Finite Group Databasebeta

← C31×#8419 #21465178

Subgroups — C31×#8419 #21465178

Order 7936.

778 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3522C211yesmaximal, characteristic
2C31×#1980C211yesmaximal, characteristic, complemented
3C31×#1763C211yesmaximal, characteristic, complemented
4C31×#1530C211yesmaximal, characteristic, complemented
5C31×#1524C211yesmaximal, characteristic, complemented
6C31×#1509C211yesmaximal, characteristic, complemented
7C31×#1491C211yesmaximal, characteristic, complemented
8C31×#559—21no—
9C31×#559C411yescharacteristic, complemented
10Q82×C31C411yescharacteristic, complemented
11C31×#554—21no—
12C31×#554C411yescharacteristic, complemented
13C31×#550C411yescharacteristic, complemented
14C31×#548—21no—
15C31×#547—22no—
16C31×#547—21no—
17C31×#542—22no—
18C31×#542—21no—
19C31×#532—21no—
20C31×#532C2211yescharacteristic
21C31×#529—21no—
22C31×#529C2211yescharacteristic
23C31×#520—21no—
24C2×C4×Q8×C31C2211yescharacteristic
25C31×#432C2211yescharacteristic
26C2×C31×#104—22no—
27C2×C31×#103—21no—
28C4×M16×C31C2211yescharacteristic
29C2×C31×#97C2211yescharacteristic
30C4×C4⋊C4×C31C2211yescharacteristic
31C31×#340—21no—
32C31×#330—21no—
33C31×#329—21no—
34C31×#329—22no—
35C31×#328—21no—
36C31×#327—21no—
37C31×Q8⋊C8—21no—
38C31×Q8⋊C8—21no—
39C2×#41×C31—22no—
40C2×#41×C31—21no—
41C2×#41×C31D412yes—
42C22×Q8×C31D411yescharacteristic
43C2×M16×C31—21no—
44C22×C8×C31—21no—
45C31×C4⋊Q8—22no—
46C31×C4⋊Q8—21no—
47C31×C4⋊Q8—21no—
48C31×C4⋊Q8C2×C411yescharacteristic
49C31×#127—21no—
50C31×#127C2×C411yescharacteristic
51C31×#126—22no—
52C31×#126—21no—
53C31×#125—21no—
54C31×#125D411yescharacteristic
55C31×#124—22no—
56C31×#124—22no—
57C31×#124—21no—
58C31×#124—21no—
59C31×#124D411yescharacteristic
60C31×#124D412yes—
61C31×#123—22no—
62C31×#123—22no—
63C31×#123—22no—
64C31×#123—22no—
65C31×#123—21no—
66C31×C22⋊Q8—22no—
67C31×C22⋊Q8—22no—
68C31×C22⋊Q8—22no—
69C31×C22⋊Q8—22no—
70C31×C22⋊Q8—21no—
71C31×C22⋊Q8D411yescharacteristic
72C31×#121—22no—
73C31×#121—22no—
74C31×#121—21no—
75C31×#121—21no—
76C31×#121—21no—
77C31×#121D411yescharacteristic
78C4×Q8×C31—21no—
79C4×Q8×C31—21no—
80C4×Q8×C31—21no—
81C4×Q8×C31—21no—
82C4×Q8×C31C2×C411yescharacteristic
83C4×Q8×C31C2×C411yescharacteristic
84C4×Q8×C31C2×C411yescharacteristic
85C4×Q8×C31D411yescharacteristic
86C4×Q8×C31D411yescharacteristic
87C4×D4×C31—22no—
88C4×D4×C31—22no—
89C4×D4×C31—22no—
90C4×D4×C31—21no—
91C4×D4×C31—21no—
92C4×D4×C31—21no—
93C4×D4×C31—21no—
94C4×D4×C31—21no—
95C4×D4×C31—21no—
96C4×D4×C31—21no—
97C4×D4×C31C2×C411yescharacteristic
98C31×#117—22no—
99C31×#117—21no—
100C2×C4⋊C4×C31—22no—
101C2×C4⋊C4×C31—21no—
102C2×C4⋊C4×C31—21no—
103C2×C4⋊C4×C31—21no—
104C2×C4⋊C4×C31D411yescharacteristic
105C2×C42×C31—21no—
106C2×C42×C31—21no—
107C2×C42×C31C2311yescharacteristic
108C31×C4⋊C8—21no—
109C31×#104—44no—
110C31×#103—24no—
111C31×C22⋊C8—21no—
112C31×#97—21no—
113C31×#97—21no—
114C31×#97—21no—
115C31×#97—21no—
116C4×C8×C31—21no—
117C31×#95—21no—
118#41×C31—42no—
119#41×C31—42no—
120#41×C31—42no—
121#41×C31—42no—
122#41×C31—82no—
123#41×C31—82no—
124C2×Q8×C31—22no—
125C2×Q8×C31—21no—
126C2×Q8×C31—22no—
