Finite Group Databasebeta

← C31×#9924 #21466683

Subgroups — C31×#9924 #21466683

Order 7936.

890 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#451C211yesmaximal, characteristic, complemented
2C31×#2626C211yesmaximal, characteristic
3C2×C31×#393C211yesmaximal, characteristic
4C31×#1787C211yesmaximal, characteristic, complemented
5C2×C31×#328C211yesmaximal, characteristic, complemented
6C2×C31×#323C211yesmaximal, characteristic
7C31×#1294C211yesmaximal, characteristic, complemented
8C23×Q8×C31C2211yescharacteristic
9C22×Q16×C31—21no—
10C2×C31×C22⋊Q8C2211yescharacteristic
11C22×C22⋊C4×C31—21no—
12C22×C22⋊C4×C31C2211yescharacteristic
13C31×#451—22no—
14C31×#451—22no—
15C2×C31×#103—21no—
16C2×C31×#103—22no—
17C2×C31×C22⋊C8C2211yescharacteristic
18C2×C31×C22⋊C8C2211yescharacteristic
19C31×#399—21no—
20C31×#396—22no—
21C31×#393—21no—
22C31×#393—22no—
23C31×#393—22no—
24C31×#385—21no—
25C31×#380C2211yescharacteristic
26C31×#379—21no—
27C2×C31×#95—21no—
28C2×C31×#95C2211yescharacteristic
29C31×#328—22no—
30C31×#323—22no—
31C25×C31D411yescharacteristic
32C22×Q8×C31—21no—
33C22×Q8×C31—21no—
34C22×Q8×C31—21no—
35C22×Q8×C31—21no—
36C22×Q8×C31—21no—
37C22×Q8×C31—44no—
38C22×Q8×C31D412yes—
39C22×Q8×C31D411yescharacteristic
40C23×C4×C31—21no—
41C23×C4×C31—21no—
42C23×C4×C31D411yescharacteristic
43C23×C4×C31C2311yescharacteristic
44C2×Q16×C31—24no—
45C2×Q16×C31—44no—
46C22×C8×C31—21no—
47C22×C8×C31—21no—
48C31×C22⋊Q8—21no—
49C31×C22⋊Q8—21no—
50C31×C22⋊Q8—21no—
51C31×C22⋊Q8—21no—
52C2×C4⋊C4×C31—21no—
53C2×C4⋊C4×C31—22no—
54C2×C4⋊C4×C31D411yescharacteristic
55C2×C4⋊C4×C31D412yes—
56C2×C22⋊C4×C31—24no—
57C2×C22⋊C4×C31—21no—
58C2×C22⋊C4×C31—21no—
59C2×C22⋊C4×C31—21no—
60C2×C22⋊C4×C31—21no—
61C2×C22⋊C4×C31—21no—
62C2×C22⋊C4×C31—21no—
63C2×C22⋊C4×C31—21no—
64C2×C22⋊C4×C31—21no—
65C2×C22⋊C4×C31—22no—
66C2×C22⋊C4×C31—22no—
67C2×C22⋊C4×C31—44no—
68C31×#103—24no—
69C31×#103—44no—
70C31×C22⋊C8—21no—
71C31×C22⋊C8—21no—
72C31×C22⋊C8—21no—
73C31×C22⋊C8D412yes—
74C31×#95—22no—
75C31×#95—24no—
76C31×#95—21no—
77C31×#95—21no—
78C31×#95D412yes—
79C24×C31—21no—
80C24×C31—21no—
81C24×C31—21no—
82C24×C31—21no—
83C24×C31—21no—
84C24×C31—21no—
85C24×C31—42no—
86C24×C31—42no—
87C24×C31Q1612yescomplemented
88C24×C31C2×D411yescharacteristic
89C2×Q8×C31—24no—
90C2×Q8×C31—24no—
91C2×Q8×C31—22no—
92C2×Q8×C31—21no—
93C2×Q8×C31—22no—
94C2×Q8×C31—21no—
95C2×Q8×C31—44no—
96C2×Q8×C31—42no—
97C2×Q8×C31—42no—
98C2×Q8×C31—44no—
99C2×Q8×C31—42no—
100C2×Q8×C31—44no—
101C2×Q8×C31—44no—
102C2×Q8×C31—41no—
103C2×Q8×C31—41no—
104C22×C4×C31—24no—
105C22×C4×C31—22no—
106C22×C4×C31—21no—
107C22×C4×C31—22no—
108C22×C4×C31—22no—
