Finite Group Databasebeta

← C31×#15928 #21472687

Subgroups — C31×#15928 #21472687

Order 7936.

1566 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×#126C211yesmaximal, characteristic
2C2×C31×#400C211yesmaximal, characteristic, complemented
3C2×C31×#397C211yesmaximal, characteristic
4C2×C31×#384C211yesmaximal, characteristic, complemented
5C31×#1445C211yesmaximal, characteristic, complemented
6C31×#1441C212yesmaximal, complemented
7C31×#1439C212yesmaximal
8C31×#1439C212yesmaximal
9C31×#1437C211yesmaximal, characteristic
10C31×#1436C212yesmaximal, complemented
11C31×#1436C211yesmaximal, characteristic, complemented
12C2×C31×#126—24no—
13C2×C31×#126—24no—
14C2×C31×#126—24no—
15C22×C4⋊C4×C31C2211yescharacteristic
16C22×C4⋊C4×C31C2211yescharacteristic
17C22×C4⋊C4×C31C2212yes—
18C22×C4⋊C4×C31C2212yes—
19C22×C4⋊C4×C31C2211yescharacteristic
20C22×C4⋊C4×C31C2211yescharacteristic
21C22×C22⋊C4×C31C2211yescharacteristic
22C22×C22⋊C4×C31C2211yescharacteristic
23C22×C22⋊C4×C31C2212yes—
24C22×C22⋊C4×C31C2211yescharacteristic
25C22×C22⋊C4×C31C2212yes—
26C22×C42×C31C2211yescharacteristic
27C22×C42×C31C2211yescharacteristic
28C31×#400—24no—
29C31×#397—24no—
30C31×#384—24no—
31C2×C31×#95C2212yes—
32C2×C31×#95C2211yescharacteristic
33C2×C31×#95C2212yes—
34C2×C31×#95C2211yescharacteristic
35C2×C31×#95C2212yes—
36C2×C31×#95C2212yes—
37C2×C31×#95C2212yes—
38C2×C31×#95C2212yes—
39C2×C31×#95C2212yes—
40C2×C31×#95C2212yes—
41C25×C31C2311yescharacteristic
42C23×C4×C31C2312yes—
43C23×C4×C31C2312yes—
44C23×C4×C31C2312yes—
45C23×C4×C31C2311yescharacteristic
46C23×C4×C31C2311yescharacteristic
47C23×C4×C31C2311yescharacteristic
48C23×C4×C31C2311yescharacteristic
49C23×C4×C31C2311yescharacteristic
50C23×C4×C31C2311yescharacteristic
51C23×C4×C31C2312yes—
52C31×#126—232no—
53C2×C4⋊C4×C31—22no—
54C2×C4⋊C4×C31—22no—
55C2×C4⋊C4×C31—22no—
56C2×C4⋊C4×C31—22no—
57C2×C4⋊C4×C31—22no—
58C2×C4⋊C4×C31—22no—
59C2×C4⋊C4×C31—22no—
60C2×C4⋊C4×C31—22no—
61C2×C4⋊C4×C31—22no—
62C2×C4⋊C4×C31—24no—
63C2×C4⋊C4×C31—24no—
64C2×C4⋊C4×C31—22no—
65C2×C4⋊C4×C31—24no—
66C2×C4⋊C4×C31—24no—
67C2×C4⋊C4×C31—42no—
68C2×C4⋊C4×C31—42no—
69C2×C4⋊C4×C31D414yes—
70C2×C4⋊C4×C31D414yes—
71C2×C22⋊C4×C31—22no—
72C2×C22⋊C4×C31—22no—
73C2×C22⋊C4×C31—22no—
74C2×C22⋊C4×C31—22no—
75C2×C22⋊C4×C31—22no—
76C2×C22⋊C4×C31—22no—
77C2×C22⋊C4×C31—22no—
78C2×C22⋊C4×C31—22no—
79C2×C22⋊C4×C31—24no—
80C2×C22⋊C4×C31—24no—
81C2×C22⋊C4×C31—24no—
82C2×C22⋊C4×C31—24no—
83C2×C22⋊C4×C31—42no—
84C2×C22⋊C4×C31—42no—
85C2×C22⋊C4×C31D414yes—
86C2×C42×C31—22no—
87C2×C42×C31—22no—
88C2×C42×C31—22no—
89C2×C42×C31—22no—
90C2×C42×C31—24no—
91C31×#95—22no—
92C31×#95—24no—
93C31×#95—22no—
94C31×#95—24no—
95C31×#95—24no—
96C31×#95—24no—
97C31×#95—24no—
98C31×#95—24no—
99C31×#95—42no—
100C31×#95—42no—
101C24×C31—21no—
102C24×C31—21no—
103C24×C31—22no—
104C24×C31—22no—
105C24×C31—22no—
106C24×C31—22no—
107C24×C31—22no—
108C24×C31—22no—
109C24×C31C2×D412yes—
110C24×C31C2411yescharacteristic
111C22×C4×C31—22no—
112C22×C4×C31—22no—
113C22×C4×C31—22no—
114C22×C4×C31—22no—
115C22×C4×C31—22no—
116C22×C4×C31—22no—
