Finite Group Databasebeta

← C31×#16693 #21473452

Subgroups — C31×#16693 #21473452

Order 7936.

1514 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#390C211yesmaximal, characteristic
2C2×C31×#383C211yesmaximal, characteristic, complemented
3C31×#1444C211yesmaximal, characteristic, complemented
4C31×#1442C211yesmaximal, characteristic, complemented
5C31×#1440C212yesmaximal, complemented
6C31×#1440C212yesmaximal, complemented
7C31×#1439C212yesmaximal, complemented
8C31×#1439C212yesmaximal, complemented
9C31×#1438C211yesmaximal, characteristic
10C31×#1438C211yesmaximal, characteristic
11C31×#1434C211yesmaximal, characteristic, complemented
12C23×D4×C31C2211yescharacteristic
13C22×C4⋊C4×C31C2211yescharacteristic
14C22×C4⋊C4×C31C2212yes—
15C22×C22⋊C4×C31C2212yes—
16C22×C22⋊C4×C31C2212yes—
17C22×C22⋊C4×C31C2212yes—
18C22×C22⋊C4×C31C2212yes—
19C22×C22⋊C4×C31C2212yes—
20C22×C22⋊C4×C31C2212yes—
21C22×C42×C31C2211yescharacteristic
22C31×#390—24no—
23C31×#383—24no—
24C2×C31×#95C2211yescharacteristic
25C2×C31×#95C2211yescharacteristic
26C2×C31×#95C2212yes—
27C2×C31×#95C2211yescharacteristic
28C2×C31×#95C2211yescharacteristic
29C2×C31×#95C2211yescharacteristic
30C2×C31×#95C2211yescharacteristic
31C2×C31×#95C2211yescharacteristic
32C2×C31×#95C2211yescharacteristic
33C2×C31×#95C2212yes—
34C2×C31×#95C2212yes—
35C2×C31×#95C2212yes—
36C2×C31×#95C2212yes—
37C25×C31C2312yes—
38C22×D4×C31—24no—
39C22×D4×C31—22no—
40C22×D4×C31—22no—
41C22×D4×C31—22no—
42C22×D4×C31—42no—
43C23×C4×C31C2311yescharacteristic
44C23×C4×C31C2312yes—
45C23×C4×C31C2311yescharacteristic
46C23×C4×C31C2311yescharacteristic
47C23×C4×C31C2311yescharacteristic
48C23×C4×C31C2312yes—
49C23×C4×C31C2311yescharacteristic
50C23×C4×C31C2311yescharacteristic
51C23×C4×C31C2312yes—
52C23×C4×C31C2311yescharacteristic
53C2×C4⋊C4×C31—22no—
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—24no—
63C2×C22⋊C4×C31—24no—
64C2×C22⋊C4×C31—42no—
65C2×C22⋊C4×C31—42no—
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—
76C2×C42×C31—22no—
77C2×C42×C31—22no—
78C2×C42×C31D414yes—
79C31×#95—22no—
80C31×#95—22no—
81C31×#95—22no—
82C31×#95—22no—
83C31×#95—22no—
84C31×#95—22no—
85C31×#95—22no—
86C31×#95—24no—
87C31×#95—22no—
88C31×#95—42no—
89C31×#95—42no—
90C31×#95—42no—
91C31×#95—42no—
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×C31—22no—
103C24×C31—22no—
104C24×C31—22no—
105C24×C31—22no—
106C24×C31#4114yes—
107C24×C31C2411yescharacteristic
108C2×D4×C31—28no—
109C2×D4×C31—48no—
110C2×D4×C31—48no—
111C2×D4×C31—44no—
112C2×D4×C31—44no—
113C22×C4×C31—22no—
114C22×C4×C31—22no—
115C22×C4×C31—22no—
116C22×C4×C31—22no—
117C22×C4×C31—21no—
118C22×C4×C31—21no—
119C22×C4×C31—21no—
120C22×C4×C31—21no—
121C22×C4×C31—22no—
122C22×C4×C31—21no—
123C22×C4×C31—22no—
124C22×C4×C31—22no—
125C22×C4×C31—21no—
126C22×C4×C31—21no—
127C22×C4×C31—22no—
128C22×C4×C31—22no—
129C22×C4×C31—22no—
130C22×C4×C31—21no—
131C22×C4×C31—21no—
132C22×C4×C31—21no—
133C22×C4×C31—21no—
134C22×C4×C31—21no—
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—21no—
