Finite Group Databasebeta

← C31×#9939 #21466698

Subgroups — C31×#9939 #21466698

Order 7936.

824 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 102 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×#3006C211yesmaximal, characteristic
2C31×#2591C211yesmaximal, characteristic, complemented
3C31×#2406C211yesmaximal, characteristic, complemented
4C31×#1789C211yesmaximal, characteristic, complemented
5C31×#1510C211yesmaximal, characteristic, complemented
6C31×#1463C211yesmaximal, characteristic, complemented
7C31×#1284C211yesmaximal, characteristic, complemented
8C22×#41×C31C2211yescharacteristic
9C2×C31×AGL(1,ℤ/8)—21no—
10C2×C31×#123—21no—
11C2×C31×#121C2211yescharacteristic
12C22×C4⋊C4×C31C2211yescharacteristic
13C31×#449—22no—
14C31×#448—21no—
15C31×#447—21no—
16C2×C31×#104—22no—
17C31×#417—21no—
18C31×#407C2211yescharacteristic
19C31×#407C2211yescharacteristic
20C31×#400—21no—
21C31×#395—21no—
22C31×#394—21no—
23C31×#394—21no—
24C31×#388—22no—
25C31×#386—21no—
26C31×#386—21no—
27C31×#386—21no—
28C31×#385—22no—
29C31×#381C2211yescharacteristic
30C4×C22⋊C4×C31C2211yescharacteristic
31C31×#327—22no—
32C31×#324—22no—
33C2×#41×C31—21no—
34C2×#41×C31—21no—
35C2×#41×C31—21no—
36C2×#41×C31—21no—
37C2×#41×C31—44no—
38C22×Q8×C31D411yescharacteristic
39C22×D4×C31—21no—
40C22×D4×C31—21no—
41C22×D4×C31D411yescharacteristic
42C22×D4×C31D411yescharacteristic
43C23×C4×C31—21no—
44C23×C4×C31D411yescharacteristic
45C23×C4×C31D411yescharacteristic
46C23×C4×C31C2311yescharacteristic
47C31×AGL(1,ℤ/8)—44no—
48C2×SD16×C31—22no—
49C2×D8×C31—22no—
50C2×M16×C31—21no—
51C2×M16×C31—21no—
52C31×#123—44no—
53C31×#121—21no—
54C31×#121—21no—
55C31×#121—21no—
56C31×#121—21no—
57C2×C4⋊C4×C31—22no—
58C2×C4⋊C4×C31—21no—
59C2×C4⋊C4×C31—21no—
60C2×C4⋊C4×C31—21no—
61C2×C4⋊C4×C31—21no—
62C2×C4⋊C4×C31—21no—
63C2×C4⋊C4×C31—21no—
64C2×C4⋊C4×C31D411yescharacteristic
65C2×C22⋊C4×C31—22no—
66C2×C22⋊C4×C31—22no—
67C2×C22⋊C4×C31—22no—
68C2×C22⋊C4×C31—21no—
69C2×C22⋊C4×C31—21no—
70C2×C22⋊C4×C31—21no—
71C2×C22⋊C4×C31—22no—
72C2×C42×C31D412yes—
73C31×#104—44no—
74C31×#103—22no—
75C31×#102—22no—
76C31×C22⋊C8—21no—
77C31×C22⋊C8—21no—
78C31×C22⋊C8—21no—
79C31×C22⋊C8D412yes—
80C31×#95—22no—
81C31×#95—22no—
82C31×#95—22no—
83C31×#95—21no—
84C31×#95—21no—
85C31×#95D412yes—
86C24×C31—21no—
87C24×C31—21no—
88C24×C31C2×D411yescharacteristic
89#41×C31—24no—
90#41×C31—24no—
91#41×C31—42no—
92#41×C31—42no—
93#41×C31—42no—
94#41×C31—42no—
95#41×C31—42no—
96#41×C31—42no—
97C2×Q8×C31—21no—
98C2×Q8×C31—21no—
99C2×Q8×C31—42no—
100C2×D4×C31—24no—
101C2×D4×C31—21no—
102C2×D4×C31—21no—
103C2×D4×C31—21no—
104C2×D4×C31—21no—
105C2×D4×C31—42no—
106C2×D4×C31—42no—
107C2×D4×C31—42no—
108C2×D4×C31—41no—
109C2×D4×C31—41no—
110C2×D4×C31—44no—
111C2×D4×C31—41no—
112C2×D4×C31—41no—
113C22×C4×C31—21no—
114C22×C4×C31—21no—
115C22×C4×C31—21no—
116C22×C4×C31—21no—
117C22×C4×C31—21no—
118C22×C4×C31—22no—
119C22×C4×C31—22no—
120C22×C4×C31—21no—
121C22×C4×C31—21no—
122C22×C4×C31—21no—
123C22×C4×C31—42no—
124C22×C4×C31—42no—
125C22×C4×C31—42no—
126C22×C4×C31—42no—
127C22×C4×C31—41no—
128C22×C4×C31—41no—
129C22×C4×C31—42no—
130C22×C4×C31C2×D411yescharacteristic
131C22×C4×C31C2×D411yescharacteristic
132C22×C4×C31C2×D411yescharacteristic
133C22×C4×C31C2×D411yescharacteristic
134C22×C4×C31C2×D411yescharacteristic
135C22×C4×C31#4111yescharacteristic
136SD16×C31—44no—
137D8×C31—44no—
138M16×C31—42no—
