Finite Group Databasebeta

← C41×#10294 #23786495

Subgroups — C41×#10294 #23786495

Order 10496.

640 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C41×#3229C211yesmaximal, characteristic, complemented
2C41×#3208C211yesmaximal, characteristic
3C41×#2858C211yesmaximal, characteristic
4C41×#1994C211yesmaximal, characteristic, complemented
5C41×#1546C211yesmaximal, characteristic, complemented
6C41×#1545C211yesmaximal, characteristic
7C41×#1276C211yesmaximal, characteristic, complemented
8C41×#533C2211yescharacteristic
9C2×C4⋊Q8×C41C2211yescharacteristic
10C2×#127×C41C2211yescharacteristic
11C2×#124×C41—22no—
12C41×#498—21no—
13C41×#492—21no—
14C41×#487—21no—
15C41×#486—21no—
16C41×#483—22no—
17C41×#475—22no—
18C41×#460—22no—
19C41×#450—22no—
20C41×#431C2211yescharacteristic
21C41×#431C2211yescharacteristic
22C41×#423C2211yescharacteristic
23C41×#390—22no—
24C41×#390—22no—
25C43×C41C2211yescharacteristic
26C41×#339—22no—
27C41×#332—22no—
28C41×#331—22no—
29C22×Q8×C41—21no—
30C22×D4×C41—21no—
31C22×D4×C41—21no—
32C22×D4×C41D412yes—
33C2×SD16×C41—24no—
34C2×M16×C41—21no—
35C4⋊Q8×C41—21no—
36C4⋊Q8×C41—21no—
37C4⋊Q8×C41—21no—
38C4⋊Q8×C41—21no—
39C4⋊Q8×C41D411yescharacteristic
40C4⋊Q8×C41D411yescharacteristic
41#127×C41—21no—
42#127×C41—21no—
43#127×C41—21no—
44#127×C41—21no—
45#125×C41—21no—
46#124×C41—44no—
47#124×C41—44no—
48C22⋊Q8×C41—21no—
49C22⋊Q8×C41—21no—
50C4×Q8×C41—21no—
51C4×Q8×C41—21no—
52#117×C41—21no—
53C2×C4⋊C4×C41—21no—
54C2×C4⋊C4×C41D412yes—
55C2×C22⋊C4×C41—28no—
56C2×C42×C41—21no—
57C2×C42×C41—21no—
58C2×C42×C41D412yes—
59C2×C42×C41C2311yescharacteristic
60#106×C41—24no—
61C4⋊C8×C41—21no—
62C4⋊C8×C41—21no—
63C4⋊C8×C41D412yes—
64C4⋊C8×C41D412yes—
65#103×C41—24no—
66#102×C41—24no—
67C22⋊C8×C41—21no—
68C22⋊C8×C41—21no—
69C4×C8×C41—21no—
70C4×C8×C41—21no—
71C24×C41—22no—
72C2×Q8×C41—21no—
73C2×Q8×C41—21no—
74C2×Q8×C41—41no—
75C2×Q8×C41—41no—
76C2×Q8×C41—42no—
77C2×Q8×C41—42no—
78C2×D4×C41—22no—
79C2×D4×C41—22no—
80C2×D4×C41—44no—
81C2×D4×C41—44no—
82C2×D4×C41—41no—
83C2×D4×C41—41no—
84C2×D4×C41—41no—
85C2×D4×C41—41no—
86C2×D4×C41—42no—
87C2×D4×C41—42no—
88C22×C4×C41—22no—
89C22×C4×C41—22no—
90C22×C4×C41C2×D411yescharacteristic
91C22×C4×C41C2×D411yescharacteristic
92C22×C4×C41C2×D411yescharacteristic
93SD16×C41—84no—
94M16×C41—42no—
95C2×C8×C41—22no—
96C2×C8×C41—22no—
97C2×C8×C41—41no—
98C4⋊C4×C41—21no—
99C4⋊C4×C41—22no—
100C4⋊C4×C41—21no—
101C4⋊C4×C41—22no—
102C4⋊C4×C41—21no—
103C4⋊C4×C41—21no—
104C4⋊C4×C41—41no—
105C4⋊C4×C41—41no—
106C4⋊C4×C41—41no—
107C4⋊C4×C41—41no—
108C22⋊C4×C41—416no—
109C22⋊C4×C41—41no—
110C42×C41—22no—
111C42×C41—42no—
112C42×C41—42no—
113C42×C41—41no—
114C42×C41SD1614yescomplemented
115C42×C41C2×D411yescharacteristic
116C42×C41C2×D411yescharacteristic
