Finite Group Databasebeta

← C2×#20244554 #26243677

Subgroups — C2×#20244554 #26243677

Order 12032.

1482 conjugacy classes of subgroups; only the 291 normal subgroups are shown.

normal subgroups only

This group has 1482 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
1C2×#15688984C213yesmaximal
2#20244554C218yesmaximal, direct product factor
3C2×#15688890C213yesmaximal
4C2×C47×#543C211yesmaximal, characteristic
5C24×Dic47C2211yescharacteristic
6C22×C22⋊Dic47C2213yes—
7#15688984—26no—
8#15688984C22112yes—
9C2×C47⋊#95—26no—
10#15688890C22112yes—
11C47×#543—26no—
12C47×#543C2214yes—
13C2×C22⋊Q8×C47—26no—
14C22×C22⋊C4×C47C2213yes—
15C23×Dic47—23no—
16C23×Dic47—23no—
17C23×Dic47—26no—
18C23×Dic47—23no—
19C23×Dic47Q814yescomplemented
20C23×Dic47C2312yes—
21C2×C22⋊Dic47—212no—
22C2×C22⋊Dic47—23no—
23C2×C22⋊Dic47C2316yes—
24C47⋊#95—224no—
25C25×C47C2311yescharacteristic
26C22×Q8×C47—22no—
27C23×C4×C47D416yes—
28C22⋊Q8×C47—224no—
29C22⋊Q8×C47—224no—
30C2×C4⋊C4×C47—26no—
31C2×C22⋊C4×C47—26no—
32C2×C22⋊C4×C47—26no—
33C2×C22⋊C4×C47D416yes—
34C2×C22⋊C4×C47C2316yes—
35C24×C47—44no—
36C24×C47C2×D413yes—
37C24×C47C2×D413yes—
38C24×C47C2×Q812yes—
39C24×C47#4116yes—
40C24×C47C2411yescharacteristic
41C2×Q8×C47—28no—
42C2×Q8×C47—84no—
43C22×C4×C47—26no—
44C22×C4×C47—26no—
45C22×C4×C47—412no—
46C22×C4×C47C2×D4112yes—
47C22×Dic47—212no—
48C22×Dic47—212no—
49C22×Dic47—26no—
50C22×Dic47—26no—
51C22×Dic47—26no—
52C22×Dic47—43no—
53C22×Dic47—41no—
54C22×Dic47—46no—
55C22×Dic47—46no—
56C22×Dic47—412no—
57C22×Dic47—46no—
58C22×Dic47C2×Q814yescomplemented
59C4⋊C4×C47—224no—
60C22⋊C4×C47—212no—
61C22⋊C4×C47—212no—
62C22⋊Dic47—212no—
63C22⋊Dic47—46no—
64C23×C47—212no—
65C23×C47—46no—
66C23×C47—44no—
67C23×C47—412no—
68C23×C47—42no—
69C23×C47#12013yes—
70C23×C47#12011yescharacteristic
71C23×C47C22⋊Q816yes—
72C23×C47C22⋊Q816yes—
73C23×C47#12416yes—
74C23×C47#12416yes—
75C23×C47C22×D413yes—
76C23×C47C22×Q811yescharacteristic
77C23×C47C2×#4113yes—
78Q8×C47—88no—
79C2×C4×C47—212no—
80C2×C4×C47—412no—
81C2×C4×C47—412no—
82C2×C4×C47—86no—
83C2×Dic47—212no—
84C2×Dic47—42no—
85C2×Dic47—46no—
86C2×Dic47—412no—
87C2×Dic47—44no—
88C2×Dic47—412no—
89C2×Dic47—46no—
90C2×Dic47—812no—
91C2×#2030—471nomaximal, Sylow
92C22×C47—26no—
93C22×C47—26no—
94C22×C47—212no—
95C22×C47—212no—
96C22×C47—46no—
97C22×C47—46no—
98C22×C47—41no—
99C22×C47—43no—
100C22×C47#39316yes—
101C22×C47#39312yes—
102C22×C47#45814yes—
103C22×C47C2×#12011yescharacteristic
104C22×C47C2×C22⋊Q813yes—
105C22×C47C2×#12413yes—
106C4×C47—412no—
107Dic47—44no—
108Dic47—86no—
109C2×#543D4711yescharacteristic
110C2×#381—473no—
111#2030—478no—
112C2×#342—473no—
113C2×C47—23no—
114C2×C47—23no—
115C2×C47—26no—
116C2×C47#203014yescomplemented
117C2×C47C2×#39311yescharacteristic
