Finite Group Databasebeta

← #26125572

Subgroups — #26125572

Order 12032.

460 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 119 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#20241706C212yesmaximal
2#20241334C212yesmaximal, complemented
3#20241322C212yesmaximal, complemented
4C47×#2326C211yesmaximal, characteristic
5C2×C22⋊(C47⋊C8)—22no—
6#15688849C2214yes—
7#15688847—22no—
8C2×#124×C47C411yescharacteristic
9C2×C4×Q8×C47C411yescharacteristic
10C2×C4×D4×C47C2211yescharacteristic
11C47×#391C2211yescharacteristic
12C47×#390C411yescharacteristic
13C47×#388—22no—
14C47×#385—22no—
15C47×#384C2211yescharacteristic
16C47×#376C411yescharacteristic
17C22×C47⋊C8—42no—
18C22⋊(C47⋊C8)—24no—
19C22⋊(C47⋊C8)—42no—
20C22×Q8×C47C2×C411yescharacteristic
21C22×D4×C47C2×C411yescharacteristic
22C23×C4×C47D412yes—
23#124×C47—22no—
24#124×C47C814yescomplemented
25C4×Q8×C47—42no—
26C4×D4×C47—42no—
27C2×C4⋊C4×C47—21no—
28C2×C4⋊C4×C47—22no—
29C2×C4⋊C4×C47C2×C411yescharacteristic
30C2×C4⋊C4×C47C2311yescharacteristic
31C2×C22⋊C4×C47—22no—
32C2×C22⋊C4×C47—22no—
33C2×C22⋊C4×C47D412yes—
34C2×C42×C47—21no—
35C2×C42×C47C2×C411yescharacteristic
36C2×C42×C47C2×C411yescharacteristic
37C2×C42×C47C2×C411yescharacteristic
38#95×C47—21no—
39#95×C47—21no—
40#95×C47—22no—
41C24×C47C22⋊C412yes—
42C2×Q8×C47—21no—
43C2×Q8×C47—41no—
44C2×Q8×C47—41no—
45C2×Q8×C47C2×C812yescomplemented
46C2×D4×C47—41no—
47C2×D4×C47—41no—
48C2×D4×C47C2×C812yes—
49C2×D4×C47M1612yes—
50C22×C4×C47—21no—
51C22×C4×C47—21no—
52C22×C4×C47—22no—
53C22×C4×C47—22no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—82no—
57C22×C4×C47C22⋊C411yescharacteristic
58C22×C4×C47C22⋊C411yescharacteristic
59C22×C4×C47C22×C411yescharacteristic
60C22×C4×C47C2×D412yes—
61C4⋊C4×C47—41no—
62C4⋊C4×C47—41no—
63C4⋊C4×C47—42no—
64C4⋊C4×C47—82no—
65C22⋊C4×C47—28no—
66C22⋊C4×C47—42no—
67C22⋊C4×C47—82no—
68C42×C47—22no—
69C42×C47—21no—
70C42×C47—42no—
71C42×C47—41no—
72C42×C47C2×C812yescomplemented
73C2×C47⋊C8—44no—
74C2×C47⋊C8—84no—
75C23×C47—22no—
76C23×C47—22no—
77C23×C47—42no—
78C23×C47—42no—
79C23×C47C22⋊C814yes—
80C23×C47C2×C22⋊C411yescharacteristic
81Q8×C47—42no—
82Q8×C47—42no—
83D4×C47—42no—
84D4×C47—42no—
85C2×C4×C47—21no—
86C2×C4×C47—21no—
87C2×C4×C47—22no—
88C2×C4×C47—22no—
89C2×C4×C47—22no—
90C2×C4×C47—21no—
91C2×C4×C47—41no—
92C2×C4×C47—44no—
93C2×C4×C47—41no—
94C2×C4×C47—41no—
95C2×C4×C47—41no—
96C2×C4×C47—41no—
97C2×C4×C47—42no—
98C2×C4×C47—41no—
99C2×C4×C47—41no—
100C2×C4×C47—42no—
101C2×C4×C47—41no—
102C2×C4×C47—41no—
103C2×C4×C47—41no—
104C2×C4×C47—82no—
105C2×C4×C47#9912yes—
106C2×C4×C47C22×C811yescharacteristic
107C2×C4×C47C2×M1611yescharacteristic
108C47⋊C8—84no—
109#11481—471nomaximal, Sylow
110C22×C47—21no—
111C22×C47—21no—
112C22×C47—24no—
113C22×C47—44no—
114C22×C47—42no—
115C22×C47—42no—
116C22×C47—82no—
117C22×C47C2×C22⋊C811yescharacteristic
118C22×C47C2×#9911yescharacteristic
119C22×C47#41311yescharacteristic
120C4×C47—22no—
121C4×C47—21no—
122C4×C47—21no—
123C4×C47—42no—
124C4×C47—42no—
125C4×C47—41no—
126C4×C47—42no—
127C4×C47—81no—
128C4×C47—81no—
129#2326D4711yescharacteristic
130#1459—472no—
131#1325—472no—
132#1319—472no—
133C2×C47—21no—
134C2×C47—21no—
135C2×C47—44no—
136C2×C47#145911yescharacteristic
