Finite Group Databasebeta

← C31×#32973 #21489732

Subgroups — C31×#32973 #21489732

Order 7936.

1286 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C31×H(5,𝔽2)C211yesmaximal, characteristic
2C31×#3080C212yesmaximal, complemented
3C31×#2566C211yesmaximal, characteristic, complemented
4C31×#1860C212yesmaximal, complemented
5C31×#1859C212yesmaximal, complemented
6C31×#1856C211yesmaximal, characteristic, complemented
7C31×#1856C211yesmaximal, characteristic, complemented
8C31×#1806C212yesmaximal, complemented
9C31×#1800C211yesmaximal, characteristic, complemented
10C31×#1792C211yesmaximal, characteristic
11C31×#1792C211yesmaximal, characteristic
12C2×C31×H(5,𝔽2)C2211yescharacteristic
13C2×C31×#125C2211yescharacteristic, complemented
14C31×#520—22no—
15C31×#520C2212yes—
16C31×#518—22no—
17C31×#518—22no—
18C31×#518C2211yescharacteristic
19C4×#41×C31—22no—
20C4×#41×C31C2212yes—
21C2×C4×D4×C31—22no—
22C2×C4×D4×C31—22no—
23C2×C4×D4×C31C2211yescharacteristic
24C31×#479—22no—
25C31×#479—22no—
26C31×#478—22no—
27C31×#478—22no—
28C31×#428C2212yescomplemented
29C31×#428C2212yescomplemented
30C31×#423C2212yescomplemented
31C2×C31×#104C2212yes—
32C2×C31×#104C2212yes—
33C31×#417C2212yes—
34C31×#417C2212yes—
35C2×C31×#100C2211yescharacteristic
36C2×C31×#100C2211yescharacteristic
37C31×#396—22no—
38C31×#390C2211yescharacteristic
39C31×#390C2211yescharacteristic
40C31×#385—22no—
41C31×#384C2211yescharacteristic, complemented
42C31×#382C2211yescharacteristic, complemented
43C4×C4⋊C4×C31C2211yescharacteristic, complemented
44C31×#343C2212yescomplemented
45C31×#339C2212yescomplemented
46C31×#339C2212yescomplemented
47C31×H(5,𝔽2)—24no—
48C31×H(5,𝔽2)—22no—
49C31×H(5,𝔽2)Q814yes—
50C2×#41×C31—22no—
51C2×#41×C31C2312yes—
52C22×D4×C31—22no—
53C22×D4×C31—22no—
54C22×D4×C31C2311yescharacteristic
55C23×C4×C31—22no—
56C23×C4×C31D412yes—
57C2×M16×C31C2312yes—
58C2×M16×C31C2312yes—
59C31×#125—22no—
60C31×#125—22no—
61C4×Q8×C31—22no—
62C4×Q8×C31D412yes—
63C4×Q8×C31D412yes—
64C4×D4×C31—24no—
65C4×D4×C31—24no—
66C4×D4×C31—24no—
67C4×D4×C31—24no—
68C4×D4×C31—22no—
69C4×D4×C31—22no—
70C4×D4×C31—24no—
71C4×D4×C31—22no—
72C4×D4×C31—22no—
73C4×D4×C31—22no—
74C4×D4×C31—22no—
75C4×D4×C31—22no—
76C4×D4×C31D412yes—
77C4×D4×C31D412yes—
78C31×#117—24no—
79C31×#117—22no—
80C31×#117—22no—
81C31×#117C2312yes—
82C2×C4⋊C4×C31—22no—
83C2×C4⋊C4×C31—22no—
84C2×C4⋊C4×C31—21no—
85C2×C4⋊C4×C31—21no—
86C2×C4⋊C4×C31—21no—
87C2×C4⋊C4×C31—22no—
88C2×C4⋊C4×C31C2311yescharacteristic
89C2×C4⋊C4×C31C2311yescharacteristic
90C2×C4⋊C4×C31C2311yescharacteristic
91C2×C22⋊C4×C31—22no—
92C2×C22⋊C4×C31—24no—
93C2×C22⋊C4×C31—22no—
94C2×C22⋊C4×C31—22no—
95C2×C22⋊C4×C31—22no—
96C2×C22⋊C4×C31D412yes—
97C2×C42×C31—22no—
98C2×C42×C31—22no—
99C2×C42×C31C2311yescharacteristic
100C2×C42×C31C2311yescharacteristic
101C2×C42×C31C2311yescharacteristic
102C31×#107—22no—
103C31×#107—22no—
104C31×#106—22no—
105C31×#106—22no—
106C31×C4⋊C8—22no—
107C31×C4⋊C8—22no—
108C31×#104—24no—
109C31×#104—24no—
110C31×#103—22no—
111C31×#103—22no—
112C31×#102—22no—
113C31×#102—22no—
114C31×#100—22no—
115C31×#100—22no—
116C31×#95—22no—
117C31×#95—21no—
118C24×C31—22no—
119C24×C31C2×D412yes—
120#41×C31—28no—
