Finite Group Databasebeta

← C31×#18987 #21475746

Subgroups — C31×#18987 #21475746

Order 7936.

904 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3529C211yesmaximal, characteristic
2C31×#3172C212yesmaximal, complemented
3C31×#3135C211yesmaximal, characteristic, complemented
4C31×#1664C212yesmaximal, complemented
5C31×#1662C212yesmaximal, complemented
6C31×#1661C212yesmaximal, complemented
7C31×#1659C212yesmaximal, complemented
8C31×#1525C211yesmaximal, characteristic, complemented
9C31×#1524C212yesmaximal, complemented
10C31×#563—21no—
11C31×#563C2211yescharacteristic
12C31×#562—22no—
13C31×#562C2212yes—
14C31×H(3,𝔽4)C2211yescharacteristic
15C31×#560—21no—
16C31×#560—22no—
17C31×#533C2211yescharacteristic, complemented
18C31×#532—21no—
19C31×#532C2211yescharacteristic
20C31×#529—22no—
21C31×#529C2212yes—
22C31×#525—21no—
23C31×#525—22no—
24C31×#501C2212yescomplemented
25C31×#497C2212yes—
26C31×#491—21no—
27C31×#491C2212yescomplemented
28C31×#490—21no—
29C31×#489C2212yes—
30C31×#489C2212yes—
31C31×#489C2212yescomplemented
32C31×#470—22no—
33C31×#468—22no—
34C4×Q16×C31—22no—
35C4×SD16×C31—22no—
36C31×#416—21no—
37C31×#416—21no—
38C4×M16×C31C2212yescomplemented
39C2×C31×#97C2211yescharacteristic, complemented
40C31×#330C2212yescomplemented
41C31×#330C2212yescomplemented
42C31×#329C2212yes—
43C31×#329C2212yes—
44C31×#329C2212yes—
45C31×#329C2211yescharacteristic, complemented
46C31×#329C2211yescharacteristic, complemented
47C2×#41×C31—21no—
48C2×#41×C31D411yescharacteristic
49C2×#41×C31D411yescharacteristic
50C23×C4×C31—21no—
51C2×Q16×C31—22no—
52C2×SD16×C31—22no—
53C2×M16×C31—22no—
54C22×C8×C31—21no—
55C31×C4⋊Q8—21no—
56C31×C4⋊Q8D411yescharacteristic
57C31×C4⋊Q8C2312yes—
58C31×#127—21no—
59C31×#127D411yescharacteristic
60C31×#126—22no—
61C31×#126—22no—
62C31×#126—22no—
63C31×#125—22no—
64C31×#125—21no—
65C31×#125C2312yes—
66C31×#124—21no—
67C31×#124—22no—
68C31×#124—21no—
69C31×#124C2311yescharacteristic
70C31×#124C2312yes—
71C31×#124C2311yescharacteristic
72C31×#123—22no—
73C31×#123—21no—
74C31×#123—22no—
75C31×#123—21no—
76C31×#123—22no—
77C31×#123—22no—
78C31×#123—22no—
79C31×C22⋊Q8—22no—
80C31×C22⋊Q8—22no—
81C31×C22⋊Q8—22no—
82C31×C22⋊Q8—22no—
83C31×C22⋊Q8—22no—
84C31×C22⋊Q8D412yes—
85C31×#121—22no—
86C31×#121—22no—
87C31×#121—22no—
88C31×#121—22no—
89C31×#121—22no—
90C31×#121—22no—
91C31×#121—22no—
92C31×#121D412yes—
93C31×#120—21no—
94C31×#120—22no—
95C31×#120—21no—
96C31×#120—22no—
97C4×Q8×C31—22no—
98C4×Q8×C31D412yes—
99C4×D4×C31—22no—
100C4×D4×C31—22no—
101C4×D4×C31—22no—
102C4×D4×C31—22no—
103C4×D4×C31D412yes—
104C31×#117—21no—
105C31×#117—22no—
106C31×#117—21no—
107C2×C42×C31—21no—
108C2×C42×C31C2311yescharacteristic
109C31×#107—22no—
110C31×#106—22no—
111C31×#103—22no—
112C31×#103—22no—
113C31×#103—22no—
114C31×#103—22no—
115C31×#103—22no—
116C31×#102—22no—
117C31×#102—22no—
118C31×#102—22no—
