Finite Group Databasebeta

← C31×#15222 #21471981

Subgroups — C31×#15222 #21471981

Order 7936.

1838 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#390C211yesmaximal, characteristic
2C2×C31×#390C211yesmaximal, characteristic
3C2×C31×#390C211yesmaximal, characteristic
4C2×C31×#389C211yesmaximal, characteristic, complemented
5C2×C31×#382C211yesmaximal, characteristic, complemented
6C2×C4×C4⋊C4×C31C211yesmaximal, characteristic, complemented
7C31×#1439C214yesmaximal, complemented
8C31×#1439C214yesmaximal, complemented
9C31×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×D4×C31C2211yescharacteristic
11C22×C4⋊C4×C31C2211yescharacteristic
12C22×C4⋊C4×C31C2211yescharacteristic
13C22×C4⋊C4×C31C2212yes—
14C22×C4⋊C4×C31C2211yescharacteristic
15C22×C4⋊C4×C31C2212yes—
16C22×C22⋊C4×C31C2212yes—
17C22×C22⋊C4×C31C2214yes—
18C22×C22⋊C4×C31C2214yes—
19C22×C22⋊C4×C31C2212yes—
20C22×C42×C31C2211yescharacteristic
21C22×C42×C31C2211yescharacteristic
22C22×C42×C31C2211yescharacteristic
23C31×#390—24no—
24C31×#390—24no—
25C31×#390—24no—
26C31×#389—24no—
27C31×#382—24no—
28C4×C4⋊C4×C31—24no—
29C2×C31×#95C2212yes—
30C2×C31×#95C2214yes—
31C2×C31×#95C2212yes—
32C2×C31×#95C2214yes—
33C25×C31C2312yes—
34C22×D4×C31—22no—
35C22×D4×C31—22no—
36C22×D4×C31—22no—
37C22×D4×C31—22no—
38C22×D4×C31—22no—
39C22×D4×C31—22no—
40C22×D4×C31Q814yes—
41C23×C4×C31C2312yes—
42C23×C4×C31C2312yes—
43C23×C4×C31C2311yescharacteristic
44C23×C4×C31C2312yes—
45C23×C4×C31C2312yes—
46C23×C4×C31C2312yes—
47C23×C4×C31C2311yescharacteristic
48C23×C4×C31C2311yescharacteristic
49C2×C4⋊C4×C31—24no—
50C2×C4⋊C4×C31—22no—
51C2×C4⋊C4×C31—24no—
52C2×C4⋊C4×C31—24no—
53C2×C4⋊C4×C31—22no—
54C2×C4⋊C4×C31—22no—
55C2×C4⋊C4×C31—24no—
56C2×C4⋊C4×C31—22no—
57C2×C4⋊C4×C31—22no—
58C2×C4⋊C4×C31—22no—
59C2×C4⋊C4×C31—22no—
60C2×C4⋊C4×C31—22no—
61C2×C4⋊C4×C31—44no—
62C2×C4⋊C4×C31D414yes—
63C2×C22⋊C4×C31—28no—
64C2×C22⋊C4×C31—28no—
65C2×C22⋊C4×C31—24no—
66C2×C22⋊C4×C31—24no—
67C2×C22⋊C4×C31—28no—
68C2×C22⋊C4×C31—28no—
69C2×C22⋊C4×C31—44no—
70C2×C22⋊C4×C31—44no—
71C2×C22⋊C4×C31—44no—
72C2×C22⋊C4×C31—44no—
73C2×C42×C31—22no—
74C2×C42×C31—22no—
75C2×C42×C31—22no—
76C2×C42×C31—22no—
77C2×C42×C31—22no—
78C2×C42×C31—22no—
79C2×C42×C31—22no—
80C2×C42×C31D414yes—
81C2×C42×C31D414yes—
82C31×#95—24no—
83C31×#95—28no—
84C31×#95—24no—
85C31×#95—44no—
86C24×C31—22no—
87C24×C31—22no—
88C24×C31—22no—
89C24×C31—24no—
90C24×C31—24no—
91C24×C31—22no—
92C24×C31—22no—
93C24×C31—22no—
94C24×C31—24no—
95C24×C31—24no—
96C24×C31C2×Q814yes—
97C24×C31C2411yescharacteristic
98C2×D4×C31—28no—
99C2×D4×C31—28no—
100C2×D4×C31—28no—
101C2×D4×C31—44no—
102C2×D4×C31—44no—
103C2×D4×C31—44no—
104C2×D4×C31—44no—
105C22×C4×C31—21no—
106C22×C4×C31—21no—
107C22×C4×C31—22no—
108C22×C4×C31—21no—
109C22×C4×C31—22no—
110C22×C4×C31—22no—
111C22×C4×C31—22no—
112C22×C4×C31—22no—
113C22×C4×C31—21no—
114C22×C4×C31—22no—
115C22×C4×C31—22no—
116C22×C4×C31—24no—
117C22×C4×C31—22no—
118C22×C4×C31—22no—
119C22×C4×C31—22no—
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—24no—