121#41×C31—28no—
122#41×C31—22no—
123#41×C31—22no—
124#41×C31C2×Q814yes—
125#41×C31#4114yes—
126C2×Q8×C31C2×D412yes—
127C2×D4×C31—24no—
128C2×D4×C31—28no—
129C2×D4×C31—28no—
130C2×D4×C31—28no—
131C2×D4×C31—22no—
132C2×D4×C31—28no—
133C2×D4×C31—22no—
134C2×D4×C31—22no—
135C2×D4×C31C2×D412yes—
136C2×D4×C31C2×Q812yes—
137C2×D4×C31#4112yes—
138C22×C4×C31—22no—
139C22×C4×C31—22no—
140C22×C4×C31—21no—
141C22×C4×C31—22no—
142C22×C4×C31—24no—
143C22×C4×C31—22no—
144C22×C4×C31—22no—
145C22×C4×C31—22no—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—21no—
149C22×C4×C31—21no—
150C22×C4×C31—21no—
151C22×C4×C31—82no—
152C22×C4×C31—84no—
153C22×C4×C31C2×D412yes—
154C22×C4×C31C2×D411yescharacteristic
155C22×C4×C31C2×D411yescharacteristic
156C22×C4×C31C2411yescharacteristic
157M16×C31—22no—
158M16×C31—22no—
159M16×C31—22no—
160M16×C31—22no—
161C2×C8×C31—22no—
162C2×C8×C31—22no—
163C4⋊C4×C31—24no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—22no—
169C4⋊C4×C31—42no—
170C4⋊C4×C31—41no—
171C4⋊C4×C31—42no—
172C4⋊C4×C31—41no—
173C4⋊C4×C31—82no—
174C4⋊C4×C31C2×D412yes—
175C4⋊C4×C31C2×D412yes—
176C22⋊C4×C31—24no—
177C22⋊C4×C31—24no—
178C22⋊C4×C31—22no—
179C22⋊C4×C31—22no—
180C22⋊C4×C31—22no—
181C22⋊C4×C31—24no—
182C22⋊C4×C31—44no—
183C22⋊C4×C31—82no—
184C42×C31—24no—
185C42×C31—22no—
186C42×C31—22no—
187C42×C31—22no—
188C42×C31—22no—
189C42×C31—22no—
190C42×C31C2×D412yes—
191C42×C31C2×D412yes—
192#32973C3111yesmaximal, characteristic, direct product factor, Sylow
193C23×C31—28no—
194C23×C31—22no—
195C23×C31—22no—
196C23×C31—22no—
197C23×C31—42no—
198C23×C31—42no—
199C23×C31—82no—
200C23×C31—82no—
201C23×C31C22⋊Q814yes—
202C23×C31C22×D411yescharacteristic
203Q8×C31—22no—
204Q8×C31C22⋊Q814yes—
205D4×C31—216no—
206D4×C31—24no—
207D4×C31—24no—
208D4×C31—24no—
209D4×C31—22no—
210D4×C31—24no—
211D4×C31C22⋊Q814yes—
212C2×C4×C31—28no—
213C2×C4×C31—22no—
214C2×C4×C31—24no—
215C2×C4×C31—24no—
216C2×C4×C31—22no—
217C2×C4×C31—22no—
218C2×C4×C31—22no—
219C2×C4×C31—22no—
220C2×C4×C31—41no—
221C2×C4×C31—41no—
222C2×C4×C31—44no—
223C2×C4×C31—44no—
224C2×C4×C31—41no—
225C2×C4×C31—41no—
226C2×C4×C31—42no—
227C2×C4×C31—42no—
228C2×C4×C31—41no—
229C2×C4×C31—42no—
230C2×C4×C31—41no—
231C2×C4×C31—41no—
232C2×C4×C31—41no—
233C2×C4×C31—41no—
234C2×C4×C31—82no—
235C2×C4×C31—82no—
236C2×C4×C31—82no—
237C2×C4×C31—84no—
238C2×C4×C31—82no—
239C2×C4×C31#12011yescharacteristic
240C2×C4×C31#12012yes—
241C2×C4×C31#12011yescharacteristic
242C2×C4×C31C22×D412yes—
243C2×C4×C31C22×Q811yescharacteristic
244C2×C4×C31C2×#4112yes—
245C2×C4×C31C2×#4111yescharacteristic
246C8×C31—42no—
247C8×C31—42no—
248C4×H(5,𝔽2)C2×C3111yescharacteristic
249#3080C2×C3112yescomplemented
250#2566C2×C3111yescharacteristic, complemented
251#1860C2×C3112yescomplemented
252#1859C2×C3112yescomplemented
253#1856C2×C3111yescharacteristic, complemented
254#1856C2×C3111yescharacteristic, complemented
255#1806C2×C3112yescomplemented
256#1800C2×C3111yescharacteristic, complemented
257#1792C2×C3111yescharacteristic
258#1792C2×C3111yescharacteristic
259C22×C31—21no—
260C22×C31—21no—
261C22×C31—24no—
262C22×C31—24no—
263C22×C31—44no—
264C22×C31—42no—
265C22×C31—42no—
266C22×C31—42no—
