Finite Group Databasebeta

← C31×#15850 #21472609

Subgroups — C31×#15850 #21472609

Order 7936.

1458 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×#125C211yesmaximal, characteristic
2C2×C31×#400C211yesmaximal, characteristic
3C2×C31×#398C211yesmaximal, characteristic
4C2×C31×#384C211yesmaximal, characteristic
5C31×#1445C211yesmaximal, characteristic
6C31×#1444C212yesmaximal
7C31×#1444C212yesmaximal
8C31×#1441C212yesmaximal
9C31×#1436C212yesmaximal
10C31×#1436C211yesmaximal, characteristic
11C31×#1436C211yesmaximal, characteristic
12C2×C31×#125—24no—
13C2×C31×#125—24no—
14C2×C31×#125—24no—
15C22×C4⋊C4×C31C2211yescharacteristic
16C22×C4⋊C4×C31C2211yescharacteristic
17C22×C4⋊C4×C31C2211yescharacteristic
18C22×C4⋊C4×C31C2211yescharacteristic
19C22×C4⋊C4×C31C2212yes—
20C22×C4⋊C4×C31C2212yes—
21C22×C4⋊C4×C31C2211yescharacteristic
22C22×C4⋊C4×C31C2211yescharacteristic
23C22×C4⋊C4×C31C2211yescharacteristic
24C22×C4⋊C4×C31C2212yes—
25C22×C42×C31C2211yescharacteristic
26C22×C42×C31C2211yescharacteristic
27C31×#400—24no—
28C31×#398—24no—
29C31×#384—24no—
30C2×C31×#95C2212yes—
31C2×C31×#95C2211yescharacteristic
32C2×C31×#95C2212yes—
33C2×C31×#95C2211yescharacteristic
34C2×C31×#95C2212yes—
35C2×C31×#95C2212yes—
36C2×C31×#95C2212yes—
37C2×C31×#95C2212yes—
38C2×C31×#95C2212yes—
39C2×C31×#95C2212yes—
40C2×C31×#95C2212yes—
41C23×C4×C31C2312yes—
42C23×C4×C31C2312yes—
43C23×C4×C31C2311yescharacteristic
44C23×C4×C31C2311yescharacteristic
45C23×C4×C31C2311yescharacteristic
46C23×C4×C31C2311yescharacteristic
47C23×C4×C31C2311yescharacteristic
48C23×C4×C31C2312yes—
49C23×C4×C31C2311yescharacteristic
50C23×C4×C31C2311yescharacteristic
51C23×C4×C31C2312yes—
52C31×#125—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—22no—
63C2×C4⋊C4×C31—22no—
64C2×C4⋊C4×C31—22no—
65C2×C4⋊C4×C31—22no—
66C2×C4⋊C4×C31—22no—
67C2×C4⋊C4×C31—22no—
68C2×C4⋊C4×C31—24no—
69C2×C4⋊C4×C31—22no—
70C2×C4⋊C4×C31—22no—
71C2×C4⋊C4×C31—24no—
72C2×C4⋊C4×C31—24no—
73C2×C4⋊C4×C31—22no—
74C2×C4⋊C4×C31—24no—
75C2×C4⋊C4×C31—24no—
76C2×C4⋊C4×C31—42no—
77C2×C4⋊C4×C31—42no—
78C2×C4⋊C4×C31—42no—
79C2×C4⋊C4×C31—42no—
80C2×C4⋊C4×C31D414yes—
81C2×C4⋊C4×C31D414yes—
82C2×C4⋊C4×C31D414yes—
83C2×C42×C31—22no—
84C2×C42×C31—22no—
85C2×C42×C31—22no—
86C2×C42×C31—22no—
87C2×C42×C31—24no—
88C31×#95—22no—
89C31×#95—24no—
90C31×#95—22no—
91C31×#95—24no—
92C31×#95—24no—
93C31×#95—24no—
94C31×#95—24no—
95C31×#95—24no—
96C31×#95—42no—
97C31×#95—42no—
98C31×#95Q818yes—
99C24×C31C2411yescharacteristic
100C22×C4×C31—21no—
101C22×C4×C31—21no—
102C22×C4×C31—22no—
103C22×C4×C31—22no—
104C22×C4×C31—22no—
