Finite Group Databasebeta

← C31×#15259 #21472018

Subgroups — C31×#15259 #21472018

Order 7936.

3670 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×#124C211yesmaximal, characteristic, complemented
2C22×C31×#120C211yesmaximal, characteristic
3C2×C31×#393C212yesmaximal, complemented
4C2×C31×#379C212yesmaximal, complemented
5C31×#1440C212yesmaximal, complemented
6C31×#1438C214yesmaximal, complemented
7C31×#1438C211yesmaximal, characteristic, complemented
8C31×#1437C212yesmaximal, complemented
9C26×C31C2211yescharacteristic
10C23×Q8×C31C2211yescharacteristic, complemented
11C23×D4×C31C2212yes—
12C23×D4×C31C2211yescharacteristic
13C2×C31×#124—24no—
14C2×C31×#124—28no—
15C2×C31×#120—24no—
16C2×C31×#120—28no—
17C22×C22⋊C4×C31C2211yescharacteristic
18C22×C22⋊C4×C31C2214yes—
19C22×C22⋊C4×C31C2214yes—
20C22×C22⋊C4×C31C2212yes—
21C22×C22⋊C4×C31C2212yes—
22C22×C22⋊C4×C31C2212yes—
23C22×C22⋊C4×C31C2212yescomplemented
24C22×C22⋊C4×C31C2212yescomplemented
25C22×C42×C31C2211yescharacteristic, complemented
26C31×#393—28no—
27C31×#379—28no—
28C2×C31×#95C2212yescomplemented
29C2×C31×#95C2214yescomplemented
30C2×C31×#95C2214yescomplemented
31C25×C31—22no—
32C25×C31—24no—
33C25×C31—24no—
34C25×C31—28no—
35C25×C31—44no—
36C25×C31—42no—
37C25×C31C2311yescharacteristic
38C25×C31C2311yescharacteristic
39C25×C31C2312yes—
40C22×Q8×C31—24no—
41C22×Q8×C31—22no—
42C22×Q8×C31—24no—
43C22×Q8×C31—42no—
44C22×D4×C31—24no—
45C22×D4×C31—24no—
46C22×D4×C31—24no—
47C22×D4×C31—28no—
48C22×D4×C31—24no—
49C22×D4×C31—24no—
50C22×D4×C31—42no—
51C22×D4×C31—44no—
52C22×D4×C31D414yes—
53C23×C4×C31C2314yes—
54C23×C4×C31C2312yes—
55C23×C4×C31C2312yes—
56C23×C4×C31C2312yes—
57C23×C4×C31C2311yescharacteristic
58C31×#124—232no—
59C31×#120—232no—
60C2×C22⋊C4×C31—22no—
61C2×C22⋊C4×C31—24no—
62C2×C22⋊C4×C31—24no—
63C2×C22⋊C4×C31—24no—
64C2×C22⋊C4×C31—24no—
65C2×C22⋊C4×C31—24no—
66C2×C22⋊C4×C31—24no—
67C2×C22⋊C4×C31—24no—
68C2×C22⋊C4×C31—24no—
69C2×C22⋊C4×C31—24no—
70C2×C22⋊C4×C31—28no—
71C2×C22⋊C4×C31—28no—
72C2×C22⋊C4×C31—28no—
73C2×C22⋊C4×C31—28no—
74C2×C22⋊C4×C31—44no—
75C2×C22⋊C4×C31—44no—
76C2×C22⋊C4×C31—44no—
77C2×C22⋊C4×C31—44no—
78C2×C22⋊C4×C31—42no—
79C2×C22⋊C4×C31D418yes—
80C2×C22⋊C4×C31D418yescomplemented
81C2×C42×C31—24no—
82C2×C42×C31—22no—
83C31×#95—28no—
84C31×#95—24no—
85C31×#95—44no—
86C24×C31—24no—
87C24×C31—22no—
88C24×C31—22no—
89C24×C31—22no—
90C24×C31—24no—
91C24×C31—24no—
92C24×C31—24no—
93C24×C31—24no—
94C24×C31—24no—
95C24×C31—22no—
96C24×C31—24no—
97C24×C31—28no—
98C24×C31—216no—
99C24×C31—22no—
100C24×C31—22no—
101C24×C31—24no—
102C24×C31—22no—
103C24×C31—24no—
104C24×C31—416no—
105C24×C31—416no—
106C24×C31—416no—
107C24×C31—416no—
108C24×C31—48no—
109C24×C31—44no—
110C24×C31—48no—
111C24×C31—48no—
112C24×C31—416no—
113C24×C31—48no—
114C24×C31—44no—
115C24×C31—48no—
116C24×C31C2×D412yes—
117C24×C31C2×D412yes—
118C24×C31C2×D414yes—
119C24×C31C2×D414yes—
120C24×C31#4118yes—
121C24×C31C2411yescharacteristic
122C2×Q8×C31—28no—
123C2×Q8×C31—416no—
124C2×Q8×C31—44no—
125C2×Q8×C31—44no—
126C2×D4×C31—216no—
127C2×D4×C31—28no—
128C2×D4×C31—28no—
129C2×D4×C31—44no—
130C2×D4×C31—416no—
