Finite Group Databasebeta

← #26117222

Subgroups — #26117222

Order 12032.

512 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 83 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#20254698C211yesmaximal, characteristic
2#20253510C212yesmaximal, complemented
3#20241998C212yesmaximal, complemented
4#20241258C211yesmaximal, characteristic, complemented
5C47×#1985C211yesmaximal, characteristic, complemented
6#15689673C2212yes—
7#15689436—22no—
8#15689418—22no—
9#15689404—22no—
10#15689359—22no—
11#15689323—22no—
12#15689319—22no—
13#15689262—22no—
14#15689221—24no—
15#15689180C2212yes—
16#15688862—24no—
17C2×C4×Q8×C47C2211yescharacteristic
18C2×C4⋊C8×C47C2211yescharacteristic
19C2×#97×C47C2211yescharacteristic
20C47×#336—22no—
21C2×#10625350—22no—
22C2×#10625336—24no—
23C2×D376—22no—
24#13094024D412yes—
25#13094021D412yes—
26#13094012—22no—
27#13093998—22no—
28#13093998—22no—
29C4×D188—22no—
30C4×D188—22no—
31#13093958—22no—
32#13093957—24no—
33C4⋊(C47⋊C8)—22no—
34#13093948—24no—
35#13093946—24no—
36#13093944—24no—
37C22⋊(C47⋊C8)—22no—
38#13093935D414yes—
39C22×Q8×C47D411yescharacteristic
40C22×C8×C47—21no—
41C22×C8×C47D412yes—
42C4×Q8×C47—22no—
43C4×Q8×C47—22no—
44C2×C4⋊C4×C47—21no—
45C2×C4⋊C4×C47D411yescharacteristic
46C2×C42×C47—21no—
47C2×C42×C47C2311yescharacteristic
48C4⋊C8×C47—22no—
49#97×C47—22no—
50C2×Q8×C47—22no—
51C2×Q8×C47—42no—
52C22×C4×C47—21no—
53C22×C4×C47—21no—
54C22×C4×C47C2×D411yescharacteristic
55C22×C4×C47C2×D411yescharacteristic
56C22×C4×C47#4111yescharacteristic
57C2×C22⋊D47—42no—
58#10625350—84no—
59C2×D188—22no—
60C2×D188—22no—
61C2×D188—22no—
62C2×C4×D47—42no—
63C2×Dic94—22no—
64C2×C8×C47—22no—
65C2×C8×C47—22no—
66C2×C8×C47—41no—
67C2×C8×C47—41no—
68C2×C8×C47—42no—
69C4⋊C4×C47—22no—
70C4⋊C4×C47—42no—
71C42×C47—41no—
72C42×C47—41no—
73C42×C47C2×D412yes—
74C42×C47C2×D412yes—
75#10625336—48no—
76(C2×D47)⋊C4—42no—
77(C2×D47)⋊C4—42no—
78C4⋊Dic47—22no—
79C4⋊Dic47—22no—
80C2×C47⋊C8—24no—
81D376—82no—
82C23×C47#12111yescharacteristic
83C22×D47—42no—
84Q8×C47—24no—
85Q8×C47—42no—
86C2×C4×C47—22no—
87C2×C4×C47—21no—
88C2×C4×C47—21no—
89C2×C4×C47—22no—
90C2×C4×C47—41no—
91C2×C4×C47—41no—
92C2×C4×C47—41no—
93C2×C4×C47—41no—
94C2×C4×C47—42no—
95C2×C4×C47#12011yescharacteristic
96C2×C4×C47#12012yes—
97C2×C4×C47#12112yes—
98C2×C4×C47#12711yescharacteristic
99C22⋊D47—84no—
100C2×Dic47—42no—
101D188—44no—
102D188—42no—
103D188—42no—
104D188—82no—
105C4×D47—84no—
106Dic94—82no—
107C8×C47—42no—
108C8×C47—42no—
109C47⋊C8—44no—
110#10028—471nomaximal, Sylow
111C22×C47—21no—
112C22×C47—21no—
113C22×C47#39211yescharacteristic
114C22×C47#47212yes—
115C2×D47—82no—
116C2×D47—82no—
117C2×D47—82no—
118C4×C47—21no—
119C4×C47—22no—
120C4×C47—22no—
121C4×C47—42no—
122C4×C47—81no—
123C4×C47#45511yescharacteristic
124C4×C47#45611yescharacteristic
125Dic47—82no—
126#3245—471no—
127#3112—472no—
128#1985D4711yescharacteristic, complemented
129#1530—472no—
130#1290—471no—
131C2×C47—22no—
132C2×C47#200711yescharacteristic