105C22×C4×C31—22no—
106C22×C4×C31—22no—
107C22×C4×C31—22no—
108C22×C4×C31—22no—
109C22×C4×C31—22no—
110C22×C4×C31—21no—
111C22×C4×C31—21no—
112C22×C4×C31—21no—
113C22×C4×C31—21no—
114C22×C4×C31—21no—
115C22×C4×C31—21no—
116C22×C4×C31—21no—
117C22×C4×C31—21no—
118C22×C4×C31—21no—
119C22×C4×C31—21no—
120C22×C4×C31—22no—
121C22×C4×C31—22no—
122C22×C4×C31—22no—
123C22×C4×C31—22no—
124C22×C4×C31—22no—
125C22×C4×C31—22no—
126C22×C4×C31—22no—
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×C31C2×D412yes—
150C22×C4×C31C2×D412yes—
151C22×C4×C31C2×D412yes—
152C22×C4×C31C2×D412yes—
153C22×C4×C31C2×D412yes—
154C22×C4×C31C2×D412yes—
155C22×C4×C31C2×D412yes—
156C22×C4×C31C2×D412yes—
157C22×C4×C31C2×D412yes—
158C22×C4×C31C2×Q814yes—
159C22×C4×C31C2×Q814yes—
160C22×C4×C31C2×Q814yes—
161C22×C4×C31#4114yes—
162C22×C4×C31#4114yes—
163C4⋊C4×C31—28no—
164C4⋊C4×C31—28no—
165C4⋊C4×C31—28no—
166C4⋊C4×C31—28no—
167C4⋊C4×C31—28no—
168C4⋊C4×C31—28no—
169C4⋊C4×C31—48no—
170C4⋊C4×C31—48no—
171C4⋊C4×C31—48no—
172C4⋊C4×C31—44no—
173C42×C31—28no—
174C42×C31—28no—
175#15850C3111yesmaximal, characteristic, direct product factor, Sylow
176C23×C31C22×D411yescharacteristic
177C23×C31C22×D411yescharacteristic
178C23×C31C22×D411yescharacteristic
179C23×C31C22×Q812yes—
180C23×C31C2×#4112yes—
181C23×C31C2×#4112yes—
182C23×C31H(5,𝔽2)12yes—
183C23×C312−1+412yes—
184C23×C312−1+412yes—
185C2×C4×C31—22no—
186C2×C4×C31—22no—
187C2×C4×C31—24no—
188C2×C4×C31—22no—
189C2×C4×C31—22no—
190C2×C4×C31—24no—
191C2×C4×C31—24no—
192C2×C4×C31—24no—
193C2×C4×C31—24no—
194C2×C4×C31—24no—
195C2×C4×C31—24no—
196C2×C4×C31—24no—
197C2×C4×C31—24no—
198C2×C4×C31—24no—
199C2×C4×C31—22no—
200C2×C4×C31—22no—
201C2×C4×C31—22no—
202C2×C4×C31—22no—
203C2×C4×C31—22no—
204C2×C4×C31—22no—
205C2×C4×C31—22no—
206C2×C4×C31—24no—
207C2×C4×C31—24no—
208C2×C4×C31—24no—
209C2×C4×C31—24no—
210C2×C4×C31—24no—
211C2×C4×C31—24no—
212C2×C4×C31—24no—
213C2×C4×C31—41no—
214C2×C4×C31—41no—
215C2×C4×C31—41no—
216C2×C4×C31—41no—
217C2×C4×C31—42no—
218C2×C4×C31—41no—
219C2×C4×C31—41no—
220C2×C4×C31—41no—
221C2×C4×C31—41no—
222C2×C4×C31—42no—
223C2×C4×C31—42no—
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—42no—
232C2×C4×C31—42no—
233C2×C4×C31—42no—
234C2×C4×C31—42no—
235C2×C4×C31—42no—
236C2×C4×C31—42no—
237C2×C4×C31—42no—
238C2×C4×C31—42no—
239C2×C4×C31—42no—
240C2×C4×C31—42no—
241C2×C4×C31—42no—
242C2×C4×C31—42no—
243C2×C4×C31—42no—
244C2×C4×C31—42no—
245C2×C4×C31#12014yes—
246C2×C4×C31#12518yescomplemented
247C2×C4×C31C4⋊Q814yes—
248C22×#125C2×C3111yescharacteristic
249C2×#400C2×C3111yescharacteristic
