Finite Group Databasebeta

← C31×#12716 #21469475

Subgroups — C31×#12716 #21469475

Order 7936.

976 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×#3500C211yesmaximal, characteristic
2C31×#2132C211yesmaximal, characteristic, complemented
3C31×#2126C211yesmaximal, characteristic, complemented
4C31×#2123C212yesmaximal, complemented
5C31×#1795C212yesmaximal, complemented
6C2×C31×2−1+4C411yescharacteristic, complemented
7C2×C31×H(5,𝔽2)C2211yescharacteristic
8C31×#548—22no—
9C31×#546—22no—
10C31×#546—24no—
11C31×#536—21no—
12C31×#536C2211yescharacteristic
13C31×#535—21no—
14C31×#535C411yescharacteristic, complemented
15C31×#532—21no—
16C31×#532C2211yescharacteristic
17C2×C31×#124—21no—
18C2×C31×#124C411yescharacteristic, complemented
19C4×#41×C31C411yescharacteristic, complemented
20C31×#421—22no—
21C31×#418—22no—
22C31×#413C2212yes—
23C31×#410C2212yes—
24C31×#356—21no—
25C31×#355—21no—
26C31×#354—21no—
27C31×#353—21no—
28C31×#352—24no—
29C31×2−1+4—42no—
30C31×2−1+4—42no—
31C31×H(5,𝔽2)—42no—
32C31×H(5,𝔽2)—42no—
33C2×#41×C31—21no—
34C2×#41×C31—21no—
35C2×#41×C31—21no—
36C2×#41×C31—22no—
37C2×#41×C31C2×C411yescharacteristic
38C2×#41×C31C2×C411yescharacteristic
39C2×#41×C31D412yes—
40C2×#41×C31C2311yescharacteristic
41C22×Q8×C31—21no—
42C22×Q8×C31—21no—
43C22×Q8×C31C2×C411yescharacteristic
44C22×D4×C31—21no—
45C22×D4×C31—22no—
46C22×D4×C31C2×C411yescharacteristic
47C22×D4×C31D412yes—
48C2×M16×C31—22no—
49C31×C4⋊Q8—21no—
50C31×C4⋊Q8D411yescharacteristic
51C31×#127—21no—
52C31×#127D411yescharacteristic
53C31×#124—24no—
54C31×#124—21no—
55C31×#124—22no—
56C31×#124—21no—
57C31×#124—21no—
58C31×#124—21no—
59C31×#124D411yescharacteristic
60C31×#124D411yescharacteristic
61C31×#123—24no—
62C31×#123D414yes—
63C31×C22⋊Q8—24no—
64C31×C22⋊Q8—22no—
65C31×#121—24no—
66C31×#121—24no—
67C31×#121—22no—
68C31×#121—22no—
69C31×#120—24no—
70C31×#120—22no—
71C4×Q8×C31—21no—
72C4×D4×C31—24no—
73C4×D4×C31—21no—
74C4×D4×C31—22no—
75C4×D4×C31—22no—
76C31×#117—21no—
77C31×#117—22no—
78C31×#117C2×C411yescharacteristic
79C2×C22⋊C4×C31—24no—
80C2×C22⋊C4×C31—22no—
81C2×C42×C31—21no—
82C2×C42×C31C2×C411yescharacteristic
83C31×#104—44no—
84C31×#102—44no—
85C31×#101—22no—
86C31×#100—22no—
87C31×#99—22no—
88C31×#99—22no—
89C24×C31—21no—
90C24×C31—22no—
91#41×C31—22no—
92#41×C31—22no—
93#41×C31—24no—
94#41×C31—41no—
95#41×C31—41no—
96#41×C31—42no—
97#41×C31—44no—
98#41×C31—44no—
99#41×C31—41no—
100#41×C31—41no—
101#41×C31—48no—
102#41×C31—42no—
103#41×C31—42no—
104C2×Q8×C31—22no—
105C2×Q8×C31—21no—
106C2×Q8×C31—21no—
107C2×Q8×C31—44no—
108C2×Q8×C31—41no—
109C2×Q8×C31—41no—
110C2×Q8×C31—42no—
111C2×Q8×C31—42no—
112C2×Q8×C31C22⋊C411yescharacteristic
113C2×Q8×C31C22⋊C411yescharacteristic
