Finite Group Databasebeta

← C31×#20703 #21477462

Subgroups — C31×#20703 #21477462

Order 7936.

1374 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#520C211yesmaximal, characteristic
2C2×C31×#408C211yesmaximal, characteristic
3C31×#2311C211yesmaximal, characteristic
4C31×#1825C212yesmaximal, complemented
5C31×#1825C212yesmaximal, complemented
6C31×#1824C212yesmaximal, complemented
7C31×#1823C212yesmaximal, complemented
8C31×#1736C211yesmaximal, characteristic, complemented
9C31×#1736C211yesmaximal, characteristic, complemented
10C31×#1735C211yesmaximal, characteristic, complemented
11C2×C31×#340C211yesmaximal, characteristic, complemented
12C22×#41×C31C2211yescharacteristic
13C22×M16×C31C2211yescharacteristic
14C23×C8×C31C2211yescharacteristic
15C31×#520—24no—
16C31×#520C418yescomplemented
17C2×C4×Q8×C31—21no—
18C2×C4×D4×C31—22no—
19C2×C4×D4×C31—21no—
20C2×C31×#117C2211yescharacteristic
21C2×C31×#117C2211yescharacteristic
22C2×C31×#117C2212yes—
23C22×C4⋊C4×C31C2211yescharacteristic
24C22×C4⋊C4×C31C2211yescharacteristic
25C22×C4⋊C4×C31C2212yes—
26C31×#408—24no—
27C2×C31×C22⋊C8C2212yes—
28C2×C31×C22⋊C8C2212yes—
29C31×#391—21no—
30C31×#390—21no—
31C31×#386—22no—
32C31×#380C2212yes—
33C31×#380C2212yes—
34C31×#340C2211yescharacteristic
35C31×#340C2212yes—
36C31×#340C2212yes—
37C31×#340C2212yes—
38C31×#340C2211yescharacteristic
39C31×#340C2212yes—
40C31×#340C2212yes—
41C31×#340C2211yescharacteristic
42C31×#340C2211yescharacteristic
43C31×#340C2212yes—
44C2×#41×C31—22no—
45C2×#41×C31—22no—
46C2×#41×C31—22no—
47C2×#41×C31—22no—
48C2×#41×C31C2×C414yes—
49C2×#41×C31D414yes—
50C22×Q8×C31D411yescharacteristic
51C22×D4×C31D412yes—
52C22×D4×C31D411yescharacteristic
53C23×C4×C31—21no—
54C23×C4×C31—22no—
55C23×C4×C31—21no—
56C23×C4×C31C2312yes—
57C23×C4×C31C2311yescharacteristic
58C2×M16×C31—22no—
59C2×M16×C31—22no—
60C2×M16×C31—22no—
61C22×C8×C31—22no—
62C22×C8×C31—22no—
63C22×C8×C31—22no—
64C22×C8×C31C2311yescharacteristic
65C22×C8×C31C2311yescharacteristic
66C22×C8×C31C2312yes—
67C4×Q8×C31—28no—
68C4×D4×C31—28no—
69C4×D4×C31—216no—
70C31×#117—22no—
71C31×#117—24no—
72C31×#117—22no—
73C31×#117—22no—
74C31×#117C2×C414yes—
75C31×#117C2×C418yescomplemented
76C2×C4⋊C4×C31—21no—
77C2×C4⋊C4×C31—22no—
78C2×C4⋊C4×C31—22no—
79C2×C4⋊C4×C31—24no—
80C2×C4⋊C4×C31—22no—
81C2×C4⋊C4×C31—22no—
82C2×C4⋊C4×C31—21no—
83C2×C4⋊C4×C31C2×C414yes—
84C2×C4⋊C4×C31C2×C418yescomplemented
85C2×C4⋊C4×C31C2312yes—
86C2×C4⋊C4×C31C2311yescharacteristic
87C2×C4⋊C4×C31C2311yescharacteristic
88C2×C4⋊C4×C31C2312yes—
89C2×C4⋊C4×C31C2311yescharacteristic
90C2×C4⋊C4×C31C2311yescharacteristic
91C2×C22⋊C4×C31—21no—
92C2×C22⋊C4×C31—22no—
93C2×C22⋊C4×C31—21no—
