Finite Group Databasebeta

← C31×#24073 #21480832

Subgroups — C31×#24073 #21480832

Order 7936.

1066 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#542C211yesmaximal, characteristic
2C2×C31×#484C211yesmaximal, characteristic, complemented
3C2×C31×#480C211yesmaximal, characteristic
4C31×#2317C211yesmaximal, characteristic, complemented
5C31×#2057C212yesmaximal, complemented
6C31×#2042C212yesmaximal, complemented
7C31×#1875C211yesmaximal, characteristic
8C31×#1873C211yesmaximal, characteristic, complemented
9C31×#1863C212yesmaximal, complemented
10C31×#1862C212yesmaximal, complemented
11C31×#1818C211yesmaximal, characteristic, complemented
12C31×#542—24no—
13C31×#542—24no—
14C2×C31×C4⋊Q8C2211yescharacteristic
15C2×C31×#126—21no—
16C2×C31×#124—21no—
17C2×C31×#124C2211yescharacteristic
18C2×C31×#123—21no—
19C2×C31×C22⋊Q8—21no—
20C2×C31×C22⋊Q8C2211yescharacteristic
21C2×C31×#121C2211yescharacteristic
22C2×C4×Q8×C31C2211yescharacteristic
23C2×C4×D4×C31C2211yescharacteristic
24C2×C31×#117C2211yescharacteristic
25C2×C31×#117C2211yescharacteristic
26C22×C4⋊C4×C31C2211yescharacteristic
27C31×#484—24no—
28C31×#480—24no—
29C2×C31×#107C2212yes—
30C2×C31×#107C2212yes—
31C2×C31×#103C2212yes—
32C2×C31×#103C2212yes—
33C2×C31×#102C2212yes—
34C2×C31×#102C2212yes—
35C2×C31×C22⋊C8C2211yescharacteristic
36C2×C31×C22⋊C8C2211yescharacteristic
37C31×#389C2212yes—
38C31×#389C2212yes—
39C31×#385—21no—
40C31×#385—21no—
41C31×#384—21no—
42C31×#382—21no—
43C31×#340C2212yes—
44C31×#340C2212yes—
45C31×#336C2212yes—
46C31×#336C2212yes—
47C22×Q8×C31—21no—
48C22×Q8×C31C2311yescharacteristic
49C22×D4×C31—21no—
50C22×D4×C31C2311yescharacteristic
51C23×C4×C31—21no—
52C23×C4×C31—21no—
53C23×C4×C31C2311yescharacteristic
54C22×C8×C31C2312yes—
55C22×C8×C31C2312yes—
56C31×C4⋊Q8—22no—
57C31×C4⋊Q8—22no—
58C31×#126—28no—
59C31×#124—28no—
60C31×#124—22no—
61C31×#124—22no—
62C31×#123—28no—
63C31×C22⋊Q8—28no—
64C31×C22⋊Q8—22no—
65C31×C22⋊Q8D414yes—
66C31×#121—22no—
67C31×#121D414yes—
68C4×Q8×C31—22no—
69C4×Q8×C31—22no—
70C4×D4×C31—22no—
71C4×D4×C31—22no—
72C31×#117—22no—
73C31×#117—22no—
74C31×#117—24no—
75C2×C4⋊C4×C31—21no—
76C2×C4⋊C4×C31—21no—
77C2×C4⋊C4×C31—21no—
78C2×C4⋊C4×C31—22no—
79C2×C4⋊C4×C31—22no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31—22no—
82C2×C4⋊C4×C31—22no—
83C2×C4⋊C4×C31D411yescharacteristic
84C2×C4⋊C4×C31D411yescharacteristic
85C2×C4⋊C4×C31C2312yes—
86C2×C4⋊C4×C31C2311yescharacteristic
87C2×C4⋊C4×C31C2311yescharacteristic
88C2×C4⋊C4×C31C2312yes—
89C2×C22⋊C4×C31—21no—
90C2×C22⋊C4×C31—21no—
91C2×C22⋊C4×C31—21no—
92C2×C22⋊C4×C31—21no—
93C2×C22⋊C4×C31—21no—
94C2×C22⋊C4×C31—21no—
95C2×C22⋊C4×C31D411yescharacteristic
96C2×C22⋊C4×C31D411yescharacteristic
97C2×C42×C31—21no—
98C2×C42×C31—21no—
99C2×C42×C31C2311yescharacteristic
100C2×C42×C31C2311yescharacteristic
101C31×#107—24no—
102C31×#107—24no—
103C31×#103—24no—
