Finite Group Databasebeta

← C31×#25579 #21482338

Subgroups — C31×#25579 #21482338

Order 7936.

1206 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#554C211yesmaximal, characteristic, complemented
2C2×C31×#489C211yesmaximal, characteristic, complemented
3C2×C31×#464C211yesmaximal, characteristic, complemented
4C31×#2540C211yesmaximal, characteristic
5C31×#2001C212yesmaximal, complemented
6C31×#2000C212yesmaximal, complemented
7C31×#1988C211yesmaximal, characteristic, complemented
8C31×#1955C211yesmaximal, characteristic, complemented
9C31×#1874C212yesmaximal, complemented
10C31×#1872C212yesmaximal, complemented
11C31×#1763C211yesmaximal, characteristic, complemented
12C22×Q16×C31C2211yescharacteristic
13C22×SD16×C31C2211yescharacteristic
14C31×#554—24no—
15C31×#554—24no—
16C2×C31×C4⋊Q8C2211yescharacteristic
17C2×C31×#125—21no—
18C2×C31×#124C2211yescharacteristic
19C2×C31×C22⋊Q8—22no—
20C2×C31×C22⋊Q8C2212yes—
21C2×C31×C22⋊Q8C2212yes—
22C2×C31×#121C2212yes—
23C2×C4×Q8×C31C2211yescharacteristic
24C2×C4×D4×C31—21no—
25C2×C4×D4×C31C2211yescharacteristic
26C22×C4⋊C4×C31C2212yes—
27C31×#489—24no—
28C31×#464—24no—
29C2×C31×C4⋊C8C2211yescharacteristic
30C2×C31×#103C2212yes—
31C2×C31×#103C2211yescharacteristic
32C2×C31×#103C2212yes—
33C2×C31×#102C2212yes—
34C2×C31×#102C2211yescharacteristic
35C2×C31×C22⋊C8C2212yes—
36C2×C31×C22⋊C8C2212yes—
37C2×C31×#97C2211yescharacteristic
38C31×#399—21no—
39C31×#392—21no—
40C31×#390—22no—
41C4×C4⋊C4×C31C2211yescharacteristic
42C31×#340C2212yes—
43C31×#340C2212yes—
44C31×#340C2212yes—
45C22×Q8×C31—21no—
46C22×Q8×C31C2311yescharacteristic
47C22×Q8×C31C2311yescharacteristic
48C22×D4×C31—22no—
49C22×D4×C31D411yescharacteristic
50C22×D4×C31C2311yescharacteristic
51C23×C4×C31—22no—
52C23×C4×C31C2312yes—
53C2×Q16×C31—22no—
54C2×Q16×C31—42no—
55C2×SD16×C31—22no—
56C2×SD16×C31—42no—
57C22×C8×C31C2312yes—
58C22×C8×C31C2311yescharacteristic
59C22×C8×C31C2311yescharacteristic
60C31×C4⋊Q8—22no—
61C31×C4⋊Q8D414yescomplemented
62C31×#125—28no—
63C31×#124—22no—
64C31×#124—22no—
65C31×C22⋊Q8—216no—
66C31×C22⋊Q8—24no—
67C31×C22⋊Q8—24no—
68C31×C22⋊Q8—24no—
69C31×C22⋊Q8—42no—
70C31×#121—24no—
71C31×#121—42no—
72C4×Q8×C31—22no—
73C4×Q8×C31D414yescomplemented
74C4×D4×C31—28no—
75C4×D4×C31—22no—
76C4×D4×C31—22no—
77C2×C4⋊C4×C31—22no—
78C2×C4⋊C4×C31—21no—
79C2×C4⋊C4×C31—21no—
80C2×C4⋊C4×C31—22no—
81C2×C4⋊C4×C31—24no—
82C2×C4⋊C4×C31—24no—
83C2×C4⋊C4×C31—22no—
84C2×C4⋊C4×C31D411yescharacteristic
85C2×C4⋊C4×C31C2311yescharacteristic
86C2×C4⋊C4×C31C2312yes—
87C2×C4⋊C4×C31C2312yes—
88C2×C22⋊C4×C31—22no—
89C2×C22⋊C4×C31—22no—
90C2×C22⋊C4×C31—22no—
91C2×C22⋊C4×C31—21no—
92C2×C22⋊C4×C31—21no—
93C2×C22⋊C4×C31D412yes—
94C2×C42×C31—21no—
95C2×C42×C31—21no—
96C2×C42×C31C2311yescharacteristic
97C31×C4⋊C8—22no—
98C31×#103—22no—
99C31×#103—24no—
100C31×#103—24no—
101C31×#102—22no—
102C31×#102—24no—
103C31×C22⋊C8—42no—
104C31×C22⋊C8—42no—
105C31×#97—22no—
106C31×#95—21no—
107C24×C31—21no—
108C24×C31C2×D412yes—
109C2×Q8×C31—24no—
