Finite Group Databasebeta

← #26129793

Subgroups — #26129793

Order 12032.

340 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 71 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#20245637C211yesmaximal, characteristic
2#20245428C211yesmaximal, characteristic, complemented
3#20242748C211yesmaximal, characteristic, complemented
4#20241434C211yesmaximal, characteristic, complemented
5Q8⋊(C47⋊C16)C211yesmaximal, characteristic, complemented
6#20241352C211yesmaximal, characteristic, complemented
7C47×#3396C211yesmaximal, characteristic, complemented
8C2×#13093967—21no—
9C2×#13093965—22no—
10#15689500C2211yescharacteristic
11#15689323—21no—
12C4⋊(C47⋊C16)C2211yescharacteristic
13#15688923—21no—
14C4×C47⋊C16C2211yescharacteristic
15#15688876C2211yescharacteristic
16#15688866—21no—
17C47×#559—21no—
18C47×#495—21no—
19C47×#493C2211yescharacteristic
20C47×#463—22no—
21C47×#460—21no—
22C8×Q8×C47C2211yescharacteristic
23C4×Q16×C47C2211yescharacteristic
24C4×D8×C47—21no—
25C2×#10625336—41no—
26C2×D376—21no—
27C2×D376—21no—
28#13094021—21no—
29#13093967—42no—
30#13093965—44no—
31C2×C47⋊C16—21no—
32C2×C47⋊C16—21no—
33C4⋊(C47⋊C8)—21no—
34#13093944—41no—
35C2×Q16×C47D411yescharacteristic
36C2×SD16×C47—22no—
37C2×D8×C47—21no—
38C2×D8×C47D412yes—
39#127×C47—21no—
40#124×C47—22no—
41#124×C47—41no—
42#123×C47—42no—
43C4×Q8×C47—21no—
44C4×Q8×C47D411yescharacteristic
45C4×D4×C47—22no—
46C4×D4×C47—41no—
47#107×C47D411yescharacteristic
48C4⋊C8×C47—21no—
49C4⋊C8×C47D411yescharacteristic
50#103×C47—21no—
51#102×C47—22no—
52C4×C8×C47—21no—
53C4×C8×C47C2311yescharacteristic
54C2×Q8×C47—21no—
55C2×Q8×C47D811yescharacteristic
56C2×D4×C47—22no—
57C2×D4×C47—41no—
58C2×D4×C47—41no—
59C22×C4×C47—42no—
60C2×D188—41no—
61C2×D188—41no—
62C2×D188—41no—
63Q16×C47—21no—
64Q16×C47—21no—
65SD16×C47—82no—
66D8×C47—42no—
67D8×C47—41no—
68D8×C47—41no—
69C2×C8×C47—21no—
70C2×C8×C47C2×D411yescharacteristic
71C2×C8×C47C2×D411yescharacteristic
72C4⋊C4×C47—21no—
73C4⋊C4×C47—21no—
74C4⋊C4×C47—41no—
75C4⋊C4×C47D811yescharacteristic
76C22⋊C4×C47—42no—
77C22⋊C4×C47—42no—
78C22⋊C4×C47—42no—
79C42×C47—21no—
80C42×C47—41no—
81C42×C47C2×D411yescharacteristic
82#10625336—82no—
83C2×C47⋊C8—41no—
84D376—42no—
85D376—41no—
86D376—41no—
87C47⋊C16—22no—
88C47⋊C16—41no—
89C23×C47—42no—
90C22×D47—81no—
91Q8×C47—21no—
92Q8×C47—21no—
93Q8×C47—41no—
94Q8×C47—41no—
95D4×C47—42no—
96D4×C47—42no—
97D4×C47—81no—
98D4×C47—81no—
99D4×C47—161no—
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—41no—
103C2×C4×C47—41no—
104C2×C4×C47—162no—
105C2×C4×C47#12011yescharacteristic
106C2×C4×C47C2×D811yescharacteristic
107C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
108D188—81no—
109D188—81no—
110D188—81no—
111D188—81no—
112D188—161no—
113C8×C47—21no—
114C8×C47—41no—
115C8×C47#13511yescharacteristic
116C8×C47AGL(1,ℤ/8)11yescharacteristic
117C47⋊C8—81no—
118#12329—471nomaximal, Sylow
119C22×C47—82no—
120C22×C47—82no—
121C22×C47—82no—
122C22×C47#44711yescharacteristic
123C2×D47—161no—
124C2×D47—161no—
125C2×D47—161no—
126C4×C47—21no—
127C4×C47—41no—
128C4×C47—41no—
129C4×C47—81no—
130C4×C47—81no—
131C4×C47#44911yescharacteristic
132C4×C47#45311yescharacteristic
133C4×C47#50811yescharacteristic
134C4×C47#50911yescharacteristic
135#3396D4711yescharacteristic, complemented
136#2246—471no—
137#2207—471no—
138#1668—471no—
139#1361—471no—
140Q8⋊C16—471no—
141#1332—471no—