117C42×C41C2×D411yescharacteristic
118C42×C41#4111yescharacteristic
119C23×C41—44no—
120C23×C41—44no—
121C23×C41—42no—
122C23×C41—44no—
123C23×C41#12011yescharacteristic
124Q8×C41—42no—
125Q8×C41—42no—
126Q8×C41—82no—
127Q8×C41—81no—
128Q8×C41—81no—
129D4×C41—44no—
130D4×C41—82no—
131D4×C41—82no—
132D4×C41—82no—
133D4×C41—82no—
134D4×C41—81no—
135D4×C41—81no—
136C2×C4×C41—22no—
137C2×C4×C41—22no—
138C2×C4×C41—24no—
139C2×C4×C41—42no—
140C2×C4×C41—42no—
141C2×C4×C41—41no—
142C2×C4×C41—41no—
143C2×C4×C41—42no—
144C2×C4×C41—44no—
145C2×C4×C41#12011yescharacteristic
146C2×C4×C41#12011yescharacteristic
147C2×C4×C41#12111yescharacteristic
148C2×C4×C41#12111yescharacteristic
149C2×C4×C41#12111yescharacteristic
150C2×C4×C41#12711yescharacteristic
151C2×C4×C41C2×SD1612yes—
152C2×C4×C41AGL(1,ℤ/8)12yes—
153C8×C41—42no—
154C8×C41—42no—
155C8×C41—81no—
156#10294C4111yesmaximal, characteristic, direct product factor, Sylow
157C22×C41—21no—
158C22×C41—21no—
159C22×C41—88no—
160C22×C41—82no—
161C22×C41—82no—
162C22×C41—82no—
163C22×C41#39211yescharacteristic
164C22×C41#45012yes—
165C4×C41—21no—
166C4×C41—21no—
167C4×C41—44no—
168C4×C41—81no—
169C4×C41—81no—
170C4×C41—82no—
171C4×C41#45412yes—
172C4×C41#45512yes—
173C4×C41#46012yes—
174C4×C41#49211yescharacteristic
175C4×C41#49611yescharacteristic
176#3229C2×C4111yescharacteristic, complemented
177#3208C2×C4111yescharacteristic
178#2858C2×C4111yescharacteristic
179#1994C2×C4111yescharacteristic, complemented
180#1546C2×C4111yescharacteristic, complemented
181#1545C2×C4111yescharacteristic
182#1276C2×C4111yescharacteristic, complemented
183C2×C41—22no—
184C2×C41—162no—
185C2×C41#200311yescharacteristic
186C2×C41#200411yescharacteristic
187C2×C41#201411yescharacteristic
188#533C22×C4111yescharacteristic
189C2×C4⋊Q8C22×C4111yescharacteristic
190C2×#127C22×C4111yescharacteristic
191C2×#124—22no—
192#498—21no—
193#492—21no—
194#487—21no—
195#486—21no—
196#483—22no—
197#475—22no—
198#460—22no—
199#450—22no—
200#431C22×C4111yescharacteristic
201#431C22×C4111yescharacteristic
202#423C22×C4111yescharacteristic
203#390—22no—
204#390—22no—
205C43C22×C4111yescharacteristic
206#339—22no—
207#332—22no—
208#331—22no—
209C41#1029411yescharacteristic, direct product factor, Sylow
210C22×Q8—21no—
211C22×D4—21no—
212C22×D4—21no—
213C22×D4D4×C4112yes—
214C2×SD16—24no—
215C2×M16—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8D4×C4111yescharacteristic
221C4⋊Q8D4×C4111yescharacteristic
222#127—21no—
223#127—21no—
224#127—21no—
225#127—21no—
226#125—21no—
227#124—44no—
228#124—44no—
229C22⋊Q8—21no—
230C22⋊Q8—21no—
231C4×Q8—21no—
232C4×Q8—21no—
233#117—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4D4×C4112yes—
236C2×C22⋊C4—28no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42D4×C4112yes—
