Finite Group Databasebeta

← C31×#23929 #21480688

Subgroups — C31×#23929 #21480688

Order 7936.

1824 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#537C212yesmaximal, complemented
2C2×C31×#452C212yesmaximal, complemented
3C2×C31×#449C212yesmaximal, complemented
4C31×#2001C211yesmaximal, characteristic, complemented
5C31×#2000C212yesmaximal, complemented
6C31×#1999C211yesmaximal, characteristic, complemented
7C31×#1874C212yesmaximal, complemented
8C31×#1872C212yesmaximal, complemented
9C31×#1780C211yesmaximal, characteristic, complemented
10C22×#41×C31C2212yes—
11C22×#41×C31C2211yescharacteristic
12C22×Q16×C31C2211yescharacteristic
13C22×SD16×C31C2212yes—
14C22×D8×C31C2211yescharacteristic
15C31×#537—28no—
16C31×#537—28no—
17C2×C31×C22⋊Q8—22no—
18C2×C31×C22⋊Q8C2212yes—
19C2×C31×C22⋊Q8C2212yes—
20C2×C31×#121—22no—
21C2×C31×#121—22no—
22C2×C31×#121—22no—
23C2×C31×#121C2212yes—
24C2×C31×#121C2212yes—
25C22×C4⋊C4×C31C2212yes—
26C31×#452—28no—
27C31×#449—28no—
28C2×C31×#103C2212yes—
29C2×C31×#103C2212yes—
30C2×C31×#102C2212yes—
31C2×C31×#102C2212yes—
32C2×C31×C22⋊C8C2212yes—
33C2×C31×C22⋊C8C2212yes—
34C2×C31×C22⋊C8C2211yescharacteristic
35C2×C31×C22⋊C8C2211yescharacteristic
36C31×#340C2211yescharacteristic
37C31×#340C2212yes—
38C31×#340C2211yescharacteristic
39C2×#41×C31—24no—
40C2×#41×C31—24no—
41C2×#41×C31—24no—
42C2×#41×C31—24no—
43C2×#41×C31—24no—
44C2×#41×C31—24no—
45C2×#41×C31—41no—
46C2×#41×C31—41no—
47C2×#41×C31—42no—
48C2×#41×C31—42no—
49C22×Q8×C31D411yescharacteristic
50C22×Q8×C31C2312yes—
51C22×D4×C31—22no—
52C22×D4×C31—22no—
53C22×D4×C31—22no—
54C22×D4×C31D411yescharacteristic
55C22×D4×C31D412yes—
56C22×D4×C31C2312yes—
57C23×C4×C31—22no—
58C23×C4×C31—22no—
59C23×C4×C31C2311yescharacteristic
60C23×C4×C31C2312yes—
61C2×Q16×C31—24no—
62C2×Q16×C31—41no—
63C2×SD16×C31—24no—
64C2×SD16×C31—24no—
65C2×SD16×C31—42no—
66C2×D8×C31—24no—
67C2×D8×C31—41no—
68C22×C8×C31C2311yescharacteristic
69C22×C8×C31C2312yes—
70C22×C8×C31C2311yescharacteristic
71C31×C22⋊Q8—216no—
72C31×C22⋊Q8—24no—
73C31×C22⋊Q8—24no—
74C31×C22⋊Q8—24no—
75C31×C22⋊Q8D418yescomplemented
76C31×#121—216no—
77C31×#121—216no—
78C31×#121—216no—
79C31×#121—24no—
80C31×#121—24no—
81C31×#121—24no—
82C31×#121D418yescomplemented
83C2×C4⋊C4×C31—22no—
84C2×C4⋊C4×C31—22no—
85C2×C4⋊C4×C31—22no—
86C2×C4⋊C4×C31—24no—
87C2×C4⋊C4×C31—24no—
88C2×C4⋊C4×C31C2312yes—
89C2×C4⋊C4×C31C2312yes—
90C2×C22⋊C4×C31—22no—
91C2×C22⋊C4×C31—22no—
92C2×C22⋊C4×C31—22no—
93C2×C22⋊C4×C31—22no—
94C31×#103—24no—
95C31×#103—24no—
96C31×#102—24no—
97C31×#102—24no—
98C31×C22⋊C8—24no—
99C31×C22⋊C8—24no—
100C31×C22⋊C8—41no—
101C31×C22⋊C8—41no—
102C24×C31—22no—
103C24×C31C2×D411yescharacteristic
104C24×C31C2×D412yes—
105#41×C31—44no—
106#41×C31—48no—
107#41×C31—44no—
108#41×C31—48no—
109#41×C31—48no—
110#41×C31—48no—
111#41×C31—48no—
112C2×Q8×C31—24no—
113C2×Q8×C31—22no—
114C2×Q8×C31—22no—
115C2×Q8×C31—22no—
116C2×Q8×C31—22no—
117C2×Q8×C31—22no—
118C2×Q8×C31—41no—
119C2×Q8×C31C2×D414yes—
120C2×D4×C31—28no—
121C2×D4×C31—24no—
