Finite Group Databasebeta

← C31×#19045 #21475804

Subgroups — C31×#19045 #21475804

Order 7936.

870 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3559C211yesmaximal, characteristic
2C31×#3236C211yesmaximal, characteristic, complemented
3C31×#3181C211yesmaximal, characteristic, complemented
4C31×#3135C211yesmaximal, characteristic, complemented
5C31×#1664C212yesmaximal, complemented
6C31×#1664C212yesmaximal, complemented
7C31×#1662C212yesmaximal, complemented
8C31×#1661C212yesmaximal, complemented
9C31×#1532C211yesmaximal, characteristic, complemented
10C31×#1530C211yesmaximal, characteristic, complemented
11C2×C31×#330C211yesmaximal, characteristic, complemented
12C31×#563C2212yes—
13C31×#562C2212yes—
14C31×#554—21no—
15C31×#553—21no—
16C31×#553—21no—
17C31×#547—22no—
18C31×#547—21no—
19C31×#544—21no—
20C31×#540—21no—
21C31×#539—22no—
22C31×#539—21no—
23C31×#537—21no—
24C31×#533C2211yescharacteristic
25C31×#532C2211yescharacteristic
26C31×#529C2211yescharacteristic
27C2×C31×#125C2211yescharacteristic
28C31×#501C2211yescharacteristic
29C31×#501C2212yes—
30C31×#497C2211yescharacteristic
31C31×#491—21no—
32C31×#491C2211yescharacteristic
33C31×#491C2212yes—
34C31×#490—21no—
35C31×#489C2212yes—
36C31×#489C2212yes—
37C31×#489C2211yescharacteristic
38C31×#484—21no—
39C31×#483—21no—
40C31×#479—22no—
41C31×#467—21no—
42C31×#465—21no—
43C31×#441—21no—
44C31×#440—21no—
45C31×#432C2211yescharacteristic
46C31×#416—22no—
47C2×C31×#97C2211yescharacteristic
48C2×C31×#97C2211yescharacteristic
49C31×#330C2212yes—
50C31×#330C2212yes—
51C31×#330C2212yes—
52C31×#329C2212yes—
53C31×#329C2212yes—
54C31×#329C2212yes—
55C2×#41×C31—22no—
56C2×#41×C31D412yes—
57C2×Q16×C31—21no—
58C2×SD16×C31—21no—
59C22×C8×C31—21no—
60C22×C8×C31—21no—
61C31×C4⋊Q8—21no—
62C31×C4⋊Q8—22no—
63C31×C4⋊Q8D411yescharacteristic
64C31×C4⋊Q8C2311yescharacteristic
65C31×C4⋊Q8C2311yescharacteristic
66C31×#127D411yescharacteristic
67C31×#126—22no—
68C31×#126—22no—
69C31×#126—21no—
70C31×#125—22no—
71C31×#125—21no—
72C31×#125—21no—
73C31×#125—21no—
74C31×#125—21no—
75C31×#125D412yes—
76C31×#125C2312yes—
77C31×#125C2311yescharacteristic
78C31×#124C2312yes—
79C31×#124C2311yescharacteristic
80C31×#123—22no—
81C31×#123—22no—
82C31×#123—22no—
83C31×#123—22no—
84C31×#123—21no—
85C31×#123—22no—
86C31×#123—22no—
87C31×C22⋊Q8—22no—
88C31×C22⋊Q8—22no—
89C31×C22⋊Q8—22no—
90C31×C22⋊Q8—22no—
91C31×C22⋊Q8—22no—
92C31×C22⋊Q8—22no—
93C31×C22⋊Q8—21no—
94C31×C22⋊Q8—21no—
95C31×C22⋊Q8D411yescharacteristic
96C31×#121—22no—
97C31×#121—22no—
98C31×#121—21no—
99C31×#121—21no—
100C31×#121—22no—
101C31×#121—22no—
102C31×#121—21no—
103C31×#121—21no—
104C31×#121D411yescharacteristic
105C4×Q8×C31—21no—
106C4×Q8×C31—21no—
107C4×Q8×C31D411yescharacteristic
108C4×D4×C31—22no—
109C4×D4×C31—22no—
110C4×D4×C31—21no—
111C4×D4×C31—21no—
112C4×D4×C31—21no—
113C4×D4×C31—21no—
114C4×D4×C31—21no—
115C4×D4×C31D411yescharacteristic
116C31×#117—21no—
117C31×#117—21no—
118C31×#117—21no—
119C2×C4⋊C4×C31—22no—
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—22no—
132C31×#103—22no—
133C31×#103—21no—
134C31×#103—21no—
135C31×#103—21no—
136C31×#103—22no—
137C31×#103—22no—
138C31×#102—22no—
139C31×#102—22no—
