Finite Group Databasebeta

← C31×#15212 #21471971

Subgroups — C31×#15212 #21471971

Order 7936.

1182 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#387C211yesmaximal, characteristic
2C2×C4×C4⋊C4×C31C211yesmaximal, characteristic
3C31×#1445C212yesmaximal
4C31×#1445C211yesmaximal, characteristic
5C31×#1445C211yesmaximal, characteristic
6C31×#1444C211yesmaximal, characteristic
7C31×#1442C212yesmaximal
8C31×#1442C211yesmaximal, characteristic
9C31×#1436C212yesmaximal
10C31×#1436C212yesmaximal
11C31×#1434C211yesmaximal, characteristic
12C22×C4⋊C4×C31C2211yescharacteristic
13C22×C4⋊C4×C31C2211yescharacteristic
14C22×C4⋊C4×C31C2211yescharacteristic
15C22×C4⋊C4×C31C2212yes—
16C22×C4⋊C4×C31C2211yescharacteristic
17C22×C42×C31C2211yescharacteristic
18C22×C42×C31C2212yes—
19C31×#387—24no—
20C4×C4⋊C4×C31—24no—
21C2×C31×#95C2212yes—
22C2×C31×#95C2211yescharacteristic
23C2×C31×#95C2212yes—
24C2×C31×#95C2211yescharacteristic
25C2×C31×#95C2211yescharacteristic
26C2×C31×#95C2212yes—
27C2×C31×#95C2212yes—
28C2×C31×#95C2212yes—
29C2×C31×#95C2212yes—
30C2×C31×#95C2212yes—
31C2×C31×#95C2211yescharacteristic
32C2×C31×#95C2211yescharacteristic
33C2×C31×#95C2211yescharacteristic
34C2×C31×#95C2212yes—
35C2×C31×#95C2212yes—
36C2×C31×#95C2212yes—
37C23×C4×C31C2311yescharacteristic
38C23×C4×C31C2311yescharacteristic
39C23×C4×C31C2312yes—
40C23×C4×C31C2311yescharacteristic
41C23×C4×C31C2312yes—
42C23×C4×C31C2312yes—
43C23×C4×C31C2311yescharacteristic
44C23×C4×C31C2311yescharacteristic
45C23×C4×C31C2312yes—
46C23×C4×C31C2311yescharacteristic
47C23×C4×C31C2311yescharacteristic
48C2×C4⋊C4×C31—22no—
49C2×C4⋊C4×C31—22no—
50C2×C4⋊C4×C31—22no—
51C2×C4⋊C4×C31—24no—
52C2×C4⋊C4×C31—22no—
53C2×C4⋊C4×C31—22no—
54C2×C4⋊C4×C31—22no—
55C2×C4⋊C4×C31—22no—
56C2×C4⋊C4×C31—24no—
57C2×C4⋊C4×C31—24no—
58C2×C4⋊C4×C31—24no—
59C2×C4⋊C4×C31—42no—
60C2×C4⋊C4×C31Q814yes—
61C2×C42×C31—24no—
62C2×C42×C31—24no—
63C2×C42×C31—42no—
64C2×C42×C31—42no—
65C2×C42×C31D414yes—
66C31×#95—24no—
67C31×#95—22no—
68C31×#95—22no—
69C31×#95—22no—
70C31×#95—22no—
71C31×#95—24no—
72C31×#95—24no—
73C31×#95—22no—
74C31×#95—24no—
75C31×#95—24no—
76C31×#95—24no—
77C31×#95—24no—
78C31×#95—22no—
79C31×#95—42no—
80C31×#95—42no—
81C31×#95—42no—
82C24×C31C2411yescharacteristic
83C22×C4×C31—22no—
84C22×C4×C31—22no—
85C22×C4×C31—21no—
86C22×C4×C31—21no—
87C22×C4×C31—22no—
88C22×C4×C31—21no—
89C22×C4×C31—22no—
90C22×C4×C31—22no—
91C22×C4×C31—21no—
92C22×C4×C31—22no—
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—21no—
104C22×C4×C31—21no—
105C22×C4×C31—21no—
106C22×C4×C31—22no—
107C22×C4×C31—21no—
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—22no—
115C22×C4×C31—22no—
116C22×C4×C31—22no—
117C22×C4×C31—22no—
118C22×C4×C31—21no—
119C22×C4×C31—21no—
120C22×C4×C31—22no—
121C22×C4×C31—22no—
122C22×C4×C31—22no—
123C22×C4×C31—22no—
124C22×C4×C31—22no—
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—22no—
132C22×C4×C31—21no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31C2×D412yes—
136C22×C4×C31C2×D412yes—
137C22×C4×C31C2×D412yes—
138C22×C4×C31C2×Q814yes—
139C22×C4×C31C2×Q812yes—
140C22×C4×C31#4112yes—
141C22×C4×C31#4112yes—
142C22×C4×C31#4114yes—
143C22×C4×C31#4114yes—
144C22×C4×C31#4112yes—
145C22×C4×C31#4114yes—
146C4⋊C4×C31—28no—
147C4⋊C4×C31—28no—
148C4⋊C4×C31—28no—
149C4⋊C4×C31—48no—
150C4⋊C4×C31—44no—
151C42×C31—28no—
152C42×C31—48no—
153#15212C3111yesmaximal, characteristic, direct product factor, Sylow