114C2×Q8×C31C2×D411yescharacteristic
115C2×D4×C31—21no—
116C2×D4×C31—24no—
117C2×D4×C31—22no—
118C2×D4×C31—21no—
119C2×D4×C31—22no—
120C2×D4×C31—24no—
121C2×D4×C31—22no—
122C2×D4×C31—21no—
123C2×D4×C31—22no—
124C2×D4×C31—41no—
125C2×D4×C31—41no—
126C2×D4×C31—42no—
127C2×D4×C31—44no—
128C2×D4×C31—48no—
129C2×D4×C31C22⋊C411yescharacteristic
130C2×D4×C31C22⋊C412yes—
131C2×D4×C31C22⋊C411yescharacteristic
132C2×D4×C31C2×D411yescharacteristic
133C2×D4×C31C2×D412yes—
134C22×C4×C31—21no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—21no—
139C22×C4×C31—21no—
140C22×C4×C31—22no—
141C22×C4×C31C22⋊C412yes—
142C22×C4×C31C22×C411yescharacteristic
143C22×C4×C31C2×D412yes—
144M16×C31—42no—
145M16×C31—42no—
146C2×C8×C31—42no—
147C4⋊C4×C31—22no—
148C4⋊C4×C31—21no—
149C4⋊C4×C31—22no—
150C4⋊C4×C31—24no—
151C4⋊C4×C31—42no—
152C22⋊C4×C31—24no—
153C22⋊C4×C31—24no—
154C22⋊C4×C31—24no—
155C22⋊C4×C31—22no—
156C22⋊C4×C31—22no—
157C22⋊C4×C31—24no—
158C22⋊C4×C31—22no—
159C22⋊C4×C31—42no—
160C22⋊C4×C31C22⋊C412yes—
161C42×C31—22no—
162C42×C31—21no—
163C42×C31—21no—
164C42×C31—42no—
165C42×C31C22⋊C411yescharacteristic
166C42×C31C22⋊C411yescharacteristic
167#12716C3111yesmaximal, characteristic, direct product factor, Sylow
168C23×C31—21no—
169C23×C31—22no—
170C23×C31—21no—
171C23×C31—44no—
172C23×C31—42no—
173C23×C31—82no—
174C23×C31—84no—
175C23×C31C2×C22⋊C411yescharacteristic
176C23×C31#12012yes—
177Q8×C31—21no—
178Q8×C31—41no—
179Q8×C31—41no—
180Q8×C31—42no—
181Q8×C31—44no—
182Q8×C31—41no—
183Q8×C31—41no—
184Q8×C31#9912yes—
185D4×C31—24no—
186D4×C31—21no—
187D4×C31—22no—
188D4×C31—22no—
189D4×C31—42no—
190D4×C31—41no—
191D4×C31—44no—
192D4×C31—44no—
193D4×C31—41no—
194D4×C31—42no—
195D4×C31—42no—
196D4×C31—44no—
197D4×C31—82no—
198D4×C31—84no—
199D4×C31#9912yes—
200C2×C4×C31—24no—
201C2×C4×C31—21no—
202C2×C4×C31—22no—
203C2×C4×C31—22no—
204C2×C4×C31—21no—
205C2×C4×C31—21no—
206C2×C4×C31—21no—
207C2×C4×C31—22no—
208C2×C4×C31—21no—
209C2×C4×C31—21no—
210C2×C4×C31—22no—
211C2×C4×C31—44no—
212C2×C4×C31—44no—
213C2×C4×C31—44no—
214C2×C4×C31—41no—
215C2×C4×C31—41no—
216C2×C4×C31—42no—
217C2×C4×C31—81no—
218C2×C4×C31—82no—
219C2×C4×C31—82no—
220C2×C4×C31C2×C22⋊C411yescharacteristic
221C2×C4×C31C2×C22⋊C411yescharacteristic
222C2×C4×C31#12012yes—
223C8×C31—82no—
224#3500C2×C3111yescharacteristic
225#2132C2×C3111yescharacteristic, complemented
226#2126C2×C3111yescharacteristic, complemented
227#2123C2×C3112yescomplemented
228#1795C2×C3112yescomplemented
229C22×C31—21no—
230C22×C31—22no—
231C22×C31—22no—
232C22×C31—41no—
233C22×C31—42no—
234C22×C31—42no—
235C22×C31—42no—
236C22×C31—44no—
237C22×C31—41no—
238C22×C31—42no—
239C22×C31—164no—
240C22×C31—162no—
241C22×C31#37911yescharacteristic
242C4×C31—22no—
