Finite Group Databasebeta

← C2×C31×#1513 #21474314

Subgroups — C2×C31×#1513 #21474314

Order 7936.

1510 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#525C211yesmaximal, characteristic
2C2×C31×#407C211yesmaximal, characteristic, complemented
3C2×C31×#380C211yesmaximal, characteristic, complemented
4C31×#1513C218yesmaximal, direct product factor
5C2×C31×#328C212yesmaximal, complemented
6C2×C31×#327C212yesmaximal, complemented
7C22×#41×C31C411yescharacteristic, complemented
8C24×C4×C31C2211yescharacteristic
9C22×M16×C31—21no—
10C31×#525—22no—
11C31×#525—22no—
12C31×#525C414yescomplemented
13C31×#525C2214yes—
14C2×C31×#123—21no—
15C2×C31×C22⋊Q8C411yescharacteristic, complemented
16C2×C31×C22⋊Q8C2211yescharacteristic
17C2×C31×#121C411yescharacteristic, complemented
18C2×C31×#121C2211yescharacteristic
19C2×C31×#120—21no—
20C2×C4×D4×C31—21no—
21C2×C4×D4×C31—21no—
22C2×C31×#117C411yescharacteristic, complemented
23C22×C4⋊C4×C31—21no—
24C22×C22⋊C4×C31—21no—
25C31×#407—22no—
26C31×#407C2214yescomplemented
27C2×C31×C22⋊C8—21no—
28C2×C31×C22⋊C8C2212yes—
29C31×#380—22no—
30C31×#380C2214yescomplemented
31C2×C31×#95C2212yes—
32C31×#328C2218yescomplemented
33C31×#327C2218yescomplemented
34C25×C31D411yescharacteristic
35C2×#41×C31—22no—
36C2×#41×C31—44no—
37C2×#41×C31C2×C414yescomplemented
38C22×Q8×C31C2×C411yescharacteristic
39C22×D4×C31—21no—
40C22×D4×C31—21no—
41C22×D4×C31—21no—
42C22×D4×C31C2×C411yescharacteristic
43C23×C4×C31—21no—
44C23×C4×C31—22no—
45C23×C4×C31—44no—
46C23×C4×C31C2×C411yescharacteristic
47C23×C4×C31C2×C411yescharacteristic
48C23×C4×C31C2×C412yes—
49C23×C4×C31D411yescharacteristic
50C23×C4×C31D411yescharacteristic
51C23×C4×C31D412yes—
52C23×C4×C31D411yescharacteristic
53C23×C4×C31D412yes—
54C23×C4×C31C2311yescharacteristic
55C23×C4×C31C2312yes—
56C2×M16×C31—24no—
57C2×M16×C31—44no—
58C22×C8×C31—22no—
59C31×#123—24no—
60C31×#123—24no—
61C31×C22⋊Q8—22no—
62C31×C22⋊Q8—21no—
63C31×C22⋊Q8—21no—
64C31×C22⋊Q8C2×C412yes—
65C31×C22⋊Q8C2×C414yescomplemented
66C31×C22⋊Q8C2312yes—
67C31×#121—22no—
68C31×#121—21no—
69C31×#121—21no—
70C31×#121C2×C412yes—
71C31×#121C2×C414yescomplemented
72C31×#121C2312yes—
73C31×#120—24no—
74C31×#120—24no—
75C4×D4×C31—24no—
76C4×D4×C31—24no—
77C4×D4×C31—24no—
78C4×D4×C31—24no—
79C31×#117—22no—
80C31×#117C2×C414yescomplemented
81C2×C4⋊C4×C31—21no—
82C2×C4⋊C4×C31—21no—
83C2×C4⋊C4×C31—24no—
84C2×C4⋊C4×C31—44no—
85C2×C4⋊C4×C31C2×C411yescharacteristic
86C2×C4⋊C4×C31C2×C411yescharacteristic
87C2×C22⋊C4×C31—21no—
88C2×C22⋊C4×C31—21no—
89C2×C22⋊C4×C31—21no—
90C2×C22⋊C4×C31—21no—
91C2×C22⋊C4×C31—21no—
92C2×C22⋊C4×C31—24no—
93C2×C22⋊C4×C31—44no—
94C2×C42×C31—21no—
95C31×C22⋊C8—24no—
96C31×C22⋊C8—22no—
97C31×C22⋊C8C2314yes—
98C31×#95—22no—
99C31×#95C2314yes—
100C24×C31—22no—
101C24×C31—22no—
102C24×C31—21no—
103C24×C31—42no—
104C24×C31—42no—
