Finite Group Databasebeta

← C31×#23952 #21480711

Subgroups — C31×#23952 #21480711

Order 7936.

1326 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3493C211yesmaximal, characteristic, complemented
2C31×#3003C212yesmaximal, complemented
3C31×#2309C211yesmaximal, characteristic, complemented
4C31×#2027C212yesmaximal, complemented
5C31×#2006C212yesmaximal, complemented
6C31×#1897C212yesmaximal, complemented
7C31×#1883C214yesmaximal, complemented
8C31×#1822C211yesmaximal, characteristic, complemented
9C23×D4×C31C2211yescharacteristic
10C2×C31×AGL(1,ℤ/8)C2214yes—
11C31×#556—22no—
12C31×#555—22no—
13C31×#553—22no—
14C31×#552—22no—
15C31×#544C2211yescharacteristic, complemented
16C31×#538—22no—
17C31×#537C2211yescharacteristic, complemented
18C31×#529—22no—
19C31×#525C2212yes—
20C31×#520C2212yescomplemented
21C2×C31×#117C2211yescharacteristic
22C2×C31×#117C2211yescharacteristic
23C31×#450—22no—
24C31×#450—22no—
25C31×#447—22no—
26C31×#447—22no—
27C2×C31×#104C2214yescomplemented
28C31×#418C2214yes—
29C31×#407C2212yes—
30C31×#390—22no—
31C31×#390C2212yes—
32C31×#390C2212yes—
33C31×#379—22no—
34C31×#343C2212yescomplemented
35C31×#343C2212yescomplemented
36C31×#339C2214yescomplemented
37C25×C31—21no—
38C2×#41×C31C2312yes—
39C22×D4×C31—24no—
40C22×D4×C31—21no—
41C22×D4×C31—22no—
42C22×D4×C31—22no—
43C22×D4×C31—42no—
44C22×D4×C31C2312yes—
45C23×C4×C31C2311yescharacteristic
46C31×AGL(1,ℤ/8)—28no—
47C31×AGL(1,ℤ/8)—44no—
48C2×SD16×C31—24no—
49C2×D8×C31—24no—
50C2×M16×C31C2314yes—
51C31×C4⋊Q8—22no—
52C31×#126—24no—
53C31×#126—24no—
54C31×#125—22no—
55C31×#125—22no—
56C31×#124—22no—
57C31×#124—24no—
58C31×#123—24no—
59C31×#123—22no—
60C31×C22⋊Q8—22no—
61C31×C22⋊Q8—22no—
62C31×C22⋊Q8D412yes—
63C31×C22⋊Q8D412yes—
64C31×#121—24no—
65C31×#121—22no—
66C31×#121—22no—
67C31×#121D412yes—
68C31×#121D412yes—
69C31×#120—22no—
70C4×Q8×C31—22no—
71C4×Q8×C31—22no—
72C4×D4×C31—22no—
73C4×D4×C31—22no—
74C4×D4×C31—24no—
75C4×D4×C31—22no—
76C4×D4×C31—22no—
77C31×#117—24no—
78C31×#117—22no—
79C31×#117—22no—
80C31×#117—22no—
81C31×#117—42no—
82C31×#117C2312yes—
83C2×C4⋊C4×C31—22no—
84C2×C4⋊C4×C31—21no—
85C2×C4⋊C4×C31C2311yescharacteristic
86C2×C4⋊C4×C31C2311yescharacteristic
87C2×C22⋊C4×C31—21no—
88C2×C22⋊C4×C31—24no—
89C2×C22⋊C4×C31—24no—
90C2×C22⋊C4×C31D412yes—
91C2×C42×C31D412yes—
92C2×C42×C31C2312yes—
93C31×#104—44no—
94C31×#102—24no—
95C31×#102—24no—
96C31×C22⋊C8—22no—
97C31×C22⋊C8—22no—
98C24×C31—24no—
99C24×C31—22no—
100C24×C31—42no—
101C24×C31—42no—
102C24×C31—82no—
103C24×C31—82no—
104C24×C31C2×D411yescharacteristic
105#41×C31—24no—
106#41×C31—24no—
107C2×Q8×C31—22no—
108C2×Q8×C31C2×D412yes—
109C2×D4×C31—24no—
110C2×D4×C31—22no—
111C2×D4×C31—22no—
112C2×D4×C31—22no—
113C2×D4×C31—22no—
114C2×D4×C31—24no—
