Finite Group Databasebeta

← #26225244

Subgroups — #26225244

Order 12032.

1294 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 1294 conjugacy classes of subgroups — too many to draw. Only its 235 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×#15689699C211yesmaximal, characteristic
2#20245973C212yesmaximal, complemented
3#20243845C218yesmaximal, complemented
4#20243465C211yesmaximal, characteristic, complemented
5#20243305C212yesmaximal
6C47×#1820C211yesmaximal, characteristic, complemented
7C22×#10625350C2211yescharacteristic
8C23×Dic94C2211yescharacteristic
9#15689699—28no—
10C2×#13094024C412yescomplemented
11C2×#13094012C412yescomplemented
12C2×#13094008C2214yes—
13C2×#13094004C414yescomplemented
14C2×#13093980C2214yes—
15C2×#13093938C2218yes—
16#15689027—28no—
17#15688986C2212yes—
18#15688984C2214yes—
19#15688890C2214yes—
20C2×#117×C47C2211yescharacteristic
21C47×#381C2212yes—
22C47×#342C2214yes—
23C2×#10625350—24no—
24C2×#10625350—44no—
25C22×Dic94—44no—
26C22×Dic94C2×C411yescharacteristic
27C22×Dic94C2×C412yes—
28C22×Dic94C2×C412yes—
29C22×Dic94D418yes—
30C22×D188C2×C411yescharacteristic
31C22×C22⋊D47C2312yes—
32C23×Dic47C2312yes—
33C22×C4×D47C2×C412yes—
34#13094024—28no—
35#13094012—28no—
36#13094008—28no—
37#13094008—28no—
38#13094004—216no—
39#13093980—48no—
40C2×C4⋊Dic47C2×C418yes—
41C2×Dic47⋊C4—24no—
42C2×C22⋊Dic47C2314yes—
43C2×(C2×D47)⋊C4C2×C418yes—
44C2×C4×Dic47—24no—
45#13093938—48no—
46C47⋊#95—24no—
47C47⋊#95—28no—
48C23×C4×C47C2311yescharacteristic
49#117×C47—24no—
50C2×C4⋊C4×C47C2×C412yes—
51C2×C22⋊C4×C47C2314yes—
52C2×C22⋊C4×C47C2312yes—
53C2×C42×C47C2×C412yes—
54#95×C47—24no—
55C24×C47C2411yescharacteristic
56C23×D47C22×C412yes—
57C22×C4×C47—24no—
58C22×C4×C47—24no—
59C22×C4×C47C22×C411yescharacteristic
60C22×C4×C47C22×C411yescharacteristic
61C22×C4×C47C22×C414yes—
62C22×C4×C47C2×D414yes—
63C2×C22⋊D47—24no—
64C2×C22⋊D47—28no—
65C22×Dic47—28no—
66C22×Dic47—24no—
67C22×Dic47C22×C412yes—
68C22×Dic47C22×C414yes—
69C22×Dic47C2×D414yes—
70C22×Dic47C2×D414yes—
71#10625350—416no—
72C2×D188—22no—
73C2×D188—42no—
74C2×C4×D47—24no—
75C2×C4×D47—44no—
76C2×Dic94—22no—
77C2×Dic94—44no—
78C2×Dic94—44no—
79C2×Dic94—48no—
80C2×Dic94—48no—
81C2×Dic94—42no—
82C2×Dic94C22⋊C418yes—
83C2×Dic94C22⋊C418yes—
84C4⋊C4×C47—24no—
85C22⋊C4×C47—24no—
86C22⋊C4×C47—28no—
87C42×C47—24no—
88C22⋊Dic47—28no—
89(C2×D47)⋊C4—216no—
90C4⋊Dic47—216no—
91Dic47⋊C4—48no—
92C4×Dic47—48no—
93C23×C47—42no—
94C23×C47C23×C411yescharacteristic
95C23×C47C22×D412yes—
96C23×C47H(5,𝔽2)14yes—
97C22×D47—22no—
98C22×D47—24no—
99C22×D47—44no—
100C2×C4×C47—22no—
101C2×C4×C47—22no—
102C2×C4×C47—28no—
103C2×C4×C47—24no—
104C2×C4×C47—44no—
105C2×C4×C47—48no—
106C2×C4×C47—44no—
107C2×C4×C47C2×C22⋊C412yes—
108C2×C4×C47C2×C22⋊C412yes—
109C22⋊D47—48no—
110C2×Dic47—28no—
111C2×Dic47—24no—
112C2×Dic47—24no—
113C2×Dic47—28no—
114C2×Dic47—22no—
115C2×Dic47—24no—
116C2×Dic47—48no—
117C2×Dic47—44no—
118C2×Dic47C2×C22⋊C418yes—
119D188—44no—
120C4×D47—48no—
121Dic94—48no—
122Dic94—48no—
123Dic94—44no—
124#21751—471nomaximal, Sylow
125C22×C47—24no—
126C22×C47—22no—
127C22×C47—44no—
128C22×C47C22×C22⋊C411yescharacteristic
129C22×C47#51812yes—
130C22×C47#53414yes—
131C2×D47—42no—
132C2×D47—44no—
133C2×D47—84no—
134C4×C47—22no—
135C4×C47—22no—
136C4×C47—44no—
137C4×C47—84no—
138Dic47—28no—
139Dic47—42no—
140Dic47—84no—
141C2×#536—471no—
142#2310—472no—
143#1899—478no—
144#1820—471no—