140C31×#102—21no—
141C31×C22⋊C8—21no—
142C31×#97—21no—
143C31×#97—21no—
144C31×#97C2312yes—
145C31×#97C2311yescharacteristic
146C31×#97C2311yescharacteristic
147C31×#97C2312yes—
148#41×C31—44no—
149#41×C31—84no—
150C2×Q8×C31—22no—
151C2×Q8×C31—21no—
152C2×Q8×C31—21no—
153C2×Q8×C31C2×D412yes—
154C2×Q8×C31C2×D411yescharacteristic
155C2×D4×C31—22no—
156C2×D4×C31—21no—
157C2×D4×C31—22no—
158C2×D4×C31—22no—
159C2×D4×C31—21no—
160C2×D4×C31—21no—
161C2×D4×C31C2×D412yes—
162C2×D4×C31C2×D411yescharacteristic
163C22×C4×C31—21no—
164C22×C4×C31—22no—
165C22×C4×C31—22no—
166C22×C4×C31—22no—
167C22×C4×C31—21no—
168C22×C4×C31—21no—
169C22×C4×C31—22no—
170C22×C4×C31C2×D411yescharacteristic
171C22×C4×C31C2×D411yescharacteristic
172C22×C4×C31C2×D411yescharacteristic
173Q16×C31—42no—
174SD16×C31—42no—
175C2×C8×C31—21no—
176C2×C8×C31—22no—
177C2×C8×C31—21no—
178C2×C8×C31—22no—
179C2×C8×C31—42no—
180C2×C8×C31—42no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—21no—
187C4⋊C4×C31—21no—
188C4⋊C4×C31—21no—
189C4⋊C4×C31—22no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—21no—
192C4⋊C4×C31—21no—
193C4⋊C4×C31—22no—
194C4⋊C4×C31—22no—
195C4⋊C4×C31—21no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31—21no—
198C4⋊C4×C31—21no—
199C4⋊C4×C31—21no—
200C4⋊C4×C31—21no—
201C4⋊C4×C31—41no—
202C4⋊C4×C31—41no—
203C4⋊C4×C31C2×D412yes—
204C4⋊C4×C31C2×D411yescharacteristic
205C4⋊C4×C31C2×D412yes—
206C4⋊C4×C31C2×D411yescharacteristic
207C22⋊C4×C31—22no—
208C22⋊C4×C31—22no—
209C22⋊C4×C31—22no—
210C22⋊C4×C31—22no—
211C22⋊C4×C31—22no—
212C22⋊C4×C31—21no—
213C22⋊C4×C31—21no—
214C22⋊C4×C31—21no—
215C22⋊C4×C31—21no—
216C22⋊C4×C31—22no—
217C22⋊C4×C31—22no—
218C22⋊C4×C31—21no—
219C22⋊C4×C31—41no—
220C42×C31—22no—
221C42×C31—21no—
222C42×C31—41no—
223C42×C31C2×D411yescharacteristic
224C42×C31C2×D411yescharacteristic
225C42×C31C2×D411yescharacteristic
226C42×C31C2411yescharacteristic
227#19045C3111yesmaximal, characteristic, direct product factor, Sylow
228C23×C31—22no—
229C23×C31—21no—
230C23×C31#12011yescharacteristic
231Q8×C31—41no—
232Q8×C31—42no—
233Q8×C31—41no—
234Q8×C31—41no—
235Q8×C31—42no—
236Q8×C31—82no—
237D4×C31—41no—
238D4×C31—42no—
239D4×C31—42no—
240D4×C31—44no—
241D4×C31—42no—
242D4×C31—81no—
243D4×C31—82no—
244D4×C31—82no—
245C2×C4×C31—22no—
246C2×C4×C31—21no—
247C2×C4×C31—21no—
248C2×C4×C31—22no—
249C2×C4×C31—22no—
250C2×C4×C31—21no—
251C2×C4×C31—22no—
252C2×C4×C31—41no—
253C2×C4×C31—41no—
254C2×C4×C31—42no—
255C2×C4×C31—42no—
256C2×C4×C31—41no—
257C2×C4×C31—41no—
258C2×C4×C31—82no—
259C2×C4×C31—81no—
260C2×C4×C31—82no—
261C2×C4×C31—82no—
262C2×C4×C31—82no—
263C2×C4×C31#12011yescharacteristic
264C2×C4×C31#12011yescharacteristic
265C2×C4×C31#12011yescharacteristic
266C2×C4×C31C22×D411yescharacteristic
267C2×C4×C31C22×D411yescharacteristic
268C2×C4×C31C22×D411yescharacteristic
269C8×C31—22no—
270C8×C31—42no—
271C8×C31—42no—
272C8×C31—41no—
273#3559C2×C3111yescharacteristic
274#3236C2×C3111yescharacteristic, complemented
275#3181C2×C3111yescharacteristic, complemented
276#3135C2×C3111yescharacteristic, complemented
277#1664C2×C3112yescomplemented
278#1664C2×C3112yescomplemented
279#1662C2×C3112yescomplemented
280#1661C2×C3112yescomplemented
