Finite Group Databasebeta

← #26125775

Subgroups — #26125775

Order 12032.

348 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 71 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#20252078C211yesmaximal, characteristic
2#20245201C212yesmaximal, complemented
3#20241325C212yesmaximal, complemented
4#20241324C211yesmaximal, characteristic, complemented
5C47×#1320C211yesmaximal, characteristic, complemented
6C2×#13094032C411yescharacteristic
7C23×C47⋊C8C411yescharacteristic
8#15689634C411yescharacteristic
9#15689191—22no—
10(C22⋊D47)⋊C8—22no—
11#15689151C411yescharacteristic
12#15689091C2211yescharacteristic
13#15689081C2211yescharacteristic
14#15688943—22no—
15#15688905C2212yes—
16#15688900—22no—
17C47×#407C2211yescharacteristic
18C47×#349C2212yes—
19C2×#10625350C2×C411yescharacteristic
20#13094032—42no—
21C2×#10625324—21no—
22C22×C47⋊C8—21no—
23C22×C47⋊C8—21no—
24C22×C47⋊C8—42no—
25C22×C47⋊C8C2×C411yescharacteristic
26C22×C47⋊C8C2×C411yescharacteristic
27C2×C8×D47—21no—
28#13094007—21no—
29#13094003—21no—
30#13093998—21no—
31#13093993—21no—
32C4×C22⋊D47—22no—
33#13093983C2×C411yescharacteristic
34#13093964—22no—
35Dic47⋊C8—22no—
36(C2×D47)⋊C8—22no—
37#13093934D412yes—
38C8×Dic47D412yes—
39C23×C4×C47C2×C411yescharacteristic
40C2×M16×C47C2×C411yescharacteristic
41C2×M16×C47C2311yescharacteristic
42M32×C47—22no—
43C22⋊C8×C47—22no—
44C24×C47M1611yescharacteristic
45C22×C4×C47—21no—
46C22×C4×C47—21no—
47C22×C4×C47—41no—
48C22×C4×C47—41no—
49C22×C4×C47M1611yescharacteristic
50C22×C4×C47C22×C411yescharacteristic
51C2×C22⋊D47—22no—
52C2×C22⋊D47—21no—
53#10625350—41no—
54#10625350—41no—
55C2×D188M1611yescharacteristic
56C2×C4×D47—21no—
57C2×Dic94M1611yescharacteristic
58M16×C47—41no—
59M16×C47—41no—
60C2×C8×C47—21no—
61C2×C8×C47C2×D412yes—
62C22⋊Dic47—22no—
63C22⋊Dic47—21no—
64(C2×D47)⋊C4—22no—
65C4⋊Dic47—21no—
66Dic47⋊C4—22no—
67C4×Dic47C22⋊C412yes—
68C2×C47⋊C8—22no—
69C2×C47⋊C8—21no—
70C2×C47⋊C8—42no—
71C2×C47⋊C8—42no—
72C2×C47⋊C8—41no—
73C2×C47⋊C8C22⋊C411yescharacteristic
74C2×C47⋊C8C22⋊C411yescharacteristic
75#10625324—42no—
76C8×D47—42no—
77C16×C47—42no—
78C47⋊C16—42no—
79C23×C47—21no—
80C23×C47—41no—
81C23×C47—42no—
82C23×C47C2×M1611yescharacteristic
83C22×D47—21no—
84C2×C4×C47—21no—
85C2×C4×C47—21no—
86C2×C4×C47—21no—
87C2×C4×C47—42no—
88C2×C4×C47—41no—
89C2×C4×C47—41no—
90C2×C4×C47—41no—
91C2×C4×C47C2×C22⋊C411yescharacteristic
92C2×C4×C47C2×M1611yescharacteristic
93C22⋊D47—42no—
94C22⋊D47—44no—
95C2×Dic47—22no—
96C2×Dic47—21no—
97D188—41no—
98C4×D47—41no—
99C4×D47—41no—
100Dic94—41no—
101C8×C47—22no—
102C8×C47—41no—
103C47⋊C8—21no—
104C47⋊C8—41no—
105C47⋊C8#10012yes—
106#11537—471nomaximal, Sylow
107C22×C47—22no—
108C22×C47—21no—
109C22×C47—41no—
110C22×C47—41no—
111C22×C47—42no—
112C22×C47—42no—
113C22×C47—41no—
114C22×C47#40711yescharacteristic
115C2×D47—42no—
116C2×D47—41no—
117C4×C47—21no—
118C4×C47—41no—
119C4×C47#41011yescharacteristic
120C4×C47C2×#10011yescharacteristic
121Dic47—42no—
