Finite Group Databasebeta

← #26125514

Subgroups — #26125514

Order 12032.

398 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 91 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#20252081C211yesmaximal, characteristic
2C22⋊C16×D47C212yesmaximal, complemented
3#20241330C211yesmaximal, characteristic, complemented
4#20241313C212yesmaximal, complemented
5C47×#1323C211yesmaximal, characteristic, complemented
6C2×#13094032C411yescharacteristic
7C22×C8×D47C2211yescharacteristic
8#15689635C411yescharacteristic
9C2×C16×D47—22no—
10#15689193—22no—
11#15689188—22no—
12#15689089C2211yescharacteristic
13#15689086C411yescharacteristic
14#15689079C411yescharacteristic
15C22⋊(C47⋊C16)C2212yes—
16(C2×D47)⋊C16—22no—
17C47×#424C2211yescharacteristic
18C47×C22⋊C16C2212yes—
19C2×#10625350C2×C411yescharacteristic
20C22×C4×D47C2×C411yescharacteristic
21#13094032—42no—
22C2×#10625329C2×C411yescharacteristic
23C2×#10625324—21no—
24C22×C47⋊C8C2×C411yescharacteristic
25C2×C8×D47—21no—
26C2×C8×D47—42no—
27C2×C8×D47D412yes—
28C2×C8×D47D412yes—
29#13094008—21no—
30#13094004C812yes—
31#13093999C812yes—
32#13093994—21no—
33C4×D188—22no—
34C2×C47⋊C16—22no—
35C16×D47—44no—
36(C2×D47)⋊C8—22no—
37(C2×D47)⋊C8—22no—
38#13093935—21no—
39C4×C47⋊C8—21no—
40C2×M16×C47C2×C411yescharacteristic
41C22×C8×C47C2311yescharacteristic
42#117×C47C2×C411yescharacteristic
43C2×C16×C47—22no—
44C4⋊C8×C47—22no—
45C23×D47M1611yescharacteristic
46C22×C4×C47C22×C411yescharacteristic
47C2×C22⋊D47—21no—
48C22×Dic47M1611yescharacteristic
49#10625350—41no—
50#10625350—41no—
51C2×D188—22no—
52C2×D188C2×C811yescharacteristic
53C2×C4×D47—21no—
54C2×C4×D47—41no—
55C2×C4×D47—41no—
56C2×C4×D47C22⋊C412yes—
57C2×C4×D47C2×C812yes—
58C2×Dic94C2×C811yescharacteristic
59M16×C47—41no—
60C2×C8×C47—21no—
61C2×C8×C47—41no—
62C2×C8×C47C2×D412yes—
63C4⋊C4×C47C2×C812yes—
64C22⋊C4×C47—21no—
65C42×C47—21no—
66(C2×D47)⋊C4—22no—
67(C2×D47)⋊C4—22no—
68C4⋊Dic47—22no—
69#10625329—41no—
70C2×C47⋊C8—22no—
71C2×C47⋊C8—21no—
72C2×C47⋊C8C22⋊C411yescharacteristic
73C2×C47⋊C8C22⋊C411yescharacteristic
74#10625324—42no—
75C8×D47—24no—
76C8×D47—42no—
77C8×D47—42no—
78C16×C47—42no—
79C47⋊C16—42no—
80C23×C47C2×M1611yescharacteristic
81C22×D47—24no—
82C22×D47—21no—
83C22×D47—41no—
84C22×D47C22⋊C812yes—
85C2×C4×C47—21no—
86C2×C4×C47—21no—
87C2×C4×C47—22no—
88C2×C4×C47C2×C22⋊C411yescharacteristic
89C2×C4×C47C22×C811yescharacteristic
90C22⋊D47—42no—
91C2×Dic47—21no—
92C2×Dic47—41no—
93C2×Dic47C22⋊C812yes—
94D188—41no—
95D188—44no—
96C4×D47—22no—
97C4×D47—22no—
98C4×D47—42no—
99C4×D47—41no—
100C4×D47—41no—
101Dic94—41no—
102C8×C47—22no—
103C8×C47—41no—
104C47⋊C8—21no—
105C47⋊C8—41no—
106C47⋊C8#9912yes—
107#11468—471nomaximal, Sylow
108C22×C47—22no—
109C22×C47—21no—
110C22×C47C2×C22⋊C811yescharacteristic
111C2×D47—22no—
112C2×D47—22no—
113C2×D47—44no—
114C2×D47—42no—
115C2×D47—41no—
116C2×D47#35014yes—
117C4×C47—21no—
118C4×C47—42no—
119C4×C47C2×#9911yescharacteristic
120C4×C47#41311yescharacteristic
121Dic47—22no—
122Dic47—41no—
123#2929—471no—
