Finite Group Databasebeta

← C31×#21476 #21478235

Subgroups — C31×#21476 #21478235

Order 7936.

1322 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C31×2−1+4C211yesmaximal, characteristic
2C2×C31×#456C211yesmaximal, characteristic
3C4×C31×#137C212yesmaximal, complemented
4C31×#2327C211yesmaximal, characteristic, complemented
5C31×#1886C211yesmaximal, characteristic, complemented
6C31×#1882C212yesmaximal, complemented
7C31×#1860C212yesmaximal, complemented
8C31×#1857C211yesmaximal, characteristic, complemented
9C31×#1793C211yesmaximal, characteristic
10C31×#1759C212yesmaximal, complemented
11C4×C31×#101C211yesmaximal, characteristic, complemented
12C2×C31×2−1+4C2211yescharacteristic
13C2×C31×#137C2212yes—
14C2×C31×C4⋊Q8C2211yescharacteristic
15C31×#520—22no—
16C31×#520—22no—
17C31×#520C2212yes—
18C31×#519—22no—
19C31×#519C2211yescharacteristic
20C4×#41×C31—22no—
21C4×#41×C31—22no—
22C4×#41×C31C2212yes—
23C2×C4×Q8×C31—22no—
24C2×C4×Q8×C31C2211yescharacteristic
25C2×C4×Q8×C31C2211yescharacteristic
26C31×#456C418yescomplemented
27C31×#441—22no—
28C31×#440—22no—
29C4×Q16×C31—22no—
30C4×SD16×C31—22no—
31C31×#428C2212yes—
32C2×C31×#104C2212yes—
33C2×C31×#104C2212yes—
34C31×#419C2212yes—
35C31×#417C2212yes—
36C2×C31×#101C2211yescharacteristic
37C2×C31×#101C2211yescharacteristic
38C4×M16×C31C2212yes—
39C31×#391C2211yescharacteristic
40C31×#391C2211yescharacteristic
41C31×#387—22no—
42C31×#387—22no—
43C31×#382C2211yescharacteristic
44C31×#376C2211yescharacteristic
45C31×#343C2212yes—
46C31×#339C2212yes—
47C31×#339C2212yes—
48C31×2−1+4—24no—
49C31×2−1+4—22no—
50C31×2−1+4C2×C414yes—
51C2×#41×C31—22no—
52C2×#41×C31—22no—
53C2×#41×C31C2312yes—
54C22×Q8×C31—22no—
55C22×Q8×C31C2311yescharacteristic
56C22×Q8×C31C2311yescharacteristic
57C31×#137—24no—
58C31×#137C2×C418yescomplemented
59C2×Q16×C31—22no—
60C2×SD16×C31—22no—
61C2×M16×C31C2312yes—
62C2×M16×C31C2312yes—
63C31×C4⋊Q8—22no—
64C31×C4⋊Q8C2×C414yescomplemented
65C4×Q8×C31—22no—
66C4×Q8×C31—24no—
67C4×Q8×C31—22no—
68C4×Q8×C31—22no—
69C4×Q8×C31—22no—
70C4×Q8×C31—22no—
71C4×Q8×C31—22no—
72C4×Q8×C31—22no—
73C4×Q8×C31—22no—
74C4×Q8×C31—22no—
75C4×Q8×C31D412yes—
76C4×Q8×C31D412yes—
77C4×D4×C31—22no—
78C4×D4×C31—24no—
79C4×D4×C31—22no—
80C4×D4×C31—22no—
81C4×D4×C31—22no—
82C4×D4×C31—24no—
83C4×D4×C31—22no—
84C4×D4×C31D412yes—
85C4×D4×C31D412yes—
86C31×#117—24no—
87C31×#117—24no—
88C31×#117—22no—
89C31×#117—22no—
90C31×#117—22no—
91C31×#117C2312yes—
92C2×C4⋊C4×C31—24no—
93C2×C4⋊C4×C31—22no—
94C2×C4⋊C4×C31—22no—
95C2×C4⋊C4×C31—21no—
96C2×C4⋊C4×C31D412yes—
97C2×C4⋊C4×C31C2311yescharacteristic
98C2×C4⋊C4×C31C2311yescharacteristic
99C2×C42×C31—24no—
100C2×C42×C31—22no—
101C2×C42×C31—21no—
102C2×C42×C31D412yes—
103C2×C42×C31C2311yescharacteristic
104C2×C42×C31C2311yescharacteristic
105C2×C42×C31C2311yescharacteristic
106C31×#107—22no—
107C31×#106—22no—
108C31×#104—24no—
109C31×#104C2×C418yescomplemented
110C31×#103—22no—
111C31×#103—22no—
112C31×#103—22no—
113C31×#102—22no—
114C31×#101—22no—
115C31×#101C2×C414yescomplemented
116C31×#97—22no—
117C4×C8×C31—22no—
118C31×#95—21no—
119C31×#95—21no—
120C31×#95—22no—
121C31×#95—22no—
122C31×#95—22no—
