Finite Group Databasebeta

← #26243681

Subgroups — #26243681

Order 12032.

1378 conjugacy classes of subgroups; only the 275 normal subgroups are shown.

normal subgroups only

This group has 1378 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#20245767C213yesmaximal
2#20244554C214yesmaximal, complemented
3#20244551C214yesmaximal
4#20243121C213yesmaximal, complemented
5C47×#3458C211yesmaximal, characteristic, complemented
6C23×C4×D47C2211yescharacteristic
7C2×#13093983C2213yes—
8#15688984C2216yes—
9#15688980—26no—
10#15688978C2216yes—
11C4×(C2×D47)⋊C4—26no—
12#15688890C22112yes—
13C47×#544—26no—
14C47×#543C2214yes—
15C47×#533—26no—
16C2×#117×C47C2213yes—
17C24×D47C2311yescharacteristic
18C23×Dic47C2311yescharacteristic
19C22×C4×D47—23no—
20C22×C4×D47—23no—
21C22×C4×D47—26no—
22C22×C4×D47Q814yescomplemented
23#13093983—212no—
24C2×C4⋊Dic47—23no—
25C2×C22⋊Dic47C2316yes—
26C2×(C2×D47)⋊C4—212no—
27C2×C4×Dic47—23no—
28C47⋊#95—212no—
29C23×C4×C47C2311yescharacteristic
30C4⋊Q8×C47—212no—
31#125×C47—212no—
32C22⋊Q8×C47—224no—
33C4×Q8×C47—28no—
34#117×C47—26no—
35#117×C47—26no—
36C2×C4⋊C4×C47D416yes—
37C2×C22⋊C4×C47C2316yes—
38C2×C42×C47D416yes—
39C24×C47C2411yescharacteristic
40C23×D47—23no—
41C23×D47—23no—
42C23×D47—44no—
43C23×D47C2×Q812yes—
44C2×Q8×C47—28no—
45C22×C4×C47C2×D4112yes—
46C22×C4×C47C2×D413yes—
47C22×C4×C47C2×D413yes—
48C22×C4×C47C2×Q812yes—
49C22×C4×C47#4116yes—
50C22×Dic47—23no—
51C22×Dic47—23no—
52C22×Dic47—26no—
53C22×Dic47C2×Q812yes—
54C2×C4×D47—212no—
55C2×C4×D47—212no—
56C2×C4×D47—43no—
57C2×C4×D47—41no—
58C2×C4×D47—46no—
59C2×C4×D47—46no—
60C4⋊C4×C47—224no—
61C4⋊C4×C47—212no—
62C22⋊C4×C47—212no—
63C42×C47—26no—
64C42×C47—26no—
65C22⋊Dic47—212no—
66(C2×D47)⋊C4—424no—
67C4⋊Dic47—46no—
68C4×Dic47—46no—
69C23×C47—42no—
70C23×C47C22×D413yes—
71C23×C47C22×Q811yescharacteristic
72C23×C47C2×#4113yes—
73C22×D47—212no—
74C22×D47—26no—
75C22×D47—44no—
76C22×D47—412no—
77C22×D47—46no—
78C22×D47—41no—
79C22×D47—43no—
80Q8×C47—88no—
81C2×C4×C47—212no—
82C2×C4×C47—412no—
83C2×C4×C47#12013yes—
84C2×C4×C47#12011yescharacteristic
85C2×C4×C47C22⋊Q816yes—
86C2×C4×C47C22⋊Q816yes—
87C2×C4×C47#12416yes—
88C2×C4×C47#12416yes—
89C2×Dic47—26no—
90C2×Dic47—41no—
91C2×Dic47—43no—
92C2×Dic47—412no—
93C2×Dic47—46no—
94C4×D47—412no—
95C4×D47—44no—
96#23467—471nomaximal, Sylow
97C22×C47—26no—
98C22×C47—41no—
99C22×C47—43no—
100C22×C47C2×#12011yescharacteristic
101C22×C47C2×C22⋊Q813yes—
102C22×C47C2×#12413yes—
103C2×D47—212no—
104C2×D47—412no—
105C2×D47—46no—
106C2×D47—42no—
107C2×D47—44no—
108C4×C47—412no—
109C4×C47#39316yes—
110C4×C47#39312yes—
111Dic47—42no—
112Dic47—86no—
113#3458D4711yescharacteristic, complemented
114#2280—473no—
115#2030—474no—
116#2029—474no—
117#1740—473no—
118C2×C47—23no—
119C2×C47—23no—
120C2×C47C2×#39311yescharacteristic
