Finite Group Databasebeta

← #26125712

Subgroups — #26125712

Order 12032.

284 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 75 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#20252024C211yesmaximal, characteristic
2#20245200C211yesmaximal, characteristic, complemented
3#20241338C212yesmaximal, complemented
4#20241323C212yesmaximal
5C47×#2165C211yesmaximal, characteristic, complemented
6C2×#13093981C411yescharacteristic
7#15689185—21no—
8#15689185—21no—
9#15689182—21no—
10#15689182—21no—
11#15689146C411yescharacteristic
12#15689146C2211yescharacteristic
13#15689086C411yescharacteristic
14#15689086C2211yescharacteristic
15C4×#10625329C411yescharacteristic
16#15688944—21no—
17#15688905C2212yes—
18#15688901—21no—
19C4×M16×C47C2211yescharacteristic
20C47×#364—21no—
21C47×#349C2212yes—
22C47×#347—21no—
23C22×C4×D47C2×C411yescharacteristic
24C2×#10625329C2×C411yescharacteristic
25C2×#10625329C2×C411yescharacteristic
26#13093981—21no—
27#13093981—21no—
28#13093981—22no—
29C2×Dic47⋊C4—21no—
30C2×(C2×D47)⋊C4—21no—
31C2×C4×Dic47C2×C411yescharacteristic
32#13093964—22no—
33Dic47⋊C8—21no—
34Dic47⋊C8—21no—
35Dic47⋊C8—21no—
36Dic47⋊C8—21no—
37(C2×D47)⋊C8—21no—
38(C2×D47)⋊C8—21no—
39(C2×D47)⋊C8—21no—
40(C2×D47)⋊C8—21no—
41#13093935—21no—
42C4×C47⋊C8—21no—
43C2×M16×C47C2×C411yescharacteristic
44C2×M16×C47C2311yescharacteristic
45C2×C42×C47C2×C411yescharacteristic
46M32×C47—22no—
47#97×C47D411yescharacteristic
48#97×C47D411yescharacteristic
49C4×C8×C47D411yescharacteristic
50C4×C8×C47D411yescharacteristic
51C23×D47M1611yescharacteristic
52C22×C4×C47—21no—
53C22×C4×C47C22×C411yescharacteristic
54C22×Dic47—21no—
55C22×Dic47M1611yescharacteristic
56C2×C4×D47—21no—
57C2×C4×D47—21no—
58C2×C4×D47—41no—
59C2×C4×D47—41no—
60M16×C47—41no—
61M16×C47—41no—
62C2×C8×C47C22⋊C411yescharacteristic
63C2×C8×C47C22⋊C411yescharacteristic
64C2×C8×C47C2×D411yescharacteristic
65C2×C8×C47C2×D411yescharacteristic
66C42×C47—21no—
67C42×C47C22⋊C411yescharacteristic
68C42×C47C22⋊C411yescharacteristic
69(C2×D47)⋊C4—22no—
70(C2×D47)⋊C4—22no—
71Dic47⋊C4—22no—
72Dic47⋊C4—22no—
73C4×Dic47—21no—
74C4×Dic47M1612yes—
75#10625329—41no—
76#10625329—41no—
77C2×C47⋊C8—21no—
78C2×C47⋊C8—21no—
79C16×C47—42no—
80C47⋊C16—42no—
81C23×C47C2×M1611yescharacteristic
82C22×D47—21no—
83C22×D47—41no—
84C22×D47—81no—
85C2×C4×C47—21no—
86C2×C4×C47—21no—
87C2×C4×C47—21no—
88C2×C4×C47—21no—
89C2×C4×C47—81no—
90C2×C4×C47C2×C22⋊C411yescharacteristic
91C2×C4×C47C2×M1611yescharacteristic
92C2×Dic47—22no—
93C2×Dic47—21no—
94C2×Dic47—41no—
95C2×Dic47—81no—
96C4×D47—42no—
97C4×D47—41no—
98C4×D47—41no—
99C8×C47—21no—
100C8×C47—21no—
101C8×C47—21no—
102C8×C47—21no—
103C47⋊C8—41no—
104C47⋊C8—41no—
105#11522—471nomaximal, Sylow
106C22×C47—21no—
107C22×C47—41no—
108C22×C47#40711yescharacteristic
109C2×D47—42no—
110C2×D47—41no—
111C2×D47—82no—
112C4×C47—21no—
113C4×C47—41no—
114C4×C47—41no—
115C4×C47#41011yescharacteristic
116C4×C47#41311yescharacteristic
117Dic47—42no—
118Dic47—41no—
119#2921—471no—
120#2165—471no—