127C2×Q8×C31—41no—
128C2×Q8×C31C22⋊C411yescharacteristic
129C2×Q8×C31C22⋊C411yescharacteristic
130C2×Q8×C31C22⋊C411yescharacteristic
131C2×Q8×C31C22⋊C412yes—
132C2×D4×C31—22no—
133C2×D4×C31—22no—
134C2×D4×C31—21no—
135C2×D4×C31—22no—
136C2×D4×C31—21no—
137C2×D4×C31—21no—
138C2×D4×C31C22⋊C411yescharacteristic
139C2×D4×C31C22⋊C412yes—
140C22×C4×C31—22no—
141C22×C4×C31—21no—
142C22×C4×C31—21no—
143C22×C4×C31—22no—
144C22×C4×C31—22no—
145C22×C4×C31—21no—
146C22×C4×C31—22no—
147C22×C4×C31—41no—
148C22×C4×C31C2×D411yescharacteristic
149C22×C4×C31C2×D411yescharacteristic
150C22×C4×C31C2×D411yescharacteristic
151M16×C31—22no—
152M16×C31—41no—
153C2×C8×C31—22no—
154C2×C8×C31—41no—
155C2×C8×C31—41no—
156C2×C8×C31—41no—
157C2×C8×C31—41no—
158C4⋊C4×C31—22no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—21no—
163C4⋊C4×C31—21no—
164C4⋊C4×C31—21no—
165C4⋊C4×C31—21no—
166C4⋊C4×C31—21no—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—21no—
170C4⋊C4×C31—21no—
171C4⋊C4×C31—21no—
172C4⋊C4×C31—21no—
173C4⋊C4×C31—22no—
174C4⋊C4×C31—41no—
175C4⋊C4×C31C22⋊C411yescharacteristic
176C4⋊C4×C31C22⋊C411yescharacteristic
177C4⋊C4×C31C22⋊C411yescharacteristic
178C4⋊C4×C31C22⋊C411yescharacteristic
179C22⋊C4×C31—22no—
180C22⋊C4×C31—22no—
181C22⋊C4×C31—22no—
182C22⋊C4×C31—22no—
183C22⋊C4×C31—22no—
184C22⋊C4×C31—21no—
185C22⋊C4×C31—21no—
186C22⋊C4×C31—22no—
187C22⋊C4×C31—21no—
188C22⋊C4×C31—21no—
189C22⋊C4×C31—22no—
190C22⋊C4×C31—21no—
191C42×C31—22no—
192C42×C31—21no—
193C42×C31—21no—
194C42×C31—21no—
195C42×C31—21no—
196C42×C31—21no—
197C42×C31—42no—
198C42×C31C22×C411yescharacteristic
199C42×C31C2×D411yescharacteristic
200C42×C31C2×D411yescharacteristic
201C42×C31C2×D411yescharacteristic
202#8419C3111yesmaximal, characteristic, direct product factor, Sylow
203C23×C31—22no—
204C23×C31—21no—
205C23×C31#12011yescharacteristic
206Q8×C31—21no—
207Q8×C31—21no—
208Q8×C31—21no—
209Q8×C31—42no—
210Q8×C31—41no—
211Q8×C31—42no—
212Q8×C31—82no—
213Q8×C31#10412yescomplemented
214D4×C31—41no—
215D4×C31—42no—
216D4×C31—42no—
217D4×C31—44no—
218D4×C31—42no—
219D4×C31—41no—
220D4×C31—42no—
221D4×C31—41no—
222D4×C31—41no—
223D4×C31—41no—
224D4×C31—82no—
225C2×C4×C31—21no—
226C2×C4×C31—21no—
227C2×C4×C31—21no—
228C2×C4×C31—21no—
229C2×C4×C31—21no—
230C2×C4×C31—21no—
231C2×C4×C31—21no—
232C2×C4×C31—21no—
233C2×C4×C31—21no—
234C2×C4×C31—22no—
235C2×C4×C31—21no—
236C2×C4×C31—21no—
237C2×C4×C31—41no—
238C2×C4×C31—42no—
239C2×C4×C31—44no—
240C2×C4×C31—41no—
241C2×C4×C31—41no—
242C2×C4×C31—41no—
243C2×C4×C31—41no—
244C2×C4×C31—42no—
245C2×C4×C31—82no—
246C2×C4×C31—82no—
247C2×C4×C31—82no—
248C2×C4×C31—81no—
249C2×C4×C31—81no—
250C2×C4×C31C2×C22⋊C411yescharacteristic
251C2×C4×C31C2×C22⋊C411yescharacteristic
252C2×C4×C31C2×C22⋊C411yescharacteristic
253C2×C4×C31#12011yescharacteristic
254C2×C4×C31#12011yescharacteristic
255C2×C4×C31#12011yescharacteristic
256C8×C31—42no—
257C8×C31—42no—
258C8×C31—81no—
259#3522C2×C3111yescharacteristic
260#1980C2×C3111yescharacteristic, complemented
261#1763C2×C3111yescharacteristic, complemented
262#1530C2×C3111yescharacteristic, complemented
263#1524C2×C3111yescharacteristic, complemented