127C22×C4×C31—22no—
128C22×C4×C31—22no—
129C22×C4×C31—22no—
130C22×C4×C31—22no—
131C22×C4×C31—22no—
132C22×C4×C31—24no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—24no—
136C22×C4×C31—24no—
137C22×C4×C31—24no—
138C22×C4×C31C2×D414yes—
139C22×C4×C31C2×D414yes—
140C22×C4×C31C2×D412yes—
141C22×C4×C31C2×D412yes—
142C22×C4×C31C2×D412yes—
143C22×C4×C31C2×D412yes—
144C22×C4×C31C2×D412yes—
145C22×C4×C31C2×Q812yes—
146C22×C4×C31#4114yes—
147C22×C4×C31#4114yes—
148C22×C4×C31#4112yes—
149C22×C4×C31#4112yes—
150C22×C4×C31#4112yes—
151C4⋊C4×C31—216no—
152C4⋊C4×C31—28no—
153C4⋊C4×C31—48no—
154C4⋊C4×C31—44no—
155C4⋊C4×C31—44no—
156C22⋊C4×C31—416no—
157C22⋊C4×C31—48no—
158C22⋊C4×C31—416no—
159C22⋊C4×C31—48no—
160C42×C31—28no—
161C42×C31—28no—
162C42×C31—44no—
163#15222C3111yesmaximal, characteristic, direct product factor, Sylow
164C23×C31—24no—
165C23×C31—24no—
166C23×C31—24no—
167C23×C31—28no—
168C23×C31—28no—
169C23×C31—42no—
170C23×C31—42no—
171C23×C31—42no—
172C23×C31—44no—
173C23×C31—42no—
174C23×C31—42no—
175C23×C31—44no—
176C23×C31—42no—
177C23×C31—42no—
178C23×C31—44no—
179C23×C31—42no—
180C23×C31—44no—
181C23×C31—44no—
182C23×C31—44no—
183C23×C31—44no—
184C23×C31—44no—
185C23×C31—44no—
186C23×C31—44no—
187C23×C31C22×D411yescharacteristic
188C23×C31C22×D411yescharacteristic
189C23×C31C22×D411yescharacteristic
190C23×C31C22×Q811yescharacteristic
191C23×C31C2×#4111yescharacteristic
192C23×C31C2×#4111yescharacteristic
193C23×C31C2×#4111yescharacteristic
194C23×C31C2×#4112yes—
195C23×C31C2×#4112yes—
196C23×C31H(5,𝔽2)12yes—
197C23×C31H(5,𝔽2)12yes—
198D4×C31—416no—
199C2×C4×C31—22no—
200C2×C4×C31—24no—
201C2×C4×C31—24no—
202C2×C4×C31—22no—
203C2×C4×C31—22no—
204C2×C4×C31—24no—
205C2×C4×C31—24no—
206C2×C4×C31—24no—
207C2×C4×C31—24no—
208C2×C4×C31—22no—
209C2×C4×C31—24no—
210C2×C4×C31—22no—
211C2×C4×C31—24no—
212C2×C4×C31—28no—
213C2×C4×C31—24no—
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—44no—
227C2×C4×C31—42no—
228C2×C4×C31—42no—
229C2×C4×C31—42no—
230C2×C4×C31—44no—
231C2×C4×C31—42no—
232C2×C4×C31—42no—
233C2×C4×C31—42no—
234C2×C4×C31—44no—
235C2×C4×C31—44no—
236C2×C4×C31—44no—
237C2×C4×C31#12114yes—
238C2×C4×C31#12114yes—
239C2×C4×C31#12114yes—
240C2×C4×C31C22⋊Q814yes—
241C2×C4×C31C22⋊Q814yes—
242C2×C4×C31C22⋊Q814yes—
243C2×C4×C31#12414yes—
244C2×C4×C31#12614yes—
245C2×#390C2×C3111yescharacteristic
246C2×#390C2×C3111yescharacteristic
247C2×#390C2×C3111yescharacteristic
248C2×#389C2×C3111yescharacteristic, complemented
249C2×#382C2×C3111yescharacteristic, complemented
250C2×C4×C4⋊C4C2×C3111yescharacteristic, complemented
251#1439C2×C3114yescomplemented
252#1439C2×C3114yescomplemented
253C4⋊#95C2×C3111yescharacteristic, complemented
254C22×C31—44no—
255C22×C31—44no—
256C22×C31—48no—
257C22×C31—44no—
258C22×C31—48no—
259C22×C31—82no—
260C22×C31—82no—
261C22×C31—84no—
262C22×C31—84no—
263C22×C31—82no—
264C22×C31—82no—
265C22×C31C2×#12111yescharacteristic
266C22×C31C2×#12111yescharacteristic
267C22×C31C2×#12111yescharacteristic
268C22×C31C2×C22⋊Q811yescharacteristic
