Finite Group Databasebeta

← C31×#32772 #21489531

Subgroups — C31×#32772 #21489531

Order 7936.

774 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2857C211yesmaximal, characteristic, complemented
2C31×#2850C211yesmaximal, characteristic
3C31×#2095C211yesmaximal, characteristic, complemented
4C31×#2094C211yesmaximal, characteristic, complemented
5C31×#2085C212yesmaximal, complemented
6C31×#2042C212yesmaximal, complemented
7C31×#2035C212yesmaximal, complemented
8C31×#2033C212yesmaximal
9C31×#2024C211yesmaximal, characteristic, complemented
10C31×#2023C211yesmaximal, characteristic, complemented
11C31×#1817C211yesmaximal, characteristic, complemented
12C2×C31×C4⋊Q8C2211yescharacteristic
13C2×C31×#124C2211yescharacteristic
14C2×C31×C22⋊Q8—21no—
15C2×C31×C22⋊Q8C2211yescharacteristic
16C2×C31×#121C2211yescharacteristic
17C22×C42×C31C2211yescharacteristic
18C2×C31×#107C2212yes—
19C2×C31×#106C2212yes—
20C2×C31×#103C2212yes—
21C2×C31×#103C2212yes—
22C2×C31×#102C2212yes—
23C2×C31×#102C2212yes—
24C2×C31×C22⋊C8C2212yes—
25C31×#391—21no—
26C31×#391C2211yescharacteristic
27C31×#390C2211yescharacteristic
28C31×#389—21no—
29C31×#389C2212yes—
30C31×#388—21no—
31C31×#388—21no—
32C31×#386—21no—
33C31×#386—21no—
34C31×#384C2211yescharacteristic
35C31×#382C2211yescharacteristic
36C31×#380—21no—
37C31×#380C2212yes—
38C31×#340C2212yes—
39C31×#340C2212yes—
40C31×#336C2212yes—
41C31×#336C2211yescharacteristic
42C31×#336C2211yescharacteristic
43C22×Q8×C31—21no—
44C22×Q8×C31C2311yescharacteristic
45C22×D4×C31—21no—
46C22×D4×C31C2311yescharacteristic
47C23×C4×C31D411yescharacteristic
48C23×C4×C31D411yescharacteristic
49C23×C4×C31C2311yescharacteristic
50C22×C8×C31C2312yes—
51C22×C8×C31C2312yes—
52C31×C4⋊Q8—42no—
53C31×#124—42no—
54C31×C22⋊Q8—42no—
55C31×C22⋊Q8—44no—
56C31×#121—42no—
57C2×C4⋊C4×C31—21no—
58C2×C4⋊C4×C31—22no—
59C2×C4⋊C4×C31—21no—
60C2×C4⋊C4×C31—21no—
61C2×C4⋊C4×C31—21no—
62C2×C4⋊C4×C31C2312yes—
63C2×C4⋊C4×C31C2312yes—
64C2×C4⋊C4×C31C2311yescharacteristic
65C2×C4⋊C4×C31C2311yescharacteristic
66C2×C22⋊C4×C31—22no—
67C2×C22⋊C4×C31—21no—
68C2×C22⋊C4×C31—21no—
69C2×C22⋊C4×C31—21no—
70C2×C22⋊C4×C31—21no—
71C2×C22⋊C4×C31—21no—
72C2×C22⋊C4×C31—21no—
73C2×C42×C31—42no—
74C2×C42×C31D411yescharacteristic
75C2×C42×C31D411yescharacteristic
76C2×C42×C31C2311yescharacteristic
77C2×C42×C31C2311yescharacteristic
78C31×#107—42no—
79C31×#106—42no—
80C31×#103—42no—
81C31×#103—42no—
82C31×#102—42no—
83C31×#102—42no—
84C31×C22⋊C8—42no—
85C31×#95—22no—
86C31×#95—22no—
87C31×#95—21no—
88C31×#95—21no—
89C31×#95—21no—
90C31×#95—21no—
91C24×C31—21no—
92C24×C31C2×D411yescharacteristic
93C2×Q8×C31—22no—
94C2×Q8×C31—22no—
95C2×Q8×C31—21no—
96C2×Q8×C31—21no—
97C2×Q8×C31—42no—
