Finite Group Databasebeta

← C31×#19091 #21475850

Subgroups — C31×#19091 #21475850

Order 7936.

854 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3561C211yesmaximal, characteristic
2C31×#3240C211yesmaximal, characteristic, complemented
3C31×#3237C211yesmaximal, characteristic, complemented
4C31×#3215C211yesmaximal, characteristic, complemented
5C31×#1664C212yesmaximal, complemented
6C31×#1662C212yesmaximal, complemented
7C31×#1661C212yesmaximal, complemented
8C31×#1660C212yesmaximal, complemented
9C31×#1537C211yesmaximal, characteristic, complemented
10C31×#1535C211yesmaximal, characteristic, complemented
11C31×#1533C211yesmaximal, characteristic, complemented
12C31×#563C2212yes—
13C31×#562C2212yes—
14C31×#556—21no—
15C31×#551—22no—
16D4×Q8×C31—21no—
17C31×#548—21no—
18C31×#546—21no—
19C31×#543—21no—
20C31×#542—21no—
21C31×#542—22no—
22C31×#538—21no—
23C31×#536—21no—
24C31×#533C2211yescharacteristic
25C31×#533C2211yescharacteristic
26C31×#529C2211yescharacteristic
27C2×C31×#124C2211yescharacteristic
28C31×#501C2211yescharacteristic, complemented
29C31×#501C2212yescomplemented
30C31×#501C2211yescharacteristic, complemented
31C31×#497C2212yes—
32C31×#491C2212yescomplemented
33C31×#489C2212yes—
34C31×#489C2211yescharacteristic, complemented
35C31×#489C2211yescharacteristic, complemented
36C31×#485—21no—
37C31×#485—22no—
38C31×#482—21no—
39C31×#479—21no—
40C31×#478—21no—
41C31×#477—21no—
42C31×#475—21no—
43C31×#464—22no—
44C31×#451—21no—
45C31×#450—21no—
46C31×#432C2211yescharacteristic
47C31×#432C2211yescharacteristic
48C31×#432C2211yescharacteristic
49C31×#330C2212yescomplemented
50C31×#330C2212yescomplemented
51C31×#330C2212yescomplemented
52C31×#329C2212yes—
53C31×#329C2212yes—
54C31×#329C2212yescomplemented
55C2×#41×C31—21no—
56C22×Q8×C31—21no—
57C22×Q8×C31D411yescharacteristic
58C22×D4×C31D411yescharacteristic
59C2×Q16×C31—22no—
60C2×SD16×C31—22no—
61C31×C4⋊Q8—21no—
62C31×C4⋊Q8—22no—
63C31×C4⋊Q8D411yescharacteristic
64C31×C4⋊Q8D411yescharacteristic
65C31×C4⋊Q8C2311yescharacteristic
66C31×C4⋊Q8C2311yescharacteristic
67C31×#126—22no—
68C31×#126—22no—
69C31×#126—22no—
70C31×#126—21no—
71C31×#125—21no—
72C31×#125C2312yes—
73C31×#125C2311yescharacteristic
74C31×#124—21no—
75C31×#124—22no—
76C31×#124—22no—
77C31×#124—21no—
78C31×#124—21no—
79C31×#124D412yes—
80C31×#124C2312yes—
81C31×#124C2311yescharacteristic
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—21no—
90C31×C22⋊Q8—22no—
91C31×C22⋊Q8—22no—
92C31×C22⋊Q8—22no—
93C31×C22⋊Q8—21no—
94C31×C22⋊Q8—21no—
95C31×C22⋊Q8—22no—
96C31×C22⋊Q8—22no—
97C31×C22⋊Q8—21no—
98C31×C22⋊Q8—21no—
99C31×C22⋊Q8D411yescharacteristic
100C31×#121—21no—
101C31×#121—22no—
102C31×#121—22no—
103C31×#121D411yescharacteristic
104C31×#120—22no—
105C4×Q8×C31—21no—
106C4×Q8×C31—21no—
