Finite Group Databasebeta

← #26113885

Subgroups — #26113885

Order 12032.

556 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 103 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#20252455C211yesmaximal, characteristic, complemented
3#20242057C212yesmaximal, complemented
4#20241277C212yesmaximal, complemented
5C47×#1560C211yesmaximal, characteristic, complemented
6C2×#13094049—22no—
7C2×#13094021C2211yescharacteristic
8#15689513C414yes—
9#15689499—21no—
10#15689497—21no—
11#15689201—22no—
12D4×C47⋊C8—22no—
13#15689176C2212yes—
14#15689090—22no—
15C4×#10625329C2211yescharacteristic
16#15688866—22no—
17#15688866—22no—
18C47×#532C2211yescharacteristic
19C47×#423C2212yes—
20C47×#333—21no—
21C47×#331—21no—
22C22×D188—21no—
23C22×D188—22no—
24#13094049—44no—
25#13094049—44no—
26C2×#10625336—24no—
27C2×#10625334—24no—
28C2×#10625329—21no—
29C2×#10625329—22no—
30C22×C47⋊C8—22no—
31#13094021—21no—
32#13094021—22no—
33#13094021C2×C412yes—
34C4⋊(C47⋊C8)D412yes—
35C4⋊(C47⋊C8)D412yes—
36C22⋊(C47⋊C8)—24no—
37C22⋊(C47⋊C8)—22no—
38#13093935C2×C412yes—
39C4×C47⋊C8—21no—
40C4×C47⋊C8—22no—
41C2×#41×C47D412yes—
42C2×M16×C47—22no—
43C4⋊Q8×C47D411yescharacteristic
44#127×C47D411yescharacteristic
45#124×C47C2×C412yes—
46#121×C47—22no—
47C4×D4×C47—22no—
48C2×C42×C47C2311yescharacteristic
49C4⋊C8×C47—22no—
50C4⋊C8×C47—22no—
51C23×D47—41no—
52#41×C47—42no—
53#41×C47—42no—
54C2×Q8×C47C22⋊C412yes—
55C2×D4×C47—22no—
56C2×D4×C47—22no—
57C2×D4×C47C22⋊C412yes—
58C22×C4×C47—22no—
59C22×C4×C47C2×D411yescharacteristic
60C22×C4×C47#4112yes—
61C2×D188—22no—
62C2×D188—24no—
63C2×D188—42no—
64C2×D188—42no—
65C2×D188—41no—
66C2×D188—42no—
67C2×D188—82no—
68C2×D188—82no—
69M16×C47—82no—
70C2×C8×C47—42no—
71C4⋊C4×C47—22no—
72C22⋊C4×C47—24no—
73C42×C47C22×C411yescharacteristic
74C42×C47C2×D411yescharacteristic
75C42×C47C2×D412yes—
76#10625336—48no—
77#10625334—48no—
78#10625329—22no—
79#10625329—22no—
80#10625329—82no—
81C2×C47⋊C8—22no—
82C2×C47⋊C8—22no—
83C2×C47⋊C8—22no—
84C2×C47⋊C8—82no—
85C23×C47—22no—
86C23×C47#12311yescharacteristic
87C22×D47—42no—
88C22×D47—82no—
89C22×D47—84no—
90Q8×C47—42no—
91D4×C47—42no—
92D4×C47—42no—
93D4×C47—42no—
94D4×C47—82no—
95C2×C4×C47—21no—
96C2×C4×C47—21no—
97C2×C4×C47—22no—
98C2×C4×C47—22no—
99C2×C4×C47—22no—
100C2×C4×C47—42no—
101C2×C4×C47—42no—
102C2×C4×C47C2×C22⋊C411yescharacteristic
103C2×C4×C47C4×D412yes—
104C2×C4×C47#12011yescharacteristic
105C2×C4×C47#12112yes—
106D188—44no—
107D188—84no—
108D188—81no—
109D188—81no—
110D188—82no—
111D188—82no—
112C8×C47—82no—
113C47⋊C8—24no—
114C47⋊C8—42no—
115C47⋊C8—42no—
116#9437—471nomaximal, Sylow
117C22×C47—21no—
118C22×C47—22no—
119C22×C47—42no—
120C22×C47—44no—
121C22×C47#38611yescharacteristic
122C2×D47—82no—
123C2×D47—84no—
124C2×D47—164no—
125C4×C47—21no—
126C4×C47—22no—
127C4×C47—42no—
128C4×C47#45311yescharacteristic
129C4×C47#45311yescharacteristic
130C4×C47#47112yes—
131C4×C47#47212yes—
132#3151—471no—
133#2979—471no—
134#1560D4711yescharacteristic, complemented
135#1545—472no—
136#1299—472no—
137C2×C47—41no—
138C2×C47—82no—