264#1509C2×C3111yescharacteristic, complemented
265#1491C2×C3111yescharacteristic, complemented
266C22×C31—21no—
267C22×C31—21no—
268C22×C31—42no—
269C22×C31—41no—
270C22×C31—42no—
271C22×C31—41no—
272C22×C31—42no—
273C22×C31—41no—
274C22×C31#37911yescharacteristic
275C22×C31#45311yescharacteristic
276C22×C31#45511yescharacteristic
277C4×C31—21no—
278C4×C31—21no—
279C4×C31—21no—
280C4×C31—21no—
281C4×C31—21no—
282C4×C31—22no—
283C4×C31—41no—
284C4×C31—41no—
285C4×C31—42no—
286C4×C31—41no—
287C4×C31—41no—
288C4×C31—41no—
289C4×C31—41no—
290C4×C31—82no—
291C4×C31C2×#10411yescharacteristic
292C4×C31#42111yescharacteristic
293#559—21no—
294#559C4×C3111yescharacteristic, complemented
295Q82C4×C3111yescharacteristic, complemented
296#554—21no—
297#554C4×C3111yescharacteristic, complemented
298#550C4×C3111yescharacteristic, complemented
299#548—21no—
300#547—22no—
301#547—21no—
302#542—22no—
303#542—21no—
304#532—21no—
305#532C22×C3111yescharacteristic
306#529—21no—
307#529C22×C3111yescharacteristic
308#520—21no—
309C2×C4×Q8C22×C3111yescharacteristic
310#432C22×C3111yescharacteristic
311C2×#104—22no—
312C2×#103—21no—
313C4×M16C22×C3111yescharacteristic
314C2×#97C22×C3111yescharacteristic
315C4×C4⋊C4C22×C3111yescharacteristic
316#340—21no—
317#330—21no—
318#329—21no—
319#329—22no—
320#328—21no—
321#327—21no—
322Q8⋊C8—21no—
323Q8⋊C8—21no—
324C2×C31—21no—
325C2×C31—21no—
326C2×C31—82no—
327C2×C31—81no—
328C2×C31#178911yescharacteristic
329C2×C31#179211yescharacteristic
330C2×C31#179611yescharacteristic
331C2×#41—22no—
332C2×#41—21no—
333C2×#41D4×C3112yes—
334C22×Q8D4×C3111yescharacteristic
335C2×M16—21no—
336C22×C8—21no—
337C4⋊Q8—22no—
338C4⋊Q8—21no—
339C4⋊Q8—21no—
340C4⋊Q8C2×C4×C3111yescharacteristic
341#127—21no—
342#127C2×C4×C3111yescharacteristic
343#126—22no—
344#126—21no—
345#125—21no—
346#125D4×C3111yescharacteristic
347#124—22no—
348#124—22no—
349#124—21no—
350#124—21no—
351#124D4×C3111yescharacteristic
352#124D4×C3112yes—
353#123—22no—
354#123—22no—
355#123—22no—
356#123—22no—
357#123—21no—
358C22⋊Q8—22no—
359C22⋊Q8—22no—
360C22⋊Q8—22no—
361C22⋊Q8—22no—
362C22⋊Q8—21no—
363C22⋊Q8D4×C3111yescharacteristic
364#121—22no—
365#121—22no—
366#121—21no—
367#121—21no—
368#121—21no—
369#121D4×C3111yescharacteristic
370C4×Q8—21no—
371C4×Q8—21no—
372C4×Q8—21no—
373C4×Q8—21no—
374C4×Q8C2×C4×C3111yescharacteristic
375C4×Q8C2×C4×C3111yescharacteristic
376C4×Q8C2×C4×C3111yescharacteristic
377C4×Q8D4×C3111yescharacteristic
378C4×Q8D4×C3111yescharacteristic
379C4×D4—22no—
380C4×D4—22no—
381C4×D4—22no—
382C4×D4—21no—
383C4×D4—21no—
384C4×D4—21no—
385C4×D4—21no—
386C4×D4—21no—
387C4×D4—21no—
388C4×D4—21no—
389C4×D4C2×C4×C3111yescharacteristic
390#117—22no—
391#117—21no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4D4×C3111yescharacteristic
397C2×C42—21no—
398C2×C42—21no—
399C2×C42C23×C3111yescharacteristic
400C4⋊C8—21no—
401#104—44no—
402#103—24no—
403C22⋊C8—21no—
404#97—21no—
405#97—21no—
406#97—21no—
407#97—21no—
408C4×C8—21no—
409#95—21no—
410C31#841911yescharacteristic, direct product factor, Sylow
411#41—42no—
412#41—42no—
413#41—42no—
414#41—42no—
415#41—82no—
416#41—82no—
417C2×Q8—22no—
418C2×Q8—21no—
