Finite Group Databasebeta

← C31×#19046 #21475805

Subgroups — C31×#19046 #21475805

Order 7936.

836 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3563C211yesmaximal, characteristic, complemented
2C31×#3234C211yesmaximal, characteristic, complemented
3C31×#3180C211yesmaximal, characteristic
4C31×#3135C211yesmaximal, characteristic, complemented
5C31×#1664C212yesmaximal, complemented
6C31×#1663C212yesmaximal, complemented
7C31×#1658C212yesmaximal, complemented
8C31×#1657C212yesmaximal, complemented
9C31×#1532C211yesmaximal, characteristic, complemented
10C31×#1531C211yesmaximal, characteristic, complemented
11C2×C31×#330C211yesmaximal, 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×#532C2211yescharacteristic
26C31×#529C2211yescharacteristic
27C2×C31×#125C2211yescharacteristic
28C31×#501C2211yescharacteristic
29C31×#501C2212yes—
30C31×#496C2211yescharacteristic
31C31×#491C2212yes—
32C31×#490C2212yes—
33C31×#490C2211yescharacteristic
34C31×#490C2212yes—
35C31×#489—21no—
36C31×#489—21no—
37C31×#489C2211yescharacteristic
38C31×#481—21no—
39C31×#480—21no—
40C31×#478—22no—
41C31×#466—21no—
42C31×#465—21no—
43C31×#442—21no—
44C31×#440—21no—
45C31×#432C2211yescharacteristic
46C31×#416—22no—
47C2×C31×#97C2211yescharacteristic
48C2×C31×#97C2211yescharacteristic
49C31×#330C2212yes—
50C31×#330C2212yes—
51C31×#330C2212yes—
52C31×#330C2212yes—
53C31×#330C2212yes—
54C31×#329C2212yes—
55C2×#41×C31—21no—
56C2×SD16×C31—21no—
57C2×D8×C31—21no—
58C22×C8×C31—21no—
59C22×C8×C31—21no—
60C31×C4⋊Q8—22no—
61C31×C4⋊Q8—21no—
62C31×C4⋊Q8—22no—
63C31×C4⋊Q8D411yescharacteristic
64C31×C4⋊Q8D411yescharacteristic
65C31×C4⋊Q8C2311yescharacteristic
66C31×#127C2311yescharacteristic
67C31×#126—22no—
68C31×#126—22no—
69C31×#126—22no—
70C31×#126—22no—
71C31×#126—21no—
72C31×#125—22no—
73C31×#125—22no—
74C31×#125—21no—
75C31×#125—22no—
76C31×#125—22no—
77C31×#125—22no—
78C31×#125—22no—
79C31×#125—22no—
80C31×#125—22no—
81C31×#125—21no—
82C31×#125—21no—
83C31×#125D412yes—
84C31×#125C2312yes—
85C31×#125C2312yes—
86C31×#125C2311yescharacteristic
87C31×#124—21no—
88C31×#124C2311yescharacteristic
89C31×#123—22no—
90C31×#123—22no—
91C31×#123—21no—
92C31×C22⋊Q8—22no—
93C31×C22⋊Q8—21no—
94C31×C22⋊Q8—21no—
95C31×C22⋊Q8—22no—
96C31×C22⋊Q8—21no—
97C31×C22⋊Q8—21no—
98C31×C22⋊Q8—22no—
99C31×C22⋊Q8—22no—
100C31×C22⋊Q8D411yescharacteristic
101C31×#121—21no—
102C31×#121—21no—
103C31×#121D411yescharacteristic
104C4×Q8×C31—21no—
105C4×Q8×C31—22no—
106C4×Q8×C31—21no—
107C4×Q8×C31—21no—
108C4×Q8×C31—21no—
109C4×Q8×C31D411yescharacteristic
110C4×D4×C31—22no—
111C4×D4×C31—21no—
112C4×D4×C31—21no—
113C4×D4×C31—21no—
114C4×D4×C31D411yescharacteristic
115C31×#117—22no—
116C31×#117—21no—
117C31×#117—22no—
118C31×#117—21no—
119C31×#117D412yes—
120C2×C4⋊C4×C31—21no—
121C2×C4⋊C4×C31—21no—
122C2×C4⋊C4×C31—21no—
123C2×C4⋊C4×C31D411yescharacteristic
124C2×C4⋊C4×C31D411yescharacteristic
125C2×C42×C31C2311yescharacteristic
126C31×#107—21no—
127C31×#107—22no—
128C31×#106—21no—
129C31×#106—22no—
130C31×C4⋊C8—21no—
131C31×#103—21no—
132C31×#103—22no—
133C31×#103—22no—
134C31×#102—21no—
135C31×#102—22no—
136C31×#102—22no—