94C2×C22⋊C4×C31—22no—
95C2×C22⋊C4×C31—22no—
96C2×C42×C31—21no—
97C2×C42×C31—22no—
98C2×C42×C31—21no—
99C31×C22⋊C8—24no—
100C31×C22⋊C8—24no—
101C31×#95—22no—
102C31×#95—22no—
103C24×C31C2×D412yes—
104C24×C31C2×D411yescharacteristic
105#41×C31—28no—
106#41×C31—28no—
107#41×C31—28no—
108#41×C31—28no—
109C2×Q8×C31—22no—
110C2×Q8×C31—22no—
111C2×Q8×C31C22⋊C414yes—
112C2×D4×C31—24no—
113C2×D4×C31—24no—
114C2×D4×C31—22no—
115C2×D4×C31—22no—
116C2×D4×C31C22⋊C418yes—
117C2×D4×C31C22⋊C414yes—
118C22×C4×C31—22no—
119C22×C4×C31—22no—
120C22×C4×C31—22no—
121C22×C4×C31—22no—
122C22×C4×C31—21no—
123C22×C4×C31—21no—
124C22×C4×C31—22no—
125C22×C4×C31—24no—
126C22×C4×C31—28no—
127C22×C4×C31—21no—
128C22×C4×C31—21no—
129C22×C4×C31—22no—
130C22×C4×C31—42no—
131C22×C4×C31—81no—
132C22×C4×C31—82no—
133C22×C4×C31—81no—
134C22×C4×C31—82no—
135C22×C4×C31—81no—
136C22×C4×C31—81no—
137C22×C4×C31C22×C414yes—
138C22×C4×C31C22×C412yes—
139C22×C4×C31C2×D412yes—
140C22×C4×C31C2×D414yes—
141C22×C4×C31C2×D411yescharacteristic
142C22×C4×C31C2×D412yes—
143C22×C4×C31#4112yes—
144C22×C4×C31#4112yes—
145C22×C4×C31#4112yes—
146C22×C4×C31#4112yes—
147C22×C4×C31C2411yescharacteristic
148M16×C31—44no—
149C2×C8×C31—21no—
150C2×C8×C31—21no—
151C2×C8×C31—22no—
152C2×C8×C31—21no—
153C2×C8×C31—21no—
154C2×C8×C31—21no—
155C2×C8×C31—21no—
156C2×C8×C31—22no—
157C2×C8×C31—22no—
158C2×C8×C31—44no—
159C4⋊C4×C31—24no—
160C4⋊C4×C31—28no—
161C4⋊C4×C31—21no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—21no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—21no—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—21no—
170C4⋊C4×C31—42no—
171C4⋊C4×C31C22×C414yes—
172C4⋊C4×C31C22×C412yes—
173C4⋊C4×C31C22×C412yes—
174C22⋊C4×C31—24no—
175C22⋊C4×C31—28no—
176C22⋊C4×C31—42no—
177C22⋊C4×C31—82no—
178C22⋊C4×C31—82no—
179C42×C31—24no—
180C42×C31—28no—
181C42×C31—42no—
182#20703C3111yesmaximal, characteristic, direct product factor, Sylow
183C23×C31—21no—
184C23×C31—21no—
185C23×C31—21no—
186C23×C31—21no—
187C23×C31—22no—
188C23×C31—22no—
189C23×C31—22no—
190C23×C31—22no—
191C23×C31—22no—
192C23×C31—22no—
193C23×C31C2×C22⋊C414yes—
194C23×C31C2×C22⋊C412yes—
195C23×C31#12312yes—
196C23×C31#12312yes—
197C23×C31C22×D411yescharacteristic
198Q8×C31—28no—
199D4×C31—216no—
200D4×C31—28no—
201C2×C4×C31—22no—
202C2×C4×C31—22no—
203C2×C4×C31—24no—
204C2×C4×C31—24no—
205C2×C4×C31—24no—
206C2×C4×C31—24no—
207C2×C4×C31—22no—
208C2×C4×C31—22no—
209C2×C4×C31—24no—
210C2×C4×C31—42no—
211C2×C4×C31—41no—
212C2×C4×C31—41no—
213C2×C4×C31—42no—
214C2×C4×C31—41no—
215C2×C4×C31—41no—
216C2×C4×C31—41no—
217C2×C4×C31—41no—
218C2×C4×C31—42no—
219C2×C4×C31—41no—
220C2×C4×C31—42no—
221C2×C4×C31—41no—