123#41×C31—28no—
124#41×C31—28no—
125#41×C31—28no—
126#41×C31—28no—
127#41×C31—22no—
128#41×C31—22no—
129#41×C31C22×C414yes—
130#41×C31#4114yes—
131C2×Q8×C31—22no—
132C2×Q8×C31—28no—
133C2×Q8×C31—28no—
134C2×Q8×C31—22no—
135C2×Q8×C31—22no—
136C2×Q8×C31—21no—
137C2×Q8×C31—22no—
138C2×Q8×C31—22no—
139C2×Q8×C31C22×C412yes—
140C2×Q8×C31C22×C412yes—
141C2×Q8×C31C2×D412yes—
142C2×Q8×C31#4112yes—
143C2×D4×C31—22no—
144C2×D4×C31—22no—
145C2×D4×C31C2×D412yes—
146C22×C4×C31—24no—
147C22×C4×C31—22no—
148C22×C4×C31—21no—
149C22×C4×C31—21no—
150C22×C4×C31—24no—
151C22×C4×C31—22no—
152C22×C4×C31—22no—
153C22×C4×C31—21no—
154C22×C4×C31—21no—
155C22×C4×C31C2×D412yes—
156C22×C4×C31C2×D412yes—
157C22×C4×C31C2×D411yescharacteristic
158C22×C4×C31C2×D411yescharacteristic
159C22×C4×C31C2411yescharacteristic
160Q16×C31—28no—
161SD16×C31—28no—
162M16×C31—22no—
163M16×C31—22no—
164M16×C31—22no—
165M16×C31C22×C414yes—
166C2×C8×C31—22no—
167C2×C8×C31—22no—
168C4⋊C4×C31—24no—
169C4⋊C4×C31—22no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—28no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—24no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—41no—
178C4⋊C4×C31—41no—
179C4⋊C4×C31C2×D412yes—
180C4⋊C4×C31C2×D412yes—
181C22⋊C4×C31—24no—
182C22⋊C4×C31—22no—
183C22⋊C4×C31—22no—
184C22⋊C4×C31—22no—
185C42×C31—24no—
186C42×C31—24no—
187C42×C31—22no—
188C42×C31—22no—
189C42×C31—22no—
190C42×C31—22no—
191C42×C31—22no—
192C42×C31—21no—
193C42×C31—22no—
194C42×C31—42no—
195C42×C31C22×C412yes—
196C42×C31C2×D412yes—
197C42×C31C2×D412yes—
198#21476C3111yesmaximal, characteristic, direct product factor, Sylow
199C23×C31—22no—
200C23×C31C22×D411yescharacteristic
201Q8×C31—28no—
202Q8×C31—24no—
203Q8×C31—22no—
204Q8×C31—24no—
205Q8×C31—24no—
206Q8×C31—42no—
207Q8×C31C4×D414yes—
208D4×C31—28no—
209D4×C31—24no—
210D4×C31—24no—
211D4×C31—22no—
212D4×C31C4×D414yes—
213C2×C4×C31—216no—
214C2×C4×C31—22no—
215C2×C4×C31—22no—
216C2×C4×C31—22no—
217C2×C4×C31—24no—
218C2×C4×C31—24no—
219C2×C4×C31—22no—
220C2×C4×C31—22no—
221C2×C4×C31—22no—
222C2×C4×C31—22no—
223C2×C4×C31—22no—
224C2×C4×C31—41no—
225C2×C4×C31—41no—
226C2×C4×C31—44no—
227C2×C4×C31—41no—
228C2×C4×C31—41no—
229C2×C4×C31—41no—
230C2×C4×C31—41no—
231C2×C4×C31—42no—
232C2×C4×C31—41no—
233C2×C4×C31—41no—
234C2×C4×C31—42no—
235C2×C4×C31—41no—
236C2×C4×C31—41no—
237C2×C4×C31—82no—
238C2×C4×C31—84no—
239C2×C4×C31C4×D414yes—
240C2×C4×C31#12011yescharacteristic
241C2×C4×C31#12011yescharacteristic
242C2×C4×C31#12012yes—
243C2×C4×C31C23×C411yescharacteristic
244C2×C4×C31C22×D412yes—
245C2×C4×C31C2×#4112yes—
246C2×C4×C31C2×#4111yescharacteristic
247C8×C31—24no—
248C8×C31—42no—
249C4×2−1+4C2×C3111yescharacteristic
250C2×#456C2×C3111yescharacteristic
251C4×#137C2×C3112yescomplemented
252#2327C2×C3111yescharacteristic, complemented
253#1886C2×C3111yescharacteristic, complemented
254#1882C2×C3112yescomplemented
255#1860C2×C3112yescomplemented
256#1857C2×C3111yescharacteristic, complemented
257#1793C2×C3111yescharacteristic
258#1759C2×C3112yescomplemented
259C4×#101C2×C3111yescharacteristic, complemented
260C22×C31—21no—
261C22×C31—21no—
262C22×C31—24no—
263C22×C31—42no—
