Finite Group Databasebeta

← #26134483

Subgroups — #26134483

Order 12032.

438 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 115 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#20252084C211yesmaximal, characteristic
2#20245154C211yesmaximal, characteristic, complemented
3#20245015C212yesmaximal, complemented
4#20244998C212yesmaximal, complemented
5C47×#2110C211yesmaximal, characteristic, complemented
6C2×#13094030C2211yescharacteristic
7C22×C8×D47C2211yescharacteristic
8#15689635C411yescharacteristic
9C2×#13093964C2212yes—
10#15689196—22no—
11C8×D188—22no—
12#15689150C411yescharacteristic
13#15689091C411yescharacteristic
14#15689086C411yescharacteristic
15#15688944—22no—
16#15688906—24no—
17(C2×D47)⋊C16—24no—
18C2×M32×C47C2212yes—
19C47×#405C2211yescharacteristic
20C47×#347—22no—
21C2×#10625350C2×C411yescharacteristic
22C22×C4×D47C2×C411yescharacteristic
23#13094030—22no—
24#13094030—22no—
25C2×#10625329C2×C411yescharacteristic
26C2×#10625324—21no—
27C22×C47⋊C8C2×C411yescharacteristic
28C2×C8×D47—22no—
29C2×C8×D47—21no—
30C2×C8×D47—21no—
31C2×C8×D47—21no—
32C2×C8×D47—22no—
33#13094008—21no—
34#13094004—21no—
35#13093999—21no—
36#13093994—21no—
37C4×D188—22no—
38#13093964—22no—
39#13093964—22no—
40C2×C47⋊C16—22no—
41C4⋊(C47⋊C8)—22no—
42(C2×D47)⋊C8—22no—
43(C2×D47)⋊C8—22no—
44C2×M16×C47C2×C411yescharacteristic
45C22×C8×C47C2311yescharacteristic
46#117×C47C2×C411yescharacteristic
47M32×C47—22no—
48M32×C47—22no—
49C2×C16×C47D414yescomplemented
50#97×C47D412yes—
51C4×C8×C47D412yes—
52C23×D47M1611yescharacteristic
53C22×C4×C47C22×C411yescharacteristic
54C2×C22⋊D47—21no—
55C22×Dic47M1611yescharacteristic
56#10625350—22no—
57#10625350—22no—
58C2×D188—22no—
59C2×D188M1611yescharacteristic
60C2×C4×D47—21no—
61C2×C4×D47—21no—
62C2×C4×D47—22no—
63C2×C4×D47—21no—
64C2×C4×D47—22no—
65C2×Dic94M1611yescharacteristic
66M16×C47—22no—
67C2×C8×C47C22⋊C412yes—
68C2×C8×C47C2×D411yescharacteristic
69C2×C8×C47C2×D412yes—
70C2×C8×C47C2×D411yescharacteristic
71C2×C8×C47#4112yes—
72C4⋊C4×C47—21no—
73C22⋊C4×C47—21no—
74C42×C47C22⋊C412yes—
75(C2×D47)⋊C4—22no—
76(C2×D47)⋊C4—22no—
77C4⋊Dic47—22no—
78#10625329—22no—
79C2×C47⋊C8—21no—
80C2×C47⋊C8—21no—
81C2×C47⋊C8—22no—
82#10625324—42no—
83C8×D47—42no—
84C8×D47—41no—
85C8×D47—41no—
86C8×D47—41no—
87C8×D47—41no—
88C16×C47—24no—
89C47⋊C16—42no—
90C23×C47C2×M1611yescharacteristic
91C22×D47—21no—
92C22×D47—22no—
93C22×D47—21no—
94C22×D47—42no—
95C2×C4×C47—22no—
96C2×C4×C47C2×C22⋊C411yescharacteristic
97C2×C4×C47C4×D412yes—
98C2×C4×C47C2×M1611yescharacteristic
99C2×C4×C47#13112yes—
100C22⋊D47—42no—
101C2×Dic47—21no—
102C2×Dic47—21no—
103C2×Dic47—22no—
104D188—22no—
105D188—44no—
106C4×D47—41no—
107C4×D47—41no—
108C4×D47—41no—
109C4×D47—41no—
110C4×D47—42no—
111Dic94—22no—
112C8×C47—22no—
113C8×C47#12112yes—
114C8×C47#12411yescharacteristic
115C8×C47#12711yescharacteristic
116C47⋊C8—41no—
117C47⋊C8—41no—
118#13200—471nomaximal, Sylow
119C22×C47—41no—
120C22×C47#40711yescharacteristic
121C22×C47#43512yes—
122C2×D47—22no—
123C2×D47—24no—
124C2×D47—41no—
125C2×D47—42no—
126C2×D47—41no—
127C2×D47—42no—
128C2×D47—41no—
129C4×C47—42no—
130C4×C47#39011yescharacteristic
131C4×C47#40811yescharacteristic
132C4×C47#43512yes—
133Dic47—41no—
134Dic47—41no—
135#2929—471no—
136#2153—471no—
137#2117—472no—
138#2112—472no—
139#2110D4711yescharacteristic, complemented
