Finite Group Databasebeta

← #26243686

Subgroups — #26243686

Order 12032.

1506 conjugacy classes of subgroups; only the 275 normal subgroups are shown.

normal subgroups only

This group has 1506 conjugacy classes of subgroups — too many to draw. Only its 275 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#20245963C213yesmaximal
2#20244554C214yesmaximal, complemented
3#20244551C214yesmaximal, complemented
4C2×#15688890C213yesmaximal, complemented
5C2×C47×#543C211yesmaximal, characteristic, complemented
6C23×C22⋊D47C2211yescharacteristic
7C22×C22⋊Dic47C2213yes—
8#15689002—26no—
9#15689000—26no—
10#15688984C2216yes—
11#15688978C2216yes—
12#15688890C22112yes—
13C47×#543—26no—
14C47×#543C2214yes—
15C2×C22⋊Q8×C47—26no—
16C22×C22⋊C4×C47C2213yes—
17C24×D47C2311yescharacteristic
18C22×C22⋊D47—23no—
19C22×C22⋊D47—23no—
20C22×C22⋊D47—26no—
21C22×C22⋊D47Q814yescomplemented
22C23×Dic47—23no—
23C23×Dic47C2311yescharacteristic
24C2×C22⋊Dic47—212no—
25C2×C22⋊Dic47—23no—
26C2×C22⋊Dic47C2316yes—
27C2×(C2×D47)⋊C4—212no—
28C47⋊#95—212no—
29C25×C47C2311yescharacteristic
30C22×Q8×C47—22no—
31C23×C4×C47D416yes—
32C22⋊Q8×C47—224no—
33C22⋊Q8×C47—224no—
34C2×C4⋊C4×C47—26no—
35C2×C22⋊C4×C47—26no—
36C2×C22⋊C4×C47—26no—
37C2×C22⋊C4×C47D416yes—
38C2×C22⋊C4×C47C2316yes—
39C24×C47—82no—
40C24×C47C2×D413yes—
41C24×C47C2×D413yes—
42C24×C47C2×Q812yes—
43C24×C47#4116yes—
44C24×C47C2411yescharacteristic
45C23×D47—23no—
46C23×D47—23no—
47C23×D47—82no—
48C23×D47C2×Q812yescomplemented
49C2×Q8×C47—28no—
50C2×Q8×C47—84no—
51C22×C4×C47—26no—
52C22×C4×C47—26no—
53C22×C4×C47—86no—
54C22×C4×C47C2×D4112yes—
55C2×C22⋊D47—212no—
56C2×C22⋊D47—212no—
57C2×C22⋊D47—43no—
58C2×C22⋊D47—41no—
59C2×C22⋊D47—46no—
60C2×C22⋊D47—46no—
61C22×Dic47—23no—
62C22×Dic47—23no—
63C22×Dic47—26no—
64C22×Dic47—46no—
65C22×Dic47—86no—
66C22×Dic47C2×Q812yescomplemented
67C4⋊C4×C47—224no—
68C22⋊C4×C47—212no—
69C22⋊C4×C47—212no—
70C22⋊Dic47—212no—
71C22⋊Dic47—46no—
72(C2×D47)⋊C4—812no—
73C23×C47—46no—
74C23×C47—46no—
75C23×C47—42no—
76C23×C47—82no—
77C23×C47—86no—
78C23×C47#12013yes—
79C23×C47#12011yescharacteristic
80C23×C47C22⋊Q816yes—
81C23×C47C22⋊Q816yes—
82C23×C47#12416yes—
83C23×C47#12416yes—
84C23×C47C22×D413yes—
85C23×C47C22×Q811yescharacteristic
86C23×C47C2×#4113yes—
87C22×D47—26no—
88C22×D47—412no—
89C22×D47—41no—
90C22×D47—43no—
91C22×D47—82no—
92C22×D47—86no—
93Q8×C47—88no—
94C2×C4×C47—212no—
95C2×C4×C47—412no—
96C2×C4×C47—412no—
97C2×C4×C47—86no—
98C22⋊D47—412no—
99C22⋊D47—44no—
100C2×Dic47—26no—
101C2×Dic47—41no—
102C2×Dic47—43no—
103C2×Dic47—412no—
104C2×Dic47—46no—
105C2×Dic47—812no—
106#23468—471nomaximal, Sylow
107C22×C47—26no—
108C22×C47—22no—
109C22×C47—26no—
110C22×C47—46no—
111C22×C47—41no—
112C22×C47—43no—
113C22×C47—46no—
114C22×C47—46no—
115C22×C47—46no—
116C22×C47#39316yes—
117C22×C47#39312yes—
118C22×C47C2×#12011yescharacteristic
119C22×C47C2×C22⋊Q813yes—
120C22×C47C2×#12413yes—
121C2×D47—412no—
122C2×D47—42no—
123C2×D47—86no—
124C2×D47—82no—
125C4×C47—412no—
126Dic47—42no—
127Dic47—86no—
128C2×#543D4711yescharacteristic, complemented
129#2308—473no—
130#2030—474no—
131#2029—474no—
132C2×#342—473no—
133C2×C47—23no—
134C2×C47—23no—
