Finite Group Databasebeta

← C31×#24057 #21480816

Subgroups — C31×#24057 #21480816

Order 7936.

1392 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3485C211yesmaximal, characteristic
2C31×#2853C212yesmaximal, complemented
3C31×#2297C211yesmaximal, characteristic, complemented
4C31×#2044C214yesmaximal, complemented
5C31×#1899C212yesmaximal, complemented
6C31×#1867C212yesmaximal, complemented
7C31×#1790C212yesmaximal, complemented
8C31×#1782C211yesmaximal, characteristic, complemented
9C2×C31×H(5,𝔽2)C2211yescharacteristic
10C22×#41×C31C2211yescharacteristic
11C31×#547—24no—
12C31×#546—24no—
13C31×#536—21no—
14C31×#534—21no—
15C31×#534C2212yes—
16C31×#525—21no—
17C2×C31×#123—21no—
18C2×C31×#123C2212yes—
19C31×#518C2211yescharacteristic
20C2×C31×#117C2211yescharacteristic, complemented
21C22×C42×C31C2211yescharacteristic, complemented
22C2×C31×#99C2212yes—
23C2×C31×#99C2214yes—
24C2×C31×#99C2212yescomplemented
25C31×#391—21no—
26C31×#390—21no—
27C31×#388—24no—
28C31×#387—24no—
29C31×#386C2214yes—
30C31×#386C2212yes—
31C31×#386C2212yescomplemented
32C31×#385C2214yescomplemented
33C31×#381C2212yescomplemented
34C4×C22⋊C4×C31—21no—
35C4×C22⋊C4×C31—21no—
36C31×#342C2214yescomplemented
37C31×H(5,𝔽2)—24no—
38C31×H(5,𝔽2)—42no—
39C2×#41×C31—24no—
40C2×#41×C31—21no—
41C2×#41×C31—24no—
42C2×#41×C31—22no—
43C2×#41×C31—42no—
44C22×Q8×C31D411yescharacteristic
45C22×D4×C31—21no—
46C22×D4×C31—22no—
47C22×D4×C31D411yescharacteristic
48C22×D4×C31C2312yes—
49C22×D4×C31C2311yescharacteristic
50C22×D4×C31C2311yescharacteristic
51C23×C4×C31D411yescharacteristic
52C23×C4×C31D411yescharacteristic
53C23×C4×C31C2312yes—
54C23×C4×C31C2311yescharacteristic
55C31×C4⋊Q8—22no—
56C31×#124—22no—
57C31×#123—24no—
58C31×#123—22no—
59C31×#123—22no—
60C31×#123—22no—
61C31×#123—22no—
62C31×#123—24no—
63C31×#123—22no—
64C31×C22⋊Q8—22no—
65C31×C22⋊Q8—24no—
66C31×#121—22no—
67C31×#121—24no—
68C31×#121—24no—
69C31×#120—22no—
70C31×#120—22no—
71C31×#120—22no—
72C4×D4×C31—24no—
73C4×D4×C31—24no—
74C31×#117—21no—
75C31×#117—42no—
76C2×C4⋊C4×C31—22no—
77C2×C4⋊C4×C31—21no—
78C2×C4⋊C4×C31—24no—
79C2×C4⋊C4×C31D414yes—
80C2×C22⋊C4×C31—22no—
81C2×C22⋊C4×C31—21no—
82C2×C22⋊C4×C31—22no—
83C2×C22⋊C4×C31—22no—
84C2×C22⋊C4×C31—24no—
85C2×C22⋊C4×C31D414yes—
86C2×C22⋊C4×C31C2314yes—
87C2×C22⋊C4×C31C2312yes—
88C2×C22⋊C4×C31C2312yes—
89C2×C42×C31—21no—
90C2×C42×C31—24no—
91C2×C42×C31—21no—
92C2×C42×C31—21no—
93C31×#99—24no—
94C31×#99—44no—
95C31×#99—42no—
96C31×#95—22no—
97C31×#95—22no—
98C31×#95—24no—
99C31×#95—24no—
100C31×#95—24no—
101C24×C31—21no—
102C24×C31C2×D412yes—
103C24×C31C2×D411yescharacteristic
104C24×C31C2×D411yescharacteristic
105C24×C31C2411yescharacteristic
106#41×C31—416no—
107#41×C31—48no—
108#41×C31—42no—
109#41×C31—42no—
110C2×Q8×C31—21no—
111C2×Q8×C31—22no—
112C2×Q8×C31—22no—
113C2×Q8×C31—41no—
114C2×D4×C31—22no—
115C2×D4×C31—22no—
116C2×D4×C31—22no—
117C2×D4×C31—21no—
