Finite Group Databasebeta

← C31×#12713 #21469472

Subgroups — C31×#12713 #21469472

Order 7936.

986 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3485C211yesmaximal, characteristic
2C31×#2135C211yesmaximal, characteristic, complemented
3C31×#2129C211yesmaximal, characteristic, complemented
4C31×#2120C212yesmaximal, complemented
5C31×#1795C212yesmaximal, complemented
6C2×C31×H(5,𝔽2)C2211yescharacteristic
7C22×#41×C31C411yescharacteristic, complemented
8C31×#547—22no—
9C31×#547—22no—
10C31×#546—22no—
11C31×#546—22no—
12C31×#536C411yescharacteristic, complemented
13C31×#536C2211yescharacteristic
14C31×#534—22no—
15C31×#525C411yescharacteristic, complemented
16C31×#525C2211yescharacteristic
17C2×C31×#123—22no—
18C31×#518C411yescharacteristic, complemented
19C31×#421—22no—
20C31×#418—22no—
21C31×#413C2212yes—
22C31×#410C2212yes—
23C31×#356—22no—
24C31×#354—22no—
25C31×#352—22no—
26C31×AΣL(1,𝔽16)—22no—
27C31×H(5,𝔽2)—44no—
28C2×#41×C31—22no—
29C2×#41×C31—21no—
30C2×#41×C31—22no—
31C2×#41×C31—44no—
32C2×#41×C31C2×C411yescharacteristic
33C2×#41×C31D412yes—
34C2×#41×C31C2311yescharacteristic
35C22×Q8×C31C2×C411yescharacteristic
36C22×D4×C31—21no—
37C22×D4×C31—22no—
38C22×D4×C31—21no—
39C22×D4×C31C2×C411yescharacteristic
40C22×D4×C31D412yes—
41C23×C4×C31—21no—
42C23×C4×C31C2×C411yescharacteristic
43C2×M16×C31—22no—
44C31×C4⋊Q8—21no—
45C31×C4⋊Q8D412yes—
46C31×#124—21no—
47C31×#124D412yes—
48C31×#123—24no—
49C31×#123—24no—
50C31×#123—24no—
51C31×#123—22no—
52C31×#123—21no—
53C31×#123—22no—
54C31×#123D412yes—
55C31×C22⋊Q8—21no—
56C31×C22⋊Q8—22no—
57C31×C22⋊Q8—21no—
58C31×C22⋊Q8—22no—
59C31×#121—24no—
60C31×#121—24no—
61C31×#121—21no—
62C31×#121—21no—
63C31×#120—24no—
64C31×#120—21no—
65C31×#120D412yes—
66C4×D4×C31—22no—
67C4×D4×C31—22no—
68C4×D4×C31—22no—
69C4×D4×C31—22no—
70C31×#117—22no—
71C31×#117C2×C411yescharacteristic
72C31×#117C2×C411yescharacteristic
73C2×C4⋊C4×C31—24no—
74C2×C22⋊C4×C31—24no—
75C2×C22⋊C4×C31—22no—
76C31×#104—44no—
77C31×#102—44no—
78C31×#101—22no—
79C31×#100—22no—
80C31×#99—22no—
81C31×#99—22no—
82C24×C31—21no—
83C24×C31—22no—
84C24×C31C22⋊C411yescharacteristic
85#41×C31—24no—
86#41×C31—42no—
87#41×C31—42no—
88#41×C31—42no—
89#41×C31—48no—
90#41×C31—44no—
91#41×C31—44no—
92#41×C31—44no—
93C2×Q8×C31—21no—
94C2×Q8×C31—42no—
95C2×Q8×C31C22⋊C412yes—
96C2×Q8×C31C22⋊C411yescharacteristic
97C2×Q8×C31C2×D411yescharacteristic
98C2×D4×C31—24no—
99C2×D4×C31—24no—
100C2×D4×C31—24no—
101C2×D4×C31—21no—
102C2×D4×C31—22no—
103C2×D4×C31—22no—
104C2×D4×C31—22no—
105C2×D4×C31—21no—
106C2×D4×C31—22no—
107C2×D4×C31—42no—
108C2×D4×C31—44no—
109C2×D4×C31—44no—
110C2×D4×C31—48no—
111C2×D4×C31C22⋊C411yescharacteristic
112C2×D4×C31C22⋊C412yes—
113C2×D4×C31C2×D411yescharacteristic
114C2×D4×C31C2×D412yes—
115C22×C4×C31—24no—
116C22×C4×C31—21no—
117C22×C4×C31—24no—
118C22×C4×C31—21no—
119C22×C4×C31—21no—
120C22×C4×C31—21no—
121C22×C4×C31—22no—
122C22×C4×C31—42no—
123C22×C4×C31—44no—
124C22×C4×C31C22⋊C411yescharacteristic
125C22×C4×C31C22×C411yescharacteristic
126C22×C4×C31C2×D412yes—
127M16×C31—42no—
128M16×C31—42no—
129C2×C8×C31—42no—
130C4⋊C4×C31—22no—
131C4⋊C4×C31—24no—
132C4⋊C4×C31—21no—
133C4⋊C4×C31—21no—
134C4⋊C4×C31—22no—
135C4⋊C4×C31—24no—
136C4⋊C4×C31—42no—
137C22⋊C4×C31—22no—
138C22⋊C4×C31—24no—
139C22⋊C4×C31—24no—