139C2×C47#188711yescharacteristic
140C2×C47#190712yes—
141D47—162no—
142C2×AGL(1,ℤ/8)—942no—
143#532C2×D4711yescharacteristic
144C2×#127—471no—
145#496—474no—
146#493—941no—
147#492—941no—
148#436—942no—
149C8×D4—942no—
150#431—472no—
151#423C2×D4712yes—
152#408—942no—
153C4×M16—471no—
154#333—21no—
155#331—21no—
156#331—942no—
157#331—942no—
158C47#943711yescharacteristic, complemented, Sylow
159C2×#41D18812yes—
160C22×D4—941no—
161C22×D4—942no—
162AGL(1,ℤ/8)—1884no—
163AGL(1,ℤ/8)—1884no—
164C2×SD16—944no—
165C2×D8—944no—
166C2×M16—22no—
167C2×M16—941no—
168C2×M16—942no—
169C22×C8—942no—
170C4⋊Q8C22⋊D4711yescharacteristic
171#127—472no—
172#127—942no—
173#127—941no—
174#127C22⋊D4711yescharacteristic
175#124C4×D4712yes—
176#121—22no—
177C4×D4—22no—
178C2×C42C22×D4711yescharacteristic
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—472no—
182C4⋊C8—472no—
183C22⋊C8—944no—
184C22⋊C8—942no—
185#97—472no—
186C4×C8—942no—
187C4×C8—941no—
188C24—1881no—
189#41—42no—
190#41—42no—
191C2×Q8(C2×D47)⋊C412yes—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—942no—
195C2×D4—944no—
196C2×D4—3762no—
197C2×D4—3762no—
198C2×D4—1882no—
199C2×D4—1882no—
200C2×D4—1881no—
201C2×D4—1882no—
202C2×D4(C2×D47)⋊C412yes—
203C22×C4—22no—
204C22×C4C2×D18811yescharacteristic
205C22×C4#1062535212yes—
206SD16—1888no—
207D8—1888no—
208M16—82no—
209M16—942no—
210M16—942no—
211M16—3762no—
212C2×C8—42no—
213C2×C8—942no—
214C2×C8—942no—
215C2×C8—942no—
216C2×C8—3762no—
217C4⋊C4—22no—
218C22⋊C4—24no—
219C42C2×C4×D4711yescharacteristic
220C42D4×D4712yes—
221C42C2×C22⋊D4711yescharacteristic
222C23—22no—
223C23—3764no—
224C23—3762no—
225C23—1882no—
226C23#1309400811yescharacteristic
227Q8—42no—
228D4—42no—
229D4—42no—
230D4—42no—
231D4—82no—
232D4—3764no—
233D4—3761no—
234D4—3761no—
235D4—3762no—
236D4—3762no—
237D4—1884no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4C2×(C2×D47)⋊C411yescharacteristic
246C2×C4(C22⋊D47)⋊C412yes—
247C2×C4#1309399411yescharacteristic
248C2×C4#1309399612yes—
249C8—82no—
250C8—944no—
251C8—1882no—
252C8—1882no—
253C22—21no—
254C22—22no—
255C22—42no—
256C22—44no—
257C22—3762no—
258C22—3764no—
259C22—7524no—
260C22#1568900011yescharacteristic
261C4—21no—
262C4—22no—
263C4—42no—
264C4#1568929311yescharacteristic
265C4#1568929311yescharacteristic
266C4#1568938812yes—
267C4#1568939112yes—
268C2—41no—
269C2—82no—
270C2—7522no—
271C2#2024378411yescharacteristic
272C2#2024389412yes—
subgroupG/H
centerC22#15689000
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1560C2
socleC22×C47#386
prime radical#20253841C2
cosocleC2×C42×C47C23
hypercenter#1560D47
hypocenterC47#9437
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1560

Complements

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

NH ≅ G/Ntype
2. #20252455C2semidirect
3. #20242057C2semidirect
4. #20241277C2semidirect
5. C47×#1560C2semidirect
134. #1560D47semidirect
158. C47#9437semidirect

Maximal subgroups

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

subgroupindex
#202538412
#202524552
#202420572
#202412772
C47×#15602
#943747