Finite Group Databasebeta

← C31×#23937 #21480696

Subgroups — C31×#23937 #21480696

Order 7936.

1304 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#529C211yesmaximal, characteristic
2C2×C31×#451C211yesmaximal, characteristic, complemented
3C2×C31×#448C211yesmaximal, characteristic
4C2×C31×#391C211yesmaximal, characteristic, complemented
5C31×#2026C212yesmaximal, complemented
6C31×#2015C212yesmaximal, complemented
7C31×#1878C212yesmaximal, complemented
8C31×#1877C212yesmaximal, complemented
9C31×#1874C211yesmaximal, characteristic
10C31×#1872C211yesmaximal, characteristic, complemented
11C31×#1817C211yesmaximal, characteristic, complemented
12C23×Q8×C31C2211yescharacteristic
13C22×Q16×C31C2212yes—
14C22×SD16×C31C2212yes—
15C31×#529—24no—
16C31×#529—24no—
17C2×C31×#126—21no—
18C2×C31×#125C2211yescharacteristic
19C2×C31×#124C2211yescharacteristic
20C2×C31×#123—21no—
21C2×C31×C22⋊Q8C2211yescharacteristic
22C2×C31×#121C2211yescharacteristic
23C2×C4×Q8×C31C2211yescharacteristic
24C2×C4×D4×C31—21no—
25C2×C4×D4×C31C2211yescharacteristic
26C2×C31×#117—21no—
27C22×C4⋊C4×C31C2211yescharacteristic
28C22×C42×C31C2211yescharacteristic
29C31×#451—24no—
30C31×#448—24no—
31C2×C31×#103C2212yes—
32C2×C31×#103C2212yes—
33C2×C31×#103C2212yes—
34C2×C31×#102C2212yes—
35C2×C31×C22⋊C8C2211yescharacteristic
36C2×C31×C22⋊C8C2211yescharacteristic
37C31×#391—21no—
38C31×#391—21no—
39C31×#391C2212yes—
40C31×#391C2212yes—
41C2×C31×#95—21no—
42C2×C31×#95—21no—
43C31×#340C2212yes—
44C31×#340C2212yes—
45C31×#336C2212yes—
46C31×#336C2212yes—
47C22×Q8×C31—24no—
48C22×Q8×C31—21no—
49C22×Q8×C31—22no—
50C22×Q8×C31—22no—
51C22×Q8×C31—42no—
52C22×Q8×C31C2311yescharacteristic
53C22×Q8×C31C2312yes—
54C22×D4×C31—21no—
55C22×D4×C31C2311yescharacteristic
56C23×C4×C31—21no—
57C23×C4×C31—21no—
58C23×C4×C31—21no—
59C23×C4×C31D411yescharacteristic
60C23×C4×C31D411yescharacteristic
61C23×C4×C31C2311yescharacteristic
62C2×Q16×C31—24no—
63C2×Q16×C31—24no—
64C2×Q16×C31—42no—
65C2×SD16×C31—24no—
66C2×SD16×C31—24no—
67C2×SD16×C31—42no—
68C22×C8×C31C2312yes—
69C22×C8×C31C2312yes—
70C31×#126—28no—
71C31×#125—22no—
72C31×#125—22no—
73C31×#124—22no—
74C31×#124—22no—
75C31×#123—28no—
76C31×C22⋊Q8—22no—
77C31×C22⋊Q8D414yes—
78C31×#121—22no—
79C31×#121D414yes—
80C4×Q8×C31—22no—
81C4×Q8×C31—22no—
82C4×D4×C31—28no—
83C4×D4×C31—22no—
84C4×D4×C31—22no—
85C31×#117—28no—
86C2×C4⋊C4×C31—21no—
87C2×C4⋊C4×C31—21no—
88C2×C4⋊C4×C31—21no—
89C2×C4⋊C4×C31—21no—
90C2×C4⋊C4×C31—21no—
91C2×C4⋊C4×C31—42no—
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×C42×C31—22no—
101C2×C42×C31—22no—
102C2×C42×C31—21no—
103C2×C42×C31D411yescharacteristic
104C2×C42×C31D411yescharacteristic
105C2×C42×C31C2311yescharacteristic
106C2×C42×C31C2311yescharacteristic
107C31×#103—24no—
108C31×#103—24no—
109C31×#103—42no—
110C31×#102—42no—
111C31×C22⋊C8—22no—
112C31×C22⋊C8—22no—
113C31×#95—22no—
114C31×#95—22no—
115C31×#95—22no—
116C31×#95—22no—
117C24×C31—21no—
118C24×C31C2×D411yescharacteristic
119C2×Q8×C31—24no—
120C2×Q8×C31—21no—
121C2×Q8×C31—22no—
122C2×Q8×C31—22no—
123C2×Q8×C31—22no—
124C2×Q8×C31—22no—
125C2×Q8×C31—22no—
126C2×Q8×C31—22no—
127C2×Q8×C31—22no—
128C2×Q8×C31—48no—
129C2×Q8×C31—48no—