222C2×C4×C31—42no—
223C2×C4×C31—41no—
224C2×C4×C31—41no—
225C2×C4×C31—84no—
226C2×C4×C31—88no—
227C2×C4×C31—84no—
228C2×C4×C31C2×C22⋊C414yes—
229C2×C4×C31C2×C22⋊C412yes—
230C2×C4×C31#11714yes—
231C2×C4×C31#11714yes—
232C2×C4×C31#12314yes—
233C2×C4×C31#12314yes—
234C2×C4×C31#12312yes—
235C2×C4×C31#12312yes—
236C2×C4×C31C23×C411yescharacteristic
237C2×C4×C31C22×D411yescharacteristic
238C2×C4×C31C2×#4111yescharacteristic
239C2×C4×C31C2×#4111yescharacteristic
240C2×C4×C31C2×#4111yescharacteristic
241C2×C4×C31C2×#4111yescharacteristic
242C8×C31—41no—
243C8×C31—41no—
244C8×C31—42no—
245C2×#520C2×C3111yescharacteristic
246C2×#408C2×C3111yescharacteristic
247#2311C2×C3111yescharacteristic
248#1825C2×C3112yescomplemented
249#1825C2×C3112yescomplemented
250#1824C2×C3112yescomplemented
251#1823C2×C3112yescomplemented
252#1736C2×C3111yescharacteristic, complemented
253#1736C2×C3111yescharacteristic, complemented
254#1735C2×C3111yescharacteristic, complemented
255C2×#340C2×C3111yescharacteristic, complemented
256C22×C31—22no—
257C22×C31—22no—
258C22×C31—22no—
259C22×C31—22no—
260C22×C31—22no—
261C22×C31—22no—
262C22×C31—24no—
263C22×C31—24no—
264C22×C31—24no—
265C22×C31—42no—
266C22×C31—42no—
267C22×C31—42no—
268C22×C31—42no—
269C22×C31#38114yes—
270C22×C31C22×C22⋊C411yescharacteristic
271C22×C31C2×#12311yescharacteristic
272C22×C31C2×#12311yescharacteristic
273C22×C31#57611yescharacteristic
274C22×C31#57711yescharacteristic
275C22×C31#57711yescharacteristic
276C22×C31#57811yescharacteristic
277C4×C31—24no—
278C4×C31—42no—
279C4×C31—44no—
280C4×C31—42no—
281C4×C31—82no—
282C4×C31—81no—
283C4×C31—81no—
284C4×C31#38114yes—
285C4×C31C2×#11711yescharacteristic
286C4×C31C2×#11711yescharacteristic
287C4×C31C2×#12311yescharacteristic
288C4×C31C2×#12311yescharacteristic
289C22×#41C22×C3111yescharacteristic
290C22×M16C22×C3111yescharacteristic
291C23×C8C22×C3111yescharacteristic
292#520—24no—
293#520C4×C3118yescomplemented
294C2×C4×Q8—21no—
295C2×C4×D4—22no—
296C2×C4×D4—21no—
297C2×#117C22×C3111yescharacteristic
298C2×#117C22×C3111yescharacteristic
299C2×#117C22×C3112yes—
300C22×C4⋊C4C22×C3111yescharacteristic
301C22×C4⋊C4C22×C3111yescharacteristic
302C22×C4⋊C4C22×C3112yes—
303#408—24no—
304C2×C22⋊C8C22×C3112yes—
305C2×C22⋊C8C22×C3112yes—
306#391—21no—
307#390—21no—
308#386—22no—
309#380C22×C3112yes—
310#380C22×C3112yes—
311#340C22×C3111yescharacteristic
312#340C22×C3112yes—
313#340C22×C3112yes—
314#340C22×C3112yes—
315#340C22×C3111yescharacteristic
316#340C22×C3112yes—
317#340C22×C3112yes—
318#340C22×C3111yescharacteristic
319#340C22×C3111yescharacteristic
320#340C22×C3112yes—
321C2×C31—24no—
322C2×C31—44no—
