Finite Group Databasebeta

← C31×#17109 #21473868

Subgroups — C31×#17109 #21473868

Order 7936.

1634 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#397C212yesmaximal, complemented
2C2×C31×#396C212yesmaximal, complemented
3C2×C31×#394C212yesmaximal
4C31×#1444C211yesmaximal, characteristic, complemented
5C31×#1444C211yesmaximal, characteristic, complemented
6C31×#1441C212yesmaximal, complemented
7C31×#1439C212yesmaximal, complemented
8C31×#1439C212yesmaximal, complemented
9C31×#1438C211yesmaximal, characteristic
10C23×D4×C31C2211yescharacteristic
11C22×C4⋊C4×C31C2212yes—
12C22×C4⋊C4×C31C2211yescharacteristic
13C22×C4⋊C4×C31C2212yes—
14C22×C4⋊C4×C31C2212yes—
15C22×C4⋊C4×C31C2211yescharacteristic
16C22×C22⋊C4×C31C2212yes—
17C22×C22⋊C4×C31C2212yes—
18C22×C22⋊C4×C31C2212yes—
19C22×C22⋊C4×C31C2212yes—
20C22×C22⋊C4×C31C2212yes—
21C22×C22⋊C4×C31C2212yes—
22C31×#397—28no—
23C31×#396—28no—
24C31×#394—28no—
25C2×C31×#95C2212yes—
26C2×C31×#95C2212yes—
27C2×C31×#95C2212yes—
28C2×C31×#95C2212yes—
29C2×C31×#95C2211yescharacteristic
30C2×C31×#95C2211yescharacteristic
31C2×C31×#95C2211yescharacteristic
32C2×C31×#95C2212yes—
33C2×C31×#95C2211yescharacteristic
34C25×C31C2312yes—
35C22×D4×C31—24no—
36C22×D4×C31—22no—
37C22×D4×C31—41no—
38C22×D4×C31—41no—
39C22×D4×C31—42no—
40C23×C4×C31C2312yes—
41C23×C4×C31C2312yes—
42C23×C4×C31C2311yescharacteristic
43C23×C4×C31C2312yes—
44C23×C4×C31C2312yes—
45C23×C4×C31C2312yes—
46C23×C4×C31C2311yescharacteristic
47C23×C4×C31C2311yescharacteristic
48C2×C4⋊C4×C31—24no—
49C2×C4⋊C4×C31—24no—
50C2×C4⋊C4×C31—24no—
51C2×C4⋊C4×C31—24no—
52C2×C4⋊C4×C31—24no—
53C2×C4⋊C4×C31—24no—
54C2×C4⋊C4×C31—41no—
55C2×C4⋊C4×C31—42no—
56C2×C4⋊C4×C31—42no—
57C2×C4⋊C4×C31—42no—
58C2×C4⋊C4×C31—42no—
59C2×C4⋊C4×C31—41no—
60C2×C4⋊C4×C31—42no—
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—24no—
67C2×C22⋊C4×C31—24no—
68C2×C22⋊C4×C31—24no—
69C2×C22⋊C4×C31—24no—
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—
76C2×C22⋊C4×C31—42no—
77C2×C22⋊C4×C31—42no—
78C2×C22⋊C4×C31D418yes—
79C31×#95—24no—
80C31×#95—24no—
81C31×#95—24no—
82C31×#95—24no—
83C31×#95—22no—
84C31×#95—42no—
85C31×#95—41no—
86C31×#95—41no—
87C31×#95D414yescomplemented
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—22no—
101C24×C31—22no—
102C24×C31C2×D414yes—
103C24×C31C2411yescharacteristic
104C2×D4×C31—48no—
105C2×D4×C31—44no—
106C2×D4×C31—44no—
107C2×D4×C31—48no—
108C2×D4×C31—44no—
109C22×C4×C31—22no—
110C22×C4×C31—22no—
111C22×C4×C31—22no—
112C22×C4×C31—22no—
113C22×C4×C31—22no—
114C22×C4×C31—22no—
115C22×C4×C31—22no—
116C22×C4×C31—22no—
117C22×C4×C31—22no—
118C22×C4×C31—22no—
119C22×C4×C31—21no—
120C22×C4×C31—22no—
121C22×C4×C31—21no—
122C22×C4×C31—21no—
123C22×C4×C31—21no—
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—21no—
136C22×C4×C31—21no—
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×C31—22no—
153C22×C4×C31C2×D414yes—
154C22×C4×C31C2×D414yes—
155C22×C4×C31C2×D414yes—
156C22×C4×C31C2×D412yes—
157C22×C4×C31#4112yes—
158C22×C4×C31#4112yes—
159C4⋊C4×C31—44no—
160C4⋊C4×C31—48no—
161C4⋊C4×C31—48no—
162C4⋊C4×C31—48no—
163C4⋊C4×C31—44no—
164C22⋊C4×C31—48no—
165C22⋊C4×C31—48no—
166C22⋊C4×C31—48no—
167C22⋊C4×C31—48no—
168C22⋊C4×C31—48no—
169C22⋊C4×C31—48no—
170#17109C3111yesmaximal, characteristic, direct product factor, Sylow
171C23×C31—24no—
172C23×C31—24no—
173C23×C31—24no—
174C23×C31—24no—
175C23×C31—24no—
176C23×C31—24no—
177C23×C31—24no—
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×C31—42no—
204C23×C31—42no—
205C23×C31—42no—
206C23×C31C22×D412yes—
207C23×C31C22×D411yescharacteristic