131C2×D4×C31—416no—
132C2×D4×C31—416no—
133C2×D4×C31—48no—
134C2×D4×C31—48no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—24no—
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—24no—
145C22×C4×C31—24no—
146C22×C4×C31—24no—
147C22×C4×C31—24no—
148C22×C4×C31—24no—
149C22×C4×C31—24no—
150C22×C4×C31—24no—
151C22×C4×C31—24no—
152C22×C4×C31—24no—
153C22×C4×C31—24no—
154C22×C4×C31—24no—
155C22×C4×C31—24no—
156C22×C4×C31C2×D414yes—
157C22×C4×C31C2×D414yes—
158C22×C4×C31C2×D414yes—
159C22×C4×C31C2×D414yes—
160C22×C4×C31C2×D412yes—
161C22⋊C4×C31—28no—
162C22⋊C4×C31—216no—
163C22⋊C4×C31—216no—
164C22⋊C4×C31—216no—
165C22⋊C4×C31—416no—
166C22⋊C4×C31—416no—
167C22⋊C4×C31—48no—
168C22⋊C4×C31—48no—
169C42×C31—28no—
170#15259C3111yesmaximal, characteristic, direct product factor, Sylow
171C23×C31—22no—
172C23×C31—24no—
173C23×C31—24no—
174C23×C31—28no—
175C23×C31—22no—
176C23×C31—24no—
177C23×C31—24no—
178C23×C31—28no—
179C23×C31—28no—
180C23×C31—28no—
181C23×C31—28no—
182C23×C31—28no—
183C23×C31—28no—
184C23×C31—28no—
185C23×C31—28no—
186C23×C31—28no—
187C23×C31—28no—
188C23×C31—22no—
189C23×C31—22no—
190C23×C31—22no—
191C23×C31—28no—
192C23×C31—416no—
193C23×C31—432no—
194C23×C31—432no—
195C23×C31—464no—
196C23×C31—464no—
197C23×C31—432no—
198C23×C31—42no—
199C23×C31—42no—
200C23×C31—42no—
201C23×C31—42no—
202C23×C31—44no—
203C23×C31—42no—
204C23×C31—42no—
205C23×C31—44no—
206C23×C31—44no—
207C23×C31—44no—
208C23×C31—44no—
209C23×C31—44no—
210C23×C31—44no—
211C23×C31—42no—
212C23×C31—42no—
213C23×C31—44no—
214C23×C31—42no—
215C23×C31—42no—
216C23×C31—42no—
217C23×C31—42no—
218C23×C31—44no—
219C23×C31—44no—
220C23×C31—44no—
221C23×C31—44no—
222C23×C31—42no—
223C23×C31—42no—
224C23×C31#12014yes—
225C23×C31#12418yescomplemented
226C23×C31C22×D411yescharacteristic
227C23×C31C22×D412yes—
228C23×C31C22×D412yes—
229C23×C31C2×#4114yes—
230C23×C31H(5,𝔽2)12yes—
231C23×C31H(5,𝔽2)14yes—
232Q8×C31—416no—
233D4×C31—416no—
234D4×C31—432no—
235C2×C4×C31—22no—
236C2×C4×C31—22no—
237C2×C4×C31—28no—
238C2×C4×C31—24no—
239C2×C4×C31—24no—
240C2×C4×C31—24no—
241C2×C4×C31—28no—
242C2×C4×C31—28no—
243C2×C4×C31—28no—
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—44no—
251C2×C4×C31—44no—
252C2×C4×C31—44no—
253C2×C4×C31—44no—
254C2×C4×C31—44no—
255C2×C4×C31—44no—
256C2×C4×C31—44no—
257C2×C4×C31—44no—
258C2×C4×C31—44no—
259C2×C4×C31—44no—
260C2×C4×C31#12014yes—
261C22×#124C2×C3111yescharacteristic, complemented
262C22×#120C2×C3111yescharacteristic
263C2×#393C2×C3112yescomplemented
264C2×#379C2×C3112yescomplemented
265#1440C2×C3112yescomplemented
266#1438C2×C3114yescomplemented
267#1438C2×C3111yescharacteristic, complemented
268#1437C2×C3112yescomplemented
269C22×C31—24no—
270C22×C31—28no—
271C22×C31—28no—
272C22×C31—216no—
273C22×C31—464no—
274C22×C31—44no—
275C22×C31—48no—
276C22×C31—48no—
277C22×C31—48no—
278C22×C31—48no—
279C22×C31—44no—
280C22×C31—48no—
281C22×C31—44no—
282C22×C31—44no—
283C22×C31—48no—
284C22×C31—48no—
285C22×C31—42no—
286C22×C31—44no—
287C22×C31—44no—
288C22×C31—44no—
289C22×C31—84no—
290C22×C31—84no—
291C22×C31C2×#12011yescharacteristic
