Finite Group Databasebeta

← #26108794

Subgroups — #26108794

Order 12032.

690 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 107 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×#15689735C211yesmaximal, characteristic
2#20243465C211yesmaximal, characteristic
3#20241922C214yesmaximal
4C47×#1822C211yesmaximal, characteristic
5#15689735—28no—
6C2×#13094024C411yescharacteristic, complemented
7C2×#13094017—22no—
8C2×#13094016C411yescharacteristic, complemented
9C2×#13094014—22no—
10C2×#13094003C2212yes—
11C2×C4×Dic94C412yescomplemented
12#15688986C2212yes—
13#15688890—22no—
14#15688861—28no—
15C2×#117×C47C2211yescharacteristic
16C47×#407C2212yes—
17C47×#343—22no—
18C22×Dic94C2×C412yes—
19#13094024—24no—
20#13094017—216no—
21#13094016—24no—
22#13094014—216no—
23#13094003—24no—
24#13094003D418yes—
25C4×Dic94—24no—
26C4×Dic94—24no—
27C2×C4⋊Dic47—22no—
28C2×C4⋊Dic47C2×C412yes—
29C2×Dic47⋊C4—22no—
30C2×Dic47⋊C4—22no—
31C2×Dic47⋊C4—22no—
32C2×C22⋊Dic47—22no—
33C2×C22⋊Dic47—22no—
34C2×C4×Dic47—22no—
35C47⋊#95—24no—
36C23×C4×C47C2311yescharacteristic
37C2×M16×C47—22no—
38#117×C47—24no—
39C2×C4⋊C4×C47D412yes—
40C2×C22⋊C4×C47D412yes—
41C2×C42×C47C2×C411yescharacteristic
42C2×C42×C47C2×C411yescharacteristic
43C22⋊C8×C47—24no—
44C24×C47C2×D411yescharacteristic
45C22×C4×C47—42no—
46C22×C4×C47C22⋊C412yes—
47C22×C4×C47C22⋊C412yes—
48C22×C4×C47C22×C411yescharacteristic
49C22×C4×C47C2×D411yescharacteristic
50C22×C4×C47C2×D414yes—
51C22×Dic47—22no—
52C22×Dic47—22no—
53C22×Dic47—42no—
54C2×Dic94—22no—
55C2×Dic94—44no—
56C2×Dic94C22⋊C414yes—
57M16×C47—44no—
58C2×C8×C47—42no—
59C4⋊C4×C47—24no—
60C22⋊C4×C47—24no—
61C42×C47—22no—
62C42×C47—22no—
63C22⋊Dic47—28no—
64C22⋊Dic47—44no—
65C4⋊Dic47—28no—
66C4⋊Dic47—22no—
67C4⋊Dic47C22⋊C414yes—
68Dic47⋊C4—28no—
69Dic47⋊C4—28no—
70Dic47⋊C4—28no—
71C4×Dic47—28no—
72C23×C47—22no—
73C23×C47—42no—
74C23×C47C2×C22⋊C411yescharacteristic
75C23×C47#12012yes—
76C2×C4×C47—22no—
77C2×C4×C47—22no—
78C2×C4×C47—22no—
79C2×C4×C47—22no—
80C2×C4×C47—44no—
81C2×C4×C47—42no—
82C2×C4×C47—42no—
83C2×C4×C47C2×C22⋊C412yes—
84C2×C4×C47#12012yes—
85C2×Dic47—24no—
86C2×Dic47—24no—
87C2×Dic47—44no—
88C2×Dic47—44no—
89C2×Dic47—82no—
90C2×Dic47—84no—
91Dic94—44no—
92Dic94—44no—
93C8×C47—82no—
94#8521—471nomaximal, Sylow
95C22×C47—22no—
96C22×C47—24no—
97C22×C47—22no—
98C22×C47—82no—
99C22×C47#37911yescharacteristic
100C22×C47#41012yes—
101C4×C47—22no—
102C4×C47—22no—
103C4×C47—42no—
104C4×C47—42no—
105Dic47—44no—
106Dic47—44no—
107Dic47—162no—
108C2×#541—471no—
109#1822D4711yescharacteristic
110#1820—471no—
111#1511—474no—
112C2×C47—22no—
113C2×C47—42no—