122Dic47—41no—
123#2929—471no—
124#2166—472no—
125#1321—472no—
126#1320—471no—
127#1320D4711yescharacteristic, complemented
128C2×C47—21no—
129C2×C47—42no—
130C2×C47—41no—
131C2×C47#146611yescharacteristic
132D47—81no—
133C2×#131—471no—
134C23×C8—471no—
135#525—471no—
136#435—942no—
137C8×D4—942no—
138#424—471no—
139#408—471no—
140#407C2×D4711yescharacteristic
141#405—471no—
142#364—942no—
143#349—472no—
144#349C2×D4712yes—
145#347—942no—
146C47#1153711yescharacteristic, complemented, Sylow
147C2×#41—471no—
148C23×C4C4×D4711yescharacteristic
149#131—1882no—
150C2×M16—941no—
151C2×M16C4×D4711yescharacteristic
152C2×M16C22×D4711yescharacteristic
153C22×C8—471no—
154C22×C8—471no—
155C22×C8—941no—
156C22×C8—941no—
157C22×C8—941no—
158C22×C8—1882no—
159#123—941no—
160C22⋊Q8—941no—
161#121—941no—
162#120—941no—
163C4×D4—942no—
164#117—471no—
165M32—22no—
166M32—942no—
167C4⋊C8—942no—
168C22⋊C8—22no—
169C22⋊C8—942no—
170#97—472no—
171C4×C8—472no—
172C24#1062532411yescharacteristic
173#41—1881no—
174#41—1881no—
175C2×Q8—471no—
176C2×D4—471no—
177C2×D4—942no—
178C2×D4—941no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—941no—
184C22×C4#1062532411yescharacteristic
185C22×C4C2×C4×D4711yescharacteristic
186M16—41no—
187M16—41no—
188M16—1882no—
189C2×C8—21no—
190C2×C8—471no—
191C2×C8—471no—
192C2×C8—942no—
193C2×C8—941no—
194C2×C8—1882no—
195C2×C8—1882no—
196C2×C8—1882no—
197C2×C8—1881no—
198C2×C8D4×D4712yes—
199C4⋊C4—942no—
200C4⋊C4—941no—
201C22⋊C4—942no—
202C22⋊C4—942no—
203C22⋊C4—941no—
204C42—472no—
205C16—42no—
206C16—1882no—
207C23—21no—
208C23—41no—
209C23—42no—
210C23—941no—
211C23C2×#1062532411yescharacteristic
212Q8—1881no—
213D4—1881no—
214D4—1884no—
215D4—1882no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—942no—
224C2×C4—941no—
225C2×C4—1881no—
226C2×C4—1881no—
227C2×C4C22⋊C4×D4711yescharacteristic
228C2×C4M16×D4711yescharacteristic
229C8—22no—
230C8—41no—
231C8—472no—
232C8—941no—
233C8—1881no—
234C22—22no—
235C22—21no—
236C22—41no—
237C22—41no—
238C22—42no—
239C22—42no—
240C22—41no—
241C22—1882no—
242C22—1881no—
243C22#1568908511yescharacteristic
244C4—21no—
245C4—41no—
246C4—1882no—
247C4—1881no—
248C4#1568909511yescharacteristic
249C4#100×D4711yescharacteristic
250C2—21no—
251C2—42no—
252C2—41no—
253C2—3761no—
254C2#2024172711yescharacteristic
subgroupG/H
centerC4#15689095
commutatorC22×C4×C47C22×C4
FrattiniC2×M16C22×D47
FittingC47×#1320C2
socleC2×C47#1466
prime radical#15689634C4
cosocleC2×M16×C47C23
hypercenter#1320D47
hypocenterC47#11537
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1320

Complements

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

NH ≅ G/Ntype
2. #20245201C2semidirect
3. #20241325C2semidirect
4. #20241324C2semidirect
5. C47×#1320C2semidirect
127. #1320D47semidirect
146. C47#11537semidirect

Maximal subgroups

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

subgroupindex
#202520782
#202452012
#202413252
#202413242
C47×#13202
#1153747