119C31×#97—21no—
120C31×#97C2312yes—
121C31×#97C2312yes—
122C31×#97C2312yes—
123C4×C8×C31—22no—
124C24×C31—21no—
125#41×C31—42no—
126#41×C31—42no—
127#41×C31—82no—
128C2×Q8×C31—21no—
129C2×Q8×C31—22no—
130C2×Q8×C31C2×D411yescharacteristic
131C2×Q8×C31C2×D411yescharacteristic
132C2×Q8×C31C2×D412yes—
133C2×D4×C31—22no—
134C2×D4×C31—22no—
135C2×D4×C31—21no—
136C2×D4×C31—22no—
137C2×D4×C31—22no—
138C2×D4×C31—21no—
139C2×D4×C31—21no—
140C2×D4×C31—22no—
141C2×D4×C31C2×D411yescharacteristic
142C2×D4×C31C2×D411yescharacteristic
143C2×D4×C31C2×D412yes—
144C22×C4×C31—22no—
145C22×C4×C31—22no—
146C22×C4×C31—21no—
147C22×C4×C31—21no—
148C22×C4×C31—21no—
149C22×C4×C31—22no—
150C22×C4×C31—82no—
151C22×C4×C31C2×D411yescharacteristic
152C22×C4×C31C2×D412yes—
153Q16×C31—44no—
154SD16×C31—44no—
155M16×C31—44no—
156C2×C8×C31—22no—
157C2×C8×C31—22no—
158C2×C8×C31—22no—
159C2×C8×C31—42no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—22no—
169C4⋊C4×C31—22no—
170C4⋊C4×C31—42no—
171C4⋊C4×C31C2×D412yes—
172C4⋊C4×C31C2×D412yes—
173C22⋊C4×C31—21no—
174C22⋊C4×C31—21no—
175C22⋊C4×C31—22no—
176C22⋊C4×C31—22no—
177C22⋊C4×C31—22no—
178C22⋊C4×C31—22no—
179C22⋊C4×C31—22no—
180C22⋊C4×C31—22no—
181C22⋊C4×C31—22no—
182C22⋊C4×C31—22no—
183C22⋊C4×C31—22no—
184C22⋊C4×C31—22no—
185C22⋊C4×C31—22no—
186C22⋊C4×C31—41no—
187C42×C31—21no—
188C42×C31—21no—
189C42×C31—22no—
190C42×C31—41no—
191C42×C31C2×D411yescharacteristic
192C42×C31C2×D412yes—
193C42×C31C2411yescharacteristic
194#18987C3111yesmaximal, characteristic, direct product factor, Sylow
195C23×C31—21no—
196C23×C31—22no—
197C23×C31—21no—
198C23×C31—81no—
199C23×C31—82no—
200C23×C31#12011yescharacteristic
201Q8×C31—42no—
202Q8×C31—41no—
203Q8×C31—41no—
204Q8×C31—42no—
205Q8×C31—42no—
206Q8×C31—81no—
207D4×C31—42no—
208D4×C31—41no—
209D4×C31—41no—
210D4×C31—82no—
211D4×C31—82no—
212D4×C31—81no—
213D4×C31—81no—
214D4×C31—82no—
215D4×C31—82no—
216D4×C31—82no—
217D4×C31—81no—
218C2×C4×C31—22no—
219C2×C4×C31—22no—
220C2×C4×C31—22no—
221C2×C4×C31—21no—
222C2×C4×C31—21no—
223C2×C4×C31—22no—
224C2×C4×C31—42no—
225C2×C4×C31—42no—
226C2×C4×C31—44no—
227C2×C4×C31—42no—
228C2×C4×C31—42no—
229C2×C4×C31—82no—
230C2×C4×C31—82no—
231C2×C4×C31—81no—
232C2×C4×C31#12012yes—
233C2×C4×C31#12011yescharacteristic
234C2×C4×C31C22×D412yes—
235C2×C4×C31C22×D411yescharacteristic
236C8×C31—24no—
237C8×C31—42no—
238C8×C31—42no—
239#3529C2×C3111yescharacteristic
240#3172C2×C3112yescomplemented
241#3135C2×C3111yescharacteristic, complemented
242#1664C2×C3112yescomplemented
243#1662C2×C3112yescomplemented
244#1661C2×C3112yescomplemented
245#1659C2×C3112yescomplemented
246#1525C2×C3111yescharacteristic, complemented
247#1524C2×C3112yescomplemented
248C22×C31—21no—
249C22×C31—22no—
250C22×C31—41no—
251C22×C31—42no—
252C22×C31—41no—
253C22×C31—42no—
254C22×C31—42no—
255C22×C31—42no—
256C22×C31—42no—
257C22×C31—84no—
