Finite Group Databasebeta

← C31×#23593 #21480352

Subgroups — C31×#23593 #21480352

Order 7936.

1118 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×#3460C211yesmaximal, characteristic
2C4×C31×#137C212yesmaximal, complemented
3C31×#2326C211yesmaximal, characteristic, complemented
4C31×#2060C211yesmaximal, characteristic, complemented
5C31×#2039C211yesmaximal, characteristic, complemented
6C31×#2037C212yesmaximal, complemented
7C31×#2028C211yesmaximal, characteristic, complemented
8C31×#2007C211yesmaximal, characteristic, complemented
9C31×#2004C212yesmaximal, complemented
10C31×#1759C212yesmaximal, complemented
11C4×C31×#100C211yesmaximal, characteristic, complemented
12C2×C31×#137C2212yes—
13C31×#544C2211yescharacteristic
14C31×#542—22no—
15C31×#540—22no—
16C31×#538—22no—
17C31×#537C2211yescharacteristic
18C31×#535C2212yes—
19C31×#532—22no—
20C31×#529—22no—
21C2×C31×#124C2211yescharacteristic
22C31×#525—22no—
23C4×#41×C31C2212yes—
24C2×C4×Q8×C31C2211yescharacteristic
25C2×C4×D4×C31C2211yescharacteristic
26C31×#441—22no—
27C31×#440—22no—
28C4×Q16×C31—22no—
29C4×SD16×C31—22no—
30C31×#428C2212yes—
31C2×C31×#104C2212yes—
32C2×C31×#104C2212yes—
33C31×#419C2212yes—
34C31×#417C2212yes—
35C2×C31×#100C2212yes—
36C4×M16×C31C2212yes—
37C31×#391C2211yescharacteristic
38C31×#390C2211yescharacteristic
39C31×#388—22no—
40C31×#385—22no—
41C31×#384C2211yescharacteristic
42C31×#376C2211yescharacteristic
43C31×#343C2211yescharacteristic
44C31×#343C2211yescharacteristic
45C31×#339C2212yes—
46C31×#339C2212yes—
47C2×#41×C31—22no—
48C2×#41×C31C2312yes—
49C22×Q8×C31C2311yescharacteristic
50C22×D4×C31C2311yescharacteristic
51C23×C4×C31D412yes—
52C31×#137—24no—
53C31×#137—24no—
54C2×Q16×C31—22no—
55C2×SD16×C31—22no—
56C2×M16×C31C2312yes—
57C2×M16×C31C2312yes—
58C31×C4⋊Q8—22no—
59C31×#127—22no—
60C31×#126—24no—
61C31×#125—22no—
62C31×#124—24no—
63C31×#124—22no—
64C31×#124—22no—
65C31×#124—22no—
66C31×#123—24no—
67C31×#123—24no—
68C31×C22⋊Q8—22no—
69C31×C22⋊Q8—22no—
70C31×C22⋊Q8—22no—
71C31×#121—22no—
72C31×#121—24no—
73C31×#121—22no—
74C31×#121—22no—
75C31×#121—22no—
76C31×#121—22no—
77C31×#120—24no—
78C4×Q8×C31—22no—
79C4×Q8×C31—22no—
80C4×Q8×C31D412yes—
81C4×Q8×C31D412yes—
82C4×D4×C31—24no—
83C4×D4×C31—22no—
84C4×D4×C31—22no—
85C4×D4×C31—22no—
86C4×D4×C31—22no—
87C4×D4×C31D412yes—
88C4×D4×C31D412yes—
89C31×#117—22no—
90C31×#117—22no—
91C31×#117C2312yes—
92C2×C4⋊C4×C31—21no—
93C2×C4⋊C4×C31—22no—
94C2×C4⋊C4×C31C2311yescharacteristic
95C2×C4⋊C4×C31C2311yescharacteristic
96C2×C22⋊C4×C31—22no—
97C2×C22⋊C4×C31—22no—
98C2×C22⋊C4×C31D412yes—
99C2×C42×C31—22no—
100C2×C42×C31—21no—
101C2×C42×C31C2311yescharacteristic
102C2×C42×C31C2311yescharacteristic
103C2×C42×C31C2311yescharacteristic
104C31×#107—22no—
105C31×#106—22no—
106C31×#104—24no—
107C31×#104—24no—
108C31×#103—22no—
109C31×#103—22no—
110C31×#103—22no—
111C31×#102—22no—
112C31×#100—24no—
113C31×#97—22no—
114C4×C8×C31—22no—
115C31×#95—21no—
116C31×#95—21no—
117C31×#95—22no—
118C24×C31C2×D412yes—
