—
37C31×#485—22no—
38C31×#484—21no—
39C31×#483—21no—
40C31×#482—21no—
41C31×#479—22no—
42C31×#479—21no—
43C31×#478—21no—
44C31×#477—21no—
45C31×#475—21no—
46C31×#432C2211yescharacteristic
47C31×#432C2211yescharacteristic
48C31×#432C2211yescharacteristic
49C31×#330C2212yes—
50C31×#330C2212yes—
51C31×#330C2212yes—
52C31×#330C2212yes—
53C31×#330C2212yes—
54C31×#329C2212yes—
55C31×C4⋊Q8—22no—
56C31×C4⋊Q8—21no—
57C31×C4⋊Q8—22no—
58C31×C4⋊Q8D411yescharacteristic
59C31×C4⋊Q8D411yescharacteristic
60C31×C4⋊Q8C2311yescharacteristic
61C31×C4⋊Q8C2311yescharacteristic
62C31×#126—22no—
63C31×#126—22no—
64C31×#126—22no—
65C31×#126—22no—
66C31×#126—21no—
67C31×#125—22no—
68C31×#125—22no—
69C31×#125—21no—
70C31×#125—22no—
71C31×#125—22no—
72C31×#125—22no—
73C31×#125—22no—
74C31×#125—22no—
75C31×#125—22no—
76C31×#125—21no—
77C31×#125—21no—
78C31×#125—21no—
79C31×#125D412yes—
80C31×#125C2312yes—
81C31×#125C2312yes—
82C31×#125C2311yescharacteristic
83C31×#124C2311yescharacteristic
84C31×#123—22no—
85C31×#123—22no—
86C31×#123—21no—
87C31×C22⋊Q8—22no—
88C31×C22⋊Q8—21no—
89C31×C22⋊Q8—21no—
90C31×C22⋊Q8—22no—
91C31×C22⋊Q8—21no—
92C31×C22⋊Q8—21no—
93C31×C22⋊Q8—22no—
94C31×C22⋊Q8—22no—
95C31×C22⋊Q8—21no—
96C31×C22⋊Q8—21no—
97C31×C22⋊Q8D411yescharacteristic
98C31×#121D411yescharacteristic
99C4×Q8×C31—21no—
100C4×Q8×C31—22no—
101C4×Q8×C31—21no—
102C4×Q8×C31—21no—
103C4×Q8×C31—21no—
104C4×Q8×C31—21no—
105C4×Q8×C31—21no—
106C4×Q8×C31D411yescharacteristic
107C4×D4×C31—22no—
108C4×D4×C31—21no—
109C4×D4×C31D411yescharacteristic
110C31×#117—22no—
111C31×#117—21no—
112C31×#117—22no—
113C31×#117—21no—
114C31×#117—21no—
115C31×#117D412yes—
116C2×C4⋊C4×C31—21no—
117C2×C4⋊C4×C31—21no—
118C2×C4⋊C4×C31—21no—
119C2×C4⋊C4×C31D411yescharacteristic
120C2×C4⋊C4×C31D411yescharacteristic
121C2×C42×C31C2311yescharacteristic
122C31×#107—22no—
123C31×#107—21no—
124C31×#107—21no—
125C31×#107—22no—
126C31×#106—22no—
127C31×#106—21no—
128C31×#106—21no—
129C31×#106—22no—
130C31×C4⋊C8—21no—
131C31×C4⋊C8—21no—
132C31×C4⋊C8—21no—
133C31×#103—21no—
134C31×#103—21no—
135C31×#103—22no—
136C31×#103—22no—
137C31×#103—22no—
138C31×#103—22no—
139C31×#102—21no—
140C31×#102—21no—
141C31×C22⋊C8—21no—
142C31×C22⋊C8—21no—
143C31×C22⋊C8—21no—
144C31×#97C2311yescharacteristic
145C31×#97C2311yescharacteristic
146C31×#97C2312yes—
147C31×#97C2312yes—
148C2×Q8×C31—22no—
149C2×Q8×C31—21no—
150C2×Q8×C31—21no—
151C2×Q8×C31—21no—
152C2×Q8×C31—21no—
153C2×Q8×C31C2×D411yescharacteristic
154C2×D4×C31—21no—
155C2×D4×C31C2×D411yescharacteristic
156C22×C4×C31—21no—
157C22×C4×C31—22no—
158C22×C4×C31—21no—
159C22×C4×C31—21no—
160C22×C4×C31C2×D411yescharacteristic
161C22×C4×C31C2×D411yescharacteristic
162C22×C4×C31C2×D411yescharacteristic
