Finite Group Databasebeta

← #26118942

Subgroups — #26118942

Order 12032.

542 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 113 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#20254518C212yesmaximal
2#20251766C211yesmaximal, characteristic, complemented
3#20244216C211yesmaximal, characteristic
4#20242072C212yesmaximal, complemented
5C47×#1276C211yesmaximal, characteristic, complemented
6#15689673C2211yescharacteristic
7C2×#13094024C2212yes—
8#15689521—22no—
9#15689470—22no—
10#15689442—24no—
11#15689414—24no—
12#15689147C2211yescharacteristic
13C2×#13093952—22no—
14#15689029—24no—
15#15688989—22no—
16#15688870—24no—
17C47×#431C2212yes—
18C43×C47C2211yescharacteristic
19C2×#10625350—21no—
20C22×Dic94—22no—
21C2×#10625329—21no—
22#13094024—22no—
23#13094024—22no—
24#13094024—22no—
25#13094024—22no—
26#13094024D411yescharacteristic
27#13094021D411yescharacteristic
28#13094012—21no—
29#13093998—22no—
30C4×D188—22no—
31C2×C4⋊Dic47—22no—
32C2×C4⋊Dic47D414yes—
33#13093956—28no—
34C4⋊(C47⋊C8)—22no—
35#13093952—44no—
36#13093946—28no—
37C47⋊#95—28no—
38C2×C42×C47—22no—
39C2×C42×C47D412yes—
40C2×C42×C47C2311yescharacteristic
41C4⋊C8×C47D414yes—
42C22⋊C8×C47—22no—
43C4×C8×C47—22no—
44C22×C4×C47—22no—
45C22×C4×C47C2×D412yes—
46C22×C4×C47C2×D411yescharacteristic
47C2×C22⋊D47—41no—
48C22×Dic47—24no—
49#10625350—42no—
50#10625350—42no—
51C2×D188—22no—
52C2×D188—21no—
53C2×C4×D47—41no—
54C2×Dic94—21no—
55C2×Dic94—48no—
56C2×Dic94—42no—
57C2×Dic94—42no—
58C2×C8×C47—24no—
59C42×C47—22no—
60C42×C47—44no—
61C42×C47SD1614yes—
62C42×C47C2×D411yescharacteristic
63C42×C47C2×D412yes—
64C42×C47#4111yescharacteristic
65(C2×D47)⋊C4—42no—
66C4⋊Dic47—24no—
67C4⋊Dic47—22no—
68C4⋊Dic47—24no—
69C4⋊Dic47—42no—
70C4⋊Dic47—42no—
71#10625329—42no—
72C2×C47⋊C8—41no—
73C23×C47#12011yescharacteristic
74C22×D47—41no—
75C2×C4×C47—24no—
76C2×C4×C47—24no—
77C2×C4×C47—44no—
78C2×C4×C47#12012yes—
79C2×C4×C47#12112yes—
80C2×C4×C47#12111yescharacteristic
81C2×C4×C47#12711yescharacteristic
82C2×C4×C47C2×SD1612yes—
83C2×C4×C47#13712yes—
84C22⋊D47—82no—
85C2×Dic47—48no—
86C2×Dic47—44no—
87C2×Dic47—41no—
88D188—44no—
89D188—42no—
90C4×D47—81no—
91C4×D47—81no—
92Dic94—42no—
93Dic94—84no—
94Dic94—84no—
95C8×C47—44no—
96C47⋊C8—81no—
97#10285—471nomaximal, Sylow
98C22×C47—22no—
99C22×C47#39211yescharacteristic
100C22×C47#44812yes—
101C2×D47—82no—
102C2×D47—81no—
103C4×C47—22no—
104C4×C47—44no—
105C4×C47#45414yes—
106C4×C47#46112yes—
107C4×C47#49211yescharacteristic
108C4×C47#49711yescharacteristic
109Dic47—84no—
110Dic47—81no—
111#3227—472no—
112#2862—471no—
113#1962—471no—
114#1549—472no—
115#1276D4711yescharacteristic, complemented
