Finite Group Databasebeta

← C31×#18961 #21475720

Subgroups — C31×#18961 #21475720

Order 7936.

818 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3555C211yesmaximal, characteristic
2C31×#3238C211yesmaximal, characteristic
3C31×#3230C211yesmaximal, characteristic, complemented
4C31×#3209C211yesmaximal, characteristic, complemented
5C31×#1638C212yesmaximal, complemented
6C31×#1636C212yesmaximal, complemented
7C31×#1633C212yesmaximal, complemented
8C31×#1632C212yesmaximal, complemented
9C31×#1554C211yesmaximal, characteristic, complemented
10C31×#1497C211yesmaximal, characteristic, complemented
11C31×#1482C211yesmaximal, characteristic
12C31×#563—22no—
13Q82×C31C2211yescharacteristic
14C31×#557—21no—
15C31×#557C2211yescharacteristic
16C31×#556—21no—
17C31×#556—21no—
18C31×#556C2211yescharacteristic
19C31×#554C2211yescharacteristic
20C31×#552—21no—
21C31×#544—21no—
22C31×#542—21no—
23C31×#541—21no—
24C31×#541—21no—
25C31×#541—22no—
26C31×#533C2211yescharacteristic
27C31×#533C2211yescharacteristic
28C2×C31×C4⋊Q8C2211yescharacteristic
29C31×#529C2211yescharacteristic
30C31×#501—21no—
31C31×#500—21no—
32C31×#494C2212yes—
33C31×#489—21no—
34C31×#487—21no—
35C31×#487C2212yes—
36C31×#484—21no—
37C31×#484—21no—
38C31×#483—21no—
39C31×#480—21no—
40C31×#477C2212yes—
41C31×#477C2212yes—
42C31×#477C2212yes—
43C31×#476C2212yes—
44C31×#462—22no—
45C31×#451—22no—
46C31×#432C2211yescharacteristic
47C31×#431C2211yescharacteristic
48C31×#431C2211yescharacteristic
49C31×#333C2212yes—
50C31×#332C2212yes—
51C31×Q8⋊C8C2212yes—
52C31×Q8⋊C8C2212yes—
53C31×Q8⋊C8C2212yes—
54C31×D4⋊C8C2212yes—
55C22×Q8×C31—21no—
56C2×Q16×C31—24no—
57C31×C4⋊Q8—22no—
58C31×C4⋊Q8—21no—
59C31×C4⋊Q8—22no—
60C31×C4⋊Q8—21no—
61C31×C4⋊Q8—21no—
62C31×C4⋊Q8—21no—
63C31×C4⋊Q8—21no—
64C31×C4⋊Q8—21no—
65C31×C4⋊Q8D411yescharacteristic
66C31×C4⋊Q8D411yescharacteristic
67C31×C4⋊Q8C2312yes—
68C31×C4⋊Q8C2311yescharacteristic
69C31×C4⋊Q8C2311yescharacteristic
70C31×#126—22no—
71C31×#126—22no—
72C31×#126—22no—
73C31×#126—22no—
74C31×#126—22no—
75C31×#126—22no—
76C31×#126—21no—
77C31×#125—22no—
78C31×#125—22no—
79C31×#125—22no—
80C31×#125—22no—
81C31×#125—21no—
82C31×#125—22no—
83C31×#125—21no—
84C31×#125—21no—
85C31×#125—21no—
86C31×#125D411yescharacteristic
87C31×#124—22no—
88C31×#124—21no—
89C31×#124D411yescharacteristic
90C31×#123—22no—
91C31×#123—21no—
92C31×C22⋊Q8—22no—
93C31×C22⋊Q8—22no—
94C31×C22⋊Q8—21no—
95C31×C22⋊Q8—21no—
96C31×C22⋊Q8—22no—
97C31×C22⋊Q8—21no—
98C31×C22⋊Q8—21no—
99C31×C22⋊Q8D412yes—
100C31×C22⋊Q8D411yescharacteristic
101C31×C22⋊Q8D411yescharacteristic
102C31×C22⋊Q8D411yescharacteristic
103C31×#121D411yescharacteristic
104C4×Q8×C31—22no—
105C4×Q8×C31—21no—
106C4×Q8×C31—21no—
107C4×Q8×C31—21no—
108C4×Q8×C31—21no—
109C4×Q8×C31—21no—
110C4×Q8×C31—21no—
111C4×Q8×C31—21no—
112C4×Q8×C31D412yes—
113C4×Q8×C31C2311yescharacteristic
114C4×Q8×C31C2311yescharacteristic
115C4×Q8×C31C2311yescharacteristic
116C4×D4×C31—21no—
117C4×D4×C31—21no—
118C4×D4×C31C2311yescharacteristic
119C31×#117—22no—
