Finite Group Databasebeta

← C31×#14317 #21471076

Subgroups — C31×#14317 #21471076

Order 7936.

1558 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#399C211yesmaximal, characteristic
2C2×C31×#389C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊C4×C31C211yesmaximal, characteristic
4C31×#1445C211yesmaximal, characteristic, complemented
5C31×#1442C211yesmaximal, characteristic, complemented
6C31×#1441C212yesmaximal, complemented
7C31×#1439C212yesmaximal
8C31×#1439C211yesmaximal, characteristic
9C31×#1436C212yesmaximal, complemented
10C31×#1434C211yesmaximal, characteristic, complemented
11C31×C22⋊#95C212yesmaximal
12C24×C4×C31C2211yescharacteristic
13C22×C4⋊C4×C31C2211yescharacteristic
14C22×C4⋊C4×C31C2212yes—
15C22×C4⋊C4×C31C2211yescharacteristic
16C22×C4⋊C4×C31C2212yes—
17C22×C22⋊C4×C31C2211yescharacteristic
18C22×C22⋊C4×C31C2211yescharacteristic
19C22×C22⋊C4×C31C2212yes—
20C22×C22⋊C4×C31C2212yes—
21C22×C42×C31C2211yescharacteristic
22C22×C42×C31C2211yescharacteristic
23C31×#399C418yescomplemented
24C31×#389—24no—
25C4×C22⋊C4×C31—24no—
26C2×C31×#95C2212yes—
27C2×C31×#95C2212yes—
28C2×C31×#95C2211yescharacteristic
29C2×C31×#95C2212yes—
30C2×C31×#95C2211yescharacteristic
31C2×C31×#95C2211yescharacteristic
32C2×C31×#95C2212yes—
33C2×C31×#95C2212yes—
34C2×C31×#95C2212yes—
35C2×C31×#95C2212yes—
36C2×C31×#95C2212yes—
37C2×C31×#95C2211yescharacteristic
38C25×C31C2311yescharacteristic
39C23×C4×C31—22no—
40C23×C4×C31—22no—
41C23×C4×C31—42no—
42C23×C4×C31—42no—
43C23×C4×C31Q814yes—
44C23×C4×C31C2312yes—
45C23×C4×C31C2312yes—
46C23×C4×C31C2311yescharacteristic
47C23×C4×C31C2312yes—
48C23×C4×C31C2312yes—
49C23×C4×C31C2311yescharacteristic
50C23×C4×C31C2311yescharacteristic
51C23×C4×C31C2311yescharacteristic
52C23×C4×C31C2311yescharacteristic
53C23×C4×C31C2311yescharacteristic
54C2×C4⋊C4×C31—22no—
55C2×C4⋊C4×C31—22no—
56C2×C4⋊C4×C31—24no—
57C2×C4⋊C4×C31—22no—
58C2×C4⋊C4×C31—42no—
59C2×C4⋊C4×C31—42no—
60C2×C4⋊C4×C31—42no—
61C2×C4⋊C4×C31—42no—
62C2×C4⋊C4×C31—42no—
63C2×C4⋊C4×C31C2×C414yescomplemented
64C2×C4⋊C4×C31C2×C418yescomplemented
65C2×C22⋊C4×C31—22no—
66C2×C22⋊C4×C31—22no—
67C2×C22⋊C4×C31—22no—
68C2×C22⋊C4×C31—22no—
69C2×C22⋊C4×C31—22no—
70C2×C22⋊C4×C31—24no—
71C2×C22⋊C4×C31—42no—
72C2×C22⋊C4×C31—42no—
73C2×C22⋊C4×C31—42no—
74C2×C22⋊C4×C31—42no—
75C2×C22⋊C4×C31C2×C414yes—
76C2×C22⋊C4×C31C2×C418yes—
77C2×C42×C31—22no—
78C2×C42×C31—42no—
79C2×C42×C31—42no—
80C2×C42×C31D414yes—
81C31×#95—22no—
82C31×#95—22no—
83C31×#95—24no—
84C31×#95—24no—
85C31×#95—24no—
86C31×#95—22no—
87C31×#95—42no—
88C31×#95—42no—
89C31×#95—42no—
90C31×#95—42no—
91C31×#95—42no—
92C31×#95C2×C414yescomplemented
93C24×C31—21no—
94C24×C31—21no—
95C24×C31—21no—
