Finite Group Databasebeta

← #26108781

Subgroups — #26108781

Order 12032.

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

normal subgroups only

This group has 818 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×#15689728C211yesmaximal, characteristic
2#20243465C211yesmaximal, characteristic, complemented
3#20241918C214yesmaximal, complemented
4C47×#1822C211yesmaximal, characteristic, complemented
5#15689728—28no—
6C2×#13094021C411yescharacteristic, complemented
7C2×#13094016C411yescharacteristic, complemented
8C2×#13094009—22no—
9C2×#13093998C2212yes—
10C2×#13093997—22no—
11C2×C4×D188C412yescomplemented
12#15688986C2212yes—
13#15688890—22no—
14#15688858—28no—
15C2×#117×C47C2211yescharacteristic
16C47×#407C2212yes—
17C47×#343—22no—
18C22×D188—22no—
19C22×D188C2×C412yes—
20C22×C22⋊D47—22no—
21C22×C4×D47—22no—
22#13094021—24no—
23#13094016—24no—
24#13094009—216no—
25#13093998—24no—
26#13093998D418yes—
27#13093997—216no—
28C4×D188—24no—
29C4×D188—24no—
30C2×C4⋊Dic47C2×C412yes—
31C2×Dic47⋊C4—22no—
32C2×C22⋊Dic47—22no—
33C2×(C2×D47)⋊C4—22no—
34C2×(C2×D47)⋊C4—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
45C23×D47—22no—
46C22×C4×C47—42no—
47C22×C4×C47C22⋊C412yes—
48C22×C4×C47C22⋊C412yes—
49C22×C4×C47C22×C411yescharacteristic
50C22×C4×C47C2×D411yescharacteristic
51C22×C4×C47C2×D414yes—
52C2×C22⋊D47—28no—
53C2×C22⋊D47—84no—
54C22×Dic47—22no—
55C22×Dic47—42no—
56C2×D188—22no—
57C2×D188—28no—
58C2×D188—44no—
59C2×D188—84no—
60C2×D188C22⋊C414yes—
61C2×C4×D47—28no—
62C2×C4×D47—84no—
63M16×C47—44no—
64C2×C8×C47—42no—
65C4⋊C4×C47—24no—
66C22⋊C4×C47—24no—
67C42×C47—22no—
68C42×C47—22no—
69C22⋊Dic47—44no—
70(C2×D47)⋊C4—28no—
71(C2×D47)⋊C4—28no—
72C4⋊Dic47—22no—
73C4⋊Dic47C22⋊C414yes—
74Dic47⋊C4—28no—
75C23×C47—22no—
76C23×C47—42no—
77C23×C47C2×C22⋊C411yescharacteristic
78C23×C47#12012yes—
79C22×D47—24no—
80C22×D47—44no—
81C22×D47—44no—
82C22×D47—44no—
83C2×C4×C47—22no—
84C2×C4×C47—22no—
85C2×C4×C47—22no—
86C2×C4×C47—22no—
87C2×C4×C47—44no—
88C2×C4×C47—42no—
89C2×C4×C47—42no—
90C2×C4×C47C2×C22⋊C412yes—
91C2×C4×C47#12012yes—
92C22⋊D47—88no—
93C2×Dic47—24no—
94C2×Dic47—44no—
95C2×Dic47—82no—
96C2×Dic47—84no—
97D188—44no—
98D188—44no—
99D188—88no—
100C4×D47—88no—
101C8×C47—82no—
102#8518—471nomaximal, Sylow
103C22×C47—22no—
104C22×C47—24no—
105C22×C47—22no—
106C22×C47—82no—
107C22×C47#37911yescharacteristic
108C22×C47#41012yes—
109C2×D47—44no—
110C2×D47—44no—
111C2×D47—44no—
112C2×D47—88no—
113C4×C47—22no—
114C4×C47—22no—
115C4×C47—42no—
116C4×C47—42no—
117Dic47—44no—
118Dic47—162no—
119C2×#540—471no—
120#1822D4711yescharacteristic, complemented
121#1820—471no—
122#1510—474no—
123C2×C47—22no—
124C2×C47—42no—
125C2×C47#178211yescharacteristic
126C2×C47#179512yes—
127D47—84no—
128#540—948no—