267C22×C31—42no—
268C22×C31—82no—
269C22×C31—88no—
270C22×C31C2×#12011yescharacteristic
271C22×C31C2×C22⋊Q811yescharacteristic
272C22×C31#52511yescharacteristic
273C4×C31—24no—
274C4×C31—22no—
275C4×C31—21no—
276C4×C31—21no—
277C4×C31—42no—
278C4×C31—42no—
279C4×C31—42no—
280C4×C31—42no—
281C4×C31—81no—
282C4×C31—81no—
283C4×C31—81no—
284C4×C31C2×C22⋊Q812yes—
285C4×C31#52512yes—
286C2×H(5,𝔽2)C22×C3111yescharacteristic
287C2×#125C22×C3111yescharacteristic, complemented
288#520—22no—
289#520C22×C3112yes—
290#518—22no—
291#518—22no—
292#518C22×C3111yescharacteristic
293C4×#41—22no—
294C4×#41C22×C3112yes—
295C2×C4×D4—22no—
296C2×C4×D4—22no—
297C2×C4×D4C22×C3111yescharacteristic
298#479—22no—
299#479—22no—
300#478—22no—
301#478—22no—
302#428C22×C3112yescomplemented
303#428C22×C3112yescomplemented
304#423C22×C3112yescomplemented
305C2×#104C22×C3112yes—
306C2×#104C22×C3112yes—
307#417C22×C3112yes—
308#417C22×C3112yes—
309C2×#100C22×C3111yescharacteristic
310C2×#100C22×C3111yescharacteristic
311#396—22no—
312#390C22×C3111yescharacteristic
313#390C22×C3111yescharacteristic
314#385—22no—
315#384C22×C3111yescharacteristic, complemented
316#382C22×C3111yescharacteristic, complemented
317C4×C4⋊C4C22×C3111yescharacteristic, complemented
318#343C22×C3112yescomplemented
319#339C22×C3112yescomplemented
320#339C22×C3112yescomplemented
321C2×C31—21no—
322C2×C31—21no—
323C2×C31—44no—
324C2×C31—44no—
325C2×C31#284811yescharacteristic
326C2×C31#301811yescharacteristic
327C2×C31#301911yescharacteristic
328H(5,𝔽2)—24no—
329H(5,𝔽2)—22no—
330H(5,𝔽2)Q8×C3114yes—
331C2×#41—22no—
332C2×#41C23×C3112yes—
333C22×D4—22no—
334C22×D4—22no—
335C22×D4C23×C3111yescharacteristic
336C23×C4—22no—
337C23×C4D4×C3112yes—
338C2×M16C23×C3112yes—
339C2×M16C23×C3112yes—
340#125—22no—
341#125—22no—
342C4×Q8—22no—
343C4×Q8D4×C3112yes—
344C4×Q8D4×C3112yes—
345C4×D4—24no—
346C4×D4—24no—
347C4×D4—24no—
348C4×D4—24no—
349C4×D4—22no—
350C4×D4—22no—
351C4×D4—24no—
352C4×D4—22no—
353C4×D4—22no—
354C4×D4—22no—
355C4×D4—22no—
356C4×D4—22no—
357C4×D4D4×C3112yes—
358C4×D4D4×C3112yes—
359#117—24no—
360#117—22no—
361#117—22no—
362#117C23×C3112yes—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4C23×C3111yescharacteristic
370C2×C4⋊C4C23×C3111yescharacteristic
371C2×C4⋊C4C23×C3111yescharacteristic
372C2×C22⋊C4—22no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4D4×C3112yes—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42C23×C3111yescharacteristic
381C2×C42C23×C3111yescharacteristic
382C2×C42C23×C3111yescharacteristic
383#107—22no—
384#107—22no—
385#106—22no—
386#106—22no—
387C4⋊C8—22no—
388C4⋊C8—22no—
389#104—24no—
390#104—24no—
391#103—22no—
392#103—22no—
393#102—22no—
394#102—22no—
395#100—22no—
396#100—22no—
397#95—21no—
398#95—22no—
399C31#3297311yescharacteristic, direct product factor, Sylow
400C24—22no—
401C24C2×D4×C3112yes—
402#41—28no—
403#41—28no—
404#41—22no—
405#41—22no—
406#41C2×Q8×C3114yes—
407#41#41×C3114yes—
408C2×Q8C2×D4×C3112yes—
409C2×D4—24no—
410C2×D4—28no—
411C2×D4—28no—
412C2×D4—28no—
413C2×D4—22no—
414C2×D4—28no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4C2×D4×C3112yes—
418C2×D4C2×Q8×C3112yes—