163C2×C8×C31—21no—
164C2×C8×C31—21no—
165C2×C8×C31—22no—
166C2×C8×C31—22no—
167C4⋊C4×C31—22no—
168C4⋊C4×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—21no—
176C4⋊C4×C31—21no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—21no—
180C4⋊C4×C31—21no—
181C4⋊C4×C31—21no—
182C4⋊C4×C31—21no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—21no—
186C4⋊C4×C31—21no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—22no—
189C4⋊C4×C31—21no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—21no—
192C4⋊C4×C31—21no—
193C4⋊C4×C31—41no—
194C4⋊C4×C31—41no—
195C4⋊C4×C31C2×D412yes—
196C4⋊C4×C31C2×D412yes—
197C4⋊C4×C31C2×D412yes—
198C4⋊C4×C31C2×D412yes—
199C4⋊C4×C31C2×D411yescharacteristic
200C4⋊C4×C31C2×D411yescharacteristic
201C22⋊C4×C31—22no—
202C22⋊C4×C31—22no—
203C22⋊C4×C31—22no—
204C22⋊C4×C31—21no—
205C22⋊C4×C31—22no—
206C22⋊C4×C31—21no—
207C22⋊C4×C31—21no—
208C22⋊C4×C31—21no—
209C22⋊C4×C31—21no—
210C22⋊C4×C31—41no—
211C42×C31—22no—
212C42×C31—22no—
213C42×C31—22no—
214C42×C31—22no—
215C42×C31—21no—
216C42×C31—41no—
217C42×C31C2×D411yescharacteristic
218C42×C31C2×D411yescharacteristic
219C42×C31C2×D411yescharacteristic
220C42×C31C2411yescharacteristic
221#19093C3111yesmaximal, characteristic, direct product factor, Sylow
222C23×C31—21no—
223C23×C31#12011yescharacteristic
224Q8×C31—41no—
225Q8×C31—42no—
226Q8×C31—42no—
227Q8×C31—42no—
228Q8×C31—81no—
229Q8×C31—82no—
230D4×C31—41no—
231D4×C31—42no—
232C2×C4×C31—22no—
233C2×C4×C31—21no—
234C2×C4×C31—21no—
235C2×C4×C31—21no—
236C2×C4×C31—22no—
237C2×C4×C31—22no—
238C2×C4×C31—22no—
239C2×C4×C31—41no—
240C2×C4×C31—41no—
241C2×C4×C31—41no—
242C2×C4×C31—42no—
243C2×C4×C31—42no—
244C2×C4×C31—41no—
245C2×C4×C31—41no—
246C2×C4×C31—81no—
247C2×C4×C31—82no—
248C2×C4×C31#12011yescharacteristic
249C2×C4×C31#12011yescharacteristic
250C2×C4×C31#12011yescharacteristic
251C2×C4×C31C22×D411yescharacteristic
252C2×C4×C31C22×D411yescharacteristic
253C2×C4×C31C22×D411yescharacteristic
254C8×C31—42no—
255C8×C31—42no—
256C8×C31—41no—
257C8×C31—41no—
258#3563C2×C3111yescharacteristic
259#3240C2×C3111yescharacteristic
260#3237C2×C3111yescharacteristic, complemented
261#3236C2×C3111yescharacteristic, complemented
262#1664C2×C3112yescomplemented
263#1664C2×C3112yescomplemented
264#1663C2×C3112yescomplemented
265#1663C2×C3112yescomplemented
266#1537C2×C3111yescharacteristic, complemented
267#1535C2×C3111yescharacteristic
268#1534C2×C3111yescharacteristic, complemented
269C22×C31—21no—
270C22×C31—21no—
271C22×C31—21no—
272C22×C31—41no—
273C22×C31—41no—
274C22×C31—41no—
275C22×C31C2×#12011yescharacteristic
276C4×C31—21no—
277C4×C31—21no—
278C4×C31—21no—
279C4×C31—21no—
280C4×C31—21no—
281C4×C31—21no—
282C4×C31—41no—
