Finite Group Databasebeta

← #26125781

Subgroups — #26125781

Order 12032.

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

normal subgroups only

This group has 334 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#20252094C211yesmaximal, characteristic
2#20245201C212yesmaximal, complemented
3#20241326C211yesmaximal, characteristic, complemented
4#20241323C212yesmaximal, complemented
5C47×#1321C211yesmaximal, characteristic, complemented
6C2×#13094031C411yescharacteristic
7C22×#10625324C411yescharacteristic
8#15689630C411yescharacteristic
9#15689198—22no—
10#15689191—22no—
11#15689149C411yescharacteristic
12#15689086C2211yescharacteristic
13#15689081C2211yescharacteristic
14#15688943—22no—
15#15688905C2212yes—
16#15688900—22no—
17C47×#408C2211yescharacteristic
18C47×#349C2212yes—
19C22×C4×D47C2×C411yescharacteristic
20#13094031—42no—
21C2×#10625329—21no—
22C2×#10625329C2×C411yescharacteristic
23C2×#10625324—21no—
24C2×#10625324—21no—
25C2×#10625324—42no—
26C22×C47⋊C8—21no—
27C22×C47⋊C8C2×C411yescharacteristic
28#13094005—21no—
29#13094000—21no—
30#13093995—21no—
31#13093992—21no—
32C4×C22⋊D47—22no—
33#13093983C2×C411yescharacteristic
34#13093964—22no—
35Dic47⋊C8—22no—
36(C2×D47)⋊C8—22no—
37#13093934D412yes—
38C8×Dic47D412yes—
39C2×#41×C47C2×C411yescharacteristic
40C2×M16×C47C2311yescharacteristic
41C22×C8×C47C2×C411yescharacteristic
42M32×C47—22no—
43C22⋊C8×C47—22no—
44C23×D47M1611yescharacteristic
45#41×C47—41no—
46#41×C47—41no—
47C2×Q8×C47M1611yescharacteristic
48C2×D4×C47—21no—
49C2×D4×C47M1611yescharacteristic
50C22×C4×C47—21no—
51C22×C4×C47C22×C411yescharacteristic
52C2×C22⋊D47—22no—
53C22×Dic47M1611yescharacteristic
54C2×C4×D47—21no—
55C2×C4×D47—21no—
56C2×C4×D47—41no—
57C2×C4×D47—41no—
58M16×C47—41no—
59C2×C8×C47—21no—
60C2×C8×C47—41no—
61C2×C8×C47C2×D412yes—
62C22⋊Dic47—22no—
63C22⋊Dic47—21no—
64(C2×D47)⋊C4—22no—
65C4⋊Dic47—21no—
66Dic47⋊C4—22no—
67C4×Dic47C22⋊C412yes—
68#10625329—42no—
69#10625329—41no—
70C2×C47⋊C8—22no—
71C2×C47⋊C8—21no—
72C2×C47⋊C8—42no—
73C2×C47⋊C8C22⋊C411yescharacteristic
74C2×C47⋊C8C22⋊C411yescharacteristic
75#10625324—42no—
76#10625324—42no—
77C16×C47—42no—
78C47⋊C16—42no—
79C23×C47—21no—
80C23×C47C2×M1611yescharacteristic
81C22×D47—21no—
82C22×D47—41no—
83C22×D47—81no—
84Q8×C47—41no—
85D4×C47—41no—
86D4×C47—42no—
87C2×C4×C47—21no—
88C2×C4×C47—21no—
89C2×C4×C47—21no—
90C2×C4×C47—41no—
91C2×C4×C47—41no—
92C2×C4×C47C2×C22⋊C411yescharacteristic
93C2×C4×C47C2×M1611yescharacteristic
94C22⋊D47—82no—
95C2×Dic47—22no—
96C2×Dic47—21no—
97C2×Dic47—41no—
98C4×D47—42no—
99C4×D47—41no—
100C4×D47—41no—
101C8×C47—22no—
102C8×C47—41no—
103C47⋊C8—21no—
104C47⋊C8—41no—
105C47⋊C8#10012yes—
106#11538—471nomaximal, Sylow
107C22×C47—21no—
108C22×C47—41no—
109C22×C47—42no—
110C22×C47—41no—
111C22×C47#40711yescharacteristic
112C2×D47—41no—
113C2×D47—41no—
114C2×D47—41no—
115C2×D47—82no—
116C4×C47—21no—
117C4×C47—41no—
118C4×C47#41011yescharacteristic
119C4×C47C2×#10011yescharacteristic
120Dic47—42no—
