Finite Group Databasebeta

← C31×#18575 #21475334

Subgroups — C31×#18575 #21475334

Order 7936.

894 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×#3503C211yesmaximal, characteristic
2C31×#3255C211yesmaximal, characteristic, complemented
3C31×#3139C211yesmaximal, characteristic, complemented
4C31×#3107C211yesmaximal, characteristic
5C31×#1645C212yesmaximal, complemented
6C31×#1645C212yesmaximal, complemented
7C31×#1644C212yesmaximal, complemented
8C31×#1643C212yesmaximal, complemented
9C31×#1548C211yesmaximal, characteristic, complemented
10C31×#1489C211yesmaximal, characteristic
11C31×#1485C211yesmaximal, characteristic, complemented
12Q82×C31C2212yes—
13C31×#557—21no—
14C31×#554—22no—
15C31×#554C2212yes—
16D4×Q8×C31—22no—
17D4×Q8×C31—21no—
18C31×#544—22no—
19C31×#544—21no—
20C31×#533—21no—
21C31×#533C2211yescharacteristic
22C31×#533C2211yescharacteristic
23C2×C31×C4⋊Q8—21no—
24C2×C31×C4⋊Q8C2211yescharacteristic
25C31×#520—21no—
26C4×#41×C31C2211yescharacteristic
27C31×#491—22no—
28C31×#487C2212yes—
29C31×#487C2212yes—
30C31×#485—21no—
31C31×#475—21no—
32C31×#475C2211yescharacteristic
33C31×#475C2212yes—
34C31×#474C2211yescharacteristic
35C31×#470—22no—
36C31×#462—21no—
37C31×#462C2211yescharacteristic
38C31×#462C2212yes—
39C31×#461C2211yescharacteristic
40C31×#452—21no—
41C4×Q16×C31—21no—
42C4×SD16×C31—21no—
43C31×#431C2211yescharacteristic
44C31×#423C2211yescharacteristic
45C31×#419—22no—
46C4×M16×C31C2211yescharacteristic
47C31×#333C2212yes—
48C31×#333C2212yes—
49C31×Q8⋊C8C2212yes—
50C31×Q8⋊C8C2212yes—
51C31×Q8⋊C8C2212yes—
52C31×D4⋊C8C2212yes—
53C2×#41×C31D411yescharacteristic
54C22×Q8×C31—22no—
55C22×Q8×C31D412yes—
56C2×Q16×C31—21no—
57C2×Q16×C31—22no—
58C2×SD16×C31—21no—
59C2×M16×C31—21no—
60C2×M16×C31—21no—
61C31×C4⋊Q8—22no—
62C31×C4⋊Q8—22no—
63C31×C4⋊Q8—24no—
64C31×C4⋊Q8—22no—
65C31×C4⋊Q8—22no—
66C31×C4⋊Q8—21no—
67C31×C4⋊Q8—21no—
68C31×C4⋊Q8D411yescharacteristic
69C31×C4⋊Q8D412yes—
70C31×C4⋊Q8D411yescharacteristic
71C31×C4⋊Q8C2311yescharacteristic
72C31×C4⋊Q8C2311yescharacteristic
73C31×C4⋊Q8C2312yes—
74C31×#125—22no—
75C31×#125—24no—
76C31×#125—22no—
77C31×#125—21no—
78C31×#125—21no—
79C31×C22⋊Q8—22no—
80C31×C22⋊Q8—24no—
81C31×C22⋊Q8—24no—
82C31×C22⋊Q8—22no—
83C31×C22⋊Q8—24no—
84C31×C22⋊Q8—21no—
85C31×C22⋊Q8—21no—
86C31×C22⋊Q8—21no—
87C31×C22⋊Q8D412yes—
88C4×Q8×C31—22no—
89C4×Q8×C31—22no—
90C4×Q8×C31—22no—
91C4×Q8×C31—21no—
92C4×Q8×C31—21no—
93C4×Q8×C31—21no—
94C4×Q8×C31D411yescharacteristic
95C4×Q8×C31C2311yescharacteristic
96C4×Q8×C31C2312yes—
97C4×D4×C31—24no—
98C4×D4×C31—22no—
99C4×D4×C31—21no—
100C4×D4×C31—21no—
101C4×D4×C31D411yescharacteristic
102C4×D4×C31C2311yescharacteristic
103C31×#117—22no—
104C31×#117—21no—
105C31×#117—21no—
106C31×#117—21no—
107C31×#117D411yescharacteristic
108C2×C4⋊C4×C31—22no—
109C2×C4⋊C4×C31—21no—
110C2×C4⋊C4×C31—21no—
111C2×C4⋊C4×C31—22no—
112C2×C42×C31—21no—
113C2×C42×C31C2311yescharacteristic
114C31×#107—21no—
115C31×#106—21no—
116C31×#106—22no—
117C31×C4⋊C8—21no—
118C31×C4⋊C8C2312yes—
119C31×C4⋊C8C2311yescharacteristic
120C31×C4⋊C8C2311yescharacteristic
