Finite Group Databasebeta

← C31×#23946 #21480705

Subgroups — C31×#23946 #21480705

Order 7936.

1222 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#542C211yesmaximal, characteristic
2C2×C31×#451C211yesmaximal, characteristic, complemented
3C2×C31×#448C211yesmaximal, characteristic
4C31×#2310C211yesmaximal, characteristic, complemented
5C31×#2024C212yesmaximal, complemented
6C31×#2015C212yesmaximal, complemented
7C31×#1878C212yesmaximal, complemented
8C31×#1877C212yesmaximal, complemented
9C31×#1875C211yesmaximal, characteristic
10C31×#1873C211yesmaximal, characteristic, complemented
11C31×#1818C211yesmaximal, characteristic, complemented
12C23×Q8×C31C2211yescharacteristic
13C22×Q16×C31C2212yes—
14C22×SD16×C31C2212yes—
15C31×#542—24no—
16C31×#542—24no—
17C2×C31×C4⋊Q8C2211yescharacteristic
18C2×C31×#126—21no—
19C2×C31×#124—21no—
20C2×C31×#124C2211yescharacteristic
21C2×C31×#123—21no—
22C2×C31×C22⋊Q8—21no—
23C2×C31×C22⋊Q8C2211yescharacteristic
24C2×C31×#121C2211yescharacteristic
25C2×C4×Q8×C31C2211yescharacteristic
26C2×C4×D4×C31C2211yescharacteristic
27C2×C31×#117C2211yescharacteristic
28C2×C31×#117C2211yescharacteristic
29C31×#451—24no—
30C31×#448—24no—
31C2×C31×#103C2212yes—
32C2×C31×#103C2212yes—
33C2×C31×#103C2212yes—
34C2×C31×#102C2212yes—
35C2×C31×C22⋊C8C2211yescharacteristic
36C2×C31×C22⋊C8C2211yescharacteristic
37C31×#391—21no—
38C31×#391—21no—
39C31×#391C2212yes—
40C31×#391C2212yes—
41C31×#381—21no—
42C31×#381—21no—
43C31×#340C2212yes—
44C31×#340C2212yes—
45C31×#336C2212yes—
46C31×#336C2212yes—
47C22×Q8×C31—24no—
48C22×Q8×C31—21no—
49C22×Q8×C31—21no—
50C22×Q8×C31—22no—
51C22×Q8×C31—22no—
52C22×Q8×C31—42no—
53C22×Q8×C31C2311yescharacteristic
54C22×Q8×C31C2312yes—
55C22×D4×C31—21no—
56C22×D4×C31C2311yescharacteristic
57C23×C4×C31—21no—
58C23×C4×C31—21no—
59C23×C4×C31C2311yescharacteristic
60C2×Q16×C31—24no—
61C2×Q16×C31—24no—
62C2×Q16×C31—42no—
63C2×SD16×C31—24no—
64C2×SD16×C31—24no—
65C2×SD16×C31—42no—
66C22×C8×C31C2312yes—
67C22×C8×C31C2312yes—
68C31×C4⋊Q8—22no—
69C31×C4⋊Q8—22no—
70C31×#126—28no—
71C31×#124—28no—
72C31×#124—22no—
73C31×#124—22no—
74C31×#123—28no—
75C31×C22⋊Q8—28no—
76C31×C22⋊Q8—22no—
77C31×C22⋊Q8D414yes—
78C31×#121—22no—
79C31×#121D414yes—
80C4×Q8×C31—22no—
81C4×Q8×C31—22no—
82C4×D4×C31—22no—
83C4×D4×C31—22no—
84C31×#117—22no—
85C31×#117—22no—
86C31×#117—42no—
87C2×C4⋊C4×C31—21no—
88C2×C4⋊C4×C31—21no—
89C2×C4⋊C4×C31—21no—
90C2×C4⋊C4×C31D411yescharacteristic
91C2×C4⋊C4×C31D411yescharacteristic
92C2×C4⋊C4×C31C2312yes—
93C2×C4⋊C4×C31C2311yescharacteristic
94C2×C4⋊C4×C31C2311yescharacteristic
95C2×C22⋊C4×C31—21no—
96C2×C22⋊C4×C31—21no—
97C2×C22⋊C4×C31—21no—
98C2×C22⋊C4×C31—21no—
99C2×C22⋊C4×C31—21no—
100C2×C22⋊C4×C31—21no—
101C2×C22⋊C4×C31D411yescharacteristic
102C2×C22⋊C4×C31D411yescharacteristic
103C2×C42×C31—21no—
104C2×C42×C31—21no—
105C2×C42×C31C2311yescharacteristic
106C2×C42×C31C2311yescharacteristic
107C31×#103—24no—
108C31×#103—24no—
109C31×#103—42no—
110C31×#102—42no—
111C31×C22⋊C8—22no—
112C31×C22⋊C8—22no—
113C31×#95—22no—
