Finite Group Databasebeta

← C41×#9939 #23786140

Subgroups — C41×#9939 #23786140

Order 10496.

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
1C41×#3006C211yesmaximal, characteristic
2C41×#2591C211yesmaximal, characteristic, complemented
3C41×#2406C211yesmaximal, characteristic, complemented
4C41×#1789C211yesmaximal, characteristic, complemented
5C41×#1510C211yesmaximal, characteristic, complemented
6C41×#1463C211yesmaximal, characteristic, complemented
7C41×#1284C211yesmaximal, characteristic, complemented
8C22×#41×C41C2211yescharacteristic
9C2×AGL(1,ℤ/8)×C41—21no—
10C2×#123×C41—21no—
11C2×#121×C41C2211yescharacteristic
12C22×C4⋊C4×C41C2211yescharacteristic
13C41×#449—22no—
14C41×#448—21no—
15C41×#447—21no—
16C2×#104×C41—22no—
17C41×#417—21no—
18C41×#407C2211yescharacteristic
19C41×#407C2211yescharacteristic
20C41×#400—21no—
21C41×#395—21no—
22C41×#394—21no—
23C41×#394—21no—
24C41×#388—22no—
25C41×#386—21no—
26C41×#386—21no—
27C41×#386—21no—
28C41×#385—22no—
29C41×#381C2211yescharacteristic
30C4×C22⋊C4×C41C2211yescharacteristic
31C41×#327—22no—
32C41×#324—22no—
33C2×#41×C41—21no—
34C2×#41×C41—21no—
35C2×#41×C41—21no—
36C2×#41×C41—21no—
37C2×#41×C41—44no—
38C22×Q8×C41D411yescharacteristic
39C22×D4×C41—21no—
40C22×D4×C41—21no—
41C22×D4×C41D411yescharacteristic
42C22×D4×C41D411yescharacteristic
43C23×C4×C41—21no—
44C23×C4×C41D411yescharacteristic
45C23×C4×C41D411yescharacteristic
46C23×C4×C41C2311yescharacteristic
47AGL(1,ℤ/8)×C41—44no—
48C2×SD16×C41—22no—
49C2×D8×C41—22no—
50C2×M16×C41—21no—
51C2×M16×C41—21no—
52#123×C41—44no—
53#121×C41—21no—
54#121×C41—21no—
55#121×C41—21no—
56#121×C41—21no—
57C2×C4⋊C4×C41—22no—
58C2×C4⋊C4×C41—21no—
59C2×C4⋊C4×C41—21no—
60C2×C4⋊C4×C41—21no—
61C2×C4⋊C4×C41—21no—
62C2×C4⋊C4×C41—21no—
63C2×C4⋊C4×C41—21no—
64C2×C4⋊C4×C41D411yescharacteristic
65C2×C22⋊C4×C41—22no—
66C2×C22⋊C4×C41—22no—
67C2×C22⋊C4×C41—22no—
68C2×C22⋊C4×C41—21no—
69C2×C22⋊C4×C41—21no—
70C2×C22⋊C4×C41—21no—
71C2×C22⋊C4×C41—22no—
72C2×C42×C41D412yes—
73#104×C41—44no—
74#103×C41—22no—
75#102×C41—22no—
76C22⋊C8×C41—21no—
77C22⋊C8×C41—21no—
78C22⋊C8×C41—21no—
79C22⋊C8×C41D412yes—
80#95×C41—22no—
81#95×C41—22no—
82#95×C41—22no—
83#95×C41—21no—
84#95×C41—21no—
85#95×C41D412yes—
86C24×C41—21no—
87C24×C41—21no—
88C24×C41C2×D411yescharacteristic
89#41×C41—24no—
90#41×C41—24no—
91#41×C41—42no—
92#41×C41—42no—
93#41×C41—42no—
94#41×C41—42no—
95#41×C41—42no—
96#41×C41—42no—
97C2×Q8×C41—21no—
98C2×Q8×C41—21no—
99C2×Q8×C41—42no—
100C2×D4×C41—24no—
101C2×D4×C41—21no—
102C2×D4×C41—21no—
103C2×D4×C41—21no—
104C2×D4×C41—21no—
105C2×D4×C41—42no—
106C2×D4×C41—42no—
107C2×D4×C41—42no—
108C2×D4×C41—41no—
109C2×D4×C41—41no—
110C2×D4×C41—44no—
111C2×D4×C41—41no—
112C2×D4×C41—41no—
113C22×C4×C41—21no—
114C22×C4×C41—21no—
115C22×C4×C41—21no—
116C22×C4×C41—21no—
117C22×C4×C41—21no—
118C22×C4×C41—22no—
119C22×C4×C41—22no—
120C22×C4×C41—21no—
121C22×C4×C41—21no—
122C22×C4×C41—21no—
123C22×C4×C41—42no—
124C22×C4×C41—42no—
125C22×C4×C41—42no—
126C22×C4×C41—42no—
127C22×C4×C41—41no—
128C22×C4×C41—41no—
129C22×C4×C41—42no—
130C22×C4×C41C2×D411yescharacteristic
131C22×C4×C41C2×D411yescharacteristic