121C31×#103—24no—
122C31×#103—24no—
123C31×#103—21no—
124C31×#103—22no—
125C31×#103—21no—
126C31×#103—21no—
127C31×#103—22no—
128C31×#102—21no—
129C31×C22⋊C8—21no—
130C31×#97—21no—
131C4×C8×C31—21no—
132C4×C8×C31C2312yes—
133#41×C31—22no—
134#41×C31—22no—
135C2×Q8×C31—24no—
136C2×Q8×C31—22no—
137C2×Q8×C31—21no—
138C2×Q8×C31—21no—
139C2×Q8×C31—21no—
140C2×Q8×C31—44no—
141C2×Q8×C31—84no—
142C2×Q8×C31C2×D412yes—
143C2×Q8×C31C2×D411yescharacteristic
144C2×Q8×C31C2×D414yes—
145C2×D4×C31—21no—
146C2×D4×C31C2×D411yescharacteristic
147C22×C4×C31—21no—
148C22×C4×C31—24no—
149C22×C4×C31—22no—
150C22×C4×C31—21no—
151C22×C4×C31—21no—
152C22×C4×C31C2×D411yescharacteristic
153C22×C4×C31C2×D411yescharacteristic
154C22×C4×C31C2×D411yescharacteristic
155Q16×C31—42no—
156Q16×C31—82no—
157SD16×C31—42no—
158M16×C31—42no—
159M16×C31—81no—
160C2×C8×C31—21no—
161C2×C8×C31—21no—
162C2×C8×C31—22no—
163C2×C8×C31—22no—
164C4⋊C4×C31—24no—
165C4⋊C4×C31—24no—
166C4⋊C4×C31—24no—
167C4⋊C4×C31—24no—
168C4⋊C4×C31—22no—
169C4⋊C4×C31—21no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—21no—
172C4⋊C4×C31—21no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—21no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—21no—
180C4⋊C4×C31—21no—
181C4⋊C4×C31—21no—
182C4⋊C4×C31—21no—
183C4⋊C4×C31—41no—
184C4⋊C4×C31—41no—
185C4⋊C4×C31C2×D412yes—
186C4⋊C4×C31C2×D411yescharacteristic
187C4⋊C4×C31C2×D411yescharacteristic
188C22⋊C4×C31—24no—
189C22⋊C4×C31—22no—
190C22⋊C4×C31—21no—
191C22⋊C4×C31—21no—
192C22⋊C4×C31—21no—
193C22⋊C4×C31—41no—
194C42×C31—24no—
195C42×C31—21no—
196C42×C31—21no—
197C42×C31—21no—
198C42×C31—22no—
199C42×C31—41no—
200C42×C31C2×D411yescharacteristic
201C42×C31C2×D411yescharacteristic
202C42×C31C2×D411yescharacteristic
203C42×C31C2411yescharacteristic
204#18575C3111yesmaximal, characteristic, direct product factor, Sylow
205C23×C31—21no—
206C23×C31#12011yescharacteristic
207Q8×C31—21no—
208Q8×C31—22no—
209Q8×C31—22no—
210Q8×C31—44no—
211Q8×C31—44no—
212Q8×C31—42no—
213Q8×C31—41no—
214Q8×C31—41no—
215Q8×C31—41no—
216Q8×C31—41no—
217Q8×C31—84no—
218Q8×C31#13712yes—
219D4×C31—24no—
220D4×C31—21no—
221D4×C31#13712yes—
222C2×C4×C31—24no—
223C2×C4×C31—21no—
224C2×C4×C31—21no—
225C2×C4×C31—21no—
226C2×C4×C31—21no—
227C2×C4×C31—21no—
228C2×C4×C31—22no—
229C2×C4×C31—22no—
230C2×C4×C31—21no—
231C2×C4×C31—21no—
232C2×C4×C31—24no—
233C2×C4×C31—22no—
234C2×C4×C31—41no—
235C2×C4×C31—41no—
236C2×C4×C31—42no—
237C2×C4×C31—41no—
238C2×C4×C31—41no—
239C2×C4×C31—82no—
240C2×C4×C31—84no—
241C2×C4×C31#12011yescharacteristic
242C2×C4×C31#12011yescharacteristic
243C2×C4×C31#12011yescharacteristic
244C2×C4×C31C22×D411yescharacteristic
245C2×C4×C31C22×D411yescharacteristic
246C2×C4×C31C22×D411yescharacteristic
247C8×C31—42no—
248C8×C31—41no—
249C8×C31—41no—
250C8×C31—42no—
251#3503C2×C3111yescharacteristic
252#3255C2×C3111yescharacteristic, complemented
253#3139C2×C3111yescharacteristic, complemented
254#3107C2×C3111yescharacteristic
255#1645C2×C3112yescomplemented
256#1645C2×C3112yescomplemented
257#1644C2×C3112yescomplemented
