Finite Group Databasebeta

← C31×#24013 #21480772

Subgroups — C31×#24013 #21480772

Order 7936.

1468 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#535C211yesmaximal, characteristic
2C2×C31×#481C211yesmaximal, characteristic, complemented
3C2×C31×#480C211yesmaximal, characteristic, complemented
4C31×#2311C211yesmaximal, characteristic, complemented
5C31×#2034C212yesmaximal, complemented
6C31×#2033C212yesmaximal, complemented
7C31×#1875C211yesmaximal, characteristic, complemented
8C31×#1875C211yesmaximal, characteristic, complemented
9C31×#1854C212yesmaximal, complemented
10C31×#1786C212yesmaximal, complemented
11C31×#1780C211yesmaximal, characteristic, complemented
12C22×#41×C31C2211yescharacteristic
13C23×D4×C31C2211yescharacteristic
14C31×#535—24no—
15C31×#535—24no—
16C2×C31×#127—21no—
17C2×C31×#124—21no—
18C2×C31×#121C2211yescharacteristic
19C2×C31×#121C2211yescharacteristic
20C2×C31×#121C2211yescharacteristic
21C2×C31×#121C2211yescharacteristic
22C2×C31×#120—21no—
23C2×C31×#120—21no—
24C2×C31×#117C2211yescharacteristic
25C2×C31×#117C2211yescharacteristic, complemented
26C22×C4⋊C4×C31C2211yescharacteristic, complemented
27C31×#481—24no—
28C31×#480—24no—
29C2×C31×#107C2212yescomplemented
30C2×C31×#106C2212yescomplemented
31C2×C31×#102C2212yes—
32C2×C31×#102C2212yes—
33C2×C31×#102C2212yes—
34C2×C31×#102C2212yes—
35C2×C31×C22⋊C8C2212yes—
36C2×C31×C22⋊C8C2211yescharacteristic, complemented
37C2×C31×C22⋊C8C2211yescharacteristic, complemented
38C2×C31×C22⋊C8C2212yescomplemented
39C31×#391—21no—
40C31×#390—21no—
41C31×#386—21no—
42C31×#386—21no—
43C31×#380C2212yes—
44C31×#380C2212yescomplemented
45C31×#340C2212yescomplemented
46C31×#340C2212yescomplemented
47C25×C31—21no—
48C2×#41×C31—22no—
49C2×#41×C31—22no—
50C2×#41×C31—24no—
51C2×#41×C31—42no—
52C22×Q8×C31D411yescharacteristic
53C22×D4×C31—21no—
54C22×D4×C31—21no—
55C22×D4×C31—21no—
56C22×D4×C31—21no—
57C22×D4×C31—22no—
58C22×D4×C31—22no—
59C22×D4×C31—21no—
60C22×D4×C31—24no—
61C22×D4×C31—42no—
62C22×D4×C31D411yescharacteristic
63C22×D4×C31D411yescharacteristic
64C22×D4×C31D411yescharacteristic
65C22×D4×C31C2311yescharacteristic
66C22×D4×C31C2311yescharacteristic
67C23×C4×C31—21no—
68C23×C4×C31C2311yescharacteristic
69C23×C4×C31C2311yescharacteristic
70C23×C4×C31C2311yescharacteristic
71C22×C8×C31C2312yes—
72C22×C8×C31C2312yes—
73C31×#127—28no—
74C31×#124—28no—
75C31×#121—22no—
76C31×#121—22no—
77C31×#121—22no—
78C31×#121—22no—
79C31×#121—22no—
80C31×#121—22no—
81C31×#121D414yes—
82C31×#121D414yes—
83C31×#120—28no—
84C31×#120—28no—
85C31×#117—22no—
86C31×#117—22no—
87C31×#117—42no—
88C2×C4⋊C4×C31—22no—
89C2×C4⋊C4×C31—22no—
90C2×C4⋊C4×C31—21no—
91C2×C4⋊C4×C31C2312yes—
92C2×C4⋊C4×C31C2312yes—
93C2×C4⋊C4×C31C2311yescharacteristic
94C2×C4⋊C4×C31C2311yescharacteristic
95C2×C22⋊C4×C31—21no—
96C2×C22⋊C4×C31—21no—
97C2×C22⋊C4×C31—21no—
98C2×C22⋊C4×C31—21no—
99C2×C22⋊C4×C31—21no—