96C24×C31—21no—
97C24×C31—22no—
98C24×C31—22no—
99C24×C31—22no—
100C24×C31—22no—
101C24×C31C22×C412yes—
102C24×C31C2×Q812yes—
103C24×C31#4112yes—
104C24×C31C2411yescharacteristic
105C22×C4×C31—22no—
106C22×C4×C31—22no—
107C22×C4×C31—22no—
108C22×C4×C31—22no—
109C22×C4×C31—21no—
110C22×C4×C31—21no—
111C22×C4×C31—21no—
112C22×C4×C31—22no—
113C22×C4×C31—22no—
114C22×C4×C31—22no—
115C22×C4×C31—21no—
116C22×C4×C31—21no—
117C22×C4×C31—21no—
118C22×C4×C31—21no—
119C22×C4×C31—22no—
120C22×C4×C31—21no—
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—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—28no—
138C22×C4×C31—22no—
139C22×C4×C31—22no—
140C22×C4×C31—22no—
141C22×C4×C31—22no—
142C22×C4×C31—22no—
143C22×C4×C31—22no—
144C22×C4×C31—22no—
145C22×C4×C31—22no—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—22no—
149C22×C4×C31—22no—
150C22×C4×C31—22no—
151C22×C4×C31—22no—
152C22×C4×C31—48no—
153C22×C4×C31—44no—
154C22×C4×C31—44no—
155C22×C4×C31—48no—
156C22×C4×C31C22×C414yes—
157C22×C4×C31C22×C412yes—
158C22×C4×C31C22×C414yes—
159C22×C4×C31C22×C412yes—
160C22×C4×C31C2×D412yes—
161C22×C4×C31C2×D412yes—
162C22×C4×C31C2×D412yes—
163C22×C4×C31C2×Q812yes—
164C22×C4×C31C2×Q812yes—
165C22×C4×C31#4112yes—
166C22×C4×C31#4112yes—
167C22×C4×C31#4112yes—
168C4⋊C4×C31—28no—
169C4⋊C4×C31—48no—
170C4⋊C4×C31—44no—
171C4⋊C4×C31—48no—
172C22⋊C4×C31—28no—
173C22⋊C4×C31—28no—
174C22⋊C4×C31—48no—
175C22⋊C4×C31—48no—
176C42×C31—44no—
177C42×C31—44no—
178#14317C3111yesmaximal, characteristic, direct product factor, Sylow
179C23×C31—22no—
180C23×C31—22no—
181C23×C31—24no—
182C23×C31—24no—
183C23×C31—22no—
184C23×C31—22no—
185C23×C31—22no—
186C23×C31—22no—
187C23×C31—24no—
188C23×C31—24no—
189C23×C31—24no—
190C23×C31—24no—
191C23×C31—41no—
192C23×C31—41no—
193C23×C31—41no—
194C23×C31—41no—
195C23×C31—42no—
196C23×C31—42no—
197C23×C31—42no—
198C23×C31—42no—
199C23×C31—42no—
200C23×C31—42no—
201C23×C31C4×Q814yes—
202C23×C31C23×C411yescharacteristic
203C23×C31C22×D411yescharacteristic
204C23×C31C22×Q811yescharacteristic
205C23×C31C2×#4111yescharacteristic
206C23×C31C2×#4111yescharacteristic
207C23×C31C2×#4111yescharacteristic
208C23×C31C2×#4111yescharacteristic
209C23×C31H(5,𝔽2)12yes—
210C23×C31H(5,𝔽2)12yes—
211C23×C31H(5,𝔽2)12yes—
212C23×C312−1+412yes—
213C2×C4×C31—24no—
214C2×C4×C31—22no—
215C2×C4×C31—22no—
216C2×C4×C31—22no—
217C2×C4×C31—22no—
218C2×C4×C31—24no—
219C2×C4×C31—24no—
220C2×C4×C31—24no—
221C2×C4×C31—22no—
222C2×C4×C31—24no—
223C2×C4×C31—24no—
224C2×C4×C31—24no—
225C2×C4×C31—24no—
226C2×C4×C31—24no—
227C2×C4×C31—24no—
228C2×C4×C31—24no—
229C2×C4×C31—22no—
230C2×C4×C31—22no—
231C2×C4×C31—22no—
232C2×C4×C31—22no—
233C2×C4×C31—22no—
234C2×C4×C31—24no—