269C22×C31C2×C22⋊Q811yescharacteristic
270C22×C31C2×C22⋊Q811yescharacteristic
271C22×C31#52512yes—
272C22×C31C2×#12411yescharacteristic
273C22×C31C2×#12611yescharacteristic
274C22×C31#52912yes—
275C22×C31#53211yescharacteristic
276C22×C31#53511yescharacteristic
277C22×C31#53511yescharacteristic
278C22×C31#53512yes—
279C22×C31#54012yes—
280C22×C31#54612yes—
281C22×C31#55012yes—
282C22×C31#55312yes—
283C22×C31#55412yes—
284C22×C31#55412yes—
285C22×C31#55512yes—
286C22×C31#55912yes—
287C22×C31#55912yes—
288C4×C31—24no—
289C4×C31—42no—
290C4×C31—44no—
291C4×C31—44no—
292C4×C31—44no—
293C4×C31—82no—
294C4×C31—82no—
295C4×C31#39618yes—
296C23×D4C22×C3111yescharacteristic
297C22×C4⋊C4C22×C3111yescharacteristic
298C22×C4⋊C4C22×C3111yescharacteristic
299C22×C4⋊C4C22×C3112yes—
300C22×C4⋊C4C22×C3111yescharacteristic
301C22×C4⋊C4C22×C3112yes—
302C22×C22⋊C4C22×C3112yes—
303C22×C22⋊C4C22×C3114yes—
304C22×C22⋊C4C22×C3114yes—
305C22×C22⋊C4C22×C3112yes—
306C22×C42C22×C3111yescharacteristic
307C22×C42C22×C3111yescharacteristic
308C22×C42C22×C3111yescharacteristic
309#390—24no—
310#390—24no—
311#390—24no—
312#389—24no—
313#382—24no—
314C4×C4⋊C4—24no—
315C2×#95C22×C3112yes—
316C2×#95C22×C3114yes—
317C2×#95C22×C3112yes—
318C2×#95C22×C3114yes—
319C2×C31—84no—
320C2×C31C2×#39611yescharacteristic
321C2×C31#241412yes—
322C2×C31#245412yes—
323C2×C31#251312yes—
324C2×C31#254411yescharacteristic
325C2×C31#254811yescharacteristic
326C2×C31#256311yescharacteristic
327C2×C31#256611yescharacteristic
328C2×C31#258012yes—
329C2×C31#261911yescharacteristic
330C2×C31#285011yescharacteristic
331C25C23×C3112yes—
332C22×D4—22no—
333C22×D4—22no—
334C22×D4—22no—
335C22×D4—22no—
336C22×D4—22no—
337C22×D4—22no—
338C22×D4Q8×C3114yes—
339C23×C4C23×C3112yes—
340C23×C4C23×C3112yes—
341C23×C4C23×C3112yes—
342C23×C4C23×C3111yescharacteristic
343C23×C4C23×C3112yes—
344C23×C4C23×C3112yes—
345C23×C4C23×C3111yescharacteristic
346C23×C4C23×C3111yescharacteristic
347C2×C4⋊C4—24no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—44no—
360C2×C4⋊C4D4×C3114yes—
361C2×C22⋊C4—28no—
362C2×C22⋊C4—28no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—28no—
366C2×C22⋊C4—28no—
367C2×C22⋊C4—44no—
368C2×C22⋊C4—44no—
369C2×C22⋊C4—44no—
370C2×C22⋊C4—44no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42D4×C3114yes—
379C2×C42D4×C3114yes—
380#95—24no—
381#95—28no—
382#95—24no—
383#95—44no—
384C31#1522211yescharacteristic, direct product factor, Sylow
385C24—22no—
386C24—22no—
387C24—22no—
388C24—24no—
389C24—24no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24—24no—
394C24—24no—
395C24C2×Q8×C3114yes—
396C24C24×C3111yescharacteristic
397C2×D4—28no—
398C2×D4—28no—
399C2×D4—28no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—44no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4C2×D4×C3114yes—
438C22×C4C2×D4×C3114yes—
439C22×C4C2×D4×C3112yes—
440C22×C4C2×D4×C3112yes—
441C22×C4C2×D4×C3112yes—
442C22×C4C2×D4×C3112yes—
443C22×C4C2×D4×C3112yes—
444C22×C4C2×Q8×C3112yes—
445C22×C4#41×C3114yes—
446C22×C4#41×C3114yes—