118C2×D4×C31—22no—
119C2×D4×C31—22no—
120C2×D4×C31—22no—
121C2×D4×C31—22no—
122C2×D4×C31—22no—
123C2×D4×C31—24no—
124C2×D4×C31—22no—
125C2×D4×C31—24no—
126C2×D4×C31—24no—
127C2×D4×C31—22no—
128C2×D4×C31—41no—
129C2×D4×C31—48no—
130C2×D4×C31—42no—
131C2×D4×C31—42no—
132C2×D4×C31#4114yes—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—21no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—24no—
140C22×C4×C31—22no—
141C22×C4×C31—22no—
142C22×C4×C31—22no—
143C22×C4×C31—22no—
144C22×C4×C31—24no—
145C22×C4×C31—22no—
146C22×C4×C31—44no—
147C22×C4×C31—41no—
148C22×C4×C31—41no—
149C22×C4×C31C2×D412yes—
150C22×C4×C31C2×D411yescharacteristic
151C22×C4×C31C2×D414yes—
152C22×C4×C31C2×D411yescharacteristic
153C22×C4×C31C2×D411yescharacteristic
154C22×C4×C31C2×D411yescharacteristic
155C22×C4×C31C2×D414yes—
156C4⋊C4×C31—24no—
157C4⋊C4×C31—22no—
158C4⋊C4×C31—24no—
159C4⋊C4×C31—24no—
160C4⋊C4×C31—48no—
161C4⋊C4×C31—42no—
162C22⋊C4×C31—22no—
163C22⋊C4×C31—24no—
164C22⋊C4×C31—24no—
165C22⋊C4×C31—22no—
166C22⋊C4×C31—22no—
167C22⋊C4×C31—24no—
168C22⋊C4×C31—24no—
169C22⋊C4×C31—24no—
170C22⋊C4×C31—24no—
171C22⋊C4×C31—24no—
172C22⋊C4×C31—44no—
173C22⋊C4×C31—44no—
174C22⋊C4×C31—48no—
175C42×C31—22no—
176C42×C31—42no—
177C42×C31—42no—
178C42×C31—42no—
179#24057C3111yesmaximal, characteristic, direct product factor, Sylow
180C23×C31—21no—
181C23×C31—21no—
182C23×C31—22no—
183C23×C31—21no—
184C23×C31—21no—
185C23×C31—21no—
186C23×C31—24no—
187C23×C31—42no—
188C23×C31—42no—
189C23×C31—42no—
190C23×C31—42no—
191C23×C31—44no—
192C23×C31—82no—
193C23×C31#12312yes—
194C23×C31C22×D411yescharacteristic
195C23×C31C22×D412yes—
196C23×C31C2×#4112yes—
197C23×C31H(5,𝔽2)12yes—
198Q8×C31—44no—
199Q8×C31—42no—
200D4×C31—28no—
201D4×C31—44no—
202D4×C31—42no—
203D4×C31—44no—
204D4×C31—42no—
205D4×C31—42no—
206D4×C31—44no—
207D4×C31—44no—
208D4×C31—84no—
209C2×C4×C31—21no—
210C2×C4×C31—21no—
211C2×C4×C31—21no—
212C2×C4×C31—21no—
213C2×C4×C31—21no—
214C2×C4×C31—21no—
215C2×C4×C31—28no—
216C2×C4×C31—24no—
217C2×C4×C31—28no—
218C2×C4×C31—24no—
219C2×C4×C31—21no—
220C2×C4×C31—22no—
221C2×C4×C31—21no—
222C2×C4×C31—21no—
223C2×C4×C31—24no—
224C2×C4×C31—44no—
225C2×C4×C31—44no—
226C2×C4×C31—44no—
227C2×C4×C31—44no—
228C2×C4×C31—42no—
229C2×C4×C31—42no—
230C2×C4×C31—42no—
231C2×C4×C31—44no—
232C2×C4×C31—44no—
233C2×C4×C31—42no—
234C2×C4×C31—42no—
235C2×C4×C31—44no—
236C2×C4×C31—42no—
237C2×C4×C31—44no—
238C2×C4×C31—82no—
239C2×C4×C31#12012yes—
240C2×C4×C31#12012yes—
241C2×C4×C31#12312yes—
242#3485C2×C3111yescharacteristic
243#2853C2×C3112yescomplemented
244#2297C2×C3111yescharacteristic, complemented
245#2044C2×C3114yescomplemented
246#1899C2×C3112yescomplemented
247#1867C2×C3112yescomplemented
248#1790C2×C3112yescomplemented
249#1782C2×C3111yescharacteristic, complemented
250C22×C31—24no—
251C22×C31—22no—
252C22×C31—22no—
253C22×C31—42no—
254C22×C31—42no—
255C22×C31—42no—
