Finite Group Databasebeta

← C31×#17121 #21473880

Subgroups — C31×#17121 #21473880

Order 7936.

1550 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#397C212yesmaximal, complemented
2C2×C31×#396C212yesmaximal, complemented
3C31×#1445C211yesmaximal, characteristic, complemented
4C31×#1444C211yesmaximal, characteristic, complemented
5C31×#1442C212yesmaximal, complemented
6C31×#1439C212yesmaximal, complemented
7C31×#1439C212yesmaximal, complemented
8C31×#1438C212yesmaximal
9C31×#1438C211yesmaximal, characteristic
10C23×D4×C31C2211yescharacteristic
11C22×C4⋊C4×C31C2212yes—
12C22×C4⋊C4×C31C2212yes—
13C22×C4⋊C4×C31C2211yescharacteristic
14C22×C4⋊C4×C31C2211yescharacteristic
15C22×C22⋊C4×C31C2212yes—
16C22×C22⋊C4×C31C2212yes—
17C22×C22⋊C4×C31C2212yes—
18C22×C22⋊C4×C31C2212yes—
19C22×C22⋊C4×C31C2212yes—
20C22×C22⋊C4×C31C2212yes—
21C31×#397—28no—
22C31×#396—28no—
23C2×C31×#95C2212yes—
24C2×C31×#95C2212yes—
25C2×C31×#95C2212yes—
26C2×C31×#95C2212yes—
27C2×C31×#95C2211yescharacteristic
28C2×C31×#95C2212yes—
29C2×C31×#95C2211yescharacteristic
30C2×C31×#95C2212yes—
31C2×C31×#95C2211yescharacteristic
32C2×C31×#95C2211yescharacteristic
33C25×C31C2312yes—
34C22×D4×C31—24no—
35C22×D4×C31—22no—
36C22×D4×C31—42no—
37C22×D4×C31—41no—
38C22×D4×C31—41no—
39C23×C4×C31C2312yes—
40C23×C4×C31C2312yes—
41C23×C4×C31C2312yes—
42C23×C4×C31C2311yescharacteristic
43C23×C4×C31C2312yes—
44C23×C4×C31C2312yes—
45C23×C4×C31C2311yescharacteristic
46C23×C4×C31C2311yescharacteristic
47C2×C4⋊C4×C31—24no—
48C2×C4⋊C4×C31—24no—
49C2×C4⋊C4×C31—24no—
50C2×C4⋊C4×C31—41no—
51C2×C4⋊C4×C31—41no—
52C2×C4⋊C4×C31—42no—
53C2×C4⋊C4×C31—42no—
54C2×C4⋊C4×C31—42no—
55C2×C4⋊C4×C31—41no—
56C2×C4⋊C4×C31—42no—
57C2×C4⋊C4×C31D414yescomplemented
58C2×C22⋊C4×C31—24no—
59C2×C22⋊C4×C31—24no—
60C2×C22⋊C4×C31—24no—
61C2×C22⋊C4×C31—24no—
62C2×C22⋊C4×C31—24no—
63C2×C22⋊C4×C31—24no—
64C2×C22⋊C4×C31—24no—
65C2×C22⋊C4×C31—24no—
66C2×C22⋊C4×C31—42no—
67C2×C22⋊C4×C31—42no—
68C2×C22⋊C4×C31—42no—
69C2×C22⋊C4×C31—42no—
70C2×C22⋊C4×C31—42no—
71C2×C22⋊C4×C31—42no—
72C2×C22⋊C4×C31—42no—
73C2×C22⋊C4×C31—42no—
74C2×C22⋊C4×C31—42no—
75C2×C22⋊C4×C31—42no—
76C31×#95—24no—
77C31×#95—24no—
78C31×#95—24no—
79C31×#95—24no—
80C31×#95—41no—
81C31×#95—42no—
82C31×#95—41no—
83C31×#95—41no—
84C31×#95—42no—
85C31×#95D414yescomplemented
86C24×C31—22no—
87C24×C31—22no—
88C24×C31—22no—
89C24×C31—22no—
90C24×C31—22no—
91C24×C31—22no—
92C24×C31—22no—
93C24×C31—22no—
94C24×C31—22no—
95C24×C31—22no—
96C24×C31—22no—
97C24×C31—22no—
98C24×C31—22no—
99C24×C31—22no—
100C24×C31#4114yes—
101C24×C31C2411yescharacteristic
102C2×D4×C31—48no—
103C2×D4×C31—44no—
104C2×D4×C31—44no—
105C2×D4×C31—48no—
106C2×D4×C31—44no—
107C22×C4×C31—22no—
108C22×C4×C31—22no—
109C22×C4×C31—22no—
110C22×C4×C31—22no—
111C22×C4×C31—22no—
112C22×C4×C31—22no—
