Finite Group Databasebeta

← C31×#16825 #21473584

Subgroups — C31×#16825 #21473584

Order 7936.

1434 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#396C212yesmaximal
2C31×#1445C212yesmaximal, complemented
3C31×#1444C212yesmaximal, complemented
4C31×#1441C212yesmaximal, complemented
5C31×#1441C212yesmaximal, complemented
6C31×#1439C212yesmaximal
7C31×C22⋊#95C211yesmaximal, characteristic
8C31×C22⋊#95C211yesmaximal, characteristic
9C31×C22⋊#95C211yesmaximal, characteristic
10C24×C4×C31C2211yescharacteristic
11C22×C4⋊C4×C31C2212yes—
12C22×C4⋊C4×C31C2212yes—
13C22×C4⋊C4×C31C2212yes—
14C22×C4⋊C4×C31C2212yes—
15C22×C22⋊C4×C31C2212yes—
16C22×C22⋊C4×C31C2212yes—
17C22×C22⋊C4×C31C2211yescharacteristic
18C22×C22⋊C4×C31C2211yescharacteristic
19C31×#396—28no—
20C2×C31×#95C2212yes—
21C2×C31×#95C2212yes—
22C2×C31×#95C2211yescharacteristic
23C2×C31×#95C2212yes—
24C2×C31×#95C2212yes—
25C2×C31×#95C2211yescharacteristic
26C2×C31×#95C2211yescharacteristic
27C2×C31×#95C2212yes—
28C2×C31×#95C2212yes—
29C2×C31×#95C2211yescharacteristic
30C2×C31×#95C2212yes—
31C2×C31×#95C2212yes—
32C25×C31C2311yescharacteristic
33C23×C4×C31—41no—
34C23×C4×C31—41no—
35C23×C4×C31—42no—
36C23×C4×C31—42no—
37C23×C4×C31Q814yes—
38C23×C4×C31C2312yes—
39C23×C4×C31C2312yes—
40C23×C4×C31C2312yes—
41C23×C4×C31C2312yes—
42C23×C4×C31C2311yescharacteristic
43C23×C4×C31C2311yescharacteristic
44C23×C4×C31C2312yes—
45C23×C4×C31C2312yes—
46C2×C4⋊C4×C31—24no—
47C2×C4⋊C4×C31—24no—
48C2×C4⋊C4×C31—24no—
49C2×C4⋊C4×C31—24no—
50C2×C4⋊C4×C31—24no—
51C2×C4⋊C4×C31—24no—
52C2×C4⋊C4×C31—42no—
53C2×C4⋊C4×C31—42no—
54C2×C4⋊C4×C31—42no—
55C2×C4⋊C4×C31—42no—
56C2×C4⋊C4×C31—42no—
57C2×C4⋊C4×C31—42no—
58C2×C22⋊C4×C31—24no—
59C2×C22⋊C4×C31—24no—
60C2×C22⋊C4×C31—24no—
61C2×C22⋊C4×C31—24no—
62C2×C22⋊C4×C31—41no—
63C2×C22⋊C4×C31—41no—
64C2×C22⋊C4×C31—41no—
65C2×C22⋊C4×C31—41no—
66C2×C22⋊C4×C31—42no—
67C2×C22⋊C4×C31—42no—
68C2×C22⋊C4×C31D414yes—
69C2×C22⋊C4×C31Q814yes—
70C31×#95—24no—
71C31×#95—24no—
72C31×#95—41no—
73C31×#95—42no—
74C31×#95—42no—
75C31×#95—42no—
76C31×#95—41no—
77C31×#95—42no—
78C31×#95—42no—
79C31×#95—42no—
80C31×#95—41no—
81C31×#95—41no—
82C24×C31—22no—
83C24×C31—22no—
84C24×C31—21no—
85C24×C31—21no—
86C24×C31—21no—
87C24×C31—22no—
88C24×C31—22no—
89C24×C31—21no—
90C24×C31C2×D412yes—
91C24×C31C2×Q812yes—
92C24×C31C2×Q812yes—
93C24×C31C2411yescharacteristic
94C22×C4×C31—22no—
95C22×C4×C31—22no—
96C22×C4×C31—22no—
97C22×C4×C31—22no—
98C22×C4×C31—22no—
99C22×C4×C31—22no—
100C22×C4×C31—22no—
101C22×C4×C31—22no—
102C22×C4×C31—22no—
103C22×C4×C31—22no—
104C22×C4×C31—22no—
105C22×C4×C31—22no—
106C22×C4×C31—22no—
107C22×C4×C31—22no—
108C22×C4×C31—22no—
109C22×C4×C31—22no—
