Finite Group Databasebeta

← C31×#12718 #21469477

Subgroups — C31×#12718 #21469477

Order 7936.

1080 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×#3498C211yesmaximal, characteristic
2C31×#2132C211yesmaximal, characteristic, complemented
3C31×#2126C211yesmaximal, characteristic, complemented
4C31×#2122C212yesmaximal, complemented
5C31×#1795C212yesmaximal, complemented
6C2×C31×H(5,𝔽2)C411yescharacteristic, complemented
7C2×C31×H(5,𝔽2)C2211yescharacteristic
8C31×#550—22no—
9D42×C31—22no—
10D42×C31—24no—
11C31×#535—21no—
12C31×#535—21no—
13C31×#535C411yescharacteristic, complemented
14C31×#535C2211yescharacteristic
15C31×#532—21no—
16C31×#532C2211yescharacteristic
17C2×C31×#127—21no—
18C2×C31×#127C411yescharacteristic, complemented
19C4×#41×C31C411yescharacteristic, complemented
20C31×#421—22no—
21C31×#418—22no—
22C31×#413C2212yescomplemented
23C31×#410C2212yescomplemented
24C31×#356—21no—
25C31×#355—21no—
26C31×#354—21no—
27C31×#353—21no—
28C31×AΣL(1,𝔽16)—24no—
29C31×H(5,𝔽2)—42no—
30C31×H(5,𝔽2)—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×C31C2×C411yescharacteristic
37C2×#41×C31C2×C411yescharacteristic
38C2×#41×C31D412yes—
39C2×#41×C31C2311yescharacteristic
40C22×D4×C31—22no—
41C22×D4×C31—24no—
42C22×D4×C31—22no—
43C22×D4×C31—21no—
44C22×D4×C31—21no—
45C22×D4×C31—21no—
46C22×D4×C31—22no—
47C22×D4×C31C2×C411yescharacteristic
48C22×D4×C31C2×C411yescharacteristic
49C22×D4×C31D412yes—
50C2×M16×C31—22no—
51C31×C4⋊Q8D411yescharacteristic
52C31×#127—24no—
53C31×#127—21no—
54C31×#127—22no—
55C31×#127—21no—
56C31×#127—21no—
57C31×#127—21no—
58C31×#127D411yescharacteristic
59C31×#124—21no—
60C31×#124—21no—
61C31×#124D411yescharacteristic
62C31×#124D411yescharacteristic
63C31×#121—24no—
64C31×#121—24no—
65C31×#121—24no—
66C31×#121—22no—
67C31×#121—22no—
68C31×#121—22no—
69C31×#120—24no—
70C31×#120—24no—
71C31×#120—22no—
72C31×#120D414yes—
73C4×Q8×C31—21no—
74C4×D4×C31—24no—
75C4×D4×C31—21no—
76C4×D4×C31—22no—
77C4×D4×C31—22no—
78C31×#117—21no—
79C31×#117—22no—
80C31×#117C2×C411yescharacteristic
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—22no—
90C24×C31—21no—
91C24×C31—22no—
92C24×C31C22⋊C412yes—
93#41×C31—22no—
94#41×C31—22no—
95#41×C31—24no—
96#41×C31—41no—
97#41×C31—41no—
98#41×C31—42no—
99#41×C31—44no—
100#41×C31—44no—
101#41×C31—41no—
102#41×C31—41no—
103#41×C31—42no—
104#41×C31—42no—
105C2×Q8×C31C22⋊C411yescharacteristic
106C2×Q8×C31C22⋊C411yescharacteristic
107C2×Q8×C31C2×D411yescharacteristic
108C2×D4×C31—21no—
109C2×D4×C31—24no—
110C2×D4×C31—24no—
111C2×D4×C31—22no—
112C2×D4×C31—22no—
113C2×D4×C31—21no—
114C2×D4×C31—22no—
115C2×D4×C31—22no—
116C2×D4×C31—24no—
117C2×D4×C31—24no—
118C2×D4×C31—24no—
119C2×D4×C31—21no—
120C2×D4×C31—21no—
121C2×D4×C31—21no—
122C2×D4×C31—22no—
