Finite Group Databasebeta

← #26216253

Subgroups — #26216253

Order 12032.

1382 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 1382 conjugacy classes of subgroups — too many to draw. Only its 251 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×#15689597C211yesmaximal, characteristic
2#20252023C212yesmaximal, complemented
3#20243471C211yesmaximal, characteristic, complemented
4#20243379C214yesmaximal, complemented
5#20243308C212yesmaximal
6#20243179C214yesmaximal, complemented
7C47×#1822C211yesmaximal, characteristic, complemented
8C23×D188C2211yescharacteristic
9#15689597—28no—
10C2×C4×D188C414yes—
11C2×#13093986C2212yes—
12C2×#13093986C2212yes—
13C2×#13093983C2211yescharacteristic
14C2×C22⋊C4×D47C414yes—
15#15689181—28no—
16#15689146C2214yescomplemented
17C2×#13093940C2218yes—
18#15689087C2212yes—
19M16×Dic47C2214yescomplemented
20#15688892C2214yescomplemented
21C2×#117×C47C2211yescharacteristic
22C47×#407C2212yes—
23C47×#343C2214yescomplemented
24C24×D47C2×C412yes—
25C22×D188—24no—
26C22×D188—28no—
27C22×D188C2×C412yes—
28C22×D188C2312yes—
29C22×C4×D47C2×C414yes—
30C22×C4×D47C2×C414yes—
31C2×#10625329C2314yes—
32C4×D188—216no—
33#13093986—28no—
34#13093986—28no—
35#13093983D418yes—
36C2×C4⋊Dic47C2×C412yes—
37C2×C22⋊Dic47C2×C412yes—
38C2×(C2×D47)⋊C4C2×C414yes—
39C2×(C2×D47)⋊C4C2×C414yes—
40C22⋊C4×D47—216no—
41C2×C4×Dic47C2311yescharacteristic
42C2×C4×Dic47C2311yescharacteristic
43Dic47⋊C8—28no—
44#13093940—216no—
45C22⋊(C47⋊C8)—24no—
46#13093934—24no—
47C8×Dic47—24no—
48C23×C4×C47C2311yescharacteristic
49C2×M16×C47C2314yes—
50#117×C47—24no—
51C2×C4⋊C4×C47C2312yes—
52C2×C22⋊C4×C47C2×C412yes—
53C2×C42×C47C2×C412yes—
54C22⋊C8×C47—24no—
55C24×C47C22×C411yescharacteristic
56C23×D47—24no—
57C23×D47—28no—
58C23×D47—84no—
59C23×D47C22×C414yes—
60C22×C4×C47—24no—
61C22×C4×C47C22×C411yescharacteristic
62C22×C4×C47C22×C414yes—
63C22×C4×C47C2×D414yes—
64C22×C4×C47C2411yescharacteristic
65C22×Dic47C22×C412yes—
66C22×Dic47C22×C412yes—
67C2×D188—232no—
68C2×D188—28no—
69C2×D188—24no—
70C2×D188—24no—
71C2×D188—24no—
72C2×D188#4118yes—
73C2×C4×D47—28no—
74C2×C4×D47—28no—
75C2×C4×D47—84no—
76C2×C4×D47—84no—
77M16×C47—28no—
78C2×C8×C47—24no—
79C4⋊C4×C47—24no—
80C22⋊C4×C47—24no—
81C42×C47—24no—
82C22⋊Dic47C22⋊C418yes—
83(C2×D47)⋊C4—28no—
84(C2×D47)⋊C4—28no—
85C4⋊Dic47C22⋊C418yes—
86C4×Dic47C2×D414yes—
87C4×Dic47C2×D414yes—
88#10625329—28no—
89C2×C47⋊C8—24no—
90C23×C47—22no—
91C23×C47—42no—
92C23×C47C2×C22⋊C412yes—
93C23×C47C23×C411yescharacteristic
94C22×D47—24no—
95C22×D47—24no—
96C22×D47—216no—
97C22×D47—44no—
98C22×D47—44no—
99C22×D47—88no—
100C22×D47—816no—
101C22×D47#11718yes—
102C2×C4×C47—22no—
103C2×C4×C47—28no—
104C2×C4×C47—24no—
105C2×C4×C47—44no—
106C2×C4×C47C2×C22⋊C412yes—
107C2×C4×C47C22×D412yes—
108C2×C4×C47C2×#4114yes—
109C2×Dic47—42no—
110C2×Dic47—42no—
111C2×Dic47C2×C22⋊C414yes—
112C2×Dic47C2×C22⋊C414yes—
113D188—216no—
114D188—216no—
115C4×D47—88no—
116C4×D47—88no—
117C8×C47—44no—
118C47⋊C8—44no—
119#20729—471nomaximal, Sylow
120C22×C47—22no—
121C22×C47—24no—
122C22×C47—22no—
123C22×C47—44no—
124C22×C47C22×C22⋊C411yescharacteristic
125C22×C47C2×#11712yes—
126C2×D47—28no—
127C2×D47—48no—
128C2×D47—44no—
129C2×D47—44no—
130C2×D47—48no—
131C2×D47—816no—
132C4×C47—22no—
133C4×C47—44no—
134C4×C47#52514yes—
135Dic47—24no—
136Dic47—24no—
137C2×#518—471no—
138#2921—472no—
139#1822—471no—
140#1822D4711yescharacteristic, complemented
