Finite Group Databasebeta

← C31×#16696 #21473455

Subgroups — C31×#16696 #21473455

Order 7936.

1122 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#391C211yesmaximal, characteristic
2C2×C31×#383C211yesmaximal, characteristic
3C31×#1446C211yesmaximal, characteristic
4C31×#1446C211yesmaximal, characteristic
5C31×#1445C212yesmaximal
6C31×#1444C211yesmaximal, characteristic
7C31×#1443C212yesmaximal
8C31×#1443C212yesmaximal
9C31×#1442C212yesmaximal
10C31×#1442C211yesmaximal, characteristic
11C31×#1434C211yesmaximal, characteristic
12C23×Q8×C31C2211yescharacteristic
13C22×C4⋊C4×C31C2212yes—
14C22×C4⋊C4×C31C2211yescharacteristic
15C22×C42×C31C2211yescharacteristic
16C31×#391—24no—
17C31×#383—24no—
18C2×C31×#95C2212yes—
19C2×C31×#95C2211yescharacteristic
20C2×C31×#95C2211yescharacteristic
21C2×C31×#95C2211yescharacteristic
22C2×C31×#95C2212yes—
23C2×C31×#95C2212yes—
24C2×C31×#95C2211yescharacteristic
25C2×C31×#95C2212yes—
26C2×C31×#95C2212yes—
27C2×C31×#95C2211yescharacteristic
28C2×C31×#95C2211yescharacteristic
29C2×C31×#95C2212yes—
30C2×C31×#95C2211yescharacteristic
31C2×C31×#95C2211yescharacteristic
32C2×C31×#95C2212yes—
33C2×C31×#95C2212yes—
34C2×C31×#95C2212yes—
35C2×C31×#95C2212yes—
36C2×C31×#95C2212yes—
37C22×Q8×C31—22no—
38C22×Q8×C31—24no—
39C22×Q8×C31—22no—
40C22×Q8×C31—22no—
41C22×Q8×C31—42no—
42C23×C4×C31C2311yescharacteristic
43C23×C4×C31C2312yes—
44C23×C4×C31C2311yescharacteristic
45C23×C4×C31C2311yescharacteristic
46C23×C4×C31C2311yescharacteristic
47C23×C4×C31C2312yes—
48C23×C4×C31C2311yescharacteristic
49C23×C4×C31C2311yescharacteristic
50C23×C4×C31C2312yes—
51C23×C4×C31C2312yes—
52C23×C4×C31C2311yescharacteristic
53C2×C4⋊C4×C31—22no—
54C2×C4⋊C4×C31—24no—
55C2×C4⋊C4×C31—42no—
56C2×C4⋊C4×C31—42no—
57C2×C4⋊C4×C31—42no—
58C2×C42×C31—22no—
59C2×C42×C31—22no—
60C2×C42×C31D414yes—
61C31×#95—22no—
62C31×#95—22no—
63C31×#95—22no—
64C31×#95—22no—
65C31×#95—22no—
66C31×#95—22no—
67C31×#95—24no—
68C31×#95—24no—
69C31×#95—22no—
70C31×#95—24no—
71C31×#95—24no—
72C31×#95—24no—
73C31×#95—24no—
74C31×#95—22no—
75C31×#95—42no—
76C31×#95—42no—
77C31×#95—42no—
78C31×#95—42no—
79C31×#95—42no—
80C24×C31C2411yescharacteristic
81C2×Q8×C31—28no—
82C2×Q8×C31—48no—
83C2×Q8×C31—48no—
84C2×Q8×C31—44no—
85C2×Q8×C31—44no—
86C22×C4×C31—22no—
87C22×C4×C31—22no—
88C22×C4×C31—22no—
89C22×C4×C31—22no—
90C22×C4×C31—22no—
91C22×C4×C31—22no—
92C22×C4×C31—21no—
93C22×C4×C31—21no—
94C22×C4×C31—21no—
95C22×C4×C31—21no—
96C22×C4×C31—22no—
97C22×C4×C31—22no—
98C22×C4×C31—22no—
99C22×C4×C31—21no—
100C22×C4×C31—22no—
101C22×C4×C31—22no—
102C22×C4×C31—22no—
103C22×C4×C31—22no—
104C22×C4×C31—21no—
105C22×C4×C31—21no—
106C22×C4×C31—22no—
107C22×C4×C31—21no—
108C22×C4×C31—22no—
109C22×C4×C31—22no—
110C22×C4×C31—21no—
111C22×C4×C31—21no—
112C22×C4×C31—21no—
113C22×C4×C31—21no—
114C22×C4×C31—22no—
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—21no—
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—21no—
133C22×C4×C31—22no—
134C22×C4×C31—21no—
135C22×C4×C31—21no—
136C22×C4×C31—21no—
137C22×C4×C31—22no—
138C22×C4×C31—21no—
139C22×C4×C31—21no—
140C22×C4×C31—21no—
141C22×C4×C31—22no—
142C22×C4×C31—22no—
143C22×C4×C31—22no—
144C22×C4×C31C2×D412yes—
145C22×C4×C31C2×D412yes—
146C22×C4×C31C2×D412yes—
147C22×C4×C31#4114yes—
148C22×C4×C31#4114yes—
149C22×C4×C31#4112yes—
150C22×C4×C31#4112yes—