110C22×C4×C31—22no—
111C22×C4×C31—21no—
112C22×C4×C31—21no—
113C22×C4×C31—21no—
114C22×C4×C31—21no—
115C22×C4×C31—22no—
116C22×C4×C31—22no—
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—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—22no—
138C22×C4×C31—22no—
139C22×C4×C31—22no—
140C22×C4×C31—44no—
141C22×C4×C31—44no—
142C22×C4×C31—44no—
143C22×C4×C31—48no—
144C22×C4×C31—48no—
145C22×C4×C31C2×D414yes—
146C22×C4×C31C2×Q814yes—
147C22×C4×C31C2×Q812yes—
148C22×C4×C31C2×Q812yes—
149C22×C4×C31#4114yes—
150C22×C4×C31#4114yes—
151C4⋊C4×C31—48no—
152C4⋊C4×C31—48no—
153C4⋊C4×C31—48no—
154C4⋊C4×C31—48no—
155C22⋊C4×C31—44no—
156C22⋊C4×C31—44no—
157C22⋊C4×C31—48no—
158C22⋊C4×C31—48no—
159#16825C3111yesmaximal, characteristic, direct product factor, Sylow
160C23×C31—22no—
161C23×C31—22no—
162C23×C31—22no—
163C23×C31—22no—
164C23×C31—22no—
165C23×C31—22no—
166C23×C31—22no—
167C23×C31—22no—
168C23×C31—22no—
169C23×C31—22no—
170C23×C31—24no—
171C23×C31—24no—
172C23×C31—24no—
173C23×C31—24no—
174C23×C31—42no—
175C23×C31—42no—
176C23×C31—42no—
177C23×C31—42no—
178C23×C31—42no—
179C23×C31—42no—
180C23×C31—42no—
181C23×C31—42no—
182C23×C31C4⋊Q814yes—
183C23×C31C22×D411yescharacteristic
184C23×C31C22×Q811yescharacteristic
185C23×C31C22×Q811yescharacteristic
186C23×C31C2×#4112yes—
187C23×C31C2×#4112yes—
188C23×C31H(5,𝔽2)11yescharacteristic
189C23×C31H(5,𝔽2)11yescharacteristic
190C23×C31H(5,𝔽2)11yescharacteristic
191C23×C31H(5,𝔽2)12yes—
192C23×C31H(5,𝔽2)12yes—
193C23×C312−1+411yescharacteristic
194C2×C4×C31—24no—
195C2×C4×C31—24no—
196C2×C4×C31—24no—
197C2×C4×C31—24no—
198C2×C4×C31—22no—
199C2×C4×C31—22no—
200C2×C4×C31—22no—
201C2×C4×C31—22no—
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—24no—
209C2×C4×C31—24no—
210C2×C4×C31—24no—
211C2×C4×C31—24no—
212C2×C4×C31—416no—
213C2×C4×C31—42no—
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—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—42no—
234C2×C4×C31—42no—
235C2×C4×C31—42no—
236C2×C4×C31—42no—
237C2×C4×C31—42no—
238C2×C4×C31—42no—
239C2×C4×C31—42no—
240C2×C4×C31—42no—
241C2×C4×C31—42no—
242C2×C4×C31—42no—
243C2×C4×C31—42no—
244C2×C4×C31—42no—
245C2×C4×C31—42no—
246C2×C4×C31—42no—
247C2×#396C2×C3112yes—
248#1445C2×C3112yescomplemented
249#1444C2×C3112yescomplemented
250#1441C2×C3112yescomplemented
251#1441C2×C3112yescomplemented
252#1439C2×C3112yes—
253C22⋊#95C2×C3111yescharacteristic
254C22⋊#95C2×C3111yescharacteristic
255C22⋊#95C2×C3111yescharacteristic
256C22×C31—24no—
257C22×C31—24no—
258C22×C31—24no—
259C22×C31—42no—
260C22×C31—42no—
261C22×C31—42no—
262C22×C31—42no—
263C22×C31—44no—
264C22×C31—44no—
265C22×C31—44no—
266C22×C31—44no—
