Finite Group Databasebeta

← C31×#15110 #21471869

Subgroups — C31×#15110 #21471869

Order 7936.

1334 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#388C211yesmaximal, characteristic
2C2×C4×C4⋊C4×C31C211yesmaximal, characteristic, complemented
3C31×#1443C212yesmaximal, complemented
4C31×#1442C211yesmaximal, characteristic, complemented
5C31×#1442C211yesmaximal, characteristic, complemented
6C31×#1440C212yesmaximal
7C31×#1439C211yesmaximal, characteristic
8C31×#1439C211yesmaximal, characteristic
9C31×#1437C212yesmaximal
10C31×#1436C212yesmaximal, complemented
11C31×#1434C211yesmaximal, characteristic, complemented
12C22×C4⋊C4×C31C2211yescharacteristic
13C22×C4⋊C4×C31C2211yescharacteristic
14C22×C4⋊C4×C31C2211yescharacteristic
15C22×C22⋊C4×C31C2211yescharacteristic
16C22×C22⋊C4×C31C2211yescharacteristic
17C22×C22⋊C4×C31C2211yescharacteristic
18C22×C22⋊C4×C31C2212yes—
19C22×C22⋊C4×C31C2212yes—
20C22×C42×C31C2211yescharacteristic
21C22×C42×C31C2212yes—
22C31×#388—24no—
23C4×C4⋊C4×C31—24no—
24C2×C31×#95C2212yes—
25C2×C31×#95C2211yescharacteristic
26C2×C31×#95C2211yescharacteristic
27C2×C31×#95C2212yes—
28C2×C31×#95C2212yes—
29C2×C31×#95C2212yes—
30C2×C31×#95C2212yes—
31C2×C31×#95C2211yescharacteristic
32C2×C31×#95C2211yescharacteristic
33C2×C31×#95C2212yes—
34C2×C31×#95C2212yes—
35C2×C31×#95C2212yes—
36C2×C31×#95C2212yes—
37C25×C31C2311yescharacteristic
38C23×C4×C31C2311yescharacteristic
39C23×C4×C31C2311yescharacteristic
40C23×C4×C31C2312yes—
41C23×C4×C31C2312yes—
42C23×C4×C31C2311yescharacteristic
43C23×C4×C31C2312yes—
44C23×C4×C31C2312yes—
45C23×C4×C31C2311yescharacteristic
46C23×C4×C31C2311yescharacteristic
47C23×C4×C31C2311yescharacteristic
48C2×C4⋊C4×C31—22no—
49C2×C4⋊C4×C31—22no—
50C2×C4⋊C4×C31—24no—
51C2×C4⋊C4×C31—24no—
52C2×C4⋊C4×C31—22no—
53C2×C4⋊C4×C31—24no—
54C2×C22⋊C4×C31—22no—
55C2×C22⋊C4×C31—22no—
56C2×C22⋊C4×C31—22no—
57C2×C22⋊C4×C31—24no—
58C2×C22⋊C4×C31—22no—
59C2×C22⋊C4×C31—22no—
60C2×C22⋊C4×C31—24no—
61C2×C22⋊C4×C31—24no—
62C2×C22⋊C4×C31—22no—
63C2×C22⋊C4×C31—22no—
64C2×C22⋊C4×C31—24no—
65C2×C22⋊C4×C31—42no—
66C2×C22⋊C4×C31—42no—
67C2×C22⋊C4×C31—42no—
68C2×C42×C31—22no—
69C2×C42×C31—22no—
70C2×C42×C31—24no—
71C2×C42×C31—42no—
72C2×C42×C31—42no—
73C2×C42×C31D414yes—
74C31×#95—24no—
75C31×#95—22no—
76C31×#95—22no—
77C31×#95—24no—
78C31×#95—24no—
79C31×#95—24no—
80C31×#95—22no—
81C31×#95—22no—
82C31×#95—24no—
83C31×#95—42no—
84C31×#95—42no—
85C31×#95—42no—
86C31×#95—42no—
87C24×C31—21no—
88C24×C31—21no—
89C24×C31—22no—
90C24×C31—22no—
91C24×C31—22no—
92C24×C31—21no—
93C24×C31—21no—
94C24×C31—22no—
95C24×C31—21no—
96C24×C31—21no—
97C24×C31#4112yes—
98C24×C31C2411yescharacteristic
99C22×C4×C31—21no—
100C22×C4×C31—21no—
101C22×C4×C31—22no—
102C22×C4×C31—22no—
103C22×C4×C31—22no—
104C22×C4×C31—22no—
105C22×C4×C31—22no—
106C22×C4×C31—21no—
107C22×C4×C31—22no—
108C22×C4×C31—22no—
109C22×C4×C31—21no—
110C22×C4×C31—21no—
111C22×C4×C31—21no—
112C22×C4×C31—21no—
113C22×C4×C31—21no—
114C22×C4×C31—22no—
115C22×C4×C31—22no—
116C22×C4×C31—22no—
117C22×C4×C31—22no—
118C22×C4×C31—21no—