104C31×#103—24no—
105C31×#102—24no—
106C31×#102—24no—
107C31×C22⋊C8—22no—
108C31×C22⋊C8—22no—
109C31×#95—22no—
110C31×#95—22no—
111C24×C31—21no—
112C24×C31C2×D411yescharacteristic
113C2×Q8×C31—24no—
114C2×Q8×C31—21no—
115C2×Q8×C31—22no—
116C2×Q8×C31—22no—
117C2×Q8×C31—82no—
118C2×Q8×C31C2×D412yes—
119C2×D4×C31—24no—
120C2×D4×C31—21no—
121C2×D4×C31—22no—
122C2×D4×C31—22no—
123C2×D4×C31—82no—
124C2×D4×C31C2×D412yes—
125C22×C4×C31—22no—
126C22×C4×C31—22no—
127C22×C4×C31—22no—
128C22×C4×C31—21no—
129C22×C4×C31—21no—
130C22×C4×C31—21no—
131C22×C4×C31—24no—
132C22×C4×C31—22no—
133C22×C4×C31—42no—
134C22×C4×C31—82no—
135C22×C4×C31—82no—
136C22×C4×C31C2×D411yescharacteristic
137C22×C4×C31C2×D411yescharacteristic
138C22×C4×C31C2×D411yescharacteristic
139C22×C4×C31C2×D411yescharacteristic
140C22×C4×C31C2×D411yescharacteristic
141C22×C4×C31C2×D412yes—
142C22×C4×C31C2×D412yes—
143C22×C4×C31C2411yescharacteristic
144C2×C8×C31—22no—
145C2×C8×C31—22no—
146C2×C8×C31—22no—
147C2×C8×C31—22no—
148C2×C8×C31—22no—
149C2×C8×C31—22no—
150C4⋊C4×C31—24no—
151C4⋊C4×C31—24no—
152C4⋊C4×C31—24no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—21no—
156C4⋊C4×C31—21no—
157C4⋊C4×C31—22no—
158C4⋊C4×C31—22no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—42no—
161C4⋊C4×C31—82no—
162C4⋊C4×C31—82no—
163C4⋊C4×C31C2×D412yes—
164C4⋊C4×C31C2×D412yes—
165C4⋊C4×C31#4114yes—
166C22⋊C4×C31—24no—
167C22⋊C4×C31—24no—
168C22⋊C4×C31—24no—
169C22⋊C4×C31—24no—
170C22⋊C4×C31—24no—
171C22⋊C4×C31—22no—
172C22⋊C4×C31—22no—
173C22⋊C4×C31—24no—
174C42×C31—24no—
175C42×C31—24no—
176C42×C31—21no—
177C42×C31—21no—
178C42×C31—21no—
179C42×C31—21no—
180#24073C3111yesmaximal, characteristic, direct product factor, Sylow
181C23×C31—21no—
182C23×C31—21no—
183C23×C31—22no—
184C23×C31—22no—
185C23×C31—22no—
186C23×C31—41no—
187C23×C31—41no—
188C23×C31—42no—
189C23×C31—42no—
190C23×C31#12012yes—
191C23×C31C22×D411yescharacteristic
192Q8×C31—24no—
193Q8×C31—42no—
194Q8×C31—84no—
195D4×C31—24no—
196D4×C31—42no—
197D4×C31—84no—
198C2×C4×C31—21no—
199C2×C4×C31—21no—
200C2×C4×C31—24no—
201C2×C4×C31—24no—
202C2×C4×C31—21no—
203C2×C4×C31—21no—
204C2×C4×C31—22no—
205C2×C4×C31—21no—
206C2×C4×C31—21no—
207C2×C4×C31—22no—
208C2×C4×C31—22no—
209C2×C4×C31—24no—
210C2×C4×C31—22no—
211C2×C4×C31—22no—
212C2×C4×C31—22no—
213C2×C4×C31—22no—
214C2×C4×C31—41no—
215C2×C4×C31—41no—
216C2×C4×C31—42no—
217C2×C4×C31—42no—
218C2×C4×C31—42no—
219C2×C4×C31—42no—
220C2×C4×C31—42no—
221C2×C4×C31—42no—
222C2×C4×C31—42no—
223C2×C4×C31—84no—
224C2×C4×C31—84no—
225C2×C4×C31#12012yes—
226C2×C4×C31C22×D411yescharacteristic
227C2×C4×C31C22×D411yescharacteristic
228C2×C4×C31C2×#4112yes—
229C2×C4×C312−1+412yes—
230C8×C31—42no—
231C8×C31—42no—
232C2×#542C2×C3111yescharacteristic
233C2×#484C2×C3111yescharacteristic, complemented
234C2×#480C2×C3111yescharacteristic