323C2×C31C2×#38111yescharacteristic
324C2×C31#289711yescharacteristic
325C2×C31#289811yescharacteristic
326C2×C31#309611yescharacteristic
327C2×C31#309711yescharacteristic
328C2×C31#309811yescharacteristic
329C2×C31#309911yescharacteristic
330C2×#41—22no—
331C2×#41—22no—
332C2×#41—22no—
333C2×#41—22no—
334C2×#41C2×C4×C3114yes—
335C2×#41D4×C3114yes—
336C22×Q8D4×C3111yescharacteristic
337C22×D4D4×C3112yes—
338C22×D4D4×C3111yescharacteristic
339C23×C4—21no—
340C23×C4—22no—
341C23×C4—21no—
342C23×C4C23×C3112yes—
343C23×C4C23×C3111yescharacteristic
344C2×M16—22no—
345C2×M16—22no—
346C2×M16—22no—
347C22×C8—22no—
348C22×C8—22no—
349C22×C8—22no—
350C22×C8C23×C3111yescharacteristic
351C22×C8C23×C3111yescharacteristic
352C22×C8C23×C3112yes—
353C4×Q8—28no—
354C4×D4—28no—
355C4×D4—216no—
356#117—22no—
357#117—24no—
358#117—22no—
359#117—22no—
360#117C2×C4×C3114yes—
361#117C2×C4×C3118yescomplemented
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—24no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4C2×C4×C3114yes—
370C2×C4⋊C4C2×C4×C3118yescomplemented
371C2×C4⋊C4C23×C3112yes—
372C2×C4⋊C4C23×C3111yescharacteristic
373C2×C4⋊C4C23×C3111yescharacteristic
374C2×C4⋊C4C23×C3112yes—
375C2×C4⋊C4C23×C3111yescharacteristic
376C2×C4⋊C4C23×C3111yescharacteristic
377C2×C22⋊C4—21no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C42—21no—
383C2×C42—22no—
384C2×C42—21no—
385C22⋊C8—24no—
386C22⋊C8—24no—
387#95—22no—
388#95—22no—
389C31#2070311yescharacteristic, direct product factor, Sylow
390C24C2×D4×C3112yes—
391C24C2×D4×C3111yescharacteristic
392#41—28no—
393#41—28no—
394#41—28no—
395#41—28no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8C22⋊C4×C3114yes—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4C22⋊C4×C3118yes—
404C2×D4C22⋊C4×C3114yes—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—28no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—42no—
418C22×C4—81no—
419C22×C4—82no—
420C22×C4—81no—
421C22×C4—82no—
422C22×C4—81no—
423C22×C4—81no—
424C22×C4C22×C4×C3114yes—
425C22×C4C22×C4×C3112yes—
426C22×C4C2×D4×C3112yes—
427C22×C4C2×D4×C3114yes—
428C22×C4C2×D4×C3111yescharacteristic
429C22×C4C2×D4×C3112yes—
430C22×C4#41×C3112yes—
431C22×C4#41×C3112yes—
432C22×C4#41×C3112yes—
433C22×C4#41×C3112yes—
434C22×C4C24×C3111yescharacteristic
435M16—44no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—22no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—44no—
446C4⋊C4—24no—
447C4⋊C4—28no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—42no—
458C4⋊C4C22×C4×C3114yes—
459C4⋊C4C22×C4×C3112yes—
460C4⋊C4C22×C4×C3112yes—
461C22⋊C4—24no—
462C22⋊C4—28no—
463C22⋊C4—42no—
464C22⋊C4—82no—
465C22⋊C4—82no—
466C42—24no—
467C42—28no—