115C2×D4×C31—22no—
116C2×D4×C31—24no—
117C2×D4×C31—48no—
118C2×D4×C31—48no—
119C2×D4×C31—42no—
120C2×D4×C31—41no—
121C2×D4×C31—41no—
122C2×D4×C31C2×D412yes—
123C2×D4×C31#4114yes—
124C22×C4×C31—22no—
125C22×C4×C31—22no—
126C22×C4×C31—24no—
127C22×C4×C31—22no—
128C22×C4×C31—22no—
129C22×C4×C31C2×D411yescharacteristic
130C22×C4×C31C2×D412yes—
131C22×C4×C31C2×D412yes—
132C22×C4×C31C2×D412yes—
133C22×C4×C31C2×D412yes—
134C22×C4×C31C2411yescharacteristic
135SD16×C31—48no—
136D8×C31—48no—
137M16×C31—24no—
138M16×C31—24no—
139C2×C8×C31—24no—
140C4⋊C4×C31—24no—
141C4⋊C4×C31—24no—
142C4⋊C4×C31—22no—
143C4⋊C4×C31—22no—
144C4⋊C4×C31—22no—
145C4⋊C4×C31—22no—
146C4⋊C4×C31—22no—
147C4⋊C4×C31—22no—
148C4⋊C4×C31—42no—
149C4⋊C4×C31C2×D412yes—
150C4⋊C4×C31C2×D412yes—
151C22⋊C4×C31—24no—
152C22⋊C4×C31—24no—
153C22⋊C4×C31—24no—
154C22⋊C4×C31—22no—
155C22⋊C4×C31—22no—
156C22⋊C4×C31—22no—
157C22⋊C4×C31—22no—
158C22⋊C4×C31—22no—
159C22⋊C4×C31—42no—
160C22⋊C4×C31—84no—
161C22⋊C4×C31—84no—
162C42×C31—24no—
163C42×C31—22no—
164C42×C31—22no—
165C42×C31—22no—
166C42×C31—24no—
167#23952C3111yesmaximal, characteristic, direct product factor, Sylow
168C23×C31—21no—
169C23×C31—21no—
170C23×C31—22no—
171C23×C31—24no—
172C23×C31—41no—
173C23×C31—41no—
174C23×C31—44no—
175C23×C31—44no—
176C23×C31—48no—
177C23×C31—42no—
178C23×C31—42no—
179C23×C31—84no—
180C23×C31—84no—
181C23×C31—88no—
182C23×C31—88no—
183C23×C31#12012yes—
184C23×C31C22×D411yescharacteristic
185Q8×C31—24no—
186Q8×C31—82no—
187D4×C31—24no—
188D4×C31—28no—
189D4×C31—44no—
190D4×C31—44no—
191D4×C31—42no—
192D4×C31—42no—
193D4×C31—44no—
194D4×C31—82no—
195D4×C31—82no—
196D4×C31—82no—
197C2×C4×C31—21no—
198C2×C4×C31—21no—
199C2×C4×C31—24no—
200C2×C4×C31—24no—
201C2×C4×C31—22no—
202C2×C4×C31—22no—
203C2×C4×C31—22no—
204C2×C4×C31—22no—
205C2×C4×C31—22no—
206C2×C4×C31—44no—
207C2×C4×C31—42no—
208C2×C4×C31—42no—
209C2×C4×C31—42no—
210C2×C4×C31—42no—
211C2×C4×C31—42no—
212C2×C4×C31—84no—
213C2×C4×C31#12011yescharacteristic
214C2×C4×C31#12011yescharacteristic
215C2×C4×C31C22×D412yes—
216C2×C4×C31C2×#4112yes—
217C2×C4×C31H(5,𝔽2)12yes—
218C8×C31—44no—
219#3493C2×C3111yescharacteristic, complemented
220#3003C2×C3112yescomplemented
221#2309C2×C3111yescharacteristic, complemented
222#2027C2×C3112yescomplemented
223#2006C2×C3112yescomplemented
224#1897C2×C3112yescomplemented
225#1883C2×C3114yescomplemented
226#1822C2×C3111yescharacteristic, complemented
227C22×C31—21no—
228C22×C31—21no—
229C22×C31—22no—
230C22×C31—22no—
231C22×C31—22no—
232C22×C31—42no—
233C22×C31—42no—
234C22×C31—48no—
235C22×C31—42no—
236C22×C31—44no—
237C22×C31—42no—
238C22×C31—42no—
239C22×C31—84no—
240C22×C31—816no—
241C22×C31—82no—
242C22×C31—82no—
243C22×C31C2×#12011yescharacteristic
