Finite Group Databasebeta

← C41×#12243 #23788444

Subgroups — C41×#12243 #23788444

Order 10496.

464 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2693C211yesmaximal, characteristic
2C41×#2062C211yesmaximal, characteristic, complemented
3C41×#1608C211yesmaximal, characteristic, complemented
4C41×#1596C211yesmaximal, characteristic
5C41×#1344C211yesmaximal, characteristic, complemented
6C41×#1343C211yesmaximal, characteristic
7C41×#1326C211yesmaximal, characteristic, complemented
8C41×#543C2211yescharacteristic
9C2×#123×C41—21no—
10C41×#483—21no—
11C41×#450—21no—
12C41×#400—21no—
13C41×#397C2211yescharacteristic
14C41×#396—21no—
15C41×#392C2211yescharacteristic
16C41×#386—21no—
17C41×#386—21no—
18C41×#385—21no—
19C41×#385C2211yescharacteristic
20C41×#380—21no—
21C41×#342—21no—
22C41×#342C2211yescharacteristic
23C41×#328—21no—
24C41×#328—21no—
25C41×#323C2211yescharacteristic
26C41×#323C2211yescharacteristic
27C22×D4×C41—21no—
28C22×D4×C41D411yescharacteristic
29C23×C4×C41—21no—
30C23×C4×C41D411yescharacteristic
31C2×SD16×C41—41no—
32#123×C41—42no—
33#123×C41—42no—
34C22⋊Q8×C41—21no—
35C22⋊Q8×C41—21no—
36C22⋊Q8×C41—41no—
37C22⋊Q8×C41—41no—
38C2×C4⋊C4×C41—21no—
39C2×C4⋊C4×C41—21no—
40C2×C4⋊C4×C41—21no—
41C2×C4⋊C4×C41—41no—
42C2×C4⋊C4×C41D411yescharacteristic
43C2×C22⋊C4×C41—21no—
44C2×C22⋊C4×C41—21no—
45C2×C22⋊C4×C41—21no—
46C2×C22⋊C4×C41—21no—
47C2×C22⋊C4×C41—21no—
48C2×C22⋊C4×C41—21no—
49C2×C22⋊C4×C41—21no—
50C2×C22⋊C4×C41D411yescharacteristic
51C2×C22⋊C4×C41C2311yescharacteristic
52#106×C41—41no—
53#103×C41—41no—
54#102×C41—41no—
55C22⋊C8×C41—21no—
56C22⋊C8×C41—21no—
57#95×C41—21no—
58#95×C41—21no—
59#95×C41—21no—
60#95×C41—21no—
61#95×C41—41no—
62#95×C41D411yescharacteristic
63#95×C41D411yescharacteristic
64C24×C41—21no—
65C24×C41C2×D411yescharacteristic
66C2×Q8×C41—21no—
67C2×Q8×C41—21no—
68C2×D4×C41—41no—
69C2×D4×C41—42no—
70C2×D4×C41—42no—
71C2×D4×C41—41no—
72C2×D4×C41—41no—
73C2×D4×C41—81no—
74C22×C4×C41—21no—
75C22×C4×C41—21no—
76C22×C4×C41—21no—
77C22×C4×C41—21no—
78C22×C4×C41—21no—
79C22×C4×C41—21no—
80C22×C4×C41—21no—
81C22×C4×C41—42no—
82C22×C4×C41—41no—
83C22×C4×C41—41no—
84C22×C4×C41—42no—
85C22×C4×C41—81no—
86C22×C4×C41—81no—
87C22×C4×C41C2×D411yescharacteristic
88C22×C4×C41C2×D411yescharacteristic
89SD16×C41—82no—
90C2×C8×C41—41no—
91C2×C8×C41—41no—
92C4⋊C4×C41—21no—
93C4⋊C4×C41—21no—
94C4⋊C4×C41—41no—
95C4⋊C4×C41—42no—
96C4⋊C4×C41—42no—
97C4⋊C4×C41—41no—
98C4⋊C4×C41—41no—
99C4⋊C4×C41—81no—
100C4⋊C4×C41—82no—
101C22⋊C4×C41—21no—
102C22⋊C4×C41—22no—
103C22⋊C4×C41—41no—
104C22⋊C4×C41—42no—
105C22⋊C4×C41—41no—
106C22⋊C4×C41—41no—
107C22⋊C4×C41—41no—
108C22⋊C4×C41—41no—
109C22⋊C4×C41—41no—
110C22⋊C4×C41—81no—
111C22⋊C4×C41—81no—
112C22⋊C4×C41—81no—
113C22⋊C4×C41SD1612yescomplemented