114C31×#95—22no—
115C31×#95—22no—
116C31×#95—22no—
117C24×C31—21no—
118C24×C31C2×D411yescharacteristic
119C2×Q8×C31—24no—
120C2×Q8×C31—24no—
121C2×Q8×C31—21no—
122C2×Q8×C31—22no—
123C2×Q8×C31—22no—
124C2×Q8×C31—22no—
125C2×Q8×C31—22no—
126C2×Q8×C31—22no—
127C2×Q8×C31—22no—
128C2×Q8×C31—22no—
129C2×Q8×C31—48no—
130C2×Q8×C31—48no—
131C2×Q8×C31—42no—
132C2×Q8×C31—42no—
133C2×Q8×C31—82no—
134C2×Q8×C31C2×D412yes—
135C2×Q8×C31#4114yes—
136C2×D4×C31—24no—
137C2×D4×C31—21no—
138C2×D4×C31—22no—
139C2×D4×C31—22no—
140C2×D4×C31—82no—
141C2×D4×C31C2×D412yes—
142C22×C4×C31—22no—
143C22×C4×C31—22no—
144C22×C4×C31—24no—
145C22×C4×C31—22no—
146C22×C4×C31—21no—
147C22×C4×C31—21no—
148C22×C4×C31—21no—
149C22×C4×C31—24no—
150C22×C4×C31—22no—
151C22×C4×C31—42no—
152C22×C4×C31—42no—
153C22×C4×C31—82no—
154C22×C4×C31C2×D411yescharacteristic
155C22×C4×C31C2×D411yescharacteristic
156C22×C4×C31C2×D411yescharacteristic
157C22×C4×C31C2×D411yescharacteristic
158C22×C4×C31C2×D411yescharacteristic
159C22×C4×C31C2×D412yes—
160C22×C4×C31C2×D412yes—
161C22×C4×C31C2411yescharacteristic
162Q16×C31—48no—
163SD16×C31—48no—
164C2×C8×C31—22no—
165C2×C8×C31—22no—
166C2×C8×C31—22no—
167C2×C8×C31—22no—
168C2×C8×C31—22no—
169C2×C8×C31—22no—
170C4⋊C4×C31—24no—
171C4⋊C4×C31—24no—
172C4⋊C4×C31—24no—
173C4⋊C4×C31—22no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—21no—
176C4⋊C4×C31—21no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31C2×D412yes—
180C4⋊C4×C31C2×D412yes—
181C22⋊C4×C31—24no—
182C22⋊C4×C31—24no—
183C22⋊C4×C31—24no—
184C22⋊C4×C31—24no—
185C22⋊C4×C31—22no—
186C22⋊C4×C31—22no—
187C22⋊C4×C31—24no—
188C22⋊C4×C31—42no—
189C42×C31—24no—
190C42×C31—21no—
191C42×C31—21no—
192C42×C31—21no—
193C42×C31—21no—
194C42×C31—42no—
195#23946C3111yesmaximal, characteristic, direct product factor, Sylow
196C23×C31—21no—
197C23×C31—21no—
198C23×C31—22no—
199C23×C31—22no—
200C23×C31—22no—
201C23×C31—41no—
202C23×C31—41no—
203C23×C31—42no—
204C23×C31—42no—
205C23×C31#12012yes—
206C23×C31C22×D411yescharacteristic
207Q8×C31—24no—
208Q8×C31—24no—
209Q8×C31—24no—
210Q8×C31—48no—
211Q8×C31—42no—
212Q8×C31—42no—
213Q8×C31—42no—
214Q8×C31—84no—
215D4×C31—24no—
216D4×C31—42no—
217D4×C31—84no—
218C2×C4×C31—21no—
219C2×C4×C31—21no—
220C2×C4×C31—24no—
221C2×C4×C31—24no—
222C2×C4×C31—21no—
223C2×C4×C31—21no—
224C2×C4×C31—22no—
225C2×C4×C31—21no—
226C2×C4×C31—21no—
227C2×C4×C31—22no—
228C2×C4×C31—22no—
229C2×C4×C31—24no—
230C2×C4×C31—22no—
231C2×C4×C31—22no—
232C2×C4×C31—22no—
233C2×C4×C31—22no—
234C2×C4×C31—48no—
235C2×C4×C31—42no—
236C2×C4×C31—41no—
237C2×C4×C31—42no—
238C2×C4×C31—41no—
239C2×C4×C31—42no—
240C2×C4×C31—42no—
241C2×C4×C31—42no—
242C2×C4×C31—42no—
243C2×C4×C31—42no—
244C2×C4×C31—84no—
245C2×C4×C31#12012yes—
246C2×C4×C31C22×D411yescharacteristic
247C2×C4×C31C22×D411yescharacteristic
248C2×C4×C31C2×#4112yes—
249C2×C4×C31H(5,𝔽2)12yes—
250C8×C31—42no—
251C8×C31—42no—
252C2×#542C2×C3111yescharacteristic
253C2×#451C2×C3111yescharacteristic, complemented
254C2×#448C2×C3111yescharacteristic
255#2310C2×C3111yescharacteristic, complemented
