Finite Group Databasebeta

← C31×#16363 #21473122

Subgroups — C31×#16363 #21473122

Order 7936.

2394 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×#121C211yesmaximal, characteristic
2C2×C31×#399C212yesmaximal, complemented
3C2×C31×#394C214yesmaximal, complemented
4C2×C31×#380C211yesmaximal, characteristic, complemented
5C31×#1444C212yesmaximal, complemented
6C31×#1439C212yesmaximal, complemented
7C31×#1438C212yesmaximal, complemented
8C31×C22⋊#95C211yesmaximal, characteristic, complemented
9C23×D4×C31C2212yes—
10C23×D4×C31C2211yescharacteristic
11C24×C4×C31C2211yescharacteristic
12C2×C31×#121—24no—
13C2×C31×#121—28no—
14C22×C4⋊C4×C31C2211yescharacteristic
15C22×C4⋊C4×C31C2211yescharacteristic
16C22×C4⋊C4×C31C2214yes—
17C22×C4⋊C4×C31C2211yescharacteristic
18C22×C22⋊C4×C31C2212yes—
19C22×C22⋊C4×C31C2214yes—
20C22×C22⋊C4×C31C2212yes—
21C22×C22⋊C4×C31C2214yes—
22C22×C22⋊C4×C31C2212yes—
23C31×#399—28no—
24C31×#394—216no—
25C31×#380—24no—
26C2×C31×#95C2214yes—
27C2×C31×#95C2212yes—
28C2×C31×#95C2212yes—
29C2×C31×#95C2212yes—
30C25×C31C2312yes—
31C25×C31C2311yescharacteristic
32C22×D4×C31—24no—
33C22×D4×C31—24no—
34C22×D4×C31—24no—
35C22×D4×C31—28no—
36C22×D4×C31—24no—
37C22×D4×C31—22no—
38C22×D4×C31—24no—
39C22×D4×C31—42no—
40C22×D4×C31—44no—
41C23×C4×C31—24no—
42C23×C4×C31—24no—
43C23×C4×C31—42no—
44C23×C4×C31D414yes—
45C23×C4×C31C2312yes—
46C23×C4×C31C2312yes—
47C23×C4×C31C2314yes—
48C23×C4×C31C2312yes—
49C23×C4×C31C2311yescharacteristic
50C23×C4×C31C2311yescharacteristic
51C31×#121—232no—
52C2×C4⋊C4×C31—24no—
53C2×C4⋊C4×C31—22no—
54C2×C4⋊C4×C31—28no—
55C2×C4⋊C4×C31—24no—
56C2×C4⋊C4×C31—22no—
57C2×C4⋊C4×C31—22no—
58C2×C4⋊C4×C31—28no—
59C2×C4⋊C4×C31—44no—
60C2×C4⋊C4×C31—42no—
61C2×C22⋊C4×C31—24no—
62C2×C22⋊C4×C31—24no—
63C2×C22⋊C4×C31—28no—
64C2×C22⋊C4×C31—24no—
65C2×C22⋊C4×C31—28no—
66C2×C22⋊C4×C31—28no—
67C2×C22⋊C4×C31—28no—
68C2×C22⋊C4×C31—42no—
69C2×C22⋊C4×C31—42no—
70C2×C22⋊C4×C31—44no—
71C2×C22⋊C4×C31—44no—
72C2×C22⋊C4×C31—44no—
73C2×C22⋊C4×C31D418yes—
74C2×C22⋊C4×C31D418yescomplemented
75C31×#95—24no—
76C31×#95—24no—
77C31×#95—28no—
78C31×#95—24no—
79C24×C31—22no—
80C24×C31—22no—
81C24×C31—21no—
82C24×C31—22no—
83C24×C31—22no—
84C24×C31—22no—
85C24×C31—21no—
86C24×C31—24no—
87C24×C31—22no—
88C24×C31—22no—
89C24×C31—24no—
90C24×C31—22no—
91C24×C31—24no—
92C24×C31—24no—
93C24×C31—22no—
94C24×C31—22no—
95C24×C31—22no—
96C24×C31C2×D414yes—
97C24×C31C2×D414yes—
98C24×C31C2×D412yes—
99C24×C31C2411yescharacteristic
100C2×D4×C31—28no—
101C2×D4×C31—216no—
102C2×D4×C31—416no—
103C2×D4×C31—416no—
104C2×D4×C31—44no—
105C2×D4×C31—416no—
106C2×D4×C31—48no—
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—24no—
114C22×C4×C31—24no—
115C22×C4×C31—22no—
116C22×C4×C31—22no—
117C22×C4×C31—24no—
118C22×C4×C31—24no—
119C22×C4×C31—24no—
120C22×C4×C31—24no—
121C22×C4×C31—24no—
122C22×C4×C31—24no—
123C22×C4×C31—28no—
124C22×C4×C31—28no—
125C22×C4×C31—24no—
126C22×C4×C31—22no—
127C22×C4×C31—22no—