130C2×Q8×C31—42no—
131C2×Q8×C31—42no—
132C2×Q8×C31C2×D412yes—
133C2×Q8×C31#4114yes—
134C2×D4×C31—24no—
135C2×D4×C31—21no—
136C2×D4×C31—22no—
137C2×D4×C31—22no—
138C2×D4×C31—42no—
139C2×D4×C31—42no—
140C2×D4×C31C2×D412yes—
141C22×C4×C31—22no—
142C22×C4×C31—22no—
143C22×C4×C31—24no—
144C22×C4×C31—24no—
145C22×C4×C31—22no—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—22no—
149C22×C4×C31—22no—
150C22×C4×C31—21no—
151C22×C4×C31—21no—
152C22×C4×C31—21no—
153C22×C4×C31—22no—
154C22×C4×C31—22no—
155C22×C4×C31—41no—
156C22×C4×C31—42no—
157C22×C4×C31—42no—
158C22×C4×C31—42no—
159C22×C4×C31—42no—
160C22×C4×C31—42no—
161C22×C4×C31—42no—
162C22×C4×C31—42no—
163C22×C4×C31C2×D411yescharacteristic
164C22×C4×C31C2×D411yescharacteristic
165C22×C4×C31C2×D411yescharacteristic
166C22×C4×C31C2×D411yescharacteristic
167C22×C4×C31C2×D412yes—
168C22×C4×C31C2×D412yes—
169C22×C4×C31C2×D411yescharacteristic
170C22×C4×C31C2411yescharacteristic
171Q16×C31—48no—
172SD16×C31—48no—
173C2×C8×C31—22no—
174C2×C8×C31—22no—
175C2×C8×C31—22no—
176C2×C8×C31—22no—
177C2×C8×C31—22no—
178C2×C8×C31—22no—
179C4⋊C4×C31—24no—
180C4⋊C4×C31—24no—
181C4⋊C4×C31—24no—
182C4⋊C4×C31—24no—
183C4⋊C4×C31—21no—
184C4⋊C4×C31—21no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—42no—
188C4⋊C4×C31C2×D412yes—
189C4⋊C4×C31C2×D412yes—
190C22⋊C4×C31—24no—
191C22⋊C4×C31—24no—
192C22⋊C4×C31—24no—
193C22⋊C4×C31—24no—
194C22⋊C4×C31—24no—
195C42×C31—24no—
196C42×C31—22no—
197C42×C31—22no—
198C42×C31—21no—
199C42×C31—21no—
200C42×C31—21no—
201C42×C31—21no—
202#23937C3111yesmaximal, characteristic, direct product factor, Sylow
203C23×C31—21no—
204C23×C31—21no—
205C23×C31—21no—
206C23×C31—21no—
207C23×C31—22no—
208C23×C31—41no—
209C23×C31—41no—
210C23×C31—41no—
211C23×C31—41no—
212C23×C31—41no—
213C23×C31—41no—
214C23×C31#12012yes—
215C23×C31#13512yes—
216C23×C31#13712yes—
217C23×C31C22×D411yescharacteristic
218Q8×C31—24no—
219Q8×C31—24no—
220Q8×C31—24no—
221Q8×C31—48no—
222Q8×C31—42no—
223Q8×C31—42no—
224Q8×C31—42no—
225D4×C31—24no—
226D4×C31—48no—
227D4×C31—42no—
228C2×C4×C31—21no—
229C2×C4×C31—21no—
230C2×C4×C31—21no—
231C2×C4×C31—21no—
232C2×C4×C31—24no—
233C2×C4×C31—24no—
234C2×C4×C31—21no—
235C2×C4×C31—21no—
236C2×C4×C31—21no—
237C2×C4×C31—22no—
238C2×C4×C31—22no—
239C2×C4×C31—22no—
240C2×C4×C31—24no—
241C2×C4×C31—21no—
242C2×C4×C31—21no—
243C2×C4×C31—21no—
244C2×C4×C31—21no—
245C2×C4×C31—21no—
246C2×C4×C31—21no—
247C2×C4×C31—44no—
248C2×C4×C31—48no—
249C2×C4×C31—48no—
250C2×C4×C31—42no—
251C2×C4×C31—41no—
252C2×C4×C31—42no—
253C2×C4×C31—41no—
254C2×C4×C31—41no—
255C2×C4×C31—41no—
256C2×C4×C31—41no—
257C2×C4×C31—41no—
258C2×C4×C31—44no—
259C2×C4×C31—42no—
260C2×C4×C31—42no—
261C2×C4×C31—42no—
262C2×C4×C31—84no—
263C2×C4×C31#12012yes—
264C2×C4×C31#13712yes—
265C2×C4×C31C22×D411yescharacteristic
266C2×C4×C31C22×D411yescharacteristic
267C2×C4×C31C2×#4112yes—
268C2×C4×C31H(5,𝔽2)12yes—
269C8×C31—42no—
270C8×C31—42no—
271C2×#529C2×C3111yescharacteristic
272C2×#451C2×C3111yescharacteristic, complemented
273C2×#448C2×C3111yescharacteristic
274C2×#391C2×C3111yescharacteristic, complemented
275#2026C2×C3112yescomplemented
276#2015C2×C3112yescomplemented
