Finite Group Databasebeta

← #26114132

Subgroups — #26114132

Order 12032.

542 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 131 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#20248293C211yesmaximal, characteristic
2#20243384C211yesmaximal, characteristic, complemented
3#20241814C212yesmaximal
4#20241247C212yesmaximal, complemented
5C47×#2982C211yesmaximal, characteristic, complemented
6C2×#13094000—21no—
7C2×C4×D188C2211yescharacteristic
8#15689147C2212yes—
9C2×#13093943—21no—
10#15689053C412yes—
11#15689040C412yes—
12#15689014—22no—
13#15688989C2211yescharacteristic
14#15688980—21no—
15#15688892—21no—
16#15688852—24no—
17C2×C4×Q8×C47C2211yescharacteristic
18C8×Q8×C47—24no—
19C47×#431—22no—
20C47×#431C2212yes—
21C22×D188C2×C411yescharacteristic
22C22×C4×D47—21no—
23C2×#10625329—22no—
24#13094000—42no—
25#13094000—42no—
26C4×D188—22no—
27C4×D188—22no—
28C2×C4⋊Dic47—21no—
29C2×C4⋊Dic47C2×C411yescharacteristic
30C2×Dic47⋊C4—22no—
31C2×(C2×D47)⋊C4—22no—
32C2×(C2×D47)⋊C4—22no—
33C2×(C2×D47)⋊C4—21no—
34C4⋊(C47⋊C8)—22no—
35C4⋊(C47⋊C8)—22no—
36#13093943—42no—
37C47⋊#95—22no—
38C47⋊#95—21no—
39C47⋊#95C2×C412yes—
40C22×Q8×C47D411yescharacteristic
41C4×Q8×C47—22no—
42C4×Q8×C47—22no—
43C2×C4⋊C4×C47C2×C412yes—
44C2×C4⋊C4×C47D411yescharacteristic
45C2×C42×C47D412yes—
46C2×C42×C47C2311yescharacteristic
47C4⋊C8×C47—24no—
48C4⋊C8×C47—22no—
49C22⋊C8×C47—22no—
50C4×C8×C47—24no—
51C4×C8×C47D414yes—
52C23×D47—21no—
53C2×Q8×C47—22no—
54C2×Q8×C47—22no—
55C2×Q8×C47D812yes—
56C2×Q8×C47SD1612yes—
57C22×C4×C47C22⋊C412yes—
58C22×C4×C47C22⋊C412yes—
59C22×C4×C47C22×C411yescharacteristic
60C22×C4×C47C2×D411yescharacteristic
61C22×C4×C47C2×D411yescharacteristic
62C22×Dic47—22no—
63C22×Dic47—21no—
64C2×D188—22no—
65C2×D188—42no—
66C2×C4×D47—42no—
67C2×C4×D47—41no—
68C2×C4×D47—41no—
69C2×C4×D47—42no—
70C2×C8×C47—24no—
71C4⋊C4×C47—24no—
72C4⋊C4×C47—22no—
73C42×C47—22no—
74C42×C47—22no—
75C42×C47C2×D412yes—
76C42×C47#4112yes—
77(C2×D47)⋊C4—44no—
78(C2×D47)⋊C4—44no—
79(C2×D47)⋊C4—41no—
80(C2×D47)⋊C4—41no—
81C4⋊Dic47—22no—
82C4⋊Dic47—41no—
83C4⋊Dic47—41no—
84Dic47⋊C4—44no—
85#10625329—42no—
86#10625329—42no—
87C2×C47⋊C8—42no—
88C23×C47C2×C22⋊C411yescharacteristic
89C22×D47—22no—
90C22×D47—41no—
91C22×D47—41no—
92C22×D47—41no—
93C22×D47—41no—
94C22×D47—41no—
95C22×D47—41no—
96Q8×C47—24no—
97Q8×C47—24no—
98C2×C4×C47—22no—
99C2×C4×C47—22no—
100C2×C4×C47—22no—
101C2×C4×C47—41no—
102C2×C4×C47—41no—
103C2×C4×C47—42no—
104C2×C4×C47#10214yes—
105C2×C4×C47C4×D412yes—
106C2×C4×C47#12011yescharacteristic
107C2×C4×C47#12112yes—
108C2×C4×C47#12311yescharacteristic
109C2×C4×C47C2×D811yescharacteristic
110C2×C4×C47C2×SD1611yescharacteristic
111C2×C4×C47#13512yes—
112C2×Dic47—42no—
113C2×Dic47—42no—
114C2×Dic47—41no—
115C2×Dic47—41no—
116C2×Dic47—42no—
117C2×Dic47—41no—
118D188—24no—
119D188—42no—
120C4×D47—82no—
121C4×D47—82no—
122C8×C47—44no—
123C47⋊C8—82no—
124#9486—471nomaximal, Sylow
125C22×C47—21no—
126C22×C47—21no—
127C22×C47#38611yescharacteristic
128C22×C47C2×#10211yescharacteristic
129C22×C47#41611yescharacteristic
130C2×D47—42no—
131C2×D47—42no—
132C2×D47—42no—
133C2×D47—81no—
134C2×D47—81no—
135C2×D47—81no—
136C2×D47—81no—
137C4×C47—22no—
138C4×C47—24no—
139C4×C47—21no—
140C4×C47—21no—
141C4×C47—42no—
142C4×C47#46512yes—
143C4×C47#46612yes—
144Dic47—82no—
145Dic47—81no—
146#2982D4711yescharacteristic, complemented
147#2536—471no—
148#1801—471no—
149#1486—472no—
150#1286—472no—