116C2×C47—22no—
117C2×C47#200312yes—
118C2×C47#201311yescharacteristic
119D47—161no—
120#532—471no—
121C2×C4⋊Q8—472no—
122#498—942no—
123#486—942no—
124#481—944no—
125#476—944no—
126#431C2×D4712yes—
127#423—471no—
128C2×#104—942no—
129#391—944no—
130#382—942no—
131C43C2×D4711yescharacteristic
132#332—944no—
133C47#1028511yescharacteristic, complemented, Sylow
134C2×#41—941no—
135C22×Q8—942no—
136C2×M16—941no—
137C4⋊Q8—471no—
138C4⋊Q8—942no—
139C4⋊Q8—942no—
140C4⋊Q8—942no—
141C4⋊Q8—942no—
142#127—471no—
143#124—941no—
144#121—942no—
145C4×D4—942no—
146C2×C4⋊C4—474no—
147C2×C4⋊C4—942no—
148C2×C42—22no—
149C2×C42D18812yes—
150C2×C42C22×D4711yescharacteristic
151#106—948no—
152C4⋊C8—942no—
153C4⋊C8D18814yes—
154#104—1884no—
155#102—948no—
156C22⋊C8—22no—
157C4×C8—22no—
158#95—948no—
159#41—1882no—
160#41—1882no—
161C2×Q8—941no—
162C2×Q8—1888no—
163C2×Q8—1882no—
164C2×Q8—1882no—
165C2×D4—942no—
166C2×D4—941no—
167C2×D4—1881no—
168C22×C4—22no—
169C22×C4—944no—
170C22×C4—1881no—
171C22×C4C2×D18811yescharacteristic
172C22×C4D4×D4712yes—
173M16—1882no—
174C2×C8—24no—
175C2×C8—1881no—
176C4⋊C4—944no—
177C4⋊C4—942no—
178C4⋊C4—944no—
179C4⋊C4—1882no—
180C4⋊C4—1882no—
181C22⋊C4—1882no—
182C42—22no—
183C42—44no—
184C42#1062532514yes—
185C42C2×D18812yes—
186C42D4×D4711yescharacteristic
187C42#1062535411yescharacteristic
188C23—1881no—
189C23#1309399411yescharacteristic
190Q8—3764no—
191Q8—3764no—
192Q8—1882no—
193D4—3762no—
194D4—1884no—
195D4—1882no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—44no—
199C2×C4—3761no—
200C2×C4—3761no—
201C2×C4—1888no—
202C2×C4—1881no—
203C2×C4—1884no—
204C2×C4#1309399412yes—
205C2×C4#1309399912yes—
206C2×C4#1309399911yescharacteristic
207C2×C4#1309402111yescharacteristic
208C2×C4C2×#1062532512yes—
209C2×C4#1309405712yes—
210C8—44no—
211C8—3761no—
212C22—22no—
213C22—3762no—
214C22—3761no—
215C22#1568903311yescharacteristic
216C22#1568926412yes—
217C4—22no—
218C4—44no—
219C4—3764no—
220C4—3761no—
221C4#1568929714yes—
222C4#1568932912yes—
223C4#1568949611yescharacteristic
224C4#1568951711yescharacteristic
225C2—22no—
226C2—7521no—
227C2#2024441312yes—
228C2#2024447311yescharacteristic
subgroupG/H
centerC22#15689033
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1276C2
socleC22×C47#392
prime radical#15689673C22
hypercenter#1276D47
hypocenterC47#10285
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
2. #20251766C2semidirect
4. #20242072C2semidirect
5. C47×#1276C2semidirect
115. #1276D47semidirect
133. C47#10285semidirect

Maximal subgroups

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

subgroupindex
#202545182
#202517662
#202442162
#202420722
C47×#12762
#1028547