419C2×D4#41×C3112yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—82no—
434C22×C4—84no—
435C22×C4C2×D4×C3112yes—
436C22×C4C2×D4×C3111yescharacteristic
437C22×C4C2×D4×C3111yescharacteristic
438C22×C4C24×C3111yescharacteristic
439M16—22no—
440M16—22no—
441M16—22no—
442M16—22no—
443C2×C8—22no—
444C2×C8—22no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—42no—
452C4⋊C4—41no—
453C4⋊C4—42no—
454C4⋊C4—41no—
455C4⋊C4—82no—
456C4⋊C4C2×D4×C3112yes—
457C4⋊C4C2×D4×C3112yes—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—24no—
464C22⋊C4—44no—
465C22⋊C4—82no—
466C42—24no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42C2×D4×C3112yes—
473C42C2×D4×C3112yes—
474C23—28no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—42no—
479C23—42no—
480C23—82no—
481C23—82no—
482C23C31×C22⋊Q814yes—
483C23C22×D4×C3111yescharacteristic
484Q8—22no—
485Q8C31×C22⋊Q814yes—
486D4—216no—
487D4—24no—
488D4—24no—
489D4—24no—
490D4—22no—
491D4—24no—
492D4C31×C22⋊Q814yes—
493C2×C4—28no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—84no—
519C2×C4—82no—
520C2×C4C31×#12011yescharacteristic
521C2×C4C31×#12012yes—
522C2×C4C31×#12011yescharacteristic
523C2×C4C22×D4×C3112yes—
524C2×C4C22×Q8×C3111yescharacteristic
525C2×C4C2×#41×C3112yes—
526C2×C4C2×#41×C3111yescharacteristic
527C8—42no—
528C8—42no—
529C22—21no—
530C22—21no—
531C22—24no—
532C22—24no—
533C22—44no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—82no—
539C22—88no—
540C22C2×C31×#12011yescharacteristic
541C22C2×C31×C22⋊Q811yescharacteristic
542C22C31×#52511yescharacteristic
543C4—24no—
544C4—22no—
545C4—21no—
546C4—21no—
547C4—42no—
548C4—42no—
549C4—42no—
550C4—42no—
551C4—81no—
552C4—81no—
553C4—81no—
554C4C2×C31×C22⋊Q812yes—
555C4C31×#52512yes—
556C2—21no—
557C2—21no—
558C2—44no—
559C2—44no—
560C2C31×#284811yescharacteristic
561C2C31×#301811yescharacteristic
562C2C31×#301911yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×H(5,𝔽2)C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3080C2semidirect
3. C31×#2566C2semidirect
4. C31×#1860C2semidirect
5. C31×#1859C2semidirect
6. C31×#1856C2semidirect
7. C31×#1856C2semidirect
8. C31×#1806C2semidirect
9. C31×#1800C2semidirect
13. C2×C31×#125C22semidirect
28. C31×#428C22semidirect
29. C31×#428C22semidirect
30. C31×#423C22semidirect
41. C31×#384C22semidirect
42. C31×#382C22semidirect
43. C4×C4⋊C4×C31C22semidirect
44. C31×#343C22semidirect
45. C31×#339C22semidirect
46. C31×#339C22semidirect
192. #32973C31direct (N × H)
249. #3080C2×C31semidirect
250. #2566C2×C31semidirect
251. #1860C2×C31semidirect
252. #1859C2×C31semidirect
253. #1856C2×C31semidirect
254. #1856C2×C31semidirect
255. #1806C2×C31semidirect
256. #1800C2×C31semidirect
287. C2×#125C22×C31semidirect
302. #428C22×C31semidirect
303. #428C22×C31semidirect
304. #423C22×C31semidirect
315. #384C22×C31semidirect
316. #382C22×C31semidirect
317. C4×C4⋊C4C22×C31semidirect
318. #343C22×C31semidirect
319. #339C22×C31semidirect
320. #339C22×C31semidirect
399. C31#32973direct (N × H)

Maximal subgroups

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

subgroupindex
C4×C31×H(5,𝔽2)2
C31×#30802
C31×#25662
C31×#18602
C31×#18592
C31×#18562
C31×#18062
C31×#18002
C31×#17922
#3297331