107C4×Q8×C31—21no—
108C4×Q8×C31—21no—
109C4×Q8×C31—21no—
110C4×Q8×C31D411yescharacteristic
111C4×D4×C31—22no—
112C4×D4×C31—21no—
113C4×D4×C31—22no—
114C4×D4×C31—21no—
115C4×D4×C31D411yescharacteristic
116C31×#117—22no—
117C31×#117—21no—
118C31×#117—21no—
119C31×#117D412yes—
120C2×C22⋊C4×C31—22no—
121C2×C22⋊C4×C31—21no—
122C2×C22⋊C4×C31—21no—
123C2×C42×C31C2311yescharacteristic
124C31×#107—22no—
125C31×#107—21no—
126C31×#107—21no—
127C31×#106—22no—
128C31×#106—21no—
129C31×#106—21no—
130C31×C4⋊C8—21no—
131C31×C4⋊C8—21no—
132C31×C4⋊C8—21no—
133C31×#103—22no—
134C31×#103—21no—
135C31×#103—21no—
136C31×#103—22no—
137C31×#103—22no—
138C31×#102—22no—
139C31×#102—21no—
140C31×#102—21no—
141C31×C22⋊C8—21no—
142C31×C22⋊C8—21no—
143C31×C22⋊C8—21no—
144C31×#97C2312yes—
145C31×#97C2311yescharacteristic
146C31×#97C2311yescharacteristic
147C31×#97C2312yes—
148C24×C31—21no—
149#41×C31—82no—
150C2×Q8×C31—22no—
151C2×Q8×C31—21no—
152C2×Q8×C31—21no—
153C2×Q8×C31—21no—
154C2×Q8×C31—21no—
155C2×Q8×C31—21no—
156C2×Q8×C31—82no—
157C2×Q8×C31—81no—
158C2×Q8×C31C2×D411yescharacteristic
159C2×Q8×C31C2×D412yes—
160C2×D4×C31—21no—
161C2×D4×C31—22no—
162C2×D4×C31—22no—
163C2×D4×C31—21no—
164C2×D4×C31—81no—
165C2×D4×C31C2×D411yescharacteristic
166C2×D4×C31C2×D412yes—
167C22×C4×C31—22no—
168C22×C4×C31—22no—
169C22×C4×C31—22no—
170C22×C4×C31—21no—
171C22×C4×C31—21no—
172C22×C4×C31C2×D411yescharacteristic
173C22×C4×C31C2×D411yescharacteristic
174C22×C4×C31C2×D411yescharacteristic
175Q16×C31—82no—
176SD16×C31—82no—
177C2×C8×C31—21no—
178C2×C8×C31—21no—
179C2×C8×C31—22no—
180C2×C8×C31—22no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—21no—
186C4⋊C4×C31—21no—
187C4⋊C4×C31—21no—
188C4⋊C4×C31—21no—
189C4⋊C4×C31—21no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—22no—
192C4⋊C4×C31—22no—
193C4⋊C4×C31—21no—
194C4⋊C4×C31—21no—
195C4⋊C4×C31—21no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31—41no—
198C4⋊C4×C31—41no—
199C4⋊C4×C31C2×D412yes—
200C4⋊C4×C31C2×D412yes—
201C4⋊C4×C31C2×D411yescharacteristic
202C4⋊C4×C31C2×D411yescharacteristic
203C22⋊C4×C31—22no—
204C22⋊C4×C31—22no—
205C22⋊C4×C31—22no—
206C22⋊C4×C31—22no—
207C22⋊C4×C31—22no—
208C22⋊C4×C31—21no—
209C22⋊C4×C31—22no—
210C22⋊C4×C31—22no—
211C22⋊C4×C31—22no—
212C22⋊C4×C31—21no—
213C22⋊C4×C31—21no—
214C22⋊C4×C31—22no—
215C22⋊C4×C31—22no—
216C22⋊C4×C31—21no—
217C22⋊C4×C31—21no—
218C22⋊C4×C31—41no—
219C42×C31—22no—
220C42×C31—22no—
221C42×C31—21no—
222C42×C31—41no—
223C42×C31C2×D411yescharacteristic
224C42×C31C2×D411yescharacteristic
225C42×C31C2×D411yescharacteristic
226C42×C31C2411yescharacteristic
227#19091C3111yesmaximal, characteristic, direct product factor, Sylow
228C23×C31—22no—
229C23×C31—21no—
230C23×C31—42no—
231C23×C31—81no—
232C23×C31—81no—
233C23×C31#12011yescharacteristic
234Q8×C31—42no—