105C24×C31C22⋊C411yescharacteristic
106C24×C31C22⋊C411yescharacteristic
107C24×C31C22⋊C412yes—
108C24×C31C2×D411yescharacteristic
109C24×C31C2×D412yes—
110#41×C31—48no—
111#41×C31—48no—
112C2×Q8×C31—21no—
113C2×Q8×C31—42no—
114C2×Q8×C31C22×C412yes—
115C2×D4×C31—22no—
116C2×D4×C31—24no—
117C2×D4×C31—24no—
118C2×D4×C31—22no—
119C2×D4×C31—21no—
120C2×D4×C31—42no—
121C2×D4×C31—44no—
122C2×D4×C31—44no—
123C2×D4×C31—42no—
124C2×D4×C31—42no—
125C2×D4×C31C22×C412yes—
126C22×C4×C31—24no—
127C22×C4×C31—24no—
128C22×C4×C31—24no—
129C22×C4×C31—21no—
130C22×C4×C31—21no—
131C22×C4×C31—21no—
132C22×C4×C31—24no—
133C22×C4×C31—24no—
134C22×C4×C31—44no—
135C22×C4×C31—44no—
136C22×C4×C31—44no—
137C22×C4×C31—44no—
138C22×C4×C31—42no—
139C22×C4×C31—42no—
140C22×C4×C31—42no—
141C22×C4×C31—44no—
142C22×C4×C31—44no—
143C22×C4×C31—44no—
144C22×C4×C31—44no—
145C22×C4×C31C22⋊C412yes—
146C22×C4×C31C22⋊C412yes—
147C22×C4×C31C22⋊C412yes—
148C22×C4×C31C22⋊C411yescharacteristic
149C22×C4×C31C22⋊C412yes—
150C22×C4×C31C22⋊C412yes—
151C22×C4×C31C22⋊C411yescharacteristic
152C22×C4×C31C22×C412yes—
153C22×C4×C31C22×C411yescharacteristic
154C22×C4×C31C22×C411yescharacteristic
155C22×C4×C31C22×C412yes—
156C22×C4×C31C2×D412yes—
157C22×C4×C31C2×D412yes—
158C22×C4×C31C2×D411yescharacteristic
159C22×C4×C31C2×D411yescharacteristic
160C22×C4×C31C2×D411yescharacteristic
161C22×C4×C31C2×D412yes—
162C22×C4×C31C2411yescharacteristic
163M16×C31—48no—
164C2×C8×C31—24no—
165C2×C8×C31—44no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—24no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—21no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—48no—
172C4⋊C4×C31C22×C412yes—
173C4⋊C4×C31C22×C412yes—
174C22⋊C4×C31—24no—
175C22⋊C4×C31—22no—
176C22⋊C4×C31—24no—
177C22⋊C4×C31—22no—
178C22⋊C4×C31—22no—
179C22⋊C4×C31—24no—
180C22⋊C4×C31—22no—
181C22⋊C4×C31—48no—
182C42×C31—24no—
183C2×#1513C3111yesmaximal, characteristic, direct product factor, Sylow
184C23×C31—22no—
185C23×C31—22no—
186C23×C31—22no—
187C23×C31—22no—
188C23×C31—22no—
189C23×C31—22no—
190C23×C31—24no—
191C23×C31—22no—
192C23×C31—22no—
193C23×C31—22no—
194C23×C31—22no—
195C23×C31—22no—
196C23×C31—42no—
197C23×C31—42no—
198C23×C31—42no—
199C23×C31—42no—
200C23×C31—48no—
201C23×C31—48no—
202C23×C31—42no—
203C23×C31—42no—
204C23×C31—42no—
205C23×C31C2×C22⋊C412yes—
206C23×C31C2×C22⋊C412yes—
207C23×C31C2×C22⋊C411yescharacteristic
208C23×C31C2×C22⋊C411yescharacteristic
209C23×C31AGL(1,ℤ/8)12yes—
210C23×C31#13712yes—
211C23×C31C22×D411yescharacteristic
212Q8×C31—42no—
213Q8×C31—42no—
214D4×C31—42no—
215D4×C31—42no—
216D4×C31—48no—
217D4×C31—48no—
218D4×C31—42no—
219D4×C31—42no—
220D4×C31—44no—
221C2×C4×C31—22no—
222C2×C4×C31—22no—
223C2×C4×C31—22no—
224C2×C4×C31—24no—
225C2×C4×C31—24no—
226C2×C4×C31—24no—
227C2×C4×C31—24no—
228C2×C4×C31—24no—