119#41×C31—22no—
120#41×C31—22no—
121#41×C31—22no—
122#41×C31—22no—
123#41×C31—84no—
124C2×Q8×C31—22no—
125C2×Q8×C31—21no—
126C2×Q8×C31—22no—
127C2×Q8×C31—22no—
128C2×Q8×C31C2×D412yes—
129C2×Q8×C31#4112yescomplemented
130C2×D4×C31—22no—
131C2×D4×C31—24no—
132C2×D4×C31—24no—
133C2×D4×C31—22no—
134C2×D4×C31—22no—
135C2×D4×C31—22no—
136C2×D4×C31—21no—
137C2×D4×C31—21no—
138C2×D4×C31C2×D412yes—
139C22×C4×C31—21no—
140C22×C4×C31—21no—
141C22×C4×C31—22no—
142C22×C4×C31—22no—
143C22×C4×C31—22no—
144C22×C4×C31—22no—
145C22×C4×C31—22no—
146C22×C4×C31—21no—
147C22×C4×C31—21no—
148C22×C4×C31—82no—
149C22×C4×C31C2×D412yes—
150C22×C4×C31C2×D411yescharacteristic
151C22×C4×C31C2×D411yescharacteristic
152C22×C4×C31C2411yescharacteristic
153Q16×C31—28no—
154SD16×C31—28no—
155M16×C31—22no—
156M16×C31—22no—
157M16×C31—22no—
158M16×C31#4114yescomplemented
159C2×C8×C31—22no—
160C2×C8×C31—22no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—24no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—41no—
169C4⋊C4×C31—41no—
170C4⋊C4×C31—82no—
171C4⋊C4×C31C2×D412yes—
172C4⋊C4×C31C2×D412yes—
173C22⋊C4×C31—28no—
174C22⋊C4×C31—24no—
175C22⋊C4×C31—24no—
176C22⋊C4×C31—24no—
177C22⋊C4×C31—22no—
178C22⋊C4×C31—24no—
179C22⋊C4×C31—22no—
180C22⋊C4×C31—22no—
181C22⋊C4×C31—82no—
182C42×C31—22no—
183C42×C31—22no—
184C42×C31—22no—
185C42×C31—21no—
186C42×C31—22no—
187C42×C31—42no—
188C42×C31C2×D412yes—
189C42×C31C2×D412yes—
190C42×C31#4112yescomplemented
191#23593C3111yesmaximal, characteristic, direct product factor, Sylow
192C23×C31—22no—
193C23×C31—22no—
194C23×C31—22no—
195C23×C31—22no—
196C23×C31—42no—
197C23×C31—42no—
198C23×C31C22×D411yescharacteristic
199Q8×C31—24no—
200Q8×C31—22no—
201Q8×C31—22no—
202Q8×C31—42no—
203Q8×C31—82no—
204D4×C31—22no—
205D4×C31—22no—
206D4×C31—42no—
207D4×C31—42no—
208D4×C31—42no—
209D4×C31—42no—
210D4×C31—84no—
211D4×C31—82no—
212D4×C31—84no—
213C2×C4×C31—22no—
214C2×C4×C31—22no—
215C2×C4×C31—22no—
216C2×C4×C31—22no—
217C2×C4×C31—22no—
218C2×C4×C31—22no—
219C2×C4×C31—41no—
220C2×C4×C31—41no—
221C2×C4×C31—44no—
222C2×C4×C31—44no—
223C2×C4×C31—41no—
224C2×C4×C31—41no—
225C2×C4×C31—41no—
226C2×C4×C31—41no—
227C2×C4×C31—42no—
228C2×C4×C31—42no—
229C2×C4×C31—42no—
230C2×C4×C31—41no—
231C2×C4×C31—42no—
232C2×C4×C31—41no—
233C2×C4×C31—41no—
234C2×C4×C31—41no—
235C2×C4×C31—82no—
236C2×C4×C31—82no—
237C2×C4×C31#12011yescharacteristic
238C2×C4×C31#12012yes—
239C2×C4×C31#12011yescharacteristic
240C2×C4×C31C22×D412yes—
241C2×C4×C31C2×#4111yescharacteristic
242C2×C4×C31C2×#4112yes—
243C2×C4×C31C2×#4111yescharacteristic
244C8×C31—24no—
245C8×C31—42no—
246#3460C2×C3111yescharacteristic
247C4×#137C2×C3112yescomplemented
248#2326C2×C3111yescharacteristic, complemented
249#2060C2×C3111yescharacteristic, complemented
250#2039C2×C3111yescharacteristic, complemented
251#2037C2×C3112yescomplemented
252#2028C2×C3111yescharacteristic, complemented
253#2007C2×C3111yescharacteristic, complemented
254#2004C2×C3112yescomplemented
255#1759C2×C3112yescomplemented
