Finite Group Databasebeta

← C31×#25613 #21482372

Subgroups — C31×#25613 #21482372

Order 7936.

1116 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#542C211yesmaximal, characteristic
2C2×C31×#497C211yesmaximal, characteristic, complemented
3C2×C31×#487C211yesmaximal, characteristic, complemented
4C31×#2621C211yesmaximal, characteristic, complemented
5C31×#2001C212yesmaximal, complemented
6C31×#2000C212yesmaximal
7C31×#1993C211yesmaximal, characteristic, complemented
8C31×#1958C211yesmaximal, characteristic, complemented
9C31×#1878C212yesmaximal, complemented
10C31×#1877C212yesmaximal, complemented
11C31×#1750C211yesmaximal, characteristic, complemented
12C22×Q16×C31C2212yes—
13C22×SD16×C31C2212yes—
14C31×#542—24no—
15C31×#542—24no—
16C2×C31×C4⋊Q8C2211yescharacteristic
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×C22⋊Q8C2212yes—
25C2×C31×#121C2211yescharacteristic
26C2×C4×Q8×C31C2211yescharacteristic
27C2×C4×D4×C31C2211yescharacteristic
28C2×C31×#117C2211yescharacteristic
29C31×#497—24no—
30C31×#487—24no—
31C2×C31×#103C2212yes—
32C2×C31×#103C2212yes—
33C2×C31×#103C2212yes—
34C2×C31×#102C2212yes—
35C2×C31×C22⋊C8C2212yes—
36C2×C31×#97C2211yescharacteristic
37C2×C4×C8×C31C2211yescharacteristic
38C31×#398—21no—
39C31×#398—21no—
40C31×#397—21no—
41C31×#397—21no—
42C31×#391C2212yes—
43C31×#382C2211yescharacteristic
44C31×#340C2212yes—
45C31×#340C2212yes—
46C31×#336C2212yes—
47C22×Q8×C31—21no—
48C22×Q8×C31C2312yes—
49C22×Q8×C31C2311yescharacteristic
50C22×D4×C31—21no—
51C22×D4×C31C2311yescharacteristic
52C23×C4×C31—21no—
53C23×C4×C31C2311yescharacteristic
54C2×Q16×C31—24no—
55C2×Q16×C31—24no—
56C2×Q16×C31—42no—
57C2×SD16×C31—24no—
58C2×SD16×C31—24no—
59C2×SD16×C31—42no—
60C22×C8×C31C2312yes—
61C22×C8×C31C2312yes—
62C31×C4⋊Q8—22no—
63C31×C4⋊Q8—22no—
64C31×C4⋊Q8—22no—
65C31×C4⋊Q8D414yes—
66C31×#126—28no—
67C31×#124—28no—
68C31×#124—22no—
69C31×#124D414yes—
70C31×#123—28no—
71C31×C22⋊Q8—28no—
72C31×C22⋊Q8—22no—
73C31×C22⋊Q8—22no—
74C31×C22⋊Q8—42no—
75C31×C22⋊Q8—42no—
76C31×#121—22no—
77C31×#121—22no—
78C4×Q8×C31—22no—
79C4×Q8×C31—22no—
80C4×D4×C31—22no—
81C4×D4×C31—22no—
82C31×#117—22no—
83C31×#117—22no—
84C2×C4⋊C4×C31—21no—
85C2×C4⋊C4×C31—21no—
86C2×C4⋊C4×C31—21no—
87C2×C4⋊C4×C31—21no—
88C2×C4⋊C4×C31—21no—
89C2×C4⋊C4×C31—22no—
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—22no—
101C2×C22⋊C4×C31D411yescharacteristic
102C2×C22⋊C4×C31D411yescharacteristic
103C2×C42×C31—21no—
104C2×C42×C31C2311yescharacteristic
105C2×C42×C31C2311yescharacteristic
106C31×#103—24no—
107C31×#103—24no—
108C31×#103—42no—
109C31×#102—42no—
110C31×C22⋊C8—24no—
111C31×#97—22no—
112C4×C8×C31—22no—
113C31×#95—21no—
114C31×#95—21no—
115C31×#95—22no—
116C31×#95—22no—
117C24×C31—21no—
118C24×C31C2×D411yescharacteristic
119C2×Q8×C31—24no—
120C2×Q8×C31—21no—
121C2×Q8×C31—22no—
