Finite Group Databasebeta

← C31×#15730 #21472489

Subgroups — C31×#15730 #21472489

Order 7936.

1474 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#399C211yesmaximal, characteristic
2C2×C31×#385C211yesmaximal, characteristic
3C31×#1441C212yesmaximal, complemented
4C31×#1441C212yesmaximal, complemented
5C31×#1441C212yesmaximal, complemented
6C31×#1439C212yesmaximal
7C31×#1437C211yesmaximal, characteristic
8C31×#1436C212yesmaximal, complemented
9C31×C22⋊#95C212yesmaximal
10C24×C4×C31C2211yescharacteristic
11C22×C4⋊C4×C31C2212yes—
12C22×C4⋊C4×C31C2212yes—
13C22×C4⋊C4×C31C2212yes—
14C22×C4⋊C4×C31C2211yescharacteristic
15C22×C22⋊C4×C31C2211yescharacteristic
16C22×C22⋊C4×C31C2212yes—
17C22×C22⋊C4×C31C2211yescharacteristic
18C22×C22⋊C4×C31C2212yes—
19C22×C42×C31C2211yescharacteristic
20C31×#399—24no—
21C31×#385—24no—
22C2×C31×#95C2211yescharacteristic
23C2×C31×#95C2212yes—
24C2×C31×#95C2212yes—
25C2×C31×#95C2212yes—
26C2×C31×#95C2212yes—
27C2×C31×#95C2212yes—
28C2×C31×#95C2212yes—
29C2×C31×#95C2212yes—
30C2×C31×#95C2211yescharacteristic
31C2×C31×#95C2212yes—
32C2×C31×#95C2212yes—
33C25×C31C2311yescharacteristic
34C23×C4×C31—22no—
35C23×C4×C31—24no—
36C23×C4×C31—41no—
37C23×C4×C31—41no—
38C23×C4×C31—42no—
39C23×C4×C31C2312yes—
40C23×C4×C31C2311yescharacteristic
41C23×C4×C31C2311yescharacteristic
42C23×C4×C31C2312yes—
43C23×C4×C31C2312yes—
44C23×C4×C31C2312yes—
45C23×C4×C31C2312yes—
46C23×C4×C31C2312yes—
47C2×C4⋊C4×C31—24no—
48C2×C4⋊C4×C31—22no—
49C2×C4⋊C4×C31—24no—
50C2×C4⋊C4×C31—24no—
51C2×C4⋊C4×C31—24no—
52C2×C4⋊C4×C31—24no—
53C2×C4⋊C4×C31—24no—
54C2×C4⋊C4×C31—42no—
55C2×C4⋊C4×C31—42no—
56C2×C4⋊C4×C31—42no—
57C2×C4⋊C4×C31—41no—
58C2×C4⋊C4×C31—41no—
59C2×C22⋊C4×C31—22no—
60C2×C22⋊C4×C31—24no—
61C2×C22⋊C4×C31—24no—
62C2×C22⋊C4×C31—24no—
63C2×C22⋊C4×C31—41no—
64C2×C22⋊C4×C31—41no—
65C2×C22⋊C4×C31—42no—
66C2×C22⋊C4×C31—42no—
67C2×C22⋊C4×C31—42no—
68C2×C22⋊C4×C31D414yes—
69C2×C22⋊C4×C31Q818yes—
70C2×C42×C31—41no—
71C2×C42×C31—42no—
72C31×#95—22no—
73C31×#95—24no—
74C31×#95—24no—
75C31×#95—24no—
76C31×#95—41no—
77C31×#95—42no—
78C31×#95—42no—
79C31×#95—42no—
80C31×#95—42no—
81C31×#95—42no—
82C31×#95—42no—
83C24×C31—22no—
84C24×C31—22no—
85C24×C31—22no—
86C24×C31—22no—
87C24×C31—21no—
88C24×C31—21no—
89C24×C31—22no—
90C24×C31C2×D412yes—
91C24×C31C2×Q814yes—
92C24×C31C2411yescharacteristic
93C22×C4×C31—22no—
94C22×C4×C31—22no—
95C22×C4×C31—22no—
96C22×C4×C31—22no—
97C22×C4×C31—22no—
98C22×C4×C31—22no—
99C22×C4×C31—22no—
100C22×C4×C31—22no—
101C22×C4×C31—22no—
102C22×C4×C31—22no—
103C22×C4×C31—22no—
104C22×C4×C31—22no—
105C22×C4×C31—22no—
106C22×C4×C31—22no—
107C22×C4×C31—22no—
108C22×C4×C31—22no—
109C22×C4×C31—22no—
110C22×C4×C31—22no—