145#1820D4711yescharacteristic, complemented
146#1782—472no—
147C2×C47—24no—
148C2×C47—22no—
149C2×C47#230811yescharacteristic
150C2×C47#288412yes—
151D47—82no—
152C22×#41—471no—
153C23×Q8—471no—
154#536—948no—
155C2×C4⋊Q8—472no—
156C2×#124—472no—
157C2×#123—474no—
158C2×C22⋊Q8—474no—
159C2×#117—474no—
160C2×#117C2×D4711yescharacteristic
161C2×#99—478no—
162#391—948no—
163#381—472no—
164#381—474no—
165#381C2×D4712yes—
166#342—474no—
167#342C2×D4714yes—
168C47#2175111yescharacteristic, complemented, Sylow
169C2×#41—944no—
170C2×#41—1884no—
171C22×Q8—472no—
172C22×Q8—472no—
173C22×Q8—471no—
174C22×Q8—478no—
175C22×Q8—1884no—
176C22×D4—472no—
177C22×D4—471no—
178C23×C4—472no—
179C23×C4—472no—
180C23×C4C22×D4711yescharacteristic
181C4⋊Q8—948no—
182#124—948no—
183#123—948no—
184#123—948no—
185C22⋊Q8—9416no—
186#117—24no—
187#117—1888no—
188C2×C4⋊C4—478no—
189C2×C4⋊C4—944no—
190C2×C4⋊C4C4×D4712yes—
191C2×C22⋊C4—478no—
192C2×C22⋊C4—474no—
193C2×C22⋊C4C22×D4714yes—
194C2×C22⋊C4C22×D4712yes—
195C2×C42—944no—
196C2×C42C4×D4712yes—
197#99—1888no—
198#95—24no—
199#95—948no—
200#95—944no—
201C24—472no—
202C24C23×D4711yescharacteristic
203#41—18816no—
204C2×Q8—478no—
205C2×Q8—478no—
206C2×Q8—942no—
207C2×Q8—1884no—
208C2×Q8—1884no—
209C2×Q8—1888no—
210C2×Q8—1888no—
211C2×Q8—1882no—
212C2×D4—944no—
213C2×D4—948no—
214C2×D4—942no—
215C2×D4—1882no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—474no—
219C22×C4—474no—
220C22×C4—474no—
221C22×C4—472no—
222C22×C4—948no—
223C22×C4—944no—
224C22×C4—944no—
225C22×C4—1884no—
226C22×C4C2×C4×D4711yescharacteristic
227C22×C4C2×C4×D4711yescharacteristic
228C22×C4C2×C4×D4714yes—
229C22×C4D4×D4714yes—
230C4⋊C4—24no—
231C4⋊C4—9416no—
232C4⋊C4—1888no—
233C22⋊C4—24no—
234C22⋊C4—28no—
235C22⋊C4—9416no—
236C22⋊C4—948no—
237C42—24no—
238C42—1888no—
239C23—42no—
240C23—944no—
241C23—942no—
242C23—1884no—
243C23C22×C4×D4711yescharacteristic
244C23C2×D4×D4712yes—
245C23#1309407014yes—
246Q8—1888no—
247Q8—1888no—
248Q8—1884no—
249D4—1884no—
250D4—1888no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—28no—
254C2×C4—24no—
255C2×C4—44no—
256C2×C4—48no—
257C2×C4—44no—
258C2×C4—478no—
259C2×C4—948no—
260C2×C4—944no—
261C2×C4—944no—
262C2×C4—948no—
263C2×C4—942no—
264C2×C4—944no—
265C2×C4—1888no—
266C2×C4—1888no—
267C2×C4—1884no—
268C2×C4C22⋊C4×D4712yes—
269C2×C4C22⋊C4×D4712yes—
270C22—24no—
271C22—22no—
272C22—44no—
273C22—3764no—
274C22—1884no—
275C22—1882no—
276C22C2×C22⋊C4×D4711yescharacteristic
277C22#1568959712yes—
278C22#1568968414yes—
279C4—22no—
280C4—22no—
281C4—44no—
282C4—84no—
283C4—948no—
284C4—3764no—
285C4—1882no—
286C2—24no—
287C2—22no—
288C2—3762no—
289C2#2024596511yescharacteristic
290C2#2025185012yes—
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC47×#1820C2
socleC22×C47C22×C22⋊C4
prime radicalC22×#10625350C22
cosocleC24×C47C24
hypercenter#1820D47
hypocenterC47#21751
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

Complements

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

NH ≅ G/Ntype
2. #20245973C2semidirect
3. #20243845C2semidirect
4. #20243465C2semidirect
6. C47×#1820C2semidirect
10. C2×#13094024C4semidirect
11. C2×#13094012C4semidirect
13. C2×#13094004C4semidirect
145. #1820D47semidirect
168. C47#21751semidirect

Maximal subgroups

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

subgroupindex
C2×#156896992
#202459732
#202438452
#202434652
#202433052
C47×#18202
#2175147