121C2×C47#302512yes—
122D47—44no—
123C24×C4—471no—
124#544—26no—
125#543C2×D4714yes—
126#533—26no—
127C2×#117—473no—
128C2×#117C2×D4713yes—
129#381—476no—
130#380—946no—
131#379—476no—
132C4×C22⋊C4—946no—
133#342—4712no—
134C47#2346711yescharacteristic, complemented, Sylow
135C25—471no—
136C23×C4—471no—
137C23×C4—474no—
138C23×C4—946no—
139C23×C4—943no—
140C23×C4—943no—
141C23×C4C22×D4711yescharacteristic
142C4⋊Q8—212no—
143#125—212no—
144C22⋊Q8—224no—
145C4×Q8—28no—
146#117—26no—
147#117—26no—
148#117—9412no—
149C2×C4⋊C4—943no—
150C2×C4⋊C4C22⋊D4716yes—
151C2×C22⋊C4—476no—
152C2×C22⋊C4—9412no—
153C2×C22⋊C4C22×D4716yes—
154C2×C42—943no—
155C2×C42C22⋊D4716yes—
156#95—9412no—
157C24—472no—
158C24—943no—
159C24—943no—
160C24—1884no—
161C24C23×D4711yescharacteristic
162C2×Q8—28no—
163C22×C4—472no—
164C22×C4—943no—
165C22×C4—943no—
166C22×C4—9412no—
167C22×C4—9412no—
168C22×C4—946no—
169C22×C4—1883no—
170C22×C4—1881no—
171C22×C4—1886no—
172C22×C4—1886no—
173C22×C4Q8×D4712yes—
174C22×C4#1062535416yes—
175C22×C4C2×C22⋊D47112yes—
176C22×C4C2×C22⋊D4713yes—
177C22×C4C2×C22⋊D4713yes—
178C4⋊C4—224no—
179C4⋊C4—212no—
180C4⋊C4—1886no—
181C22⋊C4—212no—
182C22⋊C4—9412no—
183C22⋊C4—18824no—
184C42—26no—
185C42—26no—
186C42—1886no—
187C23—42no—
188C23—9412no—
189C23—946no—
190C23—1886no—
191C23—1884no—
192C23—18812no—
193C23—1881no—
194C23—1883no—
195C23C22×C22⋊D4713yes—
196C23C2×Q8×D4711yescharacteristic
197C23C2×#1062535413yes—
198Q8—88no—
199C2×C4—212no—
200C2×C4—412no—
201C2×C4—946no—
202C2×C4—18812no—
203C2×C4—1886no—
204C2×C4—1881no—
205C2×C4—1883no—
206C2×C4—18812no—
207C2×C4—1884no—
208C2×C4#1309399313yes—
209C2×C4#1309399311yescharacteristic
210C2×C4#1309400016yes—
211C2×C4#1309400016yes—
212C2×C4#1309401316yes—
213C2×C4#1309401316yes—
214C22—26no—
215C22—41no—
216C22—43no—
217C22—9412no—
218C22—18812no—
219C22—1886no—
220C22—1882no—
221C22—1884no—
222C22C2×#1309399311yescharacteristic
223C22C2×#1309400013yes—
224C22C2×#1309401313yes—
225C4—412no—
226C4—3766no—
227C4—1882no—
228C4#1568903416yes—
229C4#1568903412yes—
230C2—23no—
231C2—23no—
232C2—1884no—
233C2C2×#1568903411yescharacteristic
234C2#2025278012yes—
subgroupG/H
centerC2×C4#13093993
commutator, cosocleC24×C47C24
FrattiniC24C23×D47
FittingC47×#3458C2
socleC22×C47C2×#120
prime radicalC24×D47C23
hypercenter#3458D47
hypocenterC47#23467
LCSG ▷ C24×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #3458

Complements

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

NH ≅ G/Ntype
2. #20244554C2semidirect
4. #20243121C2semidirect
5. C47×#3458C2semidirect
22. C22×C4×D47Q8semidirect
113. #3458D47semidirect
134. C47#23467semidirect

Maximal subgroups

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

subgroupindex
#202457672
#202445542
#202445512
#202431212
C47×#34582
#2346747