122C2×D4×C31—24no—
123C2×D4×C31—28no—
124C2×D4×C31—28no—
125C2×D4×C31—24no—
126C2×D4×C31—22no—
127C2×D4×C31—22no—
128C2×D4×C31—22no—
129C2×D4×C31—22no—
130C2×D4×C31—22no—
131C2×D4×C31—42no—
132C2×D4×C31—41no—
133C2×D4×C31—44no—
134C2×D4×C31—44no—
135C2×D4×C31—82no—
136C2×D4×C31—82no—
137C2×D4×C31—82no—
138C2×D4×C31—82no—
139C2×D4×C31C2×D414yes—
140C22×C4×C31—22no—
141C22×C4×C31—22no—
142C22×C4×C31—22no—
143C22×C4×C31—22no—
144C22×C4×C31—28no—
145C22×C4×C31—22no—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—28no—
149C22×C4×C31—22no—
150C22×C4×C31—22no—
151C22×C4×C31—21no—
152C22×C4×C31—21no—
153C22×C4×C31—44no—
154C22×C4×C31—44no—
155C22×C4×C31—82no—
156C22×C4×C31—82no—
157C22×C4×C31C2×D411yescharacteristic
158C22×C4×C31C2×D412yes—
159C22×C4×C31C2×D414yes—
160C22×C4×C31C2×D414yes—
161C22×C4×C31C2411yescharacteristic
162Q16×C31—44no—
163SD16×C31—48no—
164D8×C31—44no—
165C2×C8×C31—22no—
166C2×C8×C31—22no—
167C2×C8×C31—22no—
168C2×C8×C31—22no—
169C2×C8×C31—21no—
170C2×C8×C31—22no—
171C2×C8×C31—21no—
172C4⋊C4×C31—28no—
173C4⋊C4×C31—28no—
174C4⋊C4×C31—28no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31C2×D414yes—
178C4⋊C4×C31C2×D414yes—
179C22⋊C4×C31—28no—
180C22⋊C4×C31—28no—
181C22⋊C4×C31—28no—
182C22⋊C4×C31—28no—
183#23929C3111yesmaximal, characteristic, direct product factor, Sylow
184C23×C31—22no—
185C23×C31—22no—
186C23×C31—22no—
187C23×C31—22no—
188C23×C31—22no—
189C23×C31—22no—
190C23×C31—22no—
191C23×C31—24no—
192C23×C31—21no—
193C23×C31—21no—
194C23×C31—22no—
195C23×C31—21no—
196C23×C31—42no—
197C23×C31—42no—
198C23×C31—42no—
199C23×C31—42no—
200C23×C31—42no—
201C23×C31—42no—
202C23×C31#12014yes—
203C23×C31C22×D411yescharacteristic
204Q8×C31—24no—
205Q8×C31—24no—
206Q8×C31—44no—
207Q8×C31—42no—
208Q8×C31—42no—
209D4×C31—24no—
210D4×C31—24no—
211D4×C31—48no—
212D4×C31—44no—
213D4×C31—416no—
214D4×C31—42no—
215D4×C31—42no—
216D4×C31—88no—
217D4×C31—88no—
218C2×C4×C31—22no—
219C2×C4×C31—22no—
220C2×C4×C31—24no—
221C2×C4×C31—24no—
222C2×C4×C31—24no—
223C2×C4×C31—24no—
224C2×C4×C31—24no—
225C2×C4×C31—24no—
226C2×C4×C31—24no—
227C2×C4×C31—22no—
228C2×C4×C31—21no—
229C2×C4×C31—416no—
230C2×C4×C31—42no—
231C2×C4×C31—42no—
232C2×C4×C31—42no—
233C2×C4×C31—42no—
234C2×C4×C31—42no—
235C2×C4×C31—42no—
236C2×C4×C31—41no—
237C2×C4×C31—42no—
238C2×C4×C31—41no—
239C2×C4×C31—88no—
240C2×C4×C31#12014yes—
241C2×C4×C31C22×D412yes—
242C2×C4×C31C22×D412yes—
243C2×C4×C31H(5,𝔽2)11yescharacteristic
244C2×C4×C31H(5,𝔽2)11yescharacteristic
245C8×C31—41no—
246C8×C31—42no—
247C8×C31—41no—
248C2×#537C2×C3112yescomplemented
249C2×#452C2×C3112yescomplemented
250C2×#449C2×C3112yescomplemented
251#2001C2×C3111yescharacteristic, complemented
252#2000C2×C3112yescomplemented
253#1999C2×C3111yescharacteristic, complemented
254#1874C2×C3112yescomplemented
255#1872C2×C3112yescomplemented
256#1780C2×C3111yescharacteristic, complemented
257C22×C31—24no—
258C22×C31—24no—
259C22×C31—24no—
260C22×C31—42no—
261C22×C31—42no—
262C22×C31—42no—
263C22×C31—42no—
264C22×C31—44no—