244C22×C31#52511yescharacteristic
245C22×C31#53511yescharacteristic
246C4×C31—21no—
247C4×C31—21no—
248C4×C31—42no—
249C4×C31—42no—
250C4×C31—42no—
251C4×C31—42no—
252C4×C31—42no—
253C4×C31—82no—
254C4×C31#54614yes—
255C23×D4C22×C3111yescharacteristic
256C2×AGL(1,ℤ/8)C22×C3114yes—
257#556—22no—
258#555—22no—
259#553—22no—
260#552—22no—
261#544C22×C3111yescharacteristic, complemented
262#538—22no—
263#537C22×C3111yescharacteristic, complemented
264#529—22no—
265#525C22×C3112yes—
266#520C22×C3112yescomplemented
267C2×#117C22×C3111yescharacteristic
268C2×#117C22×C3111yescharacteristic
269#450—22no—
270#450—22no—
271#447—22no—
272#447—22no—
273C2×#104C22×C3114yescomplemented
274#418C22×C3114yes—
275#407C22×C3112yes—
276#390—22no—
277#390C22×C3112yes—
278#390C22×C3112yes—
279#379—22no—
280#343C22×C3112yescomplemented
281#343C22×C3112yescomplemented
282#339C22×C3114yescomplemented
283C2×C31—21no—
284C2×C31—21no—
285C2×C31—42no—
286C2×C31—82no—
287C2×C31—84no—
288C2×C31#240411yescharacteristic
289C2×C31#302111yescharacteristic
290C2×C31#302111yescharacteristic
291C25—21no—
292C2×#41C23×C3112yes—
293C22×D4—24no—
294C22×D4—21no—
295C22×D4—22no—
296C22×D4—22no—
297C22×D4—42no—
298C22×D4C23×C3112yes—
299C23×C4C23×C3111yescharacteristic
300AGL(1,ℤ/8)—28no—
301AGL(1,ℤ/8)—44no—
302C2×SD16—24no—
303C2×D8—24no—
304C2×M16C23×C3114yes—
305C4⋊Q8—22no—
306#126—24no—
307#126—24no—
308#125—22no—
309#125—22no—
310#124—22no—
311#124—24no—
312#123—24no—
313#123—22no—
314C22⋊Q8—22no—
315C22⋊Q8—22no—
316C22⋊Q8D4×C3112yes—
317C22⋊Q8D4×C3112yes—
318#121—24no—
319#121—22no—
320#121—22no—
321#121D4×C3112yes—
322#121D4×C3112yes—
323#120—22no—
324C4×Q8—22no—
325C4×Q8—22no—
326C4×D4—22no—
327C4×D4—22no—
328C4×D4—24no—
329C4×D4—22no—
330C4×D4—22no—
331#117—24no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117—42no—
336#117C23×C3112yes—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4C23×C3111yescharacteristic
340C2×C4⋊C4C23×C3111yescharacteristic
341C2×C22⋊C4—21no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4D4×C3112yes—
345C2×C42D4×C3112yes—
346C2×C42C23×C3112yes—
347#104—44no—
348#102—24no—
349#102—24no—
350C22⋊C8—22no—
351C22⋊C8—22no—
352C31#2395211yescharacteristic, direct product factor, Sylow
353C24—24no—
354C24—22no—
355C24—42no—
356C24—42no—
357C24—82no—
358C24—82no—
359C24C2×D4×C3111yescharacteristic
360#41—24no—
361#41—24no—
362C2×Q8—22no—
363C2×Q8C2×D4×C3112yes—
364C2×D4—24no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—24no—
370C2×D4—22no—
371C2×D4—24no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—42no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4C2×D4×C3112yes—
378C2×D4#41×C3114yes—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4C2×D4×C3111yescharacteristic