109C22×C4×C31—21no—
110C22×C4×C31—21no—
111C22×C4×C31—21no—
112C22×C4×C31—21no—
113C22×C4×C31—42no—
114C22×C4×C31—42no—
115C22×C4×C31—41no—
116C22×C4×C31—41no—
117C22×C4×C31—42no—
118C22×C4×C31—42no—
119C22×C4×C31—42no—
120C22×C4×C31—41no—
121C22×C4×C31—41no—
122C22×C4×C31C2×D411yescharacteristic
123C22×C4×C31C2×D411yescharacteristic
124C22×C4×C31C2×D411yescharacteristic
125C22×C4×C31C2×D411yescharacteristic
126C22×C4×C31C2×D411yescharacteristic
127C22×C4×C31#4111yescharacteristic
128Q16×C31—48no—
129C2×C8×C31—22no—
130C2×C8×C31—42no—
131C2×C8×C31—42no—
132C2×C8×C31—41no—
133C4⋊C4×C31—21no—
134C4⋊C4×C31—21no—
135C4⋊C4×C31—42no—
136C4⋊C4×C31—41no—
137C4⋊C4×C31—41no—
138C4⋊C4×C31—82no—
139C22⋊C4×C31—48no—
140C22⋊C4×C31—41no—
141C22⋊C4×C31—41no—
142C22⋊C4×C31—41no—
143C22⋊C4×C31—41no—
144C22⋊C4×C31—41no—
145C22⋊C4×C31—41no—
146C22⋊C4×C31—41no—
147C22⋊C4×C31—41no—
148C22⋊C4×C31—82no—
149C22⋊C4×C31—82no—
150#9924C3111yesmaximal, characteristic, direct product factor, Sylow
151C23×C31—22no—
152C23×C31—22no—
153C23×C31—22no—
154C23×C31—22no—
155C23×C31—22no—
156C23×C31—44no—
157C23×C31—42no—
158C23×C31—44no—
159C23×C31—42no—
160C23×C31—41no—
161C23×C31—41no—
162C23×C31—41no—
163C23×C31—42no—
164C23×C31—42no—
165C23×C31—41no—
166C23×C31—82no—
167C23×C31—82no—
168C23×C31—81no—
169C23×C31—81no—
170C23×C31#12011yescharacteristic
171C23×C31#12011yescharacteristic
172C23×C31#12111yescharacteristic
173C23×C31C2×Q1611yescharacteristic
174C23×C31#13511yescharacteristic
175C23×C31#13711yescharacteristic
176C23×C31#13711yescharacteristic
177Q8×C31—24no—
178Q8×C31—24no—
179Q8×C31—42no—
180Q8×C31—44no—
181Q8×C31—42no—
182Q8×C31—44no—
183Q8×C31—42no—
184Q8×C31—81no—
185C2×C4×C31—22no—
186C2×C4×C31—22no—
187C2×C4×C31—24no—
188C2×C4×C31—41no—
189C2×C4×C31—41no—
190C2×C4×C31—41no—
191C2×C4×C31—41no—
192C2×C4×C31—42no—
193C2×C4×C31—44no—
194C2×C4×C31—42no—
195C2×C4×C31—44no—
196C2×C4×C31—42no—
197C2×C4×C31—42no—
198C2×C4×C31—42no—
199C2×C4×C31—41no—
200C2×C4×C31—41no—
201C2×C4×C31—41no—
202C2×C4×C31—42no—
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—82no—
210C2×C4×C31—82no—
211C2×C4×C31—81no—
212C2×C4×C31#12011yescharacteristic
213C2×C4×C31#12111yescharacteristic
214C2×C4×C31#12111yescharacteristic
215C2×C4×C31#12711yescharacteristic
216C8×C31—42no—
217C8×C31—81no—
218C2×#451C2×C3111yescharacteristic, complemented
219#2626C2×C3111yescharacteristic
220C2×#393C2×C3111yescharacteristic
221#1787C2×C3111yescharacteristic, complemented
222C2×#328C2×C3111yescharacteristic, complemented
223C2×#323C2×C3111yescharacteristic
224#1294C2×C3111yescharacteristic, complemented
225C22×C31—22no—
226C22×C31—22no—
227C22×C31—22no—