128C22×C4×C31—24no—
129C22×C4×C31—24no—
130C22×C4×C31—22no—
131C22×C4×C31—22no—
132C22×C4×C31—24no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—44no—
136C22×C4×C31—416no—
137C22×C4×C31C2×D418yes—
138C22×C4×C31C2×D414yes—
139C22×C4×C31C2×D414yes—
140C22×C4×C31C2×D412yes—
141C22×C4×C31C2×D412yes—
142C22×C4×C31#4114yes—
143C4⋊C4×C31—28no—
144C4⋊C4×C31—416no—
145C4⋊C4×C31—44no—
146C4⋊C4×C31—44no—
147C22⋊C4×C31—216no—
148C22⋊C4×C31—48no—
149C22⋊C4×C31—416no—
150C22⋊C4×C31—416no—
151C22⋊C4×C31—48no—
152#16363C3111yesmaximal, characteristic, direct product factor, Sylow
153C23×C31—22no—
154C23×C31—22no—
155C23×C31—24no—
156C23×C31—22no—
157C23×C31—22no—
158C23×C31—22no—
159C23×C31—22no—
160C23×C31—24no—
161C23×C31—28no—
162C23×C31—28no—
163C23×C31—24no—
164C23×C31—24no—
165C23×C31—24no—
166C23×C31—28no—
167C23×C31—28no—
168C23×C31—42no—
169C23×C31—42no—
170C23×C31—42no—
171C23×C31—42no—
172C23×C31—44no—
173C23×C31—44no—
174C23×C31—42no—
175C23×C31—42no—
176C23×C31—44no—
177C23×C31—42no—
178C23×C31—42no—
179C23×C31—42no—
180C23×C31—42no—
181C23×C31—42no—
182C23×C31—42no—
183C23×C31—44no—
184C23×C31—44no—
185C23×C31—44no—
186C23×C31—44no—
187C23×C31—44no—
188C23×C31—44no—
189C23×C31—44no—
190C23×C31—44no—
191C23×C31—44no—
192C23×C31#12714yes—
193C23×C31C22×D411yescharacteristic
194C23×C31C22×D412yes—
195C23×C31C22×D412yes—
196C23×C31C2×#4112yes—
197C23×C31H(5,𝔽2)11yescharacteristic
198C23×C31H(5,𝔽2)12yes—
199C23×C31H(5,𝔽2)12yes—
200C23×C312−1+411yescharacteristic
201C23×C312−1+412yes—
202D4×C31—416no—
203D4×C31—432no—
204C2×C4×C31—22no—
205C2×C4×C31—22no—
206C2×C4×C31—22no—
207C2×C4×C31—22no—
208C2×C4×C31—28no—
209C2×C4×C31—28no—
210C2×C4×C31—24no—
211C2×C4×C31—28no—
212C2×C4×C31—28no—
213C2×C4×C31—28no—
214C2×C4×C31—28no—
215C2×C4×C31—416no—
216C2×C4×C31—42no—
217C2×C4×C31—42no—
218C2×C4×C31—44no—
219C2×C4×C31—44no—
220C2×C4×C31—44no—
221C2×C4×C31—44no—
222C2×C4×C31—44no—
223C2×C4×C31—42no—
224C2×C4×C31—42no—
225C2×C4×C31—42no—
226C2×C4×C31—44no—
227C2×C4×C31—42no—
228C2×C4×C31—42no—
229C2×C4×C31—42no—
230C2×C4×C31—44no—
231C2×C4×C31—44no—
232C2×C4×C31—44no—
233C2×C4×C31#12014yes—
234C2×C4×C31#12314yes—
235C22×#121C2×C3111yescharacteristic
236C2×#399C2×C3112yescomplemented
237C2×#394C2×C3114yescomplemented
238C2×#380C2×C3111yescharacteristic, complemented
239#1444C2×C3112yescomplemented
240#1439C2×C3112yescomplemented
241#1438C2×C3112yescomplemented
242C22⋊#95C2×C3111yescharacteristic, complemented
243C22×C31—24no—
244C22×C31—24no—
245C22×C31—24no—
246C22×C31—42no—
247C22×C31—42no—
248C22×C31—42no—
249C22×C31—42no—
250C22×C31—48no—
251C22×C31—48no—
252C22×C31—44no—
253C22×C31—44no—
254C22×C31—48no—
255C22×C31—48no—
256C22×C31—44no—
257C22×C31—82no—
258C22×C31—82no—
259C22×C31—82no—
260C22×C31—84no—
261C22×C31—84no—
262C22×C31—82no—
263C22×C31C2×#12011yescharacteristic
264C22×C31C2×#12311yescharacteristic
265C22×C31C2×#12711yescharacteristic
266C22×C31#53411yescharacteristic
