Finite Group Databasebeta

← C41×#9265 #23785466

Subgroups — C41×#9265 #23785466

Order 10496.

438 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C41×#3211C211yesmaximal, characteristic
2C41×#2968C211yesmaximal, characteristic, complemented
3C41×#1532C211yesmaximal, characteristic, complemented
4C2×C41×#329C211yesmaximal, characteristic, complemented
5C41×#1469C211yesmaximal, characteristic, complemented
6C41×#1302C211yesmaximal, characteristic, complemented
7C41×#1281C211yesmaximal, characteristic, complemented
8C41×#529C2211yescharacteristic
9C2×#124×C41C2211yescharacteristic
10C41×#495—21no—
11C41×#488—21no—
12C41×#464—21no—
13C41×#463—21no—
14C41×#451C411yescharacteristic
15C41×#450C411yescharacteristic
16C41×#448C411yescharacteristic
17C41×#447C411yescharacteristic
18C41×#446—21no—
19C41×#435—21no—
20C41×#435—21no—
21C8×D4×C41—21no—
22C41×#433—21no—
23C41×#432C2211yescharacteristic
24C41×#431C2211yescharacteristic
25C41×#431C2211yescharacteristic
26C41×#405—21no—
27C2×#97×C41C2211yescharacteristic
28C2×#97×C41C2211yescharacteristic
29C41×#330—21no—
30C41×#329—21no—
31C41×#329—21no—
32C41×#329—21no—
33C41×#323—21no—
34C41×#323—21no—
35C22×Q8×C41C2×C411yescharacteristic
36C22×D4×C41C2×C411yescharacteristic
37C2×Q16×C41—21no—
38C2×SD16×C41—21no—
39C2×SD16×C41—21no—
40C2×D8×C41—21no—
41C2×M16×C41—21no—
42C22×C8×C41—21no—
43C22×C8×C41—21no—
44C22×C8×C41—21no—
45#126×C41—21no—
46#125×C41D411yescharacteristic
47#124×C41—21no—
48#124×C41—21no—
49#124×C41—21no—
50#124×C41—21no—
51#124×C41D411yescharacteristic
52#123×C41—21no—
53C22⋊Q8×C41C2×C411yescharacteristic
54#121×C41C2×C411yescharacteristic
55C4×Q8×C41D411yescharacteristic
56C4×D4×C41—21no—
57C4×D4×C41D411yescharacteristic
58#117×C41—21no—
59C2×C22⋊C4×C41—21no—
60C2×C22⋊C4×C41—21no—
61C2×C42×C41C2311yescharacteristic
62C4⋊C8×C41—21no—
63C4⋊C8×C41—21no—
64C4⋊C8×C41—21no—
65C4⋊C8×C41—21no—
66C4⋊C8×C41—21no—
67#103×C41—21no—
68#103×C41—21no—
69#102×C41—21no—
70#102×C41—21no—
71C22⋊C8×C41—21no—
72C22⋊C8×C41—21no—
73C22⋊C8×C41—21no—
74C22⋊C8×C41C2×C411yescharacteristic
75C22⋊C8×C41C2×C411yescharacteristic
76C22⋊C8×C41D411yescharacteristic
77C22⋊C8×C41D411yescharacteristic
78#97×C41—21no—
79#97×C41—21no—
80#97×C41—21no—
81#97×C41—21no—
82#97×C41—21no—
83#97×C41D411yescharacteristic
84#97×C41D411yescharacteristic
85C4×C8×C41—21no—
86C4×C8×C41—21no—
87C4×C8×C41—21no—
88C24×C41—21no—
89C2×Q8×C41—21no—
90C2×Q8×C41—21no—
91C2×Q8×C41—81no—
92C2×Q8×C41C22⋊C411yescharacteristic
93C2×D4×C41—21no—
94C2×D4×C41—21no—
95C2×D4×C41—21no—
96C2×D4×C41—81no—
97C2×D4×C41C22⋊C411yescharacteristic
98C22×C4×C41—21no—
99C22×C4×C41—21no—
100C22×C4×C41C22×C411yescharacteristic
101C22×C4×C41C2×D411yescharacteristic
102C22×C4×C41#4111yescharacteristic
103Q16×C41—81no—
104SD16×C41—81no—
105SD16×C41—81no—
106D8×C41—81no—
107M16×C41—42no—
108C2×C8×C41—21no—
109C2×C8×C41—22no—
110C2×C8×C41—21no—
111C2×C8×C41—21no—
112C2×C8×C41—21no—
113C2×C8×C41—21no—
114C2×C8×C41—21no—
115C2×C8×C41—42no—
116C2×C8×C41—42no—
117C2×C8×C41—41no—
118C2×C8×C41—41no—
119C2×C8×C41—41no—
120C4⋊C4×C41—21no—
121C4⋊C4×C41—21no—
122C4⋊C4×C41—21no—
123C4⋊C4×C41—21no—
124C4⋊C4×C41C22⋊C411yescharacteristic
125C4⋊C4×C41C22⋊C411yescharacteristic
126C22⋊C4×C41—21no—