281#1532C2×C3111yescharacteristic, complemented
282#1530C2×C3111yescharacteristic, complemented
283C2×#330C2×C3111yescharacteristic, complemented
284C22×C31—21no—
285C22×C31—21no—
286C22×C31—42no—
287C22×C31—41no—
288C22×C31—42no—
289C22×C31—41no—
290C22×C31—41no—
291C22×C31—42no—
292C22×C31#45512yes—
293C22×C31C2×#12011yescharacteristic
294C4×C31—21no—
295C4×C31—21no—
296C4×C31—21no—
297C4×C31—21no—
298C4×C31—21no—
299C4×C31—21no—
300C4×C31—42no—
301C4×C31—42no—
302C4×C31—41no—
303C4×C31—42no—
304C4×C31—41no—
305C4×C31—41no—
306C4×C31—81no—
307C4×C31—81no—
308#563C22×C3112yes—
309#562C22×C3112yes—
310#554—21no—
311#553—21no—
312#553—21no—
313#547—22no—
314#547—21no—
315#544—21no—
316#540—21no—
317#539—22no—
318#539—21no—
319#537—21no—
320#533C22×C3111yescharacteristic
321#532C22×C3111yescharacteristic
322#529C22×C3111yescharacteristic
323C2×#125C22×C3111yescharacteristic
324#501C22×C3111yescharacteristic
325#501C22×C3112yes—
326#497C22×C3111yescharacteristic
327#491—21no—
328#491C22×C3111yescharacteristic
329#491C22×C3112yes—
330#490—21no—
331#489C22×C3112yes—
332#489C22×C3112yes—
333#489C22×C3111yescharacteristic
334#484—21no—
335#483—21no—
336#479—22no—
337#467—21no—
338#465—21no—
339#441—21no—
340#440—21no—
341#432C22×C3111yescharacteristic
342#416—22no—
343C2×#97C22×C3111yescharacteristic
344C2×#97C22×C3111yescharacteristic
345#330C22×C3112yes—
346#330C22×C3112yes—
347#330C22×C3112yes—
348#329C22×C3112yes—
349#329C22×C3112yes—
350#329C22×C3112yes—
351C2×C31—22no—
352C2×C31—82no—
353C2×C31—81no—
354C2×C31C2×#45511yescharacteristic
355C2×C31#301811yescharacteristic
356C2×C31#302011yescharacteristic
357C2×#41—22no—
358C2×#41D4×C3112yes—
359C2×Q16—21no—
360C2×SD16—21no—
361C22×C8—21no—
362C22×C8—21no—
363C4⋊Q8—21no—
364C4⋊Q8—22no—
365C4⋊Q8D4×C3111yescharacteristic
366C4⋊Q8C23×C3111yescharacteristic
367C4⋊Q8C23×C3111yescharacteristic
368#127D4×C3111yescharacteristic
369#126—22no—
370#126—22no—
371#126—21no—
372#125—22no—
373#125—21no—
374#125—21no—
375#125—21no—
376#125—21no—
377#125D4×C3112yes—
378#125C23×C3112yes—
379#125C23×C3111yescharacteristic
380#124C23×C3111yescharacteristic
381#124C23×C3112yes—
382#123—22no—
383#123—22no—
384#123—22no—
385#123—22no—
386#123—21no—
387#123—22no—
388#123—22no—
389C22⋊Q8—22no—
390C22⋊Q8—22no—
391C22⋊Q8—22no—
392C22⋊Q8—22no—
393C22⋊Q8—22no—
394C22⋊Q8—22no—
395C22⋊Q8—21no—
396C22⋊Q8—21no—
397C22⋊Q8D4×C3111yescharacteristic
398#121—22no—
399#121—22no—
400#121—21no—
401#121—21no—
402#121—22no—
403#121—22no—
404#121—21no—
405#121—21no—
406#121D4×C3111yescharacteristic
407C4×Q8—21no—
408C4×Q8—21no—
409C4×Q8D4×C3111yescharacteristic
410C4×D4—22no—
411C4×D4—22no—
412C4×D4—21no—
413C4×D4—21no—
414C4×D4—21no—
415C4×D4—21no—
416C4×D4—21no—
417C4×D4D4×C3111yescharacteristic
418#117—21no—
419#117—21no—
420#117—21no—
421C2×C4⋊C4—22no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4D4×C3111yescharacteristic
426C2×C4⋊C4D4×C3111yescharacteristic
427C2×C42C23×C3111yescharacteristic
428#107—21no—
429#107—22no—
430#106—21no—
431#106—22no—
432C4⋊C8—21no—
433#103—22no—
434#103—22no—
435#103—21no—
436#103—21no—
437#103—21no—
438#103—22no—
439#103—22no—
440#102—22no—
441#102—22no—