137C31×#102—21no—
138C31×#102—21no—
139C31×#102—22no—
140C31×#102—22no—
141C31×C22⋊C8—21no—
142C31×#97—21no—
143C31×#97—21no—
144C31×#97C2311yescharacteristic
145C31×#97C2312yes—
146C31×#97C2311yescharacteristic
147C31×#97C2312yes—
148#41×C31—44no—
149C2×Q8×C31—22no—
150C2×Q8×C31—21no—
151C2×Q8×C31—21no—
152C2×Q8×C31—21no—
153C2×Q8×C31C2×D411yescharacteristic
154C2×D4×C31—21no—
155C2×D4×C31—21no—
156C2×D4×C31—21no—
157C2×D4×C31—21no—
158C2×D4×C31—41no—
159C2×D4×C31C2×D411yescharacteristic
160C22×C4×C31—21no—
161C22×C4×C31—22no—
162C22×C4×C31—21no—
163C22×C4×C31—21no—
164C22×C4×C31—41no—
165C22×C4×C31C2×D411yescharacteristic
166C22×C4×C31C2×D411yescharacteristic
167C22×C4×C31C2×D411yescharacteristic
168SD16×C31—42no—
169D8×C31—42no—
170C2×C8×C31—21no—
171C2×C8×C31—21no—
172C2×C8×C31—22no—
173C2×C8×C31—22no—
174C2×C8×C31—42no—
175C2×C8×C31—42no—
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—22no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—21no—
185C4⋊C4×C31—21no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—21no—
189C4⋊C4×C31—21no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—21no—
192C4⋊C4×C31—22no—
193C4⋊C4×C31—22no—
194C4⋊C4×C31—21no—
195C4⋊C4×C31—21no—
196C4⋊C4×C31—22no—
197C4⋊C4×C31—22no—
198C4⋊C4×C31—21no—
199C4⋊C4×C31—21no—
200C4⋊C4×C31C2×D412yes—
201C4⋊C4×C31C2×D412yes—
202C4⋊C4×C31C2×D412yes—
203C4⋊C4×C31C2×D411yescharacteristic
204C4⋊C4×C31C2×D412yes—
205C4⋊C4×C31C2×D411yescharacteristic
206C22⋊C4×C31—22no—
207C22⋊C4×C31—22no—
208C22⋊C4×C31—22no—
209C22⋊C4×C31—21no—
210C22⋊C4×C31—22no—
211C22⋊C4×C31—21no—
212C22⋊C4×C31—21no—
213C22⋊C4×C31—21no—
214C22⋊C4×C31—21no—
215C22⋊C4×C31—41no—
216C22⋊C4×C31—41no—
217C42×C31—22no—
218C42×C31—22no—
219C42×C31—22no—
220C42×C31—22no—
221C42×C31—21no—
222C42×C31C2×D411yescharacteristic
223C42×C31C2×D411yescharacteristic
224C42×C31C2×D411yescharacteristic
225C42×C31C2411yescharacteristic
226#19046C3111yesmaximal, characteristic, direct product factor, Sylow
227C23×C31—21no—
228C23×C31—41no—
229C23×C31#12011yescharacteristic
230Q8×C31—41no—
231Q8×C31—42no—
232Q8×C31—42no—
233Q8×C31—81no—
234Q8×C31—82no—
235D4×C31—41no—
236D4×C31—41no—
237D4×C31—41no—
238D4×C31—42no—
239D4×C31—42no—
240D4×C31—42no—
241D4×C31—82no—
242C2×C4×C31—22no—
243C2×C4×C31—22no—
244C2×C4×C31—21no—
245C2×C4×C31—21no—
246C2×C4×C31—22no—
247C2×C4×C31—21no—
248C2×C4×C31—22no—
249C2×C4×C31—22no—
250C2×C4×C31—41no—
251C2×C4×C31—41no—
252C2×C4×C31—42no—
253C2×C4×C31—42no—
254C2×C4×C31—41no—
255C2×C4×C31—82no—
256C2×C4×C31—81no—
257C2×C4×C31—82no—
258C2×C4×C31#12011yescharacteristic
259C2×C4×C31#12011yescharacteristic
260C2×C4×C31#12011yescharacteristic
261C2×C4×C31C22×D411yescharacteristic
262C2×C4×C31C22×D411yescharacteristic
263C2×C4×C31C22×D411yescharacteristic
264C8×C31—22no—
265C8×C31—42no—
266C8×C31—42no—
267C8×C31—41no—
268#3563C2×C3111yescharacteristic, complemented
269#3234C2×C3111yescharacteristic, complemented
270#3180C2×C3111yescharacteristic
271#3135C2×C3111yescharacteristic, complemented
272#1664C2×C3112yescomplemented
273#1663C2×C3112yescomplemented
274#1658C2×C3112yescomplemented
275#1657C2×C3112yescomplemented
