Finite Group Databasebeta

← #26117256

Subgroups — #26117256

Order 12032.

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

normal subgroups only

This group has 584 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#20253834C211yesmaximal, characteristic
2#20253419C212yesmaximal, complemented
3#20241786C212yesmaximal, complemented
4#20241219C211yesmaximal, characteristic, complemented
5C47×#1985C211yesmaximal, characteristic, complemented
6C2×#13094046—22no—
7#15689673C2212yes—
8#15689517—22no—
9#15689512—22no—
10#15689441—22no—
11#15689408—22no—
12#15689323—22no—
13#15689262—22no—
14C4×#10625336—24no—
15#15689176C2212yes—
16#15688850—24no—
17C2×C4×Q8×C47C2211yescharacteristic
18C2×C4⋊C8×C47C2211yescharacteristic
19C2×#97×C47C2211yescharacteristic
20C47×#336—22no—
21C2×#10625350—22no—
22#13094046—44no—
23#13094046—44no—
24C2×Dic188—22no—
25C2×#10625325—24no—
26C2×#10625336—24no—
27C2×D376—22no—
28#13094024D412yes—
29#13094021D412yes—
30#13094012—22no—
31#13093998—22no—
32#13093998—22no—
33C4×D188—22no—
34C4×D188—22no—
35#13093955—24no—
36C4⋊(C47⋊C8)—22no—
37#13093948—24no—
38#13093944—24no—
39C22⋊(C47⋊C8)—22no—
40C4×C47⋊C8D414yes—
41C22×Q8×C47D411yescharacteristic
42C22×C8×C47—21no—
43C22×C8×C47D412yes—
44C4×Q8×C47—22no—
45C4×Q8×C47—22no—
46C2×C4⋊C4×C47—21no—
47C2×C4⋊C4×C47D411yescharacteristic
48C2×C42×C47—21no—
49C2×C42×C47C2311yescharacteristic
50C4⋊C8×C47—22no—
51#97×C47—22no—
52C2×Q8×C47—22no—
53C2×Q8×C47—42no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47C2×D411yescharacteristic
57C22×C4×C47C2×D411yescharacteristic
58C22×C4×C47#4111yescharacteristic
59C2×C22⋊D47—42no—
60#10625350—44no—
61#10625350—44no—
62C2×D188—22no—
63C2×D188—22no—
64C2×D188—22no—
65C2×C4×D47—42no—
66C2×Dic94—22no—
67C2×C8×C47—22no—
68C2×C8×C47—22no—
69C2×C8×C47—41no—
70C2×C8×C47—42no—
71C2×C8×C47SD1614yescomplemented
72C4⋊C4×C47—22no—
73C4⋊C4×C47—42no—
74C42×C47—41no—
75C42×C47—41no—
76C42×C47C2×D412yes—
77C42×C47C2×D412yes—
78#10625336—48no—
79(C2×D47)⋊C4—42no—
80(C2×D47)⋊C4—42no—
81C4⋊Dic47—22no—
82C4⋊Dic47—22no—
83C2×C47⋊C8—24no—
84Dic188—44no—
85D376—44no—
86#10625325—48no—
87C23×C47#12111yescharacteristic
88C22×D47—42no—
89Q8×C47—24no—
90Q8×C47—42no—
91C2×C4×C47—22no—
92C2×C4×C47—41no—
93C2×C4×C47—41no—
94C2×C4×C47—41no—
95C2×C4×C47—41no—
96C2×C4×C47—42no—
97C2×C4×C47—41no—
98C2×C4×C47#12011yescharacteristic
99C2×C4×C47#12012yes—
100C2×C4×C47#12112yes—
101C2×C4×C47#12711yescharacteristic
102C2×C4×C47C2×SD1612yes—
103C2×C4×C47AGL(1,ℤ/8)12yes—
104C22⋊D47—84no—
105C2×Dic47—42no—
106D188—44no—
107D188—44no—
108D188—44no—
109C4×D47—82no—
110C4×D47—82no—
111Dic94—44no—
112C8×C47—24no—
113C8×C47—81no—
114C47⋊C8—44no—
115#10034—471nomaximal, Sylow
116C22×C47—21no—
117C22×C47—21no—
118C22×C47#39211yescharacteristic
119C22×C47#46012yes—
120C2×D47—82no—
121C2×D47—82no—
122C2×D47—82no—
123C4×C47—22no—
124C4×C47—22no—
125C4×C47—42no—
126C4×C47—81no—
127C4×C47#45012yes—
128C4×C47#49211yescharacteristic
129C4×C47#49611yescharacteristic
130Dic47—82no—
131#3150—471no—
132#3103—472no—
133#1985D4711yescharacteristic, complemented
134#1480—472no—
135#1275—471no—
136C2×C47—22no—
137C2×C47#201011yescharacteristic
138C2×C47#201411yescharacteristic