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—21no—
153C22×C4×C31—21no—
154C22×C4×C31—21no—
155C22×C4×C31—21no—
156C22×C4×C31—22no—
157C22×C4×C31—22no—
158C22×C4×C31—22no—
159C22×C4×C31—22no—
160C22×C4×C31—22no—
161C22×C4×C31—22no—
162C22×C4×C31—21no—
163C22×C4×C31—21no—
164C22×C4×C31—21no—
165C22×C4×C31C2×D412yes—
166C22×C4×C31C2×D412yes—
167C22×C4×C31C2×D412yes—
168C22×C4×C31#4114yes—
169C22×C4×C31#4112yes—
170C22×C4×C31#4112yes—
171C4⋊C4×C31—44no—
172C4⋊C4×C31—48no—
173C22⋊C4×C31—48no—
174C22⋊C4×C31—48no—
175C22⋊C4×C31—48no—
176C22⋊C4×C31—48no—
177C22⋊C4×C31—48no—
178C22⋊C4×C31—48no—
179C42×C31—28no—
180#16693C3111yesmaximal, characteristic, direct product factor, Sylow
181C23×C31—24no—
182C23×C31—24no—
183C23×C31—24no—
184C23×C31—24no—
185C23×C31—24no—
186C23×C31—24no—
187C23×C31—24no—
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×C31—42no—
207C23×C31—42no—
208C23×C31—42no—
209C23×C31—42no—
210C23×C31—42no—
211C23×C31—42no—
212C23×C31—42no—
213C23×C31—42no—
214C23×C31—42no—
215C23×C31—42no—
216C23×C31C22×D411yescharacteristic
217C23×C31C2×#4111yescharacteristic
218C23×C31C2×#4111yescharacteristic
219C23×C31C2×#4112yes—
220C23×C31C2×#4112yes—
221C23×C31H(5,𝔽2)12yes—
222C23×C31H(5,𝔽2)12yes—
223C23×C31H(5,𝔽2)11yescharacteristic
224C23×C31H(5,𝔽2)11yescharacteristic
225C23×C31H(5,𝔽2)11yescharacteristic
226C23×C312−1+411yescharacteristic
227D4×C31—416no—
228C2×C4×C31—22no—
229C2×C4×C31—22no—
230C2×C4×C31—24no—
231C2×C4×C31—24no—
232C2×C4×C31—22no—
233C2×C4×C31—22no—
234C2×C4×C31—22no—
235C2×C4×C31—22no—
236C2×C4×C31—22no—
237C2×C4×C31—22no—
238C2×C4×C31—24no—
239C2×C4×C31—24no—
240C2×C4×C31—24no—
241C2×C4×C31—42no—
242C2×C4×C31—41no—
243C2×C4×C31—41no—
244C2×C4×C31—42no—
245C2×C4×C31—41no—
246C2×C4×C31—41no—
247C2×C4×C31—42no—
248C2×C4×C31—42no—
249C2×C4×C31—42no—
250C2×C4×C31—42no—
251C2×C4×C31—41no—
252C2×C4×C31—41no—
253C2×C4×C31—41no—
254C2×C4×C31—41no—
255C2×C4×C31—42no—
256C2×C4×C31—42no—
257C2×C4×C31—42no—
258C2×C4×C31—41no—
259C2×C4×C31—41no—
260C2×C4×C31—41no—
261C2×C4×C31—41no—
262C2×C4×C31—41no—
263C2×C4×C31—41no—
264C2×C4×C31—41no—
265C2×C4×C31—41no—
266C2×C4×C31—42no—
267C2×C4×C31—42no—
268C2×C4×C31—42no—
269C2×C4×C31—42no—
270C2×C4×C31—42no—
271C2×C4×C31—42no—
272C2×C4×C31—42no—
273C2×C4×C31—42no—
274C2×C4×C31—42no—
275C2×C4×C31—42no—
276C2×C4×C31—42no—
277C2×C4×C31—42no—
278C2×C4×C31—42no—
279C2×C4×C31—42no—
280C2×C4×C31—42no—
281C2×C4×C31—42no—
282C2×C4×C31—42no—
283C2×C4×C31—42no—
284C2×C4×C31—42no—
285C2×C4×C31#12314yes—
286C2×#390C2×C3111yescharacteristic
287C2×#383C2×C3111yescharacteristic, complemented
288#1444C2×C3111yescharacteristic, complemented
289#1442C2×C3111yescharacteristic, complemented
290#1440C2×C3112yescomplemented
291#1440C2×C3112yescomplemented
292#1439C2×C3112yescomplemented
293#1439C2×C3112yescomplemented
294#1438C2×C3111yescharacteristic
295#1438C2×C3111yescharacteristic
296#1434C2×C3111yescharacteristic, complemented
297C22×C31—44no—
298C22×C31—44no—
299C22×C31—44no—
300C22×C31—44no—