292C22×C31C2×#12011yescharacteristic
293C22×C31#52512yes—
294C22×C31C2×#12412yes—
295C22×C31#53411yescharacteristic
296C22×C31#53512yes—
297C22×C31#53512yes—
298C22×C31#53814yes—
299C22×C31D4214yes—
300C22×C31#54614yes—
301C22×C31#54614yes—
302C22×C31#54614yes—
303C22×C31#55112yes—
304C22×C31H(3,𝔽4)12yes—
305C4×C31—24no—
306C4×C31—44no—
307C4×C31—44no—
308C4×C31—44no—
309C4×C31—84no—
310C26C22×C3111yescharacteristic
311C23×Q8C22×C3111yescharacteristic, complemented
312C23×D4C22×C3112yes—
313C23×D4C22×C3111yescharacteristic
314C2×#124—24no—
315C2×#124—28no—
316C2×#120—24no—
317C2×#120—28no—
318C22×C22⋊C4C22×C3111yescharacteristic
319C22×C22⋊C4C22×C3114yes—
320C22×C22⋊C4C22×C3114yes—
321C22×C22⋊C4C22×C3112yes—
322C22×C22⋊C4C22×C3112yes—
323C22×C22⋊C4C22×C3112yes—
324C22×C22⋊C4C22×C3112yescomplemented
325C22×C22⋊C4C22×C3112yescomplemented
326C22×C42C22×C3111yescharacteristic, complemented
327#393—28no—
328#379—28no—
329C2×#95C22×C3112yescomplemented
330C2×#95C22×C3114yescomplemented
331C2×#95C22×C3114yescomplemented
332C2×C31—44no—
333C2×C31—48no—
334C2×C31—82no—
335C2×C31#240311yescharacteristic
336C2×C31#240412yes—
337C2×C31#243512yes—
338C2×C31#247214yes—
339C2×C31#267014yes—
340C2×C31#268412yes—
341C25—22no—
342C25—24no—
343C25—24no—
344C25—28no—
345C25—44no—
346C25—42no—
347C25C23×C3111yescharacteristic
348C25C23×C3111yescharacteristic
349C25C23×C3112yes—
350C22×Q8—24no—
351C22×Q8—22no—
352C22×Q8—24no—
353C22×Q8—42no—
354C22×D4—24no—
355C22×D4—24no—
356C22×D4—24no—
357C22×D4—28no—
358C22×D4—24no—
359C22×D4—24no—
360C22×D4—42no—
361C22×D4—44no—
362C22×D4D4×C3114yes—
363C23×C4C23×C3114yes—
364C23×C4C23×C3112yes—
365C23×C4C23×C3112yes—
366C23×C4C23×C3112yes—
367C23×C4C23×C3111yescharacteristic
368#124—232no—
369#120—232no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4—24no—
379C2×C22⋊C4—24no—
380C2×C22⋊C4—28no—
381C2×C22⋊C4—28no—
382C2×C22⋊C4—28no—
383C2×C22⋊C4—28no—
384C2×C22⋊C4—44no—
385C2×C22⋊C4—44no—
386C2×C22⋊C4—44no—
387C2×C22⋊C4—44no—
388C2×C22⋊C4—42no—
389C2×C22⋊C4D4×C3118yes—
390C2×C22⋊C4D4×C3118yescomplemented
391C2×C42—24no—
392C2×C42—22no—
393#95—28no—
394#95—24no—
395#95—44no—
396C31#1525911yescharacteristic, direct product factor, Sylow
397C24—24no—
398C24—22no—
399C24—22no—
400C24—22no—
401C24—24no—
402C24—24no—
403C24—24no—
404C24—24no—
405C24—24no—
406C24—22no—
407C24—24no—
408C24—28no—
409C24—216no—
410C24—22no—
411C24—22no—
412C24—24no—
413C24—22no—
414C24—24no—
415C24—416no—
416C24—416no—
417C24—416no—
418C24—416no—
419C24—48no—
420C24—44no—
421C24—48no—
422C24—48no—
423C24—416no—
424C24—48no—
425C24—44no—
426C24—48no—
427C24C2×D4×C3112yes—
428C24C2×D4×C3112yes—
429C24C2×D4×C3114yes—
430C24C2×D4×C3114yes—
431C24#41×C3118yes—
432C24C24×C3111yescharacteristic
433C2×Q8—28no—
434C2×Q8—416no—
435C2×Q8—44no—
436C2×Q8—44no—
437C2×D4—216no—
438C2×D4—28no—
439C2×D4—28no—
440C2×D4—44no—
441C2×D4—416no—
442C2×D4—416no—
443C2×D4—416no—
444C2×D4—48no—
445C2×D4—48no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4C2×D4×C3114yes—
468C22×C4C2×D4×C3114yes—
469C22×C4C2×D4×C3114yes—
470C22×C4C2×D4×C3114yes—
471C22×C4C2×D4×C3112yes—
472C22⋊C4—28no—
473C22⋊C4—216no—
