Finite Group Databasebeta

← C31×#36732 #21493491

Subgroups — C31×#36732 #21493491

Order 7936.

1770 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2848C212yesmaximal
2C31×#2744C214yesmaximal, complemented
3C31×#2739C214yesmaximal, complemented
4C31×#2698C214yesmaximal, complemented
5C31×#2689C214yesmaximal, complemented
6C31×#2659C211yesmaximal, characteristic, complemented
7C31×#2646C211yesmaximal, characteristic, complemented
8C31×#2505C214yesmaximal, complemented
9C31×#2480C214yesmaximal
10C31×#2329C212yesmaximal, complemented
11C2×C31×#381C211yesmaximal, characteristic
12C24×C4×C31C2211yescharacteristic
13C2×C31×#126C2214yes—
14C2×C31×C22⋊Q8C2212yes—
15C2×C31×C22⋊Q8C2212yes—
16C2×C31×C22⋊Q8C2212yes—
17C22×C22⋊C4×C31C2212yes—
18C31×#400C2214yescomplemented
19C31×#400C2214yescomplemented
20C31×#400C2214yescomplemented
21C31×#400C2214yescomplemented
22C31×#398C2214yescomplemented
23C31×#398C2214yescomplemented
24C31×#398C2214yescomplemented
25C31×#397C2214yes—
26C31×#397C2214yes—
27C31×#397C2214yes—
28C31×#396C2214yes—
29C31×#395C2214yescomplemented
30C31×#391C2214yescomplemented
31C31×#389C2214yescomplemented
32C31×#389C2214yescomplemented
33C31×#388C2214yes—
34C31×#388C2214yes—
35C31×#388C2214yes—
36C31×#387C2214yescomplemented
37C31×#387C2214yescomplemented
38C31×#387C2214yescomplemented
39C31×#387C2214yescomplemented
40C31×#385C2214yes—
41C31×#385C2214yes—
42C31×#385C2214yes—
43C31×#385C2214yes—
44C31×#385C2214yes—
45C31×#382C2214yescomplemented
46C31×#381C2214yes—
47C31×#381C2214yes—
48C31×#380C2212yes—
49C31×#380C2212yes—
50C31×#380C2212yes—
51C31×#380C2214yes—
52C31×#379C2214yes—
53C4×C22⋊C4×C31C2214yes—
54C2×C31×#95C2211yescharacteristic
55C2×C31×#95C2211yescharacteristic
56C2×C31×#95C2212yes—
57C25×C31C2311yescharacteristic
58C22×Q8×C31C2312yes—
59C23×C4×C31—24no—
60C23×C4×C31—44no—
61C23×C4×C31C2312yes—
62C23×C4×C31C2312yes—
63C23×C4×C31C2312yes—
64C23×C4×C31C2312yes—
65C23×C4×C31C2311yescharacteristic
66C23×C4×C31C2311yescharacteristic
67C31×#126—48no—
68C31×C22⋊Q8—28no—
69C31×C22⋊Q8—44no—
70C31×C22⋊Q8—44no—
71C2×C4⋊C4×C31C2314yes—
72C2×C4⋊C4×C31C2312yes—
73C2×C4⋊C4×C31C2314yes—
74C2×C4⋊C4×C31C2314yes—
75C2×C4⋊C4×C31C2314yes—
76C2×C4⋊C4×C31C2314yes—
77C2×C4⋊C4×C31C2314yes—
78C2×C4⋊C4×C31C2312yes—
79C2×C4⋊C4×C31C2314yes—
80C2×C4⋊C4×C31C2314yes—
81C2×C4⋊C4×C31C2312yes—
82C2×C4⋊C4×C31C2314yes—
83C2×C22⋊C4×C31—44no—
84C2×C22⋊C4×C31C2314yes—
85C2×C22⋊C4×C31C2314yes—
86C2×C22⋊C4×C31C2314yes—
87C2×C22⋊C4×C31C2314yes—
88C2×C22⋊C4×C31C2314yes—
89C2×C22⋊C4×C31C2314yes—
90C2×C22⋊C4×C31C2314yes—
91C2×C22⋊C4×C31C2314yes—
92C2×C42×C31C2314yes—
93C2×C42×C31C2314yes—
94C31×#95C2314yes—
95C31×#95C2314yes—
96C31×#95C2314yes—
97C31×#95C2314yes—
98C31×#95C2314yes—
99C31×#95C2314yes—
100C31×#95C2314yes—
101C31×#95C2314yes—
102C31×#95C2314yes—
103C31×#95C2314yes—
104C31×#95C2314yes—
105C31×#95C2314yes—
106C31×#95C2314yes—
107C31×#95C2314yes—
108C31×#95C2314yes—
109C24×C31—22no—
110C24×C31—24no—
111C24×C31—24no—
112C24×C31—41no—
113C24×C31—41no—
