Finite Group Databasebeta

← C31×#13424 #21470183

Subgroups — C31×#13424 #21470183

Order 7936.

612 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2684C211yesmaximal, characteristic
2C2×C31×#354C211yesmaximal, characteristic, complemented
3C2×C31×AΣL(1,𝔽16)C211yesmaximal, characteristic
4C31×#1889C211yesmaximal, characteristic
5C31×#1885C211yesmaximal, characteristic, complemented
6C31×#1867C211yesmaximal, characteristic, complemented
7C31×#1390C211yesmaximal, characteristic, complemented
8C2×C31×#124C2211yescharacteristic
9C2×C31×#120C2211yescharacteristic
10C31×#518C2211yescharacteristic
11C31×#419—21no—
12C2×C31×#100C2211yescharacteristic
13C2×C31×#99C2211yescharacteristic
14C2×C31×#99C2211yescharacteristic
15C31×#397—21no—
16C31×#394—21no—
17C31×#393—21no—
18C31×#388—21no—
19C31×#386—21no—
20C31×#386C2211yescharacteristic
21C31×#385—21no—
22C31×#379—21no—
23C31×#379—21no—
24C31×#354—22no—
25C31×AΣL(1,𝔽16)—22no—
26C31×#342—21no—
27C31×#342—21no—
28C31×#339—21no—
29C25×C31D411yescharacteristic
30C22×Q8×C31D411yescharacteristic
31C22×D4×C31—21no—
32C22×D4×C31C2311yescharacteristic
33C23×C4×C31D411yescharacteristic
34C2×M16×C31—21no—
35C31×#124—21no—
36C31×#124—21no—
37C31×#124—21no—
38C31×#124—21no—
39C31×#120—21no—
40C31×#120—21no—
41C31×#120—21no—
42C31×#120—21no—
43C4×D4×C31—41no—
44C4×D4×C31—41no—
45C31×#117—21no—
46C2×C4⋊C4×C31—21no—
47C2×C4⋊C4×C31—41no—
48C2×C22⋊C4×C31—21no—
49C2×C22⋊C4×C31—21no—
50C2×C22⋊C4×C31—21no—
51C2×C22⋊C4×C31—21no—
52C2×C22⋊C4×C31—21no—
53C2×C22⋊C4×C31—21no—
54C2×C22⋊C4×C31—21no—
55C2×C22⋊C4×C31—21no—
56C2×C22⋊C4×C31—21no—
57C2×C22⋊C4×C31—21no—
58C2×C22⋊C4×C31—21no—
59C2×C22⋊C4×C31—21no—
60C2×C22⋊C4×C31—41no—
61C2×C22⋊C4×C31—41no—
62C2×C22⋊C4×C31D411yescharacteristic
63C2×C22⋊C4×C31D411yescharacteristic
64C2×C42×C31D411yescharacteristic
65C31×#103—41no—
66C31×#103—41no—
67C31×#100—21no—
68C31×#100—21no—
69C31×#99—21no—
70C31×#99—21no—
71C31×#99—21no—
72C31×#99D412yes—
73C31×#95—21no—
74C31×#95—21no—
75C31×#95—21no—
76C31×#95—21no—
77C31×#95—41no—
78C24×C31—21no—
79C24×C31—21no—
80C24×C31—21no—
81C24×C31—41no—
82C24×C31—41no—
83C24×C31—42no—
84C24×C31—81no—
85C24×C31C2×D411yescharacteristic
86C24×C31C2×D411yescharacteristic
87C2×Q8×C31—21no—
88C2×Q8×C31—21no—
89C2×Q8×C31—41no—
90C2×Q8×C31—41no—
91C2×D4×C31—41no—
92C2×D4×C31—41no—
93C2×D4×C31—42no—
94C2×D4×C31—41no—
95C2×D4×C31—41no—
96C2×D4×C31—81no—
97C2×D4×C31C2×D411yescharacteristic
98C2×D4×C31#4111yescharacteristic
99C2×D4×C31#4111yescharacteristic
100C2×D4×C31#4111yescharacteristic
101C22×C4×C31—21no—
102C22×C4×C31—21no—
103C22×C4×C31—21no—
104C22×C4×C31—21no—
105C22×C4×C31—21no—
106C22×C4×C31—21no—
107C22×C4×C31—21no—
108C22×C4×C31—21no—
109C22×C4×C31—21no—
110C22×C4×C31—41no—
111C22×C4×C31—41no—
112C22×C4×C31—41no—
113C22×C4×C31—81no—
114C22×C4×C31C2×D411yescharacteristic
115M16×C31—41no—
116M16×C31—41no—
117C2×C8×C31—41no—
118C4⋊C4×C31—21no—
119C4⋊C4×C31—21no—
120C4⋊C4×C31—81no—
121C4⋊C4×C31—82no—
122C22⋊C4×C31—22no—
123C22⋊C4×C31—21no—
124C22⋊C4×C31—21no—
125C22⋊C4×C31—41no—
126C22⋊C4×C31—42no—
127C22⋊C4×C31—41no—
128C22⋊C4×C31—41no—
129C22⋊C4×C31—41no—
130C22⋊C4×C31—41no—
131C22⋊C4×C31—41no—
132C22⋊C4×C31—41no—
133C22⋊C4×C31—41no—
134C22⋊C4×C31—41no—
135C22⋊C4×C31—41no—
136C22⋊C4×C31—41no—
137C22⋊C4×C31—41no—
138C22⋊C4×C31—41no—
139C22⋊C4×C31—81no—
140C22⋊C4×C31—81no—
141C22⋊C4×C31—81no—
142C22⋊C4×C31—81no—