117C22×C4×C31—21no—
118C22×C4×C31—21no—
119C22×C4×C31—21no—
120C22×C4×C31—21no—
121C22×C4×C31—21no—
122C22×C4×C31—21no—
123C22×C4×C31—21no—
124C22×C4×C31—21no—
125C22×C4×C31—21no—
126C22×C4×C31—21no—
127C22×C4×C31—22no—
128C22×C4×C31—22no—
129C22×C4×C31—22no—
130C22×C4×C31—22no—
131C22×C4×C31—22no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—22no—
140C22×C4×C31—22no—
141C22×C4×C31—22no—
142C22×C4×C31—22no—
143C22×C4×C31—22no—
144C22×C4×C31—22no—
145C22×C4×C31—22no—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—22no—
149C22×C4×C31—22no—
150C22×C4×C31—22no—
151C22×C4×C31—22no—
152C22×C4×C31—22no—
153C22×C4×C31—22no—
154C22×C4×C31—22no—
155C22×C4×C31—22no—
156C22×C4×C31—22no—
157C22×C4×C31—22no—
158C22×C4×C31C2×D412yes—
159C22×C4×C31C2×D412yes—
160C22×C4×C31C2×D412yes—
161C22×C4×C31C2×D412yes—
162C22×C4×C31C2×D412yes—
163C22×C4×C31C2×D412yes—
164C22×C4×C31C2×D412yes—
165C22×C4×C31C2×D412yes—
166C22×C4×C31#4114yes—
167C22×C4×C31#4114yes—
168C22×C4×C31#4114yes—
169C4⋊C4×C31—28no—
170C4⋊C4×C31—28no—
171C4⋊C4×C31—28no—
172C4⋊C4×C31—48no—
173C4⋊C4×C31—48no—
174C4⋊C4×C31—44no—
175C22⋊C4×C31—28no—
176C22⋊C4×C31—28no—
177C22⋊C4×C31—28no—
178C22⋊C4×C31—48no—
179C22⋊C4×C31—48no—
180C42×C31—28no—
181C42×C31—28no—
182#15928C3111yesmaximal, characteristic, direct product factor, Sylow
183C23×C31—22no—
184C23×C31—22no—
185C23×C31—22no—
186C23×C31—24no—
187C23×C31—24no—
188C23×C31—41no—
189C23×C31—42no—
190C23×C31—41no—
191C23×C31—41no—
192C23×C31—41no—
193C23×C31—42no—
194C23×C31—42no—
195C23×C31—42no—
196C23×C31—42no—
197C23×C31—42no—
198C23×C31—42no—
199C23×C31—42no—
200C23×C31—42no—
201C23×C31—42no—
202C23×C31—42no—
203C23×C31—42no—
204C23×C31C22×D411yescharacteristic
205C23×C31C22×D411yescharacteristic
206C23×C31C22×D411yescharacteristic
207C23×C31C2×#4112yes—
208C23×C31C2×#4112yes—
209C23×C31C2×#4112yes—
210C23×C31H(5,𝔽2)12yes—
211C23×C312−1+412yes—
212C23×C312−1+412yes—
213C2×C4×C31—22no—
214C2×C4×C31—24no—
215C2×C4×C31—22no—
216C2×C4×C31—24no—
217C2×C4×C31—24no—
218C2×C4×C31—24no—
219C2×C4×C31—24no—
220C2×C4×C31—24no—
221C2×C4×C31—24no—
222C2×C4×C31—24no—
223C2×C4×C31—22no—
224C2×C4×C31—22no—
225C2×C4×C31—22no—
226C2×C4×C31—22no—
227C2×C4×C31—22no—
228C2×C4×C31—22no—
229C2×C4×C31—22no—
230C2×C4×C31—24no—
231C2×C4×C31—24no—
232C2×C4×C31—24no—
233C2×C4×C31—24no—
234C2×C4×C31—41no—
235C2×C4×C31—41no—
236C2×C4×C31—41no—
237C2×C4×C31—41no—
238C2×C4×C31—41no—
239C2×C4×C31—41no—
240C2×C4×C31—41no—
241C2×C4×C31—41no—
242C2×C4×C31—42no—
243C2×C4×C31—42no—
244C2×C4×C31—42no—
245C2×C4×C31—42no—
246C2×C4×C31—42no—
247C2×C4×C31—42no—
248C2×C4×C31—42no—
249C2×C4×C31—42no—
250C2×C4×C31—42no—
251C2×C4×C31—42no—
252C2×C4×C31—42no—
253C2×C4×C31—42no—
254C2×C4×C31—42no—
255C2×C4×C31—42no—
256C2×C4×C31—42no—
257C2×C4×C31—42no—
258C2×C4×C31—42no—
259C2×C4×C31—42no—
260C2×C4×C31—42no—
261C2×C4×C31—42no—
262C2×C4×C31—42no—
263C2×C4×C31—42no—
264C2×C4×C31—42no—
265C2×C4×C31—42no—
266C2×C4×C31—42no—
267C2×C4×C31—42no—
268C2×C4×C31—42no—
269C2×C4×C31#12014yes—
270C2×C4×C31#12618yescomplemented
271C22×#126C2×C3111yescharacteristic