127C22⋊C4×C41—21no—
128C22⋊C4×C41—21no—
129C22⋊C4×C41—21no—
130C22⋊C4×C41—21no—
131C22⋊C4×C41—41no—
132C22⋊C4×C41—41no—
133C22⋊C4×C41—41no—
134C22⋊C4×C41—41no—
135C42×C41—21no—
136C42×C41C2×D411yescharacteristic
137C42×C41C2×D411yescharacteristic
138C42×C41C2×D411yescharacteristic
139C42×C41#4111yescharacteristic
140C23×C41—21no—
141C23×C41—41no—
142C23×C41—41no—
143C23×C41—41no—
144C23×C41—81no—
145C23×C41—81no—
146C23×C41C4×D411yescharacteristic
147Q8×C41—41no—
148Q8×C41—41no—
149Q8×C41—41no—
150Q8×C41—81no—
151D4×C41—41no—
152D4×C41—41no—
153D4×C41—41no—
154D4×C41—41no—
155D4×C41—41no—
156D4×C41—81no—
157C2×C4×C41—21no—
158C2×C4×C41—21no—
159C2×C4×C41—21no—
160C2×C4×C41—21no—
161C2×C4×C41—21no—
162C2×C4×C41—41no—
163C2×C4×C41—41no—
164C2×C4×C41—41no—
165C2×C4×C41—41no—
166C2×C4×C41—41no—
167C2×C4×C41—81no—
168C2×C4×C41C2×C22⋊C411yescharacteristic
169C2×C4×C41C4×D411yescharacteristic
170C2×C4×C41#12011yescharacteristic
171C2×C4×C41#12111yescharacteristic
172C2×C4×C41#12111yescharacteristic
173C2×C4×C41#12311yescharacteristic
174C8×C41—22no—
175C8×C41—42no—
176C8×C41—41no—
177C8×C41—41no—
178C8×C41—41no—
179C8×C41—41no—
180C8×C41—41no—
181C8×C41—81no—
182#9265C4111yesmaximal, characteristic, direct product factor, Sylow
183C22×C41—21no—
184C22×C41—21no—
185C22×C41—41no—
186C22×C41—41no—
187C22×C41—41no—
188C22×C41—81no—
189C22×C41—81no—
190C22×C41—81no—
191C22×C41—161no—
192C22×C41—161no—
193C22×C41#38611yescharacteristic
194C22×C41#44311yescharacteristic
195C22×C41#44411yescharacteristic
196C4×C41—21no—
197C4×C41—21no—
198C4×C41—21no—
199C4×C41—21no—
200C4×C41—21no—
201C4×C41—21no—
202C4×C41—41no—
203C4×C41—41no—
204C4×C41—41no—
205C4×C41—81no—
206#3211C2×C4111yescharacteristic
207#2968C2×C4111yescharacteristic, complemented
208#1532C2×C4111yescharacteristic, complemented
209C2×#329C2×C4111yescharacteristic, complemented
210#1469C2×C4111yescharacteristic, complemented
211#1302C2×C4111yescharacteristic, complemented
212#1281C2×C4111yescharacteristic, complemented
213C2×C41—21no—
214C2×C41—21no—
215C2×C41—81no—
216C2×C41—161no—
217C2×C41#187611yescharacteristic
218C2×C41#190611yescharacteristic
219C2×C41#191411yescharacteristic
220#529C22×C4111yescharacteristic
221C2×#124C22×C4111yescharacteristic
222#495—21no—
223#488—21no—
224#464—21no—
225#463—21no—
226#451C4×C4111yescharacteristic
227#450C4×C4111yescharacteristic
228#448C4×C4111yescharacteristic
229#447C4×C4111yescharacteristic
230#446—21no—
231#435—21no—
232#435—21no—
233C8×D4—21no—
234#433—21no—
235#432C22×C4111yescharacteristic
236#431C22×C4111yescharacteristic
237#431C22×C4111yescharacteristic
238#405—21no—
239C2×#97C22×C4111yescharacteristic
240C2×#97C22×C4111yescharacteristic
241#330—21no—
242#329—21no—
243#329—21no—
244#329—21no—
245#323—21no—
246#323—21no—
247C41#926511yescharacteristic, direct product factor, Sylow
248C22×Q8C2×C4×C4111yescharacteristic
249C22×D4C2×C4×C4111yescharacteristic
250C2×Q16—21no—
251C2×SD16—21no—
252C2×SD16—21no—
253C2×D8—21no—
254C2×M16—21no—
255C22×C8—21no—
256C22×C8—21no—
257C22×C8—21no—
258#126—21no—
259#125D4×C4111yescharacteristic
260#124—21no—
261#124—21no—
262#124—21no—
263#124—21no—
264#124D4×C4111yescharacteristic
265#123—21no—
266C22⋊Q8C2×C4×C4111yescharacteristic
