Finite Group Databasebeta

← C31×#10951 #21467710

Subgroups — C31×#10951 #21467710

Order 7936.

916 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#421C211yesmaximal, characteristic
2C31×#2896C211yesmaximal, characteristic
3C2×C31×#410C211yesmaximal, characteristic
4C31×#2873C211yesmaximal, characteristic, complemented
5C31×#1740C211yesmaximal, characteristic, complemented
6C31×#1733C211yesmaximal, characteristic, complemented
7C2×C31×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
8C22×#41×C31C2211yescharacteristic
9C22×M16×C31C2211yescharacteristic
10C22×M16×C31C2211yescharacteristic
11C2×C31×#117C2211yescharacteristic
12C2×C31×#117C2211yescharacteristic
13C2×C31×#117C2211yescharacteristic
14C2×C31×#117C2211yescharacteristic
15C31×#421—22no—
16C31×#421—22no—
17C2×C31×#104C414yescomplemented
18C31×#419C412yescomplemented
19C31×#418C412yescomplemented
20C31×#417—22no—
21C31×#416—22no—
22C31×#410—22no—
23C31×#410C414yescomplemented
24C2×C31×#99—22no—
25C31×#405—22no—
26C31×#405—22no—
27C4×M16×C31—22no—
28C31×#342—22no—
29C31×#339—22no—
30C31×H(3,ℤ/4)—22no—
31C2×#41×C31—21no—
32C2×#41×C31—21no—
33C2×#41×C31—21no—
34C2×#41×C31—21no—
35C2×#41×C31—21no—
36C2×#41×C31—21no—
37C2×#41×C31—21no—
38C2×#41×C31—21no—
39C2×#41×C31—21no—
40C2×#41×C31—21no—
41C2×#41×C31C2×C412yes—
42C2×#41×C31C2×C412yes—
43C22×Q8×C31C2×C411yescharacteristic
44C22×D4×C31—21no—
45C22×D4×C31C2×C411yescharacteristic
46C23×C4×C31—21no—
47C23×C4×C31C2311yescharacteristic
48C2×M16×C31—21no—
49C2×M16×C31—21no—
50C2×M16×C31—21no—
51C2×M16×C31—21no—
52C2×M16×C31—21no—
53C2×M16×C31—21no—
54C2×M16×C31—21no—
55C2×M16×C31C2×C412yes—
56C2×M16×C31C2×C412yes—
57C2×M16×C31D412yes—
58C2×M16×C31D412yes—
59C2×M16×C31D412yes—
60C22×C8×C31—21no—
61C22×C8×C31D412yes—
62C31×#117—21no—
63C31×#117—21no—
64C31×#117—21no—
65C31×#117—21no—
66C31×#117—21no—
67C31×#117—21no—
68C31×#117—21no—
69C31×#117—21no—
70C31×#117—21no—
71C31×#117—21no—
72C31×#117—21no—
73C31×#117—21no—
74C31×#117C2×C412yes—
75C31×#117C2×C412yes—
76C31×#117C2×C412yes—
77C31×#117D412yes—
78C2×C4⋊C4×C31—21no—
79C2×C4⋊C4×C31—21no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31—21no—
82C2×C22⋊C4×C31—21no—
83C2×C22⋊C4×C31—21no—
84C2×C22⋊C4×C31—21no—
85C2×C22⋊C4×C31D412yes—
86C2×C42×C31—21no—
87C2×C42×C31—21no—
88C2×C42×C31—21no—
89C2×C42×C31C2×C412yes—
90C31×#104—28no—
91C31×#103—24no—
92C31×#102—24no—
93C31×#99—28no—
94C31×#97—24no—
95C4×C8×C31—24no—
96C24×C31—21no—
97C24×C31C2×D411yescharacteristic
98#41×C31—22no—
99#41×C31—22no—
100#41×C31—22no—
101#41×C31—22no—
102#41×C31—22no—
103#41×C31—22no—
104#41×C31—22no—
105#41×C31—22no—
106#41×C31—41no—
107#41×C31—41no—
108#41×C31—41no—
109#41×C31—41no—
110#41×C31—41no—
111#41×C31—41no—
112#41×C31—41no—
113#41×C31—41no—
114C2×Q8×C31—21no—
115C2×Q8×C31—21no—
116C2×Q8×C31—21no—