264C22×C31—42no—
265C22×C31C2×C4×D411yescharacteristic
266C22×C31C2×#12011yescharacteristic
267C22×C31#52511yescharacteristic
268C4×C31—21no—
269C4×C31—21no—
270C4×C31—22no—
271C4×C31—24no—
272C4×C31—24no—
273C4×C31—42no—
274C4×C31—42no—
275C4×C31—42no—
276C4×C31—42no—
277C4×C31—42no—
278C4×C31—81no—
279C4×C31—81no—
280C4×C31C2×C4×D412yes—
281C4×C31#52512yes—
282C2×2−1+4C22×C3111yescharacteristic
283C2×#137C22×C3112yes—
284C2×C4⋊Q8C22×C3111yescharacteristic
285#520—22no—
286#520—22no—
287#520C22×C3112yes—
288#519—22no—
289#519C22×C3111yescharacteristic
290C4×#41—22no—
291C4×#41—22no—
292C4×#41C22×C3112yes—
293C2×C4×Q8—22no—
294C2×C4×Q8C22×C3111yescharacteristic
295C2×C4×Q8C22×C3111yescharacteristic
296#456C4×C3118yescomplemented
297#441—22no—
298#440—22no—
299C4×Q16—22no—
300C4×SD16—22no—
301#428C22×C3112yes—
302C2×#104C22×C3112yes—
303C2×#104C22×C3112yes—
304#419C22×C3112yes—
305#417C22×C3112yes—
306C2×#101C22×C3111yescharacteristic
307C2×#101C22×C3111yescharacteristic
308C4×M16C22×C3112yes—
309#391C22×C3111yescharacteristic
310#391C22×C3111yescharacteristic
311#387—22no—
312#387—22no—
313#382C22×C3111yescharacteristic
314#376C22×C3111yescharacteristic
315#343C22×C3112yes—
316#339C22×C3112yes—
317#339C22×C3112yes—
318C2×C31—21no—
319C2×C31—21no—
320C2×C31—44no—
321C2×C31C4×#12011yescharacteristic
322C2×C31#302011yescharacteristic
323C2×C31#302211yescharacteristic
3242−1+4—24no—
3252−1+4—22no—
3262−1+4C2×C4×C3114yes—
327C2×#41—22no—
328C2×#41—22no—
329C2×#41C23×C3112yes—
330C22×Q8—22no—
331C22×Q8C23×C3111yescharacteristic
332C22×Q8C23×C3111yescharacteristic
333#137—24no—
334#137C2×C4×C3118yescomplemented
335C2×Q16—22no—
336C2×SD16—22no—
337C2×M16C23×C3112yes—
338C2×M16C23×C3112yes—
339C4⋊Q8—22no—
340C4⋊Q8C2×C4×C3114yescomplemented
341C4×Q8—22no—
342C4×Q8—24no—
343C4×Q8—22no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×Q8—22no—
347C4×Q8—22no—
348C4×Q8—22no—
349C4×Q8—22no—
350C4×Q8—22no—
351C4×Q8D4×C3112yes—
352C4×Q8D4×C3112yes—
353C4×D4—22no—
354C4×D4—24no—
355C4×D4—22no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—24no—
359C4×D4—22no—
360C4×D4D4×C3112yes—
361C4×D4D4×C3112yes—
362#117—24no—
363#117—24no—
364#117—22no—
365#117—22no—
366#117—22no—
367#117C23×C3112yes—
368C2×C4⋊C4—24no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4D4×C3112yes—
373C2×C4⋊C4C23×C3111yescharacteristic
374C2×C4⋊C4C23×C3111yescharacteristic
375C2×C42—24no—
376C2×C42—22no—
377C2×C42—21no—
378C2×C42D4×C3112yes—
379C2×C42C23×C3111yescharacteristic
380C2×C42C23×C3111yescharacteristic
381C2×C42C23×C3111yescharacteristic
382#107—22no—
383#106—22no—
384#104—24no—
385#104C2×C4×C3118yescomplemented
386#103—22no—
387#103—22no—
388#103—22no—
389#102—22no—
390#101—22no—
391#101C2×C4×C3114yescomplemented
392#97—22no—
393C4×C8—22no—
394#95—21no—
395#95—21no—
396#95—22no—
397#95—22no—
398#95—22no—
399C31#2147611yescharacteristic, direct product factor, Sylow
400#41—28no—
401#41—28no—
402#41—28no—
403#41—28no—
404#41—22no—
405#41—22no—
406#41C22×C4×C3114yes—
407#41#41×C3114yes—
408C2×Q8—22no—
409C2×Q8—28no—
410C2×Q8—28no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8C22×C4×C3112yes—
