Finite Group Databasebeta

← #26192583

Subgroups — #26192583

Order 12032.

982 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 982 conjugacy classes of subgroups — too many to draw. Only its 163 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
1C2×#15689839C211yesmaximal, characteristic
2#20253483C212yesmaximal, complemented
3#20242613C218yesmaximal, complemented
4#20242114C211yesmaximal, characteristic, complemented
5#20241832C212yesmaximal, complemented
6C47×#3139C211yesmaximal, characteristic, complemented
7#15689839—24no—
8#15689839—22no—
9#15689839C2214yes—
10C2×#13094013—22no—
11C2×#13093999—22no—
12C2×C4×D188C2212yes—
13C2×#13093982—22no—
14#15689456—22no—
15#15689400C2214yescomplemented
16#15689325C2214yescomplemented
17#15689273—22no—
18#15689221—24no—
19#15689180C2212yes—
20#15688872C2214yescomplemented
21#15688852C2218yescomplemented
22C47×#533C2211yescharacteristic
23C2×C4⋊Q8×C47C2211yescharacteristic
24C47×#491—22no—
25C47×#487C2214yescomplemented
26C47×#419—22no—
27C4×M16×C47C2211yescharacteristic
28C22×D188—22no—
29C22×D188D412yes—
30C22×C4×D47—22no—
31C2×#10625336—24no—
32#13094013—28no—
33#13094013—28no—
34#13093999—28no—
35#13093999—28no—
36C4×D188—22no—
37C4×D188—22no—
38C4×D188D414yes—
39C4×D188C2314yes—
40#13093982—28no—
41#13093982—28no—
42C2×C4⋊Dic47D412yes—
43C2×(C2×D47)⋊C4—24no—
44C2×(C2×D47)⋊C4—22no—
45C2×C4×Dic47—22no—
46#13093957—24no—
47C4⋊(C47⋊C8)C2314yes—
48#13093948—24no—
49#13093944—24no—
50C22⋊(C47⋊C8)—22no—
51#13093935—22no—
52C22×Q8×C47D412yes—
53C2×M16×C47—21no—
54C4⋊Q8×C47—22no—
55C4⋊Q8×C47D412yes—
56C4⋊Q8×C47C2312yes—
57C4⋊Q8×C47C2312yes—
58#125×C47—21no—
59C22⋊Q8×C47—22no—
60C4×Q8×C47—22no—
61#117×C47—21no—
62C2×C4⋊C4×C47—22no—
63C2×C42×C47C2311yescharacteristic
64#103×C47—28no—
65#97×C47—21no—
66C4×C8×C47C2312yes—
67C23×D47—22no—
68C2×Q8×C47—22no—
69C2×Q8×C47—22no—
70C2×Q8×C47—44no—
71C2×Q8×C47C2×D414yes—
72C22×C4×C47—22no—
73C22×C4×C47C2×D412yes—
74C22×C4×C47C2×D411yescharacteristic
75C22×Dic47—22no—
76C2×D188—22no—
77C2×D188—28no—
78C2×D188—44no—
79C2×D188—84no—
80C2×D188C2×D414yes—
81C2×C4×D47—24no—
82C2×C4×D47—24no—
83C2×C4×D47—48no—
84M16×C47—42no—
85C2×C8×C47—22no—
86C4⋊C4×C47—24no—
87C4⋊C4×C47—24no—
88C4⋊C4×C47—22no—
89C4⋊C4×C47—22no—
90C4⋊C4×C47—42no—
91C22⋊C4×C47—41no—
92C42×C47—41no—
93C42×C47C2×D412yes—
94C42×C47C2×D411yescharacteristic
95C42×C47C2411yescharacteristic
96#10625336—48no—
97(C2×D47)⋊C4—216no—
98(C2×D47)⋊C4—24no—
99(C2×D47)⋊C4—24no—
100C4⋊Dic47—22no—
101C4⋊Dic47C2×D414yes—
102C4×Dic47—28no—
103C2×C47⋊C8—24no—
104C23×C47#12011yescharacteristic
105C22×D47—24no—
106C22×D47—44no—
107C22×D47—44no—
108C22×D47—44no—
109Q8×C47—44no—
110Q8×C47—44no—
111Q8×C47—82no—
112C2×C4×C47—22no—
113C2×C4×C47—24no—
114C2×C4×C47—24no—
115C2×C4×C47—44no—
116C2×C4×C47—42no—
117C2×C4×C47#12012yes—
118C2×C4×C47#12011yescharacteristic
119C2×C4×C47C22×D412yes—
120C2×C4×C47C22×D411yescharacteristic
121C2×Dic47—24no—
122C2×Dic47—44no—
123D188—24no—
124D188—24no—
125D188—44no—
126D188—88no—
127C4×D47—48no—
128C4×D47—48no—
129C8×C47—42no—
130C47⋊C8—44no—
131#18552—471nomaximal, Sylow
132C22×C47—22no—
133C22×C47#45512yes—
134C22×C47C2×#12011yescharacteristic
135C2×D47—44no—
136C2×D47—44no—
137C2×D47—44no—
138C2×D47—88no—
139C4×C47—22no—
140C4×C47—21no—
141C4×C47—21no—
142C4×C47—44no—
143C4×C47—82no—
144C4×C47#57514yes—
145Dic47—44no—
146C2×#555—471no—
147#3139D4711yescharacteristic, complemented
148#3110—472no—
149#1646—478no—
150#1558—471no—
151#1490—472no—
152C2×C47—22no—