120C31×#117—21no—
121C31×#117—21no—
122C31×#117—21no—
123C31×#117—21no—
124C2×C4⋊C4×C31—21no—
125C2×C4⋊C4×C31—21no—
126C2×C4⋊C4×C31—21no—
127C2×C42×C31C2311yescharacteristic
128C31×#107—22no—
129C31×#107—22no—
130C31×#107—22no—
131C31×#106—22no—
132C31×C4⋊C8—21no—
133C31×C4⋊C8C2312yes—
134C31×C4⋊C8C2312yes—
135C31×#103—22no—
136C31×#103—22no—
137C31×#103—22no—
138C31×#103—24no—
139C31×#102—22no—
140C31×C22⋊C8—21no—
141C31×C22⋊C8—21no—
142C31×C22⋊C8—21no—
143C31×#97—21no—
144C4×C8×C31—21no—
145C4×C8×C31C2312yes—
146C2×Q8×C31—22no—
147C2×Q8×C31—21no—
148C2×Q8×C31—21no—
149C2×Q8×C31—22no—
150C2×Q8×C31—41no—
151C2×Q8×C31—82no—
152C2×Q8×C31C2×D411yescharacteristic
153C2×Q8×C31C2×D411yescharacteristic
154C2×Q8×C31C2×D411yescharacteristic
155C2×Q8×C31C2×D412yes—
156C2×D4×C31—21no—
157C2×D4×C31C2×D411yescharacteristic
158C22×C4×C31—22no—
159C22×C4×C31—21no—
160C22×C4×C31—41no—
161C22×C4×C31C2×D411yescharacteristic
162C22×C4×C31C2×D411yescharacteristic
163C22×C4×C31C2×D411yescharacteristic
164Q16×C31—84no—
165C2×C8×C31—22no—
166C2×C8×C31—22no—
167C2×C8×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—22no—
176C4⋊C4×C31—21no—
177C4⋊C4×C31—21no—
178C4⋊C4×C31—21no—
179C4⋊C4×C31—21no—
180C4⋊C4×C31—21no—
181C4⋊C4×C31—21no—
182C4⋊C4×C31—21no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—21no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—21no—
187C4⋊C4×C31—21no—
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—41no—
195C4⋊C4×C31—41no—
196C4⋊C4×C31C2×D412yes—
197C4⋊C4×C31C2×D411yescharacteristic
198C4⋊C4×C31C2×D411yescharacteristic
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—21no—
211C22⋊C4×C31—21no—
212C42×C31—22no—
213C42×C31—22no—
214C42×C31—22no—
215C42×C31—22no—
216C42×C31—21no—
217C42×C31—21no—
218C42×C31—21no—
219C42×C31C2×D411yescharacteristic
220C42×C31C2×D411yescharacteristic
221C42×C31C2×D411yescharacteristic
222C42×C31C2411yescharacteristic
223#18961C3111yesmaximal, characteristic, direct product factor, Sylow
224C23×C31—21no—
225C23×C31#12011yescharacteristic
226Q8×C31—22no—
227Q8×C31—22no—
228Q8×C31—22no—
229Q8×C31—42no—
230Q8×C31—44no—
231Q8×C31—42no—
232Q8×C31—81no—
233Q8×C31—82no—
234Q8×C31—82no—
235D4×C31—22no—
236D4×C31—81no—
237C2×C4×C31—22no—
238C2×C4×C31—22no—
239C2×C4×C31—21no—
240C2×C4×C31—21no—
241C2×C4×C31—21no—
242C2×C4×C31—22no—
243C2×C4×C31—21no—
244C2×C4×C31—21no—
245C2×C4×C31—21no—
246C2×C4×C31—22no—
247C2×C4×C31—21no—
248C2×C4×C31—41no—
249C2×C4×C31—42no—
250C2×C4×C31—42no—
251C2×C4×C31—82no—
252C2×C4×C31—82no—
253C2×C4×C31#12011yescharacteristic
254C2×C4×C31#12011yescharacteristic
255C2×C4×C31#12011yescharacteristic
256C2×C4×C31C22×D411yescharacteristic
257C2×C4×C31C22×D411yescharacteristic
258C2×C4×C31C22×D411yescharacteristic
259C8×C31—42no—
260C8×C31—42no—
261C8×C31—42no—
262#3555C2×C3111yescharacteristic
263#3238C2×C3111yescharacteristic
264#3230C2×C3111yescharacteristic, complemented
265#3209C2×C3111yescharacteristic, complemented
266#1638C2×C3112yescomplemented
267#1636C2×C3112yescomplemented
