Finite Group Databasebeta

← C31×#36727 #21493486

Subgroups — C31×#36727 #21493486

Order 7936.

2214 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2847C212yesmaximal, complemented
2C31×#2748C214yesmaximal, complemented
3C31×#2741C214yesmaximal, complemented
4C31×#2703C214yesmaximal, complemented
5C31×#2671C214yesmaximal, complemented
6C31×#2656C211yesmaximal, characteristic, complemented
7C31×#2646C211yesmaximal, characteristic, complemented
8C31×#2443C214yesmaximal, complemented
9C31×#2424C214yesmaximal, complemented
10C31×#2325C212yesmaximal, complemented
11C2×C31×#380C211yesmaximal, characteristic, complemented
12C24×C4×C31C2211yescharacteristic
13C2×C31×#123C2214yescomplemented
14C2×C31×#121C2212yes—
15C2×C31×#121C2212yes—
16C2×C31×#121C2212yes—
17C2×C31×#121C2214yes—
18C2×C31×#121C2214yes—
19C2×C31×#120C2214yes—
20C2×C4×D4×C31C2214yes—
21C22×C4⋊C4×C31C2212yes—
22C22×C22⋊C4×C31C2212yes—
23C31×#400C2214yescomplemented
24C31×#399C2214yescomplemented
25C31×#399C2214yescomplemented
26C31×#398C2214yescomplemented
27C31×#398C2214yescomplemented
28C31×#397C2214yescomplemented
29C31×#396C2214yes—
30C31×#394C2214yes—
31C31×#394C2214yes—
32C31×#394C2214yescomplemented
33C31×#392C2214yes—
34C31×#392C2214yescomplemented
35C31×#390C2214yes—
36C31×#388C2214yes—
37C31×#388C2214yescomplemented
38C31×#388C2214yescomplemented
39C31×#387C2214yescomplemented
40C31×#387C2214yescomplemented
41C31×#386C2214yes—
42C31×#386C2214yes—
43C31×#386C2214yes—
44C31×#386C2214yes—
45C31×#386C2214yes—
46C31×#386C2214yescomplemented
47C31×#385C2214yes—
48C31×#385C2214yescomplemented
49C31×#385C2214yescomplemented
50C31×#380C2212yes—
51C31×#380C2212yes—
52C31×#380C2212yes—
53C31×#380C2214yes—
54C31×#380C2214yes—
55C2×C31×#95C2211yescharacteristic
56C2×C31×#95C2211yescharacteristic
57C25×C31C2311yescharacteristic
58C22×D4×C31C2314yes—
59C22×D4×C31C2314yes—
60C22×D4×C31C2314yes—
61C22×D4×C31C2312yes—
62C22×D4×C31C2314yes—
63C23×C4×C31—24no—
64C23×C4×C31—44no—
65C23×C4×C31C2311yescharacteristic
66C23×C4×C31C2312yes—
67C23×C4×C31C2312yes—
68C23×C4×C31C2312yes—
69C23×C4×C31C2312yes—
70C23×C4×C31C2314yes—
71C23×C4×C31C2311yescharacteristic
72C31×#123—48no—
73C31×#121—28no—
74C31×#121—44no—
75C31×#121—44no—
76C31×#121—48no—
77C31×#121—48no—
78C31×#120—48no—
79C4×D4×C31—48no—
80C2×C4⋊C4×C31—44no—
81C2×C4⋊C4×C31C2314yes—
82C2×C4⋊C4×C31C2312yes—
83C2×C4⋊C4×C31C2314yes—
84C2×C4⋊C4×C31C2314yes—
85C2×C4⋊C4×C31C2314yes—
86C2×C4⋊C4×C31C2314yes—
87C2×C4⋊C4×C31C2312yes—
88C2×C4⋊C4×C31C2314yes—
89C2×C4⋊C4×C31C2312yes—
90C2×C22⋊C4×C31—44no—
91C2×C22⋊C4×C31C2314yes—
92C2×C22⋊C4×C31C2314yes—
93C2×C22⋊C4×C31C2314yes—
94C2×C22⋊C4×C31C2314yes—
95C2×C22⋊C4×C31C2314yes—
96C2×C22⋊C4×C31C2314yes—