256C4×#100C2×C3111yescharacteristic, complemented
257C22×C31—21no—
258C22×C31—21no—
259C22×C31—44no—
260C22×C31—42no—
261C22×C31—42no—
262C22×C31—42no—
263C22×C31—42no—
264C22×C31—42no—
265C22×C31—42no—
266C22×C31—82no—
267C22×C31C2×#12011yescharacteristic
268C22×C31#52511yescharacteristic
269C22×C31#53211yescharacteristic
270C4×C31—21no—
271C4×C31—21no—
272C4×C31—22no—
273C4×C31—42no—
274C4×C31—42no—
275C4×C31—42no—
276C4×C31—42no—
277C4×C31—42no—
278C4×C31—42no—
279C4×C31—81no—
280C4×C31—81no—
281C4×C31#52512yes—
282C4×C31#53212yes—
283C2×#137C22×C3112yes—
284#544C22×C3111yescharacteristic
285#542—22no—
286#540—22no—
287#538—22no—
288#537C22×C3111yescharacteristic
289#535C22×C3112yes—
290#532—22no—
291#529—22no—
292C2×#124C22×C3111yescharacteristic
293#525—22no—
294C4×#41C22×C3112yes—
295C2×C4×Q8C22×C3111yescharacteristic
296C2×C4×D4C22×C3111yescharacteristic
297#441—22no—
298#440—22no—
299C4×Q16—22no—
300C4×SD16—22no—
301#428C22×C3112yes—
302C2×#104C22×C3112yes—
303C2×#104C22×C3112yes—
304#419C22×C3112yes—
305#417C22×C3112yes—
306C2×#100C22×C3112yes—
307C4×M16C22×C3112yes—
308#391C22×C3111yescharacteristic
309#390C22×C3111yescharacteristic
310#388—22no—
311#385—22no—
312#384C22×C3111yescharacteristic
313#376C22×C3111yescharacteristic
314#343C22×C3111yescharacteristic
315#343C22×C3111yescharacteristic
316#339C22×C3112yes—
317#339C22×C3112yes—
318C2×C31—21no—
319C2×C31—21no—
320C2×C31—82no—
321C2×C31—82no—
322C2×C31#238811yescharacteristic
323C2×C31#301911yescharacteristic
324C2×C31#302211yescharacteristic
325C2×#41—22no—
326C2×#41C23×C3112yes—
327C22×Q8C23×C3111yescharacteristic
328C22×D4C23×C3111yescharacteristic
329C23×C4D4×C3112yes—
330#137—24no—
331#137—24no—
332C2×Q16—22no—
333C2×SD16—22no—
334C2×M16C23×C3112yes—
335C2×M16C23×C3112yes—
336C4⋊Q8—22no—
337#127—22no—
338#126—24no—
339#125—22no—
340#124—24no—
341#124—22no—
342#124—22no—
343#124—22no—
344#123—24no—
345#123—24no—
346C22⋊Q8—22no—
347C22⋊Q8—22no—
348C22⋊Q8—22no—
349#121—22no—
350#121—24no—
351#121—22no—
352#121—22no—
353#121—22no—
354#121—22no—
355#120—24no—
356C4×Q8—22no—
357C4×Q8—22no—
358C4×Q8D4×C3112yes—
359C4×Q8D4×C3112yes—
360C4×D4—24no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4D4×C3112yes—
366C4×D4D4×C3112yes—
367#117—22no—
368#117—22no—
369#117C23×C3112yes—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4C23×C3111yescharacteristic
373C2×C4⋊C4C23×C3111yescharacteristic
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4D4×C3112yes—
377C2×C42—22no—
378C2×C42—21no—
379C2×C42C23×C3111yescharacteristic
380C2×C42C23×C3111yescharacteristic
381C2×C42C23×C3111yescharacteristic
382#107—22no—
383#106—22no—
384#104—24no—
385#104—24no—
386#103—22no—
387#103—22no—
388#103—22no—
389#102—22no—
390#100—24no—
391#97—22no—
392C4×C8—22no—
393#95—21no—
394#95—21no—
395#95—22no—
396C31#2359311yescharacteristic, direct product factor, Sylow
397C24C2×D4×C3112yes—
398#41—22no—
399#41—22no—
400#41—22no—
401#41—22no—
402#41—84no—
403C2×Q8—22no—