228C22×C31—22no—
229C22×C31—21no—
230C22×C31—21no—
231C22×C31—21no—
232C22×C31—21no—
233C22×C31—44no—
234C22×C31—44no—
235C22×C31—42no—
236C22×C31—82no—
237C22×C31—82no—
238C22×C31—82no—
239C22×C31—82no—
240C22×C31—81no—
241C22×C31—81no—
242C22×C31#39211yescharacteristic
243C22×C31#45111yescharacteristic
244C22×C31#45111yescharacteristic
245C22×C31#45211yescharacteristic
246C22×C31#45211yescharacteristic
247C22×C31#45512yes—
248C22×C31B(4,𝔽2)12yes—
249C22×C31#46711yescharacteristic
250C22×C31#47011yescharacteristic
251C4×C31—21no—
252C4×C31—21no—
253C4×C31—21no—
254C4×C31—21no—
255C4×C31—44no—
256C4×C31—82no—
257C4×C31—82no—
258C4×C31—81no—
259C4×C31—81no—
260C4×C31—81no—
261C23×Q8C22×C3111yescharacteristic
262C22×Q16—21no—
263C2×C22⋊Q8C22×C3111yescharacteristic
264C22×C22⋊C4—21no—
265C22×C22⋊C4C22×C3111yescharacteristic
266#451—22no—
267#451—22no—
268C2×#103—21no—
269C2×#103—22no—
270C2×C22⋊C8C22×C3111yescharacteristic
271C2×C22⋊C8C22×C3111yescharacteristic
272#399—21no—
273#396—22no—
274#393—21no—
275#393—22no—
276#393—22no—
277#385—21no—
278#380C22×C3111yescharacteristic
279#379—21no—
280C2×#95—21no—
281C2×#95C22×C3111yescharacteristic
282#328—22no—
283#323—22no—
284C2×C31—21no—
285C2×C31—21no—
286C2×C31—21no—
287C2×C31—21no—
288C2×C31—82no—
289C2×C31#160212yes—
290C2×C31#160612yes—
291C2×C31#200111yescharacteristic
292C2×C31#200411yescharacteristic
293C2×C31#202111yescharacteristic
294C25D4×C3111yescharacteristic
295C22×Q8—21no—
296C22×Q8—21no—
297C22×Q8—21no—
298C22×Q8—21no—
299C22×Q8—21no—
300C22×Q8—44no—
301C22×Q8D4×C3111yescharacteristic
302C22×Q8D4×C3112yes—
303C23×C4—21no—
304C23×C4—21no—
305C23×C4D4×C3111yescharacteristic
306C23×C4C23×C3111yescharacteristic
307C2×Q16—24no—
308C2×Q16—44no—
309C22×C8—21no—
310C22×C8—21no—
311C22⋊Q8—21no—
312C22⋊Q8—21no—
313C22⋊Q8—21no—
314C22⋊Q8—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4D4×C3111yescharacteristic
318C2×C4⋊C4D4×C3112yes—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—44no—
331#103—24no—
332#103—44no—
333C22⋊C8—21no—
334C22⋊C8—21no—
335C22⋊C8—21no—
336C22⋊C8D4×C3112yes—
337#95—24no—
338#95—22no—
339#95—21no—
340#95—21no—
341#95D4×C3112yes—
342C31#992411yescharacteristic, direct product factor, Sylow
343C24—21no—
344C24—21no—
345C24—21no—
346C24—21no—
347C24—21no—
348C24—21no—
349C24—42no—
350C24—42no—
351C24Q16×C3112yescomplemented
352C24C2×D4×C3111yescharacteristic
353C2×Q8—24no—
354C2×Q8—24no—
355C2×Q8—22no—
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—21no—
359C2×Q8—44no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—44no—
363C2×Q8—42no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—41no—