111C22×C4×C31—22no—
112C22×C4×C31—22no—
113C22×C4×C31—22no—
114C22×C4×C31—21no—
115C22×C4×C31—22no—
116C22×C4×C31—22no—
117C22×C4×C31—22no—
118C22×C4×C31—22no—
119C22×C4×C31—22no—
120C22×C4×C31—22no—
121C22×C4×C31—22no—
122C22×C4×C31—21no—
123C22×C4×C31—22no—
124C22×C4×C31—22no—
125C22×C4×C31—21no—
126C22×C4×C31—21no—
127C22×C4×C31—22no—
128C22×C4×C31—22no—
129C22×C4×C31—22no—
130C22×C4×C31—22no—
131C22×C4×C31—22no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—44no—
139C22×C4×C31—48no—
140C22×C4×C31—48no—
141C22×C4×C31—44no—
142C22×C4×C31—44no—
143C22×C4×C31C2×D412yes—
144C22×C4×C31C2×D412yes—
145C22×C4×C31C2×Q814yes—
146C22×C4×C31C2×Q814yes—
147C22×C4×C31#4114yes—
148C22×C4×C31#4114yes—
149C22×C4×C31#4114yes—
150C4⋊C4×C31—48no—
151C4⋊C4×C31—48no—
152C4⋊C4×C31—48no—
153C4⋊C4×C31—44no—
154C22⋊C4×C31—28no—
155C22⋊C4×C31—44no—
156C22⋊C4×C31—48no—
157C22⋊C4×C31—48no—
158C42×C31—44no—
159#15730C3111yesmaximal, characteristic, direct product factor, Sylow
160C23×C31—22no—
161C23×C31—24no—
162C23×C31—22no—
163C23×C31—24no—
164C23×C31—24no—
165C23×C31—24no—
166C23×C31—24no—
167C23×C31—24no—
168C23×C31—24no—
169C23×C31—24no—
170C23×C31—42no—
171C23×C31—42no—
172C23×C31—42no—
173C23×C31—42no—
174C23×C31—41no—
175C23×C31—42no—
176C23×C31—42no—
177C23×C31—41no—
178C23×C31—41no—
179C23×C31—41no—
180C23×C31C4⋊Q814yes—
181C23×C31C22×D411yescharacteristic
182C23×C31C22×Q812yes—
183C23×C31C2×#4112yes—
184C23×C31C2×#4112yes—
185C23×C31C2×#4112yes—
186C23×C31H(5,𝔽2)11yescharacteristic
187C23×C31H(5,𝔽2)11yescharacteristic
188C23×C31H(5,𝔽2)12yes—
189C23×C31H(5,𝔽2)12yes—
190C2×C4×C31—24no—
191C2×C4×C31—24no—
192C2×C4×C31—22no—
193C2×C4×C31—22no—
194C2×C4×C31—24no—
195C2×C4×C31—24no—
196C2×C4×C31—24no—
197C2×C4×C31—24no—
198C2×C4×C31—24no—
199C2×C4×C31—24no—
200C2×C4×C31—24no—
201C2×C4×C31—24no—
202C2×C4×C31—24no—
203C2×C4×C31—24no—
204C2×C4×C31—24no—
205C2×C4×C31—24no—
206C2×C4×C31—24no—
207C2×C4×C31—24no—
208C2×C4×C31—24no—
209C2×C4×C31—416no—
210C2×C4×C31—42no—
211C2×C4×C31—42no—
212C2×C4×C31—42no—
213C2×C4×C31—42no—
214C2×C4×C31—42no—
215C2×C4×C31—42no—
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—42no—
224C2×C4×C31—42no—
225C2×C4×C31—42no—
226C2×C4×C31—42no—
227C2×C4×C31—41no—
228C2×C4×C31—42no—
229C2×C4×C31—42no—
230C2×C4×C31—41no—
231C2×C4×C31—41no—
232C2×C4×C31—42no—
233C2×C4×C31—41no—
234C2×C4×C31—42no—
235C2×C4×C31—42no—
236C2×C4×C31—42no—
237C2×C4×C31—42no—
238C2×C4×C31—42no—
239C2×C4×C31—42no—
240C2×C4×C31—42no—
241C2×C4×C31—42no—
242C2×C4×C31—42no—
243C2×#399C2×C3111yescharacteristic
244C2×#385C2×C3111yescharacteristic
245#1441C2×C3112yescomplemented
246#1441C2×C3112yescomplemented
247#1441C2×C3112yescomplemented
248#1439C2×C3112yes—