153C2×C47#300512yes—
154C2×C47C2×#45511yescharacteristic
155D47—84no—
156#555—474no—
157#555—944no—
158#555—942no—
159#533C2×D4711yescharacteristic
160C2×C4⋊Q8C2×D4711yescharacteristic
161C2×#124—942no—
162C2×#121—942no—
163C2×C4×D4—472no—
164C2×#117—942no—
165#491—22no—
166#487C2×D4714yescomplemented
167#484—942no—
168#474—474no—
169#461—474no—
170#450—942no—
171#440—944no—
172#432—472no—
173#419—22no—
174C4×M16C2×D4711yescharacteristic
175#333—474no—
176D4⋊C8—478no—
177C47#1855211yescharacteristic, complemented, Sylow
178C22×Q8C22⋊D4712yes—
179C22×D4—472no—
180C22×D4—942no—
181C23×C4—942no—
182C2×SD16—944no—
183C2×M16—21no—
184C4⋊Q8—22no—
185C4⋊Q8C22⋊D4712yes—
186C4⋊Q8C22×D4712yes—
187C4⋊Q8C22×D4712yes—
188#125—21no—
189#124—948no—
190#124—948no—
191C22⋊Q8—22no—
192#121—948no—
193#121—948no—
194C4×Q8—22no—
195C4×D4—474no—
196C4×D4—474no—
197C4×D4—942no—
198C4×D4—942no—
199#117—21no—
200#117—948no—
201#117—948no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—472no—
204C2×C22⋊C4—944no—
205C2×C22⋊C4—942no—
206C2×C42—942no—
207C2×C42C22×D4711yescharacteristic
208#107—944no—
209C4⋊C8—474no—
210#103—28no—
211#103—944no—
212#102—944no—
213C22⋊C8—942no—
214#97—21no—
215#97—942no—
216C4×C8C22×D4712yes—
217C24—942no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—44no—
221C2×Q8C2×C22⋊D4714yes—
222C2×D4—474no—
223C2×D4—948no—
224C2×D4—942no—
225C2×D4—3764no—
226C2×D4—1884no—
227C22×C4—22no—
228C22×C4—942no—
229C22×C4—944no—
230C22×C4—944no—
231C22×C4—1888no—
232C22×C4D4×D4712yes—
233C22×C4C2×C22⋊D4711yescharacteristic
234SD16—1888no—
235M16—42no—
236C2×C8—22no—
237C2×C8—944no—
238C4⋊C4—24no—
239C4⋊C4—24no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—42no—
243C4⋊C4—474no—
244C4⋊C4—942no—
245C22⋊C4—41no—
246C22⋊C4—9416no—
247C22⋊C4—944no—
248C22⋊C4—944no—
249C42—41no—
250C42—948no—
251C42D4×D4712yes—
252C42C2×C22⋊D4711yescharacteristic
253C42C23×D4711yescharacteristic
254C23—944no—
255C23—1884no—
256C23—1884no—
257C23—1884no—
258C23#1309399211yescharacteristic
259Q8—44no—
260Q8—44no—
261Q8—82no—
262D4—944no—
263D4—944no—
264D4—3768no—
265D4—1884no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—944no—
272C2×C4—1888no—
273C2×C4—1884no—
274C2×C4—1888no—
275C2×C4#1309399211yescharacteristic
276C2×C4#1309399212yes—
277C2×C4C2×D4×D4712yes—
278C2×C4C22×C22⋊D4711yescharacteristic
279C8—42no—
280C8—1884no—
281C22—22no—
282C22—3768no—
283C22—1884no—
284C22—1884no—
285C22—1884no—
286C22#1568930012yes—
287C22C2×#1309399211yescharacteristic
288C4—21no—
289C4—21no—
290C4—22no—
291C4—44no—
292C4—82no—
293C4—1884no—
294C4#1568992214yes—
295C2—22no—
296C2—3764no—
297C2#2025262912yes—
298C2C2×#1568930011yescharacteristic
subgroupG/H
centerC22C2×#13093992
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3139C2
socleC22×C47C2×#120
prime radicalC2×#15689839C2
hypercenter#3139D47
hypocenterC47#18552
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3139

Complements

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

NH ≅ G/Ntype
2. #20253483C2semidirect
3. #20242613C2semidirect
4. #20242114C2semidirect
5. #20241832C2semidirect
6. C47×#3139C2semidirect
15. #15689400C22semidirect
16. #15689325C22semidirect
20. #15688872C22semidirect
21. #15688852C22semidirect
25. C47×#487C22semidirect
147. #3139D47semidirect
166. #487C2×D47semidirect
177. C47#18552semidirect

Maximal subgroups

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

subgroupindex
C2×#156898392
#202534832
#202426132
#202421142
#202418322
C47×#31392
#1855247