Finite Group Databasebeta

← C31×#9954 #21466713

Subgroups — C31×#9954 #21466713

Order 7936.

1018 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3187C211yesmaximal, characteristic, complemented
2C2×C31×#450C211yesmaximal, characteristic, complemented
3C31×#2433C211yesmaximal, characteristic
4C31×#1996C211yesmaximal, characteristic, complemented
5C31×#1786C211yesmaximal, characteristic, complemented
6C31×#1519C211yesmaximal, characteristic, complemented
7C31×#1282C211yesmaximal, characteristic, complemented
8C23×D4×C31C2211yescharacteristic
9C22×SD16×C31—21no—
10C2×C31×#127—21no—
11C2×C31×C22⋊Q8C2211yescharacteristic
12C2×C31×#121C2211yescharacteristic
13C2×C31×#120—22no—
14C31×#483—21no—
15C31×#481—21no—
16C31×#465—22no—
17C31×#450—21no—
18C31×#450—22no—
19C31×#450—22no—
20C31×#448—21no—
21C2×C31×#102—21no—
22C2×C31×#102—22no—
23C2×C31×C22⋊C8C2211yescharacteristic
24C2×C31×C22⋊C8C2211yescharacteristic
25C31×#392—22no—
26C31×#390—21no—
27C31×#380C2211yescharacteristic
28C4×C22⋊C4×C31C2211yescharacteristic
29C31×#342—22no—
30C31×#327—22no—
31C25×C31—21no—
32C22×Q8×C31D411yescharacteristic, complemented
33C22×D4×C31—21no—
34C22×D4×C31—24no—
35C22×D4×C31—21no—
36C22×D4×C31—21no—
37C22×D4×C31—21no—
38C22×D4×C31—21no—
39C22×D4×C31—21no—
40C22×D4×C31—44no—
41C22×D4×C31D411yescharacteristic
42C22×D4×C31D412yes—
43C23×C4×C31C2311yescharacteristic
44C2×SD16×C31—24no—
45C2×SD16×C31—22no—
46C2×SD16×C31—44no—
47C22×C8×C31—21no—
48C22×C8×C31D412yescomplemented
49C31×#127—44no—
50C31×C22⋊Q8—21no—
51C31×C22⋊Q8—21no—
52C31×C22⋊Q8—21no—
53C31×C22⋊Q8—21no—
54C31×#121—21no—
55C31×#121—21no—
56C31×#121—21no—
57C31×#121—21no—
58C31×#120—44no—
59C31×#120—44no—
60C2×C4⋊C4×C31—21no—
61C2×C4⋊C4×C31—21no—
62C2×C4⋊C4×C31D411yescharacteristic
63C2×C4⋊C4×C31D411yescharacteristic, complemented
64C2×C22⋊C4×C31—24no—
65C2×C22⋊C4×C31—21no—
66C2×C22⋊C4×C31—21no—
67C2×C22⋊C4×C31—21no—
68C2×C22⋊C4×C31D412yes—
69C2×C42×C31—21no—
70C31×#106—22no—
71C31×#103—22no—
72C31×#102—24no—
73C31×#102—22no—
74C31×#102—44no—
75C31×C22⋊C8—21no—
76C31×C22⋊C8—21no—
77C31×C22⋊C8—21no—
78C31×C22⋊C8D412yescomplemented
79C31×#95—21no—
80C31×#95—21no—
81C24×C31—22no—
82C24×C31—21no—
83C24×C31—44no—
84C24×C31—41no—
85C24×C31—41no—
86C24×C31—42no—
87C24×C31—42no—
88C24×C31—82no—
89C24×C31C2×D411yescharacteristic
90C2×Q8×C31—21no—
91C2×Q8×C31—21no—
92C2×Q8×C31—42no—
93C2×D4×C31—24no—
94C2×D4×C31—24no—
95C2×D4×C31—24no—
96C2×D4×C31—22no—
97C2×D4×C31—21no—
98C2×D4×C31—22no—
99C2×D4×C31—21no—
100C2×D4×C31—42no—
101C2×D4×C31—42no—
102C2×D4×C31—42no—
103C2×D4×C31—42no—
104C2×D4×C31—48no—
105C2×D4×C31—44no—
106C2×D4×C31—44no—
107C2×D4×C31—44no—
108C2×D4×C31—48no—
109C2×D4×C31—41no—
110C2×D4×C31—41no—
111C2×D4×C31—42no—
