Finite Group Databasebeta

← #26103029

Subgroups — #26103029

Order 12032.

268 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 63 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#20244216C211yesmaximal, characteristic
2#20241221C211yesmaximal, characteristic, complemented
3C47×#1276C211yesmaximal, characteristic, complemented
4#15689673C411yescharacteristic
5#15689176—21no—
6#15689147C411yescharacteristic
7C2×#13093952—21no—
8C2×#13093952—21no—
9C47×#431—21no—
10C43×C47C2211yescharacteristic
11C2×#10625350—21no—
12C2×#10625329—21no—
13#13094024C811yescharacteristic, complemented
14#13094021C811yescharacteristic, complemented
15#13094012—21no—
16#13093998—41no—
17C4×D188—41no—
18C4⋊(C47⋊C8)—22no—
19C4⋊(C47⋊C8)—41no—
20#13093952—41no—
21#13093952—41no—
22#13093952—41no—
23#13093952—41no—
24C22⋊(C47⋊C8)—41no—
25C4×C47⋊C8—41no—
26C2×C42×C47—41no—
27C2×C42×C47C2×C411yescharacteristic
28C2×C42×C47D411yescharacteristic
29C2×C42×C47D411yescharacteristic
30C4⋊C8×C47—22no—
31C22⋊C8×C47—41no—
32C4×C8×C47—41no—
33C22×C4×C47—21no—
34C22×C4×C47—21no—
35C22×C4×C47—21no—
36C22×C4×C47C22⋊C411yescharacteristic
37C2×C22⋊D47—41no—
38#10625350—41no—
39#10625350—41no—
40#10625350—41no—
41#10625350—41no—
42C2×D188—21no—
43C2×D188—41no—
44C2×C4×D47—41no—
45C2×Dic94—21no—
46C2×C8×C47—42no—
47C42×C47—21no—
48C42×C47—21no—
49C42×C47—21no—
50C42×C47—44no—
51C42×C47C2×C811yescharacteristic
52C42×C47M1611yescharacteristic
53C42×C47D811yescharacteristic
54C42×C47D811yescharacteristic
55C42×C47SD1611yescharacteristic
56C42×C47SD1611yescharacteristic
57(C2×D47)⋊C4—42no—
58C4⋊Dic47—41no—
59#10625329—41no—
60#10625329—41no—
61C2×C47⋊C8—42no—
62C2×C47⋊C8—41no—
63C23×C47#9911yescharacteristic
64C22×D47—41no—
65C2×C4×C47—21no—
66C2×C4×C47—21no—
67C2×C4×C47—21no—
68C2×C4×C47—21no—
69C2×C4×C47—21no—
70C2×C4×C47—21no—
71C2×C4×C47—42no—
72C2×C4×C47—42no—
73C2×C4×C47—42no—
74C2×C4×C47C22⋊C811yescharacteristic
75C2×C4×C47#10011yescharacteristic
76C2×C4×C47#10211yescharacteristic
77C2×C4×C47#10211yescharacteristic
78C2×C4×C47#10411yescharacteristic
79C2×C4×C47#10411yescharacteristic
80C22⋊D47—81no—
81C22⋊D47—81no—
82C2×Dic47—41no—
83D188—41no—
84D188—41no—
85D188—82no—
86C4×D47—81no—
87C4×D47—81no—
88Dic94—41no—
89Dic94—41no—
90C8×C47—82no—
91C47⋊C8—82no—
92C47⋊C8—81no—
93#7106—471nomaximal, Sylow
94C22×C47—41no—
95C22×C47#32311yescharacteristic
96C22×C47#32711yescharacteristic
97C22×C47#32711yescharacteristic
98C2×D47—82no—
99C2×D47—81no—
100C4×C47—22no—
101C4×C47—22no—
102C4×C47—22no—
103C4×C47—41no—
104C4×C47—41no—
105C4×C47—41no—
106C4×C47D4⋊C811yescharacteristic
107C4×C47D4⋊C811yescharacteristic
108C4×C47#32911yescharacteristic
109C4×C47#33111yescharacteristic
110Dic47—81no—
111#1962—471no—
112#1276—471no—
113#1276D4711yescharacteristic, complemented
114C2×C47—21no—
115C2×C47—21no—
116C2×C47—21no—