124C2×C22⋊C16—472no—
125#1323—471no—
126#1323D4711yescharacteristic, complemented
127#1315—472no—
128C2×C47—41no—
129C2×C47#145911yescharacteristic
130C2×C47C2×#35012yes—
131D47—24no—
132D47—81no—
133C2×#131—471no—
134C23×C8—471no—
135#525—471no—
136C22×C16—942no—
137#435—942no—
138C8×D4—942no—
139#424C2×D4711yescharacteristic
140#408—471no—
141#407—471no—
142#405—471no—
143C22⋊C16—472no—
144C22⋊C16—942no—
145C22⋊C16C2×D4712yes—
146C47#1146811yescharacteristic, complemented, Sylow
147C2×#41—471no—
148C23×C4—471no—
149#131—1882no—
150C2×M16—471no—
151C2×M16—941no—
152C2×M16C4×D4711yescharacteristic
153C22×C8—471no—
154C22×C8—472no—
155C22×C8—472no—
156C22×C8—941no—
157C22×C8—1882no—
158C22×C8C22×D4711yescharacteristic
159#123—941no—
160C22⋊Q8—472no—
161#121—472no—
162#120—941no—
163C4×D4—942no—
164#117C4×D4711yescharacteristic
165C2×C16—22no—
166C2×C16—942no—
167C2×C16—1884no—
168C4⋊C8—22no—
169C22⋊C8—942no—
170C22⋊C8—942no—
171#97—941no—
172C4×C8—941no—
173C24—471no—
174#41—1881no—
175#41—1881no—
176C2×Q8—471no—
177C2×D4—471no—
178C2×D4—942no—
179C2×D4—941no—
180C22×C4—471no—
181C22×C4—472no—
182C22×C4—472no—
183C22×C4—941no—
184C22×C4—1881no—
185C22×C4—1881no—
186C22×C4C2×C4×D4711yescharacteristic
187M16—41no—
188M16—1882no—
189M16—1881no—
190C2×C8—21no—
191C2×C8—41no—
192C2×C8—471no—
193C2×C8—471no—
194C2×C8—944no—
195C2×C8—942no—
196C2×C8—941no—
197C2×C8—1882no—
198C2×C8—1882no—
199C2×C8D4×D4712yes—
200C4⋊C4—942no—
201C4⋊C4C8×D4712yes—
202C22⋊C4—21no—
203C22⋊C4—942no—
204C22⋊C4—942no—
205C42—21no—
206C16—42no—
207C16—1882no—
208C23—472no—
209C23—944no—
210C23—941no—
211C23—1881no—
212C23M16×D4711yescharacteristic
213Q8—1881no—
214D4—1881no—
215D4—1884no—
216D4—1882no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—472no—
221C2×C4—942no—
222C2×C4—942no—
223C2×C4—941no—
224C2×C4—1882no—
225C2×C4—1881no—
226C2×C4—1881no—
227C2×C4—1881no—
228C2×C4C22⋊C4×D4711yescharacteristic
229C2×C4C2×C8×D4711yescharacteristic
230C8—22no—
231C8—41no—
232C8—472no—
233C8—941no—
234C8—1881no—
235C22—22no—
236C22—21no—
237C22—474no—
238C22—942no—
239C22—942no—
240C22—1884no—
241C22—1882no—
242C22—1881no—
243C22C22⋊C8×D4711yescharacteristic
244C4—21no—
245C4—42no—
246C4—942no—
247C4—1881no—
248C4#99×D4711yescharacteristic
249C4#1568910611yescharacteristic
250C2—41no—
251C2—944no—
252C2—3761no—
253C2#2024170511yescharacteristic
254C2#350×D4712yes—
subgroupG/H
centerC2×C4C22⋊C4×D47
commutatorC2×C4×C47C22×C8
FrattiniC22×C8C22×D47
FittingC47×#1323C2
socleC22×C47C2×C22⋊C8
prime radical#15689635C4
cosocleC22×C8×C47C23
hypercenter#1323D47
hypocenterC47#11468
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1323

Complements

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

NH ≅ G/Ntype
2. C22⋊C16×D47C2semidirect
3. #20241330C2semidirect
4. #20241313C2semidirect
5. C47×#1323C2semidirect
126. #1323D47semidirect
146. C47#11468semidirect

Maximal subgroups

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

subgroupindex
#202520812
C22⋊C16×D472
#202413302
#202413132
C47×#13232
#1146847