110C2×Q8×C31—21no—
111C2×Q8×C31—21no—
112C2×Q8×C31—21no—
113C2×Q8×C31—22no—
114C2×Q8×C31—22no—
115C2×Q8×C31—22no—
116C2×Q8×C31—22no—
117C2×Q8×C31—81no—
118C2×Q8×C31—81no—
119C2×Q8×C31C2×D412yes—
120C2×D4×C31—24no—
121C2×D4×C31—22no—
122C2×D4×C31—21no—
123C2×D4×C31—21no—
124C2×D4×C31—22no—
125C2×D4×C31—22no—
126C2×D4×C31—44no—
127C2×D4×C31—44no—
128C2×D4×C31—82no—
129C22×C4×C31—22no—
130C22×C4×C31—21no—
131C22×C4×C31—21no—
132C22×C4×C31—22no—
133C22×C4×C31—28no—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—22no—
140C22×C4×C31—21no—
141C22×C4×C31—82no—
142C22×C4×C31—82no—
143C22×C4×C31C2×D412yes—
144C22×C4×C31C2×D411yescharacteristic
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2411yescharacteristic
147Q16×C31—44no—
148SD16×C31—44no—
149C2×C8×C31—22no—
150C2×C8×C31—21no—
151C2×C8×C31—21no—
152C2×C8×C31—22no—
153C2×C8×C31—22no—
154C2×C8×C31—22no—
155C2×C8×C31—22no—
156C4⋊C4×C31—28no—
157C4⋊C4×C31—28no—
158C4⋊C4×C31—24no—
159C4⋊C4×C31—24no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—21no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—44no—
165C4⋊C4×C31C2×D412yes—
166C4⋊C4×C31C2×D414yes—
167C4⋊C4×C31#4114yes—
168C22⋊C4×C31—28no—
169C22⋊C4×C31—24no—
170C22⋊C4×C31—44no—
171C22⋊C4×C31—44no—
172C22⋊C4×C31—42no—
173C22⋊C4×C31—42no—
174C42×C31—24no—
175C42×C31—81no—
176C42×C31C2×D412yes—
177C42×C31C2×D412yes—
178#25579C3111yesmaximal, characteristic, direct product factor, Sylow
179C23×C31—22no—
180C23×C31—22no—
181C23×C31—22no—
182C23×C31—22no—
183C23×C31—22no—
184C23×C31—22no—
185C23×C31—41no—
186C23×C31—42no—
187C23×C31—42no—
188C23×C31—41no—
189C23×C31—41no—
190C23×C31#13714yes—
191C23×C31C22×D411yescharacteristic
192Q8×C31—22no—
193Q8×C31—22no—
194Q8×C31—42no—
195Q8×C31—41no—
196Q8×C31—41no—
197Q8×C31—42no—
198Q8×C31—84no—
199D4×C31—28no—
200D4×C31—41no—
201D4×C31—41no—
202D4×C31—42no—
203D4×C31—88no—
204C2×C4×C31—22no—
205C2×C4×C31—21no—
206C2×C4×C31—21no—
207C2×C4×C31—22no—
208C2×C4×C31—22no—
209C2×C4×C31—22no—
210C2×C4×C31—24no—
211C2×C4×C31—24no—
212C2×C4×C31—22no—
213C2×C4×C31—22no—
214C2×C4×C31—42no—
215C2×C4×C31—42no—
216C2×C4×C31—42no—
217C2×C4×C31—41no—
218C2×C4×C31—42no—
219C2×C4×C31—41no—
220C2×C4×C31—42no—
221C2×C4×C31—42no—
222C2×C4×C31—42no—
223C2×C4×C31—42no—
224C2×C4×C31—42no—
225C2×C4×C31—41no—
226C2×C4×C31—42no—
227C2×C4×C31—88no—
228C2×C4×C31#12012yes—
229C2×C4×C31#12012yes—
230C2×C4×C31C22×D411yescharacteristic
231C2×C4×C31C22×D411yescharacteristic
232C2×C4×C31C2×#4112yes—
233C2×C4×C31H(5,𝔽2)12yes—
234C8×C31—41no—
235C8×C31—41no—
236C8×C31—42no—
237C2×#554C2×C3111yescharacteristic, complemented
238C2×#489C2×C3111yescharacteristic, complemented
239C2×#464C2×C3111yescharacteristic, complemented
240#2540C2×C3111yescharacteristic
241#2001C2×C3112yescomplemented
242#2000C2×C3112yescomplemented
243#1988C2×C3111yescharacteristic, complemented
244#1955C2×C3111yescharacteristic, complemented
245#1874C2×C3112yescomplemented
246#1872C2×C3112yescomplemented
247#1763C2×C3111yescharacteristic, complemented