123C2×D4×C31—41no—
124C2×D4×C31—41no—
125C2×D4×C31—42no—
126C2×D4×C31—44no—
127C2×D4×C31—44no—
128C2×D4×C31—41no—
129C2×D4×C31—41no—
130C2×D4×C31—48no—
131C2×D4×C31—48no—
132C2×D4×C31—42no—
133C2×D4×C31—42no—
134C2×D4×C31—84no—
135C2×D4×C31—88no—
136C2×D4×C31C22⋊C411yescharacteristic
137C2×D4×C31C22⋊C412yes—
138C2×D4×C31C22⋊C411yescharacteristic
139C2×D4×C31C2×D411yescharacteristic
140C2×D4×C31C2×D412yes—
141C22×C4×C31—21no—
142C22×C4×C31—22no—
143C22×C4×C31—22no—
144C22×C4×C31—21no—
145C22×C4×C31—21no—
146C22×C4×C31—22no—
147C22×C4×C31C22×C411yescharacteristic
148C22×C4×C31C2×D412yes—
149M16×C31—42no—
150M16×C31—42no—
151C2×C8×C31—42no—
152C4⋊C4×C31—22no—
153C4⋊C4×C31—21no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—42no—
156C22⋊C4×C31—22no—
157C22⋊C4×C31—24no—
158C22⋊C4×C31—22no—
159C22⋊C4×C31—22no—
160C22⋊C4×C31—24no—
161C22⋊C4×C31—42no—
162C22⋊C4×C31C22⋊C412yes—
163C42×C31—22no—
164C42×C31—21no—
165C42×C31—21no—
166C42×C31—42no—
167C42×C31C22⋊C411yescharacteristic
168C42×C31C22⋊C411yescharacteristic
169#12718C3111yesmaximal, characteristic, direct product factor, Sylow
170C23×C31—21no—
171C23×C31—22no—
172C23×C31—22no—
173C23×C31—21no—
174C23×C31—44no—
175C23×C31—42no—
176C23×C31—44no—
177C23×C31—42no—
178C23×C31—82no—
179C23×C31—82no—
180C23×C31—84no—
181C23×C31—84no—
182C23×C31C2×C22⋊C411yescharacteristic
183C23×C31#12012yes—
184Q8×C31—21no—
185Q8×C31—41no—
186Q8×C31—41no—
187Q8×C31#9912yes—
188D4×C31—24no—
189D4×C31—21no—
190D4×C31—22no—
191D4×C31—22no—
192D4×C31—42no—
193D4×C31—41no—
194D4×C31—44no—
195D4×C31—44no—
196D4×C31—41no—
197D4×C31—42no—
198D4×C31—42no—
199D4×C31—44no—
200D4×C31—44no—
201D4×C31—48no—
202D4×C31—42no—
203D4×C31—44no—
204D4×C31—41no—
205D4×C31—41no—
206D4×C31—84no—
207D4×C31—84no—
208D4×C31—84no—
209D4×C31#9912yes—
210C2×C4×C31—24no—
211C2×C4×C31—21no—
212C2×C4×C31—21no—
213C2×C4×C31—22no—
214C2×C4×C31—22no—
215C2×C4×C31—21no—
216C2×C4×C31—21no—
217C2×C4×C31—22no—
218C2×C4×C31—21no—
219C2×C4×C31—21no—
220C2×C4×C31—44no—
221C2×C4×C31—44no—
222C2×C4×C31—42no—
223C2×C4×C31—44no—
224C2×C4×C31—41no—
225C2×C4×C31—41no—
226C2×C4×C31—82no—
227C2×C4×C31C2×C22⋊C411yescharacteristic
228C2×C4×C31C2×C22⋊C411yescharacteristic
229C2×C4×C31#12012yes—
230C8×C31—82no—
231#3498C2×C3111yescharacteristic
232#2132C2×C3111yescharacteristic, complemented
233#2126C2×C3111yescharacteristic, complemented
234#2122C2×C3112yescomplemented
235#1795C2×C3112yescomplemented
236C22×C31—22no—
237C22×C31—21no—
238C22×C31—22no—
239C22×C31—41no—
240C22×C31—42no—
241C22×C31—42no—
242C22×C31—44no—
243C22×C31—44no—
244C22×C31—42no—
245C22×C31—42no—
246C22×C31—42no—
247C22×C31—41no—
248C22×C31—164no—
249C22×C31—162no—
250C22×C31—164no—
251C22×C31—162no—
252C22×C31#37911yescharacteristic
253C4×C31—22no—
254C4×C31—22no—