256C22×C31—42no—
257C22×C31—42no—
258C22×C31—41no—
259C22×C31—42no—
260C22×C31—42no—
261C22×C31—44no—
262C22×C31—81no—
263C22×C31—81no—
264C22×C31—84no—
265C22×C31—162no—
266C22×C31C2×#12011yescharacteristic
267C22×C31C2×#12311yescharacteristic
268C22×C31#53411yescharacteristic
269C22×C31#54614yes—
270C4×C31—22no—
271C4×C31—22no—
272C4×C31—22no—
273C4×C31—42no—
274C4×C31—41no—
275C4×C31—44no—
276C4×C31—44no—
277C4×C31—82no—
278C4×C31—84no—
279C2×H(5,𝔽2)C22×C3111yescharacteristic
280C22×#41C22×C3111yescharacteristic
281#547—24no—
282#546—24no—
283#536—21no—
284#534—21no—
285#534C22×C3112yes—
286#525—21no—
287C2×#123—21no—
288C2×#123C22×C3112yes—
289#518C22×C3111yescharacteristic
290C2×#117C22×C3111yescharacteristic, complemented
291C22×C42C22×C3111yescharacteristic, complemented
292C2×#99C22×C3112yes—
293C2×#99C22×C3114yes—
294C2×#99C22×C3112yescomplemented
295#391—21no—
296#390—21no—
297#388—24no—
298#387—24no—
299#386C22×C3114yes—
300#386C22×C3112yes—
301#386C22×C3112yescomplemented
302#385C22×C3114yescomplemented
303#381C22×C3112yescomplemented
304C4×C22⋊C4—21no—
305C4×C22⋊C4—21no—
306#342C22×C3114yescomplemented
307C2×C31—21no—
308C2×C31—24no—
309C2×C31—21no—
310C2×C31—42no—
311C2×C31—81no—
312C2×C31—82no—
313C2×C31#240511yescharacteristic
314C2×C31#302512yes—
315H(5,𝔽2)—24no—
316H(5,𝔽2)—42no—
317C2×#41—24no—
318C2×#41—21no—
319C2×#41—24no—
320C2×#41—22no—
321C2×#41—42no—
322C22×Q8D4×C3111yescharacteristic
323C22×D4—21no—
324C22×D4—22no—
325C22×D4D4×C3111yescharacteristic
326C22×D4C23×C3112yes—
327C22×D4C23×C3111yescharacteristic
328C22×D4C23×C3111yescharacteristic
329C23×C4D4×C3111yescharacteristic
330C23×C4D4×C3111yescharacteristic
331C23×C4C23×C3111yescharacteristic
332C23×C4C23×C3112yes—
333C4⋊Q8—22no—
334#124—22no—
335#123—24no—
336#123—22no—
337#123—22no—
338#123—22no—
339#123—22no—
340#123—24no—
341#123—22no—
342C22⋊Q8—22no—
343C22⋊Q8—24no—
344#121—22no—
345#121—24no—
346#121—24no—
347#120—22no—
348#120—22no—
349#120—22no—
350C4×D4—24no—
351C4×D4—24no—
352#117—21no—
353#117—42no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4D4×C3114yes—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4D4×C3114yes—
364C2×C22⋊C4C23×C3112yes—
365C2×C22⋊C4C23×C3114yes—
366C2×C22⋊C4C23×C3112yes—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42—24no—
370C2×C42—21no—
371#99—24no—
372#99—44no—
373#99—42no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—24no—
378#95—24no—
379C31#2405711yescharacteristic, direct product factor, Sylow
380C24—21no—
381C24C2×D4×C3112yes—
382C24C2×D4×C3111yescharacteristic
383C24C2×D4×C3111yescharacteristic
384C24C24×C3111yescharacteristic
385#41—416no—
386#41—48no—
387#41—42no—
388#41—42no—
389C2×Q8—21no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—41no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—41no—