114C23×C41—21no—
115C23×C41—21no—
116C23×C41—22no—
117C23×C41—41no—
118C23×C41—41no—
119C23×C41—41no—
120C23×C41—41no—
121C23×C41—41no—
122C23×C41—41no—
123C23×C41—41no—
124C23×C41—41no—
125C23×C41#12011yescharacteristic
126C23×C41C2×SD1611yescharacteristic
127C23×C41AGL(1,ℤ/8)11yescharacteristic
128Q8×C41—81no—
129Q8×C41—81no—
130D4×C41—42no—
131D4×C41—81no—
132D4×C41—82no—
133D4×C41—82no—
134C2×C4×C41—22no—
135C2×C4×C41—21no—
136C2×C4×C41—21no—
137C2×C4×C41—41no—
138C2×C4×C41—41no—
139C2×C4×C41—41no—
140C2×C4×C41—41no—
141C2×C4×C41—42no—
142C2×C4×C41—41no—
143C2×C4×C41—41no—
144C2×C4×C41—41no—
145C2×C4×C41—41no—
146C2×C4×C41—41no—
147C2×C4×C41—41no—
148C2×C4×C41—42no—
149C2×C4×C41—41no—
150C2×C4×C41—41no—
151C2×C4×C41—41no—
152C2×C4×C41—41no—
153C2×C4×C41—41no—
154C2×C4×C41—41no—
155C2×C4×C41—41no—
156C2×C4×C41—41no—
157C2×C4×C41—84no—
158C2×C4×C41—81no—
159C2×C4×C41—81no—
160C2×C4×C41—81no—
161C2×C4×C41—81no—
162C2×C4×C41—81no—
163C8×C41—81no—
164C8×C41—81no—
165#12243C4111yesmaximal, characteristic, direct product factor, Sylow
166C22×C41—21no—
167C22×C41—21no—
168C22×C41—21no—
169C22×C41—21no—
170C22×C41—41no—
171C22×C41—41no—
172C22×C41—42no—
173C22×C41—42no—
174C22×C41—41no—
175C22×C41—41no—
176C22×C41—42no—
177C22×C41—81no—
178C22×C41—81no—
179C22×C41—81no—
180C22×C41—81no—
181C22×C41—81no—
182C22×C41#45011yescharacteristic
183C22×C41#45611yescharacteristic
184C22×C41B(4,𝔽2)11yescharacteristic
185C4×C41—42no—
186C4×C41—41no—
187C4×C41—41no—
188C4×C41—81no—
189C4×C41—81no—
190C4×C41—82no—
191C4×C41—81no—
192C4×C41—81no—
193C4×C41—81no—
194C4×C41—81no—
195#2693C2×C4111yescharacteristic
196#2062C2×C4111yescharacteristic, complemented
197#1608C2×C4111yescharacteristic, complemented
198#1596C2×C4111yescharacteristic
199#1344C2×C4111yescharacteristic, complemented
200#1343C2×C4111yescharacteristic
201#1326C2×C4111yescharacteristic, complemented
202C2×C41—21no—
203C2×C41—21no—
204C2×C41—41no—
205C2×C41—41no—
206C2×C41—82no—
207C2×C41#160411yescharacteristic
208C2×C41#219711yescharacteristic
209C2×C41#220011yescharacteristic
210#543C22×C4111yescharacteristic
211C2×#123—21no—
212#483—21no—
213#450—21no—
214#400—21no—
215#397C22×C4111yescharacteristic
216#396—21no—
217#392C22×C4111yescharacteristic
218#386—21no—
219#386—21no—
220#385—21no—
221#385C22×C4111yescharacteristic
222#380—21no—
223#342—21no—
224#342C22×C4111yescharacteristic
225#328—21no—
226#328—21no—
227#323C22×C4111yescharacteristic
228#323C22×C4111yescharacteristic
229C41#1224311yescharacteristic, direct product factor, Sylow
230C22×D4—21no—
231C22×D4D4×C4111yescharacteristic
232C23×C4—21no—
233C23×C4D4×C4111yescharacteristic
234C2×SD16—41no—
235#123—42no—
236#123—42no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—41no—
