Finite Group Databasebeta

← #26117136

Subgroups — #26117136

Order 12032.

696 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 105 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#20253396C212yesmaximal, complemented
2#20244369C211yesmaximal, characteristic, complemented
3#20241813C212yesmaximal, complemented
4#20241217C211yesmaximal, characteristic, complemented
5C47×#2397C211yesmaximal, characteristic
6#15689680C2211yescharacteristic
7#15689453—22no—
8#15689413—22no—
9#15689328—22no—
10#15689274—22no—
11C4×#10625335—24no—
12#15689176C2212yes—
13#15689147C2211yescharacteristic
14#15688883—22no—
15D4⋊(C47⋊C8)—24no—
16C2×C4×D4×C47—22no—
17C2×C4×D4×C47C2212yes—
18C47×#399—22no—
19C47×#392—22no—
20C47×#376C2211yescharacteristic
21C2×#10625335—24no—
22C2×#10625329—21no—
23#13094024D412yescomplemented
24#13094016—21no—
25#13094003—22no—
26C4×Dic94—22no—
27#13093983—21no—
28#13093955—24no—
29C4⋊(C47⋊C8)—22no—
30C4⋊(C47⋊C8)—22no—
31#13093947—24no—
32#13093945—24no—
33C22⋊(C47⋊C8)—22no—
34C4×C47⋊C8D414yescomplemented
35C22×D4×C47—22no—
36C22×D4×C47D412yes—
37C23×C4×C47—22no—
38C4×D4×C47—22no—
39C4×D4×C47—22no—
40C4×D4×C47—22no—
41C4×D4×C47—22no—
42C4×D4×C47—44no—
43C4×D4×C47—44no—
44C2×C4⋊C4×C47—22no—
45C2×C4⋊C4×C47D412yes—
46C2×C22⋊C4×C47—24no—
47C2×C22⋊C4×C47—22no—
48C2×C42×C47D412yes—
49C2×C42×C47C2311yescharacteristic
50#95×C47—22no—
51C24×C47—22no—
52C2×D4×C47—22no—
53C2×D4×C47—22no—
54C2×D4×C47—44no—
55C2×D4×C47—44no—
56C2×D4×C47—48no—
57C22×C4×C47—22no—
58C22×C4×C47—22no—
59C22×C4×C47—42no—
60C22×C4×C47—42no—
61C22×C4×C47—44no—
62C22×C4×C47—44no—
63C22×C4×C47C2×D412yes—
64C22×C4×C47C2×D411yescharacteristic
65C2×Dic94—22no—
66C4⋊C4×C47—22no—
67C4⋊C4×C47—22no—
68C4⋊C4×C47—44no—
69C22⋊C4×C47—48no—
70C22⋊C4×C47—42no—
71C22⋊C4×C47—42no—
72C42×C47—22no—
73C42×C47SD1614yescomplemented
74C42×C47C2×D411yescharacteristic
75C42×C47C2×D412yes—
76C42×C47#4111yescharacteristic
77C22⋊Dic47—41no—
78#10625335—48no—
79C4⋊Dic47—22no—
80C4⋊Dic47—22no—
81Dic47⋊C4—42no—
82C4×Dic47—41no—
83#10625329—42no—
84C2×C47⋊C8—24no—
85C2×C47⋊C8—41no—
86C23×C47—24no—
87C23×C47—44no—
88C23×C47—42no—
89C23×C47—42no—
90C23×C47—44no—
91C23×C47#12011yescharacteristic
92D4×C47—24no—
93D4×C47—24no—
94D4×C47—42no—
95D4×C47—42no—
96D4×C47—416no—
97C2×C4×C47—24no—
98C2×C4×C47—42no—
99C2×C4×C47—44no—
100C2×C4×C47—42no—
101C2×C4×C47—44no—
102C2×C4×C47—84no—
103C2×C4×C47—82no—
104C2×C4×C47—82no—
105C2×C4×C47#12012yes—
106C2×C4×C47#12111yescharacteristic
107C2×C4×C47#12112yes—
108C2×C4×C47#12711yescharacteristic
109C2×C4×C47C2×SD1612yes—
110C2×C4×C47AGL(1,ℤ/8)12yes—
111C2×Dic47—42no—
112Dic94—44no—
113C47⋊C8—44no—
114C47⋊C8—81no—
115#10015—471nomaximal, Sylow
116C22×C47—22no—
117C22×C47—48no—
118C22×C47—44no—
119C22×C47—84no—
120C22×C47—82no—
121C22×C47—82no—
122C22×C47#39211yescharacteristic
123C22×C47#45012yes—
124C4×C47—22no—
125C4×C47—22no—
126C4×C47—44no—
127C4×C47—82no—
128C4×C47#46012yes—
129C4×C47#49211yescharacteristic
130C4×C47#49611yescharacteristic
131Dic47—82no—
132#3101—472no—
133#2397D4711yescharacteristic
134#1995—471no—
135#1486—472no—
136#1274—471no—
137C2×C47—22no—
138C2×C47—84no—
139C2×C47#201012yes—
