Finite Group Databasebeta

← #26192398

Subgroups — #26192398

Order 12032.

982 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 982 conjugacy classes of subgroups — too many to draw. Only its 179 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
1#20253762C211yesmaximal, characteristic
2#20253518C212yesmaximal, complemented
3#20242574C218yesmaximal, complemented
4#20242102C211yesmaximal, characteristic
5#20241835C212yesmaximal, complemented
6C2×C47×#559C211yesmaximal, characteristic
7#15689680C2211yescharacteristic
8#15689487—22no—
9#15689469C2214yes—
10#15689453—22no—
11#15689408C2214yescomplemented
12#15689333C2214yescomplemented
13#15689279—22no—
14#15689228—24no—
15#15689180C2212yes—
16#15689128—22no—
17C4×#10625329C2211yescharacteristic
18#15688867C2214yes—
19Q8⋊(C47⋊C8)C2218yescomplemented
20C47×#559—22no—
21C47×#559—24no—
22C47×#559C2214yes—
23C2×#127×C47C2211yescharacteristic
24C2×#124×C47—22no—
25C2×#123×C47—22no—
26C2×C4×Q8×C47C2212yes—
27C2×C4×D4×C47—22no—
28C2×#10625337—24no—
29C2×#10625329—21no—
30#13094024C2312yes—
31#13094016—21no—
32#13094003—22no—
33C4×Dic94—22no—
34#13093983—21no—
35#13093955—24no—
36C4⋊(C47⋊C8)C2314yes—
37#13093948—24no—
38#13093947—24no—
39#13093945—28no—
40C22⋊(C47⋊C8)—22no—
41#13093935—22no—
42#13093935—21no—
43C4×C47⋊C8C2312yes—
44C22×Q8×C47D412yes—
45C22×D4×C47—22no—
46C22×D4×C47D412yes—
47C23×C4×C47—22no—
48#127×C47—22no—
49#127×C47D412yes—
50#127×C47C2312yes—
51#124×C47—28no—
52#124×C47—28no—
53#123×C47—28no—
54#123×C47—28no—
55C4×Q8×C47—22no—
56C4×Q8×C47—22no—
57C4×Q8×C47D414yes—
58C4×Q8×C47C2314yes—
59C4×D4×C47—28no—
60C4×D4×C47—28no—
61C2×C4⋊C4×C47—22no—
62C2×C4⋊C4×C47D412yes—
63C2×C22⋊C4×C47—24no—
64C2×C22⋊C4×C47—22no—
65C2×C42×C47—22no—
66C2×C42×C47C2311yescharacteristic
67C24×C47—22no—
68C2×Q8×C47—22no—
69C2×Q8×C47—44no—
70C2×Q8×C47C2×D414yes—
71C2×D4×C47—24no—
72C2×D4×C47—22no—
73C2×D4×C47—24no—
74C2×D4×C47—48no—
75C2×D4×C47—44no—
76C2×D4×C47C2×D414yes—
77C22×C4×C47—22no—
78C22×C4×C47—22no—
79C22×C4×C47—28no—
80C22×C4×C47—84no—
81C22×C4×C47C2×D412yes—
82C22×C4×C47C2×D411yescharacteristic
83C2×Dic94—22no—
84C4⋊C4×C47—24no—
85C4⋊C4×C47—22no—
86C4⋊C4×C47—24no—
87C4⋊C4×C47C2×D414yes—
88C22⋊C4×C47—216no—
89C22⋊C4×C47—28no—
90C42×C47—24no—
91C42×C47—24no—
92C42×C47C2×D411yescharacteristic
93C42×C47C2×D412yes—
94C42×C47C2411yescharacteristic
95C22⋊Dic47—41no—
96#10625337—48no—
97C4⋊Dic47—22no—
98C4⋊Dic47—22no—
99Dic47⋊C4—42no—
100C4×Dic47—41no—
101#10625329—42no—
102C2×C47⋊C8—22no—
103C2×C47⋊C8—24no—
104C23×C47—24no—
105C23×C47—48no—
106C23×C47—44no—
107C23×C47#12011yescharacteristic
108Q8×C47—24no—
109Q8×C47—24no—
110Q8×C47—44no—
111D4×C47—44no—
112D4×C47—48no—
113D4×C47—48no—
114D4×C47—82no—
115C2×C4×C47—22no—
116C2×C4×C47—24no—
117C2×C4×C47—24no—
118C2×C4×C47—24no—
119C2×C4×C47—44no—
120C2×C4×C47—44no—
121C2×C4×C47—88no—
122C2×C4×C47#12011yescharacteristic
123C2×C4×C47#12012yes—
124C2×C4×C47C22×D411yescharacteristic
125C2×C4×C47C22×D412yes—
126C2×Dic47—42no—
127Dic94—44no—
128C47⋊C8—42no—
129C47⋊C8—44no—
130#18531—471nomaximal, Sylow
131C22×C47—22no—
132C22×C47—48no—
133C22×C47—44no—
134C22×C47—88no—
135C22×C47#45512yes—
136C22×C47C2×#12011yescharacteristic
137C4×C47—22no—
138C4×C47—21no—
139C4×C47—21no—
140C4×C47—44no—
141C4×C47—44no—
142C4×C47#57514yes—
143Dic47—82no—
144C2×#559D4711yescharacteristic