113C22×C4×C31—21no—
114C22×C4×C31—21no—
115C22×C4×C31—22no—
116C22×C4×C31—21no—
117C22×C4×C31—21no—
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—22no—
125C22×C4×C31—22no—
126C22×C4×C31—22no—
127C22×C4×C31—22no—
128C22×C4×C31—21no—
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—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×C31C2×D412yes—
153C22×C4×C31C2×D414yes—
154C22×C4×C31C2×D414yes—
155C22×C4×C31C2×D412yes—
156C22×C4×C31#4112yes—
157C4⋊C4×C31—44no—
158C4⋊C4×C31—48no—
159C4⋊C4×C31—48no—
160C4⋊C4×C31—44no—
161C22⋊C4×C31—48no—
162C22⋊C4×C31—48no—
163C22⋊C4×C31—48no—
164C22⋊C4×C31—48no—
165C22⋊C4×C31—48no—
166C22⋊C4×C31—48no—
167#17121C3111yesmaximal, characteristic, direct product factor, Sylow
168C23×C31—24no—
169C23×C31—24no—
170C23×C31—24no—
171C23×C31—24no—
172C23×C31—24no—
173C23×C31—24no—
174C23×C31—24no—
175C23×C31—42no—
176C23×C31—42no—
177C23×C31—42no—
178C23×C31—42no—
179C23×C31—42no—
180C23×C31—42no—
181C23×C31—42no—
182C23×C31—42no—
183C23×C31—42no—
184C23×C31—42no—
185C23×C31—42no—
186C23×C31—42no—
187C23×C31—42no—
188C23×C31—42no—
189C23×C31—42no—
190C23×C31—42no—
191C23×C31—42no—
192C23×C31—42no—
193C23×C31—42no—
194C23×C31—42no—
195C23×C31—42no—
196C23×C31—42no—
197C23×C31—42no—
198C23×C31—42no—
199C23×C31—42no—
200C23×C31—42no—
201C23×C31—42no—
202C23×C31—42no—
203C23×C31C22×D411yescharacteristic
204C23×C31C22×D411yescharacteristic
205C23×C31C2×#4112yes—
206C23×C31C2×#4111yescharacteristic
207C23×C31H(5,𝔽2)11yescharacteristic
208C23×C31H(5,𝔽2)11yescharacteristic
209C23×C31H(5,𝔽2)12yes—
210C23×C31H(5,𝔽2)12yes—
211C23×C31H(5,𝔽2)11yescharacteristic
212C23×C312−1+411yescharacteristic
213C23×C312−1+412yes—
214D4×C31—416no—
215C2×C4×C31—24no—
216C2×C4×C31—22no—
217C2×C4×C31—24no—
218C2×C4×C31—24no—
219C2×C4×C31—22no—
220C2×C4×C31—22no—
221C2×C4×C31—24no—
222C2×C4×C31—24no—
223C2×C4×C31—24no—
224C2×C4×C31—24no—
225C2×C4×C31—24no—
226C2×C4×C31—24no—
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—41no—
236C2×C4×C31—41no—
237C2×C4×C31—41no—
238C2×C4×C31—41no—
239C2×C4×C31—42no—
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—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×#397C2×C3112yescomplemented
266C2×#396C2×C3112yescomplemented
267#1445C2×C3111yescharacteristic, complemented
268#1444C2×C3111yescharacteristic, complemented
269#1442C2×C3112yescomplemented
270#1439C2×C3112yescomplemented
271#1439C2×C3112yescomplemented
272#1438C2×C3112yes—
273#1438C2×C3111yescharacteristic
274C22×C31—44no—
275C22×C31—44no—
276C22×C31—44no—
277C22×C31—44no—
278C22×C31—44no—
279C22×C31—44no—
280C22×C31—44no—
281C22×C31—82no—
282C22×C31—82no—
283C22×C31—82no—
284C22×C31—82no—
285C22×C31—82no—
286C22×C31—82no—
287C22×C31—82no—
288C22×C31—82no—
289C22×C31#53411yescharacteristic
290C22×C31#53512yes—
