Finite Group Databasebeta

← #26109058

Subgroups — #26109058

Order 12032.

540 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 113 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#20245927C211yesmaximal, characteristic
2#20243242C212yesmaximal, complemented
3C2×#15688862C212yesmaximal
4#20241269C211yesmaximal, characteristic, complemented
5C47×#1295C211yesmaximal, characteristic, complemented
6C2×#13094012C2211yescharacteristic
7C2×#13093989C411yescharacteristic
8C2×#13093986C411yescharacteristic
9C2×Dic47⋊C8—22no—
10C2×#13093935C2212yes—
11#15689028C411yescharacteristic
12#15689024C411yescharacteristic
13#15689012—22no—
14#15688970—22no—
15C42×Dic47C2211yescharacteristic
16#15688877—22no—
17#15688862—24no—
18C2×#97×C47C2212yes—
19C47×#382C2211yescharacteristic
20C22×Dic94C2×C411yescharacteristic
21C22×D188C2×C411yescharacteristic
22C22×C4×D47—21no—
23C22×C47⋊C8—22no—
24#13094012—22no—
25#13094012D412yes—
26#13094012Q812yes—
27#13093989—42no—
28#13093986—42no—
29C2×Dic47⋊C4—21no—
30C2×(C2×D47)⋊C4—22no—
31C2×(C2×D47)⋊C4—22no—
32C2×(C2×D47)⋊C4—21no—
33C2×C4×Dic47—21no—
34C2×C4×Dic47C2×C411yescharacteristic
35C2×C4×Dic47C2×C411yescharacteristic
36C2×C4×Dic47D412yes—
37C2×C4×Dic47D412yes—
38Dic47⋊C8—28no—
39#13093935—22no—
40#13093935—22no—
41C47⋊#95—22no—
42C22×C8×C47—22no—
43C2×C4⋊C4×C47C2×C411yescharacteristic
44C2×C4⋊C4×C47C2×C411yescharacteristic
45C2×C42×C47C2311yescharacteristic
46#97×C47—22no—
47#97×C47—22no—
48#95×C47—22no—
49C23×D47—21no—
50C22×C4×C47—21no—
51C22×C4×C47—21no—
52C22×C4×C47C22×C411yescharacteristic
53C22×C4×C47C2×D412yes—
54C22×Dic47—21no—
55C22×Dic47C22⋊C411yescharacteristic
56C22×Dic47C22⋊C411yescharacteristic
57C22×Dic47C22⋊C412yes—
58C2×D188—21no—
59C2×D188—41no—
60C2×D188—41no—
61C2×D188C4⋊C412yes—
62C2×C4×D47—44no—
63C2×C4×D47—42no—
64C2×Dic94—21no—
65C2×Dic94—41no—
66C2×Dic94—41no—
67C2×Dic94C4⋊C412yes—
68C2×C8×C47—24no—
69C2×C8×C47—42no—
70C2×C8×C47—42no—
71C4⋊C4×C47—41no—
72C4⋊C4×C47—41no—
73C42×C47C2×D411yescharacteristic
74C42×C47C2×Q811yescharacteristic
75C42×C47#4112yes—
76(C2×D47)⋊C4—24no—
77(C2×D47)⋊C4—24no—
78(C2×D47)⋊C4—42no—
79(C2×D47)⋊C4—44no—
80Dic47⋊C4—42no—
81C4×Dic47—22no—
82C4×Dic47—24no—
83C4×Dic47—41no—
84C4×Dic47—42no—
85C4×Dic47—42no—
86C2×C47⋊C8—24no—
87C2×C47⋊C8—42no—
88C2×C47⋊C8—42no—
89C23×C47C2×C22⋊C411yescharacteristic
90C22×D47—22no—
91C22×D47—41no—
92C22×D47—41no—
93C22×D47—42no—
94C22×D47—42no—
95C2×C4×C47—21no—
96C2×C4×C47—21no—
97C2×C4×C47—22no—
98C2×C4×C47—22no—
99C2×C4×C47—41no—
100C2×C4×C47—41no—
101C2×C4×C47—42no—
102C2×C4×C47—42no—
103C2×C4×C47C2×C4⋊C411yescharacteristic
104C2×C4×C47#11711yescharacteristic
105C2×C4×C47#12112yes—
106C2×C4×C47C22⋊Q812yes—
107C2×Dic47—22no—
108C2×Dic47—22no—
109C2×Dic47—22no—
110C2×Dic47—22no—
111C2×Dic47—21no—
112C2×Dic47—22no—
113C2×Dic47—41no—
114C2×Dic47—41no—
115C2×Dic47#10012yes—
116C2×Dic47#10414yes—
117D188—42no—
118D188—42no—
119C4×D47—84no—
120Dic94—42no—
121Dic94—42no—
122C8×C47—44no—
123C47⋊C8—44no—
124#8583—471nomaximal, Sylow
125C22×C47#38011yescharacteristic
126C22×C47C2×#10011yescharacteristic
127C22×C47#41311yescharacteristic
128C22×C47C2×#10412yes—
129C22×C47#42112yes—
130C2×D47—42no—
131C2×D47—44no—
132C2×D47—81no—
133C2×D47—82no—
134C2×D47—81no—
135C4×C47—21no—
136C4×C47—21no—
137C4×C47—22no—
138C4×C47—22no—
139C4×C47—81no—
140C4×C47—81no—
141Dic47—24no—
142Dic47—22no—
143Dic47—42no—
144Dic47—41no—
145C4×#124—471no—