268#1633C2×C3112yescomplemented
269#1632C2×C3112yescomplemented
270#1554C2×C3111yescharacteristic, complemented
271#1497C2×C3111yescharacteristic, complemented
272#1482C2×C3111yescharacteristic
273C22×C31—21no—
274C22×C31—21no—
275C22×C31—21no—
276C22×C31—41no—
277C22×C31—41no—
278C22×C31—41no—
279C22×C31C2×#12011yescharacteristic
280C4×C31—21no—
281C4×C31—21no—
282C4×C31—21no—
283C4×C31—21no—
284C4×C31—41no—
285C4×C31—41no—
286C4×C31—41no—
287C4×C31—41no—
288C4×C31—42no—
289C4×C31—41no—
290C4×C31—42no—
291C4×C31—41no—
292C4×C31—41no—
293C4×C31—82no—
294C4×C31#57812yes—
295C4×C31#57812yes—
296#563—22no—
297Q82C22×C3111yescharacteristic
298#557—21no—
299#557C22×C3111yescharacteristic
300#556—21no—
301#556—21no—
302#556C22×C3111yescharacteristic
303#554C22×C3111yescharacteristic
304#552—21no—
305#544—21no—
306#542—21no—
307#541—21no—
308#541—21no—
309#541—22no—
310#533C22×C3111yescharacteristic
311#533C22×C3111yescharacteristic
312C2×C4⋊Q8C22×C3111yescharacteristic
313#529C22×C3111yescharacteristic
314#501—21no—
315#500—21no—
316#494C22×C3112yes—
317#489—21no—
318#487—21no—
319#487C22×C3112yes—
320#484—21no—
321#484—21no—
322#483—21no—
323#480—21no—
324#477C22×C3112yes—
325#477C22×C3112yes—
326#477C22×C3112yes—
327#476C22×C3112yes—
328#462—22no—
329#451—22no—
330#432C22×C3111yescharacteristic
331#431C22×C3111yescharacteristic
332#431C22×C3111yescharacteristic
333#333C22×C3112yes—
334#332C22×C3112yes—
335Q8⋊C8C22×C3112yes—
336Q8⋊C8C22×C3112yes—
337Q8⋊C8C22×C3112yes—
338D4⋊C8C22×C3112yes—
339C2×C31—41no—
340C2×C31—81no—
341C2×C31#301111yescharacteristic
342C2×C31#301111yescharacteristic
343C2×C31#302111yescharacteristic
344C22×Q8—21no—
345C2×Q16—24no—
346C4⋊Q8—22no—
347C4⋊Q8—21no—
348C4⋊Q8—22no—
349C4⋊Q8—21no—
350C4⋊Q8—21no—
351C4⋊Q8—21no—
352C4⋊Q8—21no—
353C4⋊Q8—21no—
354C4⋊Q8D4×C3111yescharacteristic
355C4⋊Q8D4×C3111yescharacteristic
356C4⋊Q8C23×C3112yes—
357C4⋊Q8C23×C3111yescharacteristic
358C4⋊Q8C23×C3111yescharacteristic
359#126—22no—
360#126—22no—
361#126—22no—
362#126—22no—
363#126—22no—
364#126—22no—
365#126—21no—
366#125—22no—
367#125—22no—
368#125—22no—
369#125—22no—
370#125—21no—
371#125—22no—
372#125—21no—
373#125—21no—
374#125—21no—
375#125D4×C3111yescharacteristic
376#124—22no—
377#124—21no—
378#124D4×C3111yescharacteristic
379#123—22no—
380#123—21no—
381C22⋊Q8—22no—
382C22⋊Q8—22no—
383C22⋊Q8—21no—
384C22⋊Q8—21no—
385C22⋊Q8—22no—
386C22⋊Q8—21no—
387C22⋊Q8—21no—
388C22⋊Q8D4×C3111yescharacteristic
389C22⋊Q8D4×C3111yescharacteristic
390C22⋊Q8D4×C3111yescharacteristic
391C22⋊Q8D4×C3112yes—
392#121D4×C3111yescharacteristic
393C4×Q8—22no—
394C4×Q8—21no—
395C4×Q8—21no—
396C4×Q8—21no—
397C4×Q8—21no—
398C4×Q8—21no—
399C4×Q8—21no—
400C4×Q8—21no—
401C4×Q8D4×C3112yes—
402C4×Q8C23×C3111yescharacteristic
403C4×Q8C23×C3111yescharacteristic
404C4×Q8C23×C3111yescharacteristic
405C4×D4—21no—
406C4×D4—21no—
407C4×D4C23×C3111yescharacteristic
408#117—22no—
409#117—21no—
410#117—21no—
411#117—21no—
412#117—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C42C23×C3111yescharacteristic
