Finite Group Databasebeta

← C31×#12705 #21469464

Subgroups — C31×#12705 #21469464

Order 7936.

1186 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1186 conjugacy classes of subgroups — too many to draw. Only its 146 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×#3484C211yesmaximal, characteristic
2C31×#2121C214yesmaximal, complemented
3C31×#1783C211yesmaximal, characteristic, complemented
4C31×#1781C211yesmaximal, characteristic, complemented
5C2×C31×H(5,𝔽2)C411yescharacteristic, complemented
6C23×D4×C31C2211yescharacteristic
7C31×#546—24no—
8D42×C31—24no—
9C31×#535—21no—
10C31×#534—21no—
11C31×#534C2212yes—
12C31×#525—21no—
13C2×C31×#120—21no—
14C2×C31×#120C412yescomplemented
15C31×#518C411yescharacteristic, complemented
16C2×C31×#100—21no—
17C2×C31×#100C2212yescomplemented
18C2×C31×#99—21no—
19C2×C31×#99C2212yescomplemented
20C31×#407—21no—
21C31×#379—21no—
22C31×#352—24no—
23C31×AΣL(1,𝔽16)—24no—
24C25×C31C2×C411yescharacteristic
25C25×C31D411yescharacteristic
26C31×H(5,𝔽2)—42no—
27C31×H(5,𝔽2)—42no—
28C2×#41×C31—21no—
29C2×#41×C31—22no—
30C22×D4×C31—21no—
31C22×D4×C31—22no—
32C22×D4×C31—22no—
33C22×D4×C31—22no—
34C22×D4×C31—24no—
35C22×D4×C31—42no—
36C22×D4×C31—42no—
37C22×D4×C31C2×C411yescharacteristic
38C22×D4×C31C2×C412yes—
39C22×D4×C31D411yescharacteristic
40C22×D4×C31D411yescharacteristic
41C22×D4×C31C2311yescharacteristic
42C23×C4×C31D411yescharacteristic
43C2×M16×C31—22no—
44C31×#127—22no—
45C31×#124—22no—
46C31×#123—22no—
47C31×#123—22no—
48C31×#123D414yes—
49C31×C22⋊Q8—22no—
50C31×#121—22no—
51C31×#121—24no—
52C31×#121—24no—
53C31×#121—24no—
54C31×#120—24no—
55C31×#120—22no—
56C31×#120—22no—
57C31×#120—22no—
58C31×#120—22no—
59C31×#120—24no—
60C31×#120D414yes—
61C4×D4×C31—24no—
62C4×D4×C31—24no—
63C31×#117—21no—
64C2×C22⋊C4×C31—22no—
65C2×C22⋊C4×C31—21no—
66C2×C22⋊C4×C31—22no—
67C2×C22⋊C4×C31—22no—
68C2×C22⋊C4×C31—42no—
69C2×C22⋊C4×C31C2×C412yes—
70C31×#100—24no—
71C31×#100—42no—
72C31×#99—24no—
73C31×#99—42no—
74C31×C22⋊C8—42no—
75C24×C31—21no—
76C24×C31—22no—
77C24×C31—22no—
78C24×C31—22no—
79C24×C31—41no—
80C24×C31—41no—
81C24×C31—41no—
82C24×C31—82no—
83C24×C31—82no—
84C24×C31C22⋊C411yescharacteristic
85C24×C31C22⋊C411yescharacteristic
86C24×C31C22⋊C411yescharacteristic
87C24×C31C22⋊C412yescomplemented
88C24×C31C22×C411yescharacteristic
89C24×C31C2×D411yescharacteristic
90#41×C31—42no—
91#41×C31—48no—
92#41×C31—42no—
93C2×Q8×C31—21no—
94C2×D4×C31—22no—
95C2×D4×C31—22no—
96C2×D4×C31—24no—
97C2×D4×C31—22no—
98C2×D4×C31—21no—
99C2×D4×C31—22no—
100C2×D4×C31—22no—
101C2×D4×C31—22no—
102C2×D4×C31—22no—
103C2×D4×C31—24no—
104C2×D4×C31—24no—
105C2×D4×C31—22no—
106C2×D4×C31—22no—
107C2×D4×C31—22no—
108C2×D4×C31—41no—
109C2×D4×C31—41no—
110C2×D4×C31—41no—
111C2×D4×C31—41no—
112C2×D4×C31—416no—
113C2×D4×C31—48no—
114C2×D4×C31—42no—
115C2×D4×C31—42no—
116C2×D4×C31—42no—
117C2×D4×C31—42no—
118C2×D4×C31—84no—
119C2×D4×C31—84no—
120C2×D4×C31C2×D414yes—
121C22×C4×C31—21no—
122C22×C4×C31—22no—
123C22×C4×C31—22no—
124C22×C4×C31—22no—
125C22×C4×C31—22no—
126C22×C4×C31—22no—
127C22×C4×C31—41no—
128C22×C4×C31—42no—
129C22×C4×C31C22⋊C411yescharacteristic
130C22×C4×C31C22⋊C412yescomplemented
131C22×C4×C31C2×D411yescharacteristic
132M16×C31—42no—
133M16×C31—42no—
134C2×C8×C31—42no—
135C4⋊C4×C31—22no—
136C4⋊C4×C31—24no—
137C22⋊C4×C31—24no—
138C22⋊C4×C31—22no—
139C22⋊C4×C31—24no—
140C22⋊C4×C31—22no—
141C22⋊C4×C31—24no—
142C22⋊C4×C31—24no—
143C22⋊C4×C31—42no—
144C22⋊C4×C31—42no—
145C22⋊C4×C31—84no—
146C22⋊C4×C31C22⋊C414yescomplemented
147C42×C31—22no—
