Finite Group Databasebeta

← #26102950

Subgroups — #26102950

Order 12032.

338 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 49 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
1#20244386C211yesmaximal, characteristic
2#20241312C211yesmaximal, characteristic, complemented
3C47×#1289C211yesmaximal, characteristic, complemented
4C22×D376C411yescharacteristic, complemented
5C2×#13093998—21no—
6C2×#13093946—21no—
7C22⋊(C47⋊C16)—21no—
8#15688886C411yescharacteristic, complemented
9C2×C22⋊C8×C47C2211yescharacteristic
10C47×#380—21no—
11C22×D188—21no—
12C22×C22⋊D47—41no—
13C2×D376—41no—
14C2×D376—42no—
15#13093998—42no—
16#13093998—42no—
17C2×C4⋊Dic47—21no—
18C2×(C2×D47)⋊C4—41no—
19C2×C47⋊C16—41no—
20#13093946—42no—
21C23×C4×C47D411yescharacteristic
22C22×C8×C47C2×C411yescharacteristic
23C22×C8×C47D411yescharacteristic
24C2×C4⋊C4×C47—41no—
25C2×C22⋊C4×C47—41no—
26C22⋊C8×C47—41no—
27#95×C47—22no—
28C24×C47D811yescharacteristic
29C23×D47—41no—
30C22×C4×C47—21no—
31C22×C4×C47—21no—
32C22×C4×C47—41no—
33C22×C4×C47—41no—
34C22×C4×C47—42no—
35C22×C4×C47C22⋊C411yescharacteristic
36C22×C4×C47SD1611yescharacteristic
37C2×C22⋊D47—44no—
38C2×C22⋊D47—82no—
39C2×C22⋊D47—81no—
40C2×C22⋊D47—81no—
41C22×Dic47—41no—
42C2×D188—22no—
43C2×D188—41no—
44C2×D188—41no—
45C2×D188—41no—
46C2×D188—41no—
47C2×D188—41no—
48C2×C8×C47—21no—
49C2×C8×C47—21no—
50C2×C8×C47—41no—
51C2×C8×C47—41no—
52C4⋊C4×C47—81no—
53C4⋊C4×C47—81no—
54C22⋊C4×C47—81no—
55C22⋊C4×C47—81no—
56(C2×D47)⋊C4—44no—
57C4⋊Dic47—22no—
58C4⋊Dic47—41no—
59D376—42no—
60D376—42no—
61C47⋊C16—81no—
62C23×C47—21no—
63C23×C47—21no—
64C23×C47—41no—
65C23×C47—41no—
66C23×C47#10211yescharacteristic
67C23×C47D1611yescharacteristic
68C23×C47SD3211yescharacteristic
69C22×D47—42no—
70C22×D47—81no—
71C22×D47—81no—
72C22×D47—81no—
73C22×D47—81no—
74C22×D47—81no—
75C22×D47—81no—
76C2×C4×C47—21no—
77C2×C4×C47—21no—
78C2×C4×C47—21no—
79C2×C4×C47—41no—
80C2×C4×C47—41no—
81C2×C4×C47—41no—
82C2×C4×C47—81no—
83C2×C4×C47—82no—
84C2×C4×C47—82no—
85C2×C4×C47—82no—
86C2×C4×C47#9911yescharacteristic
87C2×C4×C47#10411yescharacteristic
88C22⋊D47—84no—
89C22⋊D47—84no—
90C2×Dic47—42no—
91C2×Dic47—81no—
92C2×Dic47—81no—
93D188—42no—
94D188—42no—
95D188—81no—
96D188—81no—
97C8×C47—41no—
98C8×C47—41no—
99#7076—471nomaximal, Sylow
100C22×C47—21no—
101C22×C47—21no—
102C22×C47—21no—
103C22×C47—21no—
104C22×C47—41no—
105C22×C47—41no—
106C22×C47—41no—
107C22×C47—41no—
108C22×C47—81no—
109C22×C47#32711yescharacteristic
110C22×C47#35711yescharacteristic
111C22×C47#36011yescharacteristic
112C2×D47—82no—
113C2×D47—82no—
114C2×D47—82no—
115C2×D47—161no—
116C2×D47—161no—
117C2×D47—161no—
118C2×D47—161no—
119C4×C47—21no—
120C4×C47—41no—
121C4×C47—162no—
122C4×C47AΣL(1,𝔽16)11yescharacteristic
123C4×C47#35311yescharacteristic
124Dic47—82no—
125#1999—471no—
126#1314—471no—
127#1289D4711yescharacteristic, complemented
128C2×C47—21no—
129C2×C47—21no—
130C2×C47—41no—
131C2×C47—41no—
132C2×C47#133911yescharacteristic
133C2×C47#134311yescharacteristic
