Finite Group Databasebeta

← #26125581

Subgroups — #26125581

Order 12032.

368 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 83 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#20252087C211yesmaximal, characteristic
2#20245007C212yesmaximal, complemented
3#20241340C212yesmaximal, complemented
4#20241324C211yesmaximal, characteristic, complemented
5C47×#1320C211yesmaximal, characteristic, complemented
6C2×#13094032C2211yescharacteristic
7C22×#10625329C411yescharacteristic
8#15689634C411yescharacteristic
9M32×D47—22no—
10#15689198—22no—
11#15689197—22no—
12#15689151C2211yescharacteristic
13#15689091C411yescharacteristic
14#15689081C411yescharacteristic
15#15688905C2212yes—
16#15688904—22no—
17C47×#407C2211yescharacteristic
18C47×#349C2212yes—
19C2×#10625350C2×C411yescharacteristic
20#13094032—42no—
21C2×#10625329—21no—
22C2×#10625329—21no—
23C2×#10625329—42no—
24C2×#10625324D412yes—
25C22×C47⋊C8C2×C411yescharacteristic
26C22×C47⋊C8C2×C411yescharacteristic
27C2×C8×D47D412yes—
28#13094007C812yes—
29#13094003—21no—
30#13093998—21no—
31#13093993C812yes—
32C4×C22⋊D47—22no—
33#13093983C2×C411yescharacteristic
34#13093964—22no—
35#13093963—42no—
36C16×D47—42no—
37Dic47⋊C8—22no—
38(C2×D47)⋊C8—22no—
39#13093934—21no—
40C8×Dic47—21no—
41C23×C4×C47C2×C411yescharacteristic
42C2×M16×C47C2×C411yescharacteristic
43C2×M16×C47C2311yescharacteristic
44M32×C47—22no—
45C22⋊C8×C47—22no—
46C24×C47C2×C811yescharacteristic
47C22×C4×C47—21no—
48C22×C4×C47—21no—
49C22×C4×C47—41no—
50C22×C4×C47—41no—
51C22×C4×C47C2×C811yescharacteristic
52C22×C4×C47C22×C411yescharacteristic
53C2×C22⋊D47—22no—
54C2×C22⋊D47C2×C812yes—
55#10625350—41no—
56#10625350—41no—
57C2×D188M1611yescharacteristic
58C2×C4×D47C22⋊C412yes—
59C2×Dic94M1611yescharacteristic
60M16×C47—41no—
61M16×C47—41no—
62C2×C8×C47—21no—
63C2×C8×C47C2×D412yes—
64C22⋊Dic47—22no—
65C22⋊Dic47C2×C812yes—
66(C2×D47)⋊C4—22no—
67C4⋊Dic47—21no—
68Dic47⋊C4—22no—
69C4×Dic47—21no—
70#10625329—42no—
71#10625329—42no—
72C2×C47⋊C8—22no—
73C2×C47⋊C8—21no—
74C2×C47⋊C8—41no—
75C2×C47⋊C8C22⋊C411yescharacteristic
76C2×C47⋊C8C22⋊C411yescharacteristic
77#10625324—42no—
78C8×D47—22no—
79C8×D47—22no—
80C16×C47—42no—
81C47⋊C16—42no—
82C23×C47—21no—
83C23×C47—41no—
84C23×C47—81no—
85C23×C47C22×C811yescharacteristic
86C22×D47C22⋊C812yes—
87C2×C4×C47—21no—
88C2×C4×C47—21no—
89C2×C4×C47—21no—
90C2×C4×C47—42no—
91C2×C4×C47—41no—
92C2×C4×C47—41no—
93C2×C4×C47—41no—
94C2×C4×C47C2×C22⋊C411yescharacteristic
95C2×C4×C47C2×M1611yescharacteristic
96C22⋊D47—42no—
97C22⋊D47—82no—
98C2×Dic47—22no—
99C2×Dic47C22⋊C812yes—
100D188—41no—
101C4×D47—22no—
102C4×D47—22no—
103Dic94—41no—
104C8×C47—22no—
105C8×C47—41no—
106C47⋊C8—21no—
107C47⋊C8—41no—
108C47⋊C8#9912yes—
109#11484—471nomaximal, Sylow
110C22×C47—21no—
111C22×C47—41no—
112C22×C47—41no—
113C22×C47—41no—
114C22×C47—41no—
115C22×C47—82no—
116C22×C47C2×C22⋊C811yescharacteristic
117C2×D47—22no—
118C2×D47—42no—
119C4×C47—21no—
120C4×C47—41no—
