Finite Group Databasebeta

← C31×#15100 #21471859

Subgroups — C31×#15100 #21471859

Order 7936.

1378 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#388C211yesmaximal, characteristic
2C2×C4×C4⋊C4×C31C211yesmaximal, characteristic, complemented
3C31×#1444C211yesmaximal, characteristic, complemented
4C31×#1442C211yesmaximal, characteristic, complemented
5C31×#1442C212yesmaximal, complemented
6C31×#1439C211yesmaximal, characteristic
7C31×#1439C211yesmaximal, characteristic
8C31×#1439C212yesmaximal
9C31×#1437C212yesmaximal
10C31×#1436C212yesmaximal, complemented
11C31×#1434C211yesmaximal, characteristic, complemented
12C22×C4⋊C4×C31C2211yescharacteristic
13C22×C4⋊C4×C31C2211yescharacteristic
14C22×C4⋊C4×C31C2212yes—
15C22×C4⋊C4×C31C2211yescharacteristic
16C22×C22⋊C4×C31C2211yescharacteristic
17C22×C22⋊C4×C31C2212yes—
18C22×C22⋊C4×C31C2211yescharacteristic
19C22×C22⋊C4×C31C2212yes—
20C22×C22⋊C4×C31C2211yescharacteristic
21C22×C42×C31C2211yescharacteristic
22C22×C42×C31C2212yes—
23C31×#388—24no—
24C4×C4⋊C4×C31—24no—
25C2×C31×#95C2212yes—
26C2×C31×#95C2211yescharacteristic
27C2×C31×#95C2212yes—
28C2×C31×#95C2211yescharacteristic
29C2×C31×#95C2211yescharacteristic
30C2×C31×#95C2212yes—
31C2×C31×#95C2212yes—
32C2×C31×#95C2212yes—
33C2×C31×#95C2211yescharacteristic
34C2×C31×#95C2212yes—
35C2×C31×#95C2212yes—
36C2×C31×#95C2212yes—
37C25×C31C2311yescharacteristic
38C23×C4×C31C2311yescharacteristic
39C23×C4×C31C2311yescharacteristic
40C23×C4×C31C2312yes—
41C23×C4×C31C2311yescharacteristic
42C23×C4×C31C2311yescharacteristic
43C23×C4×C31C2312yes—
44C23×C4×C31C2312yes—
45C23×C4×C31C2312yes—
46C23×C4×C31C2311yescharacteristic
47C23×C4×C31C2311yescharacteristic
48C2×C4⋊C4×C31—22no—
49C2×C4⋊C4×C31—22no—
50C2×C4⋊C4×C31—24no—
51C2×C4⋊C4×C31—24no—
52C2×C4⋊C4×C31—22no—
53C2×C4⋊C4×C31—24no—
54C2×C4⋊C4×C31—42no—
55C2×C4⋊C4×C31—42no—
56C2×C4⋊C4×C31—42no—
57C2×C22⋊C4×C31—22no—
58C2×C22⋊C4×C31—22no—
59C2×C22⋊C4×C31—22no—
60C2×C22⋊C4×C31—22no—
61C2×C22⋊C4×C31—24no—
62C2×C22⋊C4×C31—24no—
63C2×C22⋊C4×C31—24no—
64C2×C22⋊C4×C31—22no—
65C2×C22⋊C4×C31—22no—
66C2×C22⋊C4×C31—42no—
67C2×C22⋊C4×C31—42no—
68C2×C22⋊C4×C31—42no—
69C2×C22⋊C4×C31—42no—
70C2×C22⋊C4×C31Q814yes—
71C2×C42×C31—22no—
72C2×C42×C31—22no—
73C2×C42×C31—24no—
74C2×C42×C31—42no—
75C2×C42×C31—42no—
76C2×C42×C31D414yes—
77C31×#95—24no—
78C31×#95—22no—
79C31×#95—22no—
80C31×#95—22no—
81C31×#95—24no—
82C31×#95—24no—
83C31×#95—24no—
84C31×#95—24no—
85C31×#95—22no—
86C31×#95—42no—
87C31×#95—42no—
88C31×#95—42no—
89C24×C31—21no—
90C24×C31—21no—
91C24×C31—22no—
92C24×C31—22no—
93C24×C31—21no—
94C24×C31—21no—
95C24×C31—22no—
96C24×C31—22no—
97C24×C31—21no—
98C24×C31—21no—
99C24×C31C2×Q812yes—
100C24×C31C2411yescharacteristic
101C22×C4×C31—21no—
102C22×C4×C31—21no—
103C22×C4×C31—22no—
104C22×C4×C31—22no—
105C22×C4×C31—21no—
106C22×C4×C31—21no—
107C22×C4×C31—22no—
108C22×C4×C31—22no—
109C22×C4×C31—21no—
110C22×C4×C31—22no—
111C22×C4×C31—22no—
112C22×C4×C31—22no—
113C22×C4×C31—22no—
114C22×C4×C31—22no—
115C22×C4×C31—21no—
116C22×C4×C31—21no—
117C22×C4×C31—21no—
118C22×C4×C31—22no—
119C22×C4×C31—22no—
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—21no—