248C22×C31—24no—
249C22×C31—24no—
250C22×C31—24no—
251C22×C31—42no—
252C22×C31—42no—
253C22×C31—42no—
254C22×C31—42no—
255C22×C31—81no—
256C22×C31—81no—
257C22×C31—81no—
258C22×C31—82no—
259C22×C31—82no—
260C22×C31C2×#12011yescharacteristic
261C22×C31#54612yes—
262C22×C31C2×#13712yes—
263C22×C31#57611yescharacteristic
264C22×C31#57711yescharacteristic
265C4×C31—22no—
266C4×C31—22no—
267C4×C31—42no—
268C4×C31—42no—
269C4×C31—42no—
270C4×C31—44no—
271C4×C31—44no—
272C4×C31—81no—
273C4×C31#53211yescharacteristic
274C4×C31#53511yescharacteristic
275C4×C31#55012yes—
276C22×Q16C22×C3111yescharacteristic
277C22×SD16C22×C3111yescharacteristic
278#554—24no—
279#554—24no—
280C2×C4⋊Q8C22×C3111yescharacteristic
281C2×#125—21no—
282C2×#124C22×C3111yescharacteristic
283C2×C22⋊Q8—22no—
284C2×C22⋊Q8C22×C3112yes—
285C2×C22⋊Q8C22×C3112yes—
286C2×#121C22×C3112yes—
287C2×C4×Q8C22×C3111yescharacteristic
288C2×C4×D4—21no—
289C2×C4×D4C22×C3111yescharacteristic
290C22×C4⋊C4C22×C3112yes—
291#489—24no—
292#464—24no—
293C2×C4⋊C8C22×C3111yescharacteristic
294C2×#103C22×C3112yes—
295C2×#103C22×C3111yescharacteristic
296C2×#103C22×C3112yes—
297C2×#102C22×C3112yes—
298C2×#102C22×C3111yescharacteristic
299C2×C22⋊C8C22×C3112yes—
300C2×C22⋊C8C22×C3112yes—
301C2×#97C22×C3111yescharacteristic
302#399—21no—
303#392—21no—
304#390—22no—
305C4×C4⋊C4C22×C3111yescharacteristic
306#340C22×C3112yes—
307#340C22×C3112yes—
308#340C22×C3112yes—
309C2×C31—44no—
310C2×C31—161no—
311C2×C31#243111yescharacteristic
312C2×C31#300411yescharacteristic
313C2×C31#301211yescharacteristic
314C2×C31#331412yes—
315C2×C31#331612yes—
316C22×Q8—21no—
317C22×Q8C23×C3111yescharacteristic
318C22×Q8C23×C3111yescharacteristic
319C22×D4—22no—
320C22×D4D4×C3111yescharacteristic
321C22×D4C23×C3111yescharacteristic
322C23×C4—22no—
323C23×C4C23×C3112yes—
324C2×Q16—22no—
325C2×Q16—42no—
326C2×SD16—22no—
327C2×SD16—42no—
328C22×C8C23×C3111yescharacteristic
329C22×C8C23×C3111yescharacteristic
330C22×C8C23×C3112yes—
331C4⋊Q8—22no—
332C4⋊Q8D4×C3114yescomplemented
333#125—28no—
334#124—22no—
335#124—22no—
336C22⋊Q8—216no—
337C22⋊Q8—24no—
338C22⋊Q8—24no—
339C22⋊Q8—24no—
340C22⋊Q8—42no—
341#121—24no—
342#121—42no—
343C4×Q8—22no—
344C4×Q8D4×C3114yescomplemented
345C4×D4—28no—
346C4×D4—22no—
347C4×D4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4D4×C3111yescharacteristic
356C2×C4⋊C4C23×C3111yescharacteristic
357C2×C4⋊C4C23×C3112yes—
358C2×C4⋊C4C23×C3112yes—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4D4×C3112yes—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42C23×C3111yescharacteristic
368C4⋊C8—22no—
369#103—22no—
370#103—24no—
371#103—24no—
372#102—22no—
373#102—24no—
374C22⋊C8—42no—
375C22⋊C8—42no—
376#97—22no—
377#95—21no—
378C31#2557911yescharacteristic, direct product factor, Sylow
379C24—21no—
380C24C2×D4×C3112yes—
381C2×Q8—24no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—81no—
390C2×Q8—81no—
391C2×Q8C2×D4×C3112yes—
392C2×D4—24no—