468C42—42no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23C2×C22⋊C4×C3114yes—
480C23C2×C22⋊C4×C3112yes—
481C23C31×#12312yes—
482C23C31×#12312yes—
483C23C22×D4×C3111yescharacteristic
484Q8—28no—
485D4—216no—
486D4—28no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—84no—
512C2×C4—88no—
513C2×C4—84no—
514C2×C4C2×C22⋊C4×C3114yes—
515C2×C4C2×C22⋊C4×C3112yes—
516C2×C4C31×#11714yes—
517C2×C4C31×#11714yes—
518C2×C4C31×#12314yes—
519C2×C4C31×#12314yes—
520C2×C4C31×#12312yes—
521C2×C4C31×#12312yes—
522C2×C4C23×C4×C3111yescharacteristic
523C2×C4C22×D4×C3111yescharacteristic
524C2×C4C2×#41×C3111yescharacteristic
525C2×C4C2×#41×C3111yescharacteristic
526C2×C4C2×#41×C3111yescharacteristic
527C2×C4C2×#41×C3111yescharacteristic
528C8—41no—
529C8—41no—
530C8—42no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—24no—
538C22—24no—
539C22—24no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22C31×#38114yes—
545C22C22×C22⋊C4×C3111yescharacteristic
546C22C2×C31×#12311yescharacteristic
547C22C2×C31×#12311yescharacteristic
548C22C31×#57611yescharacteristic
549C22C31×#57711yescharacteristic
550C22C31×#57711yescharacteristic
551C22C31×#57811yescharacteristic
552C4—24no—
553C4—42no—
554C4—44no—
555C4—42no—
556C4—82no—
557C4—81no—
558C4—81no—
559C4C31×#38114yes—
560C4C2×C31×#11711yescharacteristic
561C4C2×C31×#11711yescharacteristic
562C4C2×C31×#12311yescharacteristic
563C4C2×C31×#12311yescharacteristic
564C2—24no—
565C2—44no—
566C2C2×C31×#38111yescharacteristic
567C2C31×#289711yescharacteristic
568C2C31×#289811yescharacteristic
569C2C31×#309611yescharacteristic
570C2C31×#309711yescharacteristic
571C2C31×#309811yescharacteristic
572C2C31×#309911yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutatorC2×C4C23×C4×C31
Frattini, cosocleC22×C4C24×C31
prime radicalC22×#41×C31C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
4. C31×#1825C2semidirect
5. C31×#1825C2semidirect
6. C31×#1824C2semidirect
7. C31×#1823C2semidirect
8. C31×#1736C2semidirect
9. C31×#1736C2semidirect
10. C31×#1735C2semidirect
11. C2×C31×#340C2semidirect
16. C31×#520C4semidirect
75. C31×#117C2×C4semidirect
84. C2×C4⋊C4×C31C2×C4semidirect
182. #20703C31direct (N × H)
248. #1825C2×C31semidirect
249. #1825C2×C31semidirect
250. #1824C2×C31semidirect
251. #1823C2×C31semidirect
252. #1736C2×C31semidirect
253. #1736C2×C31semidirect
254. #1735C2×C31semidirect
255. C2×#340C2×C31semidirect
293. #520C4×C31semidirect
361. #117C2×C4×C31semidirect
370. C2×C4⋊C4C2×C4×C31semidirect
389. C31#20703direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5202
C2×C31×#4082
C31×#23112
C31×#18252
C31×#18242
C31×#18232
C31×#17362
C31×#17352
C2×C31×#3402
#2070331