Finite Group Databasebeta

← C31×#9040 #21465799

Subgroups — C31×#9040 #21465799

Order 7936.

750 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#451C211yesmaximal, characteristic
2C31×#2329C211yesmaximal, characteristic
3C31×#1787C211yesmaximal, characteristic
4C2×C31×#328C211yesmaximal, characteristic
5C31×#1446C211yesmaximal, characteristic
6C31×#1294C211yesmaximal, characteristic
7C31×#1280C211yesmaximal, characteristic
8C23×Q8×C31C2211yescharacteristic
9C22×Q16×C31—21no—
10C2×C31×C22⋊Q8C2211yescharacteristic
11C31×#451—22no—
12C31×#451C414yescomplemented
13C2×C31×#103—21no—
14C2×C31×#103—21no—
15C2×C31×#103—21no—
16C2×C31×C22⋊C8C2211yescharacteristic
17C2×C31×C22⋊C8C2211yescharacteristic
18C31×#391—21no—
19C31×#389—21no—
20C31×#387—21no—
21C31×#387—21no—
22C31×#385—21no—
23C31×#380C2211yescharacteristic
24C4×C22⋊C4×C31—21no—
25C2×C31×#95—21no—
26C2×C31×#95—21no—
27C2×C31×#95C2211yescharacteristic
28C2×C31×#95C2211yescharacteristic
29C31×#328—22no—
30C22×Q8×C31—21no—
31C22×Q8×C31—21no—
32C22×Q8×C31—21no—
33C22×Q8×C31—21no—
34C22×Q8×C31—21no—
35C22×Q8×C31—42no—
36C22×Q8×C31—42no—
37C22×Q8×C31C2×C412yes—
38C22×Q8×C31D411yescharacteristic
39C23×C4×C31—21no—
40C23×C4×C31—21no—
41C23×C4×C31D411yescharacteristic
42C23×C4×C31D411yescharacteristic
43C23×C4×C31C2311yescharacteristic
44C2×Q16×C31—24no—
45C2×Q16×C31—42no—
46C2×Q16×C31—42no—
47C22×C8×C31—21no—
48C22×C8×C31—21no—
49C31×C22⋊Q8—21no—
50C31×C22⋊Q8—21no—
51C31×C22⋊Q8C2×C412yes—
52C31×C22⋊Q8D412yes—
53C2×C4⋊C4×C31—21no—
54C2×C4⋊C4×C31—21no—
55C2×C4⋊C4×C31—21no—
56C2×C4⋊C4×C31D411yescharacteristic
57C2×C4⋊C4×C31D411yescharacteristic
58C2×C4⋊C4×C31D411yescharacteristic
59C2×C22⋊C4×C31—21no—
60C2×C22⋊C4×C31—21no—
61C2×C42×C31—21no—
62C2×C42×C31—21no—
63C31×#103—24no—
64C31×#103—42no—
65C31×#103—42no—
66C31×C22⋊C8—21no—
67C31×C22⋊C8—21no—
68C31×C22⋊C8—21no—
69C31×C22⋊C8C2×C412yes—
70C31×#95—21no—
71C31×#95—21no—
72C31×#95—21no—
73C31×#95—21no—
74C31×#95—21no—
75C31×#95—21no—
76C31×#95—21no—
77C31×#95—21no—
78C31×#95—21no—
79C31×#95—42no—
80C31×#95—42no—
81C24×C31C2×D411yescharacteristic
82C2×Q8×C31—21no—
83C2×Q8×C31—22no—
84C2×Q8×C31—22no—
85C2×Q8×C31—22no—
86C2×Q8×C31—22no—
87C2×Q8×C31—22no—
88C2×Q8×C31—22no—
89C2×Q8×C31—44no—
90C2×Q8×C31—41no—
91C2×Q8×C31—41no—
92C2×Q8×C31—41no—
93C2×Q8×C31—41no—
94C2×Q8×C31—44no—
95C2×Q8×C31—41no—
96C2×Q8×C31—41no—
97C2×Q8×C31—44no—
98C2×Q8×C31—44no—
99C2×Q8×C31—41no—
100C2×Q8×C31—41no—
101C2×Q8×C31C22⋊C412yes—
102C22×C4×C31—22no—
103C22×C4×C31—22no—
104C22×C4×C31—22no—
105C22×C4×C31—21no—
106C22×C4×C31—21no—
107C22×C4×C31—21no—
108C22×C4×C31—21no—
109C22×C4×C31—22no—
110C22×C4×C31—21no—
111C22×C4×C31—21no—
112C22×C4×C31—21no—
113C22×C4×C31—21no—
114C22×C4×C31—21no—
115C22×C4×C31—41no—
