Finite Group Databasebeta

← #26192396

Subgroups — #26192396

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×#15689859C211yesmaximal, characteristic
2#20253514C212yesmaximal, complemented
3#20242571C218yesmaximal, complemented
4#20242102C211yesmaximal, characteristic
5#20241836C212yesmaximal, complemented
6C47×#3138C211yesmaximal, characteristic
7#15689859—24no—
8#15689859—22no—
9#15689859C2214yes—
10C2×#13094011—22no—
11C2×#13094005—22no—
12C2×C4×Dic94C2212yes—
13C2×D4×Dic47—22no—
14#15689451—22no—
15#15689405C2214yescomplemented
16#15689330C2214yescomplemented
17#15689278—22no—
18#15689228—24no—
19#15689180C2212yes—
20#15688867C2214yes—
21#15688855C2218yescomplemented
22C47×#533C2211yescharacteristic
23C2×#127×C47C2211yescharacteristic
24C47×#490—22no—
25C47×#486C2214yes—
26C47×#418—22no—
27C4×M16×C47C2211yescharacteristic
28C22×Dic94D412yes—
29C23×Dic47—22no—
30C2×#10625337—24no—
31#13094011—28no—
32#13094011—28no—
33#13094005—28no—
34#13094005—28no—
35C4×Dic94—22no—
36C4×Dic94—22no—
37C4×Dic94D414yes—
38C4×Dic94C2314yes—
39D4×Dic47—28no—
40D4×Dic47—28no—
41C2×C4⋊Dic47D412yes—
42C2×Dic47⋊C4—22no—
43C2×C22⋊Dic47—24no—
44C2×C22⋊Dic47—22no—
45C2×C4×Dic47—22no—
46#13093955—24no—
47C4⋊(C47⋊C8)C2314yes—
48#13093948—24no—
49#13093947—24no—
50C22⋊(C47⋊C8)—22no—
51#13093935—22no—
52C22×D4×C47—22no—
53C22×D4×C47D412yes—
54C2×M16×C47—21no—
55C4⋊Q8×C47C2312yes—
56#127×C47—22no—
57#127×C47D412yes—
58#127×C47C2312yes—
59#125×C47—21no—
60C22⋊Q8×C47—22no—
61C4×Q8×C47—22no—
62#117×C47—21no—
63C2×C42×C47C2311yescharacteristic
64#102×C47—28no—
65#97×C47—21no—
66C4×C8×C47C2312yes—
67C24×C47—22no—
68C2×Q8×C47—22no—
69C2×D4×C47—24no—
70C2×D4×C47—22no—
71C2×D4×C47—24no—
72C2×D4×C47—48no—
73C2×D4×C47—44no—
74C2×D4×C47C2×D414yes—
75C22×C4×C47C2×D412yes—
76C22×C4×C47C2×D411yescharacteristic
77C22×Dic47—22no—
78C22×Dic47—22no—
79C22×Dic47—28no—
80C22×Dic47—84no—
81C2×Dic94—22no—
82C2×Dic94—44no—
83C2×Dic94C2×D414yes—
84M16×C47—42no—
85C2×C8×C47—22no—
86C4⋊C4×C47—22no—
87C4⋊C4×C47—22no—
88C4⋊C4×C47—42no—
89C22⋊C4×C47—41no—
90C42×C47—41no—
91C42×C47C2×D412yes—
92C42×C47C2×D411yescharacteristic
93C42×C47C2411yescharacteristic
94C22⋊Dic47—216no—
95C22⋊Dic47—28no—
96#10625337—48no—
97C4⋊Dic47—22no—
98C4⋊Dic47C2×D414yes—
99Dic47⋊C4—24no—
100Dic47⋊C4—24no—
101C4×Dic47—24no—
102C4×Dic47—24no—
103C2×C47⋊C8—24no—
104C23×C47—24no—
105C23×C47—48no—
106C23×C47—44no—
107C23×C47#12011yescharacteristic
108Q8×C47—44no—
109D4×C47—44no—
110D4×C47—48no—
111D4×C47—48no—
112D4×C47—82no—
113C2×C4×C47—22no—
114C2×C4×C47—24no—
115C2×C4×C47—42no—
116C2×C4×C47#12012yes—
117C2×C4×C47#12011yescharacteristic
118C2×C4×C47C22×D412yes—
119C2×C4×C47C22×D411yescharacteristic
120C2×Dic47—24no—
121C2×Dic47—24no—
122C2×Dic47—44no—
123C2×Dic47—44no—
124C2×Dic47—88no—
125Dic94—24no—
126Dic94—24no—
127Dic94—44no—
128C8×C47—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—82no—
141C4×C47#57514yes—
142Dic47—44no—
143Dic47—44no—
144C2×#559—471no—
145#3138D4711yescharacteristic
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—474no—
155#559—944no—
156#559—942no—
157#533C2×D4711yescharacteristic
158C2×#127C2×D4711yescharacteristic