249#1437C2×C3111yescharacteristic
250#1436C2×C3112yescomplemented
251C22⋊#95C2×C3112yes—
252C22×C31—24no—
253C22×C31—28no—
254C22×C31—44no—
255C22×C31—42no—
256C22×C31—42no—
257C22×C31—44no—
258C22×C31—44no—
259C22×C31—44no—
260C22×C31—44no—
261C22×C31#52511yescharacteristic
262C22×C31#52912yes—
263C22×C31C2×C4⋊Q811yescharacteristic
264C22×C31#53312yes—
265C22×C31#53411yescharacteristic
266C22×C31#53812yes—
267C22×C31#54012yes—
268C22×C31#54012yes—
269C22×C31#54312yes—
270C22×C31#54612yes—
271C22×C31#54612yes—
272C22×C31#55112yes—
273C22×C31#55111yescharacteristic
274C22×C31#55312yes—
275C22×C31#55412yes—
276C22×C31#55412yes—
277C22×C31#55412yes—
278C22×C31#55412yes—
279C22×C31#55911yescharacteristic
280C22×C31#56011yescharacteristic
281C22×C31#56011yescharacteristic
282C4×C31—42no—
283C4×C31—44no—
284C4×C31—42no—
285C4×C31—44no—
286C4×C31—44no—
287C4×C31—44no—
288C4×C31—44no—
289C4×C31—82no—
290C24×C4C22×C3111yescharacteristic
291C22×C4⋊C4C22×C3112yes—
292C22×C4⋊C4C22×C3112yes—
293C22×C4⋊C4C22×C3112yes—
294C22×C4⋊C4C22×C3111yescharacteristic
295C22×C22⋊C4C22×C3111yescharacteristic
296C22×C22⋊C4C22×C3112yes—
297C22×C22⋊C4C22×C3111yescharacteristic
298C22×C22⋊C4C22×C3112yes—
299C22×C42C22×C3111yescharacteristic
300#399—24no—
301#385—24no—
302C2×#95C22×C3111yescharacteristic
303C2×#95C22×C3112yes—
304C2×#95C22×C3112yes—
305C2×#95C22×C3112yes—
306C2×#95C22×C3112yes—
307C2×#95C22×C3112yes—
308C2×#95C22×C3112yes—
309C2×#95C22×C3112yes—
310C2×#95C22×C3112yes—
311C2×#95C22×C3111yescharacteristic
312C2×#95C22×C3112yes—
313C2×C31—44no—
314C2×C31#245712yes—
315C2×C31#252212yes—
316C2×C31#254912yes—
317C2×C31#258311yescharacteristic
318C2×C31#258912yes—
319C2×C31#269711yescharacteristic
320C2×C31#270911yescharacteristic
321C2×C31#274812yes—
322C2×C31#278012yes—
323C25C23×C3111yescharacteristic
324C23×C4—22no—
325C23×C4—24no—
326C23×C4—41no—
327C23×C4—41no—
328C23×C4—42no—
329C23×C4C23×C3112yes—
330C23×C4C23×C3111yescharacteristic
331C23×C4C23×C3111yescharacteristic
332C23×C4C23×C3112yes—
333C23×C4C23×C3112yes—
334C23×C4C23×C3112yes—
335C23×C4C23×C3112yes—
336C23×C4C23×C3112yes—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—42no—
345C2×C4⋊C4—42no—
346C2×C4⋊C4—42no—
347C2×C4⋊C4—41no—
348C2×C4⋊C4—41no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—41no—
354C2×C22⋊C4—41no—
355C2×C22⋊C4—42no—
356C2×C22⋊C4—42no—
357C2×C22⋊C4—42no—
358C2×C22⋊C4D4×C3114yes—
359C2×C22⋊C4Q8×C3118yes—
360C2×C42—41no—
361C2×C42—42no—
362#95—24no—
363#95—22no—
364#95—24no—
365#95—24no—
366#95—41no—
367#95—42no—
368#95—42no—
369#95—42no—
370#95—42no—
371#95—42no—
372#95—42no—
373C31#1573011yescharacteristic, direct product factor, Sylow
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—21no—
