Finite Group Databasebeta

← #26244419

Subgroups — #26244419

Order 12032.

1118 conjugacy classes of subgroups; only the 291 normal subgroups are shown.

normal subgroups only

This group has 1118 conjugacy classes of subgroups — too many to draw. Only its 291 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#20246008C213yesmaximal
2#20244740C218yesmaximal
3#20243121C213yesmaximal
4C47×#3458C211yesmaximal, characteristic
5C2×#13093983C2213yes—
6C22×C4⋊Dic47C2211yescharacteristic
7#15689021—26no—
8#15688995C22112yes—
9#15688992—23no—
10#15688988—23no—
11#15688890C22112yes—
12C47×#544—23no—
13C47×#544—23no—
14C47×#543C2214yes—
15C47×#533—26no—
16C2×#117×C47C2213yes—
17C23×Dic47C2312yes—
18#13093983—26no—
19#13093983Q8112yescomplemented
20C2×C4⋊Dic47—23no—
21C2×C4⋊Dic47—23no—
22C2×C4⋊Dic47—23no—
23C2×Dic47⋊C4—212no—
24C2×C22⋊Dic47C2316yes—
25C2×C4×Dic47—23no—
26C47⋊#95—212no—
27C23×C4×C47C2311yescharacteristic
28C4⋊Q8×C47—26no—
29C4⋊Q8×C47—26no—
30#125×C47—212no—
31C22⋊Q8×C47—212no—
32C22⋊Q8×C47—212no—
33C4×Q8×C47—24no—
34C4×Q8×C47—24no—
35#117×C47—26no—
36#117×C47—26no—
37C2×C4⋊C4×C47D413yes—
38C2×C4⋊C4×C47D413yes—
39C2×C22⋊C4×C47C2316yes—
40C2×C42×C47D416yes—
41C24×C47C2411yescharacteristic
42C2×Q8×C47—24no—
43C2×Q8×C47—24no—
44C22×C4×C47C2×D416yes—
45C22×C4×C47C2×D416yes—
46C22×C4×C47C2×D413yes—
47C22×C4×C47C2×D413yes—
48C22×C4×C47C2×Q816yes—
49C22×C4×C47#4112yes—
50C22×Dic47—26no—
51C22×Dic47—26no—
52C22×Dic47—82no—
53C4⋊C4×C47—212no—
54C4⋊C4×C47—212no—
55C4⋊C4×C47—26no—
56C4⋊C4×C47—26no—
57C22⋊C4×C47—212no—
58C42×C47—26no—
59C42×C47—26no—
60C22⋊Dic47—26no—
61C22⋊Dic47C2×Q8112yes—
62C4⋊Dic47—212no—
63C4⋊Dic47—41no—
64C4⋊Dic47—43no—
65Dic47⋊C4—812no—
66C4×Dic47—212no—
67C23×C47—41no—
68C23×C47—41no—
69C23×C47C22×D413yes—
70C23×C47C22×Q813yes—
71C23×C47C2×#4111yescharacteristic
72Q8×C47—84no—
73Q8×C47—84no—
74C2×C4×C47—26no—
75C2×C4×C47—26no—
76C2×C4×C47—46no—
77C2×C4×C47—46no—
78C2×C4×C47#12011yescharacteristic
79C2×C4×C47#12013yes—
80C2×C4×C47C22⋊Q816yes—
81C2×C4×C47C22⋊Q816yes—
82C2×C4×C47C4⋊Q816yes—
83C2×C4×C47C4⋊Q816yes—
84C2×Dic47—212no—
85C2×Dic47—46no—
86C2×Dic47—46no—
87C2×Dic47—412no—
88C2×Dic47—42no—
89#23566—471nomaximal, Sylow
90C22×C47—23no—
91C22×C47—23no—
92C22×C47—43no—
93C22×C47—41no—
94C22×C47C2×#12011yescharacteristic
95C22×C47C2×C22⋊Q813yes—
96C22×C47C2×C4⋊Q813yes—
97C4×C47—46no—
98C4×C47—46no—
99C4×C47#39512yes—
100C4×C47#39516yes—
101Dic47—412no—
102Dic47—82no—
103#3458D4711yescharacteristic
104#2317—473no—
105#2062—478no—
106#1740—473no—
107C2×C47—23no—
108C2×C47—23no—
109C2×C47C2×#39511yescharacteristic