229C2×C4×C31—24no—
230C2×C4×C31—22no—
231C2×C4×C31—22no—
232C2×C4×C31—22no—
233C2×C4×C31—42no—
234C2×C4×C31—42no—
235C2×C4×C31—41no—
236C2×C4×C31—42no—
237C2×C4×C31—42no—
238C2×C4×C31—41no—
239C2×C4×C31—41no—
240C2×C4×C31—41no—
241C2×C4×C31—42no—
242C2×C4×C31—42no—
243C2×C4×C31—42no—
244C2×C4×C31—48no—
245C2×C4×C31—44no—
246C2×C4×C31—44no—
247C2×C4×C31—44no—
248C2×C4×C31—88no—
249C2×C4×C31#9914yes—
250C2×C4×C31C2×C22⋊C411yescharacteristic
251C2×C4×C31C2×C22⋊C412yes—
252C2×C4×C31C2×C22⋊C411yescharacteristic
253C2×C4×C31C2×C22⋊C412yes—
254C2×C4×C31C23×C411yescharacteristic
255C2×C4×C31C22×D411yescharacteristic
256C8×C31—44no—
257C2×#525C2×C3111yescharacteristic
258C2×#407C2×C3111yescharacteristic, complemented
259C2×#380C2×C3111yescharacteristic, complemented
260#1513C2×C3118yesdirect product factor
261C2×#328C2×C3112yescomplemented
262C2×#327C2×C3112yescomplemented
263C22×C31—21no—
264C22×C31—21no—
265C22×C31—21no—
266C22×C31—21no—
267C22×C31—24no—
268C22×C31—24no—
269C22×C31—22no—
270C22×C31—22no—
271C22×C31—22no—
272C22×C31—22no—
273C22×C31—22no—
274C22×C31—22no—
275C22×C31—44no—
276C22×C31—44no—
277C22×C31—48no—
278C22×C31—44no—
279C22×C31—44no—
280C22×C31—42no—
281C22×C31—42no—
282C22×C31—42no—
283C22×C31—84no—
284C22×C31C2×#9912yes—
285C22×C31#41712yes—
286C22×C31#41712yes—
287C22×C31#42112yes—
288C22×C31C22×C22⋊C411yescharacteristic
289C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
290C22×C31C2×#13711yescharacteristic
291C4×C31—21no—
292C4×C31—21no—
293C4×C31—22no—
294C4×C31—21no—
295C4×C31—21no—
296C4×C31—42no—
297C4×C31—42no—
298C4×C31—42no—
299C4×C31—84no—
300C4×C31C2×#9914yes—
301C22×#41C4×C3111yescharacteristic, complemented
302C24×C4C22×C3111yescharacteristic
303C22×M16—21no—
304#525—22no—
305#525—22no—
306#525C4×C3114yescomplemented
307#525C22×C3114yes—
308C2×#123—21no—
309C2×C22⋊Q8C4×C3111yescharacteristic, complemented
310C2×C22⋊Q8C22×C3111yescharacteristic
311C2×#121C4×C3111yescharacteristic, complemented
312C2×#121C22×C3111yescharacteristic
313C2×#120—21no—
314C2×C4×D4—21no—
315C2×C4×D4—21no—
316C2×#117C4×C3111yescharacteristic, complemented
317C22×C4⋊C4—21no—
318C22×C22⋊C4—21no—
319#407—22no—
320#407C22×C3114yescomplemented
321C2×C22⋊C8—21no—
322C2×C22⋊C8C22×C3112yes—
323#380—22no—
324#380C22×C3114yescomplemented
325C2×#95C22×C3112yes—
326#328C22×C3118yescomplemented
327#327C22×C3118yescomplemented
328C2×C31—21no—
329C2×C31—21no—
330C2×C31—22no—
331C2×C31—42no—
332C2×C31—42no—
333C2×C31—82no—
334C2×C31#151314yesdirect product factor
335C2×C31C22×#9911yescharacteristic
336C2×C31C2×#41711yescharacteristic
337C2×C31C2×#42111yescharacteristic
338C25D4×C3111yescharacteristic
339C2×#41—22no—
340C2×#41—44no—
341C2×#41C2×C4×C3114yescomplemented
342C22×Q8C2×C4×C3111yescharacteristic
343C22×D4—21no—
344C22×D4—21no—
345C22×D4—21no—
