Finite Group Databasebeta

← C31×#9443 #21466202

Subgroups — C31×#9443 #21466202

Order 7936.

712 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2879C211yesmaximal, characteristic
2C31×#2432C211yesmaximal, characteristic
3C31×#1782C211yesmaximal, characteristic
4C31×#1511C212yesmaximal, complemented
5C31×#1284C212yesmaximal, complemented
6C22×#41×C31C2211yescharacteristic
7C22×M16×C31—21no—
8C2×C31×C4⋊Q8C411yescharacteristic, complemented
9C2×C31×#124C411yescharacteristic, complemented
10C2×C31×C22⋊Q8C2211yescharacteristic
11C2×C31×#99—21no—
12C2×C31×#99—21no—
13C31×#408—24no—
14C31×#407—21no—
15C31×#407C2212yes—
16C31×#394—21no—
17C31×#394—21no—
18C31×#391C411yescharacteristic, complemented
19C31×#390C411yescharacteristic, complemented
20C31×#385—21no—
21C31×#385—21no—
22C31×#381C2212yes—
23C4×C22⋊C4×C31C2211yescharacteristic
24C31×#328—24no—
25C2×#41×C31—22no—
26C2×#41×C31—22no—
27C2×#41×C31—42no—
28C2×#41×C31—42no—
29C22×Q8×C31C2×C411yescharacteristic
30C22×Q8×C31C2×C411yescharacteristic
31C22×D4×C31C2×C411yescharacteristic
32C22×D4×C31D411yescharacteristic
33C22×D4×C31D411yescharacteristic
34C23×C4×C31D411yescharacteristic
35C23×C4×C31D411yescharacteristic
36C23×C4×C31C2311yescharacteristic
37C2×M16×C31—22no—
38C2×M16×C31—22no—
39C2×M16×C31—42no—
40C2×M16×C31—42no—
41C22×C8×C31—22no—
42C31×C4⋊Q8—42no—
43C31×#124—42no—
44C31×C22⋊Q8—22no—
45C31×C22⋊Q8—22no—
46C2×C4⋊C4×C31—21no—
47C2×C4⋊C4×C31—21no—
48C2×C4⋊C4×C31—21no—
49C2×C4⋊C4×C31C2×C411yescharacteristic
50C2×C22⋊C4×C31—21no—
51C2×C22⋊C4×C31—21no—
52C2×C22⋊C4×C31—21no—
53C2×C22⋊C4×C31—21no—
54C2×C22⋊C4×C31—21no—
55C2×C22⋊C4×C31—21no—
56C2×C22⋊C4×C31—22no—
57C2×C42×C31C2×C411yescharacteristic
58C2×C42×C31C2×C411yescharacteristic
59C31×#99—42no—
60C31×#99—42no—
61C31×C22⋊C8—24no—
62C31×C22⋊C8—22no—
63C31×C22⋊C8D414yes—
64C31×#95—21no—
65C31×#95—21no—
66C31×#95—21no—
67C31×#95—22no—
68C31×#95—22no—
69C24×C31C22⋊C411yescharacteristic
70C24×C31C22⋊C411yescharacteristic
71C24×C31C2×D411yescharacteristic
72#41×C31—44no—
73#41×C31—44no—
74#41×C31—48no—
75C2×Q8×C31—22no—
76C2×Q8×C31—22no—
77C2×Q8×C31—22no—
78C2×Q8×C31—41no—
79C2×Q8×C31—41no—
80C2×Q8×C31—41no—
81C2×D4×C31—22no—
82C2×D4×C31—22no—
83C2×D4×C31—22no—
84C2×D4×C31—22no—
85C2×D4×C31—22no—
86C2×D4×C31—22no—
87C2×D4×C31—41no—
88C2×D4×C31—41no—
89C2×D4×C31—41no—
90C22×C4×C31—22no—
91C22×C4×C31—22no—
92C22×C4×C31—21no—
93C22×C4×C31—21no—
94C22×C4×C31—21no—
95C22×C4×C31—21no—
96C22×C4×C31—21no—
97C22×C4×C31—21no—
98C22×C4×C31—21no—
99C22×C4×C31—21no—
100C22×C4×C31—41no—
101C22×C4×C31—41no—
102C22×C4×C31—41no—
103C22×C4×C31—41no—
104C22×C4×C31—42no—
105C22×C4×C31C22⋊C411yescharacteristic
106C22×C4×C31C22⋊C411yescharacteristic
107C22×C4×C31C22×C411yescharacteristic
108C22×C4×C31C2×D411yescharacteristic
109C22×C4×C31C2×D412yes—
110C22×C4×C31#4112yes—
111M16×C31—44no—
112M16×C31—42no—
113M16×C31—42no—
114C2×C8×C31—24no—
115C2×C8×C31—44no—
116C4⋊C4×C31—22no—
117C4⋊C4×C31—42no—