277#1878C2×C3112yescomplemented
278#1877C2×C3112yescomplemented
279#1874C2×C3111yescharacteristic
280#1872C2×C3111yescharacteristic, complemented
281#1817C2×C3111yescharacteristic, complemented
282C22×C31—22no—
283C22×C31—22no—
284C22×C31—22no—
285C22×C31—22no—
286C22×C31—22no—
287C22×C31—22no—
288C22×C31—42no—
289C22×C31—42no—
290C22×C31—42no—
291C22×C31—81no—
292C22×C31—81no—
293C22×C31—81no—
294C22×C31—81no—
295C22×C31#45214yes—
296C22×C31C2×#12011yescharacteristic
297C22×C31#52511yescharacteristic
298C22×C31#53511yescharacteristic
299C22×C31C2×#13511yescharacteristic
300C22×C31C2×#13711yescharacteristic
301C22×C31C2×#13711yescharacteristic
302C22×C31#57511yescharacteristic
303C4×C31—22no—
304C4×C31—22no—
305C4×C31—44no—
306C4×C31—42no—
307C4×C31—41no—
308C4×C31—41no—
309C4×C31—42no—
310C4×C31—41no—
311C4×C31—42no—
312C4×C31—82no—
313C4×C31#54612yes—
314C4×C31#54612yes—
315C23×Q8C22×C3111yescharacteristic
316C22×Q16C22×C3112yes—
317C22×SD16C22×C3112yes—
318#529—24no—
319#529—24no—
320C2×#126—21no—
321C2×#125C22×C3111yescharacteristic
322C2×#124C22×C3111yescharacteristic
323C2×#123—21no—
324C2×C22⋊Q8C22×C3111yescharacteristic
325C2×#121C22×C3111yescharacteristic
326C2×C4×Q8C22×C3111yescharacteristic
327C2×C4×D4—21no—
328C2×C4×D4C22×C3111yescharacteristic
329C2×#117—21no—
330C22×C4⋊C4C22×C3111yescharacteristic
331C22×C42C22×C3111yescharacteristic
332#451—24no—
333#448—24no—
334C2×#103C22×C3112yes—
335C2×#103C22×C3112yes—
336C2×#103C22×C3112yes—
337C2×#102C22×C3112yes—
338C2×C22⋊C8C22×C3111yescharacteristic
339C2×C22⋊C8C22×C3111yescharacteristic
340#391—21no—
341#391—21no—
342#391C22×C3112yes—
343#391C22×C3112yes—
344C2×#95—21no—
345C2×#95—21no—
346#340C22×C3112yes—
347#340C22×C3112yes—
348#336C22×C3112yes—
349#336C22×C3112yes—
350C2×C31—24no—
351C2×C31—82no—
352C2×C31#240411yescharacteristic
353C2×C31C2×#45211yescharacteristic
354C2×C31#300711yescharacteristic
355C2×C31#310811yescharacteristic
356C2×C31#311711yescharacteristic
357C2×C31#321111yescharacteristic
358C2×C31#321811yescharacteristic
359C22×Q8—24no—
360C22×Q8—21no—
361C22×Q8—22no—
362C22×Q8—22no—
363C22×Q8—42no—
364C22×Q8C23×C3111yescharacteristic
365C22×Q8C23×C3112yes—
366C22×D4—21no—
367C22×D4C23×C3111yescharacteristic
368C23×C4—21no—
369C23×C4—21no—
370C23×C4—21no—
371C23×C4D4×C3111yescharacteristic
372C23×C4D4×C3111yescharacteristic
373C23×C4C23×C3111yescharacteristic
374C2×Q16—24no—
375C2×Q16—24no—
376C2×Q16—42no—
377C2×SD16—24no—
378C2×SD16—24no—
379C2×SD16—42no—
380C22×C8C23×C3112yes—
381C22×C8C23×C3112yes—
382#126—28no—
383#125—22no—
384#125—22no—
385#124—22no—
386#124—22no—
387#123—28no—
388C22⋊Q8—22no—
389C22⋊Q8D4×C3114yes—
390#121—22no—
391#121D4×C3114yes—
392C4×Q8—22no—
393C4×Q8—22no—
394C4×D4—28no—
395C4×D4—22no—
396C4×D4—22no—
397#117—28no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—42no—
404C2×C4⋊C4C23×C3112yes—
405C2×C4⋊C4C23×C3111yescharacteristic
406C2×C4⋊C4C23×C3111yescharacteristic
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C42—21no—
413C2×C42—22no—
414C2×C42—22no—
415C2×C42D4×C3111yescharacteristic
416C2×C42D4×C3111yescharacteristic
417C2×C42C23×C3111yescharacteristic
418C2×C42C23×C3111yescharacteristic
