Finite Group Databasebeta

← #26119367

Subgroups — #26119367

Order 12032.

454 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 97 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#20251766C211yesmaximal, characteristic, complemented
2#20244216C211yesmaximal, characteristic
3#20241949C212yesmaximal
4#20241247C212yesmaximal, complemented
5C47×#1276C211yesmaximal, characteristic, complemented
6#15689673C2211yescharacteristic
7C2×#13094024—21no—
8#15689147C2211yescharacteristic
9C2×#13093952—21no—
10C2×#13093952—21no—
11#15689029—24no—
12#15688989—21no—
13#15688989C2212yes—
14#15688858—24no—
15C47×#431C2212yes—
16C43×C47C2211yescharacteristic
17C2×#10625350—21no—
18C22×Dic94—22no—
19C2×#10625329—21no—
20#13094024—42no—
21#13094024—42no—
22#13094024Q811yescharacteristic, complemented
23#13094021Q811yescharacteristic, complemented
24#13094012—21no—
25#13093998—22no—
26C4×D188—22no—
27C2×C4⋊Dic47—22no—
28C2×C4⋊Dic47—22no—
29C4⋊(C47⋊C8)—22no—
30#13093952—42no—
31#13093952—42no—
32C47⋊#95—24no—
33C47⋊#95—22no—
34C47⋊#95D414yes—
35C2×C42×C47—22no—
36C2×C42×C47D411yescharacteristic
37C2×C42×C47D411yescharacteristic
38C2×C42×C47C2311yescharacteristic
39C4⋊C8×C47—22no—
40C22⋊C8×C47—22no—
41C4×C8×C47—22no—
42C22×C4×C47—21no—
43C22×C4×C47—21no—
44C22×C4×C47C2×D412yes—
45C22×C4×C47C2×D411yescharacteristic
46C2×C22⋊D47—41no—
47C22×Dic47—24no—
48#10625350—41no—
49#10625350—41no—
50#10625350—41no—
51#10625350—41no—
52C2×D188—21no—
53C2×D188—22no—
54C2×C4×D47—41no—
55C2×Dic94—21no—
56C2×Dic94—44no—
57C2×Dic94—48no—
58C2×C8×C47—42no—
59C42×C47—21no—
60C42×C47—21no—
61C42×C47—44no—
62C42×C47D812yes—
63C42×C47SD1612yes—
64C42×C47C2×Q811yescharacteristic
65C42×C47#4112yes—
66C42×C47#4111yescharacteristic
67(C2×D47)⋊C4—42no—
68C4⋊Dic47—22no—
69C4⋊Dic47—44no—
70C4⋊Dic47—42no—
71C4⋊Dic47—42no—
72#10625329—41no—
73#10625329—41no—
74C2×C47⋊C8—41no—
75C23×C47#12011yescharacteristic
76C22×D47—41no—
77C2×C4×C47—24no—
78C2×C4×C47—22no—
79C2×C4×C47—22no—
80C2×C4×C47—42no—
81C2×C4×C47—42no—
82C2×C4×C47C22⋊Q812yes—
83C2×C4×C47C22⋊Q811yescharacteristic
84C2×C4×C47#12412yes—
85C2×C4×C47#12411yescharacteristic
86C2×C4×C47C2×D811yescharacteristic
87C2×C4×C47C2×SD1611yescharacteristic
88C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
89C2×C4×C47#13711yescharacteristic
90C22⋊D47—81no—
91C22⋊D47—81no—
92C2×Dic47—48no—
93C2×Dic47—41no—
94C2×Dic47—44no—
95D188—41no—
96D188—41no—
97D188—82no—
98C4×D47—81no—
99C4×D47—81no—
100Dic94—41no—
101Dic94—41no—
102Dic94—88no—
103C8×C47—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#47411yescharacteristic
117C4×C47#47611yescharacteristic
118C4×C47#48611yescharacteristic
119C4×C47#48911yescharacteristic
120Dic47—84no—
121Dic47—81no—
122#2862—471no—
123#1962—471no—
124#1517—472no—
125#1286—472no—
126#1276D4711yescharacteristic, complemented
127C2×C47—21no—
128C2×C47—21no—
129C2×C47#202311yescharacteristic