379C24—21no—
380C24—22no—
381C24C2×D4×C3112yes—
382C24C2×Q8×C3114yes—
383C24C24×C3111yescharacteristic
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—44no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4C2×D4×C3112yes—
435C22×C4C2×D4×C3112yes—
436C22×C4C2×Q8×C3114yes—
437C22×C4C2×Q8×C3114yes—
438C22×C4#41×C3114yes—
439C22×C4#41×C3114yes—
440C22×C4#41×C3114yes—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—44no—
445C22⋊C4—28no—
446C22⋊C4—44no—
447C22⋊C4—48no—
448C22⋊C4—48no—
449C42—44no—
450C23—22no—
451C23—24no—
452C23—22no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—41no—
465C23—42no—
466C23—42no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23C31×C4⋊Q814yes—
471C23C22×D4×C3111yescharacteristic
472C23C22×Q8×C3112yes—
473C23C2×#41×C3112yes—
474C23C2×#41×C3112yes—
475C23C2×#41×C3112yes—
476C23C31×H(5,𝔽2)11yescharacteristic
477C23C31×H(5,𝔽2)11yescharacteristic
478C23C31×H(5,𝔽2)12yes—
479C23C31×H(5,𝔽2)12yes—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—416no—
500C2×C4—42no—
501C2×C4—42no—
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—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C22—24no—
534C22—28no—
535C22—44no—
536C22—42no—
537C22—42no—
538C22—44no—
539C22—44no—
540C22—44no—
541C22—44no—
542C22C31×#52511yescharacteristic
543C22C31×#52912yes—
544C22C2×C31×C4⋊Q811yescharacteristic
545C22C31×#53312yes—
546C22C31×#53411yescharacteristic
547C22C31×#53812yes—
548C22C31×#54012yes—
549C22C31×#54012yes—
550C22C31×#54312yes—
551C22C31×#54612yes—
552C22C31×#54612yes—
553C22C31×#55112yes—
554C22C31×#55111yescharacteristic
555C22C31×#55312yes—
556C22C31×#55412yes—
557C22C31×#55412yes—
558C22C31×#55412yes—
559C22C31×#55412yes—
560C22C31×#55911yescharacteristic
561C22C31×#56011yescharacteristic
562C22C31×#56011yescharacteristic
563C4—44no—
564C4—42no—
565C4—42no—
566C4—44no—
567C4—44no—
568C4—44no—
569C4—44no—
570C4—82no—
571C2—44no—
572C2C31×#245712yes—
573C2C31×#252212yes—
574C2C31×#254912yes—
575C2C31×#258311yescharacteristic
576C2C31×#258912yes—
577C2C31×#269711yescharacteristic
578C2C31×#270911yescharacteristic
579C2C31×#274812yes—
580C2C31×#278012yes—
subgroupG/H
center, socleC24×C31C24
commutator, Frattini, cosocleC24C24×C31
prime radicalC25×C31C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#1441C2semidirect
4. C31×#1441C2semidirect
5. C31×#1441C2semidirect
8. C31×#1436C2semidirect
159. #15730C31direct (N × H)
245. #1441C2×C31semidirect
246. #1441C2×C31semidirect
247. #1441C2×C31semidirect
250. #1436C2×C31semidirect
373. C31#15730direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3992
C2×C31×#3852
C31×#14412
C31×#14392
C31×#14372
C31×#14362
C31×C22⋊#952
#1573031