393C2×D4—22no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—82no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—28no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—82no—
414C22×C4—82no—
415C22×C4C2×D4×C3112yes—
416C22×C4C2×D4×C3111yescharacteristic
417C22×C4C2×D4×C3111yescharacteristic
418C22×C4C24×C3111yescharacteristic
419Q16—44no—
420SD16—44no—
421C2×C8—22no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C4⋊C4—28no—
429C4⋊C4—28no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—44no—
437C4⋊C4C2×D4×C3112yes—
438C4⋊C4C2×D4×C3114yes—
439C4⋊C4#41×C3114yes—
440C22⋊C4—28no—
441C22⋊C4—24no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C42—24no—
447C42—81no—
448C42C2×D4×C3112yes—
449C42C2×D4×C3112yes—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—41no—
457C23—42no—
458C23—42no—
459C23—41no—
460C23—41no—
461C23C31×#13714yes—
462C23C22×D4×C3111yescharacteristic
463Q8—22no—
464Q8—22no—
465Q8—42no—
466Q8—41no—
467Q8—41no—
468Q8—42no—
469Q8—84no—
470D4—28no—
471D4—41no—
472D4—41no—
473D4—42no—
474D4—88no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—88no—
499C2×C4C31×#12012yes—
500C2×C4C31×#12012yes—
501C2×C4C22×D4×C3111yescharacteristic
502C2×C4C22×D4×C3111yescharacteristic
503C2×C4C2×#41×C3112yes—
504C2×C4C31×H(5,𝔽2)12yes—
505C8—41no—
506C8—41no—
507C8—42no—
508C22—24no—
509C22—24no—
510C22—24no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—81no—
516C22—81no—
517C22—81no—
518C22—82no—
519C22—82no—
520C22C2×C31×#12011yescharacteristic
521C22C31×#54612yes—
522C22C2×C31×#13712yes—
523C22C31×#57611yescharacteristic
524C22C31×#57711yescharacteristic
525C4—22no—
526C4—22no—
527C4—42no—
528C4—42no—
529C4—42no—
530C4—44no—
531C4—44no—
532C4—81no—
533C4C31×#53211yescharacteristic
534C4C31×#53511yescharacteristic
535C4C31×#55012yes—
536C2—44no—
537C2—161no—
538C2C31×#243111yescharacteristic
539C2C31×#300411yescharacteristic
540C2C31×#301211yescharacteristic
541C2C31×#331412yes—
542C2C31×#331612yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#2540C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C4×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#554C2semidirect
2. C2×C31×#489C2semidirect
3. C2×C31×#464C2semidirect
5. C31×#2001C2semidirect
6. C31×#2000C2semidirect
7. C31×#1988C2semidirect
8. C31×#1955C2semidirect
9. C31×#1874C2semidirect
10. C31×#1872C2semidirect
11. C31×#1763C2semidirect
61. C31×C4⋊Q8D4semidirect
73. C4×Q8×C31D4semidirect
178. #25579C31direct (N × H)
237. C2×#554C2×C31semidirect
238. C2×#489C2×C31semidirect
239. C2×#464C2×C31semidirect
241. #2001C2×C31semidirect
242. #2000C2×C31semidirect
243. #1988C2×C31semidirect
244. #1955C2×C31semidirect
245. #1874C2×C31semidirect
246. #1872C2×C31semidirect
247. #1763C2×C31semidirect
332. C4⋊Q8D4×C31semidirect
344. C4×Q8D4×C31semidirect
378. C31#25579direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5542
C2×C31×#4892
C2×C31×#4642
C31×#25402
C31×#20012
C31×#20002
C31×#19882
C31×#19552
C31×#18742
C31×#18722
C31×#17632
#2557931