346C22×D4C2×C4×C3111yescharacteristic
347C23×C4—21no—
348C23×C4—22no—
349C23×C4—44no—
350C23×C4C2×C4×C3111yescharacteristic
351C23×C4C2×C4×C3111yescharacteristic
352C23×C4C2×C4×C3112yes—
353C23×C4D4×C3111yescharacteristic
354C23×C4D4×C3111yescharacteristic
355C23×C4D4×C3112yes—
356C23×C4D4×C3111yescharacteristic
357C23×C4D4×C3112yes—
358C23×C4C23×C3111yescharacteristic
359C23×C4C23×C3112yes—
360C2×M16—24no—
361C2×M16—44no—
362C22×C8—22no—
363#123—24no—
364#123—24no—
365C22⋊Q8—22no—
366C22⋊Q8—21no—
367C22⋊Q8—21no—
368C22⋊Q8C2×C4×C3112yes—
369C22⋊Q8C2×C4×C3114yescomplemented
370C22⋊Q8C23×C3112yes—
371#121—22no—
372#121—21no—
373#121—21no—
374#121C2×C4×C3112yes—
375#121C2×C4×C3114yescomplemented
376#121C23×C3112yes—
377#120—24no—
378#120—24no—
379C4×D4—24no—
380C4×D4—24no—
381C4×D4—24no—
382C4×D4—24no—
383#117—22no—
384#117C2×C4×C3114yescomplemented
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—24no—
388C2×C4⋊C4—44no—
389C2×C4⋊C4C2×C4×C3111yescharacteristic
390C2×C4⋊C4C2×C4×C3111yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—44no—
398C2×C42—21no—
399C22⋊C8—24no—
400C22⋊C8—22no—
401C22⋊C8C23×C3114yes—
402#95—22no—
403#95C23×C3114yes—
404C31C2×#151311yescharacteristic, direct product factor, Sylow
405C24—22no—
406C24—22no—
407C24—21no—
408C24—42no—
409C24—42no—
410C24C22⋊C4×C3111yescharacteristic
411C24C22⋊C4×C3111yescharacteristic
412C24C22⋊C4×C3112yes—
413C24C2×D4×C3111yescharacteristic
414C24C2×D4×C3112yes—
415#41—48no—
416#41—48no—
417C2×Q8—21no—
418C2×Q8—42no—
419C2×Q8C22×C4×C3112yes—
420C2×D4—22no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—42no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4C22×C4×C3112yes—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4C22⋊C4×C3111yescharacteristic
451C22×C4C22⋊C4×C3112yes—
452C22×C4C22⋊C4×C3112yes—
453C22×C4C22⋊C4×C3112yes—
454C22×C4C22⋊C4×C3112yes—
455C22×C4C22⋊C4×C3112yes—
456C22×C4C22⋊C4×C3111yescharacteristic
457C22×C4C22×C4×C3111yescharacteristic
458C22×C4C22×C4×C3112yes—
459C22×C4C22×C4×C3111yescharacteristic
460C22×C4C22×C4×C3112yes—
461C22×C4C2×D4×C3112yes—
462C22×C4C2×D4×C3111yescharacteristic
463C22×C4C2×D4×C3112yes—
464C22×C4C2×D4×C3111yescharacteristic
465C22×C4C2×D4×C3111yescharacteristic
466C22×C4C2×D4×C3112yes—
467C22×C4C24×C3111yescharacteristic
468M16—48no—
469C2×C8—24no—
470C2×C8—44no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—48no—
477C4⋊C4C22×C4×C3112yes—
478C4⋊C4C22×C4×C3112yes—
479C22⋊C4—24no—
480C22⋊C4—22no—
481C22⋊C4—24no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—24no—
485C22⋊C4—22no—
486C22⋊C4—48no—
487C42—24no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—48no—
505C23—48no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23C2×C22⋊C4×C3112yes—
510C23C2×C22⋊C4×C3112yes—
511C23C2×C22⋊C4×C3111yescharacteristic
512C23C2×C22⋊C4×C3111yescharacteristic