98C2×Q8×C31—42no—
99C2×Q8×C31—42no—
100C2×D4×C31—22no—
101C2×D4×C31—22no—
102C2×D4×C31—21no—
103C2×D4×C31—21no—
104C2×D4×C31—42no—
105C2×D4×C31—42no—
106C2×D4×C31—42no—
107C22×C4×C31—22no—
108C22×C4×C31—22no—
109C22×C4×C31—22no—
110C22×C4×C31—22no—
111C22×C4×C31—22no—
112C22×C4×C31—21no—
113C22×C4×C31—21no—
114C22×C4×C31—21no—
115C22×C4×C31—21no—
116C22×C4×C31—21no—
117C22×C4×C31—22no—
118C22×C4×C31—22no—
119C22×C4×C31—41no—
120C22×C4×C31—41no—
121C22×C4×C31—41no—
122C22×C4×C31C2×D411yescharacteristic
123C22×C4×C31C2×D411yescharacteristic
124C22×C4×C31C2×D411yescharacteristic
125C22×C4×C31C2×D411yescharacteristic
126C22×C4×C31C2×D411yescharacteristic
127C22×C4×C31C2411yescharacteristic
128C2×C8×C31—22no—
129C2×C8×C31—22no—
130C2×C8×C31—22no—
131C2×C8×C31—22no—
132C2×C8×C31—22no—
133C2×C8×C31—22no—
134C4⋊C4×C31—22no—
135C4⋊C4×C31—22no—
136C4⋊C4×C31—22no—
137C4⋊C4×C31—22no—
138C4⋊C4×C31—22no—
139C4⋊C4×C31—22no—
140C4⋊C4×C31—42no—
141C4⋊C4×C31—42no—
142C4⋊C4×C31—42no—
143C4⋊C4×C31—44no—
144C4⋊C4×C31—42no—
145C22⋊C4×C31—42no—
146C22⋊C4×C31—42no—
147C22⋊C4×C31—42no—
148C22⋊C4×C31—42no—
149C22⋊C4×C31—44no—
150C22⋊C4×C31—42no—
151C22⋊C4×C31—42no—
152C42×C31—22no—
153C42×C31—22no—
154C42×C31—41no—
155C42×C31—41no—
156#32772C3111yesmaximal, characteristic, direct product factor, Sylow
157C23×C31—21no—
158C23×C31—21no—
159C23×C31—21no—
160C23×C31—21no—
161C23×C31—21no—
162C23×C31—21no—
163C23×C31—41no—
164C23×C31—41no—
165C23×C31—41no—
166C23×C31—41no—
167C23×C31—41no—
168C23×C31—41no—
169C23×C31—41no—
170C23×C31AGL(1,ℤ/8)12yes—
171C23×C31C22×D411yescharacteristic
172Q8×C31—42no—
173Q8×C31—42no—
174Q8×C31—84no—
175D4×C31—42no—
176D4×C31—42no—
177D4×C31—84no—
178C2×C4×C31—21no—
179C2×C4×C31—21no—
180C2×C4×C31—21no—
181C2×C4×C31—21no—
182C2×C4×C31—21no—
183C2×C4×C31—21no—
184C2×C4×C31—21no—
185C2×C4×C31—21no—
186C2×C4×C31—21no—
187C2×C4×C31—21no—
188C2×C4×C31—21no—
189C2×C4×C31—21no—
190C2×C4×C31—21no—
191C2×C4×C31—44no—
192C2×C4×C31—44no—
193C2×C4×C31—42no—
194C2×C4×C31—42no—
195C2×C4×C31—41no—
196C2×C4×C31—42no—
197C2×C4×C31—41no—
198C2×C4×C31—41no—
199C2×C4×C31—41no—
200C2×C4×C31—42no—
201C2×C4×C31—41no—
202C2×C4×C31—41no—
203C2×C4×C31—42no—
204C2×C4×C31—42no—
205C2×C4×C31—42no—
206C2×C4×C31—41no—
207C2×C4×C31—41no—
208C2×C4×C31—42no—
209C2×C4×C31—41no—
210C2×C4×C31#13512yes—
211C2×C4×C31#13712yes—
212C2×C4×C31H(5,𝔽2)12yes—
213C2×C4×C31H(5,𝔽2)12yes—
214C2×C4×C312−1+411yescharacteristic
215C2×C4×C312−1+411yescharacteristic
216C8×C31—42no—
217C8×C31—42no—
218#2857C2×C3111yescharacteristic, complemented
219#2850C2×C3111yescharacteristic
220#2095C2×C3111yescharacteristic, complemented
221#2094C2×C3111yescharacteristic, complemented
222#2085C2×C3112yescomplemented