256#2024C2×C3112yescomplemented
257#2015C2×C3112yescomplemented
258#1878C2×C3112yescomplemented
259#1877C2×C3112yescomplemented
260#1875C2×C3111yescharacteristic
261#1873C2×C3111yescharacteristic, complemented
262#1818C2×C3111yescharacteristic, complemented
263C22×C31—22no—
264C22×C31—22no—
265C22×C31—22no—
266C22×C31—42no—
267C22×C31—42no—
268C22×C31—42no—
269C22×C31—42no—
270C22×C31—42no—
271C22×C31—82no—
272C22×C31—82no—
273C22×C31C2×#12011yescharacteristic
274C22×C31#52511yescharacteristic
275C22×C31#53511yescharacteristic
276C22×C31#57712yes—
277C22×C31#57812yes—
278C4×C31—22no—
279C4×C31—44no—
280C4×C31—42no—
281C4×C31—41no—
282C4×C31—41no—
283C4×C31—42no—
284C4×C31—42no—
285C4×C31—41no—
286C4×C31—41no—
287C4×C31—82no—
288C4×C31#54612yes—
289C4×C31#54612yes—
290C23×Q8C22×C3111yescharacteristic
291C22×Q16C22×C3112yes—
292C22×SD16C22×C3112yes—
293#542—24no—
294#542—24no—
295C2×C4⋊Q8C22×C3111yescharacteristic
296C2×#126—21no—
297C2×#124—21no—
298C2×#124C22×C3111yescharacteristic
299C2×#123—21no—
300C2×C22⋊Q8—21no—
301C2×C22⋊Q8C22×C3111yescharacteristic
302C2×#121C22×C3111yescharacteristic
303C2×C4×Q8C22×C3111yescharacteristic
304C2×C4×D4C22×C3111yescharacteristic
305C2×#117C22×C3111yescharacteristic
306C2×#117C22×C3111yescharacteristic
307#451—24no—
308#448—24no—
309C2×#103C22×C3112yes—
310C2×#103C22×C3112yes—
311C2×#103C22×C3112yes—
312C2×#102C22×C3112yes—
313C2×C22⋊C8C22×C3111yescharacteristic
314C2×C22⋊C8C22×C3111yescharacteristic
315#391—21no—
316#391—21no—
317#391C22×C3112yes—
318#391C22×C3112yes—
319#381—21no—
320#381—21no—
321#340C22×C3112yes—
322#340C22×C3112yes—
323#336C22×C3112yes—
324#336C22×C3112yes—
325C2×C31—42no—
326C2×C31—82no—
327C2×C31#240411yescharacteristic
328C2×C31#300911yescharacteristic
329C2×C31#301111yescharacteristic
330C2×C31#312112yes—
331C2×C31#322512yes—
332C22×Q8—24no—
333C22×Q8—21no—
334C22×Q8—21no—
335C22×Q8—22no—
336C22×Q8—22no—
337C22×Q8—42no—
338C22×Q8C23×C3111yescharacteristic
339C22×Q8C23×C3112yes—
340C22×D4—21no—
341C22×D4C23×C3111yescharacteristic
342C23×C4—21no—
343C23×C4—21no—
344C23×C4C23×C3111yescharacteristic
345C2×Q16—24no—
346C2×Q16—24no—
347C2×Q16—42no—
348C2×SD16—24no—
349C2×SD16—24no—
350C2×SD16—42no—
351C22×C8C23×C3112yes—
352C22×C8C23×C3112yes—
353C4⋊Q8—22no—
354C4⋊Q8—22no—
355#126—28no—
356#124—28no—
357#124—22no—
358#124—22no—
359#123—28no—
360C22⋊Q8—28no—
361C22⋊Q8—22no—
362C22⋊Q8D4×C3114yes—
363#121—22no—
364#121D4×C3114yes—
365C4×Q8—22no—
366C4×Q8—22no—
367C4×D4—22no—
368C4×D4—22no—
369#117—22no—
370#117—22no—
371#117—42no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4D4×C3111yescharacteristic
376C2×C4⋊C4D4×C3111yescharacteristic
377C2×C4⋊C4C23×C3112yes—
378C2×C4⋊C4C23×C3111yescharacteristic
379C2×C4⋊C4C23×C3111yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4D4×C3111yescharacteristic
387C2×C22⋊C4D4×C3111yescharacteristic
388C2×C42—21no—
389C2×C42—21no—
390C2×C42C23×C3111yescharacteristic
391C2×C42C23×C3111yescharacteristic
392#103—24no—
393#103—24no—
394#103—42no—
395#102—42no—