97C2×C22⋊C4×C31C2314yes—
98C2×C22⋊C4×C31C2314yes—
99C2×C22⋊C4×C31C2314yes—
100C2×C22⋊C4×C31C2314yes—
101C2×C22⋊C4×C31C2314yes—
102C2×C22⋊C4×C31C2314yes—
103C2×C22⋊C4×C31C2314yes—
104C2×C42×C31C2314yes—
105C31×#95C2314yes—
106C31×#95C2314yes—
107C31×#95C2314yes—
108C31×#95C2314yes—
109C31×#95C2314yes—
110C31×#95C2314yes—
111C31×#95C2314yes—
112C31×#95C2314yes—
113C31×#95C2314yes—
114C24×C31—22no—
115C24×C31—22no—
116C24×C31—22no—
117C24×C31—24no—
118C24×C31—42no—
119C24×C31C2414yes—
120C24×C31C2411yescharacteristic
121C24×C31C2411yescharacteristic
122C24×C31C2411yescharacteristic
123C2×D4×C31—28no—
124C2×D4×C31—28no—
125C2×D4×C31—28no—
126C2×D4×C31—28no—
127C2×D4×C31—28no—
128C2×D4×C31—24no—
129C2×D4×C31—44no—
130C2×D4×C31—44no—
131C2×D4×C31—42no—
132C2×D4×C31—42no—
133C2×D4×C31—44no—
134C2×D4×C31—44no—
135C2×D4×C31—44no—
136C2×D4×C31—44no—
137C2×D4×C31—44no—
138C22×C4×C31—22no—
139C22×C4×C31—24no—
140C22×C4×C31—24no—
141C22×C4×C31—24no—
142C22×C4×C31—24no—
143C22×C4×C31—24no—
144C22×C4×C31—24no—
145C22×C4×C31—24no—
146C22×C4×C31—42no—
147C22×C4×C31—42no—
148C22×C4×C31—42no—
149C22×C4×C31—42no—
150C22×C4×C31—42no—
151C22×C4×C31—42no—
152C22×C4×C31—42no—
153C22×C4×C31—44no—
154C22×C4×C31—44no—
155C22×C4×C31—44no—
156C22×C4×C31—416no—
157C22×C4×C31#4114yes—
158C22×C4×C31C2414yes—
159C22×C4×C31C2414yes—
160C22×C4×C31C2414yes—
161C22×C4×C31C2414yes—
162C22×C4×C31C2414yes—
163C22×C4×C31C2412yes—
164C22×C4×C31C2412yes—
165C4⋊C4×C31—24no—
166C4⋊C4×C31—28no—
167C4⋊C4×C31—44no—
168C4⋊C4×C31—44no—
169C4⋊C4×C31—42no—
170C4⋊C4×C31—42no—
171C4⋊C4×C31—44no—
172C4⋊C4×C31—44no—
173C4⋊C4×C31—44no—
174C22⋊C4×C31—28no—
175C22⋊C4×C31—28no—
176C22⋊C4×C31—28no—
177C22⋊C4×C31—28no—
178C22⋊C4×C31—28no—
179C22⋊C4×C31—44no—
180C22⋊C4×C31—44no—
181C22⋊C4×C31—44no—
182C22⋊C4×C31—44no—
183C22⋊C4×C31—44no—
184C22⋊C4×C31—44no—
185C22⋊C4×C31—44no—
186C22⋊C4×C31—44no—
187C42×C31—44no—
188#36727C3111yesmaximal, characteristic, direct product factor, Sylow
189C23×C31—22no—
190C23×C31—21no—
191C23×C31—21no—
192C23×C31—28no—
193C23×C31—21no—
194C23×C31—21no—
195C23×C31—21no—
196C23×C31—22no—
197C23×C31—22no—
198C23×C31—22no—
199C23×C31—22no—
200C23×C31—21no—
201C23×C31—24no—
202C23×C31—24no—
203C23×C31—24no—
204C23×C31—24no—
205C23×C31—24no—
206C23×C31—24no—
207C23×C31—24no—
208C23×C31—48no—
209C23×C31—48no—
210C23×C31—44no—
211C23×C31—44no—
212C23×C31C2×#4112yes—
213C23×C31H(5,𝔽2)12yes—
214C23×C31H(5,𝔽2)14yes—
215C23×C312−1+412yes—
216C23×C31C2511yescharacteristic
217D4×C31—416no—
218D4×C31—84no—
219D4×C31—88no—
220D4×C31—88no—
221D4×C31—88no—
222C2×C4×C31—28no—
223C2×C4×C31—22no—
224C2×C4×C31—22no—
225C2×C4×C31—22no—
226C2×C4×C31—22no—
227C2×C4×C31—24no—
228C2×C4×C31—24no—