148#12705C3111yesmaximal, characteristic, direct product factor, Sylow
149C23×C31—21no—
150C23×C31—21no—
151C23×C31—21no—
152C23×C31—21no—
153C23×C31—21no—
154C23×C31—21no—
155C23×C31—21no—
156C23×C31—21no—
157C23×C31—21no—
158C23×C31—21no—
159C23×C31—22no—
160C23×C31—21no—
161C23×C31—21no—
162C23×C31—42no—
163C23×C31—44no—
164C23×C31—44no—
165C23×C31—44no—
166C23×C31—44no—
167C23×C31—44no—
168C23×C31—42no—
169C23×C31—42no—
170C23×C31—82no—
171C23×C31—82no—
172C23×C31—82no—
173C23×C31—81no—
174C23×C31—81no—
175C23×C31—81no—
176C23×C31—81no—
177C23×C31—82no—
178C23×C31—82no—
179C23×C31—82no—
180C23×C31—82no—
181C23×C31—82no—
182C23×C31—82no—
183C23×C31—82no—
184C23×C31—82no—
185C23×C31#9912yes—
186C23×C31C2×C22⋊C412yes—
187C23×C31C2×C22⋊C411yescharacteristic
188C23×C31#12012yes—
189Q8×C31—42no—
190D4×C31—28no—
191D4×C31—44no—
192D4×C31—44no—
193D4×C31—44no—
194D4×C31—42no—
195D4×C31—44no—
196D4×C31—44no—
197D4×C31—48no—
198D4×C31—42no—
199D4×C31—42no—
200D4×C31—84no—
201D4×C31—84no—
202C2×C4×C31—21no—
203C2×C4×C31—21no—
204C2×C4×C31—21no—
205C2×C4×C31—21no—
206C2×C4×C31—24no—
207C2×C4×C31—22no—
208C2×C4×C31—44no—
209C2×C4×C31—44no—
210C2×C4×C31—42no—
211C2×C4×C31—42no—
212C2×C4×C31—82no—
213C2×C4×C31—82no—
214C2×C4×C31—82no—
215C2×C4×C31—84no—
216C2×C4×C31#9912yes—
217C2×C4×C31#12012yes—
218C8×C31—82no—
219#3484C2×C3111yescharacteristic
220#2121C2×C3114yescomplemented
221#1783C2×C3111yescharacteristic, complemented
222#1781C2×C3111yescharacteristic, complemented
223C22×C31—22no—
224C22×C31—22no—
225C22×C31—22no—
226C22×C31—22no—
227C22×C31—22no—
228C22×C31—22no—
229C22×C31—41no—
230C22×C31—42no—
231C22×C31—42no—
232C22×C31—42no—
233C22×C31—42no—
234C22×C31—42no—
235C22×C31—41no—
236C22×C31—41no—
237C22×C31—41no—
238C22×C31—44no—
239C22×C31—42no—
240C22×C31—41no—
241C22×C31—42no—
242C22×C31—41no—
243C22×C31—81no—
244C22×C31—81no—
245C22×C31—81no—
246C22×C31—81no—
247C22×C31—81no—
248C22×C31—81no—
249C22×C31—88no—
250C22×C31—88no—
251C22×C31—162no—
252C22×C31—162no—
253C22×C31#37911yescharacteristic
254C22×C31C2×#9911yescharacteristic
255C22×C31C2×#9911yescharacteristic
256C4×C31—22no—
257C4×C31—22no—
258C4×C31—44no—
259C4×C31—42no—
260C4×C31—162no—
261C2×H(5,𝔽2)C4×C3111yescharacteristic, complemented
262C23×D4C22×C3111yescharacteristic
263#546—24no—
264D42—24no—
265#535—21no—
266#534—21no—
267#534C22×C3112yes—
268#525—21no—
269C2×#120—21no—
270C2×#120C4×C3112yescomplemented
271#518C4×C3111yescharacteristic, complemented
272C2×#100—21no—
273C2×#100C22×C3112yescomplemented
274C2×#99—21no—
275C2×#99C22×C3112yescomplemented
276#407—21no—
277#379—21no—
278#352—24no—
279AΣL(1,𝔽16)—24no—
280C2×C31—21no—
281C2×C31—21no—
282C2×C31—21no—
283C2×C31—42no—
284C2×C31—42no—
285C2×C31—42no—
286C2×C31—42no—
287C2×C31—81no—
288C2×C31—82no—
289C2×C31#178111yescharacteristic
290C25C2×C4×C3111yescharacteristic
291C25D4×C3111yescharacteristic
292H(5,𝔽2)—42no—
293H(5,𝔽2)—42no—
294C2×#41—21no—
295C2×#41—22no—
296C22×D4—21no—
297C22×D4—22no—
298C22×D4—22no—
299C22×D4—22no—
300C22×D4—24no—
301C22×D4—42no—
302C22×D4—42no—
303C22×D4C2×C4×C3111yescharacteristic
304C22×D4C2×C4×C3112yes—
305C22×D4D4×C3111yescharacteristic
306C22×D4D4×C3111yescharacteristic
307C22×D4C23×C3111yescharacteristic
308C23×C4D4×C3111yescharacteristic
309C2×M16—22no—
310#127—22no—
311#124—22no—
312#123—22no—
313#123—22no—
314#123D4×C3114yes—
315C22⋊Q8—22no—
316#121—22no—
317#121—24no—
318#121—24no—
319#121—24no—
320#120—24no—
321#120—22no—
322#120—22no—
323#120—22no—
324#120—22no—
325#120—24no—
326#120D4×C3114yes—