140C22⋊C4×C31—22no—
141C22⋊C4×C31—21no—
142C22⋊C4×C31—22no—
143C22⋊C4×C31—24no—
144C22⋊C4×C31—22no—
145C22⋊C4×C31—42no—
146C22⋊C4×C31C22⋊C412yes—
147C42×C31—21no—
148C42×C31—42no—
149C42×C31C22⋊C412yes—
150#12713C3111yesmaximal, characteristic, direct product factor, Sylow
151C23×C31—24no—
152C23×C31—22no—
153C23×C31—21no—
154C23×C31—21no—
155C23×C31—44no—
156C23×C31—41no—
157C23×C31—81no—
158C23×C31—82no—
159C23×C31—84no—
160C23×C31C2×C22⋊C411yescharacteristic
161C23×C31#12012yes—
162Q8×C31—22no—
163Q8×C31—41no—
164Q8×C31—42no—
165Q8×C31—42no—
166D4×C31—24no—
167D4×C31—22no—
168D4×C31—22no—
169D4×C31—42no—
170D4×C31—41no—
171D4×C31—44no—
172D4×C31—44no—
173D4×C31—42no—
174D4×C31—48no—
175D4×C31—82no—
176D4×C31—82no—
177D4×C31#9914yes—
178C2×C4×C31—22no—
179C2×C4×C31—22no—
180C2×C4×C31—22no—
181C2×C4×C31—22no—
182C2×C4×C31—22no—
183C2×C4×C31—21no—
184C2×C4×C31—22no—
185C2×C4×C31—21no—
186C2×C4×C31—44no—
187C2×C4×C31—42no—
188C2×C4×C31—42no—
189C2×C4×C31—42no—
190C2×C4×C31—48no—
191C2×C4×C31—42no—
192C2×C4×C31—41no—
193C2×C4×C31—84no—
194C2×C4×C31—82no—
195C2×C4×C31C2×C22⋊C411yescharacteristic
196C2×C4×C31C2×C22⋊C411yescharacteristic
197C2×C4×C31#12012yes—
198C8×C31—82no—
199#3485C2×C3111yescharacteristic
200#2135C2×C3111yescharacteristic, complemented
201#2129C2×C3111yescharacteristic, complemented
202#2120C2×C3112yescomplemented
203#1795C2×C3112yescomplemented
204C22×C31—21no—
205C22×C31—22no—
206C22×C31—24no—
207C22×C31—42no—
208C22×C31—41no—
209C22×C31—44no—
210C22×C31—41no—
211C22×C31—41no—
212C22×C31—42no—
213C22×C31—41no—
214C22×C31—41no—
215C22×C31—84no—
216C22×C31—82no—
217C22×C31—164no—
218C22×C31#37911yescharacteristic
219C4×C31—22no—
220C4×C31—21no—
221C4×C31—42no—
222C4×C31—42no—
223C4×C31—41no—
224C4×C31—41no—
225C4×C31—42no—
226C4×C31—162no—
227C4×C31C2×#9912yes—
228C2×H(5,𝔽2)C22×C3111yescharacteristic
229C22×#41C4×C3111yescharacteristic, complemented
230#547—22no—
231#547—22no—
232#546—22no—
233#546—22no—
234#536C4×C3111yescharacteristic, complemented
235#536C22×C3111yescharacteristic
236#534—22no—
237#525C4×C3111yescharacteristic, complemented
238#525C22×C3111yescharacteristic
239C2×#123—22no—
240#518C4×C3111yescharacteristic, complemented
241#421—22no—
242#418—22no—
243#413C22×C3112yes—
244#410C22×C3112yes—
245#356—22no—
246#354—22no—
247#352—22no—
248AΣL(1,𝔽16)—22no—
249C2×C31—21no—
250C2×C31—42no—
251C2×C31—44no—
252C2×C31—41no—
253C2×C31—81no—
254C2×C31—81no—
255C2×C31#178111yescharacteristic
256H(5,𝔽2)—44no—
257C2×#41—22no—
258C2×#41—21no—
259C2×#41—22no—
260C2×#41—44no—
261C2×#41C2×C4×C3111yescharacteristic
262C2×#41D4×C3112yes—
263C2×#41C23×C3111yescharacteristic
264C22×Q8C2×C4×C3111yescharacteristic
265C22×D4—21no—
266C22×D4—22no—
267C22×D4—21no—
268C22×D4C2×C4×C3111yescharacteristic
269C22×D4D4×C3112yes—
270C23×C4—21no—
271C23×C4C2×C4×C3111yescharacteristic
272C2×M16—22no—
273C4⋊Q8—21no—
274C4⋊Q8D4×C3112yes—
275#124—21no—
276#124D4×C3112yes—
277#123—24no—
278#123—24no—
279#123—24no—
280#123—22no—
281#123—21no—
282#123—22no—
283#123D4×C3112yes—
284C22⋊Q8—21no—
285C22⋊Q8—22no—
286C22⋊Q8—21no—
287C22⋊Q8—22no—
288#121—24no—
289#121—24no—
290#121—21no—
291#121—21no—
292#120—24no—
293#120—21no—
294#120D4×C3112yes—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—22no—
298C4×D4—22no—
299#117—22no—