119C22×C4×C31—22no—
120C22×C4×C31—22no—
121C22×C4×C31—22no—
122C22×C4×C31—22no—
123C22×C4×C31—22no—
124C22×C4×C31—21no—
125C22×C4×C31—22no—
126C22×C4×C31—21no—
127C22×C4×C31—21no—
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—21no—
137C22×C4×C31—21no—
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—21no—
145C22×C4×C31—22no—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—22no—
149C22×C4×C31—21no—
150C22×C4×C31C2×D412yes—
151C22×C4×C31C2×D412yes—
152C22×C4×C31C2×D412yes—
153C22×C4×C31#4112yes—
154C22×C4×C31#4114yes—
155C22×C4×C31#4114yes—
156C22×C4×C31#4112yes—
157C22×C4×C31#4112yes—
158C22×C4×C31#4114yes—
159C4⋊C4×C31—28no—
160C4⋊C4×C31—28no—
161C4⋊C4×C31—44no—
162C22⋊C4×C31—28no—
163C22⋊C4×C31—48no—
164C22⋊C4×C31—48no—
165C22⋊C4×C31—44no—
166C22⋊C4×C31—44no—
167C42×C31—28no—
168C42×C31—48no—
169#15110C3111yesmaximal, characteristic, direct product factor, Sylow
170C23×C31—22no—
171C23×C31—24no—
172C23×C31—24no—
173C23×C31—22no—
174C23×C31—22no—
175C23×C31—41no—
176C23×C31—41no—
177C23×C31—41no—
178C23×C31—41no—
179C23×C31—42no—
180C23×C31—42no—
181C23×C31—42no—
182C23×C31—41no—
183C23×C31—41no—
184C23×C31—42no—
185C23×C31—42no—
186C23×C31—42no—
187C23×C31—42no—
188C23×C31—42no—
189C23×C31—41no—
190C23×C31—42no—
191C23×C31—41no—
192C23×C31—42no—
193C23×C31C22×D411yescharacteristic
194C23×C31C2×#4111yescharacteristic
195C23×C31C2×#4111yescharacteristic
196C23×C31C2×#4112yes—
197C23×C31C2×#4111yescharacteristic
198C23×C31C2×#4112yes—
199C23×C31C2×#4111yescharacteristic
200C23×C31C2×#4112yes—
201C23×C31H(5,𝔽2)12yes—
202C23×C31H(5,𝔽2)11yescharacteristic
203C23×C312−1+411yescharacteristic
204C2×C4×C31—22no—
205C2×C4×C31—22no—
206C2×C4×C31—22no—
207C2×C4×C31—24no—
208C2×C4×C31—24no—
209C2×C4×C31—22no—
210C2×C4×C31—24no—
211C2×C4×C31—24no—
212C2×C4×C31—22no—
213C2×C4×C31—22no—
214C2×C4×C31—22no—
215C2×C4×C31—24no—
216C2×C4×C31—24no—
217C2×C4×C31—24no—
218C2×C4×C31—22no—
219C2×C4×C31—24no—
220C2×C4×C31—24no—
221C2×C4×C31—24no—
222C2×C4×C31—24no—
223C2×C4×C31—42no—
224C2×C4×C31—41no—
225C2×C4×C31—41no—
226C2×C4×C31—42no—
227C2×C4×C31—42no—
228C2×C4×C31—42no—
229C2×C4×C31—42no—
230C2×C4×C31—42no—
231C2×C4×C31—42no—
232C2×C4×C31—42no—
233C2×C4×C31—41no—
234C2×C4×C31—42no—
235C2×C4×C31—41no—
236C2×C4×C31—41no—
237C2×C4×C31—41no—
238C2×C4×C31—41no—
239C2×C4×C31—41no—
240C2×C4×C31—41no—
241C2×C4×C31—41no—
242C2×C4×C31—42no—
243C2×C4×C31—42no—
244C2×C4×C31—42no—
245C2×C4×C31—42no—
246C2×C4×C31—41no—
247C2×C4×C31—41no—
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#12314yes—
263C2×C4×C31#12414yes—
264C2×#388C2×C3111yescharacteristic
265C2×C4×C4⋊C4C2×C3111yescharacteristic, complemented
266#1443C2×C3112yescomplemented
267#1442C2×C3111yescharacteristic, complemented
268#1442C2×C3111yescharacteristic, complemented
269#1440C2×C3112yes—
270#1439C2×C3111yescharacteristic
271#1439C2×C3111yescharacteristic
272#1437C2×C3112yes—
273#1436C2×C3112yescomplemented
274#1434C2×C3111yescharacteristic, complemented
275C22×C31—44no—
276C22×C31—44no—
277C22×C31—42no—
278C22×C31—42no—
279C22×C31—42no—
280C22×C31—81no—