142C2×C47—162no—
143C2×C47#162111yescharacteristic
144C2×C47#218611yescharacteristic
145C2×C47#219211yescharacteristic
146D47—321no—
147#559—21no—
148C2×SD32—941no—
149C2×D16—942no—
150#495—21no—
151#493—471no—
152#493C2×D4711yescharacteristic
153#463—22no—
154#460—21no—
155#460—941no—
156C8×Q8C2×D4711yescharacteristic
157C4×Q16C2×D4711yescharacteristic
158C4×D8—21no—
159C4⋊C16—471no—
160#357—941no—
161C4×C16—471no—
162#335—471no—
163#331—941no—
164C47#1232911yescharacteristic, complemented, Sylow
165C2×Q16C22⋊D4711yescharacteristic
166C2×SD16—22no—
167C2×SD16—1881no—
168C2×D8—21no—
169C2×D8—941no—
170C2×D8—941no—
171C2×D8C22⋊D4712yes—
172#127—21no—
173#127—941no—
174#124—22no—
175#124—41no—
176#123—42no—
177C4×Q8—21no—
178C4×Q8C22⋊D4711yescharacteristic
179C4×D4—22no—
180C4×D4—41no—
181SD32—1882no—
182D16—1884no—
183C2×C16—941no—
184C2×C16—941no—
185#107C22⋊D4711yescharacteristic
186C4⋊C8—21no—
187C4⋊C8—941no—
188C4⋊C8C22⋊D4711yescharacteristic
189#103—21no—
190#102—22no—
191#102—1881no—
192C4×C8—21no—
193C4×C8C22×D4711yescharacteristic
194C2×Q8—21no—
195C2×Q8#1062533411yescharacteristic
196C2×D4—22no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4—1881no—
200C2×D4—1881no—
201C2×D4—1881no—
202C22×C4—42no—
203Q16—21no—
204Q16—21no—
205SD16—82no—
206SD16—3762no—
207D8—42no—
208D8—41no—
209D8—41no—
210D8—1882no—
211D8—1881no—
212D8—1881no—
213C2×C8—21no—
214C2×C8—1881no—
215C2×C8C2×C22⋊D4711yescharacteristic
216C2×C8C2×C22⋊D4711yescharacteristic
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—41no—
220C4⋊C4#1062533411yescharacteristic
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C42—21no—
225C42—41no—
226C42C2×C22⋊D4711yescharacteristic
227C16—942no—
228C16—1881no—
229C23—42no—
230C23—3761no—
231Q8—21no—
232Q8—21no—
233Q8—41no—
234Q8—41no—
235D4—42no—
236D4—42no—
237D4—81no—
238D4—81no—
239D4—161no—
240D4—3761no—
241D4—3761no—
242D4—3761no—
243D4—3761no—
244D4—7521no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—162no—
250C2×C4#1309399311yescharacteristic
251C2×C4C2×#1062533411yescharacteristic
252C2×C4#1309405011yescharacteristic
253C8—21no—
254C8—41no—
255C8—3761no—
256C8#1309404411yescharacteristic
257C8#1309404911yescharacteristic
258C22—82no—
259C22—82no—
260C22—82no—
261C22—7521no—
262C22—7521no—
263C22—7521no—
264C22#1568925711yescharacteristic
265C4—21no—
266C4—41no—
267C4—41no—
268C4—81no—
269C4—81no—
270C4#1568926811yescharacteristic
271C4#1568929111yescharacteristic
272C4#1568955311yescharacteristic
273C4#1568955811yescharacteristic
274C2—162no—
275C2—15041no—
276C2#2024245311yescharacteristic
277C2#2024531111yescharacteristic
278C2#2024534511yescharacteristic
subgroupG/H
centerC22#15689257
commutator, cosocleC4×C8×C47C23
FrattiniC4×C8C22×D47
FittingC47×#3396C2
socleC22×C47#447
prime radical#20245637C2
hypercenter#3396D47
hypocenterC47#12329
LCSG ▷ C4×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #3396

Complements

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

NH ≅ G/Ntype
2. #20245428C2semidirect
3. #20242748C2semidirect
4. #20241434C2semidirect
5. Q8⋊(C47⋊C16)C2semidirect
6. #20241352C2semidirect
7. C47×#3396C2semidirect
135. #3396D47semidirect
164. C47#12329semidirect

Maximal subgroups

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

subgroupindex
#202456372
#202454282
#202427482
#202414342
Q8⋊(C47⋊C16)2
#202413522
C47×#33962
#1232947