267C22×C31#53412yes—
268C22×C31#53511yescharacteristic
269C22×C31#53611yescharacteristic
270C22×C31#53612yes—
271C22×C31#53711yescharacteristic
272C22×C31#53712yes—
273C22×C31#53712yes—
274C22×C31#54012yes—
275C22×C31#54212yes—
276C22×C31D4214yes—
277C22×C31#54614yes—
278C22×C31#54714yes—
279C22×C31#56214yes—
280C4×C31—24no—
281C4×C31—24no—
282C4×C31—48no—
283C4×C31—44no—
284C4×C31—82no—
285C4×C31—82no—
286C23×D4C22×C3112yes—
287C23×D4C22×C3111yescharacteristic
288C24×C4C22×C3111yescharacteristic
289C2×#121—24no—
290C2×#121—28no—
291C22×C4⋊C4C22×C3111yescharacteristic
292C22×C4⋊C4C22×C3111yescharacteristic
293C22×C4⋊C4C22×C3114yes—
294C22×C4⋊C4C22×C3111yescharacteristic
295C22×C22⋊C4C22×C3112yes—
296C22×C22⋊C4C22×C3114yes—
297C22×C22⋊C4C22×C3112yes—
298C22×C22⋊C4C22×C3114yes—
299C22×C22⋊C4C22×C3112yes—
300#399—28no—
301#394—216no—
302#380—24no—
303C2×#95C22×C3114yes—
304C2×#95C22×C3112yes—
305C2×#95C22×C3112yes—
306C2×#95C22×C3112yes—
307C2×C31—44no—
308C2×C31—84no—
309C2×C31#240511yescharacteristic
310C2×C31#243311yescharacteristic
311C2×C31#250111yescharacteristic
312C2×C31#261611yescharacteristic
313C2×C31#262411yescharacteristic
314C2×C31#265611yescharacteristic
315C2×C31#267114yes—
316C2×C31#267214yes—
317C2×C31#281611yescharacteristic
318C25C23×C3112yes—
319C25C23×C3111yescharacteristic
320C22×D4—24no—
321C22×D4—24no—
322C22×D4—24no—
323C22×D4—28no—
324C22×D4—24no—
325C22×D4—22no—
326C22×D4—24no—
327C22×D4—42no—
328C22×D4—44no—
329C23×C4—24no—
330C23×C4—24no—
331C23×C4—42no—
332C23×C4D4×C3114yes—
333C23×C4C23×C3112yes—
334C23×C4C23×C3112yes—
335C23×C4C23×C3114yes—
336C23×C4C23×C3112yes—
337C23×C4C23×C3111yescharacteristic
338C23×C4C23×C3111yescharacteristic
339#121—232no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—28no—
345C2×C4⋊C4—28no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—44no—
348C2×C4⋊C4—42no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—28no—
353C2×C22⋊C4—28no—
354C2×C22⋊C4—28no—
355C2×C22⋊C4—28no—
356C2×C22⋊C4—42no—
357C2×C22⋊C4—42no—
358C2×C22⋊C4—44no—
359C2×C22⋊C4—44no—
360C2×C22⋊C4—44no—
361C2×C22⋊C4D4×C3118yes—
362C2×C22⋊C4D4×C3118yescomplemented
363#95—24no—
364#95—28no—
365#95—24no—
366#95—24no—
367C31#1636311yescharacteristic, direct product factor, Sylow
368C24—22no—
369C24—22no—
370C24—21no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—21no—
375C24—24no—
376C24—22no—
377C24—22no—
378C24—24no—
379C24—22no—
380C24—24no—
381C24—24no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24C2×D4×C3114yes—
386C24C2×D4×C3114yes—
387C24C2×D4×C3112yes—
388C24C24×C3111yescharacteristic
389C2×D4—28no—
390C2×D4—216no—
391C2×D4—416no—
392C2×D4—416no—
393C2×D4—44no—
394C2×D4—416no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4—44no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—44no—
425C22×C4—416no—
426C22×C4C2×D4×C3118yes—
427C22×C4C2×D4×C3114yes—
428C22×C4C2×D4×C3114yes—
429C22×C4C2×D4×C3112yes—
430C22×C4C2×D4×C3112yes—
431C22×C4#41×C3114yes—