151C4⋊C4×C31—44no—
152C4⋊C4×C31—48no—
153C42×C31—28no—
154#16696C3111yesmaximal, characteristic, direct product factor, Sylow
155C23×C31C22×D411yescharacteristic
156C23×C31C2×#4111yescharacteristic
157C23×C31C2×#4111yescharacteristic
158C23×C31C2×#4112yes—
159C23×C31C2×#4112yes—
160C23×C31H(5,𝔽2)12yes—
161C23×C31H(5,𝔽2)11yescharacteristic
162C23×C312−1+411yescharacteristic
163C23×C312−1+412yes—
164C23×C312−1+411yescharacteristic
165C23×C312−1+411yescharacteristic
166Q8×C31—416no—
167C2×C4×C31—22no—
168C2×C4×C31—22no—
169C2×C4×C31—24no—
170C2×C4×C31—24no—
171C2×C4×C31—22no—
172C2×C4×C31—22no—
173C2×C4×C31—24no—
174C2×C4×C31—22no—
175C2×C4×C31—22no—
176C2×C4×C31—22no—
177C2×C4×C31—22no—
178C2×C4×C31—24no—
179C2×C4×C31—24no—
180C2×C4×C31—24no—
181C2×C4×C31—24no—
182C2×C4×C31—24no—
183C2×C4×C31—42no—
184C2×C4×C31—42no—
185C2×C4×C31—41no—
186C2×C4×C31—41no—
187C2×C4×C31—41no—
188C2×C4×C31—41no—
189C2×C4×C31—42no—
190C2×C4×C31—42no—
191C2×C4×C31—42no—
192C2×C4×C31—42no—
193C2×C4×C31—42no—
194C2×C4×C31—42no—
195C2×C4×C31—42no—
196C2×C4×C31—42no—
197C2×C4×C31—41no—
198C2×C4×C31—41no—
199C2×C4×C31—41no—
200C2×C4×C31—41no—
201C2×C4×C31—42no—
202C2×C4×C31—42no—
203C2×C4×C31—41no—
204C2×C4×C31—41no—
205C2×C4×C31—41no—
206C2×C4×C31—41no—
207C2×C4×C31—41no—
208C2×C4×C31—41no—
209C2×C4×C31—41no—
210C2×C4×C31—41no—
211C2×C4×C31—42no—
212C2×C4×C31—42no—
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#12314yes—
232C2×#391C2×C3111yescharacteristic
233C2×#383C2×C3111yescharacteristic
234#1446C2×C3111yescharacteristic
235#1446C2×C3111yescharacteristic
236#1445C2×C3112yes—
237#1444C2×C3111yescharacteristic
238#1443C2×C3112yes—
239#1443C2×C3112yes—
240#1442C2×C3112yes—
241#1442C2×C3111yescharacteristic
242#1434C2×C3111yescharacteristic
243C22×C31C2×#12311yescharacteristic
244C22×C31#52511yescharacteristic
245C22×C31#53211yescharacteristic
246C22×C31#53511yescharacteristic
247C22×C31#53611yescharacteristic
248C22×C31#53611yescharacteristic
249C22×C31#53711yescharacteristic
250C22×C31#53812yes—
251C22×C31#54111yescharacteristic
252C22×C31#54111yescharacteristic
253C22×C31#54112yes—
254C22×C31#54211yescharacteristic
255C22×C31#54211yescharacteristic
256C22×C31#54212yes—
257C22×C31#54212yes—
258C22×C31#55112yes—
259C22×C31#55112yes—
260C22×C31#55212yes—
261C22×C31#55212yes—
262C22×C31#55612yes—
263C22×C31#55612yes—
264C22×C31#56312yes—
265C22×C31#56312yes—
266C4×C31—24no—
267C4×C31—42no—
268C4×C31—44no—
269C4×C31—42no—
270C4×C31—44no—
271C4×C31—81no—
272C4×C31—81no—
273C4×C31—81no—
274C4×C31—82no—
275C4×C31—81no—
276C4×C31—82no—
277C23×Q8C22×C3111yescharacteristic
278C22×C4⋊C4C22×C3112yes—
279C22×C4⋊C4C22×C3111yescharacteristic
280C22×C42C22×C3111yescharacteristic
281#391—24no—
282#383—24no—
283C2×#95C22×C3112yes—
284C2×#95C22×C3111yescharacteristic
285C2×#95C22×C3111yescharacteristic
286C2×#95C22×C3111yescharacteristic
287C2×#95C22×C3112yes—
288C2×#95C22×C3112yes—
289C2×#95C22×C3111yescharacteristic
290C2×#95C22×C3112yes—
291C2×#95C22×C3112yes—
292C2×#95C22×C3111yescharacteristic
293C2×#95C22×C3111yescharacteristic
294C2×#95C22×C3112yes—
295C2×#95C22×C3111yescharacteristic
296C2×#95C22×C3111yescharacteristic
297C2×#95C22×C3112yes—
298C2×#95C22×C3112yes—
299C2×#95C22×C3112yes—
300C2×#95C22×C3112yes—
301C2×#95C22×C3112yes—
302C2×C31#248211yescharacteristic
303C2×C31#249911yescharacteristic