229C2×C4×C31—24no—
230C2×C4×C31—24no—
231C2×C4×C31—24no—
232C2×C4×C31—24no—
233C2×C4×C31—24no—
234C2×C4×C31—24no—
235C2×C4×C31—24no—
236C2×C4×C31—24no—
237C2×C4×C31—24no—
238C2×C4×C31—24no—
239C2×C4×C31—24no—
240C2×C4×C31—24no—
241C2×C4×C31—24no—
242C2×C4×C31—44no—
243C2×C4×C31—48no—
244C2×C4×C31—48no—
245C2×C4×C31—84no—
246C2×C4×C31—84no—
247C2×C4×C31—88no—
248C2×C4×C31C2×#4114yes—
249C2×C4×C31H(5,𝔽2)14yes—
250#2847C2×C3112yescomplemented
251#2748C2×C3114yescomplemented
252#2741C2×C3114yescomplemented
253#2703C2×C3114yescomplemented
254#2671C2×C3114yescomplemented
255#2656C2×C3111yescharacteristic, complemented
256#2646C2×C3111yescharacteristic, complemented
257#2443C2×C3114yescomplemented
258#2424C2×C3114yescomplemented
259#2325C2×C3112yescomplemented
260C2×#380C2×C3111yescharacteristic, complemented
261C22×C31—22no—
262C22×C31—22no—
263C22×C31—24no—
264C22×C31—24no—
265C22×C31—22no—
266C22×C31—22no—
267C22×C31—24no—
268C22×C31—24no—
269C22×C31—416no—
270C22×C31—41no—
271C22×C31—41no—
272C22×C31—41no—
273C22×C31—41no—
274C22×C31—42no—
275C22×C31—42no—
276C22×C31—44no—
277C22×C31—44no—
278C22×C31—44no—
279C22×C31—44no—
280C22×C31—44no—
281C22×C31—44no—
282C22×C31—44no—
283C22×C31#54014yes—
284C22×C31C22×#4111yescharacteristic
285C22×C31C2×H(5,𝔽2)12yes—
286C22×C31C2×H(5,𝔽2)12yes—
287C22×C31C2×H(5,𝔽2)11yescharacteristic
288C22×C31C2×2−1+411yescharacteristic
289C4×C31—24no—
290C4×C31—24no—
291C4×C31—44no—
292C4×C31—44no—
293C4×C31—44no—
294C4×C31—44no—
295C4×C31—44no—
296C24×C4C22×C3111yescharacteristic
297C2×#123C22×C3114yescomplemented
298C2×#121C22×C3112yes—
299C2×#121C22×C3112yes—
300C2×#121C22×C3112yes—
301C2×#121C22×C3114yes—
302C2×#121C22×C3114yes—
303C2×#120C22×C3114yes—
304C2×C4×D4C22×C3114yes—
305C22×C4⋊C4C22×C3112yes—
306C22×C22⋊C4C22×C3112yes—
307#400C22×C3114yescomplemented
308#399C22×C3114yescomplemented
309#399C22×C3114yescomplemented
310#398C22×C3114yescomplemented
311#398C22×C3114yescomplemented
312#397C22×C3114yescomplemented
313#396C22×C3114yes—
314#394C22×C3114yes—
315#394C22×C3114yes—
316#394C22×C3114yescomplemented
317#392C22×C3114yes—
318#392C22×C3114yescomplemented
319#390C22×C3114yes—
320#388C22×C3114yes—
321#388C22×C3114yescomplemented
322#388C22×C3114yescomplemented
323#387C22×C3114yescomplemented
324#387C22×C3114yescomplemented
325#386C22×C3114yes—
326#386C22×C3114yes—
327#386C22×C3114yes—
328#386C22×C3114yes—
329#386C22×C3114yes—
330#386C22×C3114yescomplemented
331#385C22×C3114yes—
332#385C22×C3114yescomplemented
333#385C22×C3114yescomplemented
334#380C22×C3112yes—
335#380C22×C3112yes—
336#380C22×C3112yes—
337#380C22×C3114yes—
338#380C22×C3114yes—
339C2×#95C22×C3111yescharacteristic
340C2×#95C22×C3111yescharacteristic
