Finite Group Databasebeta

← #26119377

Subgroups — #26119377

Order 12032.

438 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 97 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#20251751C211yesmaximal, characteristic, complemented
2#20244216C211yesmaximal, characteristic
3#20241949C212yesmaximal
4#20241247C212yesmaximal, complemented
5C47×#1276C211yesmaximal, characteristic, complemented
6#15689673C2211yescharacteristic
7C2×#13094016—21no—
8#15689147C2211yescharacteristic
9C2×#13093952—21no—
10C2×#13093952—21no—
11#15689021—24no—
12#15688993—21no—
13#15688989C2212yes—
14#15688858—24no—
15C47×#431C2212yes—
16C43×C47C2211yescharacteristic
17C2×#10625350—21no—
18C2×#10625329—21no—
19#13094024Q811yescharacteristic
20#13094021Q811yescharacteristic
21#13094016—42no—
22#13094016—42no—
23#13094012—21no—
24#13093998—22no—
25C4×D188—22no—
26C2×C4⋊Dic47—24no—
27C2×C4⋊Dic47—22no—
28C2×Dic47⋊C4—24no—
29C4⋊(C47⋊C8)—22no—
30#13093952—42no—
31#13093952—42no—
32C47⋊#95—22no—
33C47⋊#95D414yes—
34C2×C42×C47—22no—
35C2×C42×C47D411yescharacteristic
36C2×C42×C47D411yescharacteristic
37C2×C42×C47C2311yescharacteristic
38C4⋊C8×C47—22no—
39C22⋊C8×C47—22no—
40C4×C8×C47—22no—
41C22×C4×C47—21no—
42C22×C4×C47—21no—
43C22×C4×C47C2×D412yes—
44C22×C4×C47C2×D411yescharacteristic
45C2×C22⋊D47—41no—
46C22×Dic47—24no—
47#10625350—41no—
48#10625350—41no—
49#10625350—41no—
50#10625350—41no—
51C2×D188—21no—
52C2×D188—22no—
53C2×C4×D47—41no—
54C2×Dic94—21no—
55C2×C8×C47—42no—
56C42×C47—21no—
57C42×C47—21no—
58C42×C47—44no—
59C42×C47D812yes—
60C42×C47SD1612yes—
61C42×C47C2×Q811yescharacteristic
62C42×C47#4112yes—
63C42×C47#4111yescharacteristic
64(C2×D47)⋊C4—42no—
65C4⋊Dic47—22no—
66C4⋊Dic47—48no—
67C4⋊Dic47—42no—
68C4⋊Dic47—42no—
69Dic47⋊C4—48no—
70#10625329—41no—
71#10625329—41no—
72C2×C47⋊C8—41no—
73C23×C47#12011yescharacteristic
74C22×D47—41no—
75C2×C4×C47—24no—
76C2×C4×C47—22no—
77C2×C4×C47—22no—
78C2×C4×C47—42no—
79C2×C4×C47—42no—
80C2×C4×C47C22⋊Q812yes—
81C2×C4×C47C22⋊Q811yescharacteristic
82C2×C4×C47#12412yes—
83C2×C4×C47#12411yescharacteristic
84C2×C4×C47C2×D811yescharacteristic
85C2×C4×C47C2×SD1611yescharacteristic
86C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
87C2×C4×C47#13711yescharacteristic
88C22⋊D47—81no—
89C22⋊D47—81no—
90C2×Dic47—48no—
91C2×Dic47—41no—
92C2×Dic47—44no—
93D188—41no—
94D188—41no—
95D188—82no—
96C4×D47—81no—
97C4×D47—81no—
98Dic94—41no—
99Dic94—41no—
100C8×C47—82no—
101C47⋊C8—81no—
102#10364—471nomaximal, Sylow
103C22×C47—22no—
104C22×C47#39311yescharacteristic
105C22×C47#44711yescharacteristic
106C22×C47#44811yescharacteristic
107C2×D47—81no—
108C2×D47—82no—
109C4×C47—24no—
110C4×C47—22no—
111C4×C47—22no—
112C4×C47—42no—
113C4×C47#47411yescharacteristic
114C4×C47#47611yescharacteristic
115C4×C47#48611yescharacteristic
116C4×C47#48911yescharacteristic
117Dic47—84no—
118Dic47—81no—
119#2859—471no—
120#1962—471no—
121#1517—472no—
122#1286—472no—
123#1276D4711yescharacteristic, complemented