118C4⋊C4×C31—42no—
119C4⋊C4×C31—41no—
120C4⋊C4×C31—41no—
121C22⋊C4×C31—42no—
122C22⋊C4×C31—42no—
123C22⋊C4×C31—42no—
124C22⋊C4×C31—42no—
125C22⋊C4×C31—42no—
126C22⋊C4×C31—42no—
127C22⋊C4×C31—42no—
128C22⋊C4×C31—42no—
129C42×C31—41no—
130C42×C31—41no—
131#9443C3111yesmaximal, characteristic, direct product factor, Sylow
132C23×C31—21no—
133C23×C31—21no—
134C23×C31—21no—
135C23×C31—21no—
136C23×C31—21no—
137C23×C31—21no—
138C23×C31—21no—
139C23×C31—41no—
140C23×C31—41no—
141C23×C31—41no—
142C23×C31—41no—
143C23×C31—41no—
144C23×C31—41no—
145C23×C31#9912yes—
146C23×C31C2×C22⋊C411yescharacteristic
147C23×C31#12112yes—
148Q8×C31—44no—
149Q8×C31—82no—
150D4×C31—44no—
151D4×C31—44no—
152D4×C31—44no—
153C2×C4×C31—22no—
154C2×C4×C31—22no—
155C2×C4×C31—24no—
156C2×C4×C31—21no—
157C2×C4×C31—21no—
158C2×C4×C31—21no—
159C2×C4×C31—21no—
160C2×C4×C31—21no—
161C2×C4×C31—21no—
162C2×C4×C31—22no—
163C2×C4×C31—21no—
164C2×C4×C31—21no—
165C2×C4×C31—21no—
166C2×C4×C31—42no—
167C2×C4×C31—42no—
168C2×C4×C31—42no—
169C2×C4×C31—42no—
170C2×C4×C31—41no—
171C2×C4×C31—41no—
172C2×C4×C31—41no—
173C2×C4×C31—41no—
174C2×C4×C31—41no—
175C2×C4×C31—41no—
176C2×C4×C31—41no—
177C2×C4×C31—41no—
178C2×C4×C31—41no—
179C2×C4×C31—41no—
180C2×C4×C31—41no—
181C2×C4×C31—82no—
182C2×C4×C31—82no—
183C2×C4×C31—82no—
184C2×C4×C31#9912yes—
185C2×C4×C31C4×D412yes—
186C2×C4×C31#12011yescharacteristic
187C2×C4×C31#12311yescharacteristic
188C8×C31—44no—
189#2879C2×C3111yescharacteristic
190#2432C2×C3111yescharacteristic
191#1782C2×C3111yescharacteristic
192#1511C2×C3112yescomplemented
193#1284C2×C3112yescomplemented
194C22×C31—21no—
195C22×C31—21no—
196C22×C31—22no—
197C22×C31—22no—
198C22×C31—22no—
199C22×C31—22no—
200C22×C31—42no—
201C22×C31—42no—
202C22×C31—41no—
203C22×C31—41no—
204C22×C31—41no—
205C22×C31—41no—
206C22×C31—41no—
207C22×C31—81no—
208C22×C31—81no—
209C22×C31—81no—
210C22×C31—81no—
211C22×C31#38611yescharacteristic
212C22×C31C2×#9911yescharacteristic
213C22×C31C2×#9911yescharacteristic
214C4×C31—21no—
215C4×C31—21no—
216C4×C31—22no—
217C4×C31—22no—
218C4×C31—42no—
219C4×C31—41no—
220C4×C31—81no—
221C4×C31—81no—
222C4×C31—81no—
223C4×C31—162no—
224C22×#41C22×C3111yescharacteristic
225C22×M16—21no—
226C2×C4⋊Q8C4×C3111yescharacteristic, complemented
227C2×#124C4×C3111yescharacteristic, complemented
228C2×C22⋊Q8C22×C3111yescharacteristic
229C2×#99—21no—
230C2×#99—21no—
231#408—24no—
232#407—21no—
233#407C22×C3112yes—
234#394—21no—
235#394—21no—
236#391C4×C3111yescharacteristic, complemented
237#390C4×C3111yescharacteristic, complemented
238#385—21no—
239#385—21no—
240#381C22×C3112yes—
241C4×C22⋊C4C22×C3111yescharacteristic
242#328—24no—
243C2×C31—21no—
244C2×C31—21no—
245C2×C31—42no—
246C2×C31—42no—
247C2×C31—81no—
248C2×C31#188411yescharacteristic
249C2×C31#189811yescharacteristic
250C2×C31#190911yescharacteristic
251C2×#41—22no—
252C2×#41—22no—
253C2×#41—42no—
254C2×#41—42no—
255C22×Q8C2×C4×C3111yescharacteristic