513C23C31×AGL(1,ℤ/8)12yes—
514C23C31×#13712yes—
515C23C22×D4×C3111yescharacteristic
516Q8—42no—
517Q8—42no—
518D4—42no—
519D4—42no—
520D4—48no—
521D4—48no—
522D4—44no—
523D4—42no—
524D4—42no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—48no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—88no—
553C2×C4C31×#9914yes—
554C2×C4C2×C22⋊C4×C3111yescharacteristic
555C2×C4C2×C22⋊C4×C3112yes—
556C2×C4C2×C22⋊C4×C3111yescharacteristic
557C2×C4C2×C22⋊C4×C3112yes—
558C2×C4C23×C4×C3111yescharacteristic
559C2×C4C22×D4×C3111yescharacteristic
560C8—44no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—24no—
566C22—24no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—44no—
574C22—48no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—84no—
582C22C2×C31×#9912yes—
583C22C31×#41712yes—
584C22C31×#41712yes—
585C22C31×#42112yes—
586C22C22×C22⋊C4×C3111yescharacteristic
587C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
588C22C2×C31×#13711yescharacteristic
589C4—21no—
590C4—21no—
591C4—21no—
592C4—22no—
593C4—21no—
594C4—42no—
595C4—42no—
596C4—42no—
597C4—84no—
598C4C2×C31×#9914yes—
599C2—21no—
600C2—22no—
601C2—21no—
602C2—42no—
603C2—42no—
604C2—82no—
605C2C31×#151314yesdirect product factor
606C2C22×C31×#9911yescharacteristic
607C2C2×C31×#41711yescharacteristic
608C2C2×C31×#42111yescharacteristic
subgroupG/H
center, socleC23×C31C2×C22⋊C4
commutatorC2×C4C23×C4×C31
Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#525C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C24×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#407C2semidirect
3. C2×C31×#380C2semidirect
4. C31×#1513C2direct (N × H)
5. C2×C31×#328C2semidirect
6. C2×C31×#327C2semidirect
7. C22×#41×C31C4semidirect
12. C31×#525C4semidirect
15. C2×C31×C22⋊Q8C4semidirect
17. C2×C31×#121C4semidirect
22. C2×C31×#117C4semidirect
26. C31×#407C22semidirect
30. C31×#380C22semidirect
32. C31×#328C22semidirect
33. C31×#327C22semidirect
37. C2×#41×C31C2×C4semidirect
65. C31×C22⋊Q8C2×C4semidirect
71. C31×#121C2×C4semidirect
80. C31×#117C2×C4semidirect
183. C2×#1513C31direct (N × H)
258. C2×#407C2×C31semidirect
259. C2×#380C2×C31semidirect
260. #1513C2×C31direct (N × H)
261. C2×#328C2×C31semidirect
262. C2×#327C2×C31semidirect
301. C22×#41C4×C31semidirect
306. #525C4×C31semidirect
309. C2×C22⋊Q8C4×C31semidirect
311. C2×#121C4×C31semidirect
316. C2×#117C4×C31semidirect
320. #407C22×C31semidirect
324. #380C22×C31semidirect
326. #328C22×C31semidirect
327. #327C22×C31semidirect
334. C2×C31#1513direct (N × H)
341. C2×#41C2×C4×C31semidirect
369. C22⋊Q8C2×C4×C31semidirect
375. #121C2×C4×C31semidirect
384. #117C2×C4×C31semidirect
404. C31C2×#1513direct (N × H)
605. C2C31×#1513direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5252
C2×C31×#4072
C2×C31×#3802
C31×#15132
C2×C31×#3282
C2×C31×#3272
C2×#151331