396C22⋊C8—22no—
397C22⋊C8—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402C31#2394611yescharacteristic, direct product factor, Sylow
403C24—21no—
404C24C2×D4×C3111yescharacteristic
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—21no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—48no—
416C2×Q8—48no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—82no—
420C2×Q8C2×D4×C3112yes—
421C2×Q8#41×C3114yes—
422C2×D4—24no—
423C2×D4—21no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—82no—
427C2×D4C2×D4×C3112yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—82no—
440C22×C4C2×D4×C3111yescharacteristic
441C22×C4C2×D4×C3111yescharacteristic
442C22×C4C2×D4×C3111yescharacteristic
443C22×C4C2×D4×C3111yescharacteristic
444C22×C4C2×D4×C3111yescharacteristic
445C22×C4C2×D4×C3112yes—
446C22×C4C2×D4×C3112yes—
447C22×C4C24×C3111yescharacteristic
448Q16—48no—
449SD16—48no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4C2×D4×C3112yes—
466C4⋊C4C2×D4×C3112yes—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—24no—
474C22⋊C4—42no—
475C42—24no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—42no—
481C23—21no—
482C23—21no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—41no—
487C23—41no—
488C23—42no—
489C23—42no—
490C23C31×#12012yes—
491C23C22×D4×C3111yescharacteristic
492Q8—24no—
493Q8—24no—
494Q8—24no—
495Q8—48no—
496Q8—42no—
497Q8—42no—
498Q8—42no—
499Q8—84no—
500D4—24no—
501D4—42no—
502D4—84no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—48no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—84no—
530C2×C4C31×#12012yes—
531C2×C4C22×D4×C3111yescharacteristic
532C2×C4C22×D4×C3111yescharacteristic
533C2×C4C2×#41×C3112yes—
534C2×C4C31×H(5,𝔽2)12yes—
535C8—42no—
536C8—42no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—82no—
546C22—82no—
547C22C2×C31×#12011yescharacteristic
548C22C31×#52511yescharacteristic
549C22C31×#53511yescharacteristic
550C22C31×#57712yes—
551C22C31×#57812yes—
552C4—22no—
553C4—44no—
554C4—42no—
555C4—41no—
556C4—41no—
557C4—42no—
558C4—42no—
559C4—41no—
560C4—41no—
561C4—82no—
562C4C31×#54612yes—
563C4C31×#54612yes—
564C2—42no—
565C2—82no—
566C2C31×#240411yescharacteristic
567C2C31×#300911yescharacteristic
568C2C31×#301111yescharacteristic
569C2C31×#312112yes—
570C2C31×#322512yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C31×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#451C2semidirect
4. C31×#2310C2semidirect
5. C31×#2024C2semidirect
6. C31×#2015C2semidirect
7. C31×#1878C2semidirect
8. C31×#1877C2semidirect
10. C31×#1873C2semidirect
11. C31×#1818C2semidirect
195. #23946C31direct (N × H)
253. C2×#451C2×C31semidirect
255. #2310C2×C31semidirect
256. #2024C2×C31semidirect
257. #2015C2×C31semidirect
258. #1878C2×C31semidirect
259. #1877C2×C31semidirect
261. #1873C2×C31semidirect
262. #1818C2×C31semidirect
402. C31#23946direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5422
C2×C31×#4512
C2×C31×#4482
C31×#23102
C31×#20242
C31×#20152
C31×#18782
C31×#18772
C31×#18752
C31×#18732
C31×#18182
#2394631