367C2×Q8—41no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4C2×D4×C3111yescharacteristic
387C22×C4C2×D4×C3111yescharacteristic
388C22×C4C2×D4×C3111yescharacteristic
389C22×C4C2×D4×C3111yescharacteristic
390C22×C4C2×D4×C3111yescharacteristic
391C22×C4#41×C3111yescharacteristic
392Q16—48no—
393C2×C8—22no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—41no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—42no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—82no—
403C22⋊C4—48no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—44no—
420C23—42no—
421C23—44no—
422C23—42no—
423C23—41no—
424C23—41no—
425C23—42no—
426C23—42no—
427C23—41no—
428C23—41no—
429C23—82no—
430C23—82no—
431C23—81no—
432C23—81no—
433C23C31×#12011yescharacteristic
434C23C31×#12011yescharacteristic
435C23C31×#12111yescharacteristic
436C23C2×Q16×C3111yescharacteristic
437C23C31×#13511yescharacteristic
438C23C31×#13711yescharacteristic
439C23C31×#13711yescharacteristic
440Q8—24no—
441Q8—24no—
442Q8—42no—
443Q8—44no—
444Q8—42no—
445Q8—44no—
446Q8—42no—
447Q8—81no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—81no—
475C2×C4C31×#12011yescharacteristic
476C2×C4C31×#12111yescharacteristic
477C2×C4C31×#12111yescharacteristic
478C2×C4C31×#12711yescharacteristic
479C8—42no—
480C8—81no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—44no—
490C22—44no—
491C22—42no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22—82no—
496C22—81no—
497C22—81no—
498C22C31×#39211yescharacteristic
499C22C31×#45111yescharacteristic
500C22C31×#45111yescharacteristic
501C22C31×#45211yescharacteristic
502C22C31×#45211yescharacteristic
503C22C31×#45512yes—
504C22C31×B(4,𝔽2)12yes—
505C22C31×#46711yescharacteristic
506C22C31×#47011yescharacteristic
507C4—21no—
508C4—21no—
509C4—21no—
510C4—21no—
511C4—44no—
512C4—82no—
513C4—82no—
514C4—81no—
515C4—81no—
516C4—81no—
517C2—21no—
518C2—21no—
519C2—21no—
520C2—21no—
521C2—82no—
522C2C31×#160212yes—
523C2C31×#160612yes—
524C2C31×#200111yescharacteristic
525C2C31×#200411yescharacteristic
526C2C31×#202111yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC23×C4C23×C31
prime radicalC25×C31D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#451C2semidirect
4. C31×#1787C2semidirect
5. C2×C31×#328C2semidirect
7. C31×#1294C2semidirect
87. C24×C31Q16semidirect
150. #9924C31direct (N × H)
218. C2×#451C2×C31semidirect
221. #1787C2×C31semidirect
222. C2×#328C2×C31semidirect
224. #1294C2×C31semidirect
342. C31#9924direct (N × H)
351. C24Q16×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×#26262
C2×C31×#3932
C31×#17872
C2×C31×#3282
C2×C31×#3232
C31×#12942
#992431