447C22×C4#41×C3112yes—
448C22×C4#41×C3112yes—
449C22×C4#41×C3112yes—
450C4⋊C4—216no—
451C4⋊C4—28no—
452C4⋊C4—48no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C22⋊C4—416no—
456C22⋊C4—48no—
457C22⋊C4—416no—
458C22⋊C4—48no—
459C42—28no—
460C42—28no—
461C42—44no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—28no—
466C23—28no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—44no—
471C23—42no—
472C23—42no—
473C23—44no—
474C23—42no—
475C23—42no—
476C23—44no—
477C23—42no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23C22×D4×C3111yescharacteristic
486C23C22×D4×C3111yescharacteristic
487C23C22×D4×C3111yescharacteristic
488C23C22×Q8×C3111yescharacteristic
489C23C2×#41×C3111yescharacteristic
490C23C2×#41×C3111yescharacteristic
491C23C2×#41×C3111yescharacteristic
492C23C2×#41×C3112yes—
493C23C2×#41×C3112yes—
494C23C31×H(5,𝔽2)12yes—
495C23C31×H(5,𝔽2)12yes—
496D4—416no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4C31×#12114yes—
536C2×C4C31×#12114yes—
537C2×C4C31×#12114yes—
538C2×C4C31×C22⋊Q814yes—
539C2×C4C31×C22⋊Q814yes—
540C2×C4C31×C22⋊Q814yes—
541C2×C4C31×#12414yes—
542C2×C4C31×#12614yes—
543C22—44no—
544C22—44no—
545C22—48no—
546C22—44no—
547C22—48no—
548C22—82no—
549C22—82no—
550C22—84no—
551C22—84no—
552C22—82no—
553C22—82no—
554C22C2×C31×#12111yescharacteristic
555C22C2×C31×#12111yescharacteristic
556C22C2×C31×#12111yescharacteristic
557C22C2×C31×C22⋊Q811yescharacteristic
558C22C2×C31×C22⋊Q811yescharacteristic
559C22C2×C31×C22⋊Q811yescharacteristic
560C22C31×#52512yes—
561C22C2×C31×#12411yescharacteristic
562C22C2×C31×#12611yescharacteristic
563C22C31×#52912yes—
564C22C31×#53211yescharacteristic
565C22C31×#53511yescharacteristic
566C22C31×#53511yescharacteristic
567C22C31×#53512yes—
568C22C31×#54012yes—
569C22C31×#54612yes—
570C22C31×#55012yes—
571C22C31×#55312yes—
572C22C31×#55412yes—
573C22C31×#55412yes—
574C22C31×#55512yes—
575C22C31×#55912yes—
576C22C31×#55912yes—
577C4—24no—
578C4—42no—
579C4—44no—
580C4—44no—
581C4—44no—
582C4—82no—
583C4—82no—
584C4C31×#39618yes—
585C2—84no—
586C2C2×C31×#39611yescharacteristic
587C2C31×#241412yes—
588C2C31×#245412yes—
589C2C31×#251312yes—
590C2C31×#254411yescharacteristic
591C2C31×#254811yescharacteristic
592C2C31×#256311yescharacteristic
593C2C31×#256611yescharacteristic
594C2C31×#258012yes—
595C2C31×#261911yescharacteristic
596C2C31×#285011yescharacteristic
subgroupG/H
center, socleC24×C31C24
commutator, Frattini, cosocleC24C24×C31
prime radicalC23×D4×C31C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
4. C2×C31×#389C2semidirect
5. C2×C31×#382C2semidirect
6. C2×C4×C4⋊C4×C31C2semidirect
7. C31×#1439C2semidirect
8. C31×#1439C2semidirect
9. C31×C4⋊#95C2semidirect
163. #15222C31direct (N × H)
248. C2×#389C2×C31semidirect
249. C2×#382C2×C31semidirect
250. C2×C4×C4⋊C4C2×C31semidirect
251. #1439C2×C31semidirect
252. #1439C2×C31semidirect
253. C4⋊#95C2×C31semidirect
384. C31#15222direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3902
C2×C31×#3892
C2×C31×#3822
C2×C4×C4⋊C4×C312
C31×#14392
C31×C4⋊#952
#1522231