Finite Group Databasebeta

← #26184027

Subgroups — #26184027

Order 12032.

1202 conjugacy classes of subgroups; only the 355 normal subgroups are shown.

normal subgroups only

This group has 1202 conjugacy classes of subgroups — too many to draw. Only its 355 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#20257159C211yesmaximal, characteristic
2#20246309C211yesmaximal, characteristic, complemented
3#20241949C216yesmaximal, complemented
4#20241884C216yesmaximal, complemented
5C47×#2982C211yesmaximal, characteristic, complemented
6C2×#13094072—21no—
7#15689795C414yescomplemented
8Q8×D188C414yescomplemented
9#15689785—24no—
10#15689704—23no—
11#15689673C2216yes—
12C2×Q8×Dic47—21no—
13#15689022—23no—
14#15688989C2213yes—
15C4×C4⋊Dic47C2213yes—
16#15688858C22112yes—
17#15688858C2214yes—
18C2×C4×Q8×C47C2211yescharacteristic
19C8×Q8×C47—24no—
20C47×#431C2216yes—
21#13094072—44no—
22#13094072—44no—
23C2×#10625354—24no—
24C2×#10625350—26no—
25C2×Q8×D47—24no—
26#13094024C2×C416yes—
27#13094021C2×C416yes—
28#13094012—26no—
29#13094004—212no—
30#13093999—212no—
31#13093998—26no—
32#13093998C2314yes—
33Q8×Dic47—44no—
34C4×D188—26no—
35C4×D188C2×C4112yes—
36C2×C4⋊Dic47—23no—
37C2×C4⋊Dic47—23no—
38C2×C4×Dic47—23no—
39C47⋊#95—23no—
40C47⋊#95C2311yescharacteristic
41C47⋊#95C2313yes—
42C22×Q8×C47D411yescharacteristic
43C4×Q8×C47C2×C414yes—
44C4×Q8×C47D414yes—
45C2×C4⋊C4×C47D413yes—
46C2×C42×C47C2313yes—
47C4⋊C8×C47—26no—
48C22⋊C8×C47C2314yes—
49C4×C8×C47—26no—
50C2×Q8×C47C22⋊C414yes—
51C2×Q8×C47D814yes—
52C2×Q8×C47SD1614yes—
53C22×C4×C47C2×D413yes—
54C22×C4×C47C2×D413yes—
55C22×C4×C47C2411yescharacteristic
56C2×C22⋊D47—24no—
57C22×Dic47—21no—
58C22×Dic47—23no—
59#10625354—416no—
60Q8×D47—416no—
61#10625350—412no—
62#10625350—412no—
63C2×D188—26no—
64C2×D188—26no—
65C2×D188C22×C414yes—
66C2×C4×D47—212no—
67C2×Dic94—26no—
68C2×C8×C47—24no—
69C4⋊C4×C47C22⋊C4112yes—
70C42×C47C22×C416yes—
71C42×C47C2×D416yes—
72(C2×D47)⋊C4—212no—
73(C2×D47)⋊C4—24no—
74C4⋊Dic47—26no—
75C4⋊Dic47—46no—
76C4⋊Dic47—46no—
77C4⋊Dic47C22×C414yes—
78C4×Dic47—46no—
79C23×C47C22×D411yescharacteristic
80C22×D47—24no—
81Q8×C47#102116yes—
82C2×C4×C47—43no—
83C2×C4×C47—41no—
84C2×C4×C47C2×C22⋊C416yes—
85C2×C4×C47C2×C22⋊C416yes—
86C2×C4×C47C2×D816yes—
87C2×C4×C47C2×SD1616yes—
88C2×C4×C47C23×C411yescharacteristic
89C2×C4×C47C22×D411yescharacteristic
90C22⋊D47—44no—
91C22⋊D47—44no—
92C2×Dic47—22no—
93C2×Dic47—24no—
94C2×Dic47—41no—
95C2×Dic47—41no—
96C2×Dic47—43no—
97C2×Dic47—43no—
98C2×Dic47—43no—
99D188—44no—
100D188—412no—
101D188—86no—
102C4×D47—424no—
103Dic94—412no—
104C8×C47—44no—
105#17618—471nomaximal, Sylow
106C22×C47—21no—
107C22×C47—21no—
108C22×C47C22×C22⋊C411yescharacteristic
109C22×C47C22×D811yescharacteristic
110C22×C47C22×SD1611yescharacteristic
111C2×D47—44no—
112C2×D47—44no—
113C2×D47—44no—
114C4×C47—21no—
115C4×C47—21no—
116C4×C47—26no—
117C4×C47C2×#102112yes—
118Dic47—42no—
119Dic47—44no—
120Dic47—83no—
121#3497—471no—
122#2982D4711yescharacteristic, complemented
123#2351—471no—
124#1517—476no—
125#1502—476no—
126C2×C47—21no—
127C2×C47—21no—