154C23×C31C22×D411yescharacteristic
155C23×C31C22×Q811yescharacteristic
156C23×C31C2×#4111yescharacteristic
157C23×C31C2×#4112yes—
158C23×C31C2×#4111yescharacteristic
159C23×C31C2×#4111yescharacteristic
160C23×C31C2×#4112yes—
161C23×C31C2×#4112yes—
162C23×C31H(5,𝔽2)11yescharacteristic
163C23×C31H(5,𝔽2)11yescharacteristic
164C23×C312−1+412yes—
165C2×C4×C31—22no—
166C2×C4×C31—22no—
167C2×C4×C31—24no—
168C2×C4×C31—24no—
169C2×C4×C31—22no—
170C2×C4×C31—22no—
171C2×C4×C31—24no—
172C2×C4×C31—24no—
173C2×C4×C31—24no—
174C2×C4×C31—22no—
175C2×C4×C31—22no—
176C2×C4×C31—22no—
177C2×C4×C31—24no—
178C2×C4×C31—22no—
179C2×C4×C31—24no—
180C2×C4×C31—24no—
181C2×C4×C31—22no—
182C2×C4×C31—24no—
183C2×C4×C31—24no—
184C2×C4×C31—24no—
185C2×C4×C31—24no—
186C2×C4×C31—24no—
187C2×C4×C31—24no—
188C2×C4×C31—24no—
189C2×C4×C31—41no—
190C2×C4×C31—42no—
191C2×C4×C31—41no—
192C2×C4×C31—42no—
193C2×C4×C31—42no—
194C2×C4×C31—42no—
195C2×C4×C31—42no—
196C2×C4×C31—42no—
197C2×C4×C31—41no—
198C2×C4×C31—42no—
199C2×C4×C31—42no—
200C2×C4×C31—41no—
201C2×C4×C31—42no—
202C2×C4×C31—42no—
203C2×C4×C31—42no—
204C2×C4×C31—42no—
205C2×C4×C31—42no—
206C2×C4×C31—42no—
207C2×C4×C31—42no—
208C2×C4×C31—42no—
209C2×C4×C31—41no—
210C2×C4×C31—41no—
211C2×C4×C31—42no—
212C2×C4×C31—42no—
213C2×C4×C31—41no—
214C2×C4×C31—41no—
215C2×C4×C31—41no—
216C2×C4×C31—41no—
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×C31C22⋊Q814yes—
227C2×C4×C31#12314yes—
228C2×#387C2×C3111yescharacteristic
229C2×C4×C4⋊C4C2×C3111yescharacteristic
230#1445C2×C3112yes—
231#1445C2×C3111yescharacteristic
232#1445C2×C3111yescharacteristic
233#1444C2×C3111yescharacteristic
234#1442C2×C3112yes—
235#1442C2×C3111yescharacteristic
236#1436C2×C3112yes—
237#1436C2×C3112yes—
238#1434C2×C3111yescharacteristic
239C22×C31C2×C22⋊Q811yescharacteristic
240C22×C31C2×#12311yescharacteristic
241C22×C31#52511yescharacteristic
242C22×C31#52511yescharacteristic
243C22×C31#52912yes—
244C22×C31#52912yes—
245C22×C31#52912yes—
246C22×C31#53211yescharacteristic
247C22×C31#53312yes—
248C22×C31#53511yescharacteristic
249C22×C31#53611yescharacteristic
250C22×C31#53912yes—
251C22×C31#53912yes—
252C22×C31#55112yes—
253C22×C31#55212yes—
254C22×C31#55212yes—
255C22×C31#55312yes—
256C22×C31#55411yescharacteristic
257C22×C31#55411yescharacteristic
258C22×C31#55612yes—
259C22×C31#55712yes—
260C22×C31#55911yescharacteristic
261C22×C31#55911yescharacteristic
262C4×C31—24no—
263C4×C31—24no—
264C4×C31—44no—
265C4×C31—44no—
266C4×C31—42no—
267C4×C31—44no—
268C4×C31—42no—
269C4×C31—42no—
270C4×C31—82no—
271C4×C31—81no—
272C4×C31—81no—
273C22×C4⋊C4C22×C3111yescharacteristic
274C22×C4⋊C4C22×C3111yescharacteristic
275C22×C4⋊C4C22×C3111yescharacteristic
276C22×C4⋊C4C22×C3112yes—
277C22×C4⋊C4C22×C3111yescharacteristic
278C22×C42C22×C3111yescharacteristic
279C22×C42C22×C3112yes—
280#387—24no—
281C4×C4⋊C4—24no—
282C2×#95C22×C3112yes—
283C2×#95C22×C3111yescharacteristic
284C2×#95C22×C3112yes—
285C2×#95C22×C3111yescharacteristic
286C2×#95C22×C3111yescharacteristic
287C2×#95C22×C3112yes—
288C2×#95C22×C3112yes—
289C2×#95C22×C3112yes—
290C2×#95C22×C3112yes—
291C2×#95C22×C3112yes—
292C2×#95C22×C3111yescharacteristic
293C2×#95C22×C3111yescharacteristic
294C2×#95C22×C3111yescharacteristic
295C2×#95C22×C3112yes—
296C2×#95C22×C3112yes—
297C2×#95C22×C3112yes—
298C2×C31#239511yescharacteristic
299C2×C31#256511yescharacteristic
300C2×C31#257512yes—