143C22⋊C4×C31—82no—
144C22⋊C4×C31—82no—
145C42×C31—21no—
146C42×C31—21no—
147C42×C31—41no—
148#13424C3111yesmaximal, characteristic, direct product factor, Sylow
149C23×C31—21no—
150C23×C31—21no—
151C23×C31—21no—
152C23×C31—21no—
153C23×C31—22no—
154C23×C31—41no—
155C23×C31—41no—
156C23×C31—41no—
157C23×C31—41no—
158C23×C31—41no—
159C23×C31—41no—
160C23×C31—41no—
161C23×C31—42no—
162C23×C31—42no—
163C23×C31—42no—
164C23×C31—41no—
165C23×C31—41no—
166C23×C31—41no—
167C23×C31—81no—
168C23×C31—82no—
169C23×C31—81no—
170C23×C31—82no—
171C23×C31—84no—
172C23×C31—81no—
173C23×C31#12011yescharacteristic
174C23×C31#12111yescharacteristic
175C23×C31#12111yescharacteristic
176C23×C31#12311yescharacteristic
177C23×C31#12411yescharacteristic
178Q8×C31—42no—
179Q8×C31—81no—
180D4×C31—41no—
181D4×C31—41no—
182D4×C31—41no—
183D4×C31—41no—
184D4×C31—82no—
185D4×C31—82no—
186C2×C4×C31—21no—
187C2×C4×C31—21no—
188C2×C4×C31—42no—
189C2×C4×C31—41no—
190C2×C4×C31—41no—
191C2×C4×C31—41no—
192C2×C4×C31—41no—
193C2×C4×C31—41no—
194C2×C4×C31—41no—
195C2×C4×C31—41no—
196C2×C4×C31—41no—
197C2×C4×C31—42no—
198C2×C4×C31—41no—
199C2×C4×C31—41no—
200C2×C4×C31—41no—
201C2×C4×C31—41no—
202C2×C4×C31—41no—
203C2×C4×C31—41no—
204C2×C4×C31—41no—
205C2×C4×C31—41no—
206C2×C4×C31—41no—
207C2×C4×C31—81no—
208C2×C4×C31—81no—
209C2×C4×C31—81no—
210C2×C4×C31—81no—
211C2×C4×C31—81no—
212C2×C4×C31—81no—
213C2×C4×C31—81no—
214C2×C4×C31—81no—
215C2×C4×C31—81no—
216C2×C4×C31—81no—
217C2×C4×C31#12111yescharacteristic
218C2×C4×C31#12311yescharacteristic
219C8×C31—81no—
220#2684C2×C3111yescharacteristic
221C2×#354C2×C3111yescharacteristic, complemented
222C2×AΣL(1,𝔽16)C2×C3111yescharacteristic
223#1889C2×C3111yescharacteristic
224#1885C2×C3111yescharacteristic, complemented
225#1867C2×C3111yescharacteristic, complemented
226#1390C2×C3111yescharacteristic, complemented
227C22×C31—21no—
228C22×C31—21no—
229C22×C31—21no—
230C22×C31—21no—
231C22×C31—21no—
232C22×C31—21no—
233C22×C31—41no—
234C22×C31—41no—
235C22×C31—41no—
236C22×C31—41no—
237C22×C31—42no—
238C22×C31—42no—
239C22×C31—41no—
240C22×C31—41no—
241C22×C31—41no—
242C22×C31—41no—
243C22×C31—44no—
244C22×C31—42no—
245C22×C31—81no—
246C22×C31—81no—
247C22×C31—81no—
248C22×C31—81no—
249C22×C31—82no—
250C22×C31—82no—
251C22×C31—82no—
252C22×C31—81no—
253C22×C31—81no—
254C22×C31#39411yescharacteristic
255C22×C31B(4,𝔽2)11yescharacteristic
256C22×C31#45811yescharacteristic
257C4×C31—21no—
258C4×C31—21no—
259C4×C31—81no—
260C4×C31—81no—
261C4×C31—81no—
262C4×C31—82no—
263C4×C31—81no—
264C4×C31—81no—
265C4×C31—81no—
266C4×C31—81no—
267C4×C31—161no—
268C2×#124C22×C3111yescharacteristic
269C2×#120C22×C3111yescharacteristic
270#518C22×C3111yescharacteristic
271#419—21no—
272C2×#100C22×C3111yescharacteristic
273C2×#99C22×C3111yescharacteristic
274C2×#99C22×C3111yescharacteristic
275#397—21no—
276#394—21no—
277#393—21no—
278#388—21no—
279#386—21no—
280#386C22×C3111yescharacteristic
281#385—21no—
282#379—21no—
283#379—21no—
284#354—22no—
285AΣL(1,𝔽16)—22no—
286#342—21no—
287#342—21no—
288#339—21no—
289C2×C31—21no—
290C2×C31—21no—
291C2×C31—41no—
292C2×C31—41no—
293C2×C31—41no—
294C2×C31—41no—
295C2×C31—82no—
296C2×C31#204411yescharacteristic
297C2×C31#219711yescharacteristic
298C2×C31#219911yescharacteristic
299C25D4×C3111yescharacteristic
300C22×Q8D4×C3111yescharacteristic
301C22×D4—21no—
302C22×D4C23×C3111yescharacteristic