250C2×#398C2×C3111yescharacteristic
251C2×#384C2×C3111yescharacteristic
252#1445C2×C3111yescharacteristic
253#1444C2×C3112yes—
254#1444C2×C3112yes—
255#1441C2×C3112yes—
256#1436C2×C3112yes—
257#1436C2×C3111yescharacteristic
258#1436C2×C3111yescharacteristic
259C22×C31C2×#12011yescharacteristic
260C22×C31#52511yescharacteristic
261C22×C31C2×#12512yes—
262C22×C31C2×C4⋊Q811yescharacteristic
263C22×C31#53211yescharacteristic
264C22×C31#53511yescharacteristic
265C22×C31#53611yescharacteristic
266C22×C31#53611yescharacteristic
267C22×C31#53912yes—
268C22×C31#54112yes—
269C22×C31#54412yes—
270C22×C31#54612yes—
271C22×C31#54712yes—
272C22×C31#54712yes—
273C22×C31D4×Q812yes—
274C22×C31D4×Q812yes—
275C22×C31#55012yes—
276C22×C31#55212yes—
277C22×C31#55412yes—
278C22×C31#55712yes—
279C22×C31#56312yes—
280C4×C31—24no—
281C4×C31—24no—
282C4×C31—28no—
283C4×C31—42no—
284C4×C31—42no—
285C4×C31—42no—
286C4×C31—42no—
287C4×C31—42no—
288C4×C31—44no—
289C4×C31—82no—
290C4×C31—82no—
291C2×#125—24no—
292C2×#125—24no—
293C2×#125—24no—
294C22×C4⋊C4C22×C3111yescharacteristic
295C22×C4⋊C4C22×C3111yescharacteristic
296C22×C4⋊C4C22×C3111yescharacteristic
297C22×C4⋊C4C22×C3111yescharacteristic
298C22×C4⋊C4C22×C3112yes—
299C22×C4⋊C4C22×C3112yes—
300C22×C4⋊C4C22×C3111yescharacteristic
301C22×C4⋊C4C22×C3111yescharacteristic
302C22×C4⋊C4C22×C3111yescharacteristic
303C22×C4⋊C4C22×C3112yes—
304C22×C42C22×C3111yescharacteristic
305C22×C42C22×C3111yescharacteristic
306#400—24no—
307#398—24no—
308#384—24no—
309C2×#95C22×C3112yes—
310C2×#95C22×C3111yescharacteristic
311C2×#95C22×C3112yes—
312C2×#95C22×C3111yescharacteristic
313C2×#95C22×C3112yes—
314C2×#95C22×C3112yes—
315C2×#95C22×C3112yes—
316C2×#95C22×C3112yes—
317C2×#95C22×C3112yes—
318C2×#95C22×C3112yes—
319C2×#95C22×C3112yes—
320C2×C31#240611yescharacteristic
321C2×C31#243111yescharacteristic
322C2×C31#243411yescharacteristic
323C2×C31#247312yes—
324C2×C31#250612yes—
325C2×C31#255212yes—
326C2×C31#271312yes—
327C2×C31#272012yes—
328C2×C31#272112yes—
329C23×C4C23×C3112yes—
330C23×C4C23×C3112yes—
331C23×C4C23×C3111yescharacteristic
332C23×C4C23×C3111yescharacteristic
333C23×C4C23×C3111yescharacteristic
334C23×C4C23×C3111yescharacteristic
335C23×C4C23×C3111yescharacteristic
336C23×C4C23×C3112yes—
337C23×C4C23×C3111yescharacteristic
338C23×C4C23×C3111yescharacteristic
339C23×C4C23×C3112yes—
340#125—232no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—24no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—24no—
363C2×C4⋊C4—24no—
364C2×C4⋊C4—42no—
365C2×C4⋊C4—42no—
366C2×C4⋊C4—42no—
367C2×C4⋊C4—42no—
368C2×C4⋊C4D4×C3114yes—