291C22×C31#53712yes—
292C22×C31#53711yescharacteristic
293C22×C31#53812yes—
294C22×C31#53811yescharacteristic
295C22×C31#53912yes—
296C22×C31#53912yes—
297C22×C31#54212yes—
298C22×C31D4211yescharacteristic
299C22×C31#54612yes—
300C22×C31#54711yescharacteristic
301C22×C31#54812yes—
302C22×C31#55011yescharacteristic
303C22×C31#55111yescharacteristic
304C22×C31#55112yes—
305C22×C31#56011yescharacteristic
306C22×C31#56012yes—
307C22×C31#56011yescharacteristic
308C22×C31#56211yescharacteristic
309C22×C31#56212yes—
310C22×C31#56312yes—
311C22×C31#56311yescharacteristic
312C4×C31—44no—
313C4×C31—44no—
314C4×C31—42no—
315C4×C31—42no—
316C4×C31—42no—
317C4×C31—82no—
318C4×C31—82no—
319C4×C31—82no—
320C23×D4C22×C3111yescharacteristic
321C22×C4⋊C4C22×C3112yes—
322C22×C4⋊C4C22×C3112yes—
323C22×C4⋊C4C22×C3111yescharacteristic
324C22×C4⋊C4C22×C3111yescharacteristic
325C22×C22⋊C4C22×C3112yes—
326C22×C22⋊C4C22×C3112yes—
327C22×C22⋊C4C22×C3112yes—
328C22×C22⋊C4C22×C3112yes—
329C22×C22⋊C4C22×C3112yes—
330C22×C22⋊C4C22×C3112yes—
331#397—28no—
332#396—28no—
333C2×#95C22×C3112yes—
334C2×#95C22×C3112yes—
335C2×#95C22×C3112yes—
336C2×#95C22×C3112yes—
337C2×#95C22×C3111yescharacteristic
338C2×#95C22×C3112yes—
339C2×#95C22×C3111yescharacteristic
340C2×#95C22×C3112yes—
341C2×#95C22×C3111yescharacteristic
342C2×#95C22×C3111yescharacteristic
343C2×C31—84no—
344C2×C31#267612yes—
345C2×C31#268211yescharacteristic
346C2×C31#269311yescharacteristic
347C2×C31#270512yes—
348C2×C31#272312yes—
349C2×C31#272811yescharacteristic
350C2×C31#273511yescharacteristic
351C2×C31#281511yescharacteristic
352C2×C31#282311yescharacteristic
353C2×C31#282712yes—
354C2×C31#283011yescharacteristic
355C25C23×C3112yes—
356C22×D4—24no—
357C22×D4—22no—
358C22×D4—42no—
359C22×D4—41no—
360C22×D4—41no—
361C23×C4C23×C3112yes—
362C23×C4C23×C3112yes—
363C23×C4C23×C3112yes—
364C23×C4C23×C3111yescharacteristic
365C23×C4C23×C3112yes—
366C23×C4C23×C3112yes—
367C23×C4C23×C3111yescharacteristic
368C23×C4C23×C3111yescharacteristic
369C2×C4⋊C4—24no—
370C2×C4⋊C4—24no—
371C2×C4⋊C4—24no—
372C2×C4⋊C4—41no—
373C2×C4⋊C4—41no—
374C2×C4⋊C4—42no—
375C2×C4⋊C4—42no—
376C2×C4⋊C4—42no—
377C2×C4⋊C4—41no—
378C2×C4⋊C4—42no—
379C2×C4⋊C4D4×C3114yescomplemented
380C2×C22⋊C4—24no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—24no—
386C2×C22⋊C4—24no—
387C2×C22⋊C4—24no—
388C2×C22⋊C4—42no—
389C2×C22⋊C4—42no—
390C2×C22⋊C4—42no—
391C2×C22⋊C4—42no—
392C2×C22⋊C4—42no—
393C2×C22⋊C4—42no—
394C2×C22⋊C4—42no—
395C2×C22⋊C4—42no—
396C2×C22⋊C4—42no—
397C2×C22⋊C4—42no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95—41no—
403#95—42no—
404#95—41no—
405#95—41no—
406#95—42no—
407#95D4×C3114yescomplemented
408C31#1712111yescharacteristic, direct product factor, Sylow
409C24—22no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24#41×C3114yes—