140C2×C47—21no—
141C2×C47—22no—
142C2×C47#195511yescharacteristic
143D47—42no—
144D47—81no—
145C2×#131—471no—
146C23×C8—471no—
147#525—471no—
148C2×M32—472no—
149C2×M32C2×D4712yes—
150#435—942no—
151C8×D4—942no—
152#424—471no—
153#408—471no—
154#407—471no—
155#405C2×D4711yescharacteristic
156#364—942no—
157#350—944no—
158C22⋊C16—944no—
159#347—22no—
160C47#1320011yescharacteristic, complemented, Sylow
161C2×#41—471no—
162C23×C4—471no—
163#131—942no—
164#131—942no—
165C2×M16—471no—
166C2×M16—941no—
167C2×M16C4×D4711yescharacteristic
168C22×C8—471no—
169C22×C8—942no—
170C22×C8—941no—
171C22×C8—941no—
172C22×C8—942no—
173C22×C8—941no—
174C22×C8C22×D4711yescharacteristic
175#123—941no—
176C22⋊Q8—941no—
177#121—941no—
178#120—941no—
179C4×D4—942no—
180#117C4×D4711yescharacteristic
181M32—22no—
182M32—22no—
183M32—942no—
184M32—942no—
185C2×C16—942no—
186C2×C16D18814yescomplemented
187C4⋊C8—942no—
188C22⋊C8—942no—
189C22⋊C8—942no—
190#97C22⋊D4712yes—
191C4×C8D18812yes—
192C24—471no—
193#41—942no—
194#41—942no—
195C2×Q8—471no—
196C2×D4—471no—
197C2×D4—942no—
198C2×D4—941no—
199C22×C4—471no—
200C22×C4—941no—
201C22×C4—941no—
202C22×C4—942no—
203C22×C4—941no—
204C22×C4—942no—
205C22×C4C2×C4×D4711yescharacteristic
206M16—22no—
207M16—942no—
208M16—1882no—
209C2×C8—941no—
210C2×C8—941no—
211C2×C8—942no—
212C2×C8—1882no—
213C2×C8—1881no—
214C2×C8—1881no—
215C2×C8—1881no—
216C2×C8—1881no—
217C2×C8(C2×D47)⋊C412yes—
218C2×C8C2×D18812yes—
219C2×C8C2×D18811yescharacteristic
220C2×C8#1062535012yes—
221C2×C8C2×C22⋊D4711yescharacteristic
222C4⋊C4—21no—
223C4⋊C4—942no—
224C22⋊C4—21no—
225C22⋊C4—942no—
226C22⋊C4—942no—
227C42(C2×D47)⋊C412yes—
228C16—24no—
229C16—1882no—
230C23—941no—
231C23—942no—
232C23—941no—
233C23—1882no—
234C23M16×D4711yescharacteristic
235Q8—942no—
236D4—942no—
237D4—1884no—
238D4—1882no—
239C2×C4—22no—
240C2×C4—941no—
241C2×C4—941no—
242C2×C4—942no—
243C2×C4—1881no—
244C2×C4—1881no—
245C2×C4—1881no—
246C2×C4—1881no—
247C2×C4—1882no—
248C2×C4C2×(C2×D47)⋊C411yescharacteristic
249C2×C4C4×D18812yes—
250C2×C4M16×D4711yescharacteristic
251C2×C4#1309403212yes—
252C8—22no—
253C8—1881no—
254C8—1881no—
255C8#1309399812yes—
256C8#1309401211yescharacteristic
257C8#1309402111yescharacteristic
258C22—41no—
259C22—942no—
260C22—944no—
261C22—1882no—
262C22—1881no—
263C22—1882no—
264C22—1881no—
265C22—1881no—
266C22#1568908611yescharacteristic
267C22#1568919612yes—
268C4—42no—
269C4—1881no—
270C4—1881no—
271C4#1568902511yescharacteristic
272C4#1568909111yescharacteristic
273C4#1568919612yes—
274C2—21no—
275C2—22no—
276C2—3761no—
277C2—1882no—
278C2#2024418611yescharacteristic
subgroupG/H
centerC8#13094021
commutatorC22×C4×C47C22×C4
FrattiniC22×C8C22×D47
FittingC47×#2110C2
socleC2×C47#1955
prime radical#15689635C4
cosocleC22×C8×C47C23
hypercenter#2110D47
hypocenterC47#13200
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2110

Complements

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

NH ≅ G/Ntype
2. #20245154C2semidirect
3. #20245015C2semidirect
4. #20244998C2semidirect
5. C47×#2110C2semidirect
49. C2×C16×C47D4semidirect
139. #2110D47semidirect
160. C47#13200semidirect
186. C2×C16D188semidirect

Maximal subgroups

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

subgroupindex
#202520842
#202451542
#202450152
#202449982
C47×#21102
#1320047