132C22×C4×C41C2×D411yescharacteristic
133C22×C4×C41C2×D411yescharacteristic
134C22×C4×C41C2×D411yescharacteristic
135C22×C4×C41#4111yescharacteristic
136SD16×C41—44no—
137D8×C41—44no—
138M16×C41—42no—
139M16×C41—42no—
140C2×C8×C41—22no—
141C2×C8×C41—41no—
142C4⋊C4×C41—21no—
143C4⋊C4×C41—21no—
144C4⋊C4×C41—44no—
145C4⋊C4×C41—41no—
146C4⋊C4×C41—41no—
147C4⋊C4×C41—41no—
148C4⋊C4×C41—41no—
149C22⋊C4×C41—44no—
150C22⋊C4×C41—44no—
151C22⋊C4×C41—42no—
152C22⋊C4×C41—42no—
153C22⋊C4×C41—44no—
154C22⋊C4×C41—41no—
155C22⋊C4×C41—41no—
156C22⋊C4×C41—82no—
157C42×C41—42no—
158C23×C41—22no—
159C23×C41—22no—
160C23×C41—22no—
161C23×C41—41no—
162C23×C41—42no—
163C23×C41—41no—
164C23×C41—42no—
165C23×C41—42no—
166C23×C41—41no—
167C23×C41—42no—
168C23×C41—41no—
169C23×C41—42no—
170C23×C41#12011yescharacteristic
171C23×C41#12011yescharacteristic
172C23×C41#12111yescharacteristic
173Q8×C41—22no—
174Q8×C41—22no—
175Q8×C41—41no—
176Q8×C41—41no—
177D4×C41—22no—
178D4×C41—22no—
179D4×C41—41no—
180D4×C41—41no—
181D4×C41—42no—
182D4×C41—44no—
183D4×C41—44no—
184D4×C41—81no—
185D4×C41—82no—
186D4×C41—82no—
187C2×C4×C41—22no—
188C2×C4×C41—22no—
189C2×C4×C41—22no—
190C2×C4×C41—22no—
191C2×C4×C41—22no—
192C2×C4×C41—22no—
193C2×C4×C41—41no—
194C2×C4×C41—41no—
195C2×C4×C41—41no—
196C2×C4×C41—41no—
197C2×C4×C41—44no—
198C2×C4×C41—44no—
199C2×C4×C41—41no—
200C2×C4×C41—41no—
201C2×C4×C41—41no—
202C2×C4×C41—41no—
203C2×C4×C41—41no—
204C2×C4×C41—41no—
205C2×C4×C41—42no—
206C2×C4×C41—42no—
207C2×C4×C41—42no—
208C2×C4×C41—42no—
209C2×C4×C41—42no—
210C2×C4×C41—42no—
211C2×C4×C41—41no—
212C2×C4×C41—41no—
213C2×C4×C41—42no—
214C2×C4×C41—42no—
215C2×C4×C41—42no—
216C2×C4×C41—81no—
217C2×C4×C41#12011yescharacteristic
218C2×C4×C41#12111yescharacteristic
219C2×C4×C41#12111yescharacteristic
220C2×C4×C41#12711yescharacteristic
221C8×C41—42no—
222C8×C41—81no—
223#9939C4111yesmaximal, characteristic, direct product factor, Sylow
224C22×C41—21no—
225C22×C41—21no—
226C22×C41—21no—
227C22×C41—21no—
228C22×C41—21no—
229C22×C41—21no—
230C22×C41—21no—
231C22×C41—44no—
232C22×C41—44no—
233C22×C41—42no—
234C22×C41—82no—
235C22×C41—84no—
236C22×C41—81no—
237C22×C41—81no—
238C22×C41—81no—
239C22×C41—81no—
240C22×C41—81no—
241C22×C41#39211yescharacteristic
242C22×C41#45412yes—
243C22×C41B(4,𝔽2)12yes—
244C4×C41—21no—
245C4×C41—21no—
246C4×C41—21no—
247C4×C41—21no—
248C4×C41—42no—
249C4×C41—42no—
250C4×C41—82no—
251C4×C41—82no—
252C4×C41—81no—
253C4×C41—81no—
254#3006C2×C4111yescharacteristic
255#2591C2×C4111yescharacteristic, complemented
256#2406C2×C4111yescharacteristic, complemented
257#1789C2×C4111yescharacteristic, complemented
258#1510C2×C4111yescharacteristic, complemented
259#1463C2×C4111yescharacteristic, complemented
260#1284C2×C4111yescharacteristic, complemented
261C2×C41—22no—
262C2×C41—22no—
263C2×C41—41no—
264C2×C41—82no—
265C2×C41—161no—
266C2×C41#200311yescharacteristic
267C2×C41#201011yescharacteristic
268C2×C41#202111yescharacteristic
269C22×#41C22×C4111yescharacteristic
270C2×AGL(1,ℤ/8)—21no—
271C2×#123—21no—
272C2×#121C22×C4111yescharacteristic
273C22×C4⋊C4C22×C4111yescharacteristic
274#449—22no—
275#448—21no—
276#447—21no—
