Finite Group Databasebeta

← #26115083

Subgroups — #26115083

Order 12032.

558 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C4×#13094012C211yesmaximal, characteristic
2#20243134C211yesmaximal, characteristic, complemented
3C2×#15688853C212yesmaximal, complemented
4C2×#15688850C212yesmaximal
5C47×#1831C211yesmaximal, characteristic, complemented
6C2×#13094012C411yescharacteristic
7C2×C4×Dic94C2211yescharacteristic
8C2×C4×D188C2211yescharacteristic
9C2×C4×C47⋊C8C2212yes—
10#15689029C411yescharacteristic
11#15689025C411yescharacteristic
12#15689014—22no—
13#15688989C411yescharacteristic
14C4×(C2×D47)⋊C4—22no—
15#15688878—22no—
16#15688853—24no—
17#15688850—24no—
18C2×C4⋊C8×C47C2212yes—
19C43×C47C2211yescharacteristic
20C47×#336—22no—
21C22×Dic94C2×C411yescharacteristic
22C22×D188C2×C411yescharacteristic
23C22×C4×D47—21no—
24C22×C47⋊C8—22no—
25#13094012—22no—
26#13094012—22no—
27C4×Dic94—22no—
28C4×Dic94—22no—
29C4×D188—22no—
30C4×D188—22no—
31C2×C4⋊Dic47C2×C411yescharacteristic
32C2×C4⋊Dic47C2×C411yescharacteristic
33C2×Dic47⋊C4—21no—
34C2×(C2×D47)⋊C4—22no—
35C2×(C2×D47)⋊C4—22no—
36C2×(C2×D47)⋊C4—21no—
37C2×C4×Dic47—21no—
38C4×C47⋊C8—22no—
39C4×C47⋊C8D414yes—
40C47⋊#95—22no—
41C47⋊#95—22no—
42C22×C8×C47—22no—
43C2×C42×C47C2×C411yescharacteristic
44C2×C42×C47C2×C411yescharacteristic
45C2×C42×C47D412yes—
46C2×C42×C47D412yes—
47C2×C42×C47C2311yescharacteristic
48C4⋊C8×C47—22no—
49C4⋊C8×C47—22no—
50C23×D47—21no—
51C22×C4×C47C22⋊C412yes—
52C22×C4×C47C22⋊C412yes—
53C22×C4×C47C22×C411yescharacteristic
54C22×C4×C47C2×D411yescharacteristic
55C22×C4×C47C2×D411yescharacteristic
56C22×Dic47—21no—
57C22×Dic47—21no—
58C22×Dic47—21no—
59C2×D188—22no—
60C2×D188—22no—
61C2×D188—21no—
62C2×D188M1612yes—
63C2×C4×D47—41no—
64C2×C4×D47—41no—
65C2×C4×D47—44no—
66C2×Dic94—22no—
67C2×Dic94—22no—
68C2×Dic94—21no—
69C2×Dic94M1612yes—
70C2×C8×C47—42no—
71C2×C8×C47—44no—
72C42×C47—21no—
73C42×C47—21no—
74C42×C47—22no—
75C42×C47—24no—
76C42×C47—42no—
77C42×C47C2×D412yes—
78C42×C47#4112yes—
79(C2×D47)⋊C4—28no—
80(C2×D47)⋊C4—44no—
81(C2×D47)⋊C4—42no—
82C4⋊Dic47—22no—
83C4⋊Dic47—22no—
84Dic47⋊C4—42no—
85C4×Dic47—41no—
86C4×Dic47—41no—
87C2×C47⋊C8—24no—
88C2×C47⋊C8—44no—
89C23×C47C2×C22⋊C411yescharacteristic
90C22×D47—22no—
91C22×D47—41no—
92C22×D47—41no—
93C22×D47—42no—
94C22×D47—42no—
95C2×C4×C47—22no—
96C2×C4×C47—22no—
97C2×C4×C47—22no—
98C2×C4×C47—22no—
99C2×C4×C47—22no—
100C2×C4×C47—22no—
101C2×C4×C47—22no—
102C2×C4×C47C4×D412yes—
103C2×C4×C47#12112yes—
104C2×C4×C47#12411yescharacteristic
105C2×C4×C47#12711yescharacteristic
106C2×C4×C47C2×M1611yescharacteristic
107C2×C4×C47C2×M1611yescharacteristic
108C2×C4×C47#13112yes—
109C2×C4×C47#13514yes—
110C2×Dic47—22no—
111C2×Dic47—41no—
112C2×Dic47—41no—
113C2×Dic47—41no—
114C2×Dic47—41no—
115C2×Dic47—42no—
116C2×Dic47—42no—
117D188—24no—
118D188—42no—
119C4×D47—82no—
120C4×D47—82no—
121Dic94—24no—
122Dic94—42no—
123C8×C47—82no—
124C47⋊C8—44no—
125#9645—471nomaximal, Sylow
126C22×C47#39011yescharacteristic
127C22×C47#40711yescharacteristic
128C22×C47#40811yescharacteristic
129C22×C47#41612yes—
130C22×C47#42112yes—
131C2×D47—42no—
132C2×D47—44no—
133C2×D47—81no—
134C2×D47—81no—
135C2×D47—82no—
136C4×C47—22no—
137C4×C47—42no—
138C4×C47—42no—
139C4×C47#43512yes—
140C4×C47#43512yes—
141C4×C47#48812yes—
142C4×C47#49512yes—
143Dic47—42no—
144Dic47—81no—
145Dic47—81no—
146C4×#124—471no—
147#1831D4711yescharacteristic, complemented
148#1744—471no—
149C2×Q8⋊C8—472no—
150C2×D4⋊C8—472no—
151C2×C47#148212yes—