208C23×C31C2×#4111yescharacteristic
209C23×C31C2×#4111yescharacteristic
210C23×C31H(5,𝔽2)11yescharacteristic
211C23×C31H(5,𝔽2)12yes—
212C23×C31H(5,𝔽2)11yescharacteristic
213C23×C31H(5,𝔽2)12yes—
214C23×C312−1+411yescharacteristic
215C23×C312−1+412yes—
216C23×C312−1+411yescharacteristic
217D4×C31—416no—
218C2×C4×C31—24no—
219C2×C4×C31—24no—
220C2×C4×C31—24no—
221C2×C4×C31—22no—
222C2×C4×C31—24no—
223C2×C4×C31—22no—
224C2×C4×C31—22no—
225C2×C4×C31—24no—
226C2×C4×C31—24no—
227C2×C4×C31—24no—
228C2×C4×C31—24no—
229C2×C4×C31—24no—
230C2×C4×C31—24no—
231C2×C4×C31—24no—
232C2×C4×C31—24no—
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—41no—
242C2×C4×C31—41no—
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—41no—
250C2×C4×C31—41no—
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×#397C2×C3112yescomplemented
269C2×#396C2×C3112yescomplemented
270C2×#394C2×C3112yes—
271#1444C2×C3111yescharacteristic, complemented
272#1444C2×C3111yescharacteristic, complemented
273#1441C2×C3112yescomplemented
274#1439C2×C3112yescomplemented
275#1439C2×C3112yescomplemented
276#1438C2×C3111yescharacteristic
277C22×C31—44no—
278C22×C31—44no—
279C22×C31—44no—
280C22×C31—44no—
281C22×C31—44no—
282C22×C31—44no—
283C22×C31—44no—
284C22×C31—82no—
285C22×C31—82no—
286C22×C31—82no—
287C22×C31—82no—
288C22×C31—82no—
289C22×C31—82no—
290C22×C31—82no—
291C22×C31—82no—
292C22×C31#53412yes—
293C22×C31#53612yes—
294C22×C31#53712yes—
295C22×C31#53711yescharacteristic
296C22×C31#53712yes—
297C22×C31#53812yes—
298C22×C31#53811yescharacteristic
299C22×C31#53911yescharacteristic
300C22×C31#54212yes—
301C22×C31D4211yescharacteristic
302C22×C31D4212yes—
303C22×C31#54612yes—
304C22×C31#54711yescharacteristic
305C22×C31#54712yes—
306C22×C31#55011yescharacteristic
307C22×C31#55611yescharacteristic
308C22×C31#56011yescharacteristic
309C22×C31#56011yescharacteristic
310C22×C31#56212yes—
311C22×C31#56212yes—
312C22×C31#56311yescharacteristic
313C22×C31#56312yes—
314C22×C31#56311yescharacteristic
315C4×C31—42no—
316C4×C31—44no—
317C4×C31—42no—
318C4×C31—44no—
319C4×C31—42no—
320C4×C31—44no—
321C4×C31—82no—
322C4×C31—82no—
323C23×D4C22×C3111yescharacteristic
324C22×C4⋊C4C22×C3112yes—
325C22×C4⋊C4C22×C3111yescharacteristic
326C22×C4⋊C4C22×C3112yes—
327C22×C4⋊C4C22×C3112yes—
328C22×C4⋊C4C22×C3111yescharacteristic
329C22×C22⋊C4C22×C3112yes—
330C22×C22⋊C4C22×C3112yes—
331C22×C22⋊C4C22×C3112yes—
332C22×C22⋊C4C22×C3112yes—
333C22×C22⋊C4C22×C3112yes—
334C22×C22⋊C4C22×C3112yes—
335#397—28no—
336#396—28no—
337#394—28no—
338C2×#95C22×C3112yes—
339C2×#95C22×C3112yes—
340C2×#95C22×C3112yes—
341C2×#95C22×C3112yes—
342C2×#95C22×C3111yescharacteristic
343C2×#95C22×C3111yescharacteristic
344C2×#95C22×C3111yescharacteristic
345C2×#95C22×C3112yes—
346C2×#95C22×C3111yescharacteristic
347C2×C31—84no—
348C2×C31#266811yescharacteristic
349C2×C31#267212yes—
350C2×C31#267511yescharacteristic
351C2×C31#268011yescharacteristic
352C2×C31#268212yes—
353C2×C31#271512yes—
354C2×C31#272711yescharacteristic
355C2×C31#281412yes—
356C2×C31#282311yescharacteristic
357C2×C31#282611yescharacteristic
358C2×C31#283011yescharacteristic
359C25C23×C3112yes—
360C22×D4—24no—
361C22×D4—22no—
362C22×D4—41no—
363C22×D4—41no—
364C22×D4—42no—
365C23×C4C23×C3112yes—
366C23×C4C23×C3112yes—
367C23×C4C23×C3111yescharacteristic
368C23×C4C23×C3112yes—
369C23×C4C23×C3112yes—
370C23×C4C23×C3112yes—
371C23×C4C23×C3111yescharacteristic
372C23×C4C23×C3111yescharacteristic
373C2×C4⋊C4—24no—
374C2×C4⋊C4—24no—
375C2×C4⋊C4—24no—
376C2×C4⋊C4—24no—
377C2×C4⋊C4—24no—
378C2×C4⋊C4—24no—
379C2×C4⋊C4—41no—
380C2×C4⋊C4—42no—
381C2×C4⋊C4—42no—