159C2×#124—942no—
160C2×#123—942no—
161C2×C4×Q8—472no—
162C2×C4×D4—942no—
163#490—22no—
164#486C2×D4714yes—
165#483—942no—
166#475—474no—
167#462—474no—
168#451—942no—
169#441—944no—
170#432—472no—
171#418—22no—
172C4×M16C2×D4711yescharacteristic
173#331—474no—
174Q8⋊C8—478no—
175C47#1853111yescharacteristic, complemented, Sylow
176C22×Q8—472no—
177C22×D4—22no—
178C22×D4C22⋊D4712yes—
179C23×C4—942no—
180C2×Q16—944no—
181C2×M16—21no—
182C4⋊Q8C22×D4712yes—
183#127—22no—
184#127C22⋊D4712yes—
185#127C22×D4712yes—
186#125—21no—
187#124—948no—
188#124—948no—
189#123—948no—
190#123—948no—
191C22⋊Q8—22no—
192C4×Q8—22no—
193C4×Q8—474no—
194C4×Q8—474no—
195C4×Q8—942no—
196C4×Q8—942no—
197C4×D4—948no—
198C4×D4—948no—
199#117—21no—
200C2×C4⋊C4—472no—
201C2×C4⋊C4—942no—
202C2×C22⋊C4—944no—
203C2×C22⋊C4—942no—
204C2×C42—942no—
205C2×C42C22×D4711yescharacteristic
206#106—944no—
207C4⋊C8—474no—
208#103—944no—
209#103—944no—
210#102—28no—
211C22⋊C8—942no—
212#97—21no—
213#97—942no—
214C4×C8C22×D4712yes—
215C24—22no—
216C2×Q8—22no—
217C2×Q8—474no—
218C2×Q8—942no—
219C2×Q8—1884no—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—24no—
223C2×D4—48no—
224C2×D4—44no—
225C2×D4C2×C22⋊D4714yes—
226C22×C4—948no—
227C22×C4—942no—
228C22×C4—942no—
229C22×C4—3764no—
230C22×C4D4×D4712yes—
231C22×C4C2×C22⋊D4711yescharacteristic
232Q16—1888no—
233M16—42no—
234C2×C8—22no—
235C2×C8—944no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—42no—
239C4⋊C4—474no—
240C4⋊C4—944no—
241C4⋊C4—942no—
242C4⋊C4—944no—
243C22⋊C4—41no—
244C22⋊C4—9416no—
245C22⋊C4—948no—
246C42—41no—
247C42—944no—
248C42—944no—
249C42D4×D4712yes—
250C42C2×C22⋊D4711yescharacteristic
251C42C23×D4711yescharacteristic
252C23—24no—
253C23—48no—
254C23—44no—
255C23#1309399211yescharacteristic
256Q8—44no—
257Q8—944no—
258Q8—944no—
259Q8—1884no—
260D4—44no—
261D4—48no—
262D4—48no—
263D4—82no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—42no—
267C2×C4—944no—
268C2×C4—944no—
269C2×C4—3768no—
270C2×C4—1884no—
271C2×C4—1884no—
272C2×C4#1309399211yescharacteristic
273C2×C4#1309399212yes—
274C2×C4C2×D4×D4712yes—
275C2×C4C22×C22⋊D4711yescharacteristic
276C8—42no—
277C8—1884no—
278C22—22no—
279C22—44no—
280C22—48no—
281C22—88no—
282C22#1568930012yes—
283C22C2×#1309399211yescharacteristic
284C4—21no—
285C4—21no—
286C4—22no—
287C4—82no—
288C4—1884no—
289C4—1884no—
290C4#1568991914yes—
291C2—22no—
292C2—84no—
293C2#2025262712yes—
294C2C2×#1568930011yescharacteristic
subgroupG/H
centerC22C2×#13093992
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3138C2
socleC22×C47C2×#120
prime radicalC2×#127×C47C22
hypercenter#3138D47
hypocenterC47#18531
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3138

Complements

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

NH ≅ G/Ntype
2. #20253514C2semidirect
3. #20242571C2semidirect
5. #20241836C2semidirect
15. #15689405C22semidirect
16. #15689330C22semidirect
21. #15688855C22semidirect
175. C47#18531semidirect

Maximal subgroups

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

subgroupindex
C2×#156898592
#202535142
#202425712
#202421022
#202418362
C47×#31382
#1853147