341C2×C31—24no—
342C2×C31—42no—
343C2×C31—42no—
344C2×C31—84no—
345C2×C31C2×#54011yescharacteristic
346C2×C31#349011yescharacteristic
347C2×C31#349211yescharacteristic
348C2×C31#353612yes—
349C2×C31#353712yes—
350C25C23×C3111yescharacteristic
351C22×D4C23×C3114yes—
352C22×D4C23×C3114yes—
353C22×D4C23×C3114yes—
354C22×D4C23×C3112yes—
355C22×D4C23×C3114yes—
356C23×C4—24no—
357C23×C4—44no—
358C23×C4C23×C3111yescharacteristic
359C23×C4C23×C3112yes—
360C23×C4C23×C3112yes—
361C23×C4C23×C3112yes—
362C23×C4C23×C3112yes—
363C23×C4C23×C3114yes—
364C23×C4C23×C3111yescharacteristic
365#123—48no—
366#121—28no—
367#121—44no—
368#121—44no—
369#121—48no—
370#121—48no—
371#120—48no—
372C4×D4—48no—
373C2×C4⋊C4—44no—
374C2×C4⋊C4C23×C3114yes—
375C2×C4⋊C4C23×C3112yes—
376C2×C4⋊C4C23×C3114yes—
377C2×C4⋊C4C23×C3114yes—
378C2×C4⋊C4C23×C3112yes—
379C2×C4⋊C4C23×C3112yes—
380C2×C4⋊C4C23×C3114yes—
381C2×C4⋊C4C23×C3114yes—
382C2×C4⋊C4C23×C3114yes—
383C2×C22⋊C4—44no—
384C2×C22⋊C4C23×C3114yes—
385C2×C22⋊C4C23×C3114yes—
386C2×C22⋊C4C23×C3114yes—
387C2×C22⋊C4C23×C3114yes—
388C2×C22⋊C4C23×C3114yes—
389C2×C22⋊C4C23×C3114yes—
390C2×C22⋊C4C23×C3114yes—
391C2×C22⋊C4C23×C3114yes—
392C2×C22⋊C4C23×C3114yes—
393C2×C22⋊C4C23×C3114yes—
394C2×C22⋊C4C23×C3114yes—
395C2×C22⋊C4C23×C3114yes—
396C2×C22⋊C4C23×C3114yes—
397C2×C42C23×C3114yes—
398#95C23×C3114yes—
399#95C23×C3114yes—
400#95C23×C3114yes—
401#95C23×C3114yes—
402#95C23×C3114yes—
403#95C23×C3114yes—
404#95C23×C3114yes—
405#95C23×C3114yes—
406#95C23×C3114yes—
407C31#3672711yescharacteristic, direct product factor, Sylow
408C24—22no—
409C24—22no—
410C24—22no—
411C24—24no—
412C24—42no—
413C24C24×C3114yes—
414C24C24×C3111yescharacteristic
415C24C24×C3111yescharacteristic
416C24C24×C3111yescharacteristic
417C2×D4—28no—
418C2×D4—28no—
419C2×D4—28no—
420C2×D4—28no—
421C2×D4—28no—
422C2×D4—24no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—416no—
451C22×C4#41×C3114yes—
452C22×C4C24×C3114yes—
453C22×C4C24×C3114yes—
454C22×C4C24×C3114yes—
455C22×C4C24×C3114yes—
456C22×C4C24×C3114yes—
457C22×C4C24×C3112yes—
458C22×C4C24×C3112yes—
459C4⋊C4—24no—
460C4⋊C4—28no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C22⋊C4—28no—
469C22⋊C4—28no—
470C22⋊C4—28no—
471C22⋊C4—28no—
472C22⋊C4—28no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C42—44no—
482C23—22no—
483C23—21no—
484C23—21no—
485C23—28no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—21no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—48no—
502C23—48no—
503C23—44no—
504C23—44no—
505C23C2×#41×C3112yes—
506C23C31×H(5,𝔽2)12yes—
507C23C31×H(5,𝔽2)14yes—
508C23C31×2−1+412yes—
509C23C25×C3111yescharacteristic
510D4—416no—
511D4—84no—
512D4—88no—
513D4—88no—
514D4—88no—