281C22×C31—82no—
282C22×C31—82no—
283C22×C31—81no—
284C22×C31—81no—
285C22×C31—81no—
286C22×C31C2×#12311yescharacteristic
287C22×C31#52511yescharacteristic
288C22×C31#52511yescharacteristic
289C22×C31#52511yescharacteristic
290C22×C31C2×#12411yescharacteristic
291C22×C31#52912yes—
292C22×C31#52912yes—
293C22×C31#52912yes—
294C22×C31#52912yes—
295C22×C31#53511yescharacteristic
296C22×C31#53711yescharacteristic
297C22×C31#53812yes—
298C22×C31#53912yes—
299C22×C31#55112yes—
300C22×C31#55211yescharacteristic
301C22×C31#55312yes—
302C22×C31#55312yes—
303C22×C31#55312yes—
304C22×C31#55311yescharacteristic
305C22×C31#55511yescharacteristic
306C22×C31#55612yes—
307C22×C31#55611yescharacteristic
308C22×C31#55912yes—
309C4×C31—24no—
310C4×C31—24no—
311C4×C31—44no—
312C4×C31—44no—
313C4×C31—42no—
314C4×C31—42no—
315C4×C31—81no—
316C4×C31—82no—
317C4×C31—82no—
318C4×C31—81no—
319C22×C4⋊C4C22×C3111yescharacteristic
320C22×C4⋊C4C22×C3111yescharacteristic
321C22×C4⋊C4C22×C3111yescharacteristic
322C22×C22⋊C4C22×C3111yescharacteristic
323C22×C22⋊C4C22×C3111yescharacteristic
324C22×C22⋊C4C22×C3111yescharacteristic
325C22×C22⋊C4C22×C3112yes—
326C22×C22⋊C4C22×C3112yes—
327C22×C42C22×C3111yescharacteristic
328C22×C42C22×C3112yes—
329#388—24no—
330C4×C4⋊C4—24no—
331C2×#95C22×C3112yes—
332C2×#95C22×C3111yescharacteristic
333C2×#95C22×C3111yescharacteristic
334C2×#95C22×C3112yes—
335C2×#95C22×C3112yes—
336C2×#95C22×C3112yes—
337C2×#95C22×C3112yes—
338C2×#95C22×C3111yescharacteristic
339C2×#95C22×C3111yescharacteristic
340C2×#95C22×C3112yes—
341C2×#95C22×C3112yes—
342C2×#95C22×C3112yes—
343C2×#95C22×C3112yes—
344C2×C31—82no—
345C2×C31#239811yescharacteristic
346C2×C31#248511yescharacteristic
347C2×C31#250411yescharacteristic
348C2×C31#257312yes—
349C2×C31#259211yescharacteristic
350C2×C31#259312yes—
351C2×C31#259412yes—
352C2×C31#259512yes—
353C2×C31#259811yescharacteristic
354C2×C31#263611yescharacteristic
355C2×C31#285311yescharacteristic
356C25C23×C3111yescharacteristic
357C23×C4C23×C3111yescharacteristic
358C23×C4C23×C3111yescharacteristic
359C23×C4C23×C3112yes—
360C23×C4C23×C3112yes—
361C23×C4C23×C3111yescharacteristic
362C23×C4C23×C3111yescharacteristic
363C23×C4C23×C3112yes—
364C23×C4C23×C3112yes—
365C23×C4C23×C3111yescharacteristic
366C23×C4C23×C3111yescharacteristic
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—24no—
370C2×C4⋊C4—24no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—24no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—24no—
380C2×C22⋊C4—24no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4—42no—
385C2×C22⋊C4—42no—
386C2×C22⋊C4—42no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—24no—
390C2×C42—42no—
391C2×C42—42no—
392C2×C42D4×C3114yes—
393#95—24no—
394#95—22no—
395#95—22no—
396#95—24no—
397#95—24no—
398#95—24no—
399#95—22no—
400#95—22no—
401#95—24no—
402#95—42no—
403#95—42no—
404#95—42no—
405#95—42no—
406C31#1511011yescharacteristic, direct product factor, Sylow
407C24—21no—
408C24—21no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—21no—
413C24—21no—
414C24—22no—