139M16×C31—42no—
140C2×C8×C31—22no—
141C2×C8×C31—41no—
142C4⋊C4×C31—21no—
143C4⋊C4×C31—21no—
144C4⋊C4×C31—44no—
145C4⋊C4×C31—41no—
146C4⋊C4×C31—41no—
147C4⋊C4×C31—41no—
148C4⋊C4×C31—41no—
149C22⋊C4×C31—44no—
150C22⋊C4×C31—44no—
151C22⋊C4×C31—42no—
152C22⋊C4×C31—42no—
153C22⋊C4×C31—44no—
154C22⋊C4×C31—41no—
155C22⋊C4×C31—41no—
156C22⋊C4×C31—82no—
157C42×C31—42no—
158#9939C3111yesmaximal, characteristic, direct product factor, Sylow
159C23×C31—22no—
160C23×C31—22no—
161C23×C31—22no—
162C23×C31—41no—
163C23×C31—42no—
164C23×C31—41no—
165C23×C31—42no—
166C23×C31—42no—
167C23×C31—41no—
168C23×C31—42no—
169C23×C31—41no—
170C23×C31—42no—
171C23×C31#12011yescharacteristic
172C23×C31#12011yescharacteristic
173C23×C31#12111yescharacteristic
174Q8×C31—22no—
175Q8×C31—22no—
176Q8×C31—41no—
177Q8×C31—41no—
178D4×C31—22no—
179D4×C31—22no—
180D4×C31—41no—
181D4×C31—41no—
182D4×C31—42no—
183D4×C31—44no—
184D4×C31—44no—
185D4×C31—81no—
186D4×C31—82no—
187D4×C31—82no—
188C2×C4×C31—22no—
189C2×C4×C31—22no—
190C2×C4×C31—22no—
191C2×C4×C31—22no—
192C2×C4×C31—22no—
193C2×C4×C31—22no—
194C2×C4×C31—41no—
195C2×C4×C31—41no—
196C2×C4×C31—41no—
197C2×C4×C31—41no—
198C2×C4×C31—44no—
199C2×C4×C31—44no—
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—42no—
207C2×C4×C31—42no—
208C2×C4×C31—42no—
209C2×C4×C31—42no—
210C2×C4×C31—42no—
211C2×C4×C31—42no—
212C2×C4×C31—41no—
213C2×C4×C31—41no—
214C2×C4×C31—42no—
215C2×C4×C31—42no—
216C2×C4×C31—42no—
217C2×C4×C31—81no—
218C2×C4×C31#12011yescharacteristic
219C2×C4×C31#12111yescharacteristic
220C2×C4×C31#12111yescharacteristic
221C2×C4×C31#12711yescharacteristic
222C8×C31—42no—
223C8×C31—81no—
224#3006C2×C3111yescharacteristic
225#2591C2×C3111yescharacteristic, complemented
226#2406C2×C3111yescharacteristic, complemented
227#1789C2×C3111yescharacteristic, complemented
228#1510C2×C3111yescharacteristic, complemented
229#1463C2×C3111yescharacteristic, complemented
230#1284C2×C3111yescharacteristic, complemented
231C22×C31—21no—
232C22×C31—21no—
233C22×C31—21no—
234C22×C31—21no—
235C22×C31—21no—
236C22×C31—21no—
237C22×C31—21no—
238C22×C31—44no—
239C22×C31—44no—
240C22×C31—42no—
241C22×C31—82no—
242C22×C31—84no—
243C22×C31—81no—
244C22×C31—81no—
245C22×C31—81no—
246C22×C31—81no—
247C22×C31—81no—
248C22×C31#39211yescharacteristic
249C22×C31#45412yes—
250C22×C31B(4,𝔽2)12yes—
251C4×C31—21no—
252C4×C31—21no—
253C4×C31—21no—
254C4×C31—21no—
255C4×C31—42no—
256C4×C31—42no—
257C4×C31—82no—
258C4×C31—82no—
259C4×C31—81no—
260C4×C31—81no—
261C22×#41C22×C3111yescharacteristic
262C2×AGL(1,ℤ/8)—21no—
263C2×#123—21no—
264C2×#121C22×C3111yescharacteristic
265C22×C4⋊C4C22×C3111yescharacteristic
266#449—22no—
267#448—21no—
268#447—21no—
269C2×#104—22no—
270#417—21no—
271#407C22×C3111yescharacteristic
272#407C22×C3111yescharacteristic
273#400—21no—
274#395—21no—
275#394—21no—
276#394—21no—
277#388—22no—
278#386—21no—
279#386—21no—
280#386—21no—
281#385—22no—
282#381C22×C3111yescharacteristic
283C4×C22⋊C4C22×C3111yescharacteristic
284#327—22no—
285#324—22no—
286C2×C31—22no—
287C2×C31—22no—
288C2×C31—41no—
289C2×C31—82no—
290C2×C31—161no—
291C2×C31#200311yescharacteristic
292C2×C31#201011yescharacteristic
293C2×C31#202111yescharacteristic
294C2×#41—21no—
295C2×#41—21no—
296C2×#41—21no—
297C2×#41—21no—
298C2×#41—44no—
299C22×Q8D4×C3111yescharacteristic
300C22×D4—21no—
301C22×D4—21no—
302C22×D4D4×C3111yescharacteristic