258#1643C2×C3112yescomplemented
259#1548C2×C3111yescharacteristic, complemented
260#1489C2×C3111yescharacteristic
261#1485C2×C3111yescharacteristic, complemented
262C22×C31—21no—
263C22×C31—22no—
264C22×C31—21no—
265C22×C31—21no—
266C22×C31—41no—
267C22×C31—41no—
268C22×C31C2×#12011yescharacteristic
269C4×C31—21no—
270C4×C31—22no—
271C4×C31—21no—
272C4×C31—21no—
273C4×C31—41no—
274C4×C31—42no—
275C4×C31—41no—
276C4×C31—44no—
277C4×C31—42no—
278C4×C31—81no—
279C4×C31—81no—
280C4×C31#45512yes—
281C4×C31C2×#13711yescharacteristic
282C4×C31C2×#13711yescharacteristic
283C4×C31#57711yescharacteristic
284C4×C31#57811yescharacteristic
285Q82C22×C3112yes—
286#557—21no—
287#554—22no—
288#554C22×C3112yes—
289D4×Q8—22no—
290D4×Q8—21no—
291#544—22no—
292#544—21no—
293#533—21no—
294#533C22×C3111yescharacteristic
295#533C22×C3111yescharacteristic
296C2×C4⋊Q8—21no—
297C2×C4⋊Q8C22×C3111yescharacteristic
298#520—21no—
299C4×#41C22×C3111yescharacteristic
300#491—22no—
301#487C22×C3112yes—
302#487C22×C3112yes—
303#485—21no—
304#475—21no—
305#475C22×C3111yescharacteristic
306#475C22×C3112yes—
307#474C22×C3111yescharacteristic
308#470—22no—
309#462—21no—
310#462C22×C3111yescharacteristic
311#462C22×C3112yes—
312#461C22×C3111yescharacteristic
313#452—21no—
314C4×Q16—21no—
315C4×SD16—21no—
316#431C22×C3111yescharacteristic
317#423C22×C3111yescharacteristic
318#419—22no—
319C4×M16C22×C3111yescharacteristic
320#333C22×C3112yes—
321#333C22×C3112yes—
322Q8⋊C8C22×C3112yes—
323Q8⋊C8C22×C3112yes—
324Q8⋊C8C22×C3112yes—
325D4⋊C8C22×C3112yes—
326C2×C31—41no—
327C2×C31—42no—
328C2×C31#300411yescharacteristic
329C2×C31#301311yescharacteristic
330C2×C31C2×#45511yescharacteristic
331C2×#41D4×C3111yescharacteristic
332C22×Q8—22no—
333C22×Q8D4×C3112yes—
334C2×Q16—21no—
335C2×Q16—22no—
336C2×SD16—21no—
337C2×M16—21no—
338C2×M16—21no—
339C4⋊Q8—22no—
340C4⋊Q8—22no—
341C4⋊Q8—24no—
342C4⋊Q8—22no—
343C4⋊Q8—22no—
344C4⋊Q8—21no—
345C4⋊Q8—21no—
346C4⋊Q8D4×C3112yes—
347C4⋊Q8D4×C3111yescharacteristic
348C4⋊Q8D4×C3111yescharacteristic
349C4⋊Q8C23×C3111yescharacteristic
350C4⋊Q8C23×C3111yescharacteristic
351C4⋊Q8C23×C3112yes—
352#125—22no—
353#125—24no—
354#125—22no—
355#125—21no—
356#125—21no—
357C22⋊Q8—22no—
358C22⋊Q8—24no—
359C22⋊Q8—24no—
360C22⋊Q8—22no—
361C22⋊Q8—24no—
362C22⋊Q8—21no—
363C22⋊Q8—21no—
364C22⋊Q8—21no—
365C22⋊Q8D4×C3112yes—
366C4×Q8—22no—
367C4×Q8—22no—
368C4×Q8—22no—
369C4×Q8—21no—
370C4×Q8—21no—
371C4×Q8—21no—
372C4×Q8D4×C3111yescharacteristic
373C4×Q8C23×C3111yescharacteristic
374C4×Q8C23×C3112yes—
375C4×D4—24no—
376C4×D4—22no—
377C4×D4—21no—
378C4×D4—21no—
379C4×D4D4×C3111yescharacteristic
380C4×D4C23×C3111yescharacteristic
381#117—22no—
382#117—21no—
383#117—21no—
384#117—21no—
385#117D4×C3111yescharacteristic
386C2×C4⋊C4—22no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—22no—
390C2×C42—21no—
391C2×C42C23×C3111yescharacteristic
392#107—21no—
393#106—21no—
394#106—22no—
395C4⋊C8—21no—
396C4⋊C8C23×C3112yes—
397C4⋊C8C23×C3111yescharacteristic
398C4⋊C8C23×C3111yescharacteristic
399#103—24no—
400#103—24no—