235C2×C4×C31—24no—
236C2×C4×C31—416no—
237C2×C4×C31—41no—
238C2×C4×C31—41no—
239C2×C4×C31—41no—
240C2×C4×C31—42no—
241C2×C4×C31—42no—
242C2×C4×C31—41no—
243C2×C4×C31—41no—
244C2×C4×C31—41no—
245C2×C4×C31—42no—
246C2×C4×C31—42no—
247C2×C4×C31—42no—
248C2×C4×C31—42no—
249C2×C4×C31—42no—
250C2×C4×C31—42no—
251C2×C4×C31—42no—
252C2×C4×C31—42no—
253C2×C4×C31—42no—
254C2×C4×C31—42no—
255C2×C4×C31—42no—
256C2×C4×C31—42no—
257C2×C4×C31—42no—
258C2×C4×C31—42no—
259C2×C4×C31—42no—
260C2×C4×C31—42no—
261C2×C4×C31—42no—
262C2×C4×C31—42no—
263C2×C4×C31—42no—
264C2×C4×C31—42no—
265C2×C4×C31—42no—
266C2×C4×C31—42no—
267C2×C4×C31—42no—
268C2×C4×C31—42no—
269C2×C4×C31—42no—
270C2×C4×C31C4×D414yes—
271C2×C4×C31C22⋊Q814yes—
272C2×#399C2×C3111yescharacteristic
273C2×#389C2×C3111yescharacteristic, complemented
274C2×C4×C22⋊C4C2×C3111yescharacteristic
275#1445C2×C3111yescharacteristic, complemented
276#1442C2×C3111yescharacteristic, complemented
277#1441C2×C3112yescomplemented
278#1439C2×C3112yes—
279#1439C2×C3111yescharacteristic
280#1436C2×C3112yescomplemented
281#1434C2×C3111yescharacteristic, complemented
282C22⋊#95C2×C3112yes—
283C22×C31—28no—
284C22×C31—24no—
285C22×C31—42no—
286C22×C31—42no—
287C22×C31—44no—
288C22×C31—42no—
289C22×C31—42no—
290C22×C31—42no—
291C22×C31—42no—
292C22×C31—44no—
293C22×C31—44no—
294C22×C31C2×C4×D411yescharacteristic
295C22×C31C2×C4×Q811yescharacteristic
296C22×C31C4×#4111yescharacteristic
297C22×C31#51812yes—
298C22×C31#52012yes—
299C22×C31C2×C22⋊Q811yescharacteristic
300C22×C31#52511yescharacteristic
301C22×C31#52911yescharacteristic
302C22×C31#53311yescharacteristic
303C22×C31#53411yescharacteristic
304C22×C31#53511yescharacteristic
305C22×C31#53511yescharacteristic
306C22×C31#53611yescharacteristic
307C22×C31#53812yes—
308C22×C31#53812yes—
309C22×C31#53912yes—
310C22×C31#53912yes—
311C22×C31#53912yes—
312C22×C31#54012yes—
313C22×C31#54012yes—
314C22×C31#54212yes—
315C22×C31#54312yes—
316C22×C31#54412yes—
317C4×C31—24no—
318C4×C31—24no—
319C4×C31—42no—
320C4×C31—44no—
321C4×C31—44no—
322C4×C31—42no—
323C4×C31—42no—
324C4×C31—42no—
325C4×C31—82no—
326C4×C31—82no—
327C24×C4C22×C3111yescharacteristic
328C22×C4⋊C4C22×C3111yescharacteristic
329C22×C4⋊C4C22×C3112yes—
330C22×C4⋊C4C22×C3111yescharacteristic
331C22×C4⋊C4C22×C3112yes—
332C22×C22⋊C4C22×C3111yescharacteristic
333C22×C22⋊C4C22×C3111yescharacteristic
334C22×C22⋊C4C22×C3112yes—
335C22×C22⋊C4C22×C3112yes—
336C22×C42C22×C3111yescharacteristic
337C22×C42C22×C3111yescharacteristic
338#399C4×C3118yescomplemented
339#389—24no—
340C4×C22⋊C4—24no—
341C2×#95C22×C3112yes—
342C2×#95C22×C3112yes—
343C2×#95C22×C3111yescharacteristic
344C2×#95C22×C3112yes—
345C2×#95C22×C3111yescharacteristic