235Q8×C31—41no—
236Q8×C31—42no—
237Q8×C31—42no—
238Q8×C31—42no—
239Q8×C31—42no—
240Q8×C31—81no—
241Q8×C31—82no—
242D4×C31—42no—
243D4×C31—41no—
244D4×C31—42no—
245D4×C31—42no—
246D4×C31—82no—
247D4×C31—82no—
248C2×C4×C31—21no—
249C2×C4×C31—22no—
250C2×C4×C31—21no—
251C2×C4×C31—22no—
252C2×C4×C31—22no—
253C2×C4×C31—21no—
254C2×C4×C31—41no—
255C2×C4×C31—41no—
256C2×C4×C31—41no—
257C2×C4×C31—42no—
258C2×C4×C31—41no—
259C2×C4×C31—41no—
260C2×C4×C31—81no—
261C2×C4×C31—82no—
262C2×C4×C31—82no—
263C2×C4×C31—82no—
264C2×C4×C31#12011yescharacteristic
265C2×C4×C31#12011yescharacteristic
266C2×C4×C31#12011yescharacteristic
267C2×C4×C31C22×D411yescharacteristic
268C2×C4×C31C22×D411yescharacteristic
269C2×C4×C31C22×D411yescharacteristic
270C8×C31—42no—
271C8×C31—41no—
272C8×C31—41no—
273C8×C31—42no—
274#3561C2×C3111yescharacteristic
275#3240C2×C3111yescharacteristic, complemented
276#3237C2×C3111yescharacteristic, complemented
277#3215C2×C3111yescharacteristic, complemented
278#1664C2×C3112yescomplemented
279#1662C2×C3112yescomplemented
280#1661C2×C3112yescomplemented
281#1660C2×C3112yescomplemented
282#1537C2×C3111yescharacteristic, complemented
283#1535C2×C3111yescharacteristic, complemented
284#1533C2×C3111yescharacteristic, complemented
285C22×C31—21no—
286C22×C31—21no—
287C22×C31—21no—
288C22×C31—41no—
289C22×C31—42no—
290C22×C31—41no—
291C22×C31—42no—
292C22×C31—42no—
293C22×C31—41no—
294C22×C31—162no—
295C22×C31C2×#12011yescharacteristic
296C4×C31—21no—
297C4×C31—21no—
298C4×C31—21no—
299C4×C31—21no—
300C4×C31—21no—
301C4×C31—21no—
302C4×C31—41no—
303C4×C31—42no—
304C4×C31—42no—
305C4×C31—41no—
306C4×C31—41no—
307C4×C31—42no—
308C4×C31—81no—
309C4×C31—81no—
310#563C22×C3112yes—
311#562C22×C3112yes—
312#556—21no—
313#551—22no—
314D4×Q8—21no—
315#548—21no—
316#546—21no—
317#543—21no—
318#542—21no—
319#542—22no—
320#538—21no—
321#536—21no—
322#533C22×C3111yescharacteristic
323#533C22×C3111yescharacteristic
324#529C22×C3111yescharacteristic
325C2×#124C22×C3111yescharacteristic
326#501C22×C3111yescharacteristic, complemented
327#501C22×C3112yescomplemented
328#501C22×C3111yescharacteristic, complemented
329#497C22×C3112yes—
330#491C22×C3112yescomplemented
331#489C22×C3112yes—
332#489C22×C3111yescharacteristic, complemented
333#489C22×C3111yescharacteristic, complemented
334#485—21no—
335#485—22no—
336#482—21no—
337#479—21no—
338#478—21no—
339#477—21no—
340#475—21no—
341#464—22no—
342#451—21no—
343#450—21no—
344#432C22×C3111yescharacteristic
345#432C22×C3111yescharacteristic
346#432C22×C3111yescharacteristic
347#330C22×C3112yescomplemented
348#330C22×C3112yescomplemented
349#330C22×C3112yescomplemented
350#329C22×C3112yes—
351#329C22×C3112yes—
352#329C22×C3112yescomplemented
353C2×C31—41no—
354C2×C31—81no—
355C2×C31—82no—