255C4×C31—21no—
256C4×C31—22no—
257C4×C31—42no—
258C4×C31—41no—
259C4×C31—41no—
260C4×C31—162no—
261C4×C31C2×#9911yescharacteristic
262C4×C31C2×#9911yescharacteristic
263C2×H(5,𝔽2)C4×C3111yescharacteristic, complemented
264C2×H(5,𝔽2)C22×C3111yescharacteristic
265#550—22no—
266D42—22no—
267D42—24no—
268#535—21no—
269#535—21no—
270#535C4×C3111yescharacteristic, complemented
271#535C22×C3111yescharacteristic
272#532—21no—
273#532C22×C3111yescharacteristic
274C2×#127—21no—
275C2×#127C4×C3111yescharacteristic, complemented
276C4×#41C4×C3111yescharacteristic, complemented
277#421—22no—
278#418—22no—
279#413C22×C3112yescomplemented
280#410C22×C3112yescomplemented
281#356—21no—
282#355—21no—
283#354—21no—
284#353—21no—
285AΣL(1,𝔽16)—24no—
286C2×C31—21no—
287C2×C31—42no—
288C2×C31—42no—
289C2×C31—41no—
290C2×C31—82no—
291C2×C31—82no—
292C2×C31—81no—
293C2×C31#178111yescharacteristic
294H(5,𝔽2)—42no—
295H(5,𝔽2)—42no—
296H(5,𝔽2)—42no—
297H(5,𝔽2)—42no—
298C2×#41—21no—
299C2×#41—21no—
300C2×#41—21no—
301C2×#41C2×C4×C3111yescharacteristic
302C2×#41C2×C4×C3111yescharacteristic
303C2×#41D4×C3112yes—
304C2×#41C23×C3111yescharacteristic
305C22×D4—22no—
306C22×D4—24no—
307C22×D4—22no—
308C22×D4—21no—
309C22×D4—21no—
310C22×D4—21no—
311C22×D4—22no—
312C22×D4C2×C4×C3111yescharacteristic
313C22×D4C2×C4×C3111yescharacteristic
314C22×D4D4×C3112yes—
315C2×M16—22no—
316C4⋊Q8D4×C3111yescharacteristic
317#127—24no—
318#127—21no—
319#127—22no—
320#127—21no—
321#127—21no—
322#127—21no—
323#127D4×C3111yescharacteristic
324#124—21no—
325#124—21no—
326#124D4×C3111yescharacteristic
327#124D4×C3111yescharacteristic
328#121—24no—
329#121—24no—
330#121—24no—
331#121—22no—
332#121—22no—
333#121—22no—
334#120—24no—
335#120—24no—
336#120—22no—
337#120D4×C3114yes—
338C4×Q8—21no—
339C4×D4—24no—
340C4×D4—21no—
341C4×D4—22no—
342C4×D4—22no—
343#117—21no—
344#117—22no—
345#117C2×C4×C3111yescharacteristic
346C2×C42—21no—
347C2×C42C2×C4×C3111yescharacteristic
348#104—44no—
349#102—44no—
350#101—22no—
351#100—22no—
352#99—22no—
353#99—22no—
354C31#1271811yescharacteristic, direct product factor, Sylow
355C24—22no—
356C24—21no—
357C24—22no—
358C24C22⋊C4×C3112yes—
359#41—22no—
360#41—22no—
361#41—24no—
362#41—41no—
363#41—41no—
364#41—42no—
365#41—44no—
366#41—44no—
367#41—41no—
368#41—41no—
369#41—42no—
370#41—42no—
371C2×Q8C22⋊C4×C3111yescharacteristic
372C2×Q8C22⋊C4×C3111yescharacteristic
373C2×Q8C2×D4×C3111yescharacteristic
374C2×D4—21no—
375C2×D4—24no—
376C2×D4—24no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—21no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—24no—
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—22no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—42no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—84no—