277C2×#104—22no—
278#417—21no—
279#407C22×C4111yescharacteristic
280#407C22×C4111yescharacteristic
281#400—21no—
282#395—21no—
283#394—21no—
284#394—21no—
285#388—22no—
286#386—21no—
287#386—21no—
288#386—21no—
289#385—22no—
290#381C22×C4111yescharacteristic
291C4×C22⋊C4C22×C4111yescharacteristic
292#327—22no—
293#324—22no—
294C41#993911yescharacteristic, direct product factor, Sylow
295C2×#41—21no—
296C2×#41—21no—
297C2×#41—21no—
298C2×#41—21no—
299C2×#41—44no—
300C22×Q8D4×C4111yescharacteristic
301C22×D4—21no—
302C22×D4—21no—
303C22×D4D4×C4111yescharacteristic
304C22×D4D4×C4111yescharacteristic
305C23×C4—21no—
306C23×C4D4×C4111yescharacteristic
307C23×C4D4×C4111yescharacteristic
308C23×C4C23×C4111yescharacteristic
309AGL(1,ℤ/8)—44no—
310C2×SD16—22no—
311C2×D8—22no—
312C2×M16—21no—
313C2×M16—21no—
314#123—44no—
315#121—21no—
316#121—21no—
317#121—21no—
318#121—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4D4×C4111yescharacteristic
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C42D4×C4112yes—
335#104—44no—
336#103—22no—
337#102—22no—
338C22⋊C8—21no—
339C22⋊C8—21no—
340C22⋊C8—21no—
341C22⋊C8D4×C4112yes—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—21no—
346#95—21no—
347#95D4×C4112yes—
348C24—21no—
349C24—21no—
350C24C2×D4×C4111yescharacteristic
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×C4111yescharacteristic
393C22×C4C2×D4×C4111yescharacteristic
394C22×C4C2×D4×C4111yescharacteristic
395C22×C4C2×D4×C4111yescharacteristic
396C22×C4C2×D4×C4111yescharacteristic
397C22×C4#41×C4111yescharacteristic
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—44no—
415C22⋊C4—42no—
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—41no—
429C23—42no—
430C23—41no—
431C23—42no—
432C23#120×C4111yescharacteristic
433C23#120×C4111yescharacteristic
434C23#121×C4111yescharacteristic
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—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—81no—
479C2×C4#120×C4111yescharacteristic
480C2×C4#121×C4111yescharacteristic
481C2×C4#121×C4111yescharacteristic
482C2×C4#127×C4111yescharacteristic
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—
502C22C41×#39211yescharacteristic
503C22C41×#45412yes—
504C22C41×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—
520C2C41×#200311yescharacteristic
521C2C41×#201011yescharacteristic
522C2C41×#202111yescharacteristic
subgroupG/H
center, socleC22×C41#392
commutator, Frattini, cosocleC23×C4C23×C41
prime radicalC41×#3006C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C41 ◁ C23×C4×C41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C41×#2591C2semidirect
3. C41×#2406C2semidirect
4. C41×#1789C2semidirect
5. C41×#1510C2semidirect
6. C41×#1463C2semidirect
7. C41×#1284C2semidirect
223. #9939C41direct (N × H)
255. #2591C2×C41semidirect
256. #2406C2×C41semidirect
257. #1789C2×C41semidirect
258. #1510C2×C41semidirect
259. #1463C2×C41semidirect
260. #1284C2×C41semidirect
294. C41#9939direct (N × H)

Maximal subgroups

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

subgroupindex
C41×#30062
C41×#25912
C41×#24062
C41×#17892
C41×#15102
C41×#14632
C41×#12842
#993941