303C23×C4D4×C3111yescharacteristic
304C2×M16—21no—
305#124—21no—
306#124—21no—
307#124—21no—
308#124—21no—
309#120—21no—
310#120—21no—
311#120—21no—
312#120—21no—
313C4×D4—41no—
314C4×D4—41no—
315#117—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—41no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—41no—
331C2×C22⋊C4—41no—
332C2×C22⋊C4D4×C3111yescharacteristic
333C2×C22⋊C4D4×C3111yescharacteristic
334C2×C42D4×C3111yescharacteristic
335#103—41no—
336#103—41no—
337#100—21no—
338#100—21no—
339#99—21no—
340#99—21no—
341#99—21no—
342#99D4×C3112yes—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—41no—
348C31#1342411yescharacteristic, direct product factor, Sylow
349C24—21no—
350C24—21no—
351C24—21no—
352C24—41no—
353C24—41no—
354C24—42no—
355C24—81no—
356C24C2×D4×C3111yescharacteristic
357C24C2×D4×C3111yescharacteristic
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—42no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—81no—
368C2×D4C2×D4×C3111yescharacteristic
369C2×D4#41×C3111yescharacteristic
370C2×D4#41×C3111yescharacteristic
371C2×D4#41×C3111yescharacteristic
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—81no—
385C22×C4C2×D4×C3111yescharacteristic
386M16—41no—
387M16—41no—
388C2×C8—41no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—81no—
392C4⋊C4—82no—
393C22⋊C4—22no—
394C22⋊C4—21no—
395C22⋊C4—21no—
396C22⋊C4—41no—
397C22⋊C4—42no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—81no—
411C22⋊C4—81no—
412C22⋊C4—81no—
413C22⋊C4—81no—
414C22⋊C4—82no—
415C22⋊C4—82no—
416C42—21no—
417C42—21no—
418C42—41no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—22no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—81no—
438C23—82no—
439C23—81no—
440C23—82no—
441C23—84no—
442C23—81no—
443C23C31×#12011yescharacteristic
444C23C31×#12111yescharacteristic
445C23C31×#12111yescharacteristic
446C23C31×#12311yescharacteristic
447C23C31×#12411yescharacteristic
448Q8—42no—
449Q8—81no—
450D4—41no—
451D4—41no—
452D4—41no—
453D4—41no—
454D4—82no—
455D4—82no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4C31×#12111yescharacteristic
488C2×C4C31×#12311yescharacteristic
489C8—81no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—42no—
501C22—42no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—44no—
507C22—42no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—81no—
516C22—81no—
517C22C31×#39411yescharacteristic
518C22C31×B(4,𝔽2)11yescharacteristic
519C22C31×#45811yescharacteristic
520C4—21no—
521C4—21no—
522C4—81no—
523C4—81no—
524C4—81no—
525C4—82no—
526C4—81no—
527C4—81no—
528C4—81no—
529C4—81no—
530C4—161no—
531C2—21no—
532C2—21no—
533C2—41no—
534C2—41no—
535C2—41no—
536C2—41no—
537C2—82no—
538C2C31×#204411yescharacteristic
539C2C31×#219711yescharacteristic
540C2C31×#219911yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC22×D4C23×C31
prime radicalC2×C31×#120C22
LCSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C22×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#354C2semidirect
5. C31×#1885C2semidirect
6. C31×#1867C2semidirect
7. C31×#1390C2semidirect
148. #13424C31direct (N × H)
221. C2×#354C2×C31semidirect
224. #1885C2×C31semidirect
225. #1867C2×C31semidirect
226. #1390C2×C31semidirect
348. C31#13424direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#26842
C2×C31×#3542
C2×C31×AΣL(1,𝔽16)2
C31×#18892
C31×#18852
C31×#18672
C31×#13902
#1342431