Finite Group Databasebeta

← #26113663

Subgroups — #26113663

Order 12032.

730 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 135 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#20248334C211yesmaximal, characteristic
2#20243380C211yesmaximal, characteristic, complemented
3#20242055C212yesmaximal, complemented
4#20241290C212yesmaximal, complemented
5C47×#2972C211yesmaximal, characteristic, complemented
6C2×#13094021C2211yescharacteristic
7C2×#13094020—21no—
8C2×#13093998—22no—
9#15689147C2212yes—
10C2×#13093941—21no—
11#15689032C414yes—
12#15689024—21no—
13C4×C4⋊Dic47C2211yescharacteristic
14#15688892—21no—
15#15688866—24no—
16C2×C4×D4×C47C2211yescharacteristic
17C8×D4×C47—24no—
18C47×#431C2212yes—
19C2×C22⋊C8×C47—22no—
20C22×D188—21no—
21C22×D188—21no—
22C22×D188C2×C412yes—
23C22×C22⋊D47—24no—
24C2×#10625329—22no—
25#13094021—22no—
26#13094021—22no—
27#13094020—42no—
28#13094020—42no—
29#13093998—48no—
30C2×C4⋊Dic47—21no—
31C2×(C2×D47)⋊C4—24no—
32C2×C4×Dic47—21no—
33C4⋊(C47⋊C8)—22no—
34C4⋊(C47⋊C8)—22no—
35#13093941—42no—
36C47⋊#95C2×C412yes—
37C22×D4×C47D411yescharacteristic
38C23×C4×C47D412yes—
39C22×C8×C47—24no—
40C4×D4×C47—22no—
41C4×D4×C47—22no—
42C2×C4⋊C4×C47D411yescharacteristic
43C2×C22⋊C4×C47C2×C412yes—
44C2×C42×C47C2311yescharacteristic
45C4⋊C8×C47D414yescomplemented
46C22⋊C8×C47—24no—
47C22⋊C8×C47—22no—
48C4×C8×C47—22no—
49C24×C47C22⋊C412yes—
50C23×D47—22no—
51C2×D4×C47—22no—
52C2×D4×C47—22no—
53C2×D4×C47D812yes—
54C2×D4×C47SD1612yes—
55C22×C4×C47—22no—
56C22×C4×C47—22no—
57C22×C4×C47—44no—
58C22×C4×C47C22⋊C412yes—
59C22×C4×C47C22×C411yescharacteristic
60C22×C4×C47C2×D411yescharacteristic
61C22×C4×C47C2×D411yescharacteristic
62C2×C22⋊D47—48no—
63C2×C22⋊D47—48no—
64C2×C22⋊D47—84no—
65C22×Dic47—22no—
66C2×D188—24no—
67C2×D188—44no—
68C2×D188—41no—
69C2×D188—42no—
70C2×D188—41no—
71C2×D188—41no—
72C2×D188—41no—
73C2×D188—42no—
74C2×D188—82no—
75C2×C8×C47—24no—
76C2×C8×C47—84no—
77C4⋊C4×C47—22no—
78C22⋊C4×C47—24no—
79C42×C47C2×D412yes—
80C42×C47#4112yes—
81(C2×D47)⋊C4—48no—
82C4⋊Dic47—24no—
83C4×Dic47—41no—
84C4×Dic47—41no—
85#10625329—42no—
86#10625329—42no—
87C2×C47⋊C8—42no—
88C23×C47—22no—
89C23×C47—22no—
90C23×C47—42no—
91C23×C47—42no—
92C23×C47#10214yes—
93C23×C47C2×C22⋊C411yescharacteristic
94C22×D47—44no—
95C22×D47—44no—
96C22×D47—42no—
97C22×D47—42no—
98C22×D47—42no—
99D4×C47—24no—
100D4×C47—24no—
101C2×C4×C47—22no—
102C2×C4×C47—22no—
103C2×C4×C47—41no—
104C2×C4×C47—44no—
105C2×C4×C47—42no—
106C2×C4×C47—82no—
107C2×C4×C47C4×D412yes—
108C2×C4×C47#12011yescharacteristic
109C2×C4×C47#12112yes—
110C2×C4×C47#12311yescharacteristic
111C2×C4×C47C2×D811yescharacteristic
112C2×C4×C47C2×SD1611yescharacteristic
113C2×C4×C47#13512yes—
114C22⋊D47—816no—
115C2×Dic47—24no—
116C2×Dic47—42no—
117C2×Dic47—42no—
118D188—84no—
119D188—82no—
120D188—82no—
121D188—82no—
122D188—82no—
123C8×C47—44no—
124C47⋊C8—82no—
125#9397—471nomaximal, Sylow
126C22×C47—21no—
127C22×C47—21no—
128C22×C47—24no—
129C22×C47—44no—
130C22×C47—42no—
131C22×C47—42no—
132C22×C47—82no—
133C22×C47#38611yescharacteristic
134C22×C47C2×#10211yescharacteristic
135C22×C47#41611yescharacteristic
136C2×D47—84no—
137C2×D47—84no—
138C2×D47—82no—
139C2×D47—82no—
140C2×D47—82no—
141C4×C47—21no—
142C4×C47—22no—
143C4×C47—42no—
144C4×C47#45312yes—
145C4×C47#46512yes—
146C4×C47#46612yes—
147Dic47—44no—
148#2972D4711yescharacteristic, complemented
149#2541—471no—
150#1800—471no—
151#1545—472no—
152#1303—472no—
153C2×C47—21no—
154C2×C47—21no—
155C2×C47—44no—