240C2×C42C23×C4111yescharacteristic
241#106—24no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8D4×C4112yes—
245C4⋊C8D4×C4112yes—
246#103—24no—
247#102—24no—
248C22⋊C8—21no—
249C22⋊C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C24—22no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—44no—
262C2×D4—44no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—42no—
268C2×D4—42no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4C2×D4×C4111yescharacteristic
272C22×C4C2×D4×C4111yescharacteristic
273C22×C4C2×D4×C4111yescharacteristic
274SD16—84no—
275M16—42no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—41no—
279C4⋊C4—21no—
280C4⋊C4—22no—
281C4⋊C4—21no—
282C4⋊C4—22no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C22⋊C4—416no—
290C22⋊C4—41no—
291C42—22no—
292C42—42no—
293C42—42no—
294C42—41no—
295C42SD16×C4114yescomplemented
296C42C2×D4×C4111yescharacteristic
297C42C2×D4×C4111yescharacteristic
298C42C2×D4×C4111yescharacteristic
299C42#41×C4111yescharacteristic
300C23—44no—
301C23—44no—
302C23—42no—
303C23—44no—
304C23#120×C4111yescharacteristic
305Q8—42no—
306Q8—42no—
307Q8—82no—
308Q8—81no—
309Q8—81no—
310D4—44no—
311D4—82no—
312D4—82no—
313D4—82no—
314D4—82no—
315D4—81no—
316D4—81no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4#120×C4111yescharacteristic
327C2×C4#120×C4111yescharacteristic
328C2×C4#121×C4111yescharacteristic
329C2×C4#121×C4111yescharacteristic
330C2×C4#121×C4111yescharacteristic
331C2×C4#127×C4111yescharacteristic
332C2×C4C2×SD16×C4112yes—
333C2×C4AGL(1,ℤ/8)×C4112yes—
334C8—42no—
335C8—42no—
336C8—81no—
337C22—21no—
338C22—21no—
339C22—88no—
340C22—82no—
341C22—82no—
342C22—82no—
343C22C41×#39211yescharacteristic
344C22C41×#45012yes—
345C4—21no—
346C4—21no—
347C4—44no—
348C4—82no—
349C4—81no—
350C4—81no—
351C4C41×#45412yes—
352C4C41×#45512yes—
353C4C41×#46012yes—
354C4C41×#49211yescharacteristic
355C4C41×#49611yescharacteristic
356C2—22no—
357C2—162no—
358C2C41×#200311yescharacteristic
359C2C41×#200411yescharacteristic
360C2C41×#201411yescharacteristic
subgroupG/H
center, socleC22×C41#392
commutator, Frattini, cosocleC2×C42C23×C41
prime radicalC2×#127×C41C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C41 ◁ C2×C42×C41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C41×#3229C2semidirect
4. C41×#1994C2semidirect
5. C41×#1546C2semidirect
7. C41×#1276C2semidirect
114. C42×C41SD16semidirect
156. #10294C41direct (N × H)
176. #3229C2×C41semidirect
179. #1994C2×C41semidirect
180. #1546C2×C41semidirect
182. #1276C2×C41semidirect
209. C41#10294direct (N × H)
295. C42SD16×C41semidirect

Maximal subgroups

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

subgroupindex
C41×#32292
C41×#32082
C41×#28582
C41×#19942
C41×#15462
C41×#15452
C41×#12762
#1029441