114C24×C31C2412yes—
115C24×C31C2411yescharacteristic
116C2×Q8×C31—24no—
117C2×Q8×C31—42no—
118C2×Q8×C31—42no—
119C22×C4×C31—22no—
120C22×C4×C31—24no—
121C22×C4×C31—24no—
122C22×C4×C31—24no—
123C22×C4×C31—24no—
124C22×C4×C31—24no—
125C22×C4×C31—24no—
126C22×C4×C31—42no—
127C22×C4×C31—42no—
128C22×C4×C31—42no—
129C22×C4×C31—42no—
130C22×C4×C31—42no—
131C22×C4×C31—42no—
132C22×C4×C31—42no—
133C22×C4×C31—42no—
134C22×C4×C31—416no—
135C22×C4×C31#4114yes—
136C22×C4×C31C2414yes—
137C22×C4×C31C2414yes—
138C22×C4×C31C2411yescharacteristic
139C22×C4×C31C2411yescharacteristic
140C22×C4×C31C2414yes—
141C22×C4×C31C2414yes—
142C22×C4×C31C2414yes—
143C22×C4×C31C2414yes—
144C22×C4×C31C2412yes—
145C4⋊C4×C31—24no—
146C4⋊C4×C31—28no—
147C4⋊C4×C31—44no—
148C4⋊C4×C31—44no—
149C4⋊C4×C31—44no—
150C4⋊C4×C31—42no—
151C4⋊C4×C31—42no—
152C4⋊C4×C31—44no—
153C4⋊C4×C31—44no—
154C4⋊C4×C31—44no—
155C4⋊C4×C31—44no—
156C4⋊C4×C31—44no—
157C22⋊C4×C31—28no—
158C22⋊C4×C31—28no—
159C22⋊C4×C31—28no—
160C22⋊C4×C31—28no—
161C22⋊C4×C31—28no—
162C22⋊C4×C31—44no—
163C22⋊C4×C31—44no—
164C22⋊C4×C31—44no—
165C42×C31—44no—
166C42×C31—44no—
167#36732C3111yesmaximal, characteristic, direct product factor, Sylow
168C23×C31—22no—
169C23×C31—21no—
170C23×C31—21no—
171C23×C31—28no—
172C23×C31—22no—
173C23×C31—22no—
174C23×C31—22no—
175C23×C31—22no—
176C23×C31—22no—
177C23×C31—22no—
178C23×C31—48no—
179C23×C31—48no—
180C23×C31—44no—
181C23×C31—44no—
182C23×C31C2×#4112yes—
183C23×C31H(5,𝔽2)14yes—
184C23×C312−1+414yes—
185C23×C31C2511yescharacteristic
186Q8×C31—84no—
187C2×C4×C31—22no—
188C2×C4×C31—22no—
189C2×C4×C31—24no—
190C2×C4×C31—24no—
191C2×C4×C31—21no—
192C2×C4×C31—21no—
193C2×C4×C31—24no—
194C2×C4×C31—24no—
195C2×C4×C31—24no—
196C2×C4×C31—24no—
197C2×C4×C31—24no—
198C2×C4×C31—24no—
199C2×C4×C31—22no—
200C2×C4×C31—22no—
201C2×C4×C31—24no—
202C2×C4×C31—24no—
203C2×C4×C31—24no—
204C2×C4×C31—24no—
205C2×C4×C31—24no—
206C2×C4×C31—24no—
207C2×C4×C31—24no—
208C2×C4×C31—24no—
209C2×C4×C31—24no—
210C2×C4×C31—24no—
211C2×C4×C31—44no—
212C2×C4×C31—48no—
213C2×C4×C31—48no—
214C2×C4×C31—44no—
215C2×C4×C31—84no—
216C2×C4×C31—84no—
217C2×C4×C31C2×#4114yes—
218#2848C2×C3112yes—
219#2744C2×C3114yescomplemented
220#2739C2×C3114yescomplemented
221#2698C2×C3114yescomplemented
222#2689C2×C3114yescomplemented
223#2659C2×C3111yescharacteristic, complemented
224#2646C2×C3111yescharacteristic, complemented
225#2505C2×C3114yescomplemented
226#2480C2×C3114yes—
227#2329C2×C3112yescomplemented
228C2×#381C2×C3111yescharacteristic
229C22×C31—22no—
230C22×C31—22no—
231C22×C31—24no—
232C22×C31—24no—
233C22×C31—24no—
234C22×C31—24no—
235C22×C31—24no—
236C22×C31—416no—
237C22×C31—42no—
238C22×C31—42no—
239C22×C31—42no—
240C22×C31—42no—
241C22×C31#54114yes—
242C22×C31C22×#4111yescharacteristic
243C22×C31C2×H(5,𝔽2)12yes—
244C22×C31C2×2−1+412yes—
245C22×C31#58511yescharacteristic
246C22×C31#58511yescharacteristic
247C4×C31—24no—
248C4×C31—41no—
249C4×C31—41no—
250C4×C31—44no—
251C4×C31—44no—
252C4×C31—44no—
253C4×C31—44no—
254C4×C31—44no—