272C2×#400C2×C3111yescharacteristic, complemented
273C2×#397C2×C3111yescharacteristic
274C2×#384C2×C3111yescharacteristic, complemented
275#1445C2×C3111yescharacteristic, complemented
276#1441C2×C3112yescomplemented
277#1439C2×C3112yes—
278#1439C2×C3112yes—
279#1437C2×C3111yescharacteristic
280#1436C2×C3112yescomplemented
281#1436C2×C3111yescharacteristic, complemented
282C22×C31—42no—
283C22×C31—42no—
284C22×C31—42no—
285C22×C31—44no—
286C22×C31—44no—
287C22×C31—82no—
288C22×C31—82no—
289C22×C31—82no—
290C22×C31—82no—
291C22×C31C2×#12011yescharacteristic
292C22×C31#52511yescharacteristic
293C22×C31C2×#12612yes—
294C22×C31#53211yescharacteristic
295C22×C31#53211yescharacteristic
296C22×C31#53511yescharacteristic
297C22×C31#53611yescharacteristic
298C22×C31#53611yescharacteristic
299C22×C31#53912yes—
300C22×C31#54112yes—
301C22×C31#54212yes—
302C22×C31#54612yes—
303C22×C31#54712yes—
304C22×C31#54712yes—
305C22×C31#54712yes—
306C22×C31#54812yes—
307C22×C31#55012yes—
308C22×C31#55212yes—
309C22×C31#55312yes—
310C22×C31#55612yes—
311C22×C31#56312yes—
312C4×C31—24no—
313C4×C31—28no—
314C4×C31—42no—
315C4×C31—42no—
316C4×C31—42no—
317C4×C31—42no—
318C4×C31—42no—
319C4×C31—82no—
320C4×C31—82no—
321C4×C31—82no—
322C2×#126—24no—
323C2×#126—24no—
324C2×#126—24no—
325C22×C4⋊C4C22×C3111yescharacteristic
326C22×C4⋊C4C22×C3111yescharacteristic
327C22×C4⋊C4C22×C3112yes—
328C22×C4⋊C4C22×C3112yes—
329C22×C4⋊C4C22×C3111yescharacteristic
330C22×C4⋊C4C22×C3111yescharacteristic
331C22×C22⋊C4C22×C3111yescharacteristic
332C22×C22⋊C4C22×C3111yescharacteristic
333C22×C22⋊C4C22×C3112yes—
334C22×C22⋊C4C22×C3111yescharacteristic
335C22×C22⋊C4C22×C3112yes—
336C22×C42C22×C3111yescharacteristic
337C22×C42C22×C3111yescharacteristic
338#400—24no—
339#397—24no—
340#384—24no—
341C2×#95C22×C3112yes—
342C2×#95C22×C3111yescharacteristic
343C2×#95C22×C3112yes—
344C2×#95C22×C3111yescharacteristic
345C2×#95C22×C3112yes—
346C2×#95C22×C3112yes—
347C2×#95C22×C3112yes—
348C2×#95C22×C3112yes—
349C2×#95C22×C3112yes—
350C2×#95C22×C3112yes—
351C2×C31—82no—
352C2×C31#240611yescharacteristic
353C2×C31#243111yescharacteristic
354C2×C31#243211yescharacteristic
355C2×C31#248112yes—
356C2×C31#249612yes—
357C2×C31#251212yes—
358C2×C31#271612yes—
359C2×C31#271812yes—
360C2×C31#272112yes—
361C25C23×C3111yescharacteristic
362C23×C4C23×C3112yes—
363C23×C4C23×C3112yes—
364C23×C4C23×C3112yes—
365C23×C4C23×C3111yescharacteristic
366C23×C4C23×C3111yescharacteristic
367C23×C4C23×C3111yescharacteristic
368C23×C4C23×C3111yescharacteristic
369C23×C4C23×C3111yescharacteristic
370C23×C4C23×C3111yescharacteristic
371C23×C4C23×C3112yes—
372#126—232no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—24no—
386C2×C4⋊C4—24no—
387C2×C4⋊C4—42no—
388C2×C4⋊C4—42no—
389C2×C4⋊C4D4×C3114yes—
390C2×C4⋊C4D4×C3114yes—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—24no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—24no—
403C2×C22⋊C4—42no—
404C2×C22⋊C4—42no—
405C2×C22⋊C4D4×C3114yes—
406C2×C42—22no—
407C2×C42—22no—
408C2×C42—22no—
409C2×C42—22no—
410C2×C42—24no—
411#95—22no—
412#95—24no—