267C22×C31C2×C4⋊Q811yescharacteristic
268C22×C31#53311yescharacteristic
269C22×C31#53311yescharacteristic
270C22×C31#53411yescharacteristic
271C22×C31#53711yescharacteristic
272C22×C31#53812yes—
273C22×C31#53912yes—
274C22×C31#54012yes—
275C22×C31#54212yes—
276C22×C31#54311yescharacteristic
277C22×C31#54311yescharacteristic
278C22×C31#54311yescharacteristic
279C22×C31#54311yescharacteristic
280C22×C31#54411yescharacteristic
281C22×C31#54411yescharacteristic
282C22×C31#54612yes—
283C22×C31#54612yes—
284C22×C31#55312yes—
285C22×C31#55312yes—
286C22×C31#55412yes—
287C22×C31#55412yes—
288C22×C31#56012yes—
289C22×C31#56012yes—
290C4×C31—42no—
291C4×C31—42no—
292C4×C31—44no—
293C4×C31—44no—
294C4×C31—44no—
295C4×C31—44no—
296C4×C31—82no—
297C4×C31—82no—
298C24×C4C22×C3111yescharacteristic
299C22×C4⋊C4C22×C3112yes—
300C22×C4⋊C4C22×C3112yes—
301C22×C4⋊C4C22×C3112yes—
302C22×C4⋊C4C22×C3112yes—
303C22×C22⋊C4C22×C3112yes—
304C22×C22⋊C4C22×C3112yes—
305C22×C22⋊C4C22×C3111yescharacteristic
306C22×C22⋊C4C22×C3111yescharacteristic
307#396—28no—
308C2×#95C22×C3112yes—
309C2×#95C22×C3112yes—
310C2×#95C22×C3111yescharacteristic
311C2×#95C22×C3112yes—
312C2×#95C22×C3112yes—
313C2×#95C22×C3111yescharacteristic
314C2×#95C22×C3111yescharacteristic
315C2×#95C22×C3112yes—
316C2×#95C22×C3112yes—
317C2×#95C22×C3111yescharacteristic
318C2×#95C22×C3112yes—
319C2×#95C22×C3112yes—
320C2×C31—44no—
321C2×C31#254911yescharacteristic
322C2×C31#254911yescharacteristic
323C2×C31#260811yescharacteristic
324C2×C31#265911yescharacteristic
325C2×C31#266511yescharacteristic
326C2×C31#266711yescharacteristic
327C2×C31#268812yes—
328C2×C31#269112yes—
329C2×C31#278012yes—
330C2×C31#279012yes—
331C2×C31#284211yescharacteristic
332C25C23×C3111yescharacteristic
333C23×C4—41no—
334C23×C4—41no—
335C23×C4—42no—
336C23×C4—42no—
337C23×C4Q8×C3114yes—
338C23×C4C23×C3112yes—
339C23×C4C23×C3112yes—
340C23×C4C23×C3112yes—
341C23×C4C23×C3112yes—
342C23×C4C23×C3111yescharacteristic
343C23×C4C23×C3111yescharacteristic
344C23×C4C23×C3112yes—
345C23×C4C23×C3112yes—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—42no—
353C2×C4⋊C4—42no—
354C2×C4⋊C4—42no—
355C2×C4⋊C4—42no—
356C2×C4⋊C4—42no—
357C2×C4⋊C4—42no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—41no—
363C2×C22⋊C4—41no—
364C2×C22⋊C4—41no—
365C2×C22⋊C4—41no—
366C2×C22⋊C4—42no—
367C2×C22⋊C4—42no—
368C2×C22⋊C4D4×C3114yes—
369C2×C22⋊C4Q8×C3114yes—
370#95—24no—
371#95—24no—
372#95—41no—
373#95—42no—
374#95—42no—
375#95—42no—
376#95—41no—
377#95—42no—
378#95—42no—
379#95—42no—
380#95—41no—
381#95—41no—
382C31#1682511yescharacteristic, direct product factor, Sylow
383C24—22no—
384C24—22no—
385C24—21no—
386C24—21no—
387C24—21no—