276#1532C2×C3111yescharacteristic, complemented
277#1531C2×C3111yescharacteristic, complemented
278C2×#330C2×C3111yescharacteristic, complemented
279C22×C31—21no—
280C22×C31—21no—
281C22×C31—41no—
282C22×C31—41no—
283C22×C31—41no—
284C22×C31—81no—
285C22×C31—81no—
286C22×C31—81no—
287C22×C31#45412yes—
288C22×C31C2×#12011yescharacteristic
289C4×C31—21no—
290C4×C31—21no—
291C4×C31—21no—
292C4×C31—21no—
293C4×C31—21no—
294C4×C31—21no—
295C4×C31—42no—
296C4×C31—41no—
297C4×C31—42no—
298C4×C31—42no—
299C4×C31—41no—
300C4×C31—41no—
301C4×C31—42no—
302C4×C31—81no—
303#564C22×C3112yes—
304#563C22×C3112yes—
305#557—21no—
306#554—21no—
307#552—21no—
308#552—22no—
309#552—21no—
310#544—21no—
311#544—21no—
312#541—21no—
313#541—22no—
314#539—21no—
315#533C22×C3111yescharacteristic
316#532C22×C3111yescharacteristic
317#529C22×C3111yescharacteristic
318C2×#125C22×C3111yescharacteristic
319#501C22×C3111yescharacteristic
320#501C22×C3112yes—
321#496C22×C3111yescharacteristic
322#491C22×C3112yes—
323#490C22×C3112yes—
324#490C22×C3111yescharacteristic
325#490C22×C3112yes—
326#489—21no—
327#489—21no—
328#489C22×C3111yescharacteristic
329#481—21no—
330#480—21no—
331#478—22no—
332#466—21no—
333#465—21no—
334#442—21no—
335#440—21no—
336#432C22×C3111yescharacteristic
337#416—22no—
338C2×#97C22×C3111yescharacteristic
339C2×#97C22×C3111yescharacteristic
340#330C22×C3112yes—
341#330C22×C3112yes—
342#330C22×C3112yes—
343#330C22×C3112yes—
344#330C22×C3112yes—
345#329C22×C3112yes—
346C2×C31—22no—
347C2×C31—81no—
348C2×C31—161no—
349C2×C31C2×#45411yescharacteristic
350C2×C31#301811yescharacteristic
351C2×C31#302011yescharacteristic
352C2×#41—21no—
353C2×SD16—21no—
354C2×D8—21no—
355C22×C8—21no—
356C22×C8—21no—
357C4⋊Q8—22no—
358C4⋊Q8—21no—
359C4⋊Q8—22no—
360C4⋊Q8D4×C3111yescharacteristic
361C4⋊Q8D4×C3111yescharacteristic
362C4⋊Q8C23×C3111yescharacteristic
363#127C23×C3111yescharacteristic
364#126—22no—
365#126—22no—
366#126—22no—
367#126—22no—
368#126—21no—
369#125—22no—
370#125—22no—
371#125—21no—
372#125—22no—
373#125—22no—
374#125—22no—
375#125—22no—
376#125—22no—
377#125—22no—
378#125—21no—
379#125—21no—
380#125D4×C3112yes—
381#125C23×C3112yes—
382#125C23×C3111yescharacteristic
383#125C23×C3112yes—
384#124—21no—
385#124C23×C3111yescharacteristic
386#123—22no—
387#123—22no—
388#123—21no—
389C22⋊Q8—22no—
390C22⋊Q8—21no—
391C22⋊Q8—21no—
392C22⋊Q8—22no—
393C22⋊Q8—21no—
394C22⋊Q8—21no—
395C22⋊Q8—22no—
396C22⋊Q8—22no—
397C22⋊Q8D4×C3111yescharacteristic
398#121—21no—
399#121—21no—
400#121D4×C3111yescharacteristic
401C4×Q8—21no—
402C4×Q8—22no—
403C4×Q8—21no—
404C4×Q8—21no—
405C4×Q8—21no—
406C4×Q8D4×C3111yescharacteristic
407C4×D4—22no—
408C4×D4—21no—
409C4×D4—21no—
410C4×D4—21no—
411C4×D4D4×C3111yescharacteristic
412#117—22no—
413#117—21no—
414#117—22no—
415#117—21no—
416#117D4×C3112yes—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4D4×C3111yescharacteristic
421C2×C4⋊C4D4×C3111yescharacteristic
422C2×C42C23×C3111yescharacteristic
423#107—21no—
424#107—22no—
425#106—21no—
426#106—22no—
427C4⋊C8—21no—
428#103—21no—
429#103—22no—
430#103—22no—
431#102—21no—
432#102—22no—
433#102—22no—
434#102—21no—