116C22×C4×C31—41no—
117C22×C4×C31—41no—
118C22×C4×C31—41no—
119C22×C4×C31—41no—
120C22×C4×C31—41no—
121C22×C4×C31—41no—
122C22×C4×C31—41no—
123C22×C4×C31—41no—
124C22×C4×C31—41no—
125C22×C4×C31—41no—
126C22×C4×C31—41no—
127C22×C4×C31—41no—
128C22×C4×C31—41no—
129C22×C4×C31—41no—
130C22×C4×C31—41no—
131C22×C4×C31Q1612yescomplemented
132C22×C4×C31C22×C411yescharacteristic
133C22×C4×C31C2×D411yescharacteristic
134C22×C4×C31C2×D411yescharacteristic
135C22×C4×C31C2×D411yescharacteristic
136C22×C4×C31#4111yescharacteristic
137C22×C4×C31#4111yescharacteristic
138Q16×C31—48no—
139C2×C8×C31—22no—
140C2×C8×C31—41no—
141C2×C8×C31—41no—
142C2×C8×C31—41no—
143C2×C8×C31—41no—
144C2×C8×C31—41no—
145C4⋊C4×C31—22no—
146C4⋊C4×C31—22no—
147C4⋊C4×C31—21no—
148C4⋊C4×C31—41no—
149C4⋊C4×C31—41no—
150C4⋊C4×C31—81no—
151C4⋊C4×C31—81no—
152C4⋊C4×C31C22⋊C412yes—
153C22⋊C4×C31—22no—
154C22⋊C4×C31—22no—
155C22⋊C4×C31—41no—
156C22⋊C4×C31—41no—
157C42×C31—81no—
158C42×C31—81no—
159#9040C3111yesmaximal, characteristic, direct product factor, Sylow
160C23×C31—21no—
161C23×C31—21no—
162C23×C31—21no—
163C23×C31—21no—
164C23×C31C4×D411yescharacteristic
165C23×C31#12011yescharacteristic
166C23×C31#12111yescharacteristic
167C23×C31C2×Q1611yescharacteristic
168C23×C31#13511yescharacteristic
169C23×C31#13711yescharacteristic
170C23×C31#13711yescharacteristic
171Q8×C31—24no—
172Q8×C31—24no—
173Q8×C31—42no—
174Q8×C31—42no—
175Q8×C31—42no—
176Q8×C31—42no—
177Q8×C31—42no—
178Q8×C31—42no—
179Q8×C31—42no—
180Q8×C31—81no—
181C2×C4×C31—21no—
182C2×C4×C31—21no—
183C2×C4×C31—21no—
184C2×C4×C31—21no—
185C2×C4×C31—24no—
186C2×C4×C31—22no—
187C2×C4×C31—22no—
188C2×C4×C31—22no—
189C2×C4×C31—41no—
190C2×C4×C31—41no—
191C2×C4×C31—41no—
192C2×C4×C31—41no—
193C2×C4×C31—42no—
194C2×C4×C31—42no—
195C2×C4×C31—42no—
196C2×C4×C31—42no—
197C2×C4×C31—41no—
198C2×C4×C31—41no—
199C2×C4×C31—41no—
200C2×C4×C31—41no—
201C2×C4×C31—41no—
202C2×C4×C31—41no—
203C2×C4×C31—42no—
204C2×C4×C31—42no—
205C2×C4×C31—42no—
206C2×C4×C31—42no—
207C2×C4×C31—42no—
208C2×C4×C31—42no—
209C2×C4×C31—42no—
210C2×C4×C31—41no—
211C2×C4×C31—41no—
212C2×C4×C31—41no—
213C2×C4×C31—41no—
214C2×C4×C31—41no—
215C2×C4×C31—41no—
216C2×C4×C31—41no—
217C2×C4×C31—41no—
218C2×C4×C31—41no—
219C2×C4×C31—81no—
220C2×C4×C31—81no—
221C2×C4×C31—81no—
222C2×C4×C31—81no—
223C2×C4×C31—81no—
224C2×C4×C31—81no—
225C2×C4×C31C2×C22⋊C411yescharacteristic
226C2×C4×C31C4×D411yescharacteristic
227C2×C4×C31#12111yescharacteristic
228C2×C4×C31#12311yescharacteristic
229C8×C31—42no—
230C8×C31—81no—
231C2×#451C2×C3111yescharacteristic
232#2329C2×C3111yescharacteristic
233#1787C2×C3111yescharacteristic
234C2×#328C2×C3111yescharacteristic
235#1446C2×C3111yescharacteristic
236#1294C2×C3111yescharacteristic