112C2×D4×C31—42no—
113C2×D4×C31—41no—
114C2×D4×C31—41no—
115C2×D4×C31—84no—
116C22×C4×C31—22no—
117C22×C4×C31—21no—
118C22×C4×C31—21no—
119C22×C4×C31—21no—
120C22×C4×C31—42no—
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×C31#4111yescharacteristic
128SD16×C31—48no—
129SD16×C31—82no—
130C2×C8×C31—22no—
131C2×C8×C31—22no—
132C2×C8×C31—42no—
133C2×C8×C31—42no—
134C4⋊C4×C31—21no—
135C4⋊C4×C31—21no—
136C4⋊C4×C31—21no—
137C4⋊C4×C31—21no—
138C4⋊C4×C31—42no—
139C4⋊C4×C31—41no—
140C4⋊C4×C31—41no—
141C22⋊C4×C31—22no—
142C22⋊C4×C31—48no—
143C22⋊C4×C31—41no—
144C22⋊C4×C31—41no—
145C22⋊C4×C31—41no—
146C22⋊C4×C31—41no—
147C22⋊C4×C31—42no—
148C22⋊C4×C31SD1614yescomplemented
149C42×C31—42no—
150#9954C3111yesmaximal, characteristic, direct product factor, Sylow
151C23×C31—24no—
152C23×C31—41no—
153C23×C31—41no—
154C23×C31—44no—
155C23×C31—44no—
156C23×C31—42no—
157C23×C31—44no—
158C23×C31—42no—
159C23×C31—42no—
160C23×C31—42no—
161C23×C31—44no—
162C23×C31—41no—
163C23×C31—42no—
164C23×C31—82no—
165C23×C31—82no—
166C23×C31—88no—
167C23×C31—88no—
168C23×C31—82no—
169C23×C31—81no—
170C23×C31—81no—
171C23×C31#12011yescharacteristic
172C23×C31#12111yescharacteristic
173C23×C31#12711yescharacteristic
174C23×C31C2×SD1612yes—
175C23×C31AGL(1,ℤ/8)12yes—
176Q8×C31—42no—
177Q8×C31—42no—
178D4×C31—24no—
179D4×C31—24no—
180D4×C31—42no—
181D4×C31—42no—
182D4×C31—44no—
183D4×C31—44no—
184D4×C31—42no—
185D4×C31—81no—
186D4×C31—816no—
187D4×C31—82no—
188D4×C31—81no—
189D4×C31—81no—
190C2×C4×C31—22no—
191C2×C4×C31—22no—
192C2×C4×C31—22no—
193C2×C4×C31—24no—
194C2×C4×C31—41no—
195C2×C4×C31—41no—
196C2×C4×C31—41no—
197C2×C4×C31—42no—
198C2×C4×C31—42no—
199C2×C4×C31—41no—
200C2×C4×C31—41no—
201C2×C4×C31—41no—
202C2×C4×C31—42no—
203C2×C4×C31—81no—
204C2×C4×C31—81no—
205C2×C4×C31—81no—
206C2×C4×C31—81no—
207C2×C4×C31—81no—
208C2×C4×C31#12011yescharacteristic
209C2×C4×C31#12011yescharacteristic
210C2×C4×C31#12111yescharacteristic
211C2×C4×C31#12111yescharacteristic
212C8×C31—42no—
213C8×C31—42no—
214#3187C2×C3111yescharacteristic, complemented
215C2×#450C2×C3111yescharacteristic, complemented
216#2433C2×C3111yescharacteristic
217#1996C2×C3111yescharacteristic, complemented
218#1786C2×C3111yescharacteristic, complemented
219#1519C2×C3111yescharacteristic, complemented
220#1282C2×C3111yescharacteristic, complemented
221C22×C31—21no—
222C22×C31—21no—
223C22×C31—21no—
224C22×C31—21no—
225C22×C31—21no—
226C22×C31—21no—
227C22×C31—42no—
228C22×C31—42no—
229C22×C31—48no—
230C22×C31—44no—
231C22×C31—84no—
232C22×C31—84no—
233C22×C31—88no—
234C22×C31—84no—
235C22×C31—84no—
236C22×C31—82no—
237C22×C31—82no—
238C22×C31—81no—
239C22×C31—81no—
240C22×C31—81no—
241C22×C31#39211yescharacteristic