267#121C2×C4×C4111yescharacteristic
268C4×Q8D4×C4111yescharacteristic
269C4×D4—21no—
270C4×D4D4×C4111yescharacteristic
271#117—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C42C23×C4111yescharacteristic
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280#103—21no—
281#103—21no—
282#102—21no—
283#102—21no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C22⋊C8C2×C4×C4111yescharacteristic
288C22⋊C8C2×C4×C4111yescharacteristic
289C22⋊C8D4×C4111yescharacteristic
290C22⋊C8D4×C4111yescharacteristic
291#97—21no—
292#97—21no—
293#97—21no—
294#97—21no—
295#97—21no—
296#97D4×C4111yescharacteristic
297#97D4×C4111yescharacteristic
298C4×C8—21no—
299C4×C8—21no—
300C4×C8—21no—
301C24—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—81no—
305C2×Q8C22⋊C4×C4111yescharacteristic
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—81no—
310C2×D4C22⋊C4×C4111yescharacteristic
311C22×C4—21no—
312C22×C4—21no—
313C22×C4C22×C4×C4111yescharacteristic
314C22×C4C2×D4×C4111yescharacteristic
315C22×C4#41×C4111yescharacteristic
316Q16—81no—
317SD16—81no—
318SD16—81no—
319D8—81no—
320M16—42no—
321C2×C8—21no—
322C2×C8—22no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4C22⋊C4×C4111yescharacteristic
338C4⋊C4C22⋊C4×C4111yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C42—21no—
349C42C2×D4×C4111yescharacteristic
350C42C2×D4×C4111yescharacteristic
351C42C2×D4×C4111yescharacteristic
352C42#41×C4111yescharacteristic
353C23—21no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—81no—
358C23—81no—
359C23C4×D4×C4111yescharacteristic
360Q8—41no—
361Q8—41no—
362Q8—41no—
363Q8—81no—
364D4—41no—
365D4—41no—
366D4—41no—
367D4—41no—
368D4—41no—
369D4—81no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—81no—
381C2×C4C2×C22⋊C4×C4111yescharacteristic
382C2×C4C4×D4×C4111yescharacteristic
383C2×C4#120×C4111yescharacteristic
384C2×C4#121×C4111yescharacteristic
385C2×C4#121×C4111yescharacteristic
386C2×C4#123×C4111yescharacteristic
387C8—22no—
388C8—42no—
389C8—41no—
390C8—41no—
391C8—41no—
392C8—41no—
393C8—41no—
394C8—81no—
395C22—21no—
396C22—21no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—161no—
404C22—161no—
405C22C41×#38611yescharacteristic
406C22C41×#44311yescharacteristic
407C22C41×#44411yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—81no—
418C2—21no—
419C2—21no—
420C2—81no—
421C2—161no—
422C2C41×#187611yescharacteristic
423C2C41×#190611yescharacteristic
424C2C41×#191411yescharacteristic
subgroupG/H
center, socleC22×C41#386
commutatorC22×C4C22×C4×C41
Frattini, cosocleC2×C42C23×C41
prime radicalC41×#447C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C41 ◁ C2×C42×C41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C41×#2968C2semidirect
3. C41×#1532C2semidirect
4. C2×C41×#329C2semidirect
5. C41×#1469C2semidirect
6. C41×#1302C2semidirect
7. C41×#1281C2semidirect
182. #9265C41direct (N × H)
207. #2968C2×C41semidirect
208. #1532C2×C41semidirect
209. C2×#329C2×C41semidirect
210. #1469C2×C41semidirect
211. #1302C2×C41semidirect
212. #1281C2×C41semidirect
247. C41#9265direct (N × H)

Maximal subgroups

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

subgroupindex
C41×#32112
C41×#29682
C41×#15322
C2×C41×#3292
C41×#14692
C41×#13022
C41×#12812
#926541