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—82no—
130C2×Q8×C31C2×D412yes—
131C2×D4×C31—24no—
132C2×D4×C31—21no—
133C2×D4×C31—22no—
134C2×D4×C31—22no—
135C2×D4×C31—82no—
136C2×D4×C31C2×D412yes—
137C22×C4×C31—21no—
138C22×C4×C31—21no—
139C22×C4×C31—21no—
140C22×C4×C31—24no—
141C22×C4×C31—21no—
142C22×C4×C31—21no—
143C22×C4×C31—22no—
144C22×C4×C31—22no—
145C22×C4×C31—22no—
146C22×C4×C31—22no—
147C22×C4×C31—82no—
148C22×C4×C31C2×D411yescharacteristic
149C22×C4×C31C2×D411yescharacteristic
150C22×C4×C31C2×D411yescharacteristic
151C22×C4×C31C2×D411yescharacteristic
152C22×C4×C31C2×D411yescharacteristic
153C22×C4×C31C2411yescharacteristic
154Q16×C31—48no—
155SD16×C31—48no—
156C2×C8×C31—22no—
157C2×C8×C31—22no—
158C2×C8×C31—22no—
159C2×C8×C31—22no—
160C2×C8×C31—22no—
161C2×C8×C31#4114yes—
162C4⋊C4×C31—24no—
163C4⋊C4×C31—24no—
164C4⋊C4×C31—24no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—22no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—44no—
172C4⋊C4×C31—42no—
173C4⋊C4×C31—42no—
174C4⋊C4×C31C2×D412yes—
175C4⋊C4×C31C2×D412yes—
176C22⋊C4×C31—24no—
177C22⋊C4×C31—24no—
178C22⋊C4×C31—24no—
179C22⋊C4×C31—24no—
180C22⋊C4×C31—24no—
181C22⋊C4×C31—22no—
182C22⋊C4×C31—22no—
183C22⋊C4×C31—44no—
184C42×C31—24no—
185C42×C31—21no—
186C42×C31—21no—
187C42×C31C2×D412yes—
188C42×C31C2×D412yes—
189#25613C3111yesmaximal, characteristic, direct product factor, Sylow
190C23×C31—21no—
191C23×C31—22no—
192C23×C31—21no—
193C23×C31—21no—
194C23×C31—22no—
195C23×C31—22no—
196C23×C31—41no—
197C23×C31—41no—
198C23×C31—42no—
199C23×C31—42no—
200C23×C31C22×D411yescharacteristic
201Q8×C31—24no—
202Q8×C31—42no—
203Q8×C31—42no—
204Q8×C31—42no—
205Q8×C31—42no—
206Q8×C31—42no—
207Q8×C31—84no—
208D4×C31—24no—
209D4×C31—42no—
210D4×C31—84no—
211C2×C4×C31—22no—
212C2×C4×C31—22no—
213C2×C4×C31—21no—
214C2×C4×C31—22no—
215C2×C4×C31—21no—
216C2×C4×C31—22no—
217C2×C4×C31—21no—
218C2×C4×C31—22no—
219C2×C4×C31—21no—
220C2×C4×C31—21no—
221C2×C4×C31—22no—
222C2×C4×C31—22no—
223C2×C4×C31—41no—
224C2×C4×C31—41no—
225C2×C4×C31—42no—
226C2×C4×C31—42no—
227C2×C4×C31—42no—
228C2×C4×C31—42no—
229C2×C4×C31—42no—
230C2×C4×C31—42no—
231C2×C4×C31—42no—
232C2×C4×C31—42no—
233C2×C4×C31—42no—
234C2×C4×C31—42no—
235C2×C4×C31—84no—
236C2×C4×C31#12012yes—
237C2×C4×C31#12012yes—
238C2×C4×C31C22×D411yescharacteristic
239C2×C4×C31C22×D411yescharacteristic
240C2×C4×C31C2×#4112yes—
241C2×C4×C31H(5,𝔽2)12yes—
242C8×C31—24no—
243C8×C31—42no—
244C2×#542C2×C3111yescharacteristic
245C2×#497C2×C3111yescharacteristic, complemented
246C2×#487C2×C3111yescharacteristic, complemented
247#2621C2×C3111yescharacteristic, complemented
248#2001C2×C3112yescomplemented
249#2000C2×C3112yes—
250#1993C2×C3111yescharacteristic, complemented
251#1958C2×C3111yescharacteristic, complemented
252#1878C2×C3112yescomplemented
253#1877C2×C3112yescomplemented