304C2×C31#260411yescharacteristic
305C2×C31#262511yescharacteristic
306C2×C31#263711yescharacteristic
307C2×C31#266211yescharacteristic
308C2×C31#275812yes—
309C2×C31#275912yes—
310C2×C31#276212yes—
311C2×C31#276412yes—
312C2×C31#282111yescharacteristic
313C22×Q8—22no—
314C22×Q8—24no—
315C22×Q8—22no—
316C22×Q8—22no—
317C22×Q8—42no—
318C23×C4C23×C3111yescharacteristic
319C23×C4C23×C3112yes—
320C23×C4C23×C3111yescharacteristic
321C23×C4C23×C3111yescharacteristic
322C23×C4C23×C3111yescharacteristic
323C23×C4C23×C3112yes—
324C23×C4C23×C3111yescharacteristic
325C23×C4C23×C3111yescharacteristic
326C23×C4C23×C3112yes—
327C23×C4C23×C3112yes—
328C23×C4C23×C3111yescharacteristic
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—42no—
332C2×C4⋊C4—42no—
333C2×C4⋊C4—42no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42D4×C3114yes—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—24no—
344#95—24no—
345#95—22no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—22no—
351#95—42no—
352#95—42no—
353#95—42no—
354#95—42no—
355#95—42no—
356C31#1669611yescharacteristic, direct product factor, Sylow
357C24C24×C3111yescharacteristic
358C2×Q8—28no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×Q8—44no—
362C2×Q8—44no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4C2×D4×C3112yes—
422C22×C4C2×D4×C3112yes—
423C22×C4C2×D4×C3112yes—
424C22×C4#41×C3114yes—
425C22×C4#41×C3114yes—
426C22×C4#41×C3112yes—
427C22×C4#41×C3112yes—
428C4⋊C4—44no—
429C4⋊C4—48no—
430C42—28no—
431C23C22×D4×C3111yescharacteristic
432C23C2×#41×C3111yescharacteristic
433C23C2×#41×C3111yescharacteristic
434C23C2×#41×C3112yes—
435C23C2×#41×C3112yes—
436C23C31×H(5,𝔽2)12yes—
437C23C31×H(5,𝔽2)11yescharacteristic
438C23C31×2−1+411yescharacteristic
439C23C31×2−1+412yes—
440C23C31×2−1+411yescharacteristic
441C23C31×2−1+411yescharacteristic
442Q8—416no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4C31×#12314yes—
508C22C2×C31×#12311yescharacteristic
509C22C31×#52511yescharacteristic
510C22C31×#53211yescharacteristic
511C22C31×#53511yescharacteristic
512C22C31×#53611yescharacteristic
513C22C31×#53611yescharacteristic
514C22C31×#53711yescharacteristic
515C22C31×#53812yes—
516C22C31×#54111yescharacteristic
517C22C31×#54111yescharacteristic
518C22C31×#54112yes—
519C22C31×#54211yescharacteristic
520C22C31×#54211yescharacteristic
521C22C31×#54212yes—
522C22C31×#54212yes—
523C22C31×#55112yes—
524C22C31×#55112yes—
525C22C31×#55212yes—
526C22C31×#55212yes—
527C22C31×#55612yes—
528C22C31×#55612yes—
529C22C31×#56312yes—
530C22C31×#56312yes—
531C4—24no—
532C4—42no—
533C4—44no—
534C4—42no—
535C4—44no—
536C4—81no—
537C4—81no—
538C4—81no—
539C4—82no—
540C4—81no—
541C4—82no—
542C2C31×#248211yescharacteristic
543C2C31×#249911yescharacteristic
544C2C31×#260411yescharacteristic
545C2C31×#262511yescharacteristic
546C2C31×#263711yescharacteristic
547C2C31×#266211yescharacteristic
548C2C31×#275812yes—
549C2C31×#275912yes—
550C2C31×#276212yes—
551C2C31×#276412yes—
552C2C31×#282111yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C31C24
commutator, Frattini, cosocleC24C24×C31
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
154. #16696C31direct (N × H)
356. C31#16696direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3912
C2×C31×#3832
C31×#14462
C31×#14452
C31×#14442
C31×#14432
C31×#14422
C31×#14342
#1669631