474C22⋊C4—216no—
475C22⋊C4—216no—
476C22⋊C4—416no—
477C22⋊C4—416no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C42—28no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—28no—
485C23—22no—
486C23—24no—
487C23—24no—
488C23—28no—
489C23—28no—
490C23—28no—
491C23—28no—
492C23—28no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—28no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—28no—
502C23—416no—
503C23—432no—
504C23—432no—
505C23—464no—
506C23—464no—
507C23—432no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—44no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—42no—
533C23—42no—
534C23C31×#12014yes—
535C23C31×#12418yescomplemented
536C23C22×D4×C3111yescharacteristic
537C23C22×D4×C3112yes—
538C23C22×D4×C3112yes—
539C23C2×#41×C3114yes—
540C23C31×H(5,𝔽2)12yes—
541C23C31×H(5,𝔽2)14yes—
542Q8—416no—
543D4—416no—
544D4—432no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4C31×#12014yes—
571C22—24no—
572C22—28no—
573C22—28no—
574C22—216no—
575C22—464no—
576C22—44no—
577C22—48no—
578C22—48no—
579C22—48no—
580C22—48no—
581C22—44no—
582C22—48no—
583C22—44no—
584C22—44no—
585C22—48no—
586C22—48no—
587C22—42no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—84no—
592C22—84no—
593C22C2×C31×#12011yescharacteristic
594C22C2×C31×#12011yescharacteristic
595C22C31×#52512yes—
596C22C2×C31×#12412yes—
597C22C31×#53411yescharacteristic
598C22C31×#53512yes—
599C22C31×#53512yes—
600C22C31×#53814yes—
601C22D42×C3114yes—
602C22C31×#54614yes—
603C22C31×#54614yes—
604C22C31×#54614yes—
605C22C31×#55112yes—
606C22C31×H(3,𝔽4)12yes—
607C4—24no—
608C4—44no—
609C4—44no—
610C4—44no—
611C4—84no—
612C2—44no—
613C2—48no—
614C2—82no—
615C2C31×#240311yescharacteristic
616C2C31×#240412yes—
617C2C31×#243512yes—
618C2C31×#247214yes—
619C2C31×#267014yes—
620C2C31×#268412yes—
subgroupG/H
center, socleC24×C31C24
commutator, Frattini, cosocleC24C24×C31
prime radicalC22×C31×#120C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C31×#124C2semidirect
3. C2×C31×#393C2semidirect
4. C2×C31×#379C2semidirect
5. C31×#1440C2semidirect
6. C31×#1438C2semidirect
7. C31×#1438C2semidirect
8. C31×#1437C2semidirect
10. C23×Q8×C31C22semidirect
23. C22×C22⋊C4×C31C22semidirect
24. C22×C22⋊C4×C31C22semidirect
25. C22×C42×C31C22semidirect
28. C2×C31×#95C22semidirect
29. C2×C31×#95C22semidirect
30. C2×C31×#95C22semidirect
80. C2×C22⋊C4×C31D4semidirect
170. #15259C31direct (N × H)
225. C23×C31#124semidirect
261. C22×#124C2×C31semidirect
263. C2×#393C2×C31semidirect
264. C2×#379C2×C31semidirect
265. #1440C2×C31semidirect
266. #1438C2×C31semidirect
267. #1438C2×C31semidirect
268. #1437C2×C31semidirect
311. C23×Q8C22×C31semidirect
324. C22×C22⋊C4C22×C31semidirect
325. C22×C22⋊C4C22×C31semidirect
326. C22×C42C22×C31semidirect
329. C2×#95C22×C31semidirect
330. C2×#95C22×C31semidirect
331. C2×#95C22×C31semidirect
390. C2×C22⋊C4D4×C31semidirect
396. C31#15259direct (N × H)
535. C23C31×#124semidirect

Maximal subgroups

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

subgroupindex
C22×C31×#1242
C22×C31×#1202
C2×C31×#3932
C2×C31×#3792
C31×#14402
C31×#14382
C31×#14372
#1525931