256C22×Q8C2×C4×C3111yescharacteristic
257C22×D4C2×C4×C3111yescharacteristic
258C22×D4D4×C3111yescharacteristic
259C22×D4D4×C3111yescharacteristic
260C23×C4D4×C3111yescharacteristic
261C23×C4D4×C3111yescharacteristic
262C23×C4C23×C3111yescharacteristic
263C2×M16—22no—
264C2×M16—22no—
265C2×M16—42no—
266C2×M16—42no—
267C22×C8—22no—
268C4⋊Q8—42no—
269#124—42no—
270C22⋊Q8—22no—
271C22⋊Q8—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4C2×C4×C3111yescharacteristic
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—22no—
283C2×C42C2×C4×C3111yescharacteristic
284C2×C42C2×C4×C3111yescharacteristic
285#99—42no—
286#99—42no—
287C22⋊C8—24no—
288C22⋊C8—22no—
289C22⋊C8D4×C3114yes—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—22no—
294#95—22no—
295C31#944311yescharacteristic, direct product factor, Sylow
296C24C22⋊C4×C3111yescharacteristic
297C24C22⋊C4×C3111yescharacteristic
298C24C2×D4×C3111yescharacteristic
299#41—44no—
300#41—44no—
301#41—48no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4C22⋊C4×C3111yescharacteristic
333C22×C4C22⋊C4×C3111yescharacteristic
334C22×C4C22×C4×C3111yescharacteristic
335C22×C4C2×D4×C3111yescharacteristic
336C22×C4C2×D4×C3112yes—
337C22×C4#41×C3112yes—
338M16—44no—
339M16—42no—
340M16—42no—
341C2×C8—24no—
342C2×C8—44no—
343C4⋊C4—22no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—42no—
354C22⋊C4—42no—
355C22⋊C4—42no—
356C42—41no—
357C42—41no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23C31×#9912yes—
372C23C2×C22⋊C4×C3111yescharacteristic
373C23C31×#12112yes—
374Q8—44no—
375Q8—82no—
376D4—44no—
377D4—44no—
378D4—44no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4C31×#9912yes—
411C2×C4C4×D4×C3112yes—
412C2×C4C31×#12011yescharacteristic
413C2×C4C31×#12311yescharacteristic
414C8—44no—
415C22—21no—
416C22—21no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—42no—
422C22—42no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22C31×#38611yescharacteristic
433C22C2×C31×#9911yescharacteristic
434C22C2×C31×#9911yescharacteristic
435C4—21no—
436C4—21no—
437C4—22no—
438C4—22no—
439C4—42no—
440C4—41no—
441C4—81no—
442C4—81no—
443C4—81no—
444C4—162no—
445C2—21no—
446C2—21no—
447C2—42no—
448C2—42no—
449C2—81no—
450C2C31×#188411yescharacteristic
451C2C31×#189811yescharacteristic
452C2C31×#190911yescharacteristic
subgroupG/H
center, socleC22×C31#386
commutatorC22×C4C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC22×#41×C31C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
4. C31×#1511C2semidirect
5. C31×#1284C2semidirect
8. C2×C31×C4⋊Q8C4semidirect
9. C2×C31×#124C4semidirect
18. C31×#391C4semidirect
19. C31×#390C4semidirect
131. #9443C31direct (N × H)
192. #1511C2×C31semidirect
193. #1284C2×C31semidirect
226. C2×C4⋊Q8C4×C31semidirect
227. C2×#124C4×C31semidirect
236. #391C4×C31semidirect
237. #390C4×C31semidirect
295. C31#9443direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28792
C31×#24322
C31×#17822
C31×#15112
C31×#12842
#944331