145#3138—471no—
146#3113—472no—
147#1640—478no—
148#1555—471no—
149#1491—472no—
150C2×C47—22no—
151C2×C47—84no—
152C2×C47#300512yes—
153C2×C47C2×#45511yescharacteristic
154#559—22no—
155#559—24no—
156#559C2×D4714yes—
157#533—471no—
158C2×#127C2×D4711yescharacteristic
159C2×#124—22no—
160C2×#123—22no—
161C2×C4×Q8C2×D4712yes—
162C2×C4×D4—22no—
163#490—942no—
164#486—474no—
165#483—942no—
166#475—474no—
167#462—474no—
168#451—942no—
169#441—944no—
170#432—472no—
171#418—942no—
172C4×M16—471no—
173#331—474no—
174Q8⋊C8—478no—
175C47#1853111yescharacteristic, complemented, Sylow
176C22×Q8C22⋊D4712yes—
177C22×D4—22no—
178C22×D4C22⋊D4712yes—
179C23×C4—22no—
180C2×Q16—944no—
181C2×M16—941no—
182C4⋊Q8—472no—
183#127—22no—
184#127C22⋊D4712yes—
185#127C22×D4712yes—
186#125—941no—
187#124—28no—
188#124—28no—
189#123—28no—
190#123—28no—
191C22⋊Q8—942no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—942no—
195C4×Q8C22⋊D4714yes—
196C4×Q8C22×D4714yes—
197C4×D4—28no—
198C4×D4—28no—
199#117—941no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4C22⋊D4712yes—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—22no—
204C2×C42—22no—
205C2×C42C22×D4711yescharacteristic
206#106—944no—
207C4⋊C8—474no—
208#103—944no—
209#103—944no—
210#102—948no—
211C22⋊C8—942no—
212#97—941no—
213#97—942no—
214C4×C8—472no—
215C24—22no—
216C2×Q8—22no—
217C2×Q8—44no—
218C2×Q8—942no—
219C2×Q8C2×C22⋊D4714yes—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—24no—
223C2×D4—48no—
224C2×D4—44no—
225C2×D4C2×C22⋊D4714yes—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—28no—
229C22×C4—84no—
230C22×C4C2×C22⋊D4712yes—
231C22×C4C2×C22⋊D4711yescharacteristic
232Q16—1888no—
233M16—1882no—
234C2×C8—942no—
235C2×C8—944no—
236C4⋊C4—24no—
237C4⋊C4—22no—
238C4⋊C4—24no—
239C4⋊C4—942no—
240C4⋊C4—942no—
241C4⋊C4—1882no—
242C4⋊C4C2×C22⋊D4714yes—
243C22⋊C4—216no—
244C22⋊C4—28no—
245C22⋊C4—1881no—
246C42—24no—
247C42—24no—
248C42—1881no—
249C42C2×C22⋊D4711yescharacteristic
250C42C2×C22⋊D4712yes—
251C42C23×D4711yescharacteristic
252C23—24no—
253C23—48no—
254C23—44no—
255C23#1309399311yescharacteristic
256Q8—24no—
257Q8—24no—
258Q8—44no—
259Q8—1884no—
260D4—44no—
261D4—48no—
262D4—48no—
263D4—82no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—88no—
271C2×C4—1882no—
272C2×C4#1309399312yes—
273C2×C4#1309399311yescharacteristic
274C2×C4C22×C22⋊D4711yescharacteristic
275C2×C4C22×C22⋊D4712yes—
276C8—1882no—
277C8—1884no—
278C22—22no—
279C22—44no—
280C22—48no—
281C22—88no—
282C22#1568930112yes—
283C22C2×#1309399311yescharacteristic
284C4—21no—
285C4—21no—
286C4—22no—
287C4—44no—
288C4—44no—
289C4—3762no—
290C4#1568992114yes—
291C2—22no—
292C2—84no—
293C2#2025263012yes—
294C2C2×#1568930111yescharacteristic
subgroupG/H
centerC22C2×#13093993
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×C47×#559C2
socleC22×C47C2×#120
prime radicalC2×#127×C47C22
hypercenterC2×#559D47
hypocenterC47#18531
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
2. #20253518C2semidirect
3. #20242574C2semidirect
5. #20241835C2semidirect
11. #15689408C22semidirect
12. #15689333C22semidirect
19. Q8⋊(C47⋊C8)C22semidirect
175. C47#18531semidirect

Maximal subgroups

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

subgroupindex
#202537622
#202535182
#202425742
#202421022
#202418352
C2×C47×#5592
#1853147