424C24C24×C3111yescharacteristic
425C2×D4—48no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—48no—
429C2×D4—44no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
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—22no—
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×C4C2×D4×C3112yes—
476C22×C4C2×D4×C3114yes—
477C22×C4C2×D4×C3114yes—
478C22×C4C2×D4×C3112yes—
479C22×C4#41×C3112yes—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23C22×D4×C3111yescharacteristic
526C23C22×D4×C3111yescharacteristic
527C23C2×#41×C3112yes—
528C23C2×#41×C3111yescharacteristic
529C23C31×H(5,𝔽2)11yescharacteristic
530C23C31×H(5,𝔽2)11yescharacteristic
531C23C31×H(5,𝔽2)12yes—
532C23C31×H(5,𝔽2)12yes—
533C23C31×H(5,𝔽2)11yescharacteristic
534C23C31×2−1+411yescharacteristic
535C23C31×2−1+412yes—
536D4—416no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
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×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22C31×#53411yescharacteristic
603C22C31×#53512yes—
604C22C31×#53712yes—
605C22C31×#53711yescharacteristic
606C22C31×#53812yes—
607C22C31×#53811yescharacteristic
608C22C31×#53912yes—
609C22C31×#53912yes—
610C22C31×#54212yes—
611C22D42×C3111yescharacteristic
612C22C31×#54612yes—
613C22C31×#54711yescharacteristic
614C22C31×#54812yes—
615C22C31×#55011yescharacteristic
616C22C31×#55111yescharacteristic
617C22C31×#55112yes—
618C22C31×#56011yescharacteristic
619C22C31×#56012yes—
620C22C31×#56011yescharacteristic
621C22C31×#56211yescharacteristic
622C22C31×#56212yes—
623C22C31×#56312yes—
624C22C31×#56311yescharacteristic
625C4—44no—
626C4—44no—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—82no—
631C4—82no—
632C4—82no—
633C2—84no—
634C2C31×#267612yes—
635C2C31×#268211yescharacteristic
636C2C31×#269311yescharacteristic
637C2C31×#270512yes—
638C2C31×#272312yes—
639C2C31×#272811yescharacteristic
640C2C31×#273511yescharacteristic
641C2C31×#281511yescharacteristic
642C2C31×#282311yescharacteristic
643C2C31×#282712yes—
644C2C31×#283011yescharacteristic
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
1. C2×C31×#397C2semidirect
2. C2×C31×#396C2semidirect
3. C31×#1445C2semidirect
4. C31×#1444C2semidirect
5. C31×#1442C2semidirect
6. C31×#1439C2semidirect
7. C31×#1439C2semidirect
57. C2×C4⋊C4×C31D4semidirect
85. C31×#95D4semidirect
167. #17121C31direct (N × H)
265. C2×#397C2×C31semidirect
266. C2×#396C2×C31semidirect
267. #1445C2×C31semidirect
268. #1444C2×C31semidirect
269. #1442C2×C31semidirect
270. #1439C2×C31semidirect
271. #1439C2×C31semidirect
379. C2×C4⋊C4D4×C31semidirect
407. #95D4×C31semidirect
408. C31#17121direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3972
C2×C31×#3962
C31×#14452
C31×#14442
C31×#14422
C31×#14392
C31×#14382
#1712131