117C2×Q8×C31—21no—
118C2×Q8×C31—21no—
119C2×Q8×C31C4212yes—
120C2×D4×C31—22no—
121C2×D4×C31—22no—
122C2×D4×C31—21no—
123C2×D4×C31—21no—
124C2×D4×C31—21no—
125C2×D4×C31—21no—
126C2×D4×C31—21no—
127C2×D4×C31—41no—
128C2×D4×C31—41no—
129C2×D4×C31—41no—
130C2×D4×C31—41no—
131C2×D4×C31C4212yes—
132C22×C4×C31—21no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—41no—
138C22×C4×C31—41no—
139C22×C4×C31—41no—
140C22×C4×C31—41no—
141C22×C4×C31—41no—
142C22×C4×C31—41no—
143C22×C4×C31C22×C411yescharacteristic
144C22×C4×C31C22×C411yescharacteristic
145C22×C4×C31C22×C411yescharacteristic
146C22×C4×C31C2×D411yescharacteristic
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31C2×D411yescharacteristic
149C22×C4×C31C2×D411yescharacteristic
150C22×C4×C31C2×D411yescharacteristic
151C22×C4×C31#4111yescharacteristic
152C22×C4×C31#4111yescharacteristic
153C22×C4×C31#4111yescharacteristic
154C22×C4×C31#4111yescharacteristic
155C22×C4×C31#4111yescharacteristic
156C22×C4×C31#4111yescharacteristic
157M16×C31—22no—
158M16×C31—22no—
159M16×C31—22no—
160M16×C31—22no—
161M16×C31—22no—
162M16×C31—22no—
163M16×C31—22no—
164M16×C31C22⋊C414yes—
165C2×C8×C31—22no—
166C2×C8×C31—22no—
167C2×C8×C31—22no—
168C2×C8×C31—22no—
169C2×C8×C31—22no—
170C2×C8×C31C22⋊C414yes—
171C4⋊C4×C31—22no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—22no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—41no—
176C4⋊C4×C31—41no—
177C4⋊C4×C31—41no—
178C4⋊C4×C31—41no—
179C22⋊C4×C31—22no—
180C22⋊C4×C31—22no—
181C22⋊C4×C31—22no—
182C22⋊C4×C31—41no—
183C22⋊C4×C31—41no—
184C22⋊C4×C31—41no—
185C22⋊C4×C31—41no—
186C22⋊C4×C31C22⋊C414yes—
187C42×C31—22no—
188C42×C31—22no—
189C42×C31—22no—
190C42×C31—22no—
191C42×C31—22no—
192C42×C31—22no—
193C42×C31—41no—
194C42×C31—41no—
195#10951C3111yesmaximal, characteristic, direct product factor, Sylow
196C23×C31—22no—
197C23×C31—41no—
198C23×C31—41no—
199C23×C31—81no—
200C23×C31—82no—
201C23×C31C2×C22⋊C411yescharacteristic
202C23×C31C4×D411yescharacteristic
203C23×C31C4×D411yescharacteristic
204C23×C31#12111yescharacteristic
205C23×C31#12111yescharacteristic
206C23×C31#12411yescharacteristic
207C23×C31#12711yescharacteristic
208Q8×C31—22no—
209Q8×C31—22no—
210Q8×C31—41no—
211Q8×C31—41no—
212D4×C31—22no—
213D4×C31—22no—
214D4×C31—42no—
215D4×C31—42no—
216D4×C31—42no—
217D4×C31—42no—
218D4×C31—41no—
219D4×C31—41no—
220C2×C4×C31—22no—
221C2×C4×C31—24no—
222C2×C4×C31—41no—
223C2×C4×C31—41no—
224C2×C4×C31—42no—
225C2×C4×C31—42no—
226C2×C4×C31—42no—
227C2×C4×C31—42no—
228C2×C4×C31—42no—
229C2×C4×C31—42no—
230C2×C4×C31—42no—
231C2×C4×C31—82no—
232C2×C4×C31—82no—
233C2×C4×C31—82no—
234C2×C4×C31—82no—
235C2×C4×C31C2×C4211yescharacteristic
236C2×C4×C31C2×C22⋊C411yescharacteristic
237C2×C4×C31C2×C22⋊C411yescharacteristic
238C2×C4×C31#11711yescharacteristic