303C22×D4D4×C3111yescharacteristic
304C23×C4—21no—
305C23×C4D4×C3111yescharacteristic
306C23×C4D4×C3111yescharacteristic
307C23×C4C23×C3111yescharacteristic
308AGL(1,ℤ/8)—44no—
309C2×SD16—22no—
310C2×D8—22no—
311C2×M16—21no—
312C2×M16—21no—
313#123—44no—
314#121—21no—
315#121—21no—
316#121—21no—
317#121—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4D4×C3111yescharacteristic
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C42D4×C3112yes—
334#104—44no—
335#103—22no—
336#102—22no—
337C22⋊C8—21no—
338C22⋊C8—21no—
339C22⋊C8—21no—
340C22⋊C8D4×C3112yes—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—21no—
345#95—21no—
346#95D4×C3112yes—
347C31#993911yescharacteristic, direct product factor, Sylow
348C24—21no—
349C24—21no—
350C24C2×D4×C3111yescharacteristic
351#41—24no—
352#41—24no—
353#41—42no—
354#41—42no—
355#41—42no—
356#41—42no—
357#41—42no—
358#41—42no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—42no—
362C2×D4—24no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—44no—
373C2×D4—41no—
374C2×D4—41no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4C2×D4×C3111yescharacteristic
393C22×C4C2×D4×C3111yescharacteristic
394C22×C4C2×D4×C3111yescharacteristic
395C22×C4C2×D4×C3111yescharacteristic
396C22×C4C2×D4×C3111yescharacteristic
397C22×C4#41×C3111yescharacteristic
398SD16—44no—
399D8—44no—
400M16—42no—
401M16—42no—
402C2×C8—22no—
403C2×C8—41no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—44no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—44no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—82no—
419C42—42no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—41no—
424C23—42no—
425C23—41no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—41no—
430C23—42no—
431C23—41no—
432C23C31×#12011yescharacteristic
433C23C31×#12011yescharacteristic
434C23C31×#12111yescharacteristic
435Q8—22no—
436Q8—22no—
437Q8—41no—
438Q8—41no—
439D4—22no—
440D4—22no—
441D4—41no—
442D4—41no—
443D4—42no—
444D4—44no—
445D4—44no—
446D4—81no—
447D4—82no—
448D4—82no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—81no—
479C2×C4C31×#12011yescharacteristic
480C2×C4C31×#12111yescharacteristic
481C2×C4C31×#12111yescharacteristic
482C2×C4C31×#12711yescharacteristic
483C8—42no—
484C8—81no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—44no—
493C22—44no—
494C22—42no—
495C22—82no—
496C22—84no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22C31×#39211yescharacteristic
503C22C31×#45412yes—
504C22C31×B(4,𝔽2)12yes—
505C4—21no—
506C4—21no—
507C4—21no—
508C4—21no—
509C4—42no—
510C4—42no—
511C4—82no—
512C4—82no—
513C4—81no—
514C4—81no—
515C2—22no—
516C2—22no—
517C2—41no—
518C2—82no—
519C2—161no—
520C2C31×#200311yescharacteristic
521C2C31×#201011yescharacteristic
522C2C31×#202111yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC23×C4C23×C31
prime radicalC31×#3006C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2591C2semidirect
3. C31×#2406C2semidirect
4. C31×#1789C2semidirect
5. C31×#1510C2semidirect
6. C31×#1463C2semidirect
7. C31×#1284C2semidirect
158. #9939C31direct (N × H)
225. #2591C2×C31semidirect
226. #2406C2×C31semidirect
227. #1789C2×C31semidirect
228. #1510C2×C31semidirect
229. #1463C2×C31semidirect
230. #1284C2×C31semidirect
347. C31#9939direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#30062
C31×#25912
C31×#24062
C31×#17892
C31×#15102
C31×#14632
C31×#12842
#993931