152C2×C47#148812yes—
153C2×C47#195511yescharacteristic
154C2×C47#195811yescharacteristic
155C2×C47#196311yescharacteristic
156D47—82no—
157C2×#124—471no—
158C2×C4×Q8—471no—
159C2×C4×D4—471no—
160C2×C4⋊C8C2×D4712yes—
161C2×C4×C8—472no—
162#391—471no—
163#390—471no—
164#388—942no—
165#382—471no—
166C4×C22⋊C4—942no—
167C43C2×D4711yescharacteristic
168#336—22no—
169#336—942no—
170Q8⋊C8—944no—
171D4⋊C8—944no—
172C47#964511yescharacteristic, complemented, Sylow
173C22×Q8—471no—
174C22×D4—471no—
175C23×C4—941no—
176C22×C8—22no—
177C22×C8—942no—
178#124—942no—
179#124—942no—
180C4×Q8—942no—
181C4×Q8—942no—
182C4×D4—942no—
183C4×D4—942no—
184C2×C4⋊C4—471no—
185C2×C4⋊C4—471no—
186C2×C4⋊C4—941no—
187C2×C22⋊C4—942no—
188C2×C22⋊C4—942no—
189C2×C22⋊C4—941no—
190C2×C42—941no—
191C2×C42C4×D4711yescharacteristic
192C2×C42C4×D4711yescharacteristic
193C2×C42D18812yes—
194C2×C42C22⋊D4712yes—
195C2×C42C22×D4711yescharacteristic
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4×C8—474no—
199C4×C8—942no—
200#95—942no—
201#95—942no—
202C24—941no—
203C2×Q8—472no—
204C2×Q8—941no—
205C2×Q8—942no—
206C2×Q8—942no—
207C2×D4—472no—
208C2×D4—942no—
209C2×D4—941no—
210C2×D4—942no—
211C22×C4—941no—
212C22×C4—941no—
213C22×C4—941no—
214C22×C4—1881no—
215C22×C4—1881no—
216C22×C4—1884no—
217C22×C4(C2×D47)⋊C412yes—
218C22×C4(C2×D47)⋊C412yes—
219C22×C4C2×C4×D4711yescharacteristic
220C22×C4C2×D18811yescharacteristic
221C22×C4C2×C22⋊D4711yescharacteristic
222C2×C8—42no—
223C2×C8—44no—
224C2×C8—944no—
225C2×C8—1884no—
226C4⋊C4—942no—
227C4⋊C4—942no—
228C4⋊C4—1882no—
229C22⋊C4—948no—
230C22⋊C4—1884no—
231C22⋊C4—1882no—
232C42—21no—
233C42—21no—
234C42—22no—
235C42—24no—
236C42—42no—
237C42—1881no—
238C42—1881no—
239C42D4×D4712yes—
240C42#1062535412yes—
241C23—942no—
242C23—1881no—
243C23—1881no—
244C23—1882no—
245C23—1882no—
246C23C2×(C2×D47)⋊C411yescharacteristic
247Q8—944no—
248Q8—1882no—
249D4—944no—
250D4—1882no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—942no—
259C2×C4—3762no—
260C2×C4—3762no—
261C2×C4—1881no—
262C2×C4—1881no—
263C2×C4—1882no—
264C2×C4—1882no—
265C2×C4—1881no—
266C2×C4—1881no—
267C2×C4#1309398612yes—
268C2×C4#1309399912yes—
269C2×C4#1309401311yescharacteristic
270C2×C4#1309402011yescharacteristic
271C2×C4M16×D4711yescharacteristic
272C2×C4M16×D4711yescharacteristic
273C2×C4#1309403012yes—
274C2×C4#1309404414yes—
275C8—82no—
276C8—1884no—
277C22—3762no—
278C22—3761no—
279C22—3761no—
280C22—1884no—
281C22—1882no—
282C22#1568902411yescharacteristic
283C22#1568908611yescharacteristic
284C22#1568908911yescharacteristic
285C22#1568911712yes—
286C22#1568913912yes—
287C4—22no—
288C4—42no—
289C4—42no—
290C4—3761no—
291C4—3761no—
292C4—1882no—
293C4#1568919312yes—
294C4#1568919312yes—
295C4#1568948012yes—
296C4#1568950512yes—
297C2—3762no—
298C2#2024179412yes—
299C2#2024182012yes—
300C2#2024418511yescharacteristic
301C2#2024419711yescharacteristic
302C2#2024421911yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1831C2
socleC23×C47C2×C22⋊C4
prime radicalC22×D188C2×C4
cosocleC2×C42×C47C23
hypercenter#1831D47
hypocenterC47#9645
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C43 ◁ #1831

Complements

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

NH ≅ G/Ntype
2. #20243134C2semidirect
3. C2×#15688853C2semidirect
5. C47×#1831C2semidirect
147. #1831D47semidirect
172. C47#9645semidirect

Maximal subgroups

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

subgroupindex
C4×#130940122
#202431342
C2×#156888532
C2×#156888502
C47×#18312
#964547