134C22×C4×C31—21no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—22no—
140C22×C4×C31—21no—
141C22×C4×C31—21no—
142C22×C4×C31—22no—
143C22×C4×C31—22no—
144C22×C4×C31—21no—
145C22×C4×C31—22no—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—22no—
149C22×C4×C31—21no—
150C22×C4×C31C2×D412yes—
151C22×C4×C31C2×D412yes—
152C22×C4×C31C2×D412yes—
153C22×C4×C31C2×Q812yes—
154C22×C4×C31C2×Q812yes—
155C22×C4×C31#4114yes—
156C22×C4×C31#4114yes—
157C22×C4×C31#4114yes—
158C22×C4×C31#4112yes—
159C22×C4×C31#4112yes—
160C22×C4×C31#4112yes—
161C4⋊C4×C31—28no—
162C4⋊C4×C31—28no—
163C4⋊C4×C31—44no—
164C4⋊C4×C31—48no—
165C22⋊C4×C31—28no—
166C22⋊C4×C31—48no—
167C22⋊C4×C31—44no—
168C22⋊C4×C31—48no—
169C22⋊C4×C31—44no—
170C42×C31—28no—
171C42×C31—48no—
172#15100C3111yesmaximal, characteristic, direct product factor, Sylow
173C23×C31—22no—
174C23×C31—24no—
175C23×C31—22no—
176C23×C31—24no—
177C23×C31—22no—
178C23×C31—41no—
179C23×C31—41no—
180C23×C31—41no—
181C23×C31—42no—
182C23×C31—42no—
183C23×C31—42no—
184C23×C31—41no—
185C23×C31—41no—
186C23×C31—41no—
187C23×C31—42no—
188C23×C31—42no—
189C23×C31—42no—
190C23×C31—42no—
191C23×C31—42no—
192C23×C31—41no—
193C23×C31—41no—
194C23×C31—42no—
195C23×C31—42no—
196C23×C31C22×D411yescharacteristic
197C23×C31C22×Q811yescharacteristic
198C23×C31C2×#4111yescharacteristic
199C23×C31C2×#4111yescharacteristic
200C23×C31C2×#4111yescharacteristic
201C23×C31C2×#4112yes—
202C23×C31C2×#4112yes—
203C23×C31C2×#4112yes—
204C23×C31H(5,𝔽2)11yescharacteristic
205C23×C31H(5,𝔽2)11yescharacteristic
206C23×C31H(5,𝔽2)12yes—
207C2×C4×C31—22no—
208C2×C4×C31—22no—
209C2×C4×C31—22no—
210C2×C4×C31—24no—
211C2×C4×C31—24no—
212C2×C4×C31—22no—
213C2×C4×C31—22no—
214C2×C4×C31—22no—
215C2×C4×C31—22no—
216C2×C4×C31—22no—
217C2×C4×C31—22no—
218C2×C4×C31—22no—
219C2×C4×C31—24no—
220C2×C4×C31—24no—
221C2×C4×C31—24no—
222C2×C4×C31—22no—
223C2×C4×C31—22no—
224C2×C4×C31—24no—
225C2×C4×C31—24no—
226C2×C4×C31—24no—
227C2×C4×C31—24no—
228C2×C4×C31—24no—
229C2×C4×C31—24no—
230C2×C4×C31—24no—
231C2×C4×C31—41no—
232C2×C4×C31—42no—
233C2×C4×C31—42no—
234C2×C4×C31—42no—
235C2×C4×C31—42no—
236C2×C4×C31—41no—
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—42no—
246C2×C4×C31—42no—
247C2×C4×C31—42no—
248C2×C4×C31—41no—
249C2×C4×C31—41no—
250C2×C4×C31—42no—
251C2×C4×C31—42no—
252C2×C4×C31—41no—
253C2×C4×C31—41no—
254C2×C4×C31—41no—
255C2×C4×C31—41no—
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×C31C22⋊Q814yes—
266C2×C4×C31#12314yes—
267C2×#388C2×C3111yescharacteristic
268C2×C4×C4⋊C4C2×C3111yescharacteristic, complemented
269#1444C2×C3111yescharacteristic, complemented
270#1442C2×C3111yescharacteristic, complemented
271#1442C2×C3112yescomplemented
272#1439C2×C3111yescharacteristic
273#1439C2×C3111yescharacteristic
274#1439C2×C3112yes—
275#1437C2×C3112yes—
276#1436C2×C3112yescomplemented
277#1434C2×C3111yescharacteristic, complemented
278C22×C31—44no—
279C22×C31—42no—
280C22×C31—44no—
281C22×C31—42no—
282C22×C31—42no—
283C22×C31—82no—
284C22×C31—82no—
285C22×C31—81no—
286C22×C31—81no—
287C22×C31—81no—
288C22×C31—81no—
289C22×C31C2×C22⋊Q811yescharacteristic