242C22×C31#45012yes—
243C22×C31#46012yes—
244C22×C31#49211yescharacteristic
245C22×C31#49611yescharacteristic
246C4×C31—21no—
247C4×C31—21no—
248C4×C31—21no—
249C4×C31—44no—
250C4×C31—81no—
251C4×C31—81no—
252C4×C31—81no—
253C4×C31—161no—
254C4×C31B(4,𝔽2)12yes—
255C23×D4C22×C3111yescharacteristic
256C22×SD16—21no—
257C2×#127—21no—
258C2×C22⋊Q8C22×C3111yescharacteristic
259C2×#121C22×C3111yescharacteristic
260C2×#120—22no—
261#483—21no—
262#481—21no—
263#465—22no—
264#450—21no—
265#450—22no—
266#450—22no—
267#448—21no—
268C2×#102—21no—
269C2×#102—22no—
270C2×C22⋊C8C22×C3111yescharacteristic
271C2×C22⋊C8C22×C3111yescharacteristic
272#392—22no—
273#390—21no—
274#380C22×C3111yescharacteristic
275C4×C22⋊C4C22×C3111yescharacteristic
276#342—22no—
277#327—22no—
278C2×C31—22no—
279C2×C31—21no—
280C2×C31—22no—
281C2×C31—21no—
282C2×C31—84no—
283C2×C31—161no—
284C2×C31#200811yescharacteristic
285C2×C31#201411yescharacteristic
286C2×C31#202111yescharacteristic
287C25—21no—
288C22×Q8D4×C3111yescharacteristic, complemented
289C22×D4—21no—
290C22×D4—24no—
291C22×D4—21no—
292C22×D4—21no—
293C22×D4—21no—
294C22×D4—21no—
295C22×D4—21no—
296C22×D4—44no—
297C22×D4D4×C3111yescharacteristic
298C22×D4D4×C3112yes—
299C23×C4C23×C3111yescharacteristic
300C2×SD16—24no—
301C2×SD16—22no—
302C2×SD16—44no—
303C22×C8—21no—
304C22×C8D4×C3112yescomplemented
305#127—44no—
306C22⋊Q8—21no—
307C22⋊Q8—21no—
308C22⋊Q8—21no—
309C22⋊Q8—21no—
310#121—21no—
311#121—21no—
312#121—21no—
313#121—21no—
314#120—44no—
315#120—44no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4D4×C3111yescharacteristic
319C2×C4⋊C4D4×C3111yescharacteristic, complemented
320C2×C22⋊C4—24no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4D4×C3112yes—
325C2×C42—21no—
326#106—22no—
327#103—22no—
328#102—24no—
329#102—22no—
330#102—44no—
331C22⋊C8—21no—
332C22⋊C8—21no—
333C22⋊C8—21no—
334C22⋊C8D4×C3112yescomplemented
335#95—21no—
336#95—21no—
337C31#995411yescharacteristic, direct product factor, Sylow
338C24—22no—
339C24—21no—
340C24—44no—
341C24—41no—
342C24—41no—
343C24—42no—
344C24—42no—
345C24—82no—
346C24C2×D4×C3111yescharacteristic
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—42no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—22no—
354C2×D4—21no—
355C2×D4—22no—
356C2×D4—21no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—48no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—48no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—84no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4C2×D4×C3111yescharacteristic
380C22×C4C2×D4×C3111yescharacteristic
381C22×C4C2×D4×C3111yescharacteristic
382C22×C4C2×D4×C3111yescharacteristic
383C22×C4C2×D4×C3111yescharacteristic
384C22×C4#41×C3111yescharacteristic