419#103—24no—
420#103—24no—
421#103—42no—
422#102—42no—
423C22⋊C8—22no—
424C22⋊C8—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429C31#2393711yescharacteristic, direct product factor, Sylow
430C24—21no—
431C24C2×D4×C3111yescharacteristic
432C2×Q8—24no—
433C2×Q8—21no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—48no—
442C2×Q8—48no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8C2×D4×C3112yes—
446C2×Q8#41×C3114yes—
447C2×D4—24no—
448C2×D4—21no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4C2×D4×C3112yes—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—41no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4C2×D4×C3111yescharacteristic
477C22×C4C2×D4×C3111yescharacteristic
478C22×C4C2×D4×C3111yescharacteristic
479C22×C4C2×D4×C3111yescharacteristic
480C22×C4C2×D4×C3112yes—
481C22×C4C2×D4×C3112yes—
482C22×C4C2×D4×C3111yescharacteristic
483C22×C4C24×C3111yescharacteristic
484Q16—48no—
485SD16—48no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—42no—
501C4⋊C4C2×D4×C3112yes—
502C4⋊C4C2×D4×C3112yes—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C42—24no—
509C42—22no—
510C42—22no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23C31×#12012yes—
527C23C31×#13512yes—
528C23C31×#13712yes—
529C23C22×D4×C3111yescharacteristic
530Q8—24no—
531Q8—24no—
532Q8—24no—
533Q8—48no—
534Q8—42no—
535Q8—42no—
536Q8—42no—
537D4—24no—
538D4—48no—
539D4—42no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—44no—
560C2×C4—48no—
561C2×C4—48no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—84no—
575C2×C4C31×#12012yes—
576C2×C4C31×#13712yes—
577C2×C4C22×D4×C3111yescharacteristic
578C2×C4C22×D4×C3111yescharacteristic
579C2×C4C2×#41×C3112yes—
580C2×C4C31×H(5,𝔽2)12yes—
581C8—42no—
582C8—42no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22C31×#45214yes—
597C22C2×C31×#12011yescharacteristic
598C22C31×#52511yescharacteristic
599C22C31×#53511yescharacteristic
600C22C2×C31×#13511yescharacteristic
601C22C2×C31×#13711yescharacteristic
602C22C2×C31×#13711yescharacteristic
603C22C31×#57511yescharacteristic
604C4—22no—
605C4—22no—
606C4—44no—
607C4—42no—
608C4—41no—
609C4—41no—
610C4—42no—
611C4—41no—
612C4—42no—
613C4—82no—
614C4C31×#54612yes—
615C4C31×#54612yes—
616C2—24no—
617C2—82no—
618C2C31×#240411yescharacteristic
619C2C2×C31×#45211yescharacteristic
620C2C31×#300711yescharacteristic
621C2C31×#310811yescharacteristic
622C2C31×#311711yescharacteristic
623C2C31×#321111yescharacteristic
624C2C31×#321811yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#451C2semidirect
4. C2×C31×#391C2semidirect
5. C31×#2026C2semidirect
6. C31×#2015C2semidirect
7. C31×#1878C2semidirect
8. C31×#1877C2semidirect
10. C31×#1872C2semidirect
11. C31×#1817C2semidirect
202. #23937C31direct (N × H)
272. C2×#451C2×C31semidirect
274. C2×#391C2×C31semidirect
275. #2026C2×C31semidirect
276. #2015C2×C31semidirect
277. #1878C2×C31semidirect
278. #1877C2×C31semidirect
280. #1872C2×C31semidirect
281. #1817C2×C31semidirect
429. C31#23937direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5292
C2×C31×#4512
C2×C31×#4482
C2×C31×#3912
C31×#20262
C31×#20152
C31×#18782
C31×#18772
C31×#18742
C31×#18722
C31×#18172
#2393731