Finite Group Databasebeta

← C31×#14641 #21471400

Subgroups — C31×#14641 #21471400

Order 7936.

1486 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×#394C211yesmaximal, characteristic
2C31×#1445C211yesmaximal, characteristic, complemented
3C31×#1442C212yesmaximal, complemented
4C31×#1440C212yesmaximal, complemented
5C31×#1439C212yesmaximal, complemented
6C31×#1439C212yesmaximal, complemented
7C31×#1438C212yesmaximal
8C31×#1437C212yesmaximal, complemented
9C31×#1436C211yesmaximal, characteristic, complemented
10C23×D4×C31C2211yescharacteristic
11C22×C4⋊C4×C31C2212yes—
12C22×C4⋊C4×C31C2211yescharacteristic
13C22×C22⋊C4×C31C2212yes—
14C22×C22⋊C4×C31C2212yes—
15C22×C22⋊C4×C31C2212yes—
16C22×C22⋊C4×C31C2212yes—
17C22×C22⋊C4×C31C2212yes—
18C22×C22⋊C4×C31C2212yes—
19C22×C42×C31C2211yescharacteristic
20C31×#394C418yescomplemented
21C2×C31×#95C2212yes—
22C2×C31×#95C2212yes—
23C2×C31×#95C2211yescharacteristic
24C2×C31×#95C2212yes—
25C2×C31×#95C2212yes—
26C2×C31×#95C2212yes—
27C2×C31×#95C2212yes—
28C2×C31×#95C2211yescharacteristic
29C2×C31×#95C2212yes—
30C2×C31×#95C2211yescharacteristic
31C2×C31×#95C2211yescharacteristic
32C25×C31C2312yes—
33C22×D4×C31—42no—
34C22×D4×C31—42no—
35C22×D4×C31—41no—
36C22×D4×C31—41no—
37C22×D4×C31C2×C414yes—
38C23×C4×C31C2312yes—
39C23×C4×C31C2312yes—
40C23×C4×C31C2312yes—
41C23×C4×C31C2311yescharacteristic
42C23×C4×C31C2311yescharacteristic
43C23×C4×C31C2312yes—
44C23×C4×C31C2311yescharacteristic
45C23×C4×C31C2312yes—
46C2×C4⋊C4×C31—41no—
47C2×C4⋊C4×C31—41no—
48C2×C4⋊C4×C31—42no—
49C2×C4⋊C4×C31—42no—
50C2×C4⋊C4×C31—42no—
51C2×C4⋊C4×C31C2×C414yes—
52C2×C22⋊C4×C31—24no—
53C2×C22⋊C4×C31—24no—
54C2×C22⋊C4×C31—42no—
55C2×C22⋊C4×C31—42no—
56C2×C22⋊C4×C31—42no—
57C2×C22⋊C4×C31—42no—
58C2×C22⋊C4×C31—42no—
59C2×C22⋊C4×C31—42no—
60C2×C22⋊C4×C31—42no—
61C2×C22⋊C4×C31—42no—
62C2×C22⋊C4×C31—42no—
63C2×C22⋊C4×C31—42no—
64C2×C22⋊C4×C31—42no—
65C2×C22⋊C4×C31—42no—
66C2×C22⋊C4×C31—42no—
67C2×C22⋊C4×C31—42no—
68C2×C22⋊C4×C31C2×C418yes—
69C2×C22⋊C4×C31C2×C418yes—
70C2×C42×C31—42no—
71C2×C42×C31—41no—
72C31×#95—24no—
73C31×#95—42no—
74C31×#95—42no—
75C31×#95—42no—
76C31×#95—42no—
77C31×#95—42no—
78C31×#95—41no—
79C31×#95—41no—
80C31×#95—41no—
81C31×#95—42no—
82C31×#95C2×C414yes—
83C24×C31—22no—
84C24×C31—22no—
85C24×C31—22no—
86C24×C31—22no—
87C24×C31—22no—
88C24×C31—22no—
89C24×C31—22no—
90C24×C31—22no—
91C24×C31—22no—
92C24×C31—22no—
93C24×C31—22no—
94C24×C31—22no—
95C24×C31—22no—
96C24×C31—22no—
97C24×C31C22×C414yes—
98C24×C31C2411yescharacteristic
99C2×D4×C31—44no—
100C2×D4×C31—48no—
101C2×D4×C31—48no—
102C2×D4×C31—44no—
103C2×D4×C31—44no—
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—22no—
111C22×C4×C31—21no—
112C22×C4×C31—21no—