415C24—21no—
416C24—21no—
417C24#41×C3112yes—
418C24C24×C3111yescharacteristic
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4C2×D4×C3112yes—
471C22×C4C2×D4×C3112yes—
472C22×C4C2×D4×C3112yes—
473C22×C4#41×C3112yes—
474C22×C4#41×C3114yes—
475C22×C4#41×C3114yes—
476C22×C4#41×C3112yes—
477C22×C4#41×C3112yes—
478C22×C4#41×C3114yes—
479C4⋊C4—28no—
480C4⋊C4—28no—
481C4⋊C4—44no—
482C22⋊C4—28no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C42—28no—
488C42—48no—
489C23—22no—
490C23—24no—
491C23—24no—
492C23—22no—
493C23—22no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—41no—
509C23—42no—
510C23—41no—
511C23—42no—
512C23C22×D4×C3111yescharacteristic
513C23C2×#41×C3111yescharacteristic
514C23C2×#41×C3111yescharacteristic
515C23C2×#41×C3112yes—
516C23C2×#41×C3111yescharacteristic
517C23C2×#41×C3112yes—
518C23C2×#41×C3111yescharacteristic
519C23C2×#41×C3112yes—
520C23C31×H(5,𝔽2)12yes—
521C23C31×H(5,𝔽2)11yescharacteristic
522C23C31×2−1+411yescharacteristic
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
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×C4C31×#12314yes—
582C2×C4C31×#12414yes—
583C22—44no—
584C22—44no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—81no—
589C22—82no—
590C22—82no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22C2×C31×#12311yescharacteristic
595C22C31×#52511yescharacteristic
596C22C31×#52511yescharacteristic
597C22C31×#52511yescharacteristic
598C22C2×C31×#12411yescharacteristic
599C22C31×#52912yes—
600C22C31×#52912yes—
601C22C31×#52912yes—
602C22C31×#52912yes—
603C22C31×#53511yescharacteristic
604C22C31×#53711yescharacteristic
605C22C31×#53812yes—
606C22C31×#53912yes—
607C22C31×#55112yes—
608C22C31×#55211yescharacteristic
609C22C31×#55312yes—
610C22C31×#55312yes—
611C22C31×#55312yes—
612C22C31×#55311yescharacteristic
613C22C31×#55511yescharacteristic
614C22C31×#55612yes—
615C22C31×#55611yescharacteristic
616C22C31×#55912yes—
617C4—24no—
618C4—24no—
619C4—44no—
620C4—44no—
621C4—42no—
622C4—42no—
623C4—81no—
624C4—82no—
625C4—82no—
626C4—81no—
627C2—82no—
628C2C31×#239811yescharacteristic
629C2C31×#248511yescharacteristic
630C2C31×#250411yescharacteristic
631C2C31×#257312yes—
632C2C31×#259211yescharacteristic
633C2C31×#259312yes—
634C2C31×#259412yes—
635C2C31×#259512yes—
636C2C31×#259811yescharacteristic
637C2C31×#263611yescharacteristic
638C2C31×#285311yescharacteristic
subgroupG/H
center, socleC24×C31C24
commutator, Frattini, cosocleC24C24×C31
prime radicalC25×C31C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C4×C4⋊C4×C31C2semidirect
3. C31×#1443C2semidirect
4. C31×#1442C2semidirect
5. C31×#1442C2semidirect
10. C31×#1436C2semidirect
11. C31×#1434C2semidirect
169. #15110C31direct (N × H)
265. C2×C4×C4⋊C4C2×C31semidirect
266. #1443C2×C31semidirect
267. #1442C2×C31semidirect
268. #1442C2×C31semidirect
273. #1436C2×C31semidirect
274. #1434C2×C31semidirect
406. C31#15110direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3882
C2×C4×C4⋊C4×C312
C31×#14432
C31×#14422
C31×#14402
C31×#14392
C31×#14372
C31×#14362
C31×#14342
#1511031