283C4×C31—42no—
284C4×C31—41no—
285C4×C31—42no—
286C4×C31—42no—
287C4×C31—42no—
288C4×C31—41no—
289C4×C31—81no—
290C4×C31—81no—
291#564C22×C3112yes—
292#563C22×C3112yes—
293#557—21no—
294#554—21no—
295#552—21no—
296#552—22no—
297#552—21no—
298#544—21no—
299#544—21no—
300#541—21no—
301#541—22no—
302#539—21no—
303#533C22×C3111yescharacteristic
304#533C22×C3111yescharacteristic
305#529C22×C3111yescharacteristic
306C2×#125C22×C3111yescharacteristic
307#501C22×C3111yescharacteristic
308#501C22×C3111yescharacteristic
309#501C22×C3112yes—
310#501C22×C3112yes—
311#491C22×C3112yes—
312#491C22×C3112yes—
313#489C22×C3111yescharacteristic
314#489C22×C3111yescharacteristic
315#485—21no—
316#485—22no—
317#484—21no—
318#483—21no—
319#482—21no—
320#479—22no—
321#479—21no—
322#478—21no—
323#477—21no—
324#475—21no—
325#432C22×C3111yescharacteristic
326#432C22×C3111yescharacteristic
327#432C22×C3111yescharacteristic
328#330C22×C3112yes—
329#330C22×C3112yes—
330#330C22×C3112yes—
331#330C22×C3112yes—
332#330C22×C3112yes—
333#329C22×C3112yes—
334C2×C31—41no—
335C2×C31—81no—
336C2×C31#302011yescharacteristic
337C2×C31#302011yescharacteristic
338C2×C31#302211yescharacteristic
339C4⋊Q8—22no—
340C4⋊Q8—21no—
341C4⋊Q8—22no—
342C4⋊Q8D4×C3111yescharacteristic
343C4⋊Q8D4×C3111yescharacteristic
344C4⋊Q8C23×C3111yescharacteristic
345C4⋊Q8C23×C3111yescharacteristic
346#126—22no—
347#126—22no—
348#126—22no—
349#126—22no—
350#126—21no—
351#125—22no—
352#125—22no—
353#125—21no—
354#125—22no—
355#125—22no—
356#125—22no—
357#125—22no—
358#125—22no—
359#125—22no—
360#125—21no—
361#125—21no—
362#125—21no—
363#125D4×C3112yes—
364#125C23×C3112yes—
365#125C23×C3111yescharacteristic
366#125C23×C3112yes—
367#124C23×C3111yescharacteristic
368#123—22no—
369#123—22no—
370#123—21no—
371C22⋊Q8—22no—
372C22⋊Q8—21no—
373C22⋊Q8—21no—
374C22⋊Q8—22no—
375C22⋊Q8—21no—
376C22⋊Q8—21no—
377C22⋊Q8—22no—
378C22⋊Q8—22no—
379C22⋊Q8—21no—
380C22⋊Q8—21no—
381C22⋊Q8D4×C3111yescharacteristic
382#121D4×C3111yescharacteristic
383C4×Q8—21no—
384C4×Q8—22no—
385C4×Q8—21no—
386C4×Q8—21no—
387C4×Q8—21no—
388C4×Q8—21no—
389C4×Q8—21no—
390C4×Q8D4×C3111yescharacteristic
391C4×D4—22no—
392C4×D4—21no—
393C4×D4D4×C3111yescharacteristic
394#117—22no—
395#117—21no—
396#117—22no—
397#117—21no—
398#117—21no—
399#117D4×C3112yes—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4D4×C3111yescharacteristic
404C2×C4⋊C4D4×C3111yescharacteristic
405C2×C42C23×C3111yescharacteristic
406#107—22no—
407#107—21no—
408#107—21no—
409#107—22no—
410#106—22no—
411#106—21no—
412#106—21no—
413#106—22no—
414C4⋊C8—21no—
415C4⋊C8—21no—
416C4⋊C8—21no—
417#103—21no—
418#103—21no—
419#103—22no—
420#103—22no—
421#103—22no—
422#103—22no—
423#102—21no—
424#102—21no—