121C4×C47C2×#9911yescharacteristic
122C4×C47#41311yescharacteristic
123Dic47—22no—
124Dic47—42no—
125#2930—471no—
126#2115—472no—
127#1328—472no—
128#1320—471no—
129#1320D4711yescharacteristic, complemented
130C2×C47—21no—
131C2×C47—41no—
132C2×C47—81no—
133C2×C47#145911yescharacteristic
134D47—42no—
135C2×#131—471no—
136C22×M16—471no—
137#525—471no—
138C2×M32—942no—
139#436—942no—
140#435—942no—
141#424—471no—
142#408—471no—
143#407C2×D4711yescharacteristic
144#405—471no—
145#349—472no—
146#349—942no—
147#349C2×D4712yes—
148C47#1148411yescharacteristic, complemented, Sylow
149C2×#41—471no—
150C23×C4C4×D4711yescharacteristic
151#131—1882no—
152C2×M16—472no—
153C2×M16—941no—
154C2×M16—941no—
155C2×M16—1882no—
156C2×M16C4×D4711yescharacteristic
157C2×M16C22×D4711yescharacteristic
158C22×C8—471no—
159C22×C8—472no—
160C22×C8—471no—
161#123—472no—
162C22⋊Q8—941no—
163#121—941no—
164#120—472no—
165C4×D4—942no—
166#117—471no—
167M32—22no—
168M32—942no—
169M32—1882no—
170C2×C16—1882no—
171C4⋊C8—942no—
172C22⋊C8—22no—
173C22⋊C8—942no—
174#97—941no—
175C4×C8—941no—
176C24C8×D4711yescharacteristic
177#41—1881no—
178#41—1881no—
179C2×Q8—471no—
180C2×D4—472no—
181C2×D4—471no—
182C2×D4—942no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—472no—
188C22×C4C8×D4711yescharacteristic
189C22×C4C2×C4×D4711yescharacteristic
190M16—41no—
191M16—41no—
192M16—1882no—
193M16—1882no—
194M16—1882no—
195C2×C8—21no—
196C2×C8—471no—
197C2×C8—471no—
198C2×C8—942no—
199C2×C8—942no—
200C2×C8—942no—
201C2×C8—941no—
202C2×C8—1881no—
203C2×C8D4×D4712yes—
204C4⋊C4—942no—
205C4⋊C4—941no—
206C22⋊C4—472no—
207C22⋊C4—942no—
208C22⋊C4—942no—
209C42—941no—
210C16—42no—
211C16—1882no—
212C23—21no—
213C23—41no—
214C23—81no—
215C23—472no—
216C23C2×C8×D4711yescharacteristic
217Q8—1881no—
218D4—3762no—
219D4—1881no—
220D4—1882no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—472no—
229C2×C4—942no—
230C2×C4—942no—
231C2×C4—942no—
232C2×C4C22⋊C4×D4711yescharacteristic
233C2×C4M16×D4711yescharacteristic
234C8—22no—
235C8—41no—
236C8—472no—
237C8—941no—
238C8—1881no—
239C22—21no—
240C22—41no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22—82no—
245C22—942no—
246C22—1882no—
247C22C22⋊C8×D4711yescharacteristic
248C4—21no—
249C4—41no—
250C4—942no—
251C4—1882no—
252C4#99×D4711yescharacteristic
253C4#1568910411yescharacteristic
254C2—21no—
255C2—41no—
256C2—81no—
257C2—1882no—
258C2#2024170311yescharacteristic
subgroupG/H
centerC4#99×D47
commutatorC23×C47C22×C8
FrattiniC2×M16C22×D47
FittingC47×#1320C2
socleC2×C47#1459
prime radical#15689634C4
cosocleC2×M16×C47C23
hypercenter#1320D47
hypocenterC47#11484
LCSG ▷ C23×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. #20245007C2semidirect
3. #20241340C2semidirect
4. #20241324C2semidirect
5. C47×#1320C2semidirect
129. #1320D47semidirect
148. C47#11484semidirect

Maximal subgroups

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

subgroupindex
#202520872
#202450072
#202413402
#202413242
C47×#13202
#1148447