369C2×C4⋊C4D4×C3114yes—
370C2×C4⋊C4D4×C3114yes—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—24no—
376#95—22no—
377#95—24no—
378#95—22no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—42no—
385#95—42no—
386#95Q8×C3118yes—
387C31#1585011yescharacteristic, direct product factor, Sylow
388C24C24×C3111yescharacteristic
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4C2×D4×C3112yes—
439C22×C4C2×D4×C3112yes—
440C22×C4C2×D4×C3112yes—
441C22×C4C2×D4×C3112yes—
442C22×C4C2×D4×C3112yes—
443C22×C4C2×D4×C3112yes—
444C22×C4C2×D4×C3112yes—
445C22×C4C2×D4×C3112yes—
446C22×C4C2×D4×C3112yes—
447C22×C4C2×Q8×C3114yes—
448C22×C4C2×Q8×C3114yes—
449C22×C4C2×Q8×C3114yes—
450C22×C4#41×C3114yes—
451C22×C4#41×C3114yes—
452C4⋊C4—28no—
453C4⋊C4—28no—
454C4⋊C4—28no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C42—28no—
463C42—28no—
464C23C22×D4×C3111yescharacteristic
465C23C22×D4×C3111yescharacteristic
466C23C22×D4×C3111yescharacteristic
467C23C22×Q8×C3112yes—
468C23C2×#41×C3112yes—
469C23C2×#41×C3112yes—
470C23C31×H(5,𝔽2)12yes—
471C23C31×2−1+412yes—
472C23C31×2−1+412yes—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4C31×#12014yes—
534C2×C4C31×#12518yescomplemented
535C2×C4C31×C4⋊Q814yes—
536C22C2×C31×#12011yescharacteristic
537C22C31×#52511yescharacteristic
538C22C2×C31×#12512yes—
539C22C2×C31×C4⋊Q811yescharacteristic
540C22C31×#53211yescharacteristic
541C22C31×#53511yescharacteristic
542C22C31×#53611yescharacteristic
543C22C31×#53611yescharacteristic
544C22C31×#53912yes—
545C22C31×#54112yes—
546C22C31×#54412yes—
547C22C31×#54612yes—
548C22C31×#54712yes—
549C22C31×#54712yes—
550C22D4×Q8×C3112yes—
551C22D4×Q8×C3112yes—
552C22C31×#55012yes—
553C22C31×#55212yes—
554C22C31×#55412yes—
555C22C31×#55712yes—
556C22C31×#56312yes—
557C4—24no—
558C4—24no—
559C4—28no—
560C4—42no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—42no—
565C4—44no—
566C4—82no—
567C4—82no—
568C2C31×#240611yescharacteristic
569C2C31×#243111yescharacteristic
570C2C31×#243411yescharacteristic
571C2C31×#247312yes—
572C2C31×#250612yes—
573C2C31×#255212yes—
574C2C31×#271312yes—
575C2C31×#272012yes—
576C2C31×#272112yes—
subgroupG/H
center, socle, prime radicalC24×C31C24
commutator, Frattini, cosocleC24C24×C31
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
175. #15850C31direct (N × H)
246. C2×C4×C31#125semidirect
387. C31#15850direct (N × H)
534. C2×C4C31×#125semidirect

Maximal subgroups

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

subgroupindex
C22×C31×#1252
C2×C31×#4002
C2×C31×#3982
C2×C31×#3842
C31×#14452
C31×#14442
C31×#14412
C31×#14362
#1585031