382C2×C4⋊C4—42no—
383C2×C4⋊C4—42no—
384C2×C4⋊C4—41no—
385C2×C4⋊C4—42no—
386C2×C22⋊C4—24no—
387C2×C22⋊C4—24no—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—24no—
390C2×C22⋊C4—24no—
391C2×C22⋊C4—24no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—24no—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—42no—
396C2×C22⋊C4—42no—
397C2×C22⋊C4—42no—
398C2×C22⋊C4—42no—
399C2×C22⋊C4—42no—
400C2×C22⋊C4—42no—
401C2×C22⋊C4—42no—
402C2×C22⋊C4—42no—
403C2×C22⋊C4D4×C3118yes—
404#95—24no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95—22no—
409#95—42no—
410#95—41no—
411#95—41no—
412#95D4×C3114yescomplemented
413C31#1710911yescharacteristic, direct product factor, Sylow
414C24—22no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24C2×D4×C3114yes—
429C24C24×C3111yescharacteristic
430C2×D4—48no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—48no—
434C2×D4—44no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
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—21no—
462C22×C4—21no—
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×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4C2×D4×C3114yes—
480C22×C4C2×D4×C3114yes—
481C22×C4C2×D4×C3114yes—
482C22×C4C2×D4×C3112yes—
483C22×C4#41×C3112yes—
484C22×C4#41×C3112yes—
485C4⋊C4—44no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
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—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23C22×D4×C3112yes—
532C23C22×D4×C3111yescharacteristic
533C23C2×#41×C3111yescharacteristic
534C23C2×#41×C3111yescharacteristic
535C23C31×H(5,𝔽2)11yescharacteristic
536C23C31×H(5,𝔽2)12yes—
537C23C31×H(5,𝔽2)11yescharacteristic
538C23C31×H(5,𝔽2)12yes—
539C23C31×2−1+411yescharacteristic
540C23C31×2−1+412yes—
541C23C31×2−1+411yescharacteristic
542D4—416no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
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—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22C31×#53412yes—
609C22C31×#53612yes—
610C22C31×#53712yes—
611C22C31×#53711yescharacteristic
612C22C31×#53712yes—
613C22C31×#53812yes—
614C22C31×#53811yescharacteristic
615C22C31×#53911yescharacteristic
616C22C31×#54212yes—
617C22D42×C3111yescharacteristic
618C22D42×C3112yes—
619C22C31×#54612yes—
620C22C31×#54711yescharacteristic
621C22C31×#54712yes—
622C22C31×#55011yescharacteristic
623C22C31×#55611yescharacteristic
624C22C31×#56011yescharacteristic
625C22C31×#56011yescharacteristic
626C22C31×#56212yes—
627C22C31×#56212yes—
628C22C31×#56311yescharacteristic
629C22C31×#56312yes—
630C22C31×#56311yescharacteristic
631C4—42no—
632C4—42no—
633C4—44no—
634C4—44no—
635C4—42no—
636C4—44no—
637C4—82no—
638C4—82no—
639C2—84no—
640C2C31×#266811yescharacteristic
641C2C31×#267212yes—
642C2C31×#267511yescharacteristic
643C2C31×#268011yescharacteristic
644C2C31×#268212yes—
645C2C31×#271512yes—
646C2C31×#272711yescharacteristic
647C2C31×#281412yes—
648C2C31×#282311yescharacteristic
649C2C31×#282611yescharacteristic
650C2C31×#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
4. C31×#1444C2semidirect
5. C31×#1444C2semidirect
6. C31×#1441C2semidirect
7. C31×#1439C2semidirect
8. C31×#1439C2semidirect
87. C31×#95D4semidirect
170. #17109C31direct (N × H)
268. C2×#397C2×C31semidirect
269. C2×#396C2×C31semidirect
271. #1444C2×C31semidirect
272. #1444C2×C31semidirect
273. #1441C2×C31semidirect
274. #1439C2×C31semidirect
275. #1439C2×C31semidirect
412. #95D4×C31semidirect
413. C31#17109direct (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
C2×C31×#3942
C31×#14442
C31×#14412
C31×#14392
C31×#14382
#1710931