113C22×C4×C31—21no—
114C22×C4×C31—22no—
115C22×C4×C31—22no—
116C22×C4×C31—22no—
117C22×C4×C31—22no—
118C22×C4×C31—22no—
119C22×C4×C31—22no—
120C22×C4×C31—21no—
121C22×C4×C31—22no—
122C22×C4×C31—22no—
123C22×C4×C31—22no—
124C22×C4×C31—21no—
125C22×C4×C31—21no—
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×C31C22×C414yes—
150C22×C4×C31C22×C412yes—
151C22×C4×C31C22×C412yes—
152C22×C4×C31C22×C412yes—
153C22×C4×C31#4114yes—
154C4⋊C4×C31—44no—
155C4⋊C4×C31—48no—
156C22⋊C4×C31—48no—
157C22⋊C4×C31—48no—
158C22⋊C4×C31—48no—
159C22⋊C4×C31—48no—
160C22⋊C4×C31—48no—
161C22⋊C4×C31—48no—
162C42×C31—44no—
163#14641C3111yesmaximal, characteristic, direct product factor, Sylow
164C23×C31—24no—
165C23×C31—24no—
166C23×C31—24no—
167C23×C31—24no—
168C23×C31—24no—
169C23×C31—24no—
170C23×C31—24no—
171C23×C31—42no—
172C23×C31—42no—
173C23×C31—42no—
174C23×C31—42no—
175C23×C31—42no—
176C23×C31—42no—
177C23×C31—42no—
178C23×C31—42no—
179C23×C31—42no—
180C23×C31—42no—
181C23×C31—42no—
182C23×C31—42no—
183C23×C31—42no—
184C23×C31—42no—
185C23×C31—42no—
186C23×C31—42no—
187C23×C31—42no—
188C23×C31—42no—
189C23×C31—42no—
190C23×C31—42no—
191C23×C31—42no—
192C23×C31—42no—
193C23×C31—42no—
194C23×C31—42no—
195C23×C31—42no—
196C23×C31—42no—
197C23×C31—42no—
198C23×C31—42no—
199C23×C31C23×C411yescharacteristic
200C23×C31C2×#4112yes—
201C23×C31H(5,𝔽2)11yescharacteristic
202C23×C31H(5,𝔽2)12yes—
203C23×C31H(5,𝔽2)12yes—
204C23×C31H(5,𝔽2)11yescharacteristic
205C23×C31H(5,𝔽2)12yes—
206C23×C31H(5,𝔽2)12yes—
207C23×C312−1+412yes—
208D4×C31—416no—
209C2×C4×C31—22no—
210C2×C4×C31—22no—
211C2×C4×C31—24no—
212C2×C4×C31—24no—
213C2×C4×C31—24no—
214C2×C4×C31—22no—
215C2×C4×C31—22no—
216C2×C4×C31—22no—
217C2×C4×C31—24no—
218C2×C4×C31—24no—
219C2×C4×C31—24no—
220C2×C4×C31—24no—
221C2×C4×C31—24no—
222C2×C4×C31—42no—
223C2×C4×C31—42no—
224C2×C4×C31—42no—
225C2×C4×C31—42no—
226C2×C4×C31—42no—
227C2×C4×C31—41no—
228C2×C4×C31—42no—
229C2×C4×C31—42no—
230C2×C4×C31—41no—
231C2×C4×C31—42no—
232C2×C4×C31—41no—
233C2×C4×C31—41no—
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—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×#394C2×C3111yescharacteristic
260#1445C2×C3111yescharacteristic, complemented
261#1442C2×C3112yescomplemented
262#1440C2×C3112yescomplemented
263#1439C2×C3112yescomplemented
264#1439C2×C3112yescomplemented
265#1438C2×C3112yes—
266#1437C2×C3112yescomplemented
267#1436C2×C3111yescharacteristic, complemented
268C22×C31—44no—
269C22×C31—44no—
270C22×C31—44no—
271C22×C31—44no—
272C22×C31—44no—
273C22×C31—44no—
274C22×C31—44no—
275C22×C31—82no—
276C22×C31—82no—
277C22×C31—82no—
278C22×C31—82no—
279C22×C31—82no—
280C22×C31—82no—
281C22×C31—82no—
282C22×C31—82no—