442#102—21no—
443C22⋊C8—21no—
444#97—21no—
445#97—21no—
446#97C23×C3112yes—
447#97C23×C3111yescharacteristic
448#97C23×C3111yescharacteristic
449#97C23×C3112yes—
450C31#1904511yescharacteristic, direct product factor, Sylow
451#41—44no—
452#41—84no—
453C2×Q8—22no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8C2×D4×C3112yes—
457C2×Q8C2×D4×C3111yescharacteristic
458C2×D4—22no—
459C2×D4—21no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4C2×D4×C3112yes—
465C2×D4C2×D4×C3111yescharacteristic
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4C2×D4×C3111yescharacteristic
474C22×C4C2×D4×C3111yescharacteristic
475C22×C4C2×D4×C3111yescharacteristic
476Q16—42no—
477SD16—42no—
478C2×C8—21no—
479C2×C8—22no—
480C2×C8—21no—
481C2×C8—22no—
482C2×C8—42no—
483C2×C8—42no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4C2×D4×C3112yes—
507C4⋊C4C2×D4×C3111yescharacteristic
508C4⋊C4C2×D4×C3112yes—
509C4⋊C4C2×D4×C3111yescharacteristic
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—21no—
518C22⋊C4—21no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—21no—
522C22⋊C4—41no—
523C42—22no—
524C42—21no—
525C42—41no—
526C42C2×D4×C3111yescharacteristic
527C42C2×D4×C3111yescharacteristic
528C42C2×D4×C3111yescharacteristic
529C42C24×C3111yescharacteristic
530C23—22no—
531C23—21no—
532C23C31×#12011yescharacteristic
533Q8—41no—
534Q8—42no—
535Q8—41no—
536Q8—41no—
537Q8—42no—
538Q8—82no—
539D4—41no—
540D4—42no—
541D4—42no—
542D4—44no—
543D4—42no—
544D4—81no—
545D4—82no—
546D4—82no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—82no—
561C2×C4—81no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4C31×#12011yescharacteristic
566C2×C4C31×#12011yescharacteristic
567C2×C4C31×#12011yescharacteristic
568C2×C4C22×D4×C3111yescharacteristic
569C2×C4C22×D4×C3111yescharacteristic
570C2×C4C22×D4×C3111yescharacteristic
571C8—22no—
572C8—42no—
573C8—42no—
574C8—41no—
575C22—21no—
576C22—21no—
577C22—42no—
578C22—41no—
579C22—42no—
580C22—41no—
581C22—41no—
582C22—42no—
583C22C31×#45512yes—
584C22C2×C31×#12011yescharacteristic
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—42no—
592C4—42no—
593C4—41no—
594C4—42no—
595C4—41no—
596C4—41no—
597C4—81no—
598C4—81no—
599C2—22no—
600C2—82no—
601C2—81no—
602C2C2×C31×#45511yescharacteristic
603C2C31×#301811yescharacteristic
604C2C31×#302011yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3559C2
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×#3236C2semidirect
3. C31×#3181C2semidirect
4. C31×#3135C2semidirect
5. C31×#1664C2semidirect
6. C31×#1664C2semidirect
7. C31×#1662C2semidirect
8. C31×#1661C2semidirect
9. C31×#1532C2semidirect
10. C31×#1530C2semidirect
11. C2×C31×#330C2semidirect
227. #19045C31direct (N × H)
274. #3236C2×C31semidirect
275. #3181C2×C31semidirect
276. #3135C2×C31semidirect
277. #1664C2×C31semidirect
278. #1664C2×C31semidirect
279. #1662C2×C31semidirect
280. #1661C2×C31semidirect
281. #1532C2×C31semidirect
282. #1530C2×C31semidirect
283. C2×#330C2×C31semidirect
450. C31#19045direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35592
C31×#32362
C31×#31812
C31×#31352
C31×#16642
C31×#16622
C31×#16612
C31×#15322
C31×#15302
C2×C31×#3302
#1904531