235#2317C2×C3111yescharacteristic, complemented
236#2057C2×C3112yescomplemented
237#2042C2×C3112yescomplemented
238#1875C2×C3111yescharacteristic
239#1873C2×C3111yescharacteristic, complemented
240#1863C2×C3112yescomplemented
241#1862C2×C3112yescomplemented
242#1818C2×C3111yescharacteristic, complemented
243C22×C31—22no—
244C22×C31—22no—
245C22×C31—22no—
246C22×C31—42no—
247C22×C31—42no—
248C22×C31—42no—
249C22×C31—42no—
250C22×C31—42no—
251C22×C31—82no—
252C22×C31—82no—
253C22×C31C2×#12011yescharacteristic
254C22×C31#52511yescharacteristic
255C22×C31#53611yescharacteristic
256C22×C31#57612yes—
257C22×C31#57812yes—
258C4×C31—22no—
259C4×C31—44no—
260C4×C31—42no—
261C4×C31—41no—
262C4×C31—41no—
263C4×C31—42no—
264C4×C31—42no—
265C4×C31—41no—
266C4×C31—41no—
267C4×C31—82no—
268C4×C31#54712yes—
269C4×C31#54712yes—
270#542—24no—
271#542—24no—
272C2×C4⋊Q8C22×C3111yescharacteristic
273C2×#126—21no—
274C2×#124—21no—
275C2×#124C22×C3111yescharacteristic
276C2×#123—21no—
277C2×C22⋊Q8—21no—
278C2×C22⋊Q8C22×C3111yescharacteristic
279C2×#121C22×C3111yescharacteristic
280C2×C4×Q8C22×C3111yescharacteristic
281C2×C4×D4C22×C3111yescharacteristic
282C2×#117C22×C3111yescharacteristic
283C2×#117C22×C3111yescharacteristic
284C22×C4⋊C4C22×C3111yescharacteristic
285#484—24no—
286#480—24no—
287C2×#107C22×C3112yes—
288C2×#107C22×C3112yes—
289C2×#103C22×C3112yes—
290C2×#103C22×C3112yes—
291C2×#102C22×C3112yes—
292C2×#102C22×C3112yes—
293C2×C22⋊C8C22×C3111yescharacteristic
294C2×C22⋊C8C22×C3111yescharacteristic
295#389C22×C3112yes—
296#389C22×C3112yes—
297#385—21no—
298#385—21no—
299#384—21no—
300#382—21no—
301#340C22×C3112yes—
302#340C22×C3112yes—
303#336C22×C3112yes—
304#336C22×C3112yes—
305C2×C31—42no—
306C2×C31—82no—
307C2×C31#240611yescharacteristic
308C2×C31#300811yescharacteristic
309C2×C31#301111yescharacteristic
310C2×C31#312412yes—
311C2×C31#324312yes—
312C22×Q8—21no—
313C22×Q8C23×C3111yescharacteristic
314C22×D4—21no—
315C22×D4C23×C3111yescharacteristic
316C23×C4—21no—
317C23×C4—21no—
318C23×C4C23×C3111yescharacteristic
319C22×C8C23×C3112yes—
320C22×C8C23×C3112yes—
321C4⋊Q8—22no—
322C4⋊Q8—22no—
323#126—28no—
324#124—28no—
325#124—22no—
326#124—22no—
327#123—28no—
328C22⋊Q8—28no—
329C22⋊Q8—22no—
330C22⋊Q8D4×C3114yes—
331#121—22no—
332#121D4×C3114yes—
333C4×Q8—22no—
334C4×Q8—22no—
335C4×D4—22no—
336C4×D4—22no—
337#117—22no—
338#117—22no—
339#117—24no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4D4×C3111yescharacteristic
349C2×C4⋊C4D4×C3111yescharacteristic
350C2×C4⋊C4C23×C3112yes—
351C2×C4⋊C4C23×C3111yescharacteristic
352C2×C4⋊C4C23×C3111yescharacteristic
353C2×C4⋊C4C23×C3112yes—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4D4×C3111yescharacteristic
361C2×C22⋊C4D4×C3111yescharacteristic
362C2×C42—21no—
363C2×C42—21no—
364C2×C42C23×C3111yescharacteristic
365C2×C42C23×C3111yescharacteristic
366#107—24no—
367#107—24no—