124C2×C47—21no—
125C2×C47—21no—
126C2×C47#202311yescharacteristic
127C2×C47#203112yes—
128D47—161no—
129#532—471no—
130C2×#125—941no—
131#431C2×D4712yes—
132#423—471no—
133C2×#104—941no—
134C2×#104—941no—
135#389—944no—
136#384—941no—
137#382—472no—
138C43C2×D4711yescharacteristic
139#327—944no—
140C47#1036411yescharacteristic, complemented, Sylow
141C2×#41—941no—
142C2×M16—941no—
143C4⋊Q8—471no—
144#127—471no—
145#125—1882no—
146#125—1882no—
147#124—941no—
148#121—942no—
149C4×D4—942no—
150C2×C4⋊C4—944no—
151C2×C4⋊C4—944no—
152C2×C4⋊C4—942no—
153C2×C42—22no—
154C2×C42D18811yescharacteristic
155C2×C42D18811yescharacteristic
156C2×C42C22×D4711yescharacteristic
157C4⋊C8—22no—
158C4⋊C8—942no—
159#104—1882no—
160#104—1882no—
161C22⋊C8—22no—
162C4×C8—22no—
163#95—474no—
164#95—942no—
165#41—1881no—
166#41—1881no—
167#41—1881no—
168#41—1881no—
169C2×Q8—941no—
170C2×D4—942no—
171C2×D4—941no—
172C2×D4—1881no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—944no—
176C22×C4—1881no—
177C22×C4C2×D18811yescharacteristic
178C22×C4D4×D4712yes—
179M16—1881no—
180M16—1881no—
181C2×C8—42no—
182C2×C8—1881no—
183C4⋊C4—942no—
184C4⋊C4—1888no—
185C4⋊C4—1888no—
186C4⋊C4—1882no—
187C4⋊C4—1882no—
188C22⋊C4—1882no—
189C42—21no—
190C42—21no—
191C42—44no—
192C42#1062532512yes—
193C42D37612yes—
194C42#1062535012yes—
195C42#1062535211yescharacteristic
196C42Q8×D4711yescharacteristic
197C23—1881no—
198C23#1309399411yescharacteristic
199Q8—1881no—
200Q8—1881no—
201D4—3762no—
202D4—3761no—
203D4—3761no—
204D4—1881no—
205D4—1881no—
206C2×C4—24no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—3761no—
212C2×C4—3761no—
213C2×C4—1888no—
214C2×C4—1884no—
215C2×C4—1881no—
216C2×C4#1309400212yes—
217C2×C4#1309400411yescharacteristic
218C2×C4#1309401012yes—
219C2×C4#1309401211yescharacteristic
220C2×C4C2×D37611yescharacteristic
221C2×C4C2×#1062532511yescharacteristic
222C2×C4#1309405211yescharacteristic
223C2×C4#1309405711yescharacteristic
224C8—82no—
225C8—3761no—
226C22—22no—
227C22—3761no—
228C22—3762no—
229C22#1568903611yescharacteristic
230C22#1568925911yescharacteristic
231C22#1568926411yescharacteristic
232C4—24no—
233C4—22no—
234C4—22no—
235C4—42no—
236C4—3764no—
237C4—3761no—
238C4#1568940411yescharacteristic
239C4#1568941411yescharacteristic
240C4#1568947011yescharacteristic
241C4#1568948311yescharacteristic
242C2—21no—
243C2—21no—
244C2—7521no—
245C2#2024452511yescharacteristic
246C2#2024456112yes—
subgroupG/H
centerC22#15689036
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1276C2
socleC22×C47#393
prime radical#15689673C22
hypercenter#1276D47
hypocenterC47#10364
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
1. #20251751C2semidirect
4. #20241247C2semidirect
5. C47×#1276C2semidirect
123. #1276D47semidirect
140. C47#10364semidirect

Maximal subgroups

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

subgroupindex
#202517512
#202442162
#202419492
#202412472
C47×#12762
#1036447