100C2×C22⋊C4×C31—21no—
101C2×C22⋊C4×C31—22no—
102C2×C22⋊C4×C31—22no—
103C2×C42×C31—21no—
104C2×C42×C31—21no—
105C31×#107—24no—
106C31×#106—24no—
107C31×#102—24no—
108C31×#102—24no—
109C31×#102—42no—
110C31×#102—42no—
111C31×C22⋊C8—22no—
112C31×C22⋊C8—22no—
113C31×C22⋊C8—24no—
114C31×C22⋊C8—42no—
115C31×#95—22no—
116C31×#95—22no—
117C24×C31—24no—
118C24×C31—21no—
119C24×C31—21no—
120C24×C31—42no—
121C24×C31—42no—
122C24×C31—82no—
123C24×C31—82no—
124C24×C31C2×D411yescharacteristic
125C24×C31C2×D411yescharacteristic
126C24×C31C2×D411yescharacteristic
127#41×C31—48no—
128#41×C31—48no—
129C2×Q8×C31—22no—
130C2×Q8×C31—22no—
131C2×Q8×C31—41no—
132C2×D4×C31—24no—
133C2×D4×C31—22no—
134C2×D4×C31—22no—
135C2×D4×C31—22no—
136C2×D4×C31—22no—
137C2×D4×C31—24no—
138C2×D4×C31—24no—
139C2×D4×C31—24no—
140C2×D4×C31—24no—
141C2×D4×C31—22no—
142C2×D4×C31—22no—
143C2×D4×C31—21no—
144C2×D4×C31—21no—
145C2×D4×C31—22no—
146C2×D4×C31—22no—
147C2×D4×C31—22no—
148C2×D4×C31—22no—
149C2×D4×C31—48no—
150C2×D4×C31—48no—
151C2×D4×C31—41no—
152C2×D4×C31—41no—
153C2×D4×C31—41no—
154C2×D4×C31—42no—
155C2×D4×C31—42no—
156C2×D4×C31—82no—
157C2×D4×C31—82no—
158C2×D4×C31—82no—
159C2×D4×C31—82no—
160C2×D4×C31C2×D412yes—
161C2×D4×C31C2×D412yes—
162C22×C4×C31—21no—
163C22×C4×C31—21no—
164C22×C4×C31—22no—
165C22×C4×C31—22no—
166C22×C4×C31—22no—
167C22×C4×C31—21no—
168C22×C4×C31—21no—
169C22×C4×C31—21no—
170C22×C4×C31—21no—
171C22×C4×C31—22no—
172C22×C4×C31—22no—
173C22×C4×C31—22no—
174C22×C4×C31—22no—
175C22×C4×C31—42no—
176C22×C4×C31—82no—
177C22×C4×C31C2×D411yescharacteristic
178C22×C4×C31C2×D411yescharacteristic
179C22×C4×C31C2×D412yes—
180C22×C4×C31C2×D412yes—
181C22×C4×C31C2×D411yescharacteristic
182C22×C4×C31#4114yes—
183C22×C4×C31C2411yescharacteristic
184C2×C8×C31—22no—
185C2×C8×C31—22no—
186C2×C8×C31—22no—
187C2×C8×C31—22no—
188C2×C8×C31—22no—
189C2×C8×C31—22no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—21no—
192C4⋊C4×C31—22no—
193C4⋊C4×C31—22no—
194C4⋊C4×C31—22no—
195C4⋊C4×C31—22no—
196C4⋊C4×C31—42no—
197C4⋊C4×C31C2×D412yes—
198C4⋊C4×C31C2×D412yes—
199C22⋊C4×C31—24no—
200C22⋊C4×C31—24no—
201C22⋊C4×C31—24no—
202C22⋊C4×C31—24no—
203C22⋊C4×C31—24no—
204C22⋊C4×C31—42no—
205C22⋊C4×C31—82no—
206C22⋊C4×C31—82no—
207C42×C31—24no—
208C42×C31—42no—
209#24013C3111yesmaximal, characteristic, direct product factor, Sylow
210C23×C31—21no—
211C23×C31—21no—
212C23×C31—21no—
213C23×C31—21no—
214C23×C31—22no—
215C23×C31—22no—
216C23×C31—22no—
217C23×C31—22no—
218C23×C31—22no—
219C23×C31—22no—
220C23×C31—21no—
221C23×C31—21no—
222C23×C31—21no—
223C23×C31—22no—
224C23×C31—22no—
225C23×C31—48no—
226C23×C31—41no—
227C23×C31—41no—
228C23×C31—41no—
229C23×C31—41no—
230C23×C31—42no—
231C23×C31—42no—
232C23×C31—42no—
233C23×C31—42no—
234C23×C31—88no—
235C23×C31—84no—
236C23×C31—84no—
237C23×C31—88no—
238C23×C31#12012yes—
239C23×C31#12312yes—
240C23×C31C22×D411yescharacteristic
241Q8×C31—44no—