237#1280C2×C3111yescharacteristic
238C22×C31—21no—
239C22×C31—21no—
240C22×C31—21no—
241C22×C31—21no—
242C22×C31—21no—
243C22×C31—21no—
244C22×C31—21no—
245C22×C31—21no—
246C22×C31—21no—
247C22×C31—21no—
248C22×C31—21no—
249C22×C31—21no—
250C22×C31#38611yescharacteristic
251C22×C31C4×Q1611yescharacteristic
252C22×C31#44111yescharacteristic
253C22×C31#45111yescharacteristic
254C22×C31#45211yescharacteristic
255C22×C31#45511yescharacteristic
256C22×C31#45611yescharacteristic
257C22×C31B(4,𝔽2)11yescharacteristic
258C22×C31B(4,𝔽2)11yescharacteristic
259C22×C31#46211yescharacteristic
260C22×C31#46411yescharacteristic
261C4×C31—21no—
262C4×C31—21no—
263C4×C31—21no—
264C4×C31—21no—
265C4×C31—42no—
266C4×C31—42no—
267C4×C31—44no—
268C4×C31—42no—
269C4×C31—82no—
270C4×C31—81no—
271C4×C31—81no—
272C4×C31—81no—
273C23×Q8C22×C3111yescharacteristic
274C22×Q16—21no—
275C2×C22⋊Q8C22×C3111yescharacteristic
276#451—22no—
277#451C4×C3114yescomplemented
278C2×#103—21no—
279C2×#103—21no—
280C2×#103—21no—
281C2×C22⋊C8C22×C3111yescharacteristic
282C2×C22⋊C8C22×C3111yescharacteristic
283#391—21no—
284#389—21no—
285#387—21no—
286#387—21no—
287#385—21no—
288#380C22×C3111yescharacteristic
289C4×C22⋊C4—21no—
290C2×#95—21no—
291C2×#95—21no—
292C2×#95C22×C3111yescharacteristic
293C2×#95C22×C3111yescharacteristic
294#328—22no—
295C2×C31—21no—
296C2×C31—21no—
297C2×C31—21no—
298C2×C31—21no—
299C2×C31#160211yescharacteristic
300C2×C31#160311yescharacteristic
301C2×C31#160511yescharacteristic
302C2×C31#160611yescharacteristic
303C2×C31#187811yescharacteristic
304C2×C31#188211yescharacteristic
305C2×C31#188911yescharacteristic
306C22×Q8—21no—
307C22×Q8—21no—
308C22×Q8—21no—
309C22×Q8—21no—
310C22×Q8—21no—
311C22×Q8—42no—
312C22×Q8—42no—
313C22×Q8C2×C4×C3112yes—
314C22×Q8D4×C3111yescharacteristic
315C23×C4—21no—
316C23×C4—21no—
317C23×C4D4×C3111yescharacteristic
318C23×C4D4×C3111yescharacteristic
319C23×C4C23×C3111yescharacteristic
320C2×Q16—24no—
321C2×Q16—42no—
322C2×Q16—42no—
323C22×C8—21no—
324C22×C8—21no—
325C22⋊Q8—21no—
326C22⋊Q8—21no—
327C22⋊Q8C2×C4×C3112yes—
328C22⋊Q8D4×C3112yes—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4D4×C3111yescharacteristic
333C2×C4⋊C4D4×C3111yescharacteristic
334C2×C4⋊C4D4×C3111yescharacteristic
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C42—21no—
338C2×C42—21no—
339#103—24no—
340#103—42no—
341#103—42no—
342C22⋊C8—21no—
343C22⋊C8—21no—
344C22⋊C8—21no—
345C22⋊C8C2×C4×C3112yes—
346#95—21no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—21no—
354#95—21no—
355#95—42no—
356#95—42no—
357C31#904011yescharacteristic, direct product factor, Sylow
358C24C2×D4×C3111yescharacteristic
359C2×Q8—21no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—44no—
367C2×Q8—41no—