258C22×C31C2×#12011yescharacteristic
259C4×C31—21no—
260C4×C31—22no—
261C4×C31—22no—
262C4×C31—41no—
263C4×C31—42no—
264C4×C31—41no—
265C4×C31—42no—
266C4×C31—42no—
267C4×C31—82no—
268C4×C31#45512yes—
269#563—21no—
270#563C22×C3111yescharacteristic
271#562—22no—
272#562C22×C3112yes—
273H(3,𝔽4)C22×C3111yescharacteristic
274#560—21no—
275#560—22no—
276#533C22×C3111yescharacteristic, complemented
277#532—21no—
278#532C22×C3111yescharacteristic
279#529—22no—
280#529C22×C3112yes—
281#525—21no—
282#525—22no—
283#501C22×C3112yescomplemented
284#497C22×C3112yes—
285#491—21no—
286#491C22×C3112yescomplemented
287#490—21no—
288#489C22×C3112yes—
289#489C22×C3112yes—
290#489C22×C3112yescomplemented
291#470—22no—
292#468—22no—
293C4×Q16—22no—
294C4×SD16—22no—
295#416—21no—
296#416—21no—
297C4×M16C22×C3112yescomplemented
298C2×#97C22×C3111yescharacteristic, complemented
299#330C22×C3112yescomplemented
300#330C22×C3112yescomplemented
301#329C22×C3112yes—
302#329C22×C3112yes—
303#329C22×C3112yes—
304#329C22×C3111yescharacteristic, complemented
305#329C22×C3111yescharacteristic, complemented
306C2×C31—41no—
307C2×C31—81no—
308C2×C31—81no—
309C2×C31—82no—
310C2×C31C2×#45511yescharacteristic
311C2×C31#301812yes—
312C2×#41—21no—
313C2×#41D4×C3111yescharacteristic
314C2×#41D4×C3111yescharacteristic
315C23×C4—21no—
316C2×Q16—22no—
317C2×SD16—22no—
318C2×M16—22no—
319C22×C8—21no—
320C4⋊Q8—21no—
321C4⋊Q8D4×C3111yescharacteristic
322C4⋊Q8C23×C3112yes—
323#127—21no—
324#127D4×C3111yescharacteristic
325#126—22no—
326#126—22no—
327#126—22no—
328#125—22no—
329#125—21no—
330#125C23×C3112yes—
331#124—21no—
332#124—22no—
333#124—21no—
334#124C23×C3112yes—
335#124C23×C3111yescharacteristic
336#124C23×C3111yescharacteristic
337#123—22no—
338#123—21no—
339#123—22no—
340#123—21no—
341#123—22no—
342#123—22no—
343#123—22no—
344C22⋊Q8—22no—
345C22⋊Q8—22no—
346C22⋊Q8—22no—
347C22⋊Q8—22no—
348C22⋊Q8—22no—
349C22⋊Q8D4×C3112yes—
350#121—22no—
351#121—22no—
352#121—22no—
353#121—22no—
354#121—22no—
355#121—22no—
356#121—22no—
357#121D4×C3112yes—
358#120—21no—
359#120—22no—
360#120—21no—
361#120—22no—
362C4×Q8—22no—
363C4×Q8D4×C3112yes—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—22no—
368C4×D4D4×C3112yes—
369#117—21no—
370#117—22no—
371#117—21no—
372C2×C42—21no—
373C2×C42C23×C3111yescharacteristic
374#107—22no—
375#106—22no—
376#103—22no—
377#103—22no—
378#103—22no—
379#103—22no—
380#103—22no—
381#102—22no—
382#102—22no—
383#102—22no—
384#97—21no—
385#97C23×C3112yes—
386#97C23×C3112yes—
387#97C23×C3112yes—
388C4×C8—22no—
389C31#1898711yescharacteristic, direct product factor, Sylow
390C24—21no—
391#41—42no—
392#41—42no—
393#41—82no—
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8C2×D4×C3111yescharacteristic
397C2×Q8C2×D4×C3111yescharacteristic
398C2×Q8C2×D4×C3112yes—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4C2×D4×C3111yescharacteristic