242D4×C31—24no—
243D4×C31—24no—
244D4×C31—44no—
245D4×C31—44no—
246D4×C31—48no—
247D4×C31—44no—
248D4×C31—42no—
249D4×C31—42no—
250D4×C31—84no—
251D4×C31—84no—
252D4×C31—84no—
253D4×C31—84no—
254C2×C4×C31—21no—
255C2×C4×C31—21no—
256C2×C4×C31—21no—
257C2×C4×C31—21no—
258C2×C4×C31—24no—
259C2×C4×C31—24no—
260C2×C4×C31—24no—
261C2×C4×C31—24no—
262C2×C4×C31—22no—
263C2×C4×C31—22no—
264C2×C4×C31—21no—
265C2×C4×C31—21no—
266C2×C4×C31—21no—
267C2×C4×C31—42no—
268C2×C4×C31—42no—
269C2×C4×C31—42no—
270C2×C4×C31—42no—
271C2×C4×C31—42no—
272C2×C4×C31—42no—
273C2×C4×C31—42no—
274C2×C4×C31—42no—
275C2×C4×C31—42no—
276C2×C4×C31—42no—
277C2×C4×C31—84no—
278C2×C4×C31#12012yes—
279C2×C4×C31#12312yes—
280C2×C4×C31C22×D411yescharacteristic
281C2×C4×C31C22×D411yescharacteristic
282C2×C4×C31C2×#4112yes—
283C2×C4×C31H(5,𝔽2)12yes—
284C8×C31—42no—
285C8×C31—42no—
286C2×#535C2×C3111yescharacteristic
287C2×#481C2×C3111yescharacteristic, complemented
288C2×#480C2×C3111yescharacteristic, complemented
289#2311C2×C3111yescharacteristic, complemented
290#2034C2×C3112yescomplemented
291#2033C2×C3112yescomplemented
292#1875C2×C3111yescharacteristic, complemented
293#1875C2×C3111yescharacteristic, complemented
294#1854C2×C3112yescomplemented
295#1786C2×C3112yescomplemented
296#1780C2×C3111yescharacteristic, complemented
297C22×C31—24no—
298C22×C31—22no—
299C22×C31—22no—
300C22×C31—22no—
301C22×C31—22no—
302C22×C31—42no—
303C22×C31—42no—
304C22×C31—42no—
305C22×C31—41no—
306C22×C31—41no—
307C22×C31—42no—
308C22×C31—41no—
309C22×C31—42no—
310C22×C31—42no—
311C22×C31—42no—
312C22×C31—41no—
313C22×C31—42no—
314C22×C31—42no—
315C22×C31—41no—
316C22×C31—42no—
317C22×C31—42no—
318C22×C31—816no—
319C22×C31—82no—
320C22×C31—82no—
321C22×C31—82no—
322C22×C31—82no—
323C22×C31C2×#12011yescharacteristic
324C22×C31C2×#12311yescharacteristic
325C22×C31#53411yescharacteristic
326C22×C31#57612yes—
327C22×C31#57712yes—
328C4×C31—22no—
329C4×C31—22no—
330C4×C31—41no—
331C4×C31—42no—
332C4×C31—42no—
333C4×C31—82no—
334C4×C31—82no—
335C4×C31#54612yes—
336C4×C31#54612yes—
337C22×#41C22×C3111yescharacteristic
338C23×D4C22×C3111yescharacteristic
339#535—24no—
340#535—24no—
341C2×#127—21no—
342C2×#124—21no—
343C2×#121C22×C3111yescharacteristic
344C2×#121C22×C3111yescharacteristic
345C2×#121C22×C3111yescharacteristic
346C2×#121C22×C3111yescharacteristic
347C2×#120—21no—
348C2×#120—21no—
349C2×#117C22×C3111yescharacteristic
350C2×#117C22×C3111yescharacteristic, complemented
351C22×C4⋊C4C22×C3111yescharacteristic, complemented
352#481—24no—
353#480—24no—
354C2×#107C22×C3112yescomplemented
355C2×#106C22×C3112yescomplemented
356C2×#102C22×C3112yes—
357C2×#102C22×C3112yes—
358C2×#102C22×C3112yes—
359C2×#102C22×C3112yes—
360C2×C22⋊C8C22×C3112yes—
361C2×C22⋊C8C22×C3111yescharacteristic, complemented
362C2×C22⋊C8C22×C3111yescharacteristic, complemented
363C2×C22⋊C8C22×C3112yescomplemented
364#391—21no—