327C4×D4—24no—
328C4×D4—24no—
329#117—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—42no—
335C2×C22⋊C4C2×C4×C3112yes—
336#100—24no—
337#100—42no—
338#99—24no—
339#99—42no—
340C22⋊C8—42no—
341C31#1270511yescharacteristic, direct product factor, Sylow
342C24—21no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—41no—
347C24—41no—
348C24—41no—
349C24—82no—
350C24—82no—
351C24C22⋊C4×C3111yescharacteristic
352C24C22⋊C4×C3111yescharacteristic
353C24C22⋊C4×C3111yescharacteristic
354C24C22⋊C4×C3112yescomplemented
355C24C22×C4×C3111yescharacteristic
356C24C2×D4×C3111yescharacteristic
357#41—42no—
358#41—48no—
359#41—42no—
360C2×Q8—21no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—24no—
364C2×D4—22no—
365C2×D4—21no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—416no—
380C2×D4—48no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—84no—
386C2×D4—84no—
387C2×D4C2×D4×C3114yes—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4C22⋊C4×C3111yescharacteristic
397C22×C4C22⋊C4×C3112yescomplemented
398C22×C4C2×D4×C3111yescharacteristic
399M16—42no—
400M16—42no—
401C2×C8—42no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—22no—
406C22⋊C4—24no—
407C22⋊C4—22no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—84no—
413C22⋊C4C22⋊C4×C3114yescomplemented
414C42—22no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—22no—
426C23—21no—
427C23—21no—
428C23—42no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—42no—
435C23—42no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—81no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23C31×#9912yes—
452C23C2×C22⋊C4×C3112yes—
453C23C2×C22⋊C4×C3111yescharacteristic
454C23C31×#12012yes—
455Q8—42no—
456D4—28no—
457D4—44no—
458D4—44no—
459D4—44no—
460D4—42no—
461D4—44no—
462D4—44no—
463D4—48no—
464D4—42no—
465D4—42no—
466D4—84no—
467D4—84no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—84no—
482C2×C4C31×#9912yes—
483C2×C4C31×#12012yes—
484C8—82no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—41no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—44no—
501C22—42no—
502C22—41no—
503C22—42no—
504C22—41no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—88no—
512C22—88no—
513C22—162no—
514C22—162no—
515C22C31×#37911yescharacteristic
516C22C2×C31×#9911yescharacteristic
517C22C2×C31×#9911yescharacteristic
518C4—22no—
519C4—22no—
520C4—44no—
521C4—42no—
522C4—162no—
523C2—21no—
524C2—21no—
525C2—21no—
526C2—42no—
527C2—42no—
528C2—42no—
529C2—42no—
530C2—81no—
531C2—82no—
532C2C31×#178111yescharacteristic
subgroupG/H
center, socleC2×C31#1781
commutatorC24C22×C4×C31
Frattini, cosocleC22×D4C23×C31
prime radicalC31×#3484C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C23×C31 ◁ C23×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2121C2semidirect
3. C31×#1783C2semidirect
4. C31×#1781C2semidirect
5. C2×C31×H(5,𝔽2)C4semidirect
14. C2×C31×#120C4semidirect
15. C31×#518C4semidirect
17. C2×C31×#100C22semidirect
19. C2×C31×#99C22semidirect
87. C24×C31C22⋊C4semidirect
130. C22×C4×C31C22⋊C4semidirect
146. C22⋊C4×C31C22⋊C4semidirect
148. #12705C31direct (N × H)
220. #2121C2×C31semidirect
221. #1783C2×C31semidirect
222. #1781C2×C31semidirect
261. C2×H(5,𝔽2)C4×C31semidirect
270. C2×#120C4×C31semidirect
271. #518C4×C31semidirect
273. C2×#100C22×C31semidirect
275. C2×#99C22×C31semidirect
341. C31#12705direct (N × H)
354. C24C22⋊C4×C31semidirect
397. C22×C4C22⋊C4×C31semidirect
413. C22⋊C4C22⋊C4×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#34842
C31×#21212
C31×#17832
C31×#17812
#1270531