133C2×C47#201011yescharacteristic
134C2×C47#201811yescharacteristic
135D47—162no—
136#532—472no—
137C2×C4×Q8C2×D4711yescharacteristic
138#480—942no—
139#477—942no—
140#474—942no—
141#466—942no—
142#460—942no—
143#459—942no—
144#448—942no—
145#440—944no—
146#432—472no—
147C2×C4⋊C8C2×D4711yescharacteristic
148C2×#97C2×D4711yescharacteristic
149#336—22no—
150#329—944no—
151C47#1002811yescharacteristic, complemented, Sylow
152C2×#41—942no—
153C22×Q8D18811yescharacteristic
154C2×SD16—944no—
155C2×D8—942no—
156C22×C8—21no—
157C22×C8C22⋊D4712yes—
158C4⋊Q8—472no—
159#127—472no—
160#124—942no—
161#121—942no—
162#121—942no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×D4—942no—
166C4×D4—942no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4D18811yescharacteristic
169C2×C42—21no—
170C2×C42C22×D4711yescharacteristic
171#107—944no—
172#107—942no—
173C4⋊C8—22no—
174C4⋊C8—942no—
175#103—944no—
176#102—944no—
177#102—944no—
178C22⋊C8—942no—
179#97—22no—
180#97—474no—
181#41—3764no—
182C2×Q8—22no—
183C2×Q8—42no—
184C2×Q8—942no—
185C2×D4—942no—
186C2×D4—942no—
187C2×D4—942no—
188C2×D4—1882no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—1882no—
192C22×C4C2×D18811yescharacteristic
193C22×C4#1062535011yescharacteristic
194C22×C4C2×C22⋊D4711yescharacteristic
195SD16—1888no—
196D8—3762no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—42no—
202C2×C8—944no—
203C4⋊C4—22no—
204C4⋊C4—42no—
205C4⋊C4—942no—
206C4⋊C4—942no—
207C22⋊C4—1882no—
208C22⋊C4—1882no—
209C42—41no—
210C42—41no—
211C42D4×D4712yes—
212C42D4×D4712yes—
213C23—1882no—
214C23#1309399811yescharacteristic
215Q8—24no—
216Q8—42no—
217Q8—3762no—
218D4—3762no—
219D4—3764no—
220D4—1884no—
221D4—1882no—
222D4—1882no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—3764no—
233C2×C4—1882no—
234C2×C4#1309399211yescharacteristic
235C2×C4#1309399412yes—
236C2×C4#1309399712yes—
237C2×C4#1309402011yescharacteristic
238C8—42no—
239C8—42no—
240C8—1884no—
241C22—21no—
242C22—21no—
243C22—3762no—
244C22—3762no—
245C22—3762no—
246C22#1568903211yescharacteristic
247C22#1568939312yes—
248C4—22no—
249C4—21no—
250C4—22no—
251C4—42no—
252C4—81no—
253C4—3762no—
254C4#1568930011yescharacteristic
255C4#1568930511yescharacteristic
256C2—22no—
257C2—7522no—
258C2#2024443411yescharacteristic
259C2#2024445511yescharacteristic
260C2#2024449711yescharacteristic
subgroupG/H
centerC22#15689032
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1985C2
socleC22×C47#392
prime radical#20254698C2
hypercenter#1985D47
hypocenterC47#10028
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1985

Complements

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

NH ≅ G/Ntype
2. #20253510C2semidirect
3. #20241998C2semidirect
4. #20241258C2semidirect
5. C47×#1985C2semidirect
128. #1985D47semidirect
151. C47#10028semidirect

Maximal subgroups

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

subgroupindex
#202546982
#202535102
#202419982
#202412582
C47×#19852
#1002847