Finite Group Databasebeta

← C31×#12780 #21469539

Subgroups — C31×#12780 #21469539

Order 7936.

756 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3551C211yesmaximal, characteristic
2C31×#2129C211yesmaximal, characteristic, complemented
3C31×#2128C211yesmaximal, characteristic, complemented
4C31×#2126C211yesmaximal, characteristic, complemented
5C31×#1910C211yesmaximal, characteristic, complemented
6C31×#1909C211yesmaximal, characteristic, complemented
7C31×#1884C211yesmaximal, characteristic, complemented
8C31×#568C2211yescharacteristic
9C31×#563C412yescomplemented
10C31×#560—21no—
11C31×#555C412yescomplemented
12C31×#553—21no—
13C31×#552—21no—
14C31×#551—21no—
15C31×#548—21no—
16C31×#546—21no—
17C31×#540—22no—
18C31×#538—22no—
19C31×#537—21no—
20C31×#536C2211yescharacteristic
21C31×#535C2211yescharacteristic
22C31×#532C2211yescharacteristic
23C31×#529—21no—
24C31×#421—21no—
25C31×#419—21no—
26C31×#418—21no—
27C31×#410C2211yescharacteristic
28C31×#410C2211yescharacteristic
29C31×#410C2211yescharacteristic
30C31×#354—21no—
31C31×#354—22no—
32C31×#353—21no—
33C31×#353—22no—
34C31×#339—21no—
35C31×H(3,ℤ/4)—21no—
36C31×H(3,ℤ/4)—21no—
37C2×#41×C31—21no—
38C2×#41×C31D411yescharacteristic
39C2×#41×C31C2311yescharacteristic
40C22×Q8×C31D411yescharacteristic
41C22×D4×C31D411yescharacteristic
42C31×#131—42no—
43C2×M16×C31—21no—
44C2×M16×C31—21no—
45C2×M16×C31—21no—
46C31×C4⋊Q8C2×C411yescharacteristic
47C31×C4⋊Q8C2×C412yes—
48C31×#127D411yescharacteristic
49C31×#127D412yes—
50C31×#126—22no—
51C31×#126—22no—
52C31×#126—22no—
53C31×#125—21no—
54C31×#125—22no—
55C31×#124—21no—
56C31×#124—22no—
57C31×#124C2×C411yescharacteristic
58C31×#124C2×C412yes—
59C31×#124D411yescharacteristic
60C31×#124D412yes—
61C31×#123—22no—
62C31×#123—22no—
63C31×#123—22no—
64C31×#123—22no—
65C31×#123—21no—
66C31×#123—21no—
67C31×C22⋊Q8—21no—
68C31×C22⋊Q8—22no—
69C31×C22⋊Q8—22no—
70C31×C22⋊Q8—21no—
71C31×C22⋊Q8—21no—
72C31×#121—22no—
73C31×#121—22no—
74C31×#121—21no—
75C31×#121—22no—
76C31×#121—22no—
77C31×#121—22no—
78C31×#121—22no—
79C31×#121—21no—
80C31×#121—21no—
81C31×#121—21no—
82C31×#121—21no—
83C31×#120—22no—
84C31×#120—21no—
85C31×#120—21no—
86C4×Q8×C31—21no—
87C4×D4×C31—22no—
88C4×D4×C31—21no—
89C4×D4×C31—22no—
90C4×D4×C31—21no—
91C4×D4×C31—21no—
92C4×D4×C31—22no—
93C31×#117—22no—
94C31×#117—22no—
95C31×#117—21no—
96C31×#117—21no—
97C31×#117—21no—
98C31×#117D411yescharacteristic
99C31×#117D411yescharacteristic
100C2×C4⋊C4×C31—21no—
101C2×C22⋊C4×C31—22no—
102C2×C42×C31D411yescharacteristic
103C31×#104—42no—
104C31×#103—42no—
105C31×#102—42no—
106C31×#99—21no—
107C31×#99—21no—
108C31×#99—21no—
109C31×#99—21no—
110C31×#99—21no—
111C31×#99—21no—
112C24×C31—21no—
113#41×C31—41no—
114#41×C31—41no—
115#41×C31—41no—
116#41×C31—41no—
117#41×C31—81no—
118#41×C31—81no—
119C2×Q8×C31—21no—
120C2×Q8×C31—41no—
121C2×Q8×C31—41no—
122C2×Q8×C31C22⋊C411yescharacteristic
123C2×Q8×C31C22⋊C412yes—
124C2×Q8×C31C22×C411yescharacteristic