114C2×C47#178211yescharacteristic
115C2×C47#179612yes—
116#541—948no—
117C2×C4⋊Q8—471no—
118C2×#126—942no—
119C2×#125—471no—
120C2×#125—942no—
121C2×C22⋊Q8—472no—
122C2×C4×Q8—472no—
123C2×#117C2×D4711yescharacteristic
124#407C2×D4712yes—
125#381—472no—
126#343—22no—
127#342—942no—
128#328—948no—
129C47#852111yescharacteristic, complemented, Sylow
130C22×Q8—472no—
131C23×C4C22×D4711yescharacteristic
132C2×M16—22no—
133C4⋊Q8—944no—
134#126—9416no—
135#125—944no—
136#125—9416no—
137C22⋊Q8—478no—
138C22⋊Q8—944no—
139C4×Q8—944no—
140C4×Q8—944no—
141#117—24no—
142C2×C4⋊C4—472no—
143C2×C4⋊C4—942no—
144C2×C4⋊C4—942no—
145C2×C4⋊C4—942no—
146C2×C4⋊C4—942no—
147C2×C4⋊C4C22⋊D4712yes—
148C2×C22⋊C4—942no—
149C2×C22⋊C4—942no—
150C2×C22⋊C4D18812yes—
151C2×C42—942no—
152C2×C42C4×D4711yescharacteristic
153C2×C42C4×D4711yescharacteristic
154C22⋊C8—24no—
155#95—944no—
156C24C2×D18811yescharacteristic
157C2×Q8—474no—
158C2×Q8—942no—
159C2×Q8—1884no—
160C22×C4—42no—
161C22×C4—942no—
162C22×C4—942no—
163C22×C4—1882no—
164C22×C4(C2×D47)⋊C412yes—
165C22×C4(C2×D47)⋊C412yes—
166C22×C4C2×C4×D4711yescharacteristic
167C22×C4D4×D4714yes—
168C22×C4C2×C22⋊D4711yescharacteristic
169M16—44no—
170C2×C8—42no—
171C4⋊C4—24no—
172C4⋊C4—474no—
173C4⋊C4—948no—
174C4⋊C4—948no—
175C4⋊C4—948no—
176C4⋊C4—942no—
177C4⋊C4—948no—
178C22⋊C4—24no—
179C22⋊C4—948no—
180C22⋊C4—1884no—
181C42—22no—
182C42—22no—
183C42—948no—
184C23—22no—
185C23—42no—
186C23C2×(C2×D47)⋊C411yescharacteristic
187C23#1309399412yes—
188Q8—1884no—
189Q8—1884no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—44no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—944no—
198C2×C4—944no—
199C2×C4—3762no—
200C2×C4—3764no—
201C2×C4—1884no—
202C2×C4—1884no—
203C2×C4C22⋊C4×D4712yes—
204C2×C4#1309399212yes—
205C8—82no—
206C22—22no—
207C22—22no—
208C22—24no—
209C22—82no—
210C22#1568897811yescharacteristic
211C22#1568909612yes—
212C4—22no—
213C4—22no—
214C4—42no—
215C4—42no—
216C4—1884no—
217C4—1884no—
218C4—7522no—
219C2—22no—
220C2—42no—
221C2#2024330511yescharacteristic
222C2#2024336412yes—
subgroupG/H
centerC22#15688978
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1822C2
socleC22×C47#379
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#1822D47
hypocenterC47#8521
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1822

Complements

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

NH ≅ G/Ntype
6. C2×#13094024C4semidirect
8. C2×#13094016C4semidirect
11. C2×C4×Dic94C4semidirect
129. C47#8521semidirect

Maximal subgroups

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

subgroupindex
C2×#156897352
#202434652
#202419222
C47×#18222
#852147