356C2×C31#301911yescharacteristic
357C2×C31#302011yescharacteristic
358C2×C31#302211yescharacteristic
359C2×#41—21no—
360C22×Q8—21no—
361C22×Q8D4×C3111yescharacteristic
362C22×D4D4×C3111yescharacteristic
363C2×Q16—22no—
364C2×SD16—22no—
365C4⋊Q8—21no—
366C4⋊Q8—22no—
367C4⋊Q8D4×C3111yescharacteristic
368C4⋊Q8D4×C3111yescharacteristic
369C4⋊Q8C23×C3111yescharacteristic
370C4⋊Q8C23×C3111yescharacteristic
371#126—22no—
372#126—22no—
373#126—22no—
374#126—21no—
375#125—21no—
376#125C23×C3111yescharacteristic
377#125C23×C3112yes—
378#124—21no—
379#124—22no—
380#124—22no—
381#124—21no—
382#124—21no—
383#124D4×C3112yes—
384#124C23×C3112yes—
385#124C23×C3111yescharacteristic
386#123—22no—
387#123—22no—
388#123—21no—
389#123—22no—
390#123—22no—
391C22⋊Q8—22no—
392C22⋊Q8—22no—
393C22⋊Q8—21no—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8—22no—
397C22⋊Q8—21no—
398C22⋊Q8—21no—
399C22⋊Q8—22no—
400C22⋊Q8—22no—
401C22⋊Q8—21no—
402C22⋊Q8—21no—
403C22⋊Q8D4×C3111yescharacteristic
404#121—21no—
405#121—22no—
406#121—22no—
407#121D4×C3111yescharacteristic
408#120—22no—
409C4×Q8—21no—
410C4×Q8—21no—
411C4×Q8—21no—
412C4×Q8—21no—
413C4×Q8—21no—
414C4×Q8D4×C3111yescharacteristic
415C4×D4—22no—
416C4×D4—21no—
417C4×D4—22no—
418C4×D4—21no—
419C4×D4D4×C3111yescharacteristic
420#117—22no—
421#117—21no—
422#117—21no—
423#117D4×C3112yes—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C42C23×C3111yescharacteristic
428#107—22no—
429#107—21no—
430#107—21no—
431#106—22no—
432#106—21no—
433#106—21no—
434C4⋊C8—21no—
435C4⋊C8—21no—
436C4⋊C8—21no—
437#103—22no—
438#103—21no—
439#103—21no—
440#103—22no—
441#103—22no—
442#102—22no—
443#102—21no—
444#102—21no—
445C22⋊C8—21no—
446C22⋊C8—21no—
447C22⋊C8—21no—
448#97C23×C3112yes—
449#97C23×C3111yescharacteristic
450#97C23×C3111yescharacteristic
451#97C23×C3112yes—
452C31#1909111yescharacteristic, direct product factor, Sylow
453C24—21no—
454#41—82no—
455C2×Q8—22no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—82no—
462C2×Q8—81no—
463C2×Q8C2×D4×C3111yescharacteristic
464C2×Q8C2×D4×C3112yes—
465C2×D4—21no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—81no—
470C2×D4C2×D4×C3111yescharacteristic
471C2×D4C2×D4×C3112yes—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4C2×D4×C3111yescharacteristic
478C22×C4C2×D4×C3111yescharacteristic
479C22×C4C2×D4×C3111yescharacteristic
480Q16—82no—
481SD16—82no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—22no—
485C2×C8—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
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—41no—
503C4⋊C4—41no—
504C4⋊C4C2×D4×C3112yes—
505C4⋊C4C2×D4×C3112yes—
506C4⋊C4C2×D4×C3111yescharacteristic
507C4⋊C4C2×D4×C3111yescharacteristic
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—21no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—21no—