401#103—21no—
402#103—22no—
403#103—21no—
404#103—21no—
405#103—22no—
406#102—21no—
407C22⋊C8—21no—
408#97—21no—
409C4×C8—21no—
410C4×C8C23×C3112yes—
411C31#1857511yescharacteristic, direct product factor, Sylow
412#41—22no—
413#41—22no—
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—44no—
420C2×Q8—84no—
421C2×Q8C2×D4×C3112yes—
422C2×Q8C2×D4×C3111yescharacteristic
423C2×Q8C2×D4×C3114yes—
424C2×D4—21no—
425C2×D4C2×D4×C3111yescharacteristic
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4C2×D4×C3111yescharacteristic
432C22×C4C2×D4×C3111yescharacteristic
433C22×C4C2×D4×C3111yescharacteristic
434Q16—42no—
435Q16—82no—
436SD16—42no—
437M16—42no—
438M16—81no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—22no—
442C2×C8—22no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4C2×D4×C3112yes—
465C4⋊C4C2×D4×C3111yescharacteristic
466C4⋊C4C2×D4×C3111yescharacteristic
467C22⋊C4—24no—
468C22⋊C4—22no—
469C22⋊C4—21no—
470C22⋊C4—21no—
471C22⋊C4—21no—
472C22⋊C4—41no—
473C42—24no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—22no—
478C42—41no—
479C42C2×D4×C3111yescharacteristic
480C42C2×D4×C3111yescharacteristic
481C42C2×D4×C3111yescharacteristic
482C42C24×C3111yescharacteristic
483C23—21no—
484C23C31×#12011yescharacteristic
485Q8—21no—
486Q8—22no—
487Q8—22no—
488Q8—44no—
489Q8—44no—
490Q8—42no—
491Q8—41no—
492Q8—41no—
493Q8—41no—
494Q8—41no—
495Q8—84no—
496Q8C31×#13712yes—
497D4—24no—
498D4—21no—
499D4C31×#13712yes—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—82no—
518C2×C4—84no—
519C2×C4C31×#12011yescharacteristic
520C2×C4C31×#12011yescharacteristic
521C2×C4C31×#12011yescharacteristic
522C2×C4C22×D4×C3111yescharacteristic
523C2×C4C22×D4×C3111yescharacteristic
524C2×C4C22×D4×C3111yescharacteristic
525C8—42no—
526C8—41no—
527C8—41no—
528C8—42no—
529C22—21no—
530C22—22no—
531C22—21no—
532C22—21no—
533C22—41no—
534C22—41no—
535C22C2×C31×#12011yescharacteristic
536C4—21no—
537C4—22no—
538C4—21no—
539C4—21no—
540C4—41no—
541C4—42no—
542C4—41no—
543C4—44no—
544C4—42no—
545C4—81no—
546C4—81no—
547C4C31×#45512yes—
548C4C2×C31×#13711yescharacteristic
549C4C2×C31×#13711yescharacteristic
550C4C31×#57711yescharacteristic
551C4C31×#57811yescharacteristic
552C2—41no—
553C2—42no—
554C2C31×#300411yescharacteristic
555C2C31×#301311yescharacteristic
556C2C2×C31×#45511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC2×#41×C31D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3255C2semidirect
3. C31×#3139C2semidirect
5. C31×#1645C2semidirect
6. C31×#1645C2semidirect
7. C31×#1644C2semidirect
8. C31×#1643C2semidirect
9. C31×#1548C2semidirect
11. C31×#1485C2semidirect
204. #18575C31direct (N × H)
252. #3255C2×C31semidirect
253. #3139C2×C31semidirect
255. #1645C2×C31semidirect
256. #1645C2×C31semidirect
257. #1644C2×C31semidirect
258. #1643C2×C31semidirect
259. #1548C2×C31semidirect
261. #1485C2×C31semidirect
411. C31#18575direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35032
C31×#32552
C31×#31392
C31×#31072
C31×#16452
C31×#16442
C31×#16432
C31×#15482
C31×#14892
C31×#14852
#1857531