121Dic47—41no—
122#2930—471no—
123#2166—472no—
124#1321—471no—
125#1321D4711yescharacteristic, complemented
126#1320—472no—
127C2×C47—21no—
128C2×C47—41no—
129C2×C47—81no—
130C2×C47#146611yescharacteristic
131D47—81no—
132C2×#131—471no—
133C22×M16—471no—
134#525—471no—
135#436—942no—
136#435—942no—
137#424—471no—
138#408C2×D4711yescharacteristic
139#407—471no—
140#405—471no—
141#364—942no—
142#349—472no—
143#349C2×D4712yes—
144#347—942no—
145C47#1153811yescharacteristic, complemented, Sylow
146C2×#41C4×D4711yescharacteristic
147C23×C4—471no—
148#131—1882no—
149C2×M16—471no—
150C2×M16—941no—
151C2×M16—941no—
152C2×M16—941no—
153C2×M16—1882no—
154C2×M16C22×D4711yescharacteristic
155C22×C8—471no—
156C22×C8—941no—
157C22×C8C4×D4711yescharacteristic
158#123—941no—
159C22⋊Q8—941no—
160#121—941no—
161#120—941no—
162C4×D4—942no—
163#117—471no—
164M32—22no—
165M32—942no—
166C4⋊C8—942no—
167C22⋊C8—22no—
168C22⋊C8—942no—
169#97—472no—
170C4×C8—472no—
171C24—471no—
172#41—41no—
173#41—41no—
174C2×Q8#1062532411yescharacteristic
175C2×D4—21no—
176C2×D4—942no—
177C2×D4#1062532411yescharacteristic
178C22×C4—21no—
179C22×C4—471no—
180C22×C4—941no—
181C22×C4—941no—
182C22×C4—1881no—
183C22×C4—1881no—
184C22×C4C2×C4×D4711yescharacteristic
185M16—41no—
186M16—1882no—
187M16—1882no—
188M16—1882no—
189M16—1881no—
190C2×C8—21no—
191C2×C8—41no—
192C2×C8—471no—
193C2×C8—471no—
194C2×C8—942no—
195C2×C8—941no—
196C2×C8—1882no—
197C2×C8D4×D4712yes—
198C4⋊C4—942no—
199C4⋊C4—941no—
200C22⋊C4—942no—
201C22⋊C4—942no—
202C22⋊C4—941no—
203C42—472no—
204C16—42no—
205C16—1882no—
206C23—21no—
207C23—941no—
208C23—3761no—
209C23—1881no—
210C23M16×D4711yescharacteristic
211Q8—41no—
212D4—41no—
213D4—42no—
214D4—3762no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—942no—
221C2×C4—941no—
222C2×C4—1882no—
223C2×C4—1881no—
224C2×C4—1881no—
225C2×C4—1881no—
226C2×C4C22⋊C4×D4711yescharacteristic
227C2×C4C2×#1062532411yescharacteristic
228C8—22no—
229C8—41no—
230C8—472no—
231C8—941no—
232C8—1881no—
233C22—21no—
234C22—41no—
235C22—42no—
236C22—41no—
237C22—3762no—
238C22—1881no—
239C22—1881no—
240C22—1881no—
241C22#1568908511yescharacteristic
242C4—21no—
243C4—41no—
244C4—1882no—
245C4—1881no—
246C4#1568909511yescharacteristic
247C4#100×D4711yescharacteristic
248C2—21no—
249C2—41no—
250C2—81no—
251C2—3761no—
252C2#2024172711yescharacteristic
subgroupG/H
centerC4#15689095
commutatorC22×C4×C47C22×C4
FrattiniC2×M16C22×D47
FittingC47×#1321C2
socleC2×C47#1466
prime radical#15689630C4
cosocleC2×M16×C47C23
hypercenter#1321D47
hypocenterC47#11538
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1321

Complements

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

NH ≅ G/Ntype
2. #20245201C2semidirect
3. #20241326C2semidirect
4. #20241323C2semidirect
5. C47×#1321C2semidirect
125. #1321D47semidirect
145. C47#11538semidirect

Maximal subgroups

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

subgroupindex
#202520942
#202452012
#202413262
#202413232
C47×#13212
#1153847