283C22×C31C4×#4111yescharacteristic
284C22×C31#51811yescharacteristic
285C22×C31#51811yescharacteristic
286C22×C31#51812yes—
287C22×C31#52012yes—
288C22×C31#53812yes—
289C22×C31#53812yes—
290C22×C31#53912yes—
291C22×C31#54012yes—
292C22×C31#54012yes—
293C22×C31#54212yes—
294C22×C31#56012yes—
295C22×C31#56012yes—
296C22×C31#56012yes—
297C22×C31#56011yescharacteristic
298C22×C31H(3,𝔽4)11yescharacteristic
299C22×C31H(3,𝔽4)11yescharacteristic
300C22×C31#56212yes—
301C22×C31#56212yes—
302C22×C31#56212yes—
303C22×C31#56311yescharacteristic
304C4×C31—44no—
305C4×C31—42no—
306C4×C31—42no—
307C4×C31—42no—
308C4×C31—44no—
309C4×C31—82no—
310C4×C31—82no—
311C4×C31—82no—
312C23×D4C22×C3111yescharacteristic
313C22×C4⋊C4C22×C3112yes—
314C22×C4⋊C4C22×C3111yescharacteristic
315C22×C22⋊C4C22×C3112yes—
316C22×C22⋊C4C22×C3112yes—
317C22×C22⋊C4C22×C3112yes—
318C22×C22⋊C4C22×C3112yes—
319C22×C22⋊C4C22×C3112yes—
320C22×C22⋊C4C22×C3112yes—
321C22×C42C22×C3111yescharacteristic
322#394C4×C3118yescomplemented
323C2×#95C22×C3112yes—
324C2×#95C22×C3112yes—
325C2×#95C22×C3111yescharacteristic
326C2×#95C22×C3112yes—
327C2×#95C22×C3112yes—
328C2×#95C22×C3112yes—
329C2×#95C22×C3112yes—
330C2×#95C22×C3111yescharacteristic
331C2×#95C22×C3112yes—
332C2×#95C22×C3111yescharacteristic
333C2×#95C22×C3111yescharacteristic
334C2×C31—84no—
335C2×C31#233311yescharacteristic
336C2×C31#233512yes—
337C2×C31#237511yescharacteristic
338C2×C31#237712yes—
339C2×C31#237911yescharacteristic
340C2×C31#282412yes—
341C2×C31#282812yes—
342C2×C31#282912yes—
343C2×C31#283712yes—
344C25C23×C3112yes—
345C22×D4—42no—
346C22×D4—42no—
347C22×D4—41no—
348C22×D4—41no—
349C22×D4C2×C4×C3114yes—
350C23×C4C23×C3112yes—
351C23×C4C23×C3112yes—
352C23×C4C23×C3112yes—
353C23×C4C23×C3111yescharacteristic
354C23×C4C23×C3112yes—
355C23×C4C23×C3111yescharacteristic
356C23×C4C23×C3111yescharacteristic
357C23×C4C23×C3112yes—
358C2×C4⋊C4—41no—
359C2×C4⋊C4—41no—
360C2×C4⋊C4—42no—
361C2×C4⋊C4—42no—
362C2×C4⋊C4—42no—
363C2×C4⋊C4C2×C4×C3114yes—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—42no—
367C2×C22⋊C4—42no—
368C2×C22⋊C4—42no—
369C2×C22⋊C4—42no—
370C2×C22⋊C4—42no—
371C2×C22⋊C4—42no—
372C2×C22⋊C4—42no—
373C2×C22⋊C4—42no—
374C2×C22⋊C4—42no—
375C2×C22⋊C4—42no—
376C2×C22⋊C4—42no—
377C2×C22⋊C4—42no—
378C2×C22⋊C4—42no—
379C2×C22⋊C4—42no—
380C2×C22⋊C4C2×C4×C3118yes—
381C2×C22⋊C4C2×C4×C3118yes—
382C2×C42—42no—
383C2×C42—41no—
384#95—24no—
385#95—42no—
386#95—42no—
387#95—42no—
388#95—42no—
389#95—42no—
390#95—41no—
391#95—41no—
392#95—41no—
393#95—42no—
394#95C2×C4×C3114yes—
395C31#1464111yescharacteristic, direct product factor, Sylow
396C24—22no—
397C24—22no—
398C24—22no—
399C24—22no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—22no—