300#117C2×C4×C3111yescharacteristic
301#117C2×C4×C3111yescharacteristic
302C2×C4⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—22no—
305#104—44no—
306#102—44no—
307#101—22no—
308#100—22no—
309#99—22no—
310#99—22no—
311C31#1271311yescharacteristic, direct product factor, Sylow
312C24—21no—
313C24—22no—
314C24C22⋊C4×C3111yescharacteristic
315#41—24no—
316#41—42no—
317#41—42no—
318#41—42no—
319#41—48no—
320#41—44no—
321#41—44no—
322#41—44no—
323C2×Q8—21no—
324C2×Q8—42no—
325C2×Q8C22⋊C4×C3111yescharacteristic
326C2×Q8C22⋊C4×C3112yes—
327C2×Q8C2×D4×C3111yescharacteristic
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—21no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—21no—
336C2×D4—22no—
337C2×D4—42no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—48no—
341C2×D4C22⋊C4×C3111yescharacteristic
342C2×D4C22⋊C4×C3112yes—
343C2×D4C2×D4×C3111yescharacteristic
344C2×D4C2×D4×C3112yes—
345C22×C4—24no—
346C22×C4—21no—
347C22×C4—24no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4C22⋊C4×C3111yescharacteristic
355C22×C4C22×C4×C3111yescharacteristic
356C22×C4C2×D4×C3112yes—
357M16—42no—
358M16—42no—
359C2×C8—42no—
360C4⋊C4—22no—
361C4⋊C4—24no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—42no—
367C22⋊C4—22no—
368C22⋊C4—24no—
369C22⋊C4—24no—
370C22⋊C4—22no—
371C22⋊C4—21no—
372C22⋊C4—22no—
373C22⋊C4—24no—
374C22⋊C4—22no—
375C22⋊C4—42no—
376C22⋊C4C22⋊C4×C3112yes—
377C42—21no—
378C42—42no—
379C42C22⋊C4×C3112yes—
380C23—24no—
381C23—22no—
382C23—21no—
383C23—21no—
384C23—44no—
385C23—41no—
386C23—81no—
387C23—82no—
388C23—84no—
389C23C2×C22⋊C4×C3111yescharacteristic
390C23C31×#12012yes—
391Q8—22no—
392Q8—41no—
393Q8—42no—
394Q8—42no—
395D4—24no—
396D4—22no—
397D4—22no—
398D4—42no—
399D4—41no—
400D4—44no—
401D4—44no—
402D4—42no—
403D4—48no—
404D4—82no—
405D4—82no—
406D4C31×#9914yes—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—48no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—84no—
423C2×C4—82no—
424C2×C4C2×C22⋊C4×C3111yescharacteristic
425C2×C4C2×C22⋊C4×C3111yescharacteristic
426C2×C4C31×#12012yes—
427C8—82no—
428C22—21no—
429C22—22no—
430C22—24no—
431C22—42no—
432C22—41no—
433C22—44no—
434C22—41no—
435C22—41no—
436C22—42no—
437C22—41no—
438C22—41no—
439C22—84no—
440C22—82no—
441C22—164no—
442C22C31×#37911yescharacteristic
443C4—22no—
444C4—21no—
445C4—42no—
446C4—42no—
447C4—41no—
448C4—41no—
449C4—42no—
450C4—162no—
451C4C2×C31×#9912yes—
452C2—21no—
453C2—42no—
454C2—44no—
455C2—41no—
456C2—81no—
457C2—81no—
458C2C31×#178111yescharacteristic
subgroupG/H
center, socleC2×C31#1781
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×#41C23×C31
prime radicalC31×#3485C2
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×#2135C2semidirect
3. C31×#2129C2semidirect
4. C31×#2120C2semidirect
5. C31×#1795C2semidirect
7. C22×#41×C31C4semidirect
12. C31×#536C4semidirect
15. C31×#525C4semidirect
18. C31×#518C4semidirect
150. #12713C31direct (N × H)
200. #2135C2×C31semidirect
201. #2129C2×C31semidirect
202. #2120C2×C31semidirect
203. #1795C2×C31semidirect
229. C22×#41C4×C31semidirect
234. #536C4×C31semidirect
237. #525C4×C31semidirect
240. #518C4×C31semidirect
311. C31#12713direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34852
C31×#21352
C31×#21292
C31×#21202
C31×#17952
#1271331