137C2×C47#212711yescharacteristic
138C2×C47#213511yescharacteristic
139C2×#124Dic4711yescharacteristic
140C2×C4×Q8Dic4711yescharacteristic
141C2×C4×D4C2×D4711yescharacteristic
142C2×C22⋊C8—942no—
143#391C2×D4711yescharacteristic
144#390Dic4711yescharacteristic
145#388—22no—
146#385—22no—
147#384C2×D4711yescharacteristic
148#376Dic4711yescharacteristic
149#324—474no—
150#323—942no—
151C47#1148111yescharacteristic, complemented, Sylow
152C22×Q8C2×Dic4711yescharacteristic
153C22×D4C2×Dic4711yescharacteristic
154C23×C4C22⋊D4712yes—
155C22×C8—1882no—
156#124—22no—
157#124C47⋊C814yescomplemented
158C4×Q8—42no—
159C4×D4—42no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4C2×Dic4711yescharacteristic
163C2×C4⋊C4C22×D4711yescharacteristic
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4C22⋊D4712yes—
167C2×C42—21no—
168C2×C42C2×Dic4711yescharacteristic
169C2×C42C2×Dic4711yescharacteristic
170C2×C42C2×Dic4711yescharacteristic
171C22⋊C8—944no—
172C22⋊C8—1882no—
173#95—21no—
174#95—21no—
175#95—22no—
176C24C22⋊Dic4712yes—
177C2×Q8—21no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8C2×C47⋊C812yescomplemented
181C2×D4—41no—
182C2×D4—41no—
183C2×D4C2×C47⋊C812yes—
184C2×D4#1062532912yes—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—82no—
192C22×C4C22⋊Dic4711yescharacteristic
193C22×C4C22⋊Dic4711yescharacteristic
194C22×C4C22×Dic4711yescharacteristic
195C22×C4C2×C22⋊D4712yes—
196C2×C8—3764no—
197C2×C8—1884no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—82no—
202C22⋊C4—28no—
203C22⋊C4—42no—
204C22⋊C4—82no—
205C42—22no—
206C42—21no—
207C42—41no—
208C42—42no—
209C42C2×C47⋊C812yescomplemented
210C23—22no—
211C23—22no—
212C23—42no—
213C23—42no—
214C23C22⋊(C47⋊C8)14yes—
215C23C2×C22⋊Dic4711yescharacteristic
216Q8—42no—
217Q8—42no—
218D4—42no—
219D4—42no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—44no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—82no—
240C2×C4#1309393912yes—
241C2×C4C22×C47⋊C811yescharacteristic
242C2×C4C2×#1062532911yescharacteristic
243C8—3764no—
244C22—21no—
245C22—21no—
246C22—24no—
247C22—44no—
248C22—42no—
249C22—42no—
250C22—82no—
251C22C2×C22⋊(C47⋊C8)11yescharacteristic
252C22C2×#1309393911yescharacteristic
253C22#1568910711yescharacteristic
254C4—21no—
255C4—21no—
256C4—22no—
257C4—42no—
258C4—42no—
259C4—42no—
260C4—41no—
261C4—81no—
262C4—81no—
263C2—21no—
264C2—21no—
265C2—44no—
266C2#2024170611yescharacteristic
267C2#2024505711yescharacteristic
268C2#2024508311yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC2×C4×C47C22×C8
FrattiniC2×C4⋊C4C22×D47
FittingC47×#2326C2
socleC22×C47C2×#99
prime radicalC22×D4×C47C2×C4
cosocleC2×C4⋊C4×C47C23
hypercenter#2326D47
hypocenterC47#11481
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2326

Complements

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

NH ≅ G/Ntype
2. #20241334C2semidirect
3. #20241322C2semidirect
24. #124×C47C8semidirect
45. C2×Q8×C47C2×C8semidirect
72. C42×C47C2×C8semidirect
151. C47#11481semidirect
157. #124C47⋊C8semidirect
180. C2×Q8C2×C47⋊C8semidirect
209. C42C2×C47⋊C8semidirect

Maximal subgroups

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

subgroupindex
#202417062
#202413342
#202413222
C47×#23262
#1148147