404C24—22no—
405C24—22no—
406C24—22no—
407C24—22no—
408C24—22no—
409C24—22no—
410C24C22×C4×C3114yes—
411C24C24×C3111yescharacteristic
412C2×D4—44no—
413C2×D4—48no—
414C2×D4—48no—
415C2×D4—44no—
416C2×D4—44no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
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—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4C22×C4×C3114yes—
463C22×C4C22×C4×C3112yes—
464C22×C4C22×C4×C3112yes—
465C22×C4C22×C4×C3112yes—
466C22×C4#41×C3114yes—
467C4⋊C4—44no—
468C4⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C42—44no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23C23×C4×C3111yescharacteristic
512C23C2×#41×C3112yes—
513C23C31×H(5,𝔽2)11yescharacteristic
514C23C31×H(5,𝔽2)12yes—
515C23C31×H(5,𝔽2)12yes—
516C23C31×H(5,𝔽2)11yescharacteristic
517C23C31×H(5,𝔽2)12yes—
518C23C31×H(5,𝔽2)12yes—
519C23C31×2−1+412yes—
520D4—416no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
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—42no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22C4×#41×C3111yescharacteristic
587C22C31×#51811yescharacteristic
588C22C31×#51811yescharacteristic
589C22C31×#51812yes—
590C22C31×#52012yes—
591C22C31×#53812yes—
592C22C31×#53812yes—
593C22C31×#53912yes—
594C22C31×#54012yes—
595C22C31×#54012yes—
596C22C31×#54212yes—
597C22C31×#56012yes—
598C22C31×#56012yes—
599C22C31×#56012yes—
600C22C31×#56011yescharacteristic
601C22C31×H(3,𝔽4)11yescharacteristic
602C22C31×H(3,𝔽4)11yescharacteristic
603C22C31×#56212yes—
604C22C31×#56212yes—
605C22C31×#56212yes—
606C22C31×#56311yescharacteristic
607C4—44no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—44no—
612C4—82no—
613C4—82no—
614C4—82no—
615C2—84no—
616C2C31×#233311yescharacteristic
617C2C31×#233512yes—
618C2C31×#237511yescharacteristic
619C2C31×#237712yes—
620C2C31×#237911yescharacteristic
621C2C31×#282412yes—
622C2C31×#282812yes—
623C2C31×#282912yes—
624C2C31×#283712yes—
subgroupG/H
center, socleC24×C31C24
commutatorC23C23×C4×C31
Frattini, cosocleC24C24×C31
prime radicalC23×D4×C31C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1445C2semidirect
3. C31×#1442C2semidirect
4. C31×#1440C2semidirect
5. C31×#1439C2semidirect
6. C31×#1439C2semidirect
8. C31×#1437C2semidirect
9. C31×#1436C2semidirect
20. C31×#394C4semidirect
163. #14641C31direct (N × H)
260. #1445C2×C31semidirect
261. #1442C2×C31semidirect
262. #1440C2×C31semidirect
263. #1439C2×C31semidirect
264. #1439C2×C31semidirect
266. #1437C2×C31semidirect
267. #1436C2×C31semidirect
322. #394C4×C31semidirect
395. C31#14641direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3942
C31×#14452
C31×#14422
C31×#14402
C31×#14392
C31×#14382
C31×#14372
C31×#14362
#1464131