413#95—22no—
414#95—24no—
415#95—24no—
416#95—24no—
417#95—24no—
418#95—24no—
419#95—42no—
420#95—42no—
421C31#1592811yescharacteristic, direct product factor, Sylow
422C24—21no—
423C24—21no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24C2×D4×C3112yes—
431C24C24×C3111yescharacteristic
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4C2×D4×C3112yes—
480C22×C4C2×D4×C3112yes—
481C22×C4C2×D4×C3112yes—
482C22×C4C2×D4×C3112yes—
483C22×C4C2×D4×C3112yes—
484C22×C4C2×D4×C3112yes—
485C22×C4C2×D4×C3112yes—
486C22×C4C2×D4×C3112yes—
487C22×C4#41×C3114yes—
488C22×C4#41×C3114yes—
489C22×C4#41×C3114yes—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—44no—
496C22⋊C4—28no—
497C22⋊C4—28no—
498C22⋊C4—28no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C42—28no—
502C42—28no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—24no—
508C23—41no—
509C23—42no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23C22×D4×C3111yescharacteristic
525C23C22×D4×C3111yescharacteristic
526C23C22×D4×C3111yescharacteristic
527C23C2×#41×C3112yes—
528C23C2×#41×C3112yes—
529C23C2×#41×C3112yes—
530C23C31×H(5,𝔽2)12yes—
531C23C31×2−1+412yes—
532C23C31×2−1+412yes—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4C31×#12014yes—
590C2×C4C31×#12618yescomplemented
591C22—42no—
592C22—42no—
593C22—42no—
594C22—44no—
595C22—44no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22C2×C31×#12011yescharacteristic
601C22C31×#52511yescharacteristic
602C22C2×C31×#12612yes—
603C22C31×#53211yescharacteristic
604C22C31×#53211yescharacteristic
605C22C31×#53511yescharacteristic
606C22C31×#53611yescharacteristic
607C22C31×#53611yescharacteristic
608C22C31×#53912yes—
609C22C31×#54112yes—
610C22C31×#54212yes—
611C22C31×#54612yes—
612C22C31×#54712yes—
613C22C31×#54712yes—
614C22C31×#54712yes—
615C22C31×#54812yes—
616C22C31×#55012yes—
617C22C31×#55212yes—
618C22C31×#55312yes—
619C22C31×#55612yes—
620C22C31×#56312yes—
621C4—24no—
622C4—28no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—42no—
628C4—82no—
629C4—82no—
630C4—82no—
631C2—82no—
632C2C31×#240611yescharacteristic
633C2C31×#243111yescharacteristic
634C2C31×#243211yescharacteristic
635C2C31×#248112yes—
636C2C31×#249612yes—
637C2C31×#251212yes—
638C2C31×#271612yes—
639C2C31×#271812yes—
640C2C31×#272112yes—
subgroupG/H
center, socleC24×C31C24
commutator, Frattini, cosocleC24C24×C31
prime radicalC25×C31C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#400C2semidirect
4. C2×C31×#384C2semidirect
5. C31×#1445C2semidirect
6. C31×#1441C2semidirect
10. C31×#1436C2semidirect
11. C31×#1436C2semidirect
182. #15928C31direct (N × H)
270. C2×C4×C31#126semidirect
272. C2×#400C2×C31semidirect
274. C2×#384C2×C31semidirect
275. #1445C2×C31semidirect
276. #1441C2×C31semidirect
280. #1436C2×C31semidirect
281. #1436C2×C31semidirect
421. C31#15928direct (N × H)
590. C2×C4C31×#126semidirect

Maximal subgroups

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

subgroupindex
C22×C31×#1262
C2×C31×#4002
C2×C31×#3972
C2×C31×#3842
C31×#14452
C31×#14412
C31×#14392
C31×#14372
C31×#14362
#1592831