435#102—21no—
436#102—22no—
437#102—22no—
438C22⋊C8—21no—
439#97—21no—
440#97—21no—
441#97C23×C3112yes—
442#97C23×C3112yes—
443#97C23×C3111yescharacteristic
444#97C23×C3111yescharacteristic
445C31#1904611yescharacteristic, direct product factor, Sylow
446#41—44no—
447C2×Q8—22no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8C2×D4×C3111yescharacteristic
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—41no—
457C2×D4C2×D4×C3111yescharacteristic
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—41no—
463C22×C4C2×D4×C3111yescharacteristic
464C22×C4C2×D4×C3111yescharacteristic
465C22×C4C2×D4×C3111yescharacteristic
466SD16—42no—
467D8—42no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—42no—
473C2×C8—42no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4C2×D4×C3112yes—
499C4⋊C4C2×D4×C3112yes—
500C4⋊C4C2×D4×C3112yes—
501C4⋊C4C2×D4×C3111yescharacteristic
502C4⋊C4C2×D4×C3112yes—
503C4⋊C4C2×D4×C3111yescharacteristic
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—21no—
508C22⋊C4—22no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—21no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—21no—
520C42C2×D4×C3111yescharacteristic
521C42C2×D4×C3111yescharacteristic
522C42C2×D4×C3111yescharacteristic
523C42C24×C3111yescharacteristic
524C23—21no—
525C23—41no—
526C23C31×#12011yescharacteristic
527Q8—41no—
528Q8—42no—
529Q8—42no—
530Q8—81no—
531Q8—82no—
532D4—41no—
533D4—41no—
534D4—41no—
535D4—42no—
536D4—42no—
537D4—42no—
538D4—82no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—82no—
553C2×C4—81no—
554C2×C4—82no—
555C2×C4C31×#12011yescharacteristic
556C2×C4C31×#12011yescharacteristic
557C2×C4C31×#12011yescharacteristic
558C2×C4C22×D4×C3111yescharacteristic
559C2×C4C22×D4×C3111yescharacteristic
560C2×C4C22×D4×C3111yescharacteristic
561C8—22no—
562C8—42no—
563C8—42no—
564C8—41no—
565C22—21no—
566C22—21no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22C31×#45412yes—
574C22C2×C31×#12011yescharacteristic
575C4—21no—
576C4—21no—
577C4—21no—
578C4—21no—
579C4—21no—
580C4—21no—
581C4—42no—
582C4—41no—
583C4—42no—
584C4—42no—
585C4—41no—
586C4—41no—
587C4—42no—
588C4—81no—
589C2—22no—
590C2—81no—
591C2—161no—
592C2C2×C31×#45411yescharacteristic
593C2C31×#301811yescharacteristic
594C2C31×#302011yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3180C2
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
1. C31×#3563C2semidirect
2. C31×#3234C2semidirect
4. C31×#3135C2semidirect
5. C31×#1664C2semidirect
6. C31×#1663C2semidirect
7. C31×#1658C2semidirect
8. C31×#1657C2semidirect
9. C31×#1532C2semidirect
10. C31×#1531C2semidirect
11. C2×C31×#330C2semidirect
226. #19046C31direct (N × H)
268. #3563C2×C31semidirect
269. #3234C2×C31semidirect
271. #3135C2×C31semidirect
272. #1664C2×C31semidirect
273. #1663C2×C31semidirect
274. #1658C2×C31semidirect
275. #1657C2×C31semidirect
276. #1532C2×C31semidirect
277. #1531C2×C31semidirect
278. C2×#330C2×C31semidirect
445. C31#19046direct (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×#32342
C31×#31802
C31×#31352
C31×#16642
C31×#16632
C31×#16582
C31×#16572
C31×#15322
C31×#15312
C2×C31×#3302
#1904631