419C2×Q8—22no—
420C2×Q8—41no—
421C2×Q8C22⋊C4×C3111yescharacteristic
422C2×Q8C22⋊C4×C3111yescharacteristic
423C2×Q8C22⋊C4×C3111yescharacteristic
424C2×Q8C22⋊C4×C3112yes—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4C22⋊C4×C3111yescharacteristic
432C2×D4C22⋊C4×C3112yes—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—41no—
441C22×C4C2×D4×C3111yescharacteristic
442C22×C4C2×D4×C3111yescharacteristic
443C22×C4C2×D4×C3111yescharacteristic
444M16—22no—
445M16—41no—
446C2×C8—22no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—41no—
468C4⋊C4C22⋊C4×C3111yescharacteristic
469C4⋊C4C22⋊C4×C3111yescharacteristic
470C4⋊C4C22⋊C4×C3111yescharacteristic
471C4⋊C4C22⋊C4×C3111yescharacteristic
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—22no—
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—22no—
483C22⋊C4—21no—
484C42—22no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—42no—
491C42C22×C4×C3111yescharacteristic
492C42C2×D4×C3111yescharacteristic
493C42C2×D4×C3111yescharacteristic
494C42C2×D4×C3111yescharacteristic
495C23—22no—
496C23—21no—
497C23C31×#12011yescharacteristic
498Q8—21no—
499Q8—21no—
500Q8—21no—
501Q8—42no—
502Q8—41no—
503Q8—42no—
504Q8—82no—
505Q8C31×#10412yescomplemented
506D4—41no—
507D4—42no—
508D4—42no—
509D4—44no—
510D4—42no—
511D4—41no—
512D4—42no—
513D4—41no—
514D4—41no—
515D4—41no—
516D4—82no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4C2×C22⋊C4×C3111yescharacteristic
543C2×C4C2×C22⋊C4×C3111yescharacteristic
544C2×C4C2×C22⋊C4×C3111yescharacteristic
545C2×C4C31×#12011yescharacteristic
546C2×C4C31×#12011yescharacteristic
547C2×C4C31×#12011yescharacteristic
548C8—42no—
549C8—42no—
550C8—81no—
551C22—21no—
552C22—21no—
553C22—42no—
554C22—41no—
555C22—42no—
556C22—41no—
557C22—42no—
558C22—41no—
559C22C31×#37911yescharacteristic
560C22C31×#45311yescharacteristic
561C22C31×#45511yescharacteristic
562C4—21no—
563C4—21no—
564C4—21no—
565C4—22no—
566C4—21no—
567C4—21no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—42no—
574C4—41no—
575C4—82no—
576C4C2×C31×#10411yescharacteristic
577C4C31×#42111yescharacteristic
578C2—21no—
579C2—21no—
580C2—82no—
581C2—81no—
582C2C31×#178911yescharacteristic
583C2C31×#179211yescharacteristic
584C2C31×#179611yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC31×#3522C2
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×#1980C2semidirect
3. C31×#1763C2semidirect
4. C31×#1530C2semidirect
5. C31×#1524C2semidirect
6. C31×#1509C2semidirect
7. C31×#1491C2semidirect
9. C31×#559C4semidirect
10. Q82×C31C4semidirect
12. C31×#554C4semidirect
13. C31×#550C4semidirect
202. #8419C31direct (N × H)
213. Q8×C31#104semidirect
260. #1980C2×C31semidirect
261. #1763C2×C31semidirect
262. #1530C2×C31semidirect
263. #1524C2×C31semidirect
264. #1509C2×C31semidirect
265. #1491C2×C31semidirect
294. #559C4×C31semidirect
295. Q82C4×C31semidirect
297. #554C4×C31semidirect
298. #550C4×C31semidirect
410. C31#8419direct (N × H)
505. Q8C31×#104semidirect

Maximal subgroups

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

subgroupindex
C31×#35222
C31×#19802
C31×#17632
C31×#15302
C31×#15242
C31×#15092
C31×#14912
#841931