118C2×C47C2×#45812yes—
119C24×C4—471no—
120#543—26no—
121#543C2×D4714yes—
122C2×C22⋊Q8—26no—
123C22×C22⋊C4—473no—
124C22×C22⋊C4C2×D4713yes—
125#381—4712no—
126#381—946no—
127C2×#95—946no—
128#342—4712no—
129C47C2×#203011yescharacteristic, complemented, Sylow
130C25C22×D4711yescharacteristic
131C22×Q8—22no—
132C23×C4—474no—
133C23×C4—472no—
134C23×C4—946no—
135C23×C4—943no—
136C23×C4—943no—
137C23×C4—943no—
138C23×C4C22⋊D4716yes—
139C22⋊Q8—224no—
140C22⋊Q8—224no—
141C2×C4⋊C4—26no—
142C2×C22⋊C4—26no—
143C2×C22⋊C4—26no—
144C2×C22⋊C4—476no—
145C2×C22⋊C4—9412no—
146C2×C22⋊C4—943no—
147C2×C22⋊C4C22⋊D4716yes—
148C2×C22⋊C4C22×D4716yes—
149#95—9424no—
150C24—44no—
151C24Q8×D4712yes—
152C24#1062535416yes—
153C24C2×C22⋊D4713yes—
154C24C2×C22⋊D4713yes—
155C24C23×D4711yescharacteristic
156C2×Q8—28no—
157C2×Q8—84no—
158C22×C4—26no—
159C22×C4—26no—
160C22×C4—412no—
161C22×C4—474no—
162C22×C4—9412no—
163C22×C4—9412no—
164C22×C4—946no—
165C22×C4—946no—
166C22×C4—946no—
167C22×C4—18812no—
168C22×C4—1886no—
169C22×C4—1883no—
170C22×C4—1881no—
171C22×C4—1886no—
172C22×C4—1886no—
173C22×C4C2×C22⋊D47112yes—
174C4⋊C4—224no—
175C22⋊C4—212no—
176C22⋊C4—212no—
177C22⋊C4—9412no—
178C22⋊C4—1886no—
179C23—212no—
180C23—44no—
181C23—412no—
182C23—46no—
183C23—42no—
184C23#1309399313yes—
185C23#1309399311yescharacteristic
186C23#1309400016yes—
187C23#1309400016yes—
188C23#1309401316yes—
189C23#1309401316yes—
190C23C22×C22⋊D4713yes—
191C23C2×Q8×D4711yescharacteristic
192C23C2×#1062535413yes—
193Q8—88no—
194C2×C4—212no—
195C2×C4—412no—
196C2×C4—412no—
197C2×C4—86no—
198C2×C4—9412no—
199C2×C4—37612no—
200C2×C4—18812no—
201C2×C4—1886no—
202C2×C4—18812no—
203C2×C4—1884no—
204C2×C4—1882no—
205C2×C4—1886no—
206C22—212no—
207C22—212no—
208C22—26no—
209C22—26no—
210C22—46no—
211C22—46no—
212C22—41no—
213C22—43no—
214C22#1568903416yes—
215C22#1568903412yes—
216C22#1568931314yes—
217C22C2×#1309399311yescharacteristic
218C22C2×#1309400013yes—
219C22C2×#1309401313yes—
220C4—412no—
221C4—3766no—
222C4—1884no—
223C2—23no—
224C2—26no—
225C2—23no—
226C2#2024455414yesdirect product factor
227C2C2×#1568903411yescharacteristic
228C2C2×#1568931312yes—
subgroupG/H
centerC23#13093993
commutator, cosocleC24×C47C24
FrattiniC24C23×D47
FittingC2×C47×#543C2
socleC23×C47#120
prime radicalC25×C47C23
hypercenterC2×#543D47
hypocenterC47C2×#2030
LCSG ▷ C24×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C25 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
2. #20244554C2direct (N × H)
19. C23×Dic47Q8semidirect
58. C22×Dic47C2×Q8semidirect
116. C2×C47#2030semidirect
129. C47C2×#2030semidirect
226. C2#20244554direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156889842
#202445542
C2×#156888902
C2×C47×#5432
C2×#203047