401C2×D4—88no—
402C2×D4C22⋊C4×C3112yes—
403C2×D4C22⋊C4×C3111yescharacteristic
404C2×D4C22⋊C4×C3111yescharacteristic
405C2×D4C2×D4×C3111yescharacteristic
406C2×D4C2×D4×C3112yes—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4C22×C4×C3111yescharacteristic
414C22×C4C2×D4×C3112yes—
415M16—42no—
416M16—42no—
417C2×C8—42no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—22no—
421C4⋊C4—42no—
422C22⋊C4—22no—
423C22⋊C4—24no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—24no—
427C22⋊C4—42no—
428C22⋊C4C22⋊C4×C3112yes—
429C42—22no—
430C42—21no—
431C42—21no—
432C42—42no—
433C42C22⋊C4×C3111yescharacteristic
434C42C22⋊C4×C3111yescharacteristic
435C23—21no—
436C23—22no—
437C23—22no—
438C23—21no—
439C23—44no—
440C23—42no—
441C23—44no—
442C23—42no—
443C23—82no—
444C23—82no—
445C23—84no—
446C23—84no—
447C23C2×C22⋊C4×C3111yescharacteristic
448C23C31×#12012yes—
449Q8—21no—
450Q8—41no—
451Q8—41no—
452Q8C31×#9912yes—
453D4—24no—
454D4—21no—
455D4—22no—
456D4—22no—
457D4—42no—
458D4—41no—
459D4—44no—
460D4—44no—
461D4—41no—
462D4—42no—
463D4—42no—
464D4—44no—
465D4—44no—
466D4—48no—
467D4—42no—
468D4—44no—
469D4—41no—
470D4—41no—
471D4—84no—
472D4—84no—
473D4—84no—
474D4C31×#9912yes—
475C2×C4—24no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—82no—
492C2×C4C2×C22⋊C4×C3111yescharacteristic
493C2×C4C2×C22⋊C4×C3111yescharacteristic
494C2×C4C31×#12012yes—
495C8—82no—
496C22—22no—
497C22—21no—
498C22—22no—
499C22—41no—
500C22—42no—
501C22—42no—
502C22—44no—
503C22—42no—
504C22—44no—
505C22—42no—
506C22—42no—
507C22—41no—
508C22—164no—
509C22—162no—
510C22—164no—
511C22—162no—
512C22C31×#37911yescharacteristic
513C4—22no—
514C4—22no—
515C4—21no—
516C4—22no—
517C4—42no—
518C4—41no—
519C4—41no—
520C4—162no—
521C4C2×C31×#9911yescharacteristic
522C4C2×C31×#9911yescharacteristic
523C2—21no—
524C2—42no—
525C2—42no—
526C2—41no—
527C2—82no—
528C2—81no—
529C2—82no—
530C2C31×#178111yescharacteristic
subgroupG/H
center, socleC2×C31#1781
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×#41C23×C31
prime radicalC31×#3498C2
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×#2122C2semidirect
5. C31×#1795C2semidirect
6. C2×C31×H(5,𝔽2)C4semidirect
13. C31×#535C4semidirect
18. C2×C31×#127C4semidirect
19. C4×#41×C31C4semidirect
22. C31×#413C22semidirect
23. C31×#410C22semidirect
169. #12718C31direct (N × H)
232. #2132C2×C31semidirect
233. #2126C2×C31semidirect
234. #2122C2×C31semidirect
235. #1795C2×C31semidirect
263. C2×H(5,𝔽2)C4×C31semidirect
270. #535C4×C31semidirect
275. C2×#127C4×C31semidirect
276. C4×#41C4×C31semidirect
279. #413C22×C31semidirect
280. #410C22×C31semidirect
354. C31#12718direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34982
C31×#21322
C31×#21262
C31×#21222
C31×#17952
#1271831