156C2×C47#188711yescharacteristic
157C2×C47#189311yescharacteristic
158C2×C47#191411yescharacteristic
159D47—162no—
160C2×#127—471no—
161C2×#127—941no—
162C2×#121—942no—
163C2×C4×D4C2×D4711yescharacteristic
164C8×D4—24no—
165#431C2×D4712yes—
166#423—472no—
167C2×#100—941no—
168C2×C22⋊C8—22no—
169#392—474no—
170#390—941no—
171C4×C4⋊C4—471no—
172#343—941no—
173#331—944no—
174C47#939711yescharacteristic, complemented, Sylow
175C22×D4—472no—
176C22×D4—944no—
177C22×D4—941no—
178C22×D4—941no—
179C22×D4D18811yescharacteristic
180C23×C4C22⋊D4712yes—
181C2×M16—942no—
182C22×C8—24no—
183#127—942no—
184#127—942no—
185#127—1882no—
186#127—1882no—
187#121—1888no—
188C4×D4—22no—
189C4×D4—22no—
190C2×C4⋊C4—941no—
191C2×C4⋊C4D18811yescharacteristic
192C2×C22⋊C4—944no—
193C2×C22⋊C4C4×D4712yes—
194C2×C42—941no—
195C2×C42C22×D4711yescharacteristic
196C4⋊C8—942no—
197C4⋊C8—942no—
198C4⋊C8C22⋊D4714yescomplemented
199#100—1882no—
200C22⋊C8—24no—
201C22⋊C8—22no—
202C4×C8—22no—
203#95—472no—
204C24—942no—
205C24(C2×D47)⋊C412yes—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—944no—
209C2×D4—3762no—
210C2×D4—3764no—
211C2×D4—1884no—
212C2×D4—1888no—
213C2×D4—1888no—
214C2×D4—1881no—
215C2×D4—1882no—
216C2×D4—1881no—
217C2×D4—1881no—
218C2×D4—1881no—
219C2×D4—1882no—
220C2×D4#1062532512yes—
221C2×D4D37612yes—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—44no—
225C22×C4—942no—
226C22×C4(C2×D47)⋊C412yes—
227C22×C4C2×C4×D4711yescharacteristic
228C22×C4C2×D18811yescharacteristic
229C22×C4C2×C22⋊D4711yescharacteristic
230M16—1882no—
231M16—1882no—
232C2×C8—24no—
233C2×C8—84no—
234C2×C8—1882no—
235C4⋊C4—22no—
236C4⋊C4—944no—
237C22⋊C4—24no—
238C22⋊C4—1888no—
239C42—1881no—
240C42—1881no—
241C42#1062535012yes—
242C42C2×C22⋊D4712yes—
243C23—22no—
244C23—22no—
245C23—42no—
246C23—42no—
247C23—1884no—
248C23—1884no—
249C23—1882no—
250C23—1882no—
251C23—1882no—
252C23#1309394614yes—
253C23C2×(C2×D47)⋊C411yescharacteristic
254D4—24no—
255D4—24no—
256D4—3764no—
257D4—3762no—
258D4—37616no—
259D4—3762no—
260D4—3762no—
261D4—3762no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—41no—
265C2×C4—44no—
266C2×C4—42no—
267C2×C4—82no—
268C2×C4—944no—
269C2×C4—1882no—
270C2×C4—1882no—
271C2×C4C4×C22⋊D4712yes—
272C2×C4#1309399311yescharacteristic
273C2×C4#1309399812yes—
274C2×C4#1309400711yescharacteristic
275C2×C4C2×D37611yescharacteristic
276C2×C4C2×#1062532511yescharacteristic
277C2×C4#1309404612yes—
278C8—44no—
279C8—3762no—
280C22—21no—
281C22—21no—
282C22—24no—
283C22—44no—
284C22—42no—
285C22—42no—
286C22—82no—
287C22—3764no—
288C22—3764no—
289C22—3762no—
290C22—3762no—
291C22—3762no—
292C22#1568900211yescharacteristic
293C22C2×#1309394611yescharacteristic
294C22#1568912011yescharacteristic
295C4—21no—
296C4—22no—
297C4—42no—
298C4—1884no—
299C4#1568929112yes—
300C4#1568935212yes—
301C4#1568935912yes—
302C2—21no—
303C2—21no—
304C2—44no—
305C2—7522no—
306C2#2024378611yescharacteristic
307C2#2024381611yescharacteristic
308C2#2024393511yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2972C2
socleC22×C47#386
prime radical#20248334C2
cosocleC2×C42×C47C23
hypercenter#2972D47
hypocenterC47#9397
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #20243380C2semidirect
3. #20242055C2semidirect
4. #20241290C2semidirect
5. C47×#2972C2semidirect
45. C4⋊C8×C47D4semidirect
148. #2972D47semidirect
174. C47#9397semidirect
198. C4⋊C8C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
#202483342
#202433802
#202420552
#202412902
C47×#29722
#939747