417#107—22no—
418#107—22no—
419#107—22no—
420#106—22no—
421C4⋊C8—21no—
422C4⋊C8C23×C3112yes—
423C4⋊C8C23×C3112yes—
424#103—22no—
425#103—22no—
426#103—22no—
427#103—24no—
428#102—22no—
429C22⋊C8—21no—
430C22⋊C8—21no—
431C22⋊C8—21no—
432#97—21no—
433C4×C8—21no—
434C4×C8C23×C3112yes—
435C31#1896111yescharacteristic, direct product factor, Sylow
436C2×Q8—22no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—22no—
440C2×Q8—41no—
441C2×Q8—82no—
442C2×Q8C2×D4×C3111yescharacteristic
443C2×Q8C2×D4×C3111yescharacteristic
444C2×Q8C2×D4×C3111yescharacteristic
445C2×Q8C2×D4×C3112yes—
446C2×D4—21no—
447C2×D4C2×D4×C3111yescharacteristic
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—41no—
451C22×C4C2×D4×C3111yescharacteristic
452C22×C4C2×D4×C3111yescharacteristic
453C22×C4C2×D4×C3111yescharacteristic
454Q16—84no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4C2×D4×C3112yes—
487C4⋊C4C2×D4×C3111yescharacteristic
488C4⋊C4C2×D4×C3111yescharacteristic
489C4⋊C4C2×D4×C3111yescharacteristic
490C4⋊C4C2×D4×C3111yescharacteristic
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—21no—
495C22⋊C4—22no—
496C22⋊C4—21no—
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—21no—
500C22⋊C4—21no—
501C22⋊C4—21no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42C2×D4×C3111yescharacteristic
510C42C2×D4×C3111yescharacteristic
511C42C2×D4×C3111yescharacteristic
512C42C24×C3111yescharacteristic
513C23—21no—
514C23C31×#12011yescharacteristic
515Q8—22no—
516Q8—22no—
517Q8—22no—
518Q8—42no—
519Q8—44no—
520Q8—42no—
521Q8—81no—
522Q8—82no—
523Q8—82no—
524D4—22no—
525D4—81no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4C31×#12011yescharacteristic
543C2×C4C31×#12011yescharacteristic
544C2×C4C31×#12011yescharacteristic
545C2×C4C22×D4×C3111yescharacteristic
546C2×C4C22×D4×C3111yescharacteristic
547C2×C4C22×D4×C3111yescharacteristic
548C8—42no—
549C8—42no—
550C8—42no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22C2×C31×#12011yescharacteristic
558C4—21no—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—42no—
567C4—41no—
568C4—42no—
569C4—41no—
570C4—41no—
571C4—82no—
572C4C31×#57812yes—
573C4C31×#57812yes—
574C2—41no—
575C2—81no—
576C2C31×#301111yescharacteristic
577C2C31×#301111yescharacteristic
578C2C31×#302111yescharacteristic
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×#3230C2semidirect
4. C31×#3209C2semidirect
5. C31×#1638C2semidirect
6. C31×#1636C2semidirect
7. C31×#1633C2semidirect
8. C31×#1632C2semidirect
9. C31×#1554C2semidirect
10. C31×#1497C2semidirect
223. #18961C31direct (N × H)
264. #3230C2×C31semidirect
265. #3209C2×C31semidirect
266. #1638C2×C31semidirect
267. #1636C2×C31semidirect
268. #1633C2×C31semidirect
269. #1632C2×C31semidirect
270. #1554C2×C31semidirect
271. #1497C2×C31semidirect
435. C31#18961direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35552
C31×#32382
C31×#32302
C31×#32092
C31×#16382
C31×#16362
C31×#16332
C31×#16322
C31×#15542
C31×#14972
C31×#14822
#1896131