243C4×C31—21no—
244C4×C31—22no—
245C4×C31—41no—
246C4×C31—42no—
247C4×C31—41no—
248C4×C31—41no—
249C4×C31—42no—
250C4×C31—162no—
251C4×C31C2×#9911yescharacteristic
252C4×C31C2×#9911yescharacteristic
253C2×2−1+4C4×C3111yescharacteristic, complemented
254C2×H(5,𝔽2)C22×C3111yescharacteristic
255#548—22no—
256#546—22no—
257#546—24no—
258#536—21no—
259#536C22×C3111yescharacteristic
260#535—21no—
261#535C4×C3111yescharacteristic, complemented
262#532—21no—
263#532C22×C3111yescharacteristic
264C2×#124—21no—
265C2×#124C4×C3111yescharacteristic, complemented
266C4×#41C4×C3111yescharacteristic, complemented
267#421—22no—
268#418—22no—
269#413C22×C3112yes—
270#410C22×C3112yes—
271#356—21no—
272#355—21no—
273#354—21no—
274#353—21no—
275#352—24no—
276C2×C31—21no—
277C2×C31—42no—
278C2×C31—42no—
279C2×C31—41no—
280C2×C31—81no—
281C2×C31—82no—
282C2×C31#178111yescharacteristic
2832−1+4—42no—
2842−1+4—42no—
285H(5,𝔽2)—42no—
286H(5,𝔽2)—42no—
287C2×#41—21no—
288C2×#41—21no—
289C2×#41—21no—
290C2×#41—22no—
291C2×#41C2×C4×C3111yescharacteristic
292C2×#41C2×C4×C3111yescharacteristic
293C2×#41D4×C3112yes—
294C2×#41C23×C3111yescharacteristic
295C22×Q8—21no—
296C22×Q8—21no—
297C22×Q8C2×C4×C3111yescharacteristic
298C22×D4—21no—
299C22×D4—22no—
300C22×D4C2×C4×C3111yescharacteristic
301C22×D4D4×C3112yes—
302C2×M16—22no—
303C4⋊Q8—21no—
304C4⋊Q8D4×C3111yescharacteristic
305#127—21no—
306#127D4×C3111yescharacteristic
307#124—24no—
308#124—21no—
309#124—22no—
310#124—21no—
311#124—21no—
312#124—21no—
313#124D4×C3111yescharacteristic
314#124D4×C3111yescharacteristic
315#123—24no—
316#123D4×C3114yes—
317C22⋊Q8—24no—
318C22⋊Q8—22no—
319#121—24no—
320#121—24no—
321#121—22no—
322#121—22no—
323#120—24no—
324#120—22no—
325C4×Q8—21no—
326C4×D4—24no—
327C4×D4—21no—
328C4×D4—22no—
329C4×D4—22no—
330#117—21no—
331#117—22no—
332#117C2×C4×C3111yescharacteristic
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C42—21no—
336C2×C42C2×C4×C3111yescharacteristic
337#104—44no—
338#102—44no—
339#101—22no—
340#100—22no—
341#99—22no—
342#99—22no—
343C31#1271611yescharacteristic, direct product factor, Sylow
344C24—21no—
345C24—22no—
346#41—22no—
347#41—22no—
348#41—24no—
349#41—41no—
350#41—41no—
351#41—42no—
352#41—44no—
353#41—44no—
354#41—41no—
355#41—41no—
356#41—48no—
357#41—42no—
358#41—42no—
359C2×Q8—22no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—41no—
363C2×Q8—41no—
364C2×Q8—44no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8C22⋊C4×C3111yescharacteristic
368C2×Q8C22⋊C4×C3111yescharacteristic
369C2×Q8C2×D4×C3111yescharacteristic
370C2×D4—21no—
371C2×D4—24no—
372C2×D4—22no—
373C2×D4—21no—
374C2×D4—22no—
375C2×D4—24no—
376C2×D4—22no—
377C2×D4—21no—
378C2×D4—22no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—42no—