141#1800—474no—
142#1783—472no—
143C4×#100—474no—
144C2×C47—22no—
145C2×C47—42no—
146C2×C47#228011yescharacteristic
147C2×C47#288812yes—
148D47—48no—
149C23×D4—471no—
150#518—948no—
151C2×C4×D4—472no—
152C2×C4×D4—472no—
153C2×C4×D4—474no—
154C2×#117—471no—
155C2×#117C2×D4711yescharacteristic
156C22×C22⋊C4—474no—
157#433—948no—
158#423—474no—
159C2×#100—478no—
160#407—472no—
161#407C2×D4712yes—
162C4×M16—474no—
163#343—474no—
164#343C2×D4714yescomplemented
165C47#2072911yescharacteristic, complemented, Sylow
166C25—472no—
167C22×D4—472no—
168C22×D4—472no—
169C22×D4—948no—
170C22×D4—944no—
171C23×C4—474no—
172C23×C4—474no—
173C23×C4C22×D4711yescharacteristic
174C2×M16—474no—
175C2×M16C22×D4714yes—
176C4×D4—9416no—
177C4×D4—948no—
178C4×D4—948no—
179#117—24no—
180#117—478no—
181C2×C4⋊C4—472no—
182C2×C4⋊C4C22×D4712yes—
183C2×C22⋊C4—472no—
184C2×C22⋊C4—474no—
185C2×C22⋊C4—474no—
186C2×C22⋊C4—9416no—
187C2×C22⋊C4C4×D4712yes—
188C2×C42—471no—
189C2×C42—471no—
190C2×C42C4×D4712yes—
191C4⋊C8—948no—
192#100—9416no—
193C22⋊C8—24no—
194C22⋊C8—944no—
195#97—944no—
196C4×C8—944no—
197C24—474no—
198C24—948no—
199C24—944no—
200C24—3764no—
201C24C2×C4×D4711yescharacteristic
202C2×D4—478no—
203C2×D4—948no—
204C2×D4—9432no—
205C2×D4—944no—
206C2×D4—944no—
207C2×D4—944no—
208C22×C4—24no—
209C22×C4—472no—
210C22×C4—472no—
211C22×C4—948no—
212C22×C4—948no—
213C22×C4—3764no—
214C22×C4—3764no—
215C22×C4C2×C4×D4711yescharacteristic
216C22×C4C2×C4×D4714yes—
217C22×C4D4×D4714yes—
218C22×C4C23×D4711yescharacteristic
219M16—28no—
220M16—948no—
221C2×C8—24no—
222C2×C8—944no—
223C4⋊C4—24no—
224C4⋊C4—478no—
225C22⋊C4—24no—
226C22⋊C4—478no—
227C22⋊C4—948no—
228C22⋊C4—948no—
229C42—24no—
230C42—474no—
231C42—474no—
232C23—22no—
233C23—42no—
234C23—478no—
235C23—9416no—
236C23—944no—
237C23—944no—
238C23—3768no—
239C23—37616no—
240C23—1884no—
241C23—1884no—
242C23C22⋊C4×D4712yes—
243C23C22×C4×D4711yescharacteristic
244D4—9416no—
245D4—9416no—
246C2×C4—22no—
247C2×C4—28no—
248C2×C4—24no—
249C2×C4—44no—
250C2×C4—474no—
251C2×C4—474no—
252C2×C4—3768no—
253C2×C4—3768no—
254C2×C4—1882no—
255C2×C4—1882no—
256C2×C4C22⋊C4×D4712yes—
257C2×C4C2×D4×D4712yes—
258C2×C4#41×D4714yes—
259C8—44no—
260C8—1884no—
261C22—22no—
262C22—22no—
263C22—24no—
264C22—44no—
265C22—948no—
266C22—37616no—
267C22—1888no—
268C22—1888no—
269C22—1884no—
270C22—1884no—
271C22C2×C22⋊C4×D4711yescharacteristic
272C22#117×D4712yes—
273C4—22no—
274C4—44no—
275C4—944no—
276C4—944no—
277C4#1568963114yes—
278C2—22no—
279C2—42no—
280C2—1888no—
281C2#2024576811yescharacteristic
282C2#2025186512yes—
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1822C2
socleC22×C47C22×C22⋊C4
prime radicalC23×D188C22
cosocleC22×C4×C47C24
hypercenter#1822D47
hypocenterC47#20729
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1822

Complements

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

NH ≅ G/Ntype
2. #20252023C2semidirect
3. #20243471C2semidirect
4. #20243379C2semidirect
6. #20243179C2semidirect
7. C47×#1822C2semidirect
16. #15689146C22semidirect
19. M16×Dic47C22semidirect
20. #15688892C22semidirect
23. C47×#343C22semidirect
140. #1822D47semidirect
164. #343C2×D47semidirect
165. C47#20729semidirect

Maximal subgroups

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

subgroupindex
C2×#156895972
#202520232
#202434712
#202433792
#202433082
#202431792
C47×#18222
#2072947