146#1767—472no—
147C2×#329—472no—
148#1295—471no—
149#1295D4711yescharacteristic, complemented
150C2×C47#152511yescharacteristic
151C2×C47#153212yes—
152C2×C47#153311yescharacteristic
153C2×C47#180011yescharacteristic
154C2×C47#180612yes—
155D47—82no—
156C2×#124—471no—
157C2×C4×Q8—471no—
158C2×C4×D4—471no—
159C2×C4⋊C8—942no—
160C2×#97—472no—
161C2×#97C2×D4712yes—
162#391—471no—
163#390—471no—
164#388—942no—
165#382C2×D4711yescharacteristic
166C4×C22⋊C4—942no—
167C43—471no—
168#336—942no—
169#329—944no—
170C47#858311yescharacteristic, complemented, Sylow
171C22×Q8—471no—
172C22×D4—471no—
173C23×C4—941no—
174C22×C8—22no—
175C22×C8—942no—
176#124—472no—
177#124—472no—
178#124—942no—
179C4×Q8—1882no—
180C4×D4—1882no—
181C2×C4⋊C4—941no—
182C2×C4⋊C4C4×D4711yescharacteristic
183C2×C4⋊C4C4×D4711yescharacteristic
184C2×C22⋊C4—942no—
185C2×C22⋊C4—941no—
186C2×C22⋊C4—942no—
187C2×C42—471no—
188C2×C42—472no—
189C2×C42—471no—
190C2×C42—472no—
191C2×C42—941no—
192C2×C42C22×D4711yescharacteristic
193C4⋊C8—948no—
194#97—22no—
195#97—22no—
196#97—942no—
197#97—942no—
198#95—22no—
199#95—942no—
200C24—941no—
201C2×Q8—472no—
202C2×Q8—941no—
203C2×Q8—1881no—
204C2×Q8—1881no—
205C2×D4—472no—
206C2×D4—941no—
207C2×D4—1881no—
208C2×D4—1881no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—472no—
212C22×C4—471no—
213C22×C4—471no—
214C22×C4—941no—
215C22×C4—1882no—
216C22×C4—1884no—
217C22×C4C2×C4×D4711yescharacteristic
218C22×C4D4×D4712yes—
219C2×C8—24no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—944no—
223C2×C8—1882no—
224C2×C8—1882no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—1882no—
228C22⋊C4—944no—
229C22⋊C4—944no—
230C22⋊C4—1884no—
231C22⋊C4—1882no—
232C42—944no—
233C42—942no—
234C42—1881no—
235C42—1882no—
236C42—1882no—
237C42#1062535012yes—
238C42D4×D4711yescharacteristic
239C42Q8×D4711yescharacteristic
240C23—942no—
241C23—1881no—
242C23—1881no—
243C23—1882no—
244C23—1882no—
245C23C22⋊C4×D4711yescharacteristic
246Q8—1882no—
247Q8—1882no—
248D4—1882no—
249D4—1882no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—472no—
259C2×C4—474no—
260C2×C4—942no—
261C2×C4—942no—
262C2×C4—942no—
263C2×C4—942no—
264C2×C4—941no—
265C2×C4—942no—
266C2×C4—3764no—
267C2×C4—1881no—
268C2×C4—1881no—
269C2×C4C4⋊C4×D4711yescharacteristic
270C2×C4#1309398111yescharacteristic
271C2×C4#1309399712yes—
272C2×C4#1309400212yes—
273C8—44no—
274C8—1884no—
275C22—3761no—
276C22—3762no—
277C22—3761no—
278C22—1882no—
279C22—1884no—
280C22#1568898211yescharacteristic
281C22#100×D4711yescharacteristic
282C22#1568910611yescharacteristic
283C22#104×D4712yes—
284C22#1568913812yes—
285C4—21no—
286C4—21no—
287C4—22no—
288C4—22no—
289C4—81no—
290C4—81no—
291C4—944no—
292C4—942no—
293C4—1881no—
294C4—1882no—
295C2—3762no—
296C2#2024197811yescharacteristic
297C2#2024200712yes—
298C2#2024201011yescharacteristic
299C2#2024337911yescharacteristic
300C2#2024340712yes—
subgroupG/H
centerC23C22⋊C4×D47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1295C2
socleC23×C47C2×C22⋊C4
prime radicalC22×D188C2×C4
cosocleC2×C42×C47C23
hypercenter#1295D47
hypocenterC47#8583
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
2. #20243242C2semidirect
4. #20241269C2semidirect
5. C47×#1295C2semidirect
149. #1295D47semidirect
170. C47#8583semidirect

Maximal subgroups

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

subgroupindex
#202459272
#202432422
C2×#156888622
#202412692
C47×#12952
#858347