365#390—21no—
366#386—21no—
367#386—21no—
368#380C22×C3112yes—
369#380C22×C3112yescomplemented
370#340C22×C3112yescomplemented
371#340C22×C3112yescomplemented
372C2×C31—42no—
373C2×C31—42no—
374C2×C31—81no—
375C2×C31—82no—
376C2×C31—82no—
377C2×C31#240511yescharacteristic
378C2×C31#300811yescharacteristic
379C2×C31#300911yescharacteristic
380C2×C31#309612yes—
381C2×C31#319812yes—
382C25—21no—
383C2×#41—22no—
384C2×#41—22no—
385C2×#41—24no—
386C2×#41—42no—
387C22×Q8D4×C3111yescharacteristic
388C22×D4—21no—
389C22×D4—21no—
390C22×D4—21no—
391C22×D4—21no—
392C22×D4—22no—
393C22×D4—22no—
394C22×D4—21no—
395C22×D4—24no—
396C22×D4—42no—
397C22×D4D4×C3111yescharacteristic
398C22×D4D4×C3111yescharacteristic
399C22×D4D4×C3111yescharacteristic
400C22×D4C23×C3111yescharacteristic
401C22×D4C23×C3111yescharacteristic
402C23×C4—21no—
403C23×C4C23×C3111yescharacteristic
404C23×C4C23×C3111yescharacteristic
405C23×C4C23×C3111yescharacteristic
406C22×C8C23×C3112yes—
407C22×C8C23×C3112yes—
408#127—28no—
409#124—28no—
410#121—22no—
411#121—22no—
412#121—22no—
413#121—22no—
414#121—22no—
415#121—22no—
416#121D4×C3114yes—
417#121D4×C3114yes—
418#120—28no—
419#120—28no—
420#117—22no—
421#117—22no—
422#117—42no—
423C2×C4⋊C4—22no—
424C2×C4⋊C4—22no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4C23×C3112yes—
427C2×C4⋊C4C23×C3112yes—
428C2×C4⋊C4C23×C3111yescharacteristic
429C2×C4⋊C4C23×C3111yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—22no—
437C2×C22⋊C4—22no—
438C2×C42—21no—
439C2×C42—21no—
440#107—24no—
441#106—24no—
442#102—24no—
443#102—24no—
444#102—42no—
445#102—42no—
446C22⋊C8—22no—
447C22⋊C8—22no—
448C22⋊C8—24no—
449C22⋊C8—42no—
450#95—22no—
451#95—22no—
452C31#2401311yescharacteristic, direct product factor, Sylow
453C24—24no—
454C24—21no—
455C24—21no—
456C24—42no—
457C24—42no—
458C24—82no—
459C24—82no—
460C24C2×D4×C3111yescharacteristic
461C24C2×D4×C3111yescharacteristic
462C24C2×D4×C3111yescharacteristic
463#41—48no—
464#41—48no—
465C2×Q8—22no—
466C2×Q8—22no—
467C2×Q8—41no—
468C2×D4—24no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—24no—
474C2×D4—24no—
475C2×D4—24no—
476C2×D4—24no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—48no—
486C2×D4—48no—
487C2×D4—41no—
488C2×D4—41no—
489C2×D4—41no—
490C2×D4—42no—
491C2×D4—42no—
492C2×D4—82no—
493C2×D4—82no—
494C2×D4—82no—
495C2×D4—82no—
496C2×D4C2×D4×C3112yes—
497C2×D4C2×D4×C3112yes—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—42no—
512C22×C4—82no—
513C22×C4C2×D4×C3111yescharacteristic
514C22×C4C2×D4×C3111yescharacteristic
515C22×C4C2×D4×C3112yes—
516C22×C4C2×D4×C3112yes—
517C22×C4C2×D4×C3111yescharacteristic
518C22×C4#41×C3114yes—
519C22×C4C24×C3111yescharacteristic
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—42no—
533C4⋊C4C2×D4×C3112yes—
534C4⋊C4C2×D4×C3112yes—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—42no—
541C22⋊C4—82no—
542C22⋊C4—82no—
543C42—24no—