254#1750C2×C3111yescharacteristic, complemented
255C22×C31—42no—
256C22×C31—41no—
257C22×C31—42no—
258C22×C31—42no—
259C22×C31—42no—
260C22×C31—41no—
261C22×C31—41no—
262C22×C31—42no—
263C22×C31—82no—
264C22×C31—82no—
265C22×C31C2×#12011yescharacteristic
266C22×C31#53211yescharacteristic
267C22×C31#53511yescharacteristic
268C22×C31#57712yes—
269C22×C31#57812yes—
270C4×C31—22no—
271C4×C31—22no—
272C4×C31—41no—
273C4×C31—42no—
274C4×C31—41no—
275C4×C31—42no—
276C4×C31—42no—
277C4×C31—41no—
278C4×C31—82no—
279C4×C31—82no—
280C4×C31#54612yes—
281C4×C31#55012yes—
282C22×Q16C22×C3112yes—
283C22×SD16C22×C3112yes—
284#542—24no—
285#542—24no—
286C2×C4⋊Q8C22×C3111yescharacteristic
287C2×C4⋊Q8C22×C3111yescharacteristic
288C2×#126—21no—
289C2×#124—21no—
290C2×#124C22×C3111yescharacteristic
291C2×#123—21no—
292C2×C22⋊Q8—21no—
293C2×C22⋊Q8C22×C3111yescharacteristic
294C2×C22⋊Q8C22×C3112yes—
295C2×#121C22×C3111yescharacteristic
296C2×C4×Q8C22×C3111yescharacteristic
297C2×C4×D4C22×C3111yescharacteristic
298C2×#117C22×C3111yescharacteristic
299#497—24no—
300#487—24no—
301C2×#103C22×C3112yes—
302C2×#103C22×C3112yes—
303C2×#103C22×C3112yes—
304C2×#102C22×C3112yes—
305C2×C22⋊C8C22×C3112yes—
306C2×#97C22×C3111yescharacteristic
307C2×C4×C8C22×C3111yescharacteristic
308#398—21no—
309#398—21no—
310#397—21no—
311#397—21no—
312#391C22×C3112yes—
313#382C22×C3111yescharacteristic
314#340C22×C3112yes—
315#340C22×C3112yes—
316#336C22×C3112yes—
317C2×C31—81no—
318C2×C31—82no—
319C2×C31#243111yescharacteristic
320C2×C31#300911yescharacteristic
321C2×C31#301111yescharacteristic
322C2×C31#317812yes—
323C2×C31#322512yes—
324C22×Q8—21no—
325C22×Q8C23×C3112yes—
326C22×Q8C23×C3111yescharacteristic
327C22×D4—21no—
328C22×D4C23×C3111yescharacteristic
329C23×C4—21no—
330C23×C4C23×C3111yescharacteristic
331C2×Q16—24no—
332C2×Q16—24no—
333C2×Q16—42no—
334C2×SD16—24no—
335C2×SD16—24no—
336C2×SD16—42no—
337C22×C8C23×C3112yes—
338C22×C8C23×C3112yes—
339C4⋊Q8—22no—
340C4⋊Q8—22no—
341C4⋊Q8—22no—
342C4⋊Q8D4×C3114yes—
343#126—28no—
344#124—28no—
345#124—22no—
346#124D4×C3114yes—
347#123—28no—
348C22⋊Q8—28no—
349C22⋊Q8—22no—
350C22⋊Q8—22no—
351C22⋊Q8—42no—
352C22⋊Q8—42no—
353#121—22no—
354#121—22no—
355C4×Q8—22no—
356C4×Q8—22no—
357C4×D4—22no—
358C4×D4—22no—
359#117—22no—
360#117—22no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4D4×C3111yescharacteristic
368C2×C4⋊C4D4×C3111yescharacteristic
369C2×C4⋊C4C23×C3112yes—
370C2×C4⋊C4C23×C3111yescharacteristic
371C2×C4⋊C4C23×C3111yescharacteristic
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4D4×C3111yescharacteristic
379C2×C22⋊C4D4×C3111yescharacteristic
380C2×C42—21no—
381C2×C42C23×C3111yescharacteristic
382C2×C42C23×C3111yescharacteristic
383#103—24no—
384#103—24no—
385#103—42no—
386#102—42no—
387C22⋊C8—24no—
388#97—22no—
389C4×C8—22no—
390#95—21no—
391#95—21no—
392#95—22no—