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8C2×D4×C3112yes—
408C2×Q8#41×C3112yescomplemented
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4C2×D4×C3112yes—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—82no—
428C22×C4C2×D4×C3112yes—
429C22×C4C2×D4×C3111yescharacteristic
430C22×C4C2×D4×C3111yescharacteristic
431C22×C4C24×C3111yescharacteristic
432Q16—28no—
433SD16—28no—
434M16—22no—
435M16—22no—
436M16—22no—
437M16#41×C3114yescomplemented
438C2×C8—22no—
439C2×C8—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—82no—
450C4⋊C4C2×D4×C3112yes—
451C4⋊C4C2×D4×C3112yes—
452C22⋊C4—28no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—22no—
457C22⋊C4—24no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—82no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—21no—
465C42—22no—
466C42—42no—
467C42C2×D4×C3112yes—
468C42C2×D4×C3112yes—
469C42#41×C3112yescomplemented
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—42no—
475C23—42no—
476C23C22×D4×C3111yescharacteristic
477Q8—24no—
478Q8—22no—
479Q8—22no—
480Q8—42no—
481Q8—82no—
482D4—22no—
483D4—22no—
484D4—42no—
485D4—42no—
486D4—42no—
487D4—42no—
488D4—84no—
489D4—82no—
490D4—84no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4C31×#12011yescharacteristic
516C2×C4C31×#12012yes—
517C2×C4C31×#12011yescharacteristic
518C2×C4C22×D4×C3112yes—
519C2×C4C2×#41×C3111yescharacteristic
520C2×C4C2×#41×C3112yes—
521C2×C4C2×#41×C3111yescharacteristic
522C8—24no—
523C8—42no—
524C22—21no—
525C22—21no—
526C22—44no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—82no—
534C22C2×C31×#12011yescharacteristic
535C22C31×#52511yescharacteristic
536C22C31×#53211yescharacteristic
537C4—21no—
538C4—21no—
539C4—22no—
540C4—42no—
541C4—42no—
542C4—42no—
543C4—42no—
544C4—42no—
545C4—42no—
546C4—81no—
547C4—81no—
548C4C31×#52512yes—
549C4C31×#53212yes—
550C2—21no—
551C2—21no—
552C2—82no—
553C2—82no—
554C2C31×#238811yescharacteristic
555C2C31×#301911yescharacteristic
556C2C31×#302211yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3460C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×C31×#137C2semidirect
3. C31×#2326C2semidirect
4. C31×#2060C2semidirect
5. C31×#2039C2semidirect
6. C31×#2037C2semidirect
7. C31×#2028C2semidirect
8. C31×#2007C2semidirect
9. C31×#2004C2semidirect
10. C31×#1759C2semidirect
11. C4×C31×#100C2semidirect
129. C2×Q8×C31#41semidirect
158. M16×C31#41semidirect
190. C42×C31#41semidirect
191. #23593C31direct (N × H)
247. C4×#137C2×C31semidirect
248. #2326C2×C31semidirect
249. #2060C2×C31semidirect
250. #2039C2×C31semidirect
251. #2037C2×C31semidirect
252. #2028C2×C31semidirect
253. #2007C2×C31semidirect
254. #2004C2×C31semidirect
255. #1759C2×C31semidirect
256. C4×#100C2×C31semidirect
396. C31#23593direct (N × H)
408. C2×Q8#41×C31semidirect
437. M16#41×C31semidirect
469. C42#41×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#34602
C4×C31×#1372
C31×#23262
C31×#20602
C31×#20392
C31×#20372
C31×#20282
C31×#20072
C31×#20042
C31×#17592
C4×C31×#1002
#2359331