140C2×C47#201411yescharacteristic
141#533—471no—
142C2×C4×D4—22no—
143C2×C4×D4C2×D4712yes—
144#483—942no—
145#476—942no—
146#461—942no—
147#450—942no—
148C4×SD16—944no—
149#431—472no—
150#423—471no—
151#399—22no—
152#392—22no—
153#376C2×D4711yescharacteristic
154#339—942no—
155D4⋊C8—944no—
156C47#1001511yescharacteristic, complemented, Sylow
157C22×D4—22no—
158C22×D4C22⋊D4712yes—
159C23×C4—22no—
160C2×SD16—944no—
161C2×M16—941no—
162C4⋊Q8—472no—
163#125—941no—
164C22⋊Q8—942no—
165C4×Q8—942no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—44no—
171C4×D4—44no—
172#117—941no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4C22⋊D4712yes—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—22no—
177C2×C42C22⋊D4712yes—
178C2×C42C22×D4711yescharacteristic
179#106—944no—
180C4⋊C8—942no—
181C4⋊C8—942no—
182#103—944no—
183#102—944no—
184C22⋊C8—942no—
185C4×C8—474no—
186#95—22no—
187C24—22no—
188C2×Q8—942no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—44no—
192C2×D4—44no—
193C2×D4—48no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—44no—
199C22×C4—44no—
200C22×C4C2×C22⋊D4712yes—
201C22×C4C2×C22⋊D4711yescharacteristic
202SD16—1888no—
203M16—1882no—
204C2×C8—944no—
205C2×C8—1881no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—44no—
209C4⋊C4—942no—
210C4⋊C4—942no—
211C4⋊C4—1882no—
212C22⋊C4—48no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—1881no—
216C42—22no—
217C42—1881no—
218C42#1062533514yescomplemented
219C42D4×D4711yescharacteristic
220C42D4×D4712yes—
221C42#1062535211yescharacteristic
222C23—24no—
223C23—44no—
224C23—42no—
225C23—44no—
226C23—42no—
227C23#1309399311yescharacteristic
228Q8—1884no—
229D4—24no—
230D4—24no—
231D4—42no—
232D4—42no—
233D4—416no—
234C2×C4—24no—
235C2×C4—42no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—42no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—84no—
242C2×C4—1882no—
243C2×C4#1309399212yes—
244C2×C4#1309399511yescharacteristic
245C2×C4#1309399612yes—
246C2×C4#1309402011yescharacteristic
247C2×C4C2×#1062533512yes—
248C2×C4#1309405012yes—
249C8—3761no—
250C8—1884no—
251C22—22no—
252C22—48no—
253C22—44no—
254C22—84no—
255C22—82no—
256C22—82no—
257C22#1568903111yescharacteristic
258C22#1568927412yes—
259C4—22no—
260C4—22no—
261C4—44no—
262C4—82no—
263C4—3762no—
264C4#1568932012yes—
265C4#1568949511yescharacteristic
266C4#1568951011yescharacteristic
267C2—22no—
268C2—84no—
269C2#2024445312yes—
270C2#2024447611yescharacteristic
subgroupG/H
centerC2×C4#13094020
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
Fitting, prime radicalC47×#2397C2
socleC22×C47#392
hypercenter#2397D47
hypocenterC47#10015
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2397

Complements

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

NH ≅ G/Ntype
1. #20253396C2semidirect
2. #20244369C2semidirect
3. #20241813C2semidirect
4. #20241217C2semidirect
23. #13094024D4semidirect
34. C4×C47⋊C8D4semidirect
73. C42×C47SD16semidirect
156. C47#10015semidirect
218. C42#10625335semidirect

Maximal subgroups

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

subgroupindex
#202533962
#202443692
#202418132
#202412172
C47×#23972
#1001547