128C2×C47#288411yescharacteristic
129C2×C47C22×#10211yescharacteristic
130C2×C47#290111yescharacteristic
131D47—84no—
132C2×2−1+4—941no—
133#550—474no—
134D4×Q8—474no—
135#548—944no—
136#537—943no—
137#532—476no—
138C2×C4×Q8—941no—
139C2×C4×Q8C2×D4711yescharacteristic
140C8×Q8—24no—
141#431C2×D4716yes—
142#389—943no—
143#382—473no—
144C4×C4⋊C4—473no—
145#327—474no—
146#327—4712no—
147C47#1761811yescharacteristic, complemented, Sylow
1482−1+4—1884no—
1492−1+4—1884no—
150C2×#41—944no—
151C2×#41—946no—
152C22×Q8—944no—
153C22×Q8D18811yescharacteristic
154C4⋊Q8—476no—
155#127—476no—
156#124—946no—
157C22⋊Q8—9412no—
158#121—474no—
159#121—946no—
160#121—9412no—
161C4×Q8—1884no—
162C4×Q8C4×D4714yes—
163C4×Q8C22⋊D4714yes—
164C4×D4—4712no—
165C4×D4—946no—
166C2×C4⋊C4—943no—
167C2×C4⋊C4—943no—
168C2×C4⋊C4D18813yes—
169C2×C42—943no—
170C2×C42C22×D4713yes—
171C4⋊C8—26no—
172C22⋊C8C22×D4714yes—
173C4×C8—26no—
174#95—471no—
175#95—473no—
176#95—943no—
177#41—18816no—
178#41—18812no—
179#41—18812no—
180C2×Q8—946no—
181C2×Q8—18816no—
182C2×Q8#1062532514yes—
183C2×Q8D37614yes—
184C2×Q8(C2×D47)⋊C414yes—
185C2×D4—474no—
186C2×D4—944no—
187C2×D4—946no—
188C2×D4—946no—
189C22×C4—941no—
190C22×C4—9412no—
191C22×C4—943no—
192C22×C4C2×D18813yes—
193C22×C4C2×D18813yes—
194C22×C4C23×D4711yescharacteristic
195C2×C8—24no—
196C4⋊C4—474no—
197C4⋊C4—946no—
198C4⋊C4—1886no—
199C4⋊C4—1886no—
200C4⋊C4(C2×D47)⋊C4112yes—
201C22⋊C4—9412no—
202C22⋊C4—944no—
203C42—1886no—
204C42C2×C4×D4716yes—
205C42C2×C22⋊D4716yes—
206C23—944no—
207C23C22×D18811yescharacteristic
208Q8—18812no—
209Q8#13093946116yes—
210D4—3766no—
211D4—1884no—
212D4—1884no—
213D4—1884no—
214D4—18812no—
215C2×C4—43no—
216C2×C4—41no—
217C2×C4—942no—
218C2×C4—944no—
219C2×C4—1883no—
220C2×C4—1883no—
221C2×C4—1883no—
222C2×C4—1881no—
223C2×C4—1881no—
224C2×C4—18824no—
225C2×C4C2×(C2×D47)⋊C416yes—
226C2×C4C2×(C2×D47)⋊C416yes—
227C2×C4C2×D37616yes—
228C2×C4C2×#1062532516yes—
229C2×C4C22×C4×D4711yescharacteristic
230C2×C4C22×C22⋊D4711yescharacteristic
231C8—44no—
232C22—21no—
233C22—21no—
234C22—1884no—
235C22—1884no—
236C22—1884no—
237C22C22×(C2×D47)⋊C411yescharacteristic
238C22C22×D37611yescharacteristic
239C22C22×#1062532511yescharacteristic
240C4—26no—
241C4—21no—
242C4—21no—
243C4—3763no—
244C4—1882no—
245C4—1884no—
246C4C2×#13093946112yes—
247C2—21no—
248C2—21no—
249C2—3764no—
250C2#2025185111yescharacteristic
251C2C22×#1309394611yescharacteristic
252C2#2025193311yescharacteristic
subgroupG/H
centerC22C22×(C2×D47)⋊C4
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2982C2
socleC22×C47C22×C22⋊C4
prime radical#20257159C2
cosocleC22×C4×C47C24
hypercenter#2982D47
hypocenterC47#17618
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. #20246309C2semidirect
3. #20241949C2semidirect
4. #20241884C2semidirect
5. C47×#2982C2semidirect
7. #15689795C4semidirect
8. Q8×D188C4semidirect
122. #2982D47semidirect
147. C47#17618semidirect

Maximal subgroups

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

subgroupindex
#202571592
#202463092
#202419492
#202418842
C47×#29822
#1761847