239C2×C4×C31#11711yescharacteristic
240C2×C4×C31#11711yescharacteristic
241C2×C4×C31C4×D411yescharacteristic
242C2×C4×C31C4×D411yescharacteristic
243C2×C4×C31C4×D411yescharacteristic
244C2×C4×C31C4×D411yescharacteristic
245C2×C4×C31C4×D411yescharacteristic
246C2×C4×C31C4×D411yescharacteristic
247C2×C4×C31C4×D411yescharacteristic
248C2×C4×C31C4×D411yescharacteristic
249C2×C4×C31C4×D411yescharacteristic
250C2×C4×C31C4×D411yescharacteristic
251C2×C4×C31#12011yescharacteristic
252C2×C4×C31#12111yescharacteristic
253C2×C4×C31#12111yescharacteristic
254C2×C4×C31#12111yescharacteristic
255C2×C4×C31#12111yescharacteristic
256C2×C4×C31#12311yescharacteristic
257C2×C4×C31#12311yescharacteristic
258C2×C4×C31#12311yescharacteristic
259C2×C4×C31#12411yescharacteristic
260C2×C4×C31#12411yescharacteristic
261C2×C4×C31#12611yescharacteristic
262C2×C4×C31#12611yescharacteristic
263C8×C31—24no—
264C8×C31—24no—
265C2×#421C2×C3111yescharacteristic
266#2896C2×C3111yescharacteristic
267C2×#410C2×C3111yescharacteristic
268#2873C2×C3111yescharacteristic, complemented
269#1740C2×C3111yescharacteristic, complemented
270#1733C2×C3111yescharacteristic, complemented
271C2×H(3,ℤ/4)C2×C3111yescharacteristic, complemented
272C22×C31—41no—
273C22×C31—41no—
274C22×C31—41no—
275C22×C31—41no—
276C22×C31—41no—
277C22×C31—41no—
278C22×C31—41no—
279C22×C31—42no—
280C22×C31—82no—
281C22×C31—82no—
282C22×C31—82no—
283C22×C31C4×C22⋊C411yescharacteristic
284C22×C31#38611yescharacteristic
285C22×C31#38811yescharacteristic
286C22×C31#38811yescharacteristic
287C22×C31#39011yescharacteristic
288C22×C31#39011yescharacteristic
289C22×C31#39011yescharacteristic
290C4×C31—44no—
291C4×C31—42no—
292C4×C31—82no—
293C4×C31—82no—
294C4×C31—82no—
295C4×C31C4×C22⋊C411yescharacteristic
296C4×C31C4×C22⋊C411yescharacteristic
297C4×C31#38611yescharacteristic
298C4×C31#38611yescharacteristic
299C4×C31#38811yescharacteristic
300C4×C31#38811yescharacteristic
301C4×C31#38811yescharacteristic
302C4×C31#38811yescharacteristic
303C22×#41C22×C3111yescharacteristic
304C22×M16C22×C3111yescharacteristic
305C22×M16C22×C3111yescharacteristic
306C2×#117C22×C3111yescharacteristic
307C2×#117C22×C3111yescharacteristic
308C2×#117C22×C3111yescharacteristic
309C2×#117C22×C3111yescharacteristic
310#421—22no—
311#421—22no—
312C2×#104C4×C3114yescomplemented
313#419C4×C3112yescomplemented
314#418C4×C3112yescomplemented
315#417—22no—
316#416—22no—
317#410—22no—
318#410C4×C3114yescomplemented
319C2×#99—22no—
320#405—22no—
321#405—22no—
322C4×M16—22no—
323#342—22no—
324#339—22no—
325H(3,ℤ/4)—22no—
326C2×C31—21no—
327C2×C31—21no—
328C2×C31—21no—
329C2×C31—21no—
330C2×C31—21no—
331C2×C31—21no—
332C2×C31—21no—
333C2×C31—82no—
334C2×C31#143711yescharacteristic
335C2×#41—21no—
336C2×#41—21no—
337C2×#41—21no—
338C2×#41—21no—
339C2×#41—21no—
340C2×#41—21no—