130C2×C47#203112yes—
131D47—161no—
132#532—471no—
133C2×C4⋊Q8—941no—
134#431C2×D4712yes—
135#423—471no—
136C2×#104—941no—
137C2×#104—941no—
138#391—944no—
139#382—472no—
140#382—941no—
141C43C2×D4711yescharacteristic
142#327—944no—
143C47#1036211yescharacteristic, complemented, Sylow
144C2×#41—941no—
145C22×Q8—942no—
146C2×M16—941no—
147C4⋊Q8—471no—
148C4⋊Q8—1882no—
149C4⋊Q8—1882no—
150#127—471no—
151#124—941no—
152#121—942no—
153C4×D4—942no—
154C2×C4⋊C4—942no—
155C2×C4⋊C4—942no—
156C2×C42—22no—
157C2×C42D18811yescharacteristic
158C2×C42D18811yescharacteristic
159C2×C42C22×D4711yescharacteristic
160C4⋊C8—22no—
161C4⋊C8—942no—
162#104—1882no—
163#104—1882no—
164C22⋊C8—22no—
165C4×C8—22no—
166#95—474no—
167#95—944no—
168#95—942no—
169#41—1881no—
170#41—1881no—
171#41—1881no—
172#41—1881no—
173C2×Q8—941no—
174C2×Q8—1884no—
175C2×Q8—1888no—
176C2×D4—942no—
177C2×D4—941no—
178C2×D4—1881no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—944no—
182C22×C4—1881no—
183C22×C4C2×D18811yescharacteristic
184C22×C4D4×D4712yes—
185M16—1881no—
186M16—1881no—
187C2×C8—42no—
188C2×C8—1881no—
189C4⋊C4—942no—
190C4⋊C4—1884no—
191C4⋊C4—1882no—
192C4⋊C4—1882no—
193C22⋊C4—1882no—
194C42—21no—
195C42—21no—
196C42—44no—
197C42#1062532512yes—
198C42D37612yes—
199C42#1062535012yes—
200C42#1062535211yescharacteristic
201C42Q8×D4711yescharacteristic
202C23—1881no—
203C23#1309399411yescharacteristic
204Q8—3768no—
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—3761no—
218C2×C4—3761no—
219C2×C4—1888no—
220C2×C4—1884no—
221C2×C4—1881no—
222C2×C4#1309400212yes—
223C2×C4#1309400411yescharacteristic
224C2×C4#1309401012yes—
225C2×C4#1309401211yescharacteristic
226C2×C4C2×D37611yescharacteristic
227C2×C4C2×#1062532511yescharacteristic
228C2×C4#1309405211yescharacteristic
229C2×C4#1309405711yescharacteristic
230C8—82no—
231C8—3761no—
232C22—22no—
233C22—3761no—
234C22—3762no—
235C22#1568903611yescharacteristic
236C22#1568925911yescharacteristic
237C22#1568926411yescharacteristic
238C4—24no—
239C4—22no—
240C4—22no—
241C4—42no—
242C4—3764no—
243C4—3761no—
244C4#1568940411yescharacteristic
245C4#1568941411yescharacteristic
246C4#1568947011yescharacteristic
247C4#1568948311yescharacteristic
248C2—21no—
249C2—21no—
250C2—7521no—
251C2#2024452511yescharacteristic
252C2#2024456112yes—
subgroupG/H
centerC22#15689036
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1276C2
socleC22×C47#393
prime radical#15689673C22
hypercenter#1276D47
hypocenterC47#10362
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
1. #20251766C2semidirect
4. #20241247C2semidirect
5. C47×#1276C2semidirect
22. #13094024Q8semidirect
23. #13094021Q8semidirect
126. #1276D47semidirect
143. C47#10362semidirect

Maximal subgroups

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

subgroupindex
#202517662
#202442162
#202419492
#202412472
C47×#12762
#1036247