223#2042C2×C3112yescomplemented
224#2035C2×C3112yescomplemented
225#2033C2×C3112yes—
226#2024C2×C3111yescharacteristic, complemented
227#2023C2×C3111yescharacteristic, complemented
228#1817C2×C3111yescharacteristic, complemented
229C22×C31—22no—
230C22×C31—22no—
231C22×C31—22no—
232C22×C31—41no—
233C22×C31—41no—
234C22×C31—41no—
235C22×C31—41no—
236C22×C31—81no—
237C22×C31—81no—
238C22×C31—81no—
239C22×C31—81no—
240C22×C31—81no—
241C22×C31—81no—
242C22×C31—81no—
243C22×C31#53411yescharacteristic
244C22×C31#53511yescharacteristic
245C22×C31#53611yescharacteristic
246C22×C31C2×#13511yescharacteristic
247C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
248C22×C31C2×#13711yescharacteristic
249C22×C31#57511yescharacteristic
250C4×C31—22no—
251C4×C31—22no—
252C4×C31—41no—
253C4×C31—41no—
254C4×C31—41no—
255C4×C31—81no—
256C4×C31—82no—
257C4×C31—81no—
258C4×C31—82no—
259C4×C31—81no—
260C4×C31#56212yes—
261C4×C31#56212yes—
262C2×C4⋊Q8C22×C3111yescharacteristic
263C2×#124C22×C3111yescharacteristic
264C2×C22⋊Q8—21no—
265C2×C22⋊Q8C22×C3111yescharacteristic
266C2×#121C22×C3111yescharacteristic
267C22×C42C22×C3111yescharacteristic
268C2×#107C22×C3112yes—
269C2×#106C22×C3112yes—
270C2×#103C22×C3112yes—
271C2×#103C22×C3112yes—
272C2×#102C22×C3112yes—
273C2×#102C22×C3112yes—
274C2×C22⋊C8C22×C3112yes—
275#391—21no—
276#391C22×C3111yescharacteristic
277#390C22×C3111yescharacteristic
278#389—21no—
279#389C22×C3112yes—
280#388—21no—
281#388—21no—
282#386—21no—
283#386—21no—
284#384C22×C3111yescharacteristic
285#382C22×C3111yescharacteristic
286#380—21no—
287#380C22×C3112yes—
288#340C22×C3112yes—
289#340C22×C3112yes—
290#336C22×C3112yes—
291#336C22×C3111yescharacteristic
292#336C22×C3111yescharacteristic
293C2×C31—42no—
294C2×C31—161no—
295C2×C31#281611yescharacteristic
296C2×C31#318811yescharacteristic
297C2×C31#319611yescharacteristic
298C2×C31#321011yescharacteristic
299C2×C31#321311yescharacteristic
300C2×C31#323311yescharacteristic
301C2×C31#323811yescharacteristic
302C22×Q8—21no—
303C22×Q8C23×C3111yescharacteristic
304C22×D4—21no—
305C22×D4C23×C3111yescharacteristic
306C23×C4D4×C3111yescharacteristic
307C23×C4D4×C3111yescharacteristic
308C23×C4C23×C3111yescharacteristic
309C22×C8C23×C3112yes—
310C22×C8C23×C3112yes—
311C4⋊Q8—42no—
312#124—42no—
313C22⋊Q8—42no—
314C22⋊Q8—44no—
315#121—42no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4C23×C3111yescharacteristic
322C2×C4⋊C4C23×C3111yescharacteristic
323C2×C4⋊C4C23×C3112yes—
324C2×C4⋊C4C23×C3112yes—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C42—42no—
333C2×C42D4×C3111yescharacteristic
334C2×C42D4×C3111yescharacteristic
335C2×C42C23×C3111yescharacteristic
336C2×C42C23×C3111yescharacteristic