518C22⋊C4—21no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—41no—
524C42—22no—
525C42—22no—
526C42—21no—
527C42—41no—
528C42C2×D4×C3111yescharacteristic
529C42C2×D4×C3111yescharacteristic
530C42C2×D4×C3111yescharacteristic
531C42C24×C3111yescharacteristic
532C23—22no—
533C23—21no—
534C23—42no—
535C23—81no—
536C23—81no—
537C23C31×#12011yescharacteristic
538Q8—42no—
539Q8—41no—
540Q8—42no—
541Q8—42no—
542Q8—42no—
543Q8—42no—
544Q8—81no—
545Q8—82no—
546D4—42no—
547D4—41no—
548D4—42no—
549D4—42no—
550D4—82no—
551D4—82no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—81no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4C31×#12011yescharacteristic
569C2×C4C31×#12011yescharacteristic
570C2×C4C31×#12011yescharacteristic
571C2×C4C22×D4×C3111yescharacteristic
572C2×C4C22×D4×C3111yescharacteristic
573C2×C4C22×D4×C3111yescharacteristic
574C8—42no—
575C8—41no—
576C8—41no—
577C8—42no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—41no—
582C22—42no—
583C22—41no—
584C22—42no—
585C22—42no—
586C22—41no—
587C22—162no—
588C22C2×C31×#12011yescharacteristic
589C4—21no—
590C4—21no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—41no—
596C4—42no—
597C4—42no—
598C4—41no—
599C4—41no—
600C4—42no—
601C4—81no—
602C4—81no—
603C2—41no—
604C2—82no—
605C2—81no—
606C2C31×#301911yescharacteristic
607C2C31×#302011yescharacteristic
608C2C31×#302211yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3561C2
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×#3240C2semidirect
3. C31×#3237C2semidirect
4. C31×#3215C2semidirect
5. C31×#1664C2semidirect
6. C31×#1662C2semidirect
7. C31×#1661C2semidirect
8. C31×#1660C2semidirect
9. C31×#1537C2semidirect
10. C31×#1535C2semidirect
11. C31×#1533C2semidirect
28. C31×#501C22semidirect
29. C31×#501C22semidirect
30. C31×#501C22semidirect
32. C31×#491C22semidirect
34. C31×#489C22semidirect
35. C31×#489C22semidirect
49. C31×#330C22semidirect
50. C31×#330C22semidirect
51. C31×#330C22semidirect
54. C31×#329C22semidirect
227. #19091C31direct (N × H)
275. #3240C2×C31semidirect
276. #3237C2×C31semidirect
277. #3215C2×C31semidirect
278. #1664C2×C31semidirect
279. #1662C2×C31semidirect
280. #1661C2×C31semidirect
281. #1660C2×C31semidirect
282. #1537C2×C31semidirect
283. #1535C2×C31semidirect
284. #1533C2×C31semidirect
326. #501C22×C31semidirect
327. #501C22×C31semidirect
328. #501C22×C31semidirect
330. #491C22×C31semidirect
332. #489C22×C31semidirect
333. #489C22×C31semidirect
347. #330C22×C31semidirect
348. #330C22×C31semidirect
349. #330C22×C31semidirect
352. #329C22×C31semidirect
452. C31#19091direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35612
C31×#32402
C31×#32372
C31×#32152
C31×#16642
C31×#16622
C31×#16612
C31×#16602
C31×#15372
C31×#15352
C31×#15332
#1909131