385SD16—48no—
386SD16—82no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—42no—
390C2×C8—42no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—42no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C22⋊C4—22no—
399C22⋊C4—48no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—42no—
405C22⋊C4SD16×C3114yescomplemented
406C42—42no—
407C23—24no—
408C23—41no—
409C23—41no—
410C23—44no—
411C23—44no—
412C23—42no—
413C23—44no—
414C23—42no—
415C23—42no—
416C23—44no—
417C23—42no—
418C23—42no—
419C23—41no—
420C23—82no—
421C23—82no—
422C23—88no—
423C23—88no—
424C23—82no—
425C23—81no—
426C23—81no—
427C23C31×#12011yescharacteristic
428C23C31×#12111yescharacteristic
429C23C31×#12711yescharacteristic
430C23C2×SD16×C3112yes—
431C23C31×AGL(1,ℤ/8)12yes—
432Q8—42no—
433Q8—42no—
434D4—24no—
435D4—24no—
436D4—42no—
437D4—42no—
438D4—44no—
439D4—44no—
440D4—42no—
441D4—81no—
442D4—816no—
443D4—82no—
444D4—81no—
445D4—81no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4C31×#12011yescharacteristic
465C2×C4C31×#12011yescharacteristic
466C2×C4C31×#12111yescharacteristic
467C2×C4C31×#12111yescharacteristic
468C8—42no—
469C8—42no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—42no—
477C22—42no—
478C22—48no—
479C22—44no—
480C22—84no—
481C22—84no—
482C22—88no—
483C22—84no—
484C22—84no—
485C22—82no—
486C22—82no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22C31×#39211yescharacteristic
491C22C31×#45012yes—
492C22C31×#46012yes—
493C22C31×#49211yescharacteristic
494C22C31×#49611yescharacteristic
495C4—21no—
496C4—21no—
497C4—21no—
498C4—44no—
499C4—81no—
500C4—81no—
501C4—81no—
502C4—161no—
503C4C31×B(4,𝔽2)12yes—
504C2—22no—
505C2—21no—
506C2—22no—
507C2—21no—
508C2—84no—
509C2—161no—
510C2C31×#200811yescharacteristic
511C2C31×#201411yescharacteristic
512C2C31×#202111yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC23×C4C23×C31
prime radicalC31×#2433C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3187C2semidirect
2. C2×C31×#450C2semidirect
4. C31×#1996C2semidirect
5. C31×#1786C2semidirect
6. C31×#1519C2semidirect
7. C31×#1282C2semidirect
32. C22×Q8×C31D4semidirect
48. C22×C8×C31D4semidirect
63. C2×C4⋊C4×C31D4semidirect
78. C31×C22⋊C8D4semidirect
148. C22⋊C4×C31SD16semidirect
150. #9954C31direct (N × H)
214. #3187C2×C31semidirect
215. C2×#450C2×C31semidirect
217. #1996C2×C31semidirect
218. #1786C2×C31semidirect
219. #1519C2×C31semidirect
220. #1282C2×C31semidirect
288. C22×Q8D4×C31semidirect
304. C22×C8D4×C31semidirect
319. C2×C4⋊C4D4×C31semidirect
334. C22⋊C8D4×C31semidirect
337. C31#9954direct (N × H)
405. C22⋊C4SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#31872
C2×C31×#4502
C31×#24332
C31×#19962
C31×#17862
C31×#15192
C31×#12822
#995431