337#107—42no—
338#106—42no—
339#103—42no—
340#103—42no—
341#102—42no—
342#102—42no—
343C22⋊C8—42no—
344#95—22no—
345#95—22no—
346#95—21no—
347#95—21no—
348#95—21no—
349#95—21no—
350C31#3277211yescharacteristic, direct product factor, Sylow
351C24—21no—
352C24C2×D4×C3111yescharacteristic
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4C2×D4×C3111yescharacteristic
383C22×C4C2×D4×C3111yescharacteristic
384C22×C4C2×D4×C3111yescharacteristic
385C22×C4C2×D4×C3111yescharacteristic
386C22×C4C2×D4×C3111yescharacteristic
387C22×C4C24×C3111yescharacteristic
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—44no—
404C4⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—44no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C42—22no—
413C42—22no—
414C42—41no—
415C42—41no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23C31×AGL(1,ℤ/8)12yes—
430C23C22×D4×C3111yescharacteristic
431Q8—42no—
432Q8—42no—
433Q8—84no—
434D4—42no—
435D4—42no—
436D4—84no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4C31×#13512yes—
470C2×C4C31×#13712yes—
471C2×C4C31×H(5,𝔽2)12yes—
472C2×C4C31×H(5,𝔽2)12yes—
473C2×C4C31×2−1+411yescharacteristic
474C2×C4C31×2−1+411yescharacteristic
475C8—42no—
476C8—42no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22—81no—
491C22C31×#53411yescharacteristic
492C22C31×#53511yescharacteristic
493C22C31×#53611yescharacteristic
494C22C2×C31×#13511yescharacteristic
495C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
496C22C2×C31×#13711yescharacteristic
497C22C31×#57511yescharacteristic
498C4—22no—
499C4—22no—
500C4—41no—
501C4—41no—
502C4—41no—
503C4—82no—
504C4—81no—
505C4—81no—
506C4—82no—
507C4—81no—
508C4C31×#56212yes—
509C4C31×#56212yes—
510C2—42no—
511C2—161no—
512C2C31×#281611yescharacteristic
513C2C31×#318811yescharacteristic
514C2C31×#319611yescharacteristic
515C2C31×#321011yescharacteristic
516C2C31×#321311yescharacteristic
517C2C31×#323311yescharacteristic
518C2C31×#323811yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2857C2semidirect
3. C31×#2095C2semidirect
4. C31×#2094C2semidirect
5. C31×#2085C2semidirect
6. C31×#2042C2semidirect
7. C31×#2035C2semidirect
9. C31×#2024C2semidirect
10. C31×#2023C2semidirect
11. C31×#1817C2semidirect
156. #32772C31direct (N × H)
218. #2857C2×C31semidirect
220. #2095C2×C31semidirect
221. #2094C2×C31semidirect
222. #2085C2×C31semidirect
223. #2042C2×C31semidirect
224. #2035C2×C31semidirect
226. #2024C2×C31semidirect
227. #2023C2×C31semidirect
228. #1817C2×C31semidirect
350. C31#32772direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28572
C31×#28502
C31×#20952
C31×#20942
C31×#20852
C31×#20422
C31×#20352
C31×#20332
C31×#20242
C31×#20232
C31×#18172
#3277231