301C2×C31#258011yescharacteristic
302C2×C31#258812yes—
303C2×C31#259012yes—
304C2×C31#259612yes—
305C2×C31#259911yescharacteristic
306C2×C31#260311yescharacteristic
307C2×C31#262311yescharacteristic
308C2×C31#285511yescharacteristic
309C23×C4C23×C3111yescharacteristic
310C23×C4C23×C3111yescharacteristic
311C23×C4C23×C3112yes—
312C23×C4C23×C3111yescharacteristic
313C23×C4C23×C3112yes—
314C23×C4C23×C3111yescharacteristic
315C23×C4C23×C3112yes—
316C23×C4C23×C3111yescharacteristic
317C23×C4C23×C3112yes—
318C23×C4C23×C3111yescharacteristic
319C23×C4C23×C3111yescharacteristic
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—42no—
332C2×C4⋊C4Q8×C3114yes—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—42no—
336C2×C42—42no—
337C2×C42D4×C3114yes—
338#95—24no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—24no—
344#95—24no—
345#95—22no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—22no—
351#95—42no—
352#95—42no—
353#95—42no—
354C31#1521211yescharacteristic, direct product factor, Sylow
355C24C24×C3111yescharacteristic
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
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—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4C2×D4×C3112yes—
409C22×C4C2×D4×C3112yes—
410C22×C4C2×D4×C3112yes—
411C22×C4C2×Q8×C3114yes—
412C22×C4C2×Q8×C3112yes—
413C22×C4#41×C3112yes—
414C22×C4#41×C3112yes—
415C22×C4#41×C3114yes—
416C22×C4#41×C3114yes—
417C22×C4#41×C3112yes—
418C22×C4#41×C3114yes—
419C4⋊C4—28no—
420C4⋊C4—28no—
421C4⋊C4—28no—
422C4⋊C4—48no—
423C4⋊C4—44no—
424C42—28no—
425C42—48no—
426C23C22×D4×C3111yescharacteristic
427C23C22×Q8×C3111yescharacteristic
428C23C2×#41×C3111yescharacteristic
429C23C2×#41×C3112yes—
430C23C2×#41×C3111yescharacteristic
431C23C2×#41×C3111yescharacteristic
432C23C2×#41×C3112yes—
433C23C2×#41×C3112yes—
434C23C31×H(5,𝔽2)11yescharacteristic
435C23C31×H(5,𝔽2)11yescharacteristic
436C23C31×2−1+412yes—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4C31×C22⋊Q814yes—
499C2×C4C31×#12314yes—
500C22C2×C31×C22⋊Q811yescharacteristic
501C22C2×C31×#12311yescharacteristic
502C22C31×#52511yescharacteristic
503C22C31×#52511yescharacteristic
504C22C31×#52912yes—
505C22C31×#52912yes—
506C22C31×#52912yes—
507C22C31×#53211yescharacteristic
508C22C31×#53312yes—
509C22C31×#53511yescharacteristic
510C22C31×#53611yescharacteristic
511C22C31×#53912yes—
512C22C31×#53912yes—
513C22C31×#55112yes—
514C22C31×#55212yes—
515C22C31×#55212yes—
516C22C31×#55312yes—
517C22C31×#55411yescharacteristic
518C22C31×#55411yescharacteristic
519C22C31×#55612yes—
520C22C31×#55712yes—
521C22C31×#55911yescharacteristic
522C22C31×#55911yescharacteristic
523C4—24no—
524C4—24no—
525C4—44no—
526C4—44no—
527C4—42no—
528C4—44no—
529C4—42no—
530C4—42no—
531C4—82no—
532C4—81no—
533C4—81no—
534C2C31×#239511yescharacteristic
535C2C31×#256511yescharacteristic
536C2C31×#257512yes—
537C2C31×#258011yescharacteristic
538C2C31×#258812yes—
539C2C31×#259012yes—
540C2C31×#259612yes—
541C2C31×#259911yescharacteristic
542C2C31×#260311yescharacteristic
543C2C31×#262311yescharacteristic
544C2C31×#285511yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C31C24
commutator, Frattini, cosocleC24C24×C31
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
153. #15212C31direct (N × H)
354. C31#15212direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3872
C2×C4×C4⋊C4×C312
C31×#14452
C31×#14442
C31×#14422
C31×#14362
C31×#14342
#1521231