121#2165D4711yescharacteristic, complemented
122#1327—472no—
123#1320—472no—
124C2×C47—21no—
125C2×C47—41no—
126C2×C47#146411yescharacteristic
127D47—81no—
128C2×#117—471no—
129#433—941no—
130#433—941no—
131#433—941no—
132#433—941no—
133#423—471no—
134#423—471no—
135#407—471no—
136#407—471no—
137C4×M16—471no—
138C4×M16C2×D4711yescharacteristic
139#364—21no—
140#364—941no—
141#349—472no—
142#349C2×D4712yes—
143#347—21no—
144#347—941no—
145C47#1152211yescharacteristic, complemented, Sylow
146C23×C4—471no—
147C2×M16—471no—
148C2×M16—471no—
149C2×M16C4×D4711yescharacteristic
150C2×M16C22×D4711yescharacteristic
151#117—942no—
152#117—941no—
153#117—941no—
154C2×C4⋊C4—941no—
155C2×C22⋊C4—941no—
156C2×C42—471no—
157C2×C42C4×D4711yescharacteristic
158M32—22no—
159M32—942no—
160C4⋊C8—941no—
161C4⋊C8—941no—
162C4⋊C8—941no—
163C4⋊C8—941no—
164C22⋊C8—941no—
165C22⋊C8—941no—
166C22⋊C8—941no—
167C22⋊C8—941no—
168#97—941no—
169#97D18811yescharacteristic
170#97C22⋊D4711yescharacteristic
171C4×C8—941no—
172C4×C8D18811yescharacteristic
173C4×C8C22⋊D4711yescharacteristic
174C24—471no—
175C22×C4—21no—
176C22×C4—471no—
177C22×C4—941no—
178C22×C4—941no—
179C22×C4—941no—
180C22×C4—1881no—
181C22×C4—1881no—
182C22×C4C2×C4×D4711yescharacteristic
183M16—41no—
184M16—41no—
185M16—1881no—
186M16—1881no—
187C2×C8—941no—
188C2×C8—941no—
189C2×C8(C2×D47)⋊C411yescharacteristic
190C2×C8(C2×D47)⋊C411yescharacteristic
191C2×C8C2×D18811yescharacteristic
192C2×C8C2×C22⋊D4711yescharacteristic
193C4⋊C4—942no—
194C4⋊C4—942no—
195C22⋊C4—942no—
196C22⋊C4—942no—
197C42—21no—
198C42—472no—
199C42—941no—
200C42(C2×D47)⋊C411yescharacteristic
201C42(C2×D47)⋊C411yescharacteristic
202C16—42no—
203C16—1882no—
204C23—941no—
205C23—3761no—
206C23—1881no—
207C23M16×D4711yescharacteristic
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—81no—
213C2×C4—942no—
214C2×C4—941no—
215C2×C4—3761no—
216C2×C4—1882no—
217C2×C4—1881no—
218C2×C4—1881no—
219C2×C4—1881no—
220C2×C4C2×(C2×D47)⋊C411yescharacteristic
221C2×C4M16×D4711yescharacteristic
222C8—21no—
223C8—21no—
224C8—21no—
225C8—21no—
226C8—1881no—
227C8—1881no—
228C22—21no—
229C22—41no—
230C22—3762no—
231C22—1882no—
232C22—1881no—
233C22#1568908611yescharacteristic
234C4—21no—
235C4—41no—
236C4—41no—
237C4—1882no—
238C4—1881no—
239C4#1568909611yescharacteristic
240C4#1568910511yescharacteristic
241C2—21no—
242C2—41no—
243C2—3761no—
244C2#2024172011yescharacteristic
subgroupG/H
centerC4#15689096
commutatorC22×C4×C47C22×C4
FrattiniC2×M16C22×D47
FittingC47×#2165C2
socleC2×C47#1464
prime radicalC23×D47M16
cosocleC2×M16×C47C23
hypercenter#2165D47
hypocenterC47#11522
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C42 ◁ C4×M16 ◁ #2165

Complements

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

NH ≅ G/Ntype
2. #20245200C2semidirect
3. #20241338C2semidirect
5. C47×#2165C2semidirect
121. #2165D47semidirect
145. C47#11522semidirect

Maximal subgroups

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

subgroupindex
#202520242
#202452002
#202413382
#202413232
C47×#21652
#1152247