255C4×C31—44no—
256C24×C4C22×C3111yescharacteristic
257C2×#126C22×C3114yes—
258C2×C22⋊Q8C22×C3112yes—
259C2×C22⋊Q8C22×C3112yes—
260C2×C22⋊Q8C22×C3112yes—
261C22×C22⋊C4C22×C3112yes—
262#400C22×C3114yescomplemented
263#400C22×C3114yescomplemented
264#400C22×C3114yescomplemented
265#400C22×C3114yescomplemented
266#398C22×C3114yescomplemented
267#398C22×C3114yescomplemented
268#398C22×C3114yescomplemented
269#397C22×C3114yes—
270#397C22×C3114yes—
271#397C22×C3114yes—
272#396C22×C3114yes—
273#395C22×C3114yescomplemented
274#391C22×C3114yescomplemented
275#389C22×C3114yescomplemented
276#389C22×C3114yescomplemented
277#388C22×C3114yes—
278#388C22×C3114yes—
279#388C22×C3114yes—
280#387C22×C3114yescomplemented
281#387C22×C3114yescomplemented
282#387C22×C3114yescomplemented
283#387C22×C3114yescomplemented
284#385C22×C3114yes—
285#385C22×C3114yes—
286#385C22×C3114yes—
287#385C22×C3114yes—
288#385C22×C3114yes—
289#382C22×C3114yescomplemented
290#381C22×C3114yes—
291#381C22×C3114yes—
292#380C22×C3112yes—
293#380C22×C3112yes—
294#380C22×C3112yes—
295#380C22×C3114yes—
296#379C22×C3114yes—
297C4×C22⋊C4C22×C3114yes—
298C2×#95C22×C3111yescharacteristic
299C2×#95C22×C3111yescharacteristic
300C2×#95C22×C3112yes—
301C2×C31—24no—
302C2×C31—44no—
303C2×C31C2×#54111yescharacteristic
304C2×C31#349011yescharacteristic
305C2×C31#349311yescharacteristic
306C2×C31#353512yes—
307C2×C31#353512yes—
308C25C23×C3111yescharacteristic
309C22×Q8C23×C3112yes—
310C23×C4—24no—
311C23×C4—44no—
312C23×C4C23×C3112yes—
313C23×C4C23×C3112yes—
314C23×C4C23×C3112yes—
315C23×C4C23×C3112yes—
316C23×C4C23×C3111yescharacteristic
317C23×C4C23×C3111yescharacteristic
318#126—48no—
319C22⋊Q8—28no—
320C22⋊Q8—44no—
321C22⋊Q8—44no—
322C2×C4⋊C4C23×C3112yes—
323C2×C4⋊C4C23×C3114yes—
324C2×C4⋊C4C23×C3112yes—
325C2×C4⋊C4C23×C3112yes—
326C2×C4⋊C4C23×C3114yes—
327C2×C4⋊C4C23×C3114yes—
328C2×C4⋊C4C23×C3114yes—
329C2×C4⋊C4C23×C3114yes—
330C2×C4⋊C4C23×C3114yes—
331C2×C4⋊C4C23×C3114yes—
332C2×C4⋊C4C23×C3114yes—
333C2×C4⋊C4C23×C3114yes—
334C2×C22⋊C4—44no—
335C2×C22⋊C4C23×C3114yes—
336C2×C22⋊C4C23×C3114yes—
337C2×C22⋊C4C23×C3114yes—
338C2×C22⋊C4C23×C3114yes—
339C2×C22⋊C4C23×C3114yes—
340C2×C22⋊C4C23×C3114yes—
341C2×C22⋊C4C23×C3114yes—
342C2×C22⋊C4C23×C3114yes—
343C2×C42C23×C3114yes—
344C2×C42C23×C3114yes—
345#95C23×C3114yes—
346#95C23×C3114yes—
347#95C23×C3114yes—
348#95C23×C3114yes—
349#95C23×C3114yes—
350#95C23×C3114yes—
351#95C23×C3114yes—
352#95C23×C3114yes—
353#95C23×C3114yes—
354#95C23×C3114yes—
355#95C23×C3114yes—
356#95C23×C3114yes—
357#95C23×C3114yes—
358#95C23×C3114yes—
359#95C23×C3114yes—
360C31#3673211yescharacteristic, direct product factor, Sylow
361C24—22no—
362C24—24no—
363C24—24no—
364C24—41no—
365C24—41no—
366C24C24×C3112yes—
367C24C24×C3111yescharacteristic
368C2×Q8—24no—
369C2×Q8—42no—
370C2×Q8—42no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—416no—
387C22×C4#41×C3114yes—
388C22×C4C24×C3114yes—
389C22×C4C24×C3114yes—
390C22×C4C24×C3111yescharacteristic
391C22×C4C24×C3111yescharacteristic