408C2×D4—48no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4#41×C3114yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—44no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4C2×D4×C3112yes—
429C22×C4C2×D4×C3111yescharacteristic
430C22×C4C2×D4×C3114yes—
431C22×C4C2×D4×C3111yescharacteristic
432C22×C4C2×D4×C3111yescharacteristic
433C22×C4C2×D4×C3111yescharacteristic
434C22×C4C2×D4×C3114yes—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—48no—
440C4⋊C4—42no—
441C22⋊C4—22no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—48no—
454C42—22no—
455C42—42no—
456C42—42no—
457C42—42no—
458C23—21no—
459C23—21no—
460C23—22no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—24no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—44no—
470C23—82no—
471C23C31×#12312yes—
472C23C22×D4×C3111yescharacteristic
473C23C22×D4×C3112yes—
474C23C2×#41×C3112yes—
475C23C31×H(5,𝔽2)12yes—
476Q8—44no—
477Q8—42no—
478D4—28no—
479D4—44no—
480D4—42no—
481D4—44no—
482D4—42no—
483D4—42no—
484D4—44no—
485D4—44no—
486D4—84no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—28no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—24no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—24no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—82no—
517C2×C4C31×#12012yes—
518C2×C4C31×#12012yes—
519C2×C4C31×#12312yes—
520C22—24no—
521C22—22no—
522C22—22no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—41no—
529C22—42no—
530C22—42no—
531C22—44no—
532C22—81no—
533C22—81no—
534C22—84no—
535C22—162no—
536C22C2×C31×#12011yescharacteristic
537C22C2×C31×#12311yescharacteristic
538C22C31×#53411yescharacteristic
539C22C31×#54614yes—
540C4—22no—
541C4—22no—
542C4—22no—
543C4—42no—
544C4—41no—
545C4—44no—
546C4—44no—
547C4—82no—
548C4—84no—
549C2—21no—
550C2—24no—
551C2—21no—
552C2—42no—
553C2—81no—
554C2—82no—
555C2C31×#240511yescharacteristic
556C2C31×#302512yes—
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC24C24×C31
prime radicalC31×#3485C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2853C2semidirect
3. C31×#2297C2semidirect
4. C31×#2044C2semidirect
5. C31×#1899C2semidirect
6. C31×#1867C2semidirect
7. C31×#1790C2semidirect
8. C31×#1782C2semidirect
20. C2×C31×#117C22semidirect
21. C22×C42×C31C22semidirect
24. C2×C31×#99C22semidirect
31. C31×#386C22semidirect
32. C31×#385C22semidirect
33. C31×#381C22semidirect
36. C31×#342C22semidirect
179. #24057C31direct (N × H)
243. #2853C2×C31semidirect
244. #2297C2×C31semidirect
245. #2044C2×C31semidirect
246. #1899C2×C31semidirect
247. #1867C2×C31semidirect
248. #1790C2×C31semidirect
249. #1782C2×C31semidirect
290. C2×#117C22×C31semidirect
291. C22×C42C22×C31semidirect
294. C2×#99C22×C31semidirect
301. #386C22×C31semidirect
302. #385C22×C31semidirect
303. #381C22×C31semidirect
306. #342C22×C31semidirect
379. C31#24057direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34852
C31×#28532
C31×#22972
C31×#20442
C31×#18992
C31×#18672
C31×#17902
C31×#17822
#2405731