515C2×C4—28no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—44no—
536C2×C4—48no—
537C2×C4—48no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4—88no—
541C2×C4C2×#41×C3114yes—
542C2×C4C31×H(5,𝔽2)14yes—
543C22—22no—
544C22—22no—
545C22—24no—
546C22—24no—
547C22—22no—
548C22—22no—
549C22—24no—
550C22—24no—
551C22—416no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—42no—
557C22—42no—
558C22—44no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22C31×#54014yes—
566C22C22×#41×C3111yescharacteristic
567C22C2×C31×H(5,𝔽2)12yes—
568C22C2×C31×H(5,𝔽2)12yes—
569C22C2×C31×H(5,𝔽2)11yescharacteristic
570C22C2×C31×2−1+411yescharacteristic
571C4—24no—
572C4—24no—
573C4—44no—
574C4—44no—
575C4—44no—
576C4—44no—
577C4—44no—
578C2—24no—
579C2—42no—
580C2—42no—
581C2—84no—
582C2C2×C31×#54011yescharacteristic
583C2C31×#349011yescharacteristic
584C2C31×#349211yescharacteristic
585C2C31×#353612yes—
586C2C31×#353712yes—
subgroupG/H
center, socleC23×C31C25
commutator, Frattini, cosocleC23C25×C31
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2847C2semidirect
2. C31×#2748C2semidirect
3. C31×#2741C2semidirect
4. C31×#2703C2semidirect
5. C31×#2671C2semidirect
6. C31×#2656C2semidirect
7. C31×#2646C2semidirect
8. C31×#2443C2semidirect
9. C31×#2424C2semidirect
10. C31×#2325C2semidirect
11. C2×C31×#380C2semidirect
13. C2×C31×#123C22semidirect
23. C31×#400C22semidirect
24. C31×#399C22semidirect
25. C31×#399C22semidirect
26. C31×#398C22semidirect
27. C31×#398C22semidirect
28. C31×#397C22semidirect
32. C31×#394C22semidirect
34. C31×#392C22semidirect
37. C31×#388C22semidirect
38. C31×#388C22semidirect
39. C31×#387C22semidirect
40. C31×#387C22semidirect
46. C31×#386C22semidirect
48. C31×#385C22semidirect
49. C31×#385C22semidirect
188. #36727C31direct (N × H)
250. #2847C2×C31semidirect
251. #2748C2×C31semidirect
252. #2741C2×C31semidirect
253. #2703C2×C31semidirect
254. #2671C2×C31semidirect
255. #2656C2×C31semidirect
256. #2646C2×C31semidirect
257. #2443C2×C31semidirect
258. #2424C2×C31semidirect
259. #2325C2×C31semidirect
260. C2×#380C2×C31semidirect
297. C2×#123C22×C31semidirect
307. #400C22×C31semidirect
308. #399C22×C31semidirect
309. #399C22×C31semidirect
310. #398C22×C31semidirect
311. #398C22×C31semidirect
312. #397C22×C31semidirect
316. #394C22×C31semidirect
318. #392C22×C31semidirect
321. #388C22×C31semidirect
322. #388C22×C31semidirect
323. #387C22×C31semidirect
324. #387C22×C31semidirect
330. #386C22×C31semidirect
332. #385C22×C31semidirect
333. #385C22×C31semidirect
407. C31#36727direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28472
C31×#27482
C31×#27412
C31×#27032
C31×#26712
C31×#26562
C31×#26462
C31×#24432
C31×#24242
C31×#23252
C2×C31×#3802
#3672731