Finite Group Databasebeta

← #26117131

Subgroups — #26117131

Order 12032.

566 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 105 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#20253419C212yesmaximal
2#20244220C211yesmaximal, characteristic
3#20241817C212yesmaximal, complemented
4#20241217C211yesmaximal, characteristic, complemented
5C47×#2399C211yesmaximal, characteristic, complemented
6#15689673C2211yescharacteristic
7#15689441—22no—
8#15689408—22no—
9#15689323—22no—
10#15689262—22no—
11C4×#10625336—24no—
12#15689176C2212yes—
13#15689147C2211yescharacteristic
14C2×#13093952—22no—
15Q8⋊(C47⋊C8)—24no—
16C2×C4×Q8×C47C2212yes—
17C47×#398—22no—
18C47×#395—22no—
19C4×C4⋊C4×C47—22no—
20C47×#376C2211yescharacteristic
21C2×#10625350—21no—
22C2×#10625336—24no—
23C2×#10625329—21no—
24#13094024D411yescharacteristic
25#13094021D411yescharacteristic
26#13094012—21no—
27#13093998—22no—
28C4×D188—22no—
29#13093955—24no—
30C4⋊(C47⋊C8)—22no—
31C4⋊(C47⋊C8)—22no—
32#13093952—44no—
33#13093948—24no—
34#13093944—24no—
35C22⋊(C47⋊C8)—22no—
36C4×C47⋊C8D414yes—
37C22×Q8×C47D412yes—
38C4×Q8×C47—22no—
39C4×Q8×C47—22no—
40C4×Q8×C47—22no—
41C4×Q8×C47—22no—
42C2×C4⋊C4×C47—24no—
43C2×C4⋊C4×C47—22no—
44C2×C4⋊C4×C47D412yes—
45C2×C42×C47—22no—
46C2×C42×C47D412yes—
47C2×C42×C47C2311yescharacteristic
48#95×C47—22no—
49#95×C47—22no—
50#95×C47—22no—
51C2×Q8×C47—22no—
52C2×Q8×C47—22no—
53C2×Q8×C47—44no—
54C22×C4×C47—22no—
55C22×C4×C47—22no—
56C22×C4×C47—22no—
57C22×C4×C47C2×D412yes—
58C22×C4×C47C2×D411yescharacteristic
59C2×C22⋊D47—41no—
60#10625350—42no—
61#10625350—42no—
62C2×D188—22no—
63C2×D188—21no—
64C2×C4×D47—41no—
65C2×Dic94—21no—
66C4⋊C4×C47—22no—
67C4⋊C4×C47—22no—
68C4⋊C4×C47—48no—
69C4⋊C4×C47—42no—
70C4⋊C4×C47—42no—
71C42×C47—22no—
72C42×C47—42no—
73C42×C47—42no—
74C42×C47SD1614yescomplemented
75C42×C47C2×D411yescharacteristic
76C42×C47C2×D412yes—
77C42×C47#4111yescharacteristic
78#10625336—48no—
79(C2×D47)⋊C4—42no—
80C4⋊Dic47—22no—
81#10625329—42no—
82C2×C47⋊C8—24no—
83C2×C47⋊C8—41no—
84C23×C47#12011yescharacteristic
85C22×D47—41no—
86Q8×C47—24no—
87Q8×C47—24no—
88Q8×C47—42no—
89Q8×C47—42no—
90C2×C4×C47—24no—
91C2×C4×C47—24no—
92C2×C4×C47—42no—
93C2×C4×C47—44no—
94C2×C4×C47—42no—
95C2×C4×C47—42no—
96C2×C4×C47—44no—
97C2×C4×C47—42no—
98C2×C4×C47#12012yes—
99C2×C4×C47#12111yescharacteristic
100C2×C4×C47#12112yes—
101C2×C4×C47#12711yescharacteristic
102C2×C4×C47C2×SD1612yes—
103C2×C4×C47#13712yes—
104C22⋊D47—82no—
105C2×Dic47—41no—
106D188—44no—
107D188—42no—
108C4×D47—81no—
109C4×D47—81no—
110Dic94—42no—
111C47⋊C8—44no—
112C47⋊C8—81no—
113#10014—471nomaximal, Sylow
114C22×C47—22no—
115C22×C47#39211yescharacteristic
116C22×C47#44812yes—
117C2×D47—82no—
118C2×D47—81no—
119C4×C47—22no—
120C4×C47—22no—
121C4×C47—44no—
122C4×C47—44no—
123C4×C47—82no—
124C4×C47#46112yes—
125C4×C47#49211yescharacteristic
126C4×C47#49711yescharacteristic
127Dic47—81no—
128#3103—472no—
129#2399D4711yescharacteristic, complemented
130#1963—471no—
131#1487—472no—
132#1274—471no—
133C2×C47—22no—
134C2×C47#201012yes—