290C22×C31C2×#12311yescharacteristic
291C22×C31#52511yescharacteristic
292C22×C31#52511yescharacteristic
293C22×C31#52912yes—
294C22×C31#52912yes—
295C22×C31#52912yes—
296C22×C31#53211yescharacteristic
297C22×C31#53312yes—
298C22×C31#53511yescharacteristic
299C22×C31#53511yescharacteristic
300C22×C31#53812yes—
301C22×C31#53812yes—
302C22×C31#55112yes—
303C22×C31#55312yes—
304C22×C31#55312yes—
305C22×C31#55311yescharacteristic
306C22×C31#55311yescharacteristic
307C22×C31#55411yescharacteristic
308C22×C31#55412yes—
309C22×C31#55411yescharacteristic
310C22×C31#55512yes—
311C22×C31#55912yes—
312C4×C31—24no—
313C4×C31—24no—
314C4×C31—42no—
315C4×C31—44no—
316C4×C31—44no—
317C4×C31—44no—
318C4×C31—42no—
319C4×C31—82no—
320C4×C31—81no—
321C4×C31—81no—
322C22×C4⋊C4C22×C3111yescharacteristic
323C22×C4⋊C4C22×C3111yescharacteristic
324C22×C4⋊C4C22×C3112yes—
325C22×C4⋊C4C22×C3111yescharacteristic
326C22×C22⋊C4C22×C3111yescharacteristic
327C22×C22⋊C4C22×C3112yes—
328C22×C22⋊C4C22×C3111yescharacteristic
329C22×C22⋊C4C22×C3112yes—
330C22×C22⋊C4C22×C3111yescharacteristic
331C22×C42C22×C3111yescharacteristic
332C22×C42C22×C3112yes—
333#388—24no—
334C4×C4⋊C4—24no—
335C2×#95C22×C3112yes—
336C2×#95C22×C3111yescharacteristic
337C2×#95C22×C3112yes—
338C2×#95C22×C3111yescharacteristic
339C2×#95C22×C3111yescharacteristic
340C2×#95C22×C3112yes—
341C2×#95C22×C3112yes—
342C2×#95C22×C3112yes—
343C2×#95C22×C3111yescharacteristic
344C2×#95C22×C3112yes—
345C2×#95C22×C3112yes—
346C2×#95C22×C3112yes—
347C2×C31—82no—
348C2×C31#239611yescharacteristic
349C2×C31#248711yescharacteristic
350C2×C31#250011yescharacteristic
351C2×C31#257412yes—
352C2×C31#258011yescharacteristic
353C2×C31#258712yes—
354C2×C31#259312yes—
355C2×C31#259512yes—
356C2×C31#260311yescharacteristic
357C2×C31#263111yescharacteristic
358C2×C31#285311yescharacteristic
359C25C23×C3111yescharacteristic
360C23×C4C23×C3111yescharacteristic
361C23×C4C23×C3111yescharacteristic
362C23×C4C23×C3112yes—
363C23×C4C23×C3111yescharacteristic
364C23×C4C23×C3111yescharacteristic
365C23×C4C23×C3112yes—
366C23×C4C23×C3112yes—
367C23×C4C23×C3112yes—
368C23×C4C23×C3111yescharacteristic
369C23×C4C23×C3111yescharacteristic
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—24no—
373C2×C4⋊C4—24no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—24no—
376C2×C4⋊C4—42no—
377C2×C4⋊C4—42no—
378C2×C4⋊C4—42no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—24no—
386C2×C22⋊C4—22no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—42no—
389C2×C22⋊C4—42no—
390C2×C22⋊C4—42no—
391C2×C22⋊C4—42no—
392C2×C22⋊C4Q8×C3114yes—
393C2×C42—22no—
394C2×C42—22no—
395C2×C42—24no—
396C2×C42—42no—
397C2×C42—42no—
398C2×C42D4×C3114yes—
399#95—24no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95—22no—
408#95—42no—
409#95—42no—
410#95—42no—
411C31#1510011yescharacteristic, direct product factor, Sylow
412C24—21no—
413C24—21no—
414C24—22no—
415C24—22no—
416C24—21no—
417C24—21no—
418C24—22no—
419C24—22no—
420C24—21no—
421C24—21no—
422C24C2×Q8×C3112yes—
423C24C24×C3111yescharacteristic
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
