Finite Group Databasebeta

← #26119363

Subgroups — #26119363

Order 12032.

454 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 101 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#20244216C211yesmaximal, characteristic
2#20241947C212yesmaximal
3#20241246C212yesmaximal, complemented
4#20241221C211yesmaximal, characteristic, complemented
5C47×#2862C211yesmaximal, characteristic, complemented
6#15689673C2211yescharacteristic
7#15689176C2212yes—
8#15689147C2211yescharacteristic
9C2×#13093952—21no—
10C2×#13093952—21no—
11#15688856—24no—
12C2×C4⋊Q8×C47—21no—
13C47×#391—24no—
14C47×#382—21no—
15C47×#382C2212yes—
16C43×C47C2211yescharacteristic
17C2×#10625350—21no—
18C2×#10625329—21no—
19#13094024Q811yescharacteristic, complemented
20#13094021Q811yescharacteristic, complemented
21#13094012—21no—
22#13093998—22no—
23C4×D188—22no—
24C4⋊(C47⋊C8)—22no—
25C4⋊(C47⋊C8)—22no—
26#13093952—42no—
27#13093952—42no—
28C22⋊(C47⋊C8)—22no—
29C4×C47⋊C8—22no—
30C22×Q8×C47—22no—
31C4⋊Q8×C47—42no—
32C4⋊Q8×C47—42no—
33C2×C4⋊C4×C47—22no—
34C2×C4⋊C4×C47—22no—
35C2×C42×C47—22no—
36C2×C42×C47D411yescharacteristic
37C2×C42×C47D411yescharacteristic
38C2×C42×C47C2311yescharacteristic
39#95×C47—24no—
40#95×C47—22no—
41#95×C47D414yes—
42C2×Q8×C47—44no—
43C2×Q8×C47—48no—
44C22×C4×C47—21no—
45C22×C4×C47—21no—
46C22×C4×C47—24no—
47C22×C4×C47C2×D412yes—
48C22×C4×C47C2×D411yescharacteristic
49C2×C22⋊D47—41no—
50#10625350—41no—
51#10625350—41no—
52#10625350—41no—
53#10625350—41no—
54C2×D188—21no—
55C2×D188—22no—
56C2×C4×D47—41no—
57C2×Dic94—21no—
58C4⋊C4×C47—44no—
59C4⋊C4×C47—42no—
60C4⋊C4×C47—42no—
61C42×C47—21no—
62C42×C47—21no—
63C42×C47—44no—
64C42×C47D812yes—
65C42×C47SD1612yes—
66C42×C47C2×Q811yescharacteristic
67C42×C47#4111yescharacteristic
68C42×C47#4112yes—
69(C2×D47)⋊C4—42no—
70C4⋊Dic47—22no—
71#10625329—41no—
72#10625329—41no—
73C2×C47⋊C8—42no—
74C2×C47⋊C8—41no—
75C23×C47#12011yescharacteristic
76C22×D47—41no—
77Q8×C47—88no—
78C2×C4×C47—24no—
79C2×C4×C47—22no—
80C2×C4×C47—22no—
81C2×C4×C47—42no—
82C2×C4×C47—42no—
83C2×C4×C47—48no—
84C2×C4×C47—44no—
85C2×C4×C47C22⋊Q812yes—
86C2×C4×C47C22⋊Q811yescharacteristic
87C2×C4×C47#12412yes—
88C2×C4×C47#12411yescharacteristic
89C2×C4×C47C2×D811yescharacteristic
90C2×C4×C47C2×SD1611yescharacteristic
91C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
92C2×C4×C47#13711yescharacteristic
93C22⋊D47—81no—
94C22⋊D47—81no—
95C2×Dic47—41no—
96D188—41no—
97D188—41no—
98D188—82no—
99C4×D47—81no—
100C4×D47—81no—
101Dic94—41no—
102Dic94—41no—
103C47⋊C8—82no—
104C47⋊C8—81no—
105#10362—471nomaximal, Sylow
106C22×C47—22no—
107C22×C47#39311yescharacteristic
108C22×C47#44711yescharacteristic
109C22×C47#44811yescharacteristic
110C2×D47—81no—
111C2×D47—82no—
112C4×C47—24no—
113C4×C47—22no—
114C4×C47—22no—
115C4×C47—42no—
116C4×C47—84no—
117C4×C47#47411yescharacteristic
118C4×C47#47611yescharacteristic
119C4×C47#48611yescharacteristic
120C4×C47#48911yescharacteristic
121Dic47—81no—
122#2862D4711yescharacteristic, complemented
123#1962—471no—
124#1517—472no—
125#1286—472no—
126#1276—471no—
127C2×C47—21no—
128C2×C47—21no—