135C2×C47#201311yescharacteristic
136D47—161no—
137#532—471no—
138C2×C4×Q8C2×D4712yes—
139#481—942no—
140#475—942no—
141#460—942no—
142#448—942no—
143C4×SD16—944no—
144#431—472no—
145#423—471no—
146C2×#104—942no—
147#398—22no—
148#395—22no—
149C4×C4⋊C4—22no—
150#376C2×D4711yescharacteristic
151Q8⋊C8—944no—
152C47#1001411yescharacteristic, complemented, Sylow
153C2×#41—941no—
154C22×Q8C22⋊D4712yes—
155C2×SD16—944no—
156C2×M16—941no—
157C4⋊Q8—471no—
158#127—471no—
159#124—941no—
160#121—942no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×D4—942no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4C22⋊D4712yes—
169C2×C42—22no—
170C2×C42C22⋊D4712yes—
171C2×C42C22×D4711yescharacteristic
172#106—944no—
173C4⋊C8—942no—
174C4⋊C8—942no—
175#104—1884no—
176#103—944no—
177#102—944no—
178C22⋊C8—942no—
179C4×C8—474no—
180#95—22no—
181#95—22no—
182#95—22no—
183#41—1882no—
184#41—1882no—
185C2×Q8—22no—
186C2×Q8—22no—
187C2×Q8—44no—
188C2×Q8—941no—
189C2×D4—942no—
190C2×D4—941no—
191C2×D4—1881no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—1881no—
196C22×C4C2×C22⋊D4712yes—
197C22×C4C2×C22⋊D4711yescharacteristic
198SD16—1888no—
199M16—1882no—
200C2×C8—944no—
201C2×C8—1881no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—48no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—942no—
208C22⋊C4—1882no—
209C42—22no—
210C42—42no—
211C42—42no—
212C42#1062533614yescomplemented
213C42D4×D4711yescharacteristic
214C42D4×D4712yes—
215C42#1062535211yescharacteristic
216C23—1881no—
217C23#1309399311yescharacteristic
218Q8—24no—
219Q8—24no—
220Q8—42no—
221Q8—42no—
222Q8—1882no—
223D4—3762no—
224D4—1884no—
225D4—1882no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—42no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—44no—
233C2×C4—42no—
234C2×C4—3761no—
235C2×C4—3761no—
236C2×C4—1881no—
237C2×C4#1309399212yes—
238C2×C4#1309399511yescharacteristic
239C2×C4#1309399612yes—
240C2×C4#1309402011yescharacteristic
241C2×C4C2×#1062533612yes—
242C2×C4#1309405512yes—
243C8—3761no—
244C8—1884no—
245C22—22no—
246C22—3762no—
247C22—3761no—
248C22#1568903111yescharacteristic
249C22#1568926212yes—
250C4—22no—
251C4—22no—
252C4—44no—
253C4—44no—
254C4—82no—
255C4—3761no—
256C4#1568932512yes—
257C4#1568949711yescharacteristic
258C4#1568951511yescharacteristic
259C2—22no—
260C2—7521no—
261C2#2024445312yes—
262C2#2024447111yescharacteristic
subgroupG/H
centerC2×C4#13094020
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2399C2
socleC22×C47#392
prime radical#15689673C22
hypercenter#2399D47
hypocenterC47#10014
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2399

Complements

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

NH ≅ G/Ntype
3. #20241817C2semidirect
4. #20241217C2semidirect
5. C47×#2399C2semidirect
74. C42×C47SD16semidirect
129. #2399D47semidirect
152. C47#10014semidirect
212. C42#10625336semidirect

Maximal subgroups

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

subgroupindex
#202534192
#202442202
#202418172
#202412172
C47×#23992
#1001447