125C2×D4×C31—22no—
126C2×D4×C31—22no—
127C2×D4×C31—21no—
128C2×D4×C31—22no—
129C2×D4×C31—22no—
130C2×D4×C31—21no—
131C2×D4×C31—21no—
132C2×D4×C31—21no—
133C2×D4×C31—41no—
134C2×D4×C31—41no—
135C2×D4×C31C22⋊C411yescharacteristic
136C2×D4×C31C22⋊C412yes—
137C2×D4×C31C2×D411yescharacteristic
138C2×D4×C31C2×D411yescharacteristic
139C2×D4×C31C2×D411yescharacteristic
140C22×C4×C31—22no—
141C22×C4×C31—22no—
142C22×C4×C31—22no—
143C22×C4×C31—21no—
144C22×C4×C31—21no—
145C22×C4×C31—21no—
146C22×C4×C31C2×D411yescharacteristic
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31C2×D411yescharacteristic
149M16×C31—22no—
150M16×C31—41no—
151M16×C31—41no—
152M16×C31—41no—
153C2×C8×C31—22no—
154C2×C8×C31—22no—
155C2×C8×C31—41no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—22no—
158C4⋊C4×C31—22no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—21no—
164C4⋊C4×C31—21no—
165C4⋊C4×C31—21no—
166C4⋊C4×C31—21no—
167C4⋊C4×C31—41no—
168C4⋊C4×C31—41no—
169C4⋊C4×C31—41no—
170C22⋊C4×C31—22no—
171C22⋊C4×C31—22no—
172C22⋊C4×C31—22no—
173C22⋊C4×C31—22no—
174C22⋊C4×C31—22no—
175C22⋊C4×C31—21no—
176C22⋊C4×C31—22no—
177C22⋊C4×C31—21no—
178C22⋊C4×C31—21no—
179C22⋊C4×C31—21no—
180C22⋊C4×C31—22no—
181C22⋊C4×C31—21no—
182C22⋊C4×C31—22no—
183C22⋊C4×C31—21no—
184C22⋊C4×C31—22no—
185C22⋊C4×C31—41no—
186C22⋊C4×C31—41no—
187C22⋊C4×C31—41no—
188C42×C31—22no—
189C42×C31—21no—
190C42×C31—41no—
191C42×C31—41no—
192C42×C31—41no—
193C42×C31C22⋊C411yescharacteristic
194C42×C31C22⋊C411yescharacteristic
195C42×C31C22⋊C412yes—
196C42×C31C22⋊C412yes—
197#12780C3111yesmaximal, characteristic, direct product factor, Sylow
198C23×C31—22no—
199C23×C31—21no—
200C23×C31—81no—
201C23×C31—82no—
202C23×C31#12011yescharacteristic
203C23×C31#12011yescharacteristic
204C23×C31#12011yescharacteristic
205Q8×C31—41no—
206Q8×C31—42no—
207Q8×C31—41no—
208Q8×C31—41no—
209Q8×C31—41no—
210D4×C31—41no—
211D4×C31—41no—
212D4×C31—41no—
213D4×C31—42no—
214D4×C31—41no—
215D4×C31—41no—
216D4×C31—41no—
217D4×C31—81no—
218D4×C31—82no—
219D4×C31—82no—
220D4×C31—82no—
221D4×C31—81no—
222D4×C31—81no—
223D4×C31—82no—
224C2×C4×C31—21no—
225C2×C4×C31—21no—
226C2×C4×C31—21no—
227C2×C4×C31—21no—
228C2×C4×C31—21no—
229C2×C4×C31—21no—
230C2×C4×C31—21no—
231C2×C4×C31—22no—
232C2×C4×C31—21no—
233C2×C4×C31—22no—
234C2×C4×C31—22no—
235C2×C4×C31—21no—
236C2×C4×C31—41no—
237C2×C4×C31—41no—
238C2×C4×C31—81no—
239C2×C4×C31—82no—
240C2×C4×C31—81no—
241C2×C4×C31—82no—
242C2×C4×C31—82no—
243C2×C4×C31—81no—
244C2×C4×C31—81no—
245C2×C4×C31—81no—
246C2×C4×C31C2×C22⋊C411yescharacteristic
247C2×C4×C31C2×C22⋊C411yescharacteristic
248C2×C4×C31C2×C22⋊C411yescharacteristic
249C2×C4×C31#12011yescharacteristic
250C8×C31—42no—
251#3551C2×C3111yescharacteristic
252#2129C2×C3111yescharacteristic, complemented
253#2128C2×C3111yescharacteristic, complemented
254#2126C2×C3111yescharacteristic, complemented
255#1910C2×C3111yescharacteristic, complemented
256#1909C2×C3111yescharacteristic, complemented