341C2×#41—21no—
342C2×#41—21no—
343C2×#41—21no—
344C2×#41—21no—
345C2×#41C2×C4×C3112yes—
346C2×#41C2×C4×C3112yes—
347C22×Q8C2×C4×C3111yescharacteristic
348C22×D4—21no—
349C22×D4C2×C4×C3111yescharacteristic
350C23×C4—21no—
351C23×C4C23×C3111yescharacteristic
352C2×M16—21no—
353C2×M16—21no—
354C2×M16—21no—
355C2×M16—21no—
356C2×M16—21no—
357C2×M16—21no—
358C2×M16—21no—
359C2×M16C2×C4×C3112yes—
360C2×M16C2×C4×C3112yes—
361C2×M16D4×C3112yes—
362C2×M16D4×C3112yes—
363C2×M16D4×C3112yes—
364C22×C8—21no—
365C22×C8D4×C3112yes—
366#117—21no—
367#117—21no—
368#117—21no—
369#117—21no—
370#117—21no—
371#117—21no—
372#117—21no—
373#117—21no—
374#117—21no—
375#117—21no—
376#117—21no—
377#117—21no—
378#117C2×C4×C3112yes—
379#117C2×C4×C3112yes—
380#117C2×C4×C3112yes—
381#117D4×C3112yes—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4D4×C3112yes—
390C2×C42—21no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42C2×C4×C3112yes—
394#104—28no—
395#103—24no—
396#102—24no—
397#99—28no—
398#97—24no—
399C4×C8—24no—
400C31#1095111yescharacteristic, direct product factor, Sylow
401C24—21no—
402C24C2×D4×C3111yescharacteristic
403#41—22no—
404#41—22no—
405#41—22no—
406#41—22no—
407#41—22no—
408#41—22no—
409#41—22no—
410#41—22no—
411#41—41no—
412#41—41no—
413#41—41no—
414#41—41no—
415#41—41no—
416#41—41no—
417#41—41no—
418#41—41no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8C42×C3112yes—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—41no—
433C2×D4—41no—
434C2×D4—41no—
435C2×D4—41no—
436C2×D4C42×C3112yes—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4C22×C4×C3111yescharacteristic
449C22×C4C22×C4×C3111yescharacteristic
450C22×C4C22×C4×C3111yescharacteristic
451C22×C4C2×D4×C3111yescharacteristic
452C22×C4C2×D4×C3111yescharacteristic
453C22×C4C2×D4×C3111yescharacteristic
454C22×C4C2×D4×C3111yescharacteristic
455C22×C4C2×D4×C3111yescharacteristic
456C22×C4#41×C3111yescharacteristic
457C22×C4#41×C3111yescharacteristic
458C22×C4#41×C3111yescharacteristic
459C22×C4#41×C3111yescharacteristic
460C22×C4#41×C3111yescharacteristic
461C22×C4#41×C3111yescharacteristic
462M16—22no—
463M16—22no—
464M16—22no—
465M16—22no—
466M16—22no—
467M16—22no—
468M16—22no—
469M16C22⋊C4×C3114yes—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8C22⋊C4×C3114yes—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—41no—
491C22⋊C4C22⋊C4×C3114yes—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—41no—
499C42—41no—
500C23—22no—
501C23—41no—
502C23—41no—
503C23—81no—
504C23—82no—
505C23C2×C22⋊C4×C3111yescharacteristic
506C23C4×D4×C3111yescharacteristic
507C23C4×D4×C3111yescharacteristic
508C23C31×#12111yescharacteristic
509C23C31×#12111yescharacteristic
510C23C31×#12411yescharacteristic
511C23C31×#12711yescharacteristic
512Q8—22no—
513Q8—22no—
514Q8—41no—
515Q8—41no—
516D4—22no—