385C22×C4C2×D4×C3112yes—
386C22×C4C2×D4×C3112yes—
387C22×C4C2×D4×C3112yes—
388C22×C4C2×D4×C3112yes—
389C22×C4C24×C3111yescharacteristic
390SD16—48no—
391D8—48no—
392M16—24no—
393M16—24no—
394C2×C8—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—42no—
404C4⋊C4C2×D4×C3112yes—
405C4⋊C4C2×D4×C3112yes—
406C22⋊C4—24no—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—42no—
415C22⋊C4—84no—
416C22⋊C4—84no—
417C42—24no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—24no—
422C23—21no—
423C23—21no—
424C23—22no—
425C23—24no—
426C23—41no—
427C23—41no—
428C23—44no—
429C23—44no—
430C23—48no—
431C23—42no—
432C23—42no—
433C23—84no—
434C23—84no—
435C23—88no—
436C23—88no—
437C23C31×#12012yes—
438C23C22×D4×C3111yescharacteristic
439Q8—24no—
440Q8—82no—
441D4—24no—
442D4—28no—
443D4—44no—
444D4—44no—
445D4—42no—
446D4—42no—
447D4—44no—
448D4—82no—
449D4—82no—
450D4—82no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—84no—
467C2×C4C31×#12011yescharacteristic
468C2×C4C31×#12011yescharacteristic
469C2×C4C22×D4×C3112yes—
470C2×C4C2×#41×C3112yes—
471C2×C4C31×H(5,𝔽2)12yes—
472C8—44no—
473C22—21no—
474C22—21no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—42no—
479C22—42no—
480C22—48no—
481C22—42no—
482C22—44no—
483C22—42no—
484C22—42no—
485C22—84no—
486C22—816no—
487C22—82no—
488C22—82no—
489C22C2×C31×#12011yescharacteristic
490C22C31×#52511yescharacteristic
491C22C31×#53511yescharacteristic
492C4—21no—
493C4—21no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—42no—
499C4—82no—
500C4C31×#54614yes—
501C2—21no—
502C2—21no—
503C2—42no—
504C2—84no—
505C2—82no—
506C2C31×#240411yescharacteristic
507C2C31×#302111yescharacteristic
508C2C31×#302111yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3493C2semidirect
2. C31×#3003C2semidirect
3. C31×#2309C2semidirect
4. C31×#2027C2semidirect
5. C31×#2006C2semidirect
6. C31×#1897C2semidirect
7. C31×#1883C2semidirect
8. C31×#1822C2semidirect
15. C31×#544C22semidirect
17. C31×#537C22semidirect
20. C31×#520C22semidirect
27. C2×C31×#104C22semidirect
34. C31×#343C22semidirect
35. C31×#343C22semidirect
36. C31×#339C22semidirect
167. #23952C31direct (N × H)
219. #3493C2×C31semidirect
220. #3003C2×C31semidirect
221. #2309C2×C31semidirect
222. #2027C2×C31semidirect
223. #2006C2×C31semidirect
224. #1897C2×C31semidirect
225. #1883C2×C31semidirect
226. #1822C2×C31semidirect
261. #544C22×C31semidirect
263. #537C22×C31semidirect
266. #520C22×C31semidirect
273. C2×#104C22×C31semidirect
280. #343C22×C31semidirect
281. #343C22×C31semidirect
282. #339C22×C31semidirect
352. C31#23952direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34932
C31×#30032
C31×#23092
C31×#20272
C31×#20062
C31×#18972
C31×#18832
C31×#18222
#2395231