301C22×C31—44no—
302C22×C31—44no—
303C22×C31—44no—
304C22×C31—82no—
305C22×C31—82no—
306C22×C31—82no—
307C22×C31—82no—
308C22×C31—82no—
309C22×C31—82no—
310C22×C31—82no—
311C22×C31—82no—
312C22×C31C2×#12311yescharacteristic
313C22×C31#52511yescharacteristic
314C22×C31#53211yescharacteristic
315C22×C31#53511yescharacteristic
316C22×C31#53511yescharacteristic
317C22×C31#53511yescharacteristic
318C22×C31#53711yescharacteristic
319C22×C31#53812yes—
320C22×C31#53812yes—
321C22×C31#53912yes—
322C22×C31#54011yescharacteristic
323C22×C31#54011yescharacteristic
324C22×C31#54211yescharacteristic
325C22×C31#54211yescharacteristic
326C22×C31#54212yes—
327C22×C31#55112yes—
328C22×C31#55112yes—
329C22×C31#55112yes—
330C22×C31#55312yes—
331C22×C31#55312yes—
332C22×C31#55512yes—
333C22×C31#56012yes—
334C22×C31#56012yes—
335C4×C31—24no—
336C4×C31—42no—
337C4×C31—44no—
338C4×C31—42no—
339C4×C31—82no—
340C4×C31—81no—
341C4×C31—81no—
342C4×C31—81no—
343C4×C31—81no—
344C4×C31—82no—
345C23×D4C22×C3111yescharacteristic
346C22×C4⋊C4C22×C3111yescharacteristic
347C22×C4⋊C4C22×C3112yes—
348C22×C22⋊C4C22×C3112yes—
349C22×C22⋊C4C22×C3112yes—
350C22×C22⋊C4C22×C3112yes—
351C22×C22⋊C4C22×C3112yes—
352C22×C22⋊C4C22×C3112yes—
353C22×C22⋊C4C22×C3112yes—
354C22×C42C22×C3111yescharacteristic
355#390—24no—
356#383—24no—
357C2×#95C22×C3111yescharacteristic
358C2×#95C22×C3111yescharacteristic
359C2×#95C22×C3112yes—
360C2×#95C22×C3111yescharacteristic
361C2×#95C22×C3111yescharacteristic
362C2×#95C22×C3111yescharacteristic
363C2×#95C22×C3111yescharacteristic
364C2×#95C22×C3111yescharacteristic
365C2×#95C22×C3111yescharacteristic
366C2×#95C22×C3112yes—
367C2×#95C22×C3112yes—
368C2×#95C22×C3112yes—
369C2×#95C22×C3112yes—
370C2×C31—84no—
371C2×C31#248211yescharacteristic
372C2×C31#250011yescharacteristic
373C2×C31#260311yescharacteristic
374C2×C31#262411yescharacteristic
375C2×C31#263611yescharacteristic
376C2×C31#266211yescharacteristic
377C2×C31#273712yes—
378C2×C31#274912yes—
379C2×C31#275612yes—
380C2×C31#275712yes—
381C2×C31#281811yescharacteristic
382C25C23×C3112yes—
383C22×D4—24no—
384C22×D4—22no—
385C22×D4—22no—
386C22×D4—22no—
387C22×D4—42no—
388C23×C4C23×C3111yescharacteristic
389C23×C4C23×C3112yes—
390C23×C4C23×C3111yescharacteristic
391C23×C4C23×C3111yescharacteristic
392C23×C4C23×C3111yescharacteristic
393C23×C4C23×C3112yes—
394C23×C4C23×C3111yescharacteristic
395C23×C4C23×C3111yescharacteristic
396C23×C4C23×C3112yes—
397C23×C4C23×C3111yescharacteristic
398C2×C4⋊C4—22no—
399C2×C4⋊C4—42no—
400C2×C4⋊C4—42no—
401C2×C4⋊C4—42no—
402C2×C4⋊C4—42no—
403C2×C22⋊C4—24no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—24no—
406C2×C22⋊C4—24no—
407C2×C22⋊C4—24no—
408C2×C22⋊C4—24no—
409C2×C22⋊C4—42no—
410C2×C22⋊C4—42no—
411C2×C22⋊C4—42no—
412C2×C22⋊C4—42no—
413C2×C22⋊C4—42no—
414C2×C22⋊C4—42no—
415C2×C22⋊C4—42no—
416C2×C22⋊C4—42no—
417C2×C22⋊C4—42no—
418C2×C22⋊C4—42no—
419C2×C22⋊C4—42no—
420C2×C22⋊C4—42no—
421C2×C42—22no—
422C2×C42—22no—
423C2×C42D4×C3114yes—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—24no—