265C22×C31—41no—
266C22×C31—41no—
267C22×C31—41no—
268C22×C31—42no—
269C22×C31—44no—
270C22×C31—42no—
271C22×C31—44no—
272C22×C31—82no—
273C22×C31—82no—
274C22×C31—82no—
275C22×C31—82no—
276C22×C31C2×#12012yes—
277C22×C31#53411yescharacteristic
278C22×C31#57611yescharacteristic
279C22×C31#57712yes—
280C22×C31#57811yescharacteristic
281C4×C31—24no—
282C4×C31—41no—
283C4×C31—44no—
284C4×C31—44no—
285C4×C31—44no—
286C4×C31D4211yescharacteristic
287C4×C31D4212yes—
288C4×C31D4211yescharacteristic
289C22×#41C22×C3112yes—
290C22×#41C22×C3111yescharacteristic
291C22×Q16C22×C3111yescharacteristic
292C22×SD16C22×C3112yes—
293C22×D8C22×C3111yescharacteristic
294#537—28no—
295#537—28no—
296C2×C22⋊Q8—22no—
297C2×C22⋊Q8C22×C3112yes—
298C2×C22⋊Q8C22×C3112yes—
299C2×#121—22no—
300C2×#121—22no—
301C2×#121—22no—
302C2×#121C22×C3112yes—
303C2×#121C22×C3112yes—
304C22×C4⋊C4C22×C3112yes—
305#452—28no—
306#449—28no—
307C2×#103C22×C3112yes—
308C2×#103C22×C3112yes—
309C2×#102C22×C3112yes—
310C2×#102C22×C3112yes—
311C2×C22⋊C8C22×C3112yes—
312C2×C22⋊C8C22×C3112yes—
313C2×C22⋊C8C22×C3111yescharacteristic
314C2×C22⋊C8C22×C3111yescharacteristic
315#340C22×C3111yescharacteristic
316#340C22×C3112yes—
317#340C22×C3111yescharacteristic
318C2×C31—44no—
319C2×C31—81no—
320C2×C31—84no—
321C2×C31#240311yescharacteristic
322C2×C31#301212yes—
323C2×C31#301312yes—
324C2×C31#320311yescharacteristic
325C2×C31#320411yescharacteristic
326C2×#41—24no—
327C2×#41—24no—
328C2×#41—24no—
329C2×#41—24no—
330C2×#41—24no—
331C2×#41—24no—
332C2×#41—41no—
333C2×#41—41no—
334C2×#41—42no—
335C2×#41—42no—
336C22×Q8D4×C3111yescharacteristic
337C22×Q8C23×C3112yes—
338C22×D4—22no—
339C22×D4—22no—
340C22×D4—22no—
341C22×D4D4×C3111yescharacteristic
342C22×D4D4×C3112yes—
343C22×D4C23×C3112yes—
344C23×C4—22no—
345C23×C4—22no—
346C23×C4C23×C3111yescharacteristic
347C23×C4C23×C3112yes—
348C2×Q16—24no—
349C2×Q16—41no—
350C2×SD16—24no—
351C2×SD16—24no—
352C2×SD16—42no—
353C2×D8—24no—
354C2×D8—41no—
355C22×C8C23×C3111yescharacteristic
356C22×C8C23×C3112yes—
357C22×C8C23×C3111yescharacteristic
358C22⋊Q8—216no—
359C22⋊Q8—24no—
360C22⋊Q8—24no—
361C22⋊Q8—24no—
362C22⋊Q8D4×C3118yescomplemented
363#121—216no—
364#121—216no—
365#121—216no—
366#121—24no—
367#121—24no—
368#121—24no—
369#121D4×C3118yescomplemented
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—24no—
374C2×C4⋊C4—24no—
375C2×C4⋊C4C23×C3112yes—
376C2×C4⋊C4C23×C3112yes—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381#103—24no—
382#103—24no—
383#102—24no—
384#102—24no—
385C22⋊C8—24no—
386C22⋊C8—24no—
387C22⋊C8—41no—
388C22⋊C8—41no—
389C31#2392911yescharacteristic, direct product factor, Sylow
390C24—22no—
391C24C2×D4×C3111yescharacteristic
392C24C2×D4×C3112yes—
393#41—44no—
394#41—48no—
395#41—44no—
396#41—48no—
397#41—48no—
398#41—48no—
399#41—48no—
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—41no—
407C2×Q8C2×D4×C3114yes—
408C2×D4—28no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—28no—
412C2×D4—28no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—42no—