425C22⋊C8—21no—
426C22⋊C8—21no—
427C22⋊C8—21no—
428#97C23×C3112yes—
429#97C23×C3112yes—
430#97C23×C3111yescharacteristic
431#97C23×C3111yescharacteristic
432C31#1909311yescharacteristic, direct product factor, Sylow
433C2×Q8—22no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8C2×D4×C3111yescharacteristic
439C2×D4—21no—
440C2×D4C2×D4×C3111yescharacteristic
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4C2×D4×C3111yescharacteristic
446C22×C4C2×D4×C3111yescharacteristic
447C22×C4C2×D4×C3111yescharacteristic
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—22no—
451C2×C8—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4C2×D4×C3112yes—
481C4⋊C4C2×D4×C3112yes—
482C4⋊C4C2×D4×C3112yes—
483C4⋊C4C2×D4×C3112yes—
484C4⋊C4C2×D4×C3111yescharacteristic
485C4⋊C4C2×D4×C3111yescharacteristic
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—21no—
490C22⋊C4—22no—
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—21no—
495C22⋊C4—41no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—21no—
501C42—41no—
502C42C2×D4×C3111yescharacteristic
503C42C2×D4×C3111yescharacteristic
504C42C2×D4×C3111yescharacteristic
505C42C24×C3111yescharacteristic
506C23—21no—
507C23C31×#12011yescharacteristic
508Q8—41no—
509Q8—42no—
510Q8—42no—
511Q8—42no—
512Q8—81no—
513Q8—82no—
514D4—41no—
515D4—42no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—81no—
531C2×C4—82no—
532C2×C4C31×#12011yescharacteristic
533C2×C4C31×#12011yescharacteristic
534C2×C4C31×#12011yescharacteristic
535C2×C4C22×D4×C3111yescharacteristic
536C2×C4C22×D4×C3111yescharacteristic
537C2×C4C22×D4×C3111yescharacteristic
538C8—42no—
539C8—42no—
540C8—41no—
541C8—41no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22C2×C31×#12011yescharacteristic
549C4—21no—
550C4—21no—
551C4—21no—
552C4—21no—
553C4—21no—
554C4—21no—
555C4—41no—
556C4—42no—
557C4—41no—
558C4—42no—
559C4—42no—
560C4—42no—
561C4—41no—
562C4—81no—
563C4—81no—
564C2—41no—
565C2—81no—
566C2C31×#302011yescharacteristic
567C2C31×#302011yescharacteristic
568C2C31×#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
3. C31×#3237C2semidirect
4. C31×#3236C2semidirect
5. C31×#1664C2semidirect
6. C31×#1664C2semidirect
7. C31×#1663C2semidirect
8. C31×#1663C2semidirect
9. C31×#1537C2semidirect
11. C31×#1534C2semidirect
221. #19093C31direct (N × H)
260. #3237C2×C31semidirect
261. #3236C2×C31semidirect
262. #1664C2×C31semidirect
263. #1664C2×C31semidirect
264. #1663C2×C31semidirect
265. #1663C2×C31semidirect
266. #1537C2×C31semidirect
268. #1534C2×C31semidirect
432. C31#19093direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35632
C31×#32402
C31×#32372
C31×#32362
C31×#16642
C31×#16632
C31×#15372
C31×#15352
C31×#15342
#1909331