117C2×C47#127011yescharacteristic
118D47—161no—
119#532—471no—
120#431—21no—
121#431—941no—
122#423—471no—
123C2×#104—941no—
124C2×#104—941no—
125C43C2×D4711yescharacteristic
126C47#710611yescharacteristic, complemented, Sylow
127C2×#41—941no—
128C2×M16—941no—
129C4⋊Q8—471no—
130#127—471no—
131#124—941no—
132#121—1881no—
133C4×D4—1881no—
134C2×C42—41no—
135C2×C42C4×D4711yescharacteristic
136C2×C42D18811yescharacteristic
137C2×C42C22⋊D4711yescharacteristic
138C4⋊C8—22no—
139C4⋊C8—942no—
140C4⋊C8—1881no—
141#104—1881no—
142#104—1881no—
143#104—1881no—
144#104—1881no—
145C22⋊C8—41no—
146C22⋊C8—1881no—
147C4×C8—41no—
148C4×C8—1881no—
149#41—1881no—
150#41—1881no—
151#41—1881no—
152#41—1881no—
153C2×Q8—941no—
154C2×D4—941no—
155C2×D4—1881no—
156C2×D4—1881no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—1881no—
161C22×C4(C2×D47)⋊C411yescharacteristic
162M16—1881no—
163M16—1881no—
164C2×C8—42no—
165C2×C8—1882no—
166C2×C8—1881no—
167C4⋊C4—1881no—
168C22⋊C4—1882no—
169C42—21no—
170C42—21no—
171C42—21no—
172C42—44no—
173C42C8×D4711yescharacteristic
174C42#1062532411yescharacteristic
175C42#1062532511yescharacteristic
176C42D37611yescharacteristic
177C42#1062533411yescharacteristic
178C42#1062533611yescharacteristic
179C23—1881no—
180C23#1309393811yescharacteristic
181Q8—1881no—
182Q8—1881no—
183D4—3762no—
184D4—3761no—
185D4—3761no—
186D4—1881no—
187D4—1881no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—3761no—
198C2×C4—3761no—
199C2×C4—1881no—
200C2×C4(C2×D47)⋊C811yescharacteristic
201C2×C4#1309394011yescharacteristic
202C2×C4#1309394411yescharacteristic
203C2×C4#1309394611yescharacteristic
204C2×C4#1309395011yescharacteristic
205C2×C4#1309395211yescharacteristic
206C8—82no—
207C8—3762no—
208C8—3761no—
209C22—41no—
210C22—3762no—
211C22—3761no—
212C22#1568884611yescharacteristic
213C22#1568885611yescharacteristic
214C22#1568885811yescharacteristic
215C4—22no—
216C4—22no—
217C4—22no—
218C4—41no—
219C4—41no—
220C4—41no—
221C4—3761no—
222C4#1568885011yescharacteristic
223C4#1568885211yescharacteristic
224C4#1568886211yescharacteristic
225C4#1568886611yescharacteristic
226C2—21no—
227C2—21no—
228C2—21no—
229C2—7521no—
230C2#2024120811yescharacteristic
subgroupG/H
centerC4#15688866
commutatorC42×C47C2×C8
FrattiniC43C2×D47
FittingC47×#1276C2
socleC2×C47#1270
prime radical#15689673C4
cosocleC43×C47C22
hypercenter#1276D47
hypocenterC47#7106
LCSG ▷ C42×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C4 ◁ C43 ◁ #1276

Complements

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

NH ≅ G/Ntype
2. #20241221C2semidirect
3. C47×#1276C2semidirect
13. #13094024C8semidirect
14. #13094021C8semidirect
113. #1276D47semidirect
126. C47#7106semidirect

Maximal subgroups

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

subgroupindex
#202442162
#202412212
C47×#12762
#710647