346C2×#95C22×C3111yescharacteristic
347C2×#95C22×C3112yes—
348C2×#95C22×C3112yes—
349C2×#95C22×C3112yes—
350C2×#95C22×C3112yes—
351C2×#95C22×C3112yes—
352C2×#95C22×C3111yescharacteristic
353C2×C31—44no—
354C2×C31C4×C22⋊Q811yescharacteristic
355C2×C31#232211yescharacteristic
356C2×C31#232811yescharacteristic
357C2×C31#232912yes—
358C2×C31#233512yes—
359C2×C31#262611yescharacteristic
360C2×C31#262811yescharacteristic
361C2×C31#263311yescharacteristic
362C2×C31#263511yescharacteristic
363C2×C31#264912yes—
364C2×C31#266712yes—
365C25C23×C3111yescharacteristic
366C23×C4—22no—
367C23×C4—22no—
368C23×C4—42no—
369C23×C4—42no—
370C23×C4Q8×C3114yes—
371C23×C4C23×C3112yes—
372C23×C4C23×C3112yes—
373C23×C4C23×C3112yes—
374C23×C4C23×C3111yescharacteristic
375C23×C4C23×C3112yes—
376C23×C4C23×C3111yescharacteristic
377C23×C4C23×C3111yescharacteristic
378C23×C4C23×C3111yescharacteristic
379C23×C4C23×C3111yescharacteristic
380C23×C4C23×C3111yescharacteristic
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—24no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—42no—
386C2×C4⋊C4—42no—
387C2×C4⋊C4—42no—
388C2×C4⋊C4—42no—
389C2×C4⋊C4—42no—
390C2×C4⋊C4C2×C4×C3114yescomplemented
391C2×C4⋊C4C2×C4×C3118yescomplemented
392C2×C22⋊C4—22no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—42no—
399C2×C22⋊C4—42no—
400C2×C22⋊C4—42no—
401C2×C22⋊C4—42no—
402C2×C22⋊C4C2×C4×C3114yes—
403C2×C22⋊C4C2×C4×C3118yes—
404C2×C42—22no—
405C2×C42—42no—
406C2×C42—42no—
407C2×C42D4×C3114yes—
408#95—22no—
409#95—22no—
410#95—24no—
411#95—24no—
412#95—24no—
413#95—22no—
414#95—42no—
415#95—42no—
416#95—42no—
417#95—42no—
418#95—42no—
419#95C2×C4×C3114yescomplemented
420C31#1431711yescharacteristic, direct product factor, Sylow
421C24—21no—
422C24—21no—
423C24—21no—
424C24—21no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24C22×C4×C3112yes—
430C24C2×Q8×C3112yes—
431C24#41×C3112yes—
432C24C24×C3111yescharacteristic
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—28no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—48no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—48no—
484C22×C4C22×C4×C3114yes—
485C22×C4C22×C4×C3112yes—
486C22×C4C22×C4×C3114yes—
487C22×C4C22×C4×C3112yes—
488C22×C4C2×D4×C3112yes—
489C22×C4C2×D4×C3112yes—
490C22×C4C2×D4×C3112yes—
491C22×C4C2×Q8×C3112yes—
492C22×C4C2×Q8×C3112yes—
493C22×C4#41×C3112yes—
494C22×C4#41×C3112yes—
495C22×C4#41×C3112yes—
496C4⋊C4—28no—
497C4⋊C4—48no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C22⋊C4—28no—
501C22⋊C4—28no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C42—44no—
505C42—44no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—24no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23C4×Q8×C3114yes—