517D4—22no—
518D4—42no—
519D4—42no—
520D4—42no—
521D4—42no—
522D4—41no—
523D4—41no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4C2×C42×C3111yescharacteristic
540C2×C4C2×C22⋊C4×C3111yescharacteristic
541C2×C4C2×C22⋊C4×C3111yescharacteristic
542C2×C4C31×#11711yescharacteristic
543C2×C4C31×#11711yescharacteristic
544C2×C4C31×#11711yescharacteristic
545C2×C4C4×D4×C3111yescharacteristic
546C2×C4C4×D4×C3111yescharacteristic
547C2×C4C4×D4×C3111yescharacteristic
548C2×C4C4×D4×C3111yescharacteristic
549C2×C4C4×D4×C3111yescharacteristic
550C2×C4C4×D4×C3111yescharacteristic
551C2×C4C4×D4×C3111yescharacteristic
552C2×C4C4×D4×C3111yescharacteristic
553C2×C4C4×D4×C3111yescharacteristic
554C2×C4C4×D4×C3111yescharacteristic
555C2×C4C31×#12011yescharacteristic
556C2×C4C31×#12111yescharacteristic
557C2×C4C31×#12111yescharacteristic
558C2×C4C31×#12111yescharacteristic
559C2×C4C31×#12111yescharacteristic
560C2×C4C31×#12311yescharacteristic
561C2×C4C31×#12311yescharacteristic
562C2×C4C31×#12311yescharacteristic
563C2×C4C31×#12411yescharacteristic
564C2×C4C31×#12411yescharacteristic
565C2×C4C31×#12611yescharacteristic
566C2×C4C31×#12611yescharacteristic
567C8—24no—
568C8—24no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—42no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22C4×C22⋊C4×C3111yescharacteristic
581C22C31×#38611yescharacteristic
582C22C31×#38811yescharacteristic
583C22C31×#38811yescharacteristic
584C22C31×#39011yescharacteristic
585C22C31×#39011yescharacteristic
586C22C31×#39011yescharacteristic
587C4—44no—
588C4—42no—
589C4—82no—
590C4—82no—
591C4—82no—
592C4C4×C22⋊C4×C3111yescharacteristic
593C4C4×C22⋊C4×C3111yescharacteristic
594C4C31×#38611yescharacteristic
595C4C31×#38611yescharacteristic
596C4C31×#38811yescharacteristic
597C4C31×#38811yescharacteristic
598C4C31×#38811yescharacteristic
599C4C31×#38811yescharacteristic
600C2—21no—
601C2—21no—
602C2—21no—
603C2—21no—
604C2—21no—
605C2—21no—
606C2—21no—
607C2—82no—
608C2C31×#143711yescharacteristic
subgroupG/H
centerC4×C31#386
commutatorC2×C4C2×C42×C31
Frattini, cosocleC23×C4C23×C31
socleC2×C31#1437
prime radicalC22×#41×C31C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
4. C31×#2873C2semidirect
5. C31×#1740C2semidirect
6. C31×#1733C2semidirect
7. C2×C31×H(3,ℤ/4)C2semidirect
17. C2×C31×#104C4semidirect
18. C31×#419C4semidirect
19. C31×#418C4semidirect
23. C31×#410C4semidirect
195. #10951C31direct (N × H)
268. #2873C2×C31semidirect
269. #1740C2×C31semidirect
270. #1733C2×C31semidirect
271. C2×H(3,ℤ/4)C2×C31semidirect
312. C2×#104C4×C31semidirect
313. #419C4×C31semidirect
314. #418C4×C31semidirect
318. #410C4×C31semidirect
400. C31#10951direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4212
C31×#28962
C2×C31×#4102
C31×#28732
C31×#17402
C31×#17332
C2×C31×H(3,ℤ/4)2
#1095131