Finite Group Databasebeta

← #26107202

Subgroups — #26107202

Order 12032.

528 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 141 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#124×Dic47C211yesmaximal, characteristic
2#20243136C212yesmaximal, complemented
3C2×#15688863C211yesmaximal, characteristic
4#20241270C212yesmaximal, complemented
5C2×C47×#329C211yesmaximal, characteristic
6C2×Q8×Dic47C411yescharacteristic
7C2×D4×Dic47C411yescharacteristic
8C2×#13093935C2212yes—
9#15689030C411yescharacteristic
10#15689026C411yescharacteristic
11#15689016—22no—
12#15688989C2211yescharacteristic
13C22⋊C4×Dic47—22no—
14C42×Dic47C2211yescharacteristic
15#15688877—22no—
16#15688877—22no—
17#15688863C414yescomplemented
18C2×#124×C47C2211yescharacteristic
19C2×#97×C47C2212yes—
20C47×#329C414yescomplemented
21C23×Dic47—21no—
22C22×C47⋊C8—22no—
23Q8×Dic47—42no—
24D4×Dic47—42no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×C22⋊Dic47—22no—
29C2×C22⋊Dic47—21no—
30C2×C4×Dic47—21no—
31C2×C4×Dic47C2×C411yescharacteristic
32C2×C4×Dic47C2×C411yescharacteristic
33C2×C4×Dic47D412yes—
34C2×C4×Dic47D412yes—
35#13093935—22no—
36#13093935C2×C414yescomplemented
37C47⋊#95—22no—
38C47⋊#95—22no—
39C22×Q8×C47C2×C411yescharacteristic
40C22×D4×C47C2×C411yescharacteristic
41C22×C8×C47—22no—
42#124×C47—22no—
43#124×C47C2×C412yes—
44#124×C47C2×C412yes—
45C2×C22⋊C4×C47—22no—
46C2×C42×C47C2311yescharacteristic
47#97×C47—22no—
48#97×C47C2×C414yescomplemented
49C24×C47—21no—
50C2×Q8×C47—21no—
51C2×Q8×C47—41no—
52C2×Q8×C47—41no—
53C2×Q8×C47C4212yes—
54C2×D4×C47—21no—
55C2×D4×C47—41no—
56C2×D4×C47—41no—
57C2×D4×C47C4212yes—
58C22×C4×C47—21no—
59C22×C4×C47C22×C411yescharacteristic
60C22×C4×C47C2×D412yes—
61C22×Dic47—21no—
62C22×Dic47—21no—
63C22×Dic47—44no—
64C22×Dic47—42no—
65C22×Dic47C22⋊C412yes—
66C22×Dic47C22⋊C411yescharacteristic
67C22×Dic47C22⋊C411yescharacteristic
68C2×C8×C47—24no—
69C2×C8×C47—42no—
70C2×C8×C47—42no—
71C22⋊C4×C47—24no—
72C22⋊C4×C47—24no—
73C42×C47C22×C411yescharacteristic
74C42×C47C22×C411yescharacteristic
75C42×C47#4112yes—
76C22⋊Dic47—44no—
77C22⋊Dic47—42no—
78C4⋊Dic47—42no—
79C4⋊Dic47—41no—
80C4⋊Dic47—41no—
81C4×Dic47—22no—
82C4×Dic47—41no—
83C4×Dic47—42no—
84C4×Dic47—42no—
85C4×Dic47—42no—
86C2×C47⋊C8—24no—
87C2×C47⋊C8—42no—
88C2×C47⋊C8—42no—
89C23×C47—22no—
90C23×C47—41no—
91C23×C47—41no—
92C23×C47—42no—
93C23×C47—42no—
94C23×C47C2×C22⋊C411yescharacteristic
95Q8×C47—42no—
96Q8×C47—42no—
97D4×C47—42no—
98D4×C47—42no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—22no—
102C2×C4×C47—22no—
103C2×C4×C47—22no—
104C2×C4×C47—41no—
105C2×C4×C47—41no—
106C2×C4×C47C2×C4211yescharacteristic
107C2×C4×C47#11711yescharacteristic
108C2×C4×C47C4×D412yes—
109C2×C4×C47C4×D412yes—
110C2×Dic47—22no—
111C2×Dic47—21no—
112C2×Dic47—22no—
113C2×Dic47—22no—
114C2×Dic47—22no—
115C2×Dic47—22no—
116C2×Dic47—41no—
117C2×Dic47—41no—
118C2×Dic47—42no—
119C2×Dic47—42no—
120C2×Dic47—84no—
121C2×Dic47#10012yes—
122C8×C47—44no—
123C47⋊C8—44no—
124#8224—471nomaximal, Sylow
125C22×C47—42no—
126C22×C47—44no—
127C22×C47—81no—
128C22×C47—81no—
129C22×C47—82no—
130C22×C47C4×C22⋊C411yescharacteristic
131C22×C47C2×#10011yescharacteristic
132C22×C47#41311yescharacteristic
133C22×C47#42112yes—
134C22×C47#42112yes—
135C4×C47—21no—
136C4×C47—21no—
137C4×C47—22no—
138C4×C47—22no—
139C4×C47—42no—
140Dic47—22no—
141Dic47—41no—
142Dic47—42no—
143Dic47—81no—
144Dic47—81no—
145C4×#124—471no—
146#1745—472no—
147C2×#329—471no—