417C2×Q8C22×C4×C3112yes—
418C2×Q8C2×D4×C3112yes—
419C2×Q8#41×C3112yes—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4C2×D4×C3112yes—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4C2×D4×C3112yes—
433C22×C4C2×D4×C3112yes—
434C22×C4C2×D4×C3111yescharacteristic
435C22×C4C2×D4×C3111yescharacteristic
436C22×C4C24×C3111yescharacteristic
437Q16—28no—
438SD16—28no—
439M16—22no—
440M16—22no—
441M16—22no—
442M16C22×C4×C3114yes—
443C2×C8—22no—
444C2×C8—22no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—28no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4C2×D4×C3112yes—
457C4⋊C4C2×D4×C3112yes—
458C22⋊C4—24no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C42—24no—
463C42—24no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—21no—
470C42—22no—
471C42—42no—
472C42C22×C4×C3112yes—
473C42C2×D4×C3112yes—
474C42C2×D4×C3112yes—
475C23—22no—
476C23C22×D4×C3111yescharacteristic
477Q8—28no—
478Q8—24no—
479Q8—22no—
480Q8—24no—
481Q8—24no—
482Q8—42no—
483Q8C4×D4×C3114yes—
484D4—28no—
485D4—24no—
486D4—24no—
487D4—22no—
488D4C4×D4×C3114yes—
489C2×C4—216no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—44no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—82no—
514C2×C4—84no—
515C2×C4C4×D4×C3114yes—
516C2×C4C31×#12011yescharacteristic
517C2×C4C31×#12011yescharacteristic
518C2×C4C31×#12012yes—
519C2×C4C23×C4×C3111yescharacteristic
520C2×C4C22×D4×C3112yes—
521C2×C4C2×#41×C3112yes—
522C2×C4C2×#41×C3111yescharacteristic
523C8—24no—
524C8—42no—
525C22—21no—
526C22—21no—
527C22—24no—
528C22—42no—
529C22—42no—
530C22C2×C4×D4×C3111yescharacteristic
531C22C2×C31×#12011yescharacteristic
532C22C31×#52511yescharacteristic
533C4—21no—
534C4—21no—
535C4—22no—
536C4—24no—
537C4—24no—
538C4—42no—
539C4—42no—
540C4—42no—
541C4—42no—
542C4—42no—
543C4—81no—
544C4—81no—
545C4C2×C4×D4×C3112yes—
546C4C31×#52512yes—
547C2—21no—
548C2—21no—
549C2—44no—
550C2C4×C31×#12011yescharacteristic
551C2C31×#302011yescharacteristic
552C2C31×#302211yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutatorC2×C4C23×C4×C31
Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×2−1+4C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C4×C31×#137C2semidirect
4. C31×#2327C2semidirect
5. C31×#1886C2semidirect
6. C31×#1882C2semidirect
7. C31×#1860C2semidirect
8. C31×#1857C2semidirect
10. C31×#1759C2semidirect
11. C4×C31×#101C2semidirect
26. C31×#456C4semidirect
58. C31×#137C2×C4semidirect
64. C31×C4⋊Q8C2×C4semidirect
109. C31×#104C2×C4semidirect
115. C31×#101C2×C4semidirect
198. #21476C31direct (N × H)
251. C4×#137C2×C31semidirect
252. #2327C2×C31semidirect
253. #1886C2×C31semidirect
254. #1882C2×C31semidirect
255. #1860C2×C31semidirect
256. #1857C2×C31semidirect
258. #1759C2×C31semidirect
259. C4×#101C2×C31semidirect
296. #456C4×C31semidirect
334. #137C2×C4×C31semidirect
340. C4⋊Q8C2×C4×C31semidirect
385. #104C2×C4×C31semidirect
391. #101C2×C4×C31semidirect
399. C31#21476direct (N × H)

Maximal subgroups

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

subgroupindex
C4×C31×2−1+42
C2×C31×#4562
C4×C31×#1372
C31×#23272
C31×#18862
C31×#18822
C31×#18602
C31×#18572
C31×#17932
C31×#17592
C4×C31×#1012
#2147631