110C2×C47#302711yescharacteristic
111C2×C47#302811yescharacteristic
112#544—23no—
113#544—23no—
114#543C2×D4714yes—
115#533—26no—
116C2×#117—473no—
117C2×#117C2×D4713yes—
118C22×C4⋊C4—471no—
119#389—946no—
120#385—4712no—
121#384—943no—
122#382—943no—
123#342—4712no—
124C47#2356611yescharacteristic, complemented, Sylow
125C23×C4—472no—
126C23×C4C22×D4711yescharacteristic
127C4⋊Q8—26no—
128C4⋊Q8—26no—
129#125—212no—
130C22⋊Q8—212no—
131C22⋊Q8—212no—
132C4×Q8—24no—
133C4×Q8—24no—
134#117—26no—
135#117—26no—
136#117—4712no—
137#117—946no—
138C2×C4⋊C4—943no—
139C2×C4⋊C4—9412no—
140C2×C4⋊C4—943no—
141C2×C4⋊C4—943no—
142C2×C4⋊C4C22⋊D4713yes—
143C2×C4⋊C4C22⋊D4713yes—
144C2×C22⋊C4—476no—
145C2×C22⋊C4C22×D4716yes—
146C2×C42—943no—
147C2×C42C22⋊D4716yes—
148#95—9412no—
149C24C23×D4711yescharacteristic
150C2×Q8—24no—
151C2×Q8—24no—
152C22×C4—946no—
153C22×C4—946no—
154C22×C4—3762no—
155C22×C4Q8×D4716yes—
156C22×C4#1062535412yes—
157C22×C4C2×C22⋊D4716yes—
158C22×C4C2×C22⋊D4716yes—
159C22×C4C2×C22⋊D4713yes—
160C22×C4C2×C22⋊D4713yes—
161C4⋊C4—212no—
162C4⋊C4—212no—
163C4⋊C4—26no—
164C4⋊C4—26no—
165C4⋊C4—9412no—
166C4⋊C4—37612no—
167C4⋊C4—1881no—
168C4⋊C4—1883no—
169C22⋊C4—212no—
170C22⋊C4—4712no—
171C22⋊C4—946no—
172C42—26no—
173C42—26no—
174C42—9412no—
175C23—41no—
176C23—41no—
177C23C22×C22⋊D4713yes—
178C23C2×Q8×D4713yes—
179C23C2×#1062535411yescharacteristic
180Q8—84no—
181Q8—84no—
182C2×C4—26no—
183C2×C4—26no—
184C2×C4—46no—
185C2×C4—46no—
186C2×C4—9412no—
187C2×C4—18812no—
188C2×C4—1882no—
189C2×C4—1886no—
190C2×C4—1886no—
191C2×C4#1309399311yescharacteristic
192C2×C4#1309399313yes—
193C2×C4#1309400016yes—
194C2×C4#1309400016yes—
195C2×C4#1309402216yes—
196C2×C4#1309402216yes—
197C22—23no—
198C22—23no—
199C22—43no—
200C22—41no—
201C22C2×#1309399311yescharacteristic
202C22C2×#1309400013yes—
203C22C2×#1309402213yes—
204C4—46no—
205C4—46no—
206C4—3762no—
207C4—18812no—
208C4#1568904212yes—
209C4#1568904216yes—
210C2—23no—
211C2—23no—
212C2C2×#1568904211yescharacteristic
213C2#2025279011yescharacteristic
214C2#2025279411yescharacteristic
subgroupG/H
centerC22C2×#13093993
commutator, prime radical, cosocleC24×C47C24
FrattiniC24C23×D47
FittingC47×#3458C2
socleC22×C47C2×#120
hypercenter#3458D47
hypocenterC47#23566
LCSG ▷ C24×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3458

Complements

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

NH ≅ G/Ntype
19. #13093983Q8semidirect
124. C47#23566semidirect

Maximal subgroups

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

subgroupindex
#202460082
#202447402
#202431212
C47×#34582
#2356647