368#103—24no—
369#103—24no—
370#102—24no—
371#102—24no—
372C22⋊C8—22no—
373C22⋊C8—22no—
374#95—22no—
375#95—22no—
376C31#2407311yescharacteristic, direct product factor, Sylow
377C24—21no—
378C24C2×D4×C3111yescharacteristic
379C2×Q8—24no—
380C2×Q8—21no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—82no—
384C2×Q8C2×D4×C3112yes—
385C2×D4—24no—
386C2×D4—21no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—82no—
390C2×D4C2×D4×C3112yes—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—42no—
400C22×C4—82no—
401C22×C4—82no—
402C22×C4C2×D4×C3111yescharacteristic
403C22×C4C2×D4×C3111yescharacteristic
404C22×C4C2×D4×C3111yescharacteristic
405C22×C4C2×D4×C3111yescharacteristic
406C22×C4C2×D4×C3111yescharacteristic
407C22×C4C2×D4×C3112yes—
408C22×C4C2×D4×C3112yes—
409C22×C4C24×C3111yescharacteristic
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—42no—
427C4⋊C4—82no—
428C4⋊C4—82no—
429C4⋊C4C2×D4×C3112yes—
430C4⋊C4C2×D4×C3112yes—
431C4⋊C4#41×C3114yes—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—24no—
440C42—24no—
441C42—24no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—21no—
446C23—21no—
447C23—21no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—41no—
452C23—41no—
453C23—42no—
454C23—42no—
455C23C31×#12012yes—
456C23C22×D4×C3111yescharacteristic
457Q8—24no—
458Q8—42no—
459Q8—84no—
460D4—24no—
461D4—42no—
462D4—84no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—84no—
489C2×C4—84no—
490C2×C4C31×#12012yes—
491C2×C4C22×D4×C3111yescharacteristic
492C2×C4C22×D4×C3111yescharacteristic
493C2×C4C2×#41×C3112yes—
494C2×C4C31×2−1+412yes—
495C8—42no—
496C8—42no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—82no—
506C22—82no—
507C22C2×C31×#12011yescharacteristic
508C22C31×#52511yescharacteristic
509C22C31×#53611yescharacteristic
510C22C31×#57612yes—
511C22C31×#57812yes—
512C4—22no—
513C4—44no—
514C4—42no—
515C4—41no—
516C4—41no—
517C4—42no—
518C4—42no—
519C4—41no—
520C4—41no—
521C4—82no—
522C4C31×#54712yes—
523C4C31×#54712yes—
524C2—42no—
525C2—82no—
526C2C31×#240611yescharacteristic
527C2C31×#300811yescharacteristic
528C2C31×#301111yescharacteristic
529C2C31×#312412yes—
530C2C31×#324312yes—
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×#484C2semidirect
4. C31×#2317C2semidirect
5. C31×#2057C2semidirect
6. C31×#2042C2semidirect
8. C31×#1873C2semidirect
9. C31×#1863C2semidirect
10. C31×#1862C2semidirect
11. C31×#1818C2semidirect
180. #24073C31direct (N × H)
233. C2×#484C2×C31semidirect
235. #2317C2×C31semidirect
236. #2057C2×C31semidirect
237. #2042C2×C31semidirect
239. #1873C2×C31semidirect
240. #1863C2×C31semidirect
241. #1862C2×C31semidirect
242. #1818C2×C31semidirect
376. C31#24073direct (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×#4842
C2×C31×#4802
C31×#23172
C31×#20572
C31×#20422
C31×#18752
C31×#18732
C31×#18632
C31×#18622
C31×#18182
#2407331