432C4⋊C4—28no—
433C4⋊C4—416no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C22⋊C4—216no—
437C22⋊C4—48no—
438C22⋊C4—416no—
439C22⋊C4—416no—
440C22⋊C4—48no—
441C23—22no—
442C23—22no—
443C23—24no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—24no—
449C23—28no—
450C23—28no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—28no—
455C23—28no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—44no—
461C23—44no—
462C23—42no—
463C23—42no—
464C23—44no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23C31×#12714yes—
481C23C22×D4×C3111yescharacteristic
482C23C22×D4×C3112yes—
483C23C22×D4×C3112yes—
484C23C2×#41×C3112yes—
485C23C31×H(5,𝔽2)11yescharacteristic
486C23C31×H(5,𝔽2)12yes—
487C23C31×H(5,𝔽2)12yes—
488C23C31×2−1+411yescharacteristic
489C23C31×2−1+412yes—
490D4—416no—
491D4—432no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—28no—
497C2×C4—28no—
498C2×C4—24no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—416no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4C31×#12014yes—
522C2×C4C31×#12314yes—
523C22—24no—
524C22—24no—
525C22—24no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—48no—
530C22—48no—
531C22—42no—
532C22—44no—
533C22—44no—
534C22—48no—
535C22—48no—
536C22—44no—
537C22—82no—
538C22—82no—
539C22—84no—
540C22—84no—
541C22—82no—
542C22—82no—
543C22C2×C31×#12011yescharacteristic
544C22C2×C31×#12311yescharacteristic
545C22C2×C31×#12711yescharacteristic
546C22C31×#53411yescharacteristic
547C22C31×#53412yes—
548C22C31×#53511yescharacteristic
549C22C31×#53612yes—
550C22C31×#53611yescharacteristic
551C22C31×#53711yescharacteristic
552C22C31×#53712yes—
553C22C31×#53712yes—
554C22C31×#54012yes—
555C22C31×#54212yes—
556C22D42×C3114yes—
557C22C31×#54614yes—
558C22C31×#54714yes—
559C22C31×#56214yes—
560C4—24no—
561C4—24no—
562C4—48no—
563C4—44no—
564C4—82no—
565C4—82no—
566C2—44no—
567C2—84no—
568C2C31×#240511yescharacteristic
569C2C31×#243311yescharacteristic
570C2C31×#250111yescharacteristic
571C2C31×#261611yescharacteristic
572C2C31×#262411yescharacteristic
573C2C31×#265611yescharacteristic
574C2C31×#267114yes—
575C2C31×#267214yes—
576C2C31×#281611yescharacteristic
subgroupG/H
center, socleC24×C31C24
commutator, Frattini, cosocleC24C24×C31
prime radicalC22×C31×#121C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#399C2semidirect
3. C2×C31×#394C2semidirect
4. C2×C31×#380C2semidirect
5. C31×#1444C2semidirect
6. C31×#1439C2semidirect
7. C31×#1438C2semidirect
8. C31×C22⋊#95C2semidirect
74. C2×C22⋊C4×C31D4semidirect
152. #16363C31direct (N × H)
236. C2×#399C2×C31semidirect
237. C2×#394C2×C31semidirect
238. C2×#380C2×C31semidirect
239. #1444C2×C31semidirect
240. #1439C2×C31semidirect
241. #1438C2×C31semidirect
242. C22⋊#95C2×C31semidirect
362. C2×C22⋊C4D4×C31semidirect
367. C31#16363direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C31×#1212
C2×C31×#3992
C2×C31×#3942
C2×C31×#3802
C31×#14442
C31×#14392
C31×#14382
C31×C22⋊#952
#1636331