393#95—22no—
394C31#2561311yescharacteristic, direct product factor, Sylow
395C24—21no—
396C24C2×D4×C3111yescharacteristic
397C2×Q8—24no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—82no—
408C2×Q8C2×D4×C3112yes—
409C2×D4—24no—
410C2×D4—21no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—82no—
414C2×D4C2×D4×C3112yes—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—82no—
426C22×C4C2×D4×C3111yescharacteristic
427C22×C4C2×D4×C3111yescharacteristic
428C22×C4C2×D4×C3111yescharacteristic
429C22×C4C2×D4×C3111yescharacteristic
430C22×C4C2×D4×C3111yescharacteristic
431C22×C4C24×C3111yescharacteristic
432Q16—48no—
433SD16—48no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8#41×C3114yes—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4C2×D4×C3112yes—
453C4⋊C4C2×D4×C3112yes—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—44no—
462C42—24no—
463C42—21no—
464C42—21no—
465C42C2×D4×C3112yes—
466C42C2×D4×C3112yes—
467C23—21no—
468C23—22no—
469C23—21no—
470C23—21no—
471C23—22no—
472C23—22no—
473C23—41no—
474C23—41no—
475C23—42no—
476C23—42no—
477C23C22×D4×C3111yescharacteristic
478Q8—24no—
479Q8—42no—
480Q8—42no—
481Q8—42no—
482Q8—42no—
483Q8—42no—
484Q8—84no—
485D4—24no—
486D4—42no—
487D4—84no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—84no—
513C2×C4C31×#12012yes—
514C2×C4C31×#12012yes—
515C2×C4C22×D4×C3111yescharacteristic
516C2×C4C22×D4×C3111yescharacteristic
517C2×C4C2×#41×C3112yes—
518C2×C4C31×H(5,𝔽2)12yes—
519C8—24no—
520C8—42no—
521C22—42no—
522C22—41no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—41no—
527C22—41no—
528C22—42no—
529C22—82no—
530C22—82no—
531C22C2×C31×#12011yescharacteristic
532C22C31×#53211yescharacteristic
533C22C31×#53511yescharacteristic
534C22C31×#57712yes—
535C22C31×#57812yes—
536C4—22no—
537C4—22no—
538C4—41no—
539C4—42no—
540C4—41no—
541C4—42no—
542C4—42no—
543C4—41no—
544C4—82no—
545C4—82no—
546C4C31×#54612yes—
547C4C31×#55012yes—
548C2—81no—
549C2—82no—
550C2C31×#243111yescharacteristic
551C2C31×#300911yescharacteristic
552C2C31×#301111yescharacteristic
553C2C31×#317812yes—
554C2C31×#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×#497C2semidirect
3. C2×C31×#487C2semidirect
4. C31×#2621C2semidirect
5. C31×#2001C2semidirect
7. C31×#1993C2semidirect
8. C31×#1958C2semidirect
9. C31×#1878C2semidirect
10. C31×#1877C2semidirect
11. C31×#1750C2semidirect
189. #25613C31direct (N × H)
245. C2×#497C2×C31semidirect
246. C2×#487C2×C31semidirect
247. #2621C2×C31semidirect
248. #2001C2×C31semidirect
250. #1993C2×C31semidirect
251. #1958C2×C31semidirect
252. #1878C2×C31semidirect
253. #1877C2×C31semidirect
254. #1750C2×C31semidirect
394. C31#25613direct (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×#4972
C2×C31×#4872
C31×#26212
C31×#20012
C31×#20002
C31×#19932
C31×#19582
C31×#18782
C31×#18772
C31×#17502
#2561331