148C2×#329D4711yescharacteristic
149#1295—472no—
150C2×C47—82no—
151C2×C47#153311yescharacteristic
152C2×C47#153512yes—
153C2×C47#153611yescharacteristic
154C2×C47C4×#10011yescharacteristic
155C2×C47#175912yes—
156C2×#124C2×D4711yescharacteristic
157C2×C4×Q8—471no—
158C2×C4×D4—471no—
159C2×#97—472no—
160C2×#97C2×D4712yes—
161#391—471no—
162#390—471no—
163#388—942no—
164#382—471no—
165C4×C22⋊C4—942no—
166C43—471no—
167#336—942no—
168#336—942no—
169#329—474no—
170#329Dic4714yescomplemented
171C47#822411yescharacteristic, complemented, Sylow
172C22×Q8C4×D4711yescharacteristic
173C22×D4C4×D4711yescharacteristic
174C23×C4—941no—
175C22×C8—22no—
176C22×C8—942no—
177#124—22no—
178#124C4×D4712yes—
179#124C2×Dic4712yes—
180C4×Q8—1882no—
181C4×D4—1882no—
182C2×C4⋊C4—471no—
183C2×C4⋊C4—471no—
184C2×C4⋊C4—941no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—941no—
187C2×C22⋊C4—942no—
188C2×C42—471no—
189C2×C42—471no—
190C2×C42—472no—
191C2×C42—472no—
192C2×C42—941no—
193C2×C42C22×D4711yescharacteristic
194#97—22no—
195#97—474no—
196#97—942no—
197#97C2×Dic4714yescomplemented
198#95—942no—
199#95—942no—
200C24—21no—
201C2×Q8—21no—
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8C4×Dic4712yes—
205C2×D4—21no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4C4×Dic4712yes—
209C22×C4—21no—
210C22×C4—471no—
211C22×C4—472no—
212C22×C4—471no—
213C22×C4—941no—
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—1881no—
226C4⋊C4—1881no—
227C4⋊C4—1882no—
228C22⋊C4—24no—
229C22⋊C4—24no—
230C22⋊C4—1884no—
231C22⋊C4—1882no—
232C42—942no—
233C42—1881no—
234C42—1882no—
235C42—1882no—
236C42—1882no—
237C42C2×C4×D4711yescharacteristic
238C42#1062535212yes—
239C42C22×Dic4711yescharacteristic
240C23—22no—
241C23—41no—
242C23—41no—
243C23—42no—
244C23—42no—
245C23C22⋊C4×D4711yescharacteristic
246Q8—42no—
247Q8—42no—
248D4—42no—
249D4—42no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—472no—
258C2×C4—942no—
259C2×C4—941no—
260C2×C4—942no—
261C2×C4—942no—
262C2×C4—942no—
263C2×C4—942no—
264C2×C4—3764no—
265C2×C4—1881no—
266C2×C4—1881no—
267C2×C4—1882no—
268C2×C4—1882no—
269C2×C4C2×C4×Dic4711yescharacteristic
270C2×C4#1309398011yescharacteristic
271C2×C4(C22⋊D47)⋊C412yes—
272C2×C4D4×Dic4712yes—
273C8—44no—
274C8—1884no—
275C22—44no—
276C22—42no—
277C22—81no—
278C22—82no—
279C22—81no—
280C22C22⋊C4×Dic4711yescharacteristic
281C22#100×D4711yescharacteristic
282C22#1568910411yescharacteristic
283C22#1568913812yes—
284C22#1568913812yes—
285C4—22no—
286C4—22no—
287C4—21no—
288C4—21no—
289C4—42no—
290C4—942no—
291C4—3761no—
292C4—3761no—
293C4—1882no—
294C4—1881no—
295C2—82no—
296C2#2024201011yescharacteristic
297C2#2024201612yes—
298C2#2024202111yescharacteristic
299C2#100×Dic4711yescharacteristic
300C2#2024320012yes—
subgroupG/H
centerC23C22⋊C4×D47
commutatorC2×C4×C47C2×C42
FrattiniC2×C42C22×D47
FittingC2×C47×#329C2
socleC23×C47C2×C22⋊C4
prime radicalC22×D4×C47C2×C4
cosocleC2×C42×C47C23
hypercenterC2×#329D47
hypocenterC47#8224
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#329

Complements

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

NH ≅ G/Ntype
2. #20243136C2semidirect
4. #20241270C2semidirect
17. #15688863C4semidirect
20. C47×#329C4semidirect
36. #13093935C2×C4semidirect
48. #97×C47C2×C4semidirect
170. #329Dic47semidirect
171. C47#8224semidirect
197. #97C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
#124×Dic472
#202431362
C2×#156888632
#202412702
C2×C47×#3292
#822447