151C2×C47—21no—
152C2×C47—21no—
153C2×C47#189111yescharacteristic
154C2×C47#189311yescharacteristic
155C2×C47#191211yescharacteristic
156D47—82no—
157C2×C22⋊Q8—941no—
158C2×C4×Q8C2×D4711yescharacteristic
159C2×C4×D4—471no—
160C8×Q8—24no—
161#431—22no—
162#431C2×D4712yes—
163#423—472no—
164C2×#101—941no—
165#398—472no—
166#394—472no—
167#388—942no—
168#382—471no—
169#380—941no—
170#343—941no—
171D4⋊C8—944no—
172C47#948611yescharacteristic, complemented, Sylow
173C22×Q8D18811yescharacteristic
174C22×D4—471no—
175C23×C4—941no—
176C2×M16—942no—
177C22⋊Q8—1882no—
178C22⋊Q8—1882no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×D4—942no—
182C4×D4—942no—
183C2×C4⋊C4—471no—
184C2×C4⋊C4—942no—
185C2×C4⋊C4—941no—
186C2×C4⋊C4C4×D4712yes—
187C2×C4⋊C4D18811yescharacteristic
188C2×C22⋊C4—942no—
189C2×C22⋊C4—942no—
190C2×C22⋊C4—941no—
191C2×C42C22⋊D4712yes—
192C2×C42C22×D4711yescharacteristic
193C4⋊C8—24no—
194C4⋊C8—22no—
195C4⋊C8—942no—
196C4⋊C8—942no—
197#101—1882no—
198C22⋊C8—22no—
199C4×C8—24no—
200C4×C8C22⋊D4714yes—
201#95—472no—
202#95—942no—
203#95—941no—
204C24—941no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8#1062532512yes—
208C2×Q8D37612yes—
209C2×D4—942no—
210C2×D4—1882no—
211C22×C4—942no—
212C22×C4—941no—
213C22×C4—1882no—
214C22×C4—1881no—
215C22×C4—1881no—
216C22×C4—1882no—
217C22×C4(C2×D47)⋊C412yes—
218C22×C4(C2×D47)⋊C412yes—
219C22×C4C2×C4×D4711yescharacteristic
220C22×C4C2×D18811yescharacteristic
221C22×C4C2×C22⋊D4711yescharacteristic
222M16—1882no—
223M16—1882no—
224C2×C8—24no—
225C2×C8—1882no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—942no—
229C4⋊C4—1884no—
230C4⋊C4—1881no—
231C4⋊C4—1881no—
232C22⋊C4—1884no—
233C22⋊C4—1884no—
234C22⋊C4—1881no—
235C22⋊C4—1881no—
236C42—22no—
237C42—22no—
238C42#1062535012yes—
239C42C2×C22⋊D4712yes—
240C23—942no—
241C23—1881no—
242C23—1881no—
243C23—1881no—
244C23—1881no—
245C23—1881no—
246C23—1881no—
247C23C2×(C2×D47)⋊C411yescharacteristic
248Q8—24no—
249Q8—24no—
250D4—944no—
251D4—1882no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—3762no—
259C2×C4—3762no—
260C2×C4—1882no—
261C2×C4—1882no—
262C2×C4—1881no—
263C2×C4—1882no—
264C2×C4—1881no—
265C2×C4—1881no—
266C2×C4#1309394614yes—
267C2×C4C4×C22⋊D4712yes—
268C2×C4#1309399311yescharacteristic
269C2×C4#1309399812yes—
270C2×C4#1309400711yescharacteristic
271C2×C4C2×D37611yescharacteristic
272C2×C4C2×#1062532511yescharacteristic
273C2×C4#1309404612yes—
274C8—44no—
275C8—3762no—
276C22—21no—
277C22—21no—
278C22—3761no—
279C22—3761no—
280C22—3761no—
281C22—3761no—
282C22—1882no—
283C22—1882no—
284C22—1882no—
285C22#1568900211yescharacteristic
286C22C2×#1309394611yescharacteristic
287C22#1568912011yescharacteristic
288C4—22no—
289C4—24no—
290C4—21no—
291C4—21no—
292C4—42no—
293C4—3762no—
294C4—3761no—
295C4#1568935212yes—
296C4#1568935912yes—
297C2—21no—
298C2—21no—
299C2—3762no—
300C2#2024380611yescharacteristic
301C2#2024381611yescharacteristic
302C2#2024392511yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2982C2
socleC22×C47#386
prime radicalC22×D188C2×C4
cosocleC2×C42×C47C23
hypercenter#2982D47
hypocenterC47#9486
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. #20243384C2semidirect
4. #20241247C2semidirect
5. C47×#2982C2semidirect
146. #2982D47semidirect
172. C47#9486semidirect

Maximal subgroups

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

subgroupindex
#202482932
#202433842
#202418142
#202412472
C47×#29822
#948647