408C2×D4C2×D4×C3111yescharacteristic
409C2×D4C2×D4×C3112yes—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—82no—
417C22×C4C2×D4×C3111yescharacteristic
418C22×C4C2×D4×C3112yes—
419Q16—44no—
420SD16—44no—
421M16—44no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—42no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—42no—
437C4⋊C4C2×D4×C3112yes—
438C4⋊C4C2×D4×C3112yes—
439C22⋊C4—21no—
440C22⋊C4—21no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—41no—
453C42—21no—
454C42—21no—
455C42—22no—
456C42—41no—
457C42C2×D4×C3112yes—
458C42C2×D4×C3111yescharacteristic
459C42C24×C3111yescharacteristic
460C23—21no—
461C23—22no—
462C23—21no—
463C23—81no—
464C23—82no—
465C23C31×#12011yescharacteristic
466Q8—42no—
467Q8—41no—
468Q8—41no—
469Q8—42no—
470Q8—42no—
471Q8—81no—
472D4—42no—
473D4—41no—
474D4—41no—
475D4—82no—
476D4—82no—
477D4—81no—
478D4—81no—
479D4—82no—
480D4—82no—
481D4—82no—
482D4—81no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—81no—
497C2×C4C31×#12012yes—
498C2×C4C31×#12011yescharacteristic
499C2×C4C22×D4×C3112yes—
500C2×C4C22×D4×C3111yescharacteristic
501C8—24no—
502C8—42no—
503C8—42no—
504C22—21no—
505C22—22no—
506C22—41no—
507C22—42no—
508C22—41no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—84no—
514C22C2×C31×#12011yescharacteristic
515C4—21no—
516C4—22no—
517C4—22no—
518C4—41no—
519C4—42no—
520C4—41no—
521C4—42no—
522C4—42no—
523C4—82no—
524C4C31×#45512yes—
525C2—41no—
526C2—81no—
527C2—82no—
528C2—81no—
529C2C2×C31×#45511yescharacteristic
530C2C31×#301812yes—
subgroupG/H
centerC2×C4×C31#120
commutator, Frattini, cosocleC42C24×C31
socleC22×C31C2×#120
prime radicalC31×#3529C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3172C2semidirect
3. C31×#3135C2semidirect
4. C31×#1664C2semidirect
5. C31×#1662C2semidirect
6. C31×#1661C2semidirect
7. C31×#1659C2semidirect
8. C31×#1525C2semidirect
9. C31×#1524C2semidirect
17. C31×#533C22semidirect
24. C31×#501C22semidirect
27. C31×#491C22semidirect
31. C31×#489C22semidirect
38. C4×M16×C31C22semidirect
39. C2×C31×#97C22semidirect
40. C31×#330C22semidirect
41. C31×#330C22semidirect
45. C31×#329C22semidirect
46. C31×#329C22semidirect
194. #18987C31direct (N × H)
240. #3172C2×C31semidirect
241. #3135C2×C31semidirect
242. #1664C2×C31semidirect
243. #1662C2×C31semidirect
244. #1661C2×C31semidirect
245. #1659C2×C31semidirect
246. #1525C2×C31semidirect
247. #1524C2×C31semidirect
276. #533C22×C31semidirect
283. #501C22×C31semidirect
286. #491C22×C31semidirect
290. #489C22×C31semidirect
297. C4×M16C22×C31semidirect
298. C2×#97C22×C31semidirect
299. #330C22×C31semidirect
300. #330C22×C31semidirect
304. #329C22×C31semidirect
305. #329C22×C31semidirect
389. C31#18987direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35292
C31×#31722
C31×#31352
C31×#16642
C31×#16622
C31×#16612
C31×#16592
C31×#15252
C31×#15242
#1898731