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—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
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—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4C2×D4×C3112yes—
474C22×C4C2×D4×C3112yes—
475C22×C4C2×D4×C3112yes—
476C22×C4C2×Q8×C3112yes—
477C22×C4C2×Q8×C3112yes—
478C22×C4#41×C3114yes—
479C22×C4#41×C3114yes—
480C22×C4#41×C3114yes—
481C22×C4#41×C3112yes—
482C22×C4#41×C3112yes—
483C22×C4#41×C3112yes—
484C4⋊C4—28no—
485C4⋊C4—28no—
486C4⋊C4—44no—
487C4⋊C4—48no—
488C22⋊C4—28no—
489C22⋊C4—48no—
490C22⋊C4—44no—
491C22⋊C4—48no—
492C22⋊C4—44no—
493C42—28no—
494C42—48no—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—24no—
499C23—22no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—42no—
518C23C22×D4×C3111yescharacteristic
519C23C22×Q8×C3111yescharacteristic
520C23C2×#41×C3111yescharacteristic
521C23C2×#41×C3111yescharacteristic
522C23C2×#41×C3111yescharacteristic
523C23C2×#41×C3112yes—
524C23C2×#41×C3112yes—
525C23C2×#41×C3112yes—
526C23C31×H(5,𝔽2)11yescharacteristic
527C23C31×H(5,𝔽2)11yescharacteristic
528C23C31×H(5,𝔽2)12yes—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
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—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
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×C4C31×C22⋊Q814yes—
588C2×C4C31×#12314yes—
589C22—44no—
590C22—42no—
591C22—44no—
592C22—42no—
593C22—42no—
594C22—82no—
595C22—82no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22C2×C31×C22⋊Q811yescharacteristic
601C22C2×C31×#12311yescharacteristic
602C22C31×#52511yescharacteristic
603C22C31×#52511yescharacteristic
604C22C31×#52912yes—
605C22C31×#52912yes—
606C22C31×#52912yes—
607C22C31×#53211yescharacteristic
608C22C31×#53312yes—
609C22C31×#53511yescharacteristic
610C22C31×#53511yescharacteristic
611C22C31×#53812yes—
612C22C31×#53812yes—
613C22C31×#55112yes—
614C22C31×#55312yes—
615C22C31×#55312yes—
616C22C31×#55311yescharacteristic
617C22C31×#55311yescharacteristic
618C22C31×#55411yescharacteristic
619C22C31×#55412yes—
620C22C31×#55411yescharacteristic
621C22C31×#55512yes—
622C22C31×#55912yes—
623C4—24no—
624C4—24no—
625C4—42no—
626C4—44no—
627C4—44no—
628C4—44no—
629C4—42no—
630C4—82no—
631C4—81no—
632C4—81no—
633C2—82no—
634C2C31×#239611yescharacteristic
635C2C31×#248711yescharacteristic
636C2C31×#250011yescharacteristic
637C2C31×#257412yes—
638C2C31×#258011yescharacteristic
639C2C31×#258712yes—
640C2C31×#259312yes—
641C2C31×#259512yes—
642C2C31×#260311yescharacteristic
643C2C31×#263111yescharacteristic
644C2C31×#285311yescharacteristic
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×C4×C4⋊C4×C31C2semidirect
3. C31×#1444C2semidirect
4. C31×#1442C2semidirect
5. C31×#1442C2semidirect
10. C31×#1436C2semidirect
11. C31×#1434C2semidirect
172. #15100C31direct (N × H)
268. C2×C4×C4⋊C4C2×C31semidirect
269. #1444C2×C31semidirect
270. #1442C2×C31semidirect
271. #1442C2×C31semidirect
276. #1436C2×C31semidirect
277. #1434C2×C31semidirect
411. C31#15100direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3882
C2×C4×C4⋊C4×C312
C31×#14442
C31×#14422
C31×#14392
C31×#14372
C31×#14362
C31×#14342
#1510031