544C42—42no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—22no—
559C23—22no—
560C23—48no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—88no—
570C23—84no—
571C23—84no—
572C23—88no—
573C23C31×#12012yes—
574C23C31×#12312yes—
575C23C22×D4×C3111yescharacteristic
576Q8—44no—
577D4—24no—
578D4—24no—
579D4—44no—
580D4—44no—
581D4—48no—
582D4—44no—
583D4—42no—
584D4—42no—
585D4—84no—
586D4—84no—
587D4—84no—
588D4—84no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—84no—
613C2×C4C31×#12012yes—
614C2×C4C31×#12312yes—
615C2×C4C22×D4×C3111yescharacteristic
616C2×C4C22×D4×C3111yescharacteristic
617C2×C4C2×#41×C3112yes—
618C2×C4C31×H(5,𝔽2)12yes—
619C8—42no—
620C8—42no—
621C22—24no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—41no—
630C22—41no—
631C22—42no—
632C22—41no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—41no—
637C22—42no—
638C22—42no—
639C22—41no—
640C22—42no—
641C22—42no—
642C22—816no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22C2×C31×#12011yescharacteristic
648C22C2×C31×#12311yescharacteristic
649C22C31×#53411yescharacteristic
650C22C31×#57612yes—
651C22C31×#57712yes—
652C4—22no—
653C4—22no—
654C4—41no—
655C4—42no—
656C4—42no—
657C4—82no—
658C4—82no—
659C4C31×#54612yes—
660C4C31×#54612yes—
661C2—42no—
662C2—42no—
663C2—81no—
664C2—82no—
665C2—82no—
666C2C31×#240511yescharacteristic
667C2C31×#300811yescharacteristic
668C2C31×#300911yescharacteristic
669C2C31×#309612yes—
670C2C31×#319812yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#535C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#481C2semidirect
3. C2×C31×#480C2semidirect
4. C31×#2311C2semidirect
5. C31×#2034C2semidirect
6. C31×#2033C2semidirect
7. C31×#1875C2semidirect
8. C31×#1875C2semidirect
9. C31×#1854C2semidirect
10. C31×#1786C2semidirect
11. C31×#1780C2semidirect
25. C2×C31×#117C22semidirect
26. C22×C4⋊C4×C31C22semidirect
29. C2×C31×#107C22semidirect
30. C2×C31×#106C22semidirect
36. C2×C31×C22⋊C8C22semidirect
37. C2×C31×C22⋊C8C22semidirect
38. C2×C31×C22⋊C8C22semidirect
44. C31×#380C22semidirect
45. C31×#340C22semidirect
46. C31×#340C22semidirect
209. #24013C31direct (N × H)
287. C2×#481C2×C31semidirect
288. C2×#480C2×C31semidirect
289. #2311C2×C31semidirect
290. #2034C2×C31semidirect
291. #2033C2×C31semidirect
292. #1875C2×C31semidirect
293. #1875C2×C31semidirect
294. #1854C2×C31semidirect
295. #1786C2×C31semidirect
296. #1780C2×C31semidirect
350. C2×#117C22×C31semidirect
351. C22×C4⋊C4C22×C31semidirect
354. C2×#107C22×C31semidirect
355. C2×#106C22×C31semidirect
361. C2×C22⋊C8C22×C31semidirect
362. C2×C22⋊C8C22×C31semidirect
363. C2×C22⋊C8C22×C31semidirect
369. #380C22×C31semidirect
370. #340C22×C31semidirect
371. #340C22×C31semidirect
452. C31#24013direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5352
C2×C31×#4812
C2×C31×#4802
C31×#23112
C31×#20342
C31×#20332
C31×#18752
C31×#18542
C31×#17862
C31×#17802
#2401331