382C2×D4—44no—
383C2×D4—48no—
384C2×D4C22⋊C4×C3112yes—
385C2×D4C22⋊C4×C3111yescharacteristic
386C2×D4C22⋊C4×C3111yescharacteristic
387C2×D4C2×D4×C3111yescharacteristic
388C2×D4C2×D4×C3112yes—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4C22⋊C4×C3112yes—
397C22×C4C22×C4×C3111yescharacteristic
398C22×C4C2×D4×C3112yes—
399M16—42no—
400M16—42no—
401C2×C8—42no—
402C4⋊C4—22no—
403C4⋊C4—21no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—42no—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—24no—
413C22⋊C4—22no—
414C22⋊C4—42no—
415C22⋊C4C22⋊C4×C3112yes—
416C42—22no—
417C42—21no—
418C42—21no—
419C42—42no—
420C42C22⋊C4×C3111yescharacteristic
421C42C22⋊C4×C3111yescharacteristic
422C23—21no—
423C23—22no—
424C23—21no—
425C23—44no—
426C23—42no—
427C23—82no—
428C23—84no—
429C23C2×C22⋊C4×C3111yescharacteristic
430C23C31×#12012yes—
431Q8—21no—
432Q8—41no—
433Q8—41no—
434Q8—42no—
435Q8—44no—
436Q8—41no—
437Q8—41no—
438Q8C31×#9912yes—
439D4—24no—
440D4—21no—
441D4—22no—
442D4—22no—
443D4—42no—
444D4—41no—
445D4—44no—
446D4—44no—
447D4—41no—
448D4—42no—
449D4—42no—
450D4—44no—
451D4—82no—
452D4—84no—
453D4C31×#9912yes—
454C2×C4—24no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—81no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4C2×C22⋊C4×C3111yescharacteristic
475C2×C4C2×C22⋊C4×C3111yescharacteristic
476C2×C4C31×#12012yes—
477C8—82no—
478C22—21no—
479C22—22no—
480C22—22no—
481C22—41no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—44no—
486C22—41no—
487C22—42no—
488C22—164no—
489C22—162no—
490C22C31×#37911yescharacteristic
491C4—22no—
492C4—21no—
493C4—22no—
494C4—41no—
495C4—42no—
496C4—41no—
497C4—41no—
498C4—42no—
499C4—162no—
500C4C2×C31×#9911yescharacteristic
501C4C2×C31×#9911yescharacteristic
502C2—21no—
503C2—42no—
504C2—42no—
505C2—41no—
506C2—82no—
507C2—81no—
508C2C31×#178111yescharacteristic
subgroupG/H
center, socleC2×C31#1781
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×#41C23×C31
prime radicalC31×#3500C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C2×C4×C31 ◁ C2×C31×H(5,𝔽2) ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2132C2semidirect
3. C31×#2126C2semidirect
4. C31×#2123C2semidirect
5. C31×#1795C2semidirect
6. C2×C31×2−1+4C4semidirect
14. C31×#535C4semidirect
18. C2×C31×#124C4semidirect
19. C4×#41×C31C4semidirect
167. #12716C31direct (N × H)
225. #2132C2×C31semidirect
226. #2126C2×C31semidirect
227. #2123C2×C31semidirect
228. #1795C2×C31semidirect
253. C2×2−1+4C4×C31semidirect
261. #535C4×C31semidirect
265. C2×#124C4×C31semidirect
266. C4×#41C4×C31semidirect
343. C31#12716direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35002
C31×#21322
C31×#21262
C31×#21232
C31×#17952
#1271631