257#1884C2×C3111yescharacteristic, complemented
258C22×C31—21no—
259C22×C31—21no—
260C22×C31—21no—
261C22×C31—41no—
262C22×C31—41no—
263C22×C31—41no—
264C22×C31—41no—
265C22×C31—42no—
266C22×C31—41no—
267C22×C31—42no—
268C22×C31—41no—
269C22×C31—42no—
270C22×C31—162no—
271C22×C31#37911yescharacteristic
272C4×C31—21no—
273C4×C31—21no—
274C4×C31—21no—
275C4×C31—41no—
276C4×C31—42no—
277C4×C31—42no—
278C4×C31—41no—
279C4×C31—41no—
280C4×C31—42no—
281C4×C31—161no—
282C4×C31—161no—
283C4×C31—161no—
284C4×C31B(4,𝔽2)11yescharacteristic
285C4×C31#45811yescharacteristic
286#568C22×C3111yescharacteristic
287#563C4×C3112yescomplemented
288#560—21no—
289#555C4×C3112yescomplemented
290#553—21no—
291#552—21no—
292#551—21no—
293#548—21no—
294#546—21no—
295#540—22no—
296#538—22no—
297#537—21no—
298#536C22×C3111yescharacteristic
299#535C22×C3111yescharacteristic
300#532C22×C3111yescharacteristic
301#529—21no—
302#421—21no—
303#419—21no—
304#418—21no—
305#410C22×C3111yescharacteristic
306#410C22×C3111yescharacteristic
307#410C22×C3111yescharacteristic
308#354—21no—
309#354—22no—
310#353—21no—
311#353—22no—
312#339—21no—
313H(3,ℤ/4)—21no—
314H(3,ℤ/4)—21no—
315C2×C31—21no—
316C2×C31—41no—
317C2×C31—41no—
318C2×C31—41no—
319C2×C31—81no—
320C2×C31—82no—
321C2×C31#179011yescharacteristic
322C2×#41—21no—
323C2×#41D4×C3111yescharacteristic
324C2×#41C23×C3111yescharacteristic
325C22×Q8D4×C3111yescharacteristic
326C22×D4D4×C3111yescharacteristic
327#131—42no—
328C2×M16—21no—
329C2×M16—21no—
330C2×M16—21no—
331C4⋊Q8C2×C4×C3111yescharacteristic
332C4⋊Q8C2×C4×C3112yes—
333#127D4×C3111yescharacteristic
334#127D4×C3112yes—
335#126—22no—
336#126—22no—
337#126—22no—
338#125—21no—
339#125—22no—
340#124—21no—
341#124—22no—
342#124C2×C4×C3111yescharacteristic
343#124C2×C4×C3112yes—
344#124D4×C3111yescharacteristic
345#124D4×C3112yes—
346#123—22no—
347#123—22no—
348#123—22no—
349#123—22no—
350#123—21no—
351#123—21no—
352C22⋊Q8—21no—
353C22⋊Q8—22no—
354C22⋊Q8—22no—
355C22⋊Q8—21no—
356C22⋊Q8—21no—
357#121—22no—
358#121—22no—
359#121—21no—
360#121—22no—
361#121—22no—
362#121—22no—
363#121—22no—
364#121—21no—
365#121—21no—
366#121—21no—
367#121—21no—
368#120—22no—
369#120—21no—
370#120—21no—
371C4×Q8—21no—
372C4×D4—22no—
373C4×D4—21no—
374C4×D4—22no—
375C4×D4—21no—
376C4×D4—21no—
377C4×D4—22no—
378#117—22no—
379#117—22no—
380#117—21no—
381#117—21no—
382#117—21no—
383#117D4×C3111yescharacteristic
384#117D4×C3111yescharacteristic
385C2×C4⋊C4—21no—
386C2×C22⋊C4—22no—
387C2×C42D4×C3111yescharacteristic
388#104—42no—
389#103—42no—
390#102—42no—
391#99—21no—
392#99—21no—
393#99—21no—
394#99—21no—
395#99—21no—
396#99—21no—
397C31#1278011yescharacteristic, direct product factor, Sylow
398C24—21no—
399#41—41no—
400#41—41no—
401#41—41no—
402#41—41no—
403#41—81no—
404#41—81no—
405C2×Q8—21no—
406C2×Q8—41no—
407C2×Q8—41no—
408C2×Q8C22⋊C4×C3111yescharacteristic
409C2×Q8C22⋊C4×C3112yes—
410C2×Q8C22×C4×C3111yescharacteristic