129C2×C47#202311yescharacteristic
130C2×C47#203112yes—
131D47—161no—
132#532—471no—
133C2×C4⋊Q8—21no—
134#431—472no—
135#423—471no—
136C2×#104—941no—
137C2×#104—941no—
138#391—24no—
139#382—21no—
140#382C2×D4712yes—
141C43C2×D4711yescharacteristic
142#327—944no—
143C47#1036211yescharacteristic, complemented, Sylow
144C2×#41—941no—
145C22×Q8—22no—
146C2×M16—941no—
147C4⋊Q8—42no—
148C4⋊Q8—42no—
149C4⋊Q8—471no—
150#127—471no—
151#124—941no—
152#121—942no—
153C4×D4—942no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C42—22no—
157C2×C42C22⋊D4711yescharacteristic
158C2×C42C22⋊D4711yescharacteristic
159C2×C42C22×D4711yescharacteristic
160C4⋊C8—942no—
161C4⋊C8—942no—
162#104—1882no—
163#104—1882no—
164C22⋊C8—942no—
165C4×C8—942no—
166#95—24no—
167#95—22no—
168#95C22⋊D4714yes—
169#41—1881no—
170#41—1881no—
171#41—1881no—
172#41—1881no—
173C2×Q8—44no—
174C2×Q8—48no—
175C2×Q8—941no—
176C2×D4—942no—
177C2×D4—941no—
178C2×D4—1881no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—24no—
182C22×C4—1881no—
183C22×C4C2×C22⋊D4712yes—
184C22×C4C2×C22⋊D4711yescharacteristic
185M16—1881no—
186M16—1881no—
187C2×C8—1882no—
188C2×C8—1881no—
189C4⋊C4—44no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—942no—
193C22⋊C4—1882no—
194C42—21no—
195C42—21no—
196C42—44no—
197C42#1062533412yes—
198C42#1062533612yes—
199C42Q8×D4711yescharacteristic
200C42#1062535411yescharacteristic
201C42#1062535412yes—
202C23—1881no—
203C23#1309399311yescharacteristic
204Q8—88no—
205Q8—1881no—
206Q8—1881no—
207D4—3762no—
208D4—3761no—
209D4—3761no—
210D4—1881no—
211D4—1881no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—48no—
218C2×C4—44no—
219C2×C4—3761no—
220C2×C4—3761no—
221C2×C4—1881no—
222C2×C4#1309400012yes—
223C2×C4#1309400011yescharacteristic
224C2×C4#1309401312yes—
225C2×C4#1309401311yescharacteristic
226C2×C4C2×#1062533411yescharacteristic
227C2×C4C2×#1062533611yescharacteristic
228C2×C4#1309405011yescharacteristic
229C2×C4#1309405511yescharacteristic
230C8—3762no—
231C8—3761no—
232C22—22no—
233C22—3761no—
234C22—3762no—
235C22#1568903411yescharacteristic
236C22#1568925711yescharacteristic
237C22#1568926211yescharacteristic
238C4—24no—
239C4—22no—
240C4—22no—
241C4—42no—
242C4—84no—
243C4—3761no—
244C4#1568940011yescharacteristic
245C4#1568941011yescharacteristic
246C4#1568947111yescharacteristic
247C4#1568948511yescharacteristic
248C2—21no—
249C2—21no—
250C2—7521no—
251C2#2024452111yescharacteristic
252C2#2024455712yes—
subgroupG/H
centerC22#15689034
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2862C2
socleC22×C47#393
prime radical#15689673C22
hypercenter#2862D47
hypocenterC47#10362
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2862

Complements

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

NH ≅ G/Ntype
3. #20241246C2semidirect
4. #20241221C2semidirect
5. C47×#2862C2semidirect
19. #13094024Q8semidirect
20. #13094021Q8semidirect
122. #2862D47semidirect
143. C47#10362semidirect

Maximal subgroups

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

subgroupindex
#202442162
#202419472
#202412462
#202412212
C47×#28622
#1036247