368C2×Q8—41no—
369C2×Q8—41no—
370C2×Q8—41no—
371C2×Q8—44no—
372C2×Q8—41no—
373C2×Q8—41no—
374C2×Q8—44no—
375C2×Q8—44no—
376C2×Q8—41no—
377C2×Q8—41no—
378C2×Q8C22⋊C4×C3112yes—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4Q16×C3112yescomplemented
409C22×C4C22×C4×C3111yescharacteristic
410C22×C4C2×D4×C3111yescharacteristic
411C22×C4C2×D4×C3111yescharacteristic
412C22×C4C2×D4×C3111yescharacteristic
413C22×C4#41×C3111yescharacteristic
414C22×C4#41×C3111yescharacteristic
415Q16—48no—
416C2×C8—22no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—21no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—81no—
428C4⋊C4—81no—
429C4⋊C4C22⋊C4×C3112yes—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C42—81no—
435C42—81no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23C4×D4×C3111yescharacteristic
441C23C31×#12011yescharacteristic
442C23C31×#12111yescharacteristic
443C23C2×Q16×C3111yescharacteristic
444C23C31×#13511yescharacteristic
445C23C31×#13711yescharacteristic
446C23C31×#13711yescharacteristic
447Q8—24no—
448Q8—24no—
449Q8—42no—
450Q8—42no—
451Q8—42no—
452Q8—42no—
453Q8—42no—
454Q8—42no—
455Q8—42no—
456Q8—81no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4C2×C22⋊C4×C3111yescharacteristic
502C2×C4C4×D4×C3111yescharacteristic
503C2×C4C31×#12111yescharacteristic
504C2×C4C31×#12311yescharacteristic
505C8—42no—
506C8—81no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22C31×#38611yescharacteristic
520C22C4×Q16×C3111yescharacteristic
521C22C31×#44111yescharacteristic
522C22C31×#45111yescharacteristic
523C22C31×#45211yescharacteristic
524C22C31×#45511yescharacteristic
525C22C31×#45611yescharacteristic
526C22C31×B(4,𝔽2)11yescharacteristic
527C22C31×B(4,𝔽2)11yescharacteristic
528C22C31×#46211yescharacteristic
529C22C31×#46411yescharacteristic
530C4—21no—
531C4—21no—
532C4—21no—
533C4—21no—
534C4—42no—
535C4—42no—
536C4—44no—
537C4—42no—
538C4—82no—
539C4—81no—
540C4—81no—
541C4—81no—
542C2—21no—
543C2—21no—
544C2—21no—
545C2—21no—
546C2C31×#160211yescharacteristic
547C2C31×#160311yescharacteristic
548C2C31×#160511yescharacteristic
549C2C31×#160611yescharacteristic
550C2C31×#187811yescharacteristic
551C2C31×#188211yescharacteristic
552C2C31×#188911yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutatorC22×C4C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC24×C31C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C23×C4×C31 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
12. C31×#451C4semidirect
131. C22×C4×C31Q16semidirect
159. #9040C31direct (N × H)
277. #451C4×C31semidirect
357. C31#9040direct (N × H)
408. C22×C4Q16×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#4512
C31×#23292
C31×#17872
C2×C31×#3282
C31×#14462
C31×#12942
C31×#12802
#904031