Finite Group Databasebeta

← #26113805

Subgroups — #26113805

Order 12032.

540 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 99 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#20253841C211yesmaximal, characteristic
2#20242101C212yesmaximal, complemented
3#20242023C211yesmaximal, characteristic, complemented
4#20241259C212yesmaximal, complemented
5C47×#2980C211yesmaximal, characteristic, complemented
6C2×#13094049—22no—
7C2×#13094021C2211yescharacteristic
8#15689513—22no—
9#15689499C412yes—
10#15689497C412yes—
11#15689180C2212yes—
12C4×#10625329C2211yescharacteristic
13#15688866—24no—
14#15688863—22no—
15C47×#532C2211yescharacteristic
16C47×#435—24no—
17C47×#432C2212yes—
18C47×#408—22no—
19C22×D188—21no—
20C22×D188—22no—
21#13094049—44no—
22#13094049—44no—
23C2×#10625336—24no—
24C2×#10625334—24no—
25C2×#10625329—21no—
26#13094021—21no—
27#13094021—22no—
28#13094021C2×C412yes—
29C4⋊(C47⋊C8)—22no—
30C22⋊(C47⋊C8)—22no—
31#13093935—21no—
32#13093935—22no—
33C4×C47⋊C8C2×C412yes—
34C2×#41×C47D412yes—
35C2×M16×C47—22no—
36C22×C8×C47—22no—
37C4⋊Q8×C47C2×C411yescharacteristic
38#127×C47C2×C411yescharacteristic
39#124×C47D412yes—
40#121×C47—22no—
41C4×D4×C47—22no—
42C2×C42×C47C2311yescharacteristic
43C4⋊C8×C47D414yes—
44C22⋊C8×C47—24no—
45C22⋊C8×C47—22no—
46#97×C47—22no—
47C23×D47—41no—
48#41×C47—42no—
49#41×C47—42no—
50C2×Q8×C47C22⋊C412yes—
51C2×D4×C47—22no—
52C2×D4×C47—22no—
53C2×D4×C47C22⋊C412yes—
54C22×C4×C47—22no—
55C22×C4×C47C2×D411yescharacteristic
56C22×C4×C47#4112yes—
57C2×D188—22no—
58C2×D188—24no—
59C2×D188—42no—
60C2×D188—42no—
61C2×D188—41no—
62C2×D188—42no—
63C2×D188—84no—
64M16×C47—82no—
65C2×C8×C47—24no—
66C2×C8×C47—82no—
67C4⋊C4×C47—22no—
68C22⋊C4×C47—24no—
69C42×C47C22×C411yescharacteristic
70C42×C47C2×D411yescharacteristic
71C42×C47C2×D412yes—
72#10625336—48no—
73#10625334—48no—
74#10625329—24no—
75C2×C47⋊C8—22no—
76C2×C47⋊C8—42no—
77C23×C47—22no—
78C23×C47#12311yescharacteristic
79C22×D47—42no—
80C22×D47—81no—
81C22×D47—81no—
82C22×D47—84no—
83Q8×C47—42no—
84D4×C47—42no—
85D4×C47—82no—
86D4×C47—82no—
87C2×C4×C47—22no—
88C2×C4×C47—22no—
89C2×C4×C47—42no—
90C2×C4×C47—42no—
91C2×C4×C47—42no—
92C2×C4×C47C2×C22⋊C411yescharacteristic
93C2×C4×C47C4×D412yes—
94C2×C4×C47#12011yescharacteristic
95C2×C4×C47#12112yes—
96D188—44no—
97D188—84no—
98D188—84no—
99D188—82no—
100C8×C47—44no—
101C47⋊C8—24no—
102C47⋊C8—82no—
103#9423—471nomaximal, Sylow
104C22×C47—21no—
105C22×C47—22no—
106C22×C47—42no—
107C22×C47—44no—
108C22×C47#38611yescharacteristic
109C2×D47—82no—
110C2×D47—84no—
111C2×D47—162no—
112C2×D47—162no—
113C4×C47—21no—
114C4×C47—22no—
115C4×C47—22no—
116C4×C47—42no—
117C4×C47#45312yes—
118#3151—471no—
119#2980D4711yescharacteristic, complemented
120#1555—472no—
121#1536—471no—
122#1291—472no—
123C2×C47—41no—
124C2×C47—82no—