420C2×D4—41no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—82no—
424C2×D4—82no—
425C2×D4—82no—
426C2×D4—82no—
427C2×D4C2×D4×C3114yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—28no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—28no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4C2×D4×C3111yescharacteristic
446C22×C4C2×D4×C3112yes—
447C22×C4C2×D4×C3114yes—
448C22×C4C2×D4×C3114yes—
449C22×C4C24×C3111yescharacteristic
450Q16—44no—
451SD16—48no—
452D8—44no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—21no—
458C2×C8—22no—
459C2×C8—21no—
460C4⋊C4—28no—
461C4⋊C4—28no—
462C4⋊C4—28no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4C2×D4×C3114yes—
466C4⋊C4C2×D4×C3114yes—
467C22⋊C4—28no—
468C22⋊C4—28no—
469C22⋊C4—28no—
470C22⋊C4—28no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—21no—
480C23—21no—
481C23—22no—
482C23—21no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23C31×#12014yes—
490C23C22×D4×C3111yescharacteristic
491Q8—24no—
492Q8—24no—
493Q8—44no—
494Q8—42no—
495Q8—42no—
496D4—24no—
497D4—24no—
498D4—48no—
499D4—44no—
500D4—416no—
501D4—42no—
502D4—42no—
503D4—88no—
504D4—88no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—416no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—88no—
527C2×C4C31×#12014yes—
528C2×C4C22×D4×C3112yes—
529C2×C4C22×D4×C3112yes—
530C2×C4C31×H(5,𝔽2)11yescharacteristic
531C2×C4C31×H(5,𝔽2)11yescharacteristic
532C8—41no—
533C8—42no—
534C8—41no—
535C22—24no—
536C22—24no—
537C22—24no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—44no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—42no—
547C22—44no—
548C22—42no—
549C22—44no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22C2×C31×#12012yes—
555C22C31×#53411yescharacteristic
556C22C31×#57611yescharacteristic
557C22C31×#57712yes—
558C22C31×#57811yescharacteristic
559C4—24no—
560C4—41no—
561C4—44no—
562C4—44no—
563C4—44no—
564C4D42×C3111yescharacteristic
565C4D42×C3112yes—
566C4D42×C3111yescharacteristic
567C2—44no—
568C2—81no—
569C2—84no—
570C2C31×#240311yescharacteristic
571C2C31×#301212yes—
572C2C31×#301312yes—
573C2C31×#320311yescharacteristic
574C2C31×#320411yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#537C2semidirect
2. C2×C31×#452C2semidirect
3. C2×C31×#449C2semidirect
4. C31×#2001C2semidirect
5. C31×#2000C2semidirect
6. C31×#1999C2semidirect
7. C31×#1874C2semidirect
8. C31×#1872C2semidirect
9. C31×#1780C2semidirect
75. C31×C22⋊Q8D4semidirect
82. C31×#121D4semidirect
183. #23929C31direct (N × H)
248. C2×#537C2×C31semidirect
249. C2×#452C2×C31semidirect
250. C2×#449C2×C31semidirect
251. #2001C2×C31semidirect
252. #2000C2×C31semidirect
253. #1999C2×C31semidirect
254. #1874C2×C31semidirect
255. #1872C2×C31semidirect
256. #1780C2×C31semidirect
362. C22⋊Q8D4×C31semidirect
369. #121D4×C31semidirect
389. C31#23929direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5372
C2×C31×#4522
C2×C31×#4492
C31×#20012
C31×#20002
C31×#19992
C31×#18742
C31×#18722
C31×#17802
#2392931