139C2×C47#201711yescharacteristic
140D47—162no—
141C2×#135—942no—
142#532—472no—
143C2×C4×Q8C2×D4711yescharacteristic
144#497—942no—
145#496—942no—
146#481—942no—
147#475—942no—
148#460—942no—
149#448—942no—
150C4×SD16—944no—
151#431—472no—
152C2×C4⋊C8C2×D4711yescharacteristic
153C2×#97C2×D4711yescharacteristic
154#336—22no—
155D4⋊C8—944no—
156C47#1003411yescharacteristic, complemented, Sylow
157C2×#41—942no—
158C22×Q8D18811yescharacteristic
159#135—1884no—
160#135—1884no—
161C2×Q16—942no—
162C2×SD16—944no—
163C2×SD16—944no—
164C2×D8—942no—
165C22×C8—21no—
166C22×C8C22⋊D4712yes—
167C4⋊Q8—472no—
168#127—472no—
169#124—942no—
170#121—942no—
171#121—942no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×D4—942no—
175C4×D4—942no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4D18811yescharacteristic
178C2×C42—21no—
179C2×C42C22×D4711yescharacteristic
180#106—944no—
181C4⋊C8—22no—
182C4⋊C8—942no—
183#103—944no—
184#102—944no—
185C22⋊C8—942no—
186#97—22no—
187C4×C8—474no—
188#41—1884no—
189#41—1884no—
190C2×Q8—22no—
191C2×Q8—42no—
192C2×Q8—942no—
193C2×D4—942no—
194C2×D4—942no—
195C2×D4—942no—
196C2×D4—1882no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—1882no—
200C22×C4C2×D18811yescharacteristic
201C22×C4#1062535011yescharacteristic
202C22×C4C2×C22⋊D4711yescharacteristic
203Q16—1884no—
204SD16—1888no—
205SD16—1888no—
206D8—1884no—
207C2×C8—22no—
208C2×C8—22no—
209C2×C8—41no—
210C2×C8—42no—
211C2×C8—944no—
212C2×C8#1062533614yescomplemented
213C4⋊C4—22no—
214C4⋊C4—42no—
215C4⋊C4—942no—
216C4⋊C4—942no—
217C22⋊C4—1882no—
218C22⋊C4—1882no—
219C42—41no—
220C42—41no—
221C42D4×D4712yes—
222C42D4×D4712yes—
223C23—1882no—
224C23#1309399811yescharacteristic
225Q8—24no—
226Q8—42no—
227Q8—1884no—
228D4—3764no—
229D4—1884no—
230D4—1884no—
231D4—1884no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—3762no—
240C2×C4—3762no—
241C2×C4—1882no—
242C2×C4#1309399211yescharacteristic
243C2×C4#1309399412yes—
244C2×C4#1309399712yes—
245C2×C4#1309402011yescharacteristic
246C2×C4C2×#1062533612yes—
247C2×C4#1309404912yes—
248C8—24no—
249C8—81no—
250C8—1884no—
251C22—21no—
252C22—21no—
253C22—3762no—
254C22—3762no—
255C22—3762no—
256C22#1568903211yescharacteristic
257C22#1568932312yes—
258C4—22no—
259C4—22no—
260C4—42no—
261C4—81no—
262C4—3762no—
263C4#1568927312yes—
264C4#1568949711yescharacteristic
265C4#1568951311yescharacteristic
266C2—22no—
267C2—7522no—
268C2#2024445511yescharacteristic
269C2#2024447511yescharacteristic
270C2#2024449211yescharacteristic
subgroupG/H
centerC2×C4#13094020
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1985C2
socleC22×C47#392
prime radical#20253834C2
hypercenter#1985D47
hypocenterC47#10034
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1985

Complements

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

NH ≅ G/Ntype
2. #20253419C2semidirect
3. #20241786C2semidirect
4. #20241219C2semidirect
5. C47×#1985C2semidirect
71. C2×C8×C47SD16semidirect
133. #1985D47semidirect
156. C47#10034semidirect
212. C2×C8#10625336semidirect

Maximal subgroups

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

subgroupindex
#202538342
#202534192
#202417862
#202412192
C47×#19852
#1003447