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—41no—
420C2×D4—41no—
421C2×D4C22⋊C4×C3111yescharacteristic
422C2×D4C22⋊C4×C3112yes—
423C2×D4C2×D4×C3111yescharacteristic
424C2×D4C2×D4×C3111yescharacteristic
425C2×D4C2×D4×C3111yescharacteristic
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4C2×D4×C3111yescharacteristic
433C22×C4C2×D4×C3111yescharacteristic
434C22×C4C2×D4×C3111yescharacteristic
435M16—22no—
436M16—41no—
437M16—41no—
438M16—41no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—41no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—21no—
462C22⋊C4—22no—
463C22⋊C4—21no—
464C22⋊C4—21no—
465C22⋊C4—21no—
466C22⋊C4—22no—
467C22⋊C4—21no—
468C22⋊C4—22no—
469C22⋊C4—21no—
470C22⋊C4—22no—
471C22⋊C4—41no—
472C22⋊C4—41no—
473C22⋊C4—41no—
474C42—22no—
475C42—21no—
476C42—41no—
477C42—41no—
478C42—41no—
479C42C22⋊C4×C3111yescharacteristic
480C42C22⋊C4×C3111yescharacteristic
481C42C22⋊C4×C3112yes—
482C42C22⋊C4×C3112yes—
483C23—22no—
484C23—21no—
485C23—81no—
486C23—82no—
487C23C31×#12011yescharacteristic
488C23C31×#12011yescharacteristic
489C23C31×#12011yescharacteristic
490Q8—41no—
491Q8—42no—
492Q8—41no—
493Q8—41no—
494Q8—41no—
495D4—41no—
496D4—41no—
497D4—41no—
498D4—42no—
499D4—41no—
500D4—41no—
501D4—41no—
502D4—81no—
503D4—82no—
504D4—82no—
505D4—82no—
506D4—81no—
507D4—81no—
508D4—82no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—81no—
524C2×C4—82no—
525C2×C4—81no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4C2×C22⋊C4×C3111yescharacteristic
532C2×C4C2×C22⋊C4×C3111yescharacteristic
533C2×C4C2×C22⋊C4×C3111yescharacteristic
534C2×C4C31×#12011yescharacteristic
535C8—42no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—42no—
544C22—41no—
545C22—42no—
546C22—41no—
547C22—42no—
548C22—162no—
549C22C31×#37911yescharacteristic
550C4—21no—
551C4—21no—
552C4—21no—
553C4—41no—
554C4—42no—
555C4—42no—
556C4—41no—
557C4—41no—
558C4—42no—
559C4—161no—
560C4—161no—
561C4—161no—
562C4C31×B(4,𝔽2)11yescharacteristic
563C4C31×#45811yescharacteristic
564C2—21no—
565C2—41no—
566C2—41no—
567C2—41no—
568C2—82no—
569C2—81no—
570C2C31×#179011yescharacteristic
subgroupG/H
center, socleC2×C31#1790
commutatorC2×Q8C22×C4×C31
Frattini, cosocleC2×#41C23×C31
prime radicalC31×#3551C2
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C2×C4×C31 ◁ C2×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2129C2semidirect
3. C31×#2128C2semidirect
4. C31×#2126C2semidirect
5. C31×#1910C2semidirect
6. C31×#1909C2semidirect
7. C31×#1884C2semidirect
9. C31×#563C4semidirect
11. C31×#555C4semidirect
197. #12780C31direct (N × H)
252. #2129C2×C31semidirect
253. #2128C2×C31semidirect
254. #2126C2×C31semidirect
255. #1910C2×C31semidirect
256. #1909C2×C31semidirect
257. #1884C2×C31semidirect
287. #563C4×C31semidirect
289. #555C4×C31semidirect
397. C31#12780direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35512
C31×#21292
C31×#21282
C31×#21262
C31×#19102
C31×#19092
C31×#18842
#1278031