135C2×C47—22no—
136C2×C47—26no—
137C2×C47C2×#39311yescharacteristic
138C2×C47#302612yes—
139D47—82no—
140C23×D4—471no—
141#543—26no—
142#543C2×D4714yes—
143C2×C22⋊Q8—26no—
144C22×C22⋊C4—473no—
145C22×C22⋊C4C2×D4713yes—
146#386—946no—
147#386—946no—
148#381—476no—
149#379—476no—
150#342—4712no—
151C47#2346811yescharacteristic, complemented, Sylow
152C25—471no—
153C25C22×D4711yescharacteristic
154C22×Q8—22no—
155C22×D4—474no—
156C22×D4—946no—
157C22×D4—943no—
158C22×D4—943no—
159C23×C4—471no—
160C23×C4—943no—
161C23×C4C22⋊D4716yes—
162C22⋊Q8—224no—
163C22⋊Q8—224no—
164C2×C4⋊C4—26no—
165C2×C22⋊C4—26no—
166C2×C22⋊C4—26no—
167C2×C22⋊C4—476no—
168C2×C22⋊C4—9412no—
169C2×C22⋊C4—9412no—
170C2×C22⋊C4—943no—
171C2×C22⋊C4C22⋊D4716yes—
172C2×C22⋊C4C22×D4716yes—
173#95—9412no—
174C24—82no—
175C24—472no—
176C24—943no—
177C24—943no—
178C24—3762no—
179C24Q8×D4712yes—
180C24#1062535416yes—
181C24C2×C22⋊D4713yes—
182C24C2×C22⋊D4713yes—
183C24C23×D4711yescharacteristic
184C2×Q8—28no—
185C2×Q8—84no—
186C2×D4—9412no—
187C2×D4—9412no—
188C2×D4—1883no—
189C2×D4—1881no—
190C2×D4—1886no—
191C2×D4—1886no—
192C22×C4—26no—
193C22×C4—26no—
194C22×C4—86no—
195C22×C4—472no—
196C22×C4—943no—
197C22×C4—943no—
198C22×C4—946no—
199C22×C4—3766no—
200C22×C4—1886no—
201C22×C4C2×C22⋊D47112yes—
202C4⋊C4—224no—
203C22⋊C4—212no—
204C22⋊C4—212no—
205C22⋊C4—9412no—
206C22⋊C4—37612no—
207C22⋊C4—1886no—
208C23—46no—
209C23—46no—
210C23—42no—
211C23—82no—
212C23—86no—
213C23—946no—
214C23—3766no—
215C23—3762no—
216C23—18812no—
217C23—1881no—
218C23—1883no—
219C23#1309399313yes—
220C23#1309399311yescharacteristic
221C23#1309400016yes—
222C23#1309400016yes—
223C23#1309401316yes—
224C23#1309401316yes—
225C23C22×C22⋊D4713yes—
226C23C2×Q8×D4711yescharacteristic
227C23C2×#1062535413yes—
228Q8—88no—
229D4—18812no—
230D4—1884no—
231C2×C4—212no—
232C2×C4—412no—
233C2×C4—412no—
234C2×C4—86no—
235C2×C4—946no—
236C2×C4—37612no—
237C2×C4—18812no—
238C2×C4—1886no—
239C2×C4—1881no—
240C2×C4—1883no—
241C22—22no—
242C22—26no—
243C22—26no—
244C22—46no—
245C22—41no—
246C22—43no—
247C22—46no—
248C22—46no—
249C22—46no—
250C22—3766no—
251C22—3762no—
252C22—18812no—
253C22—1882no—
254C22#1568903416yes—
255C22#1568903412yes—
256C22C2×#1309399311yescharacteristic
257C22C2×#1309400013yes—
258C22C2×#1309401313yes—
259C4—412no—
260C4—3766no—
261C4—1882no—
262C2—23no—
263C2—23no—
264C2—22no—
265C2—26no—
266C2—3762no—
267C2C2×#1568903411yescharacteristic
268C2#2025278512yes—
subgroupG/H
centerC22C2×#13093993
commutator, cosocleC24×C47C24
FrattiniC24C23×D47
FittingC2×C47×#543C2
socleC22×C47C2×#120
prime radicalC23×C22⋊D47C22
hypercenterC2×#543D47
hypocenterC47#23468
LCSG ▷ C24×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C25 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
2. #20244554C2semidirect
3. #20244551C2semidirect
4. C2×#15688890C2semidirect
5. C2×C47×#543C2semidirect
21. C22×C22⋊D47Q8semidirect
48. C23×D47C2×Q8semidirect
66. C22×Dic47C2×Q8semidirect
128. C2×#543D47semidirect
151. C47#23468semidirect

Maximal subgroups

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

subgroupindex
#202459632
#202445542
#202445512
C2×#156888902
C2×C47×#5432
#2346847