125C2×C47#188711yescharacteristic
126C2×C47#190312yes—
127D47—162no—
128C2×AGL(1,ℤ/8)—942no—
129#532C2×D4711yescharacteristic
130C2×#127—471no—
131#496—942no—
132#493—472no—
133#492—472no—
134#435—24no—
135#432—472no—
136#432C2×D4712yes—
137#408—22no—
138C4×M16—471no—
139#331—944no—
140#329—942no—
141C47#942311yescharacteristic, complemented, Sylow
142C2×#41C22⋊D4712yes—
143C22×D4—942no—
144C22×D4—941no—
145AGL(1,ℤ/8)—1884no—
146AGL(1,ℤ/8)—1884no—
147C2×SD16—944no—
148C2×D8—944no—
149C2×M16—22no—
150C2×M16—941no—
151C22×C8—22no—
152C4⋊Q8C4×D4711yescharacteristic
153#127—472no—
154#127—941no—
155#127—942no—
156#127C4×D4711yescharacteristic
157#124D18812yes—
158#121—22no—
159C4×D4—22no—
160C2×C42C22×D4711yescharacteristic
161C4⋊C8—942no—
162C4⋊C8C22⋊D4714yes—
163C22⋊C8—24no—
164C22⋊C8—22no—
165C22⋊C8—942no—
166#97—22no—
167#97—941no—
168#97—942no—
169C4×C8—472no—
170C24—1881no—
171#41—42no—
172#41—42no—
173C2×Q8(C2×D47)⋊C412yes—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—942no—
177C2×D4—944no—
178C2×D4—3764no—
179C2×D4—1882no—
180C2×D4—1882no—
181C2×D4—1881no—
182C2×D4—1882no—
183C2×D4(C2×D47)⋊C412yes—
184C22×C4—22no—
185C22×C4#1062535012yes—
186C22×C4C2×C22⋊D4711yescharacteristic
187SD16—1888no—
188D8—1888no—
189M16—82no—
190M16—944no—
191C2×C8—24no—
192C2×C8—82no—
193C2×C8—942no—
194C2×C8—1882no—
195C4⋊C4—22no—
196C22⋊C4—24no—
197C42C2×C4×D4711yescharacteristic
198C42C2×D18811yescharacteristic
199C42C2×C22⋊D4712yes—
200C23—22no—
201C23—3764no—
202C23—3761no—
203C23—3761no—
204C23—1882no—
205C23#1309400711yescharacteristic
206Q8—42no—
207D4—42no—
208D4—82no—
209D4—82no—
210D4—3764no—
211D4—3764no—
212D4—3762no—
213D4—1884no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4C2×(C2×D47)⋊C411yescharacteristic
220C2×C4C4×C22⋊D4712yes—
221C2×C4#1309399311yescharacteristic
222C2×C4#1309399812yes—
223C8—44no—
224C8—944no—
225C8—3762no—
226C22—21no—
227C22—22no—
228C22—42no—
229C22—44no—
230C22—3762no—
231C22—3764no—
232C22—7522no—
233C22—7522no—
234C22#1568900211yescharacteristic
235C4—21no—
236C4—22no—
237C4—22no—
238C4—42no—
239C4#1568929112yes—
240C2—41no—
241C2—82no—
242C2—7522no—
243C2#2024378611yescharacteristic
244C2#2024386812yes—
subgroupG/H
centerC22#15689002
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2980C2
socleC22×C47#386
prime radical#20253841C2
cosocleC2×C42×C47C23
hypercenter#2980D47
hypocenterC47#9423
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2980

Complements

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

NH ≅ G/Ntype
2. #20242101C2semidirect
3. #20242023C2semidirect
4. #20241259C2semidirect
5. C47×#2980C2semidirect
119. #2980D47semidirect
141. C47#9423semidirect

Maximal subgroups

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

subgroupindex
#202538412
#202421012
#202420232
#202412592
C47×#29802
#942347