529C23C23×C4×C3111yescharacteristic
530C23C22×D4×C3111yescharacteristic
531C23C22×Q8×C3111yescharacteristic
532C23C2×#41×C3111yescharacteristic
533C23C2×#41×C3111yescharacteristic
534C23C2×#41×C3111yescharacteristic
535C23C2×#41×C3111yescharacteristic
536C23C31×H(5,𝔽2)12yes—
537C23C31×H(5,𝔽2)12yes—
538C23C31×H(5,𝔽2)12yes—
539C23C31×2−1+412yes—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—416no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4C4×D4×C3114yes—
598C2×C4C31×C22⋊Q814yes—
599C22—28no—
600C22—24no—
601C22—42no—
602C22—42no—
603C22—44no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—44no—
609C22—44no—
610C22C2×C4×D4×C3111yescharacteristic
611C22C2×C4×Q8×C3111yescharacteristic
612C22C4×#41×C3111yescharacteristic
613C22C31×#51812yes—
614C22C31×#52012yes—
615C22C2×C31×C22⋊Q811yescharacteristic
616C22C31×#52511yescharacteristic
617C22C31×#52911yescharacteristic
618C22C31×#53311yescharacteristic
619C22C31×#53411yescharacteristic
620C22C31×#53511yescharacteristic
621C22C31×#53511yescharacteristic
622C22C31×#53611yescharacteristic
623C22C31×#53812yes—
624C22C31×#53812yes—
625C22C31×#53912yes—
626C22C31×#53912yes—
627C22C31×#53912yes—
628C22C31×#54012yes—
629C22C31×#54012yes—
630C22C31×#54212yes—
631C22C31×#54312yes—
632C22C31×#54412yes—
633C4—24no—
634C4—24no—
635C4—42no—
636C4—44no—
637C4—44no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—82no—
642C4—82no—
643C2—44no—
644C2C4×C31×C22⋊Q811yescharacteristic
645C2C31×#232211yescharacteristic
646C2C31×#232811yescharacteristic
647C2C31×#232912yes—
648C2C31×#233512yes—
649C2C31×#262611yescharacteristic
650C2C31×#262811yescharacteristic
651C2C31×#263311yescharacteristic
652C2C31×#263511yescharacteristic
653C2C31×#264912yes—
654C2C31×#266712yes—
subgroupG/H
center, socleC24×C31C24
commutatorC23C23×C4×C31
Frattini, cosocleC24C24×C31
prime radicalC25×C31C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#389C2semidirect
4. C31×#1445C2semidirect
5. C31×#1442C2semidirect
6. C31×#1441C2semidirect
9. C31×#1436C2semidirect
10. C31×#1434C2semidirect
23. C31×#399C4semidirect
63. C2×C4⋊C4×C31C2×C4semidirect
64. C2×C4⋊C4×C31C2×C4semidirect
92. C31×#95C2×C4semidirect
178. #14317C31direct (N × H)
273. C2×#389C2×C31semidirect
275. #1445C2×C31semidirect
276. #1442C2×C31semidirect
277. #1441C2×C31semidirect
280. #1436C2×C31semidirect
281. #1434C2×C31semidirect
338. #399C4×C31semidirect
390. C2×C4⋊C4C2×C4×C31semidirect
391. C2×C4⋊C4C2×C4×C31semidirect
419. #95C2×C4×C31semidirect
420. C31#14317direct (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×#3892
C2×C4×C22⋊C4×C312
C31×#14452
C31×#14422
C31×#14412
C31×#14392
C31×#14362
C31×#14342
C31×C22⋊#952
#1431731