134C2×C47#134711yescharacteristic
135D47—162no—
136C22×D8—471no—
137C2×#121—941no—
138C2×#102—941no—
139C2×C22⋊C8C2×D4711yescharacteristic
140#380—21no—
141C22⋊C16—941no—
142#340—471no—
143C47#707611yescharacteristic, complemented, Sylow
144C22×D4—941no—
145C22×D4—1881no—
146C23×C4C22⋊D4711yescharacteristic
147C2×D8—1881no—
148C2×D8—1882no—
149C22×C8C4×D4711yescharacteristic
150C22×C8D18811yescharacteristic
151#121—1882no—
152#121—1882no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—941no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—1881no—
157C2×C16—1881no—
158#102—1882no—
159C22⋊C8—41no—
160#95—22no—
161C24—1881no—
162C24#1062533411yescharacteristic
163C2×D4—942no—
164C2×D4—3762no—
165C2×D4—3761no—
166C2×D4—3761no—
167C2×D4—1881no—
168C2×D4—1881no—
169C2×D4—1881no—
170C2×D4—1884no—
171C2×D4—1881no—
172C2×D4—1881no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—1881no—
179C22×C4(C2×D47)⋊C411yescharacteristic
180C22×C4#1062533611yescharacteristic
181D8—1882no—
182D8—1882no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—41no—
186C2×C8—41no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C4⋊C4—942no—
190C4⋊C4—1881no—
191C22⋊C4—81no—
192C22⋊C4—81no—
193C22⋊C4—1884no—
194C16—3761no—
195C23—21no—
196C23—21no—
197C23—41no—
198C23—41no—
199C23—3761no—
200C23—3761no—
201C23—3761no—
202C23—3761no—
203C23—3761no—
204C23—3761no—
205C23—1882no—
206C23#1309394411yescharacteristic
207C23#1309396511yescharacteristic
208C23#1309396711yescharacteristic
209D4—3761no—
210D4—3764no—
211D4—3764no—
212D4—3761no—
213D4—1882no—
214D4—1882no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—81no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—3761no—
226C2×C4—3761no—
227C2×C4—1882no—
228C2×C4#1309393811yescharacteristic
229C2×C4#1309395211yescharacteristic
230C8—41no—
231C8—41no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22—81no—
241C22—3762no—
242C22—3762no—
243C22—3762no—
244C22—7521no—
245C22—7521no—
246C22—7521no—
247C22—7521no—
248C22#1568885611yescharacteristic
249C22#1568892311yescharacteristic
250C22#1568893211yescharacteristic
251C4—21no—
252C4—41no—
253C4—162no—
254C4—3762no—
255C4#1568890911yescharacteristic
256C4#1568891511yescharacteristic
257C2—21no—
258C2—21no—
259C2—41no—
260C2—41no—
261C2—7522no—
262C2#2024137211yescharacteristic
263C2#2024138411yescharacteristic
264C2#2024139611yescharacteristic
subgroupG/H
centerC22#15688856
commutatorC22×C8×C47C2×C4
FrattiniC2×C22⋊C8C2×D47
FittingC47×#1289C2
socleC22×C47#327
prime radical#20244386C2
cosocleC2×C22⋊C8×C47C22
hypercenter#1289D47
hypocenterC47#7076
LCSG ▷ C22×C8×C47 ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #1289

Complements

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

NH ≅ G/Ntype
2. #20241312C2semidirect
3. C47×#1289C2semidirect
4. C22×D376C4semidirect
8. #15688886C4semidirect
127. #1289D47semidirect
143. C47#7076semidirect

Maximal subgroups

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

subgroupindex
#202443862
#202413122
C47×#12892
#707647