432#95—22no—
433#95—42no—
434#95—42no—
435#95—42no—
436#95—42no—
437C31#1669311yescharacteristic, direct product factor, Sylow
438C24—22no—
439C24—22no—
440C24—22no—
441C24—22no—
442C24—22no—
443C24—22no—
444C24—22no—
445C24—22no—
446C24—22no—
447C24—22no—
448C24—22no—
449C24—22no—
450C24—22no—
451C24—22no—
452C24#41×C3114yes—
453C24C24×C3111yescharacteristic
454C2×D4—28no—
455C2×D4—48no—
456C2×D4—48no—
457C2×D4—44no—
458C2×D4—44no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4C2×D4×C3112yes—
512C22×C4C2×D4×C3112yes—
513C22×C4C2×D4×C3112yes—
514C22×C4#41×C3114yes—
515C22×C4#41×C3112yes—
516C22×C4#41×C3112yes—
517C4⋊C4—44no—
518C4⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C42—28no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23C22×D4×C3111yescharacteristic
562C23C2×#41×C3111yescharacteristic
563C23C2×#41×C3111yescharacteristic
564C23C2×#41×C3112yes—
565C23C2×#41×C3112yes—
566C23C31×H(5,𝔽2)12yes—
567C23C31×H(5,𝔽2)12yes—
568C23C31×H(5,𝔽2)11yescharacteristic
569C23C31×H(5,𝔽2)11yescharacteristic
570C23C31×H(5,𝔽2)11yescharacteristic
571C23C31×2−1+411yescharacteristic
572D4—416no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4C31×#12314yes—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22C2×C31×#12311yescharacteristic
647C22C31×#52511yescharacteristic
648C22C31×#53211yescharacteristic
649C22C31×#53511yescharacteristic
650C22C31×#53511yescharacteristic
651C22C31×#53511yescharacteristic
652C22C31×#53711yescharacteristic
653C22C31×#53812yes—
654C22C31×#53812yes—
655C22C31×#53912yes—
656C22C31×#54011yescharacteristic
657C22C31×#54011yescharacteristic
658C22C31×#54211yescharacteristic
659C22C31×#54211yescharacteristic
660C22C31×#54212yes—
661C22C31×#55112yes—
662C22C31×#55112yes—
663C22C31×#55112yes—
664C22C31×#55312yes—
665C22C31×#55312yes—
666C22C31×#55512yes—
667C22C31×#56012yes—
668C22C31×#56012yes—
669C4—24no—
670C4—42no—
671C4—44no—
672C4—42no—
673C4—81no—
674C4—81no—
675C4—81no—
676C4—82no—
677C4—82no—
678C4—81no—
679C2—84no—
680C2C31×#248211yescharacteristic
681C2C31×#250011yescharacteristic
682C2C31×#260311yescharacteristic
683C2C31×#262411yescharacteristic
684C2C31×#263611yescharacteristic
685C2C31×#266211yescharacteristic
686C2C31×#273712yes—
687C2C31×#274912yes—
688C2C31×#275612yes—
689C2C31×#275712yes—
690C2C31×#281811yescharacteristic
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
2. C2×C31×#383C2semidirect
3. C31×#1444C2semidirect
4. C31×#1442C2semidirect
5. C31×#1440C2semidirect
6. C31×#1440C2semidirect
7. C31×#1439C2semidirect
8. C31×#1439C2semidirect
11. C31×#1434C2semidirect
180. #16693C31direct (N × H)
287. C2×#383C2×C31semidirect
288. #1444C2×C31semidirect
289. #1442C2×C31semidirect
290. #1440C2×C31semidirect
291. #1440C2×C31semidirect
292. #1439C2×C31semidirect
293. #1439C2×C31semidirect
296. #1434C2×C31semidirect
437. C31#16693direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3902
C2×C31×#3832
C31×#14442
C31×#14422
C31×#14402
C31×#14392
C31×#14382
C31×#14342
#1669331