388C24—22no—
389C24—22no—
390C24—21no—
391C24C2×D4×C3112yes—
392C24C2×Q8×C3112yes—
393C24C2×Q8×C3112yes—
394C24C24×C3111yescharacteristic
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
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—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4C2×D4×C3114yes—
447C22×C4C2×Q8×C3114yes—
448C22×C4C2×Q8×C3112yes—
449C22×C4C2×Q8×C3112yes—
450C22×C4#41×C3114yes—
451C22×C4#41×C3114yes—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23C31×C4⋊Q814yes—
483C23C22×D4×C3111yescharacteristic
484C23C22×Q8×C3111yescharacteristic
485C23C22×Q8×C3111yescharacteristic
486C23C2×#41×C3112yes—
487C23C2×#41×C3112yes—
488C23C31×H(5,𝔽2)11yescharacteristic
489C23C31×H(5,𝔽2)11yescharacteristic
490C23C31×H(5,𝔽2)11yescharacteristic
491C23C31×H(5,𝔽2)12yes—
492C23C31×H(5,𝔽2)12yes—
493C23C31×2−1+411yescharacteristic
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—416no—
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—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C22—24no—
548C22—24no—
549C22—24no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—44no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22C2×C31×C4⋊Q811yescharacteristic
559C22C31×#53311yescharacteristic
560C22C31×#53311yescharacteristic
561C22C31×#53411yescharacteristic
562C22C31×#53711yescharacteristic
563C22C31×#53812yes—
564C22C31×#53912yes—
565C22C31×#54012yes—
566C22C31×#54212yes—
567C22C31×#54311yescharacteristic
568C22C31×#54311yescharacteristic
569C22C31×#54311yescharacteristic
570C22C31×#54311yescharacteristic
571C22C31×#54411yescharacteristic
572C22C31×#54411yescharacteristic
573C22C31×#54612yes—
574C22C31×#54612yes—
575C22C31×#55312yes—
576C22C31×#55312yes—
577C22C31×#55412yes—
578C22C31×#55412yes—
579C22C31×#56012yes—
580C22C31×#56012yes—
581C4—42no—
582C4—42no—
583C4—44no—
584C4—44no—
585C4—44no—
586C4—44no—
587C4—82no—
588C4—82no—
589C2—44no—
590C2C31×#254911yescharacteristic
591C2C31×#254911yescharacteristic
592C2C31×#260811yescharacteristic
593C2C31×#265911yescharacteristic
594C2C31×#266511yescharacteristic
595C2C31×#266711yescharacteristic
596C2C31×#268812yes—
597C2C31×#269112yes—
598C2C31×#278012yes—
599C2C31×#279012yes—
600C2C31×#284211yescharacteristic
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. C31×#1445C2semidirect
3. C31×#1444C2semidirect
4. C31×#1441C2semidirect
5. C31×#1441C2semidirect
159. #16825C31direct (N × H)
248. #1445C2×C31semidirect
249. #1444C2×C31semidirect
250. #1441C2×C31semidirect
251. #1441C2×C31semidirect
382. C31#16825direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3962
C31×#14452
C31×#14442
C31×#14412
C31×#14392
C31×C22⋊#952
#1682531