240C22⋊Q8—41no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4D4×C4111yescharacteristic
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4D4×C4111yescharacteristic
254C2×C22⋊C4C23×C4111yescharacteristic
255#106—41no—
256#103—41no—
257#102—41no—
258C22⋊C8—21no—
259C22⋊C8—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—41no—
265#95D4×C4111yescharacteristic
266#95D4×C4111yescharacteristic
267C24—21no—
268C24C2×D4×C4111yescharacteristic
269C2×Q8—21no—
270C2×Q8—21no—
271C2×D4—41no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—81no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—81no—
289C22×C4—81no—
290C22×C4C2×D4×C4111yescharacteristic
291C22×C4C2×D4×C4111yescharacteristic
292SD16—82no—
293C2×C8—41no—
294C2×C8—41no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—81no—
303C4⋊C4—82no—
304C22⋊C4—21no—
305C22⋊C4—22no—
306C22⋊C4—41no—
307C22⋊C4—42no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—81no—
314C22⋊C4—81no—
315C22⋊C4—81no—
316C22⋊C4SD16×C4112yescomplemented
317C23—21no—
318C23—21no—
319C23—22no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23#120×C4111yescharacteristic
329C23C2×SD16×C4111yescharacteristic
330C23AGL(1,ℤ/8)×C4111yescharacteristic
331Q8—81no—
332Q8—81no—
333D4—42no—
334D4—81no—
335D4—82no—
336D4—82no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—84no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C8—81no—
367C8—81no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—41no—
373C22—41no—
374C22—42no—
375C22—42no—
376C22—41no—
377C22—41no—
378C22—42no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22C41×#45011yescharacteristic
385C22C41×#45611yescharacteristic
386C22C41×B(4,𝔽2)11yescharacteristic
387C4—42no—
388C4—41no—
389C4—41no—
390C4—81no—
391C4—81no—
392C4—82no—
393C4—81no—
394C4—81no—
395C4—81no—
396C4—81no—
397C2—21no—
398C2—21no—
399C2—41no—
400C2—41no—
401C2—82no—
402C2C41×#160411yescharacteristic
403C2C41×#219711yescharacteristic
404C2C41×#220011yescharacteristic
subgroupG/H
center, socleC22×C41B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C41
prime radicalC41×#392C22
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C41 ◁ C23×C41 ◁ C2×C22⋊C4×C41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C41×#2062C2semidirect
3. C41×#1608C2semidirect
5. C41×#1344C2semidirect
7. C41×#1326C2semidirect
113. C22⋊C4×C41SD16semidirect
165. #12243C41direct (N × H)
196. #2062C2×C41semidirect
197. #1608C2×C41semidirect
199. #1344C2×C41semidirect
201. #1326C2×C41semidirect
229. C41#12243direct (N × H)
316. C22⋊C4SD16×C41semidirect

Maximal subgroups

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

subgroupindex
C41×#26932
C41×#20622
C41×#16082
C41×#15962
C41×#13442
C41×#13432
C41×#13262
#1224341