392C22×C4C24×C3114yes—
393C22×C4C24×C3114yes—
394C22×C4C24×C3114yes—
395C22×C4C24×C3114yes—
396C22×C4C24×C3112yes—
397C4⋊C4—24no—
398C4⋊C4—28no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C22⋊C4—28no—
410C22⋊C4—28no—
411C22⋊C4—28no—
412C22⋊C4—28no—
413C22⋊C4—28no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C42—44no—
418C42—44no—
419C23—22no—
420C23—21no—
421C23—21no—
422C23—28no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—48no—
430C23—48no—
431C23—44no—
432C23—44no—
433C23C2×#41×C3112yes—
434C23C31×H(5,𝔽2)14yes—
435C23C31×2−1+414yes—
436C23C25×C3111yescharacteristic
437Q8—84no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—44no—
463C2×C4—48no—
464C2×C4—48no—
465C2×C4—44no—
466C2×C4—84no—
467C2×C4—84no—
468C2×C4C2×#41×C3114yes—
469C22—22no—
470C22—22no—
471C22—24no—
472C22—24no—
473C22—24no—
474C22—24no—
475C22—24no—
476C22—416no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22C31×#54114yes—
482C22C22×#41×C3111yescharacteristic
483C22C2×C31×H(5,𝔽2)12yes—
484C22C2×C31×2−1+412yes—
485C22C31×#58511yescharacteristic
486C22C31×#58511yescharacteristic
487C4—24no—
488C4—41no—
489C4—41no—
490C4—44no—
491C4—44no—
492C4—44no—
493C4—44no—
494C4—44no—
495C4—44no—
496C2—24no—
497C2—44no—
498C2C2×C31×#54111yescharacteristic
499C2C31×#349011yescharacteristic
500C2C31×#349311yescharacteristic
501C2C31×#353512yes—
502C2C31×#353512yes—
subgroupG/H
center, socleC23×C31C25
commutator, Frattini, cosocleC23C25×C31
prime radicalC25×C31C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2744C2semidirect
3. C31×#2739C2semidirect
4. C31×#2698C2semidirect
5. C31×#2689C2semidirect
6. C31×#2659C2semidirect
7. C31×#2646C2semidirect
8. C31×#2505C2semidirect
10. C31×#2329C2semidirect
18. C31×#400C22semidirect
19. C31×#400C22semidirect
20. C31×#400C22semidirect
21. C31×#400C22semidirect
22. C31×#398C22semidirect
23. C31×#398C22semidirect
24. C31×#398C22semidirect
29. C31×#395C22semidirect
30. C31×#391C22semidirect
31. C31×#389C22semidirect
32. C31×#389C22semidirect
36. C31×#387C22semidirect
37. C31×#387C22semidirect
38. C31×#387C22semidirect
39. C31×#387C22semidirect
45. C31×#382C22semidirect
167. #36732C31direct (N × H)
219. #2744C2×C31semidirect
220. #2739C2×C31semidirect
221. #2698C2×C31semidirect
222. #2689C2×C31semidirect
223. #2659C2×C31semidirect
224. #2646C2×C31semidirect
225. #2505C2×C31semidirect
227. #2329C2×C31semidirect
262. #400C22×C31semidirect
263. #400C22×C31semidirect
264. #400C22×C31semidirect
265. #400C22×C31semidirect
266. #398C22×C31semidirect
267. #398C22×C31semidirect
268. #398C22×C31semidirect
273. #395C22×C31semidirect
274. #391C22×C31semidirect
275. #389C22×C31semidirect
276. #389C22×C31semidirect
280. #387C22×C31semidirect
281. #387C22×C31semidirect
282. #387C22×C31semidirect
283. #387C22×C31semidirect
289. #382C22×C31semidirect
360. C31#36732direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28482
C31×#27442
C31×#27392
C31×#26982
C31×#26892
C31×#26592
C31×#26462
C31×#25052
C31×#24802
C31×#23292
C2×C31×#3812
#3673231