129C2×#127—471no—
130C2×#125—471no—
131C2×#123—942no—
132C2×#121—472no—
133C2×#121—942no—
134C2×C4×D4—472no—
135C2×#117C2×D4711yescharacteristic
136#407C2×D4712yes—
137#381—472no—
138#343—22no—
139#342—942no—
140#327—948no—
141C47#851811yescharacteristic, complemented, Sylow
142C22×D4—472no—
143C22×D4—942no—
144C22×D4—942no—
145C23×C4—942no—
146C23×C4C22×D4711yescharacteristic
147C2×M16—22no—
148#127—944no—
149#125—944no—
150#123—9416no—
151#121—478no—
152#121—9416no—
153#121—944no—
154C4×D4—944no—
155C4×D4—944no—
156#117—24no—
157C2×C4⋊C4—472no—
158C2×C4⋊C4—942no—
159C2×C4⋊C4C22⋊D4712yes—
160C2×C22⋊C4—942no—
161C2×C22⋊C4—942no—
162C2×C22⋊C4—942no—
163C2×C22⋊C4D18812yes—
164C2×C42C4×D4711yescharacteristic
165C2×C42C4×D4711yescharacteristic
166C22⋊C8—24no—
167#95—944no—
168C24—942no—
169C24C2×D18811yescharacteristic
170C2×D4—474no—
171C2×D4—948no—
172C2×D4—948no—
173C2×D4—942no—
174C2×D4—3764no—
175C2×D4—3764no—
176C2×D4—1884no—
177C22×C4—42no—
178C22×C4—948no—
179C22×C4—942no—
180C22×C4—3764no—
181C22×C4—1882no—
182C22×C4(C2×D47)⋊C412yes—
183C22×C4(C2×D47)⋊C412yes—
184C22×C4C2×C4×D4711yescharacteristic
185C22×C4D4×D4714yes—
186C22×C4C2×C22⋊D4711yescharacteristic
187M16—44no—
188C2×C8—42no—
189C4⋊C4—24no—
190C4⋊C4—474no—
191C4⋊C4—948no—
192C4⋊C4—942no—
193C22⋊C4—24no—
194C22⋊C4—948no—
195C22⋊C4—948no—
196C22⋊C4—1884no—
197C42—22no—
198C42—22no—
199C23—22no—
200C23—42no—
201C23—944no—
202C23—1884no—
203C23—1884no—
204C23—1884no—
205C23C2×(C2×D47)⋊C411yescharacteristic
206C23#1309399412yes—
207D4—3768no—
208D4—3768no—
209D4—1884no—
210D4—1884no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—44no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—944no—
219C2×C4—3768no—
220C2×C4—3762no—
221C2×C4—3764no—
222C2×C4—1884no—
223C2×C4C22⋊C4×D4712yes—
224C2×C4#1309399212yes—
225C8—82no—
226C22—22no—
227C22—22no—
228C22—24no—
229C22—82no—
230C22—3768no—
231C22—1884no—
232C22—1884no—
233C22—1884no—
234C22#1568897811yescharacteristic
235C22#1568909612yes—
236C4—22no—
237C4—22no—
238C4—42no—
239C4—42no—
240C4—1884no—
241C4—7522no—
242C2—22no—
243C2—42no—
244C2—3764no—
245C2#2024330511yescharacteristic
246C2#2024335912yes—
subgroupG/H
centerC22#15688978
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1822C2
socleC22×C47#379
prime radicalC2×#15689728C2
cosocleC23×C4×C47C23
hypercenter#1822D47
hypocenterC47#8518
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
2. #20243465C2semidirect
3. #20241918C2semidirect
4. C47×#1822C2semidirect
6. C2×#13094021C4semidirect
7. C2×#13094016C4semidirect
11. C2×C4×D188C4semidirect
120. #1822D47semidirect
141. C47#8518semidirect

Maximal subgroups

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

subgroupindex
C2×#156897282
#202434652
#202419182
C47×#18222
#851847