Finite Group Databasebeta

← #26129807

Subgroups — #26129807

Order 12032.

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

normal subgroups only

This group has 328 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#20245645C211yesmaximal, characteristic, complemented
2#20245423C211yesmaximal, characteristic
3#20242748C211yesmaximal, characteristic, complemented
4#20241437C211yesmaximal, characteristic, complemented
5Q8⋊(C47⋊C16)C211yesmaximal, characteristic, complemented
6#20241349C211yesmaximal, characteristic, complemented
7C47×#3393C211yesmaximal, characteristic, complemented
8C2×#13093967—22no—
9C2×#13093965—21no—
10#15689500C2211yescharacteristic
11#15689323—21no—
12C4⋊(C47⋊C16)C2211yescharacteristic
13#15688923—21no—
14C4×C47⋊C16C2211yescharacteristic
15#15688876C2211yescharacteristic
16#15688866—21no—
17C47×#556—21no—
18C47×#495—21no—
19C47×#494C2211yescharacteristic
20C47×#464—22no—
21C47×#461—21no—
22C8×Q8×C47C2211yescharacteristic
23C4×Q16×C47—21no—
24C4×D8×C47C2211yescharacteristic
25C2×#10625336—41no—
26C2×D376—21no—
27C2×D376—21no—
28#13094021—21no—
29#13093967—44no—
30#13093965—42no—
31C2×C47⋊C16—21no—
32C2×C47⋊C16—21no—
33C4⋊(C47⋊C8)—21no—
34#13093944—41no—
35C2×Q16×C47—21no—
36C2×Q16×C47D412yes—
37C2×SD16×C47—22no—
38C2×D8×C47D411yescharacteristic
39C4⋊Q8×C47—21no—
40#126×C47—42no—
41#124×C47—22no—
42C22⋊Q8×C47—41no—
43C4×Q8×C47—22no—
44C4×Q8×C47D411yescharacteristic
45C4×D4×C47—21no—
46#117×C47—41no—
47#107×C47D411yescharacteristic
48C4⋊C8×C47—21no—
49C4⋊C8×C47D411yescharacteristic
50#103×C47—22no—
51#102×C47—21no—
52C4×C8×C47—21no—
53C4×C8×C47C2311yescharacteristic
54C2×Q8×C47—22no—
55C2×Q8×C47D811yescharacteristic
56C2×D4×C47—21no—
57C22×C4×C47—41no—
58C2×D188—41no—
59C2×D188—41no—
60C2×D188—41no—
61Q16×C47—42no—
62Q16×C47—41no—
63Q16×C47—41no—
64SD16×C47—82no—
65D8×C47—21no—
66D8×C47—21no—
67C2×C8×C47—21no—
68C2×C8×C47C2×D411yescharacteristic
69C2×C8×C47C2×D411yescharacteristic
70C4⋊C4×C47—21no—
71C4⋊C4×C47—21no—
72C4⋊C4×C47—42no—
73C4⋊C4×C47—41no—
74C4⋊C4×C47—41no—
75C4⋊C4×C47D811yescharacteristic
76C22⋊C4×C47—42no—
77C22⋊C4×C47—42no—
78C22⋊C4×C47—41no—
79C42×C47—21no—
80C42×C47—42no—
81C42×C47C2×D411yescharacteristic
82#10625336—82no—
83C2×C47⋊C8—41no—
84D376—42no—
85D376—41no—
86D376—41no—
87C47⋊C16—22no—
88C47⋊C16—41no—
89C23×C47—41no—
90C22×D47—81no—
91Q8×C47—21no—
92Q8×C47—21no—
93Q8×C47—42no—
94Q8×C47—42no—
95D4×C47—41no—
96D4×C47—41no—
97C2×C4×C47—21no—
98C2×C4×C47—21no—
99C2×C4×C47—41no—
100C2×C4×C47—42no—
101C2×C4×C47—81no—
102C2×C4×C47—81no—
103C2×C4×C47#12011yescharacteristic
104C2×C4×C47C2×D811yescharacteristic
105C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
106D188—81no—
107D188—81no—
108D188—81no—
109D188—81no—
110D188—161no—
111C8×C47—21no—
112C8×C47—41no—
113C8×C47#13511yescharacteristic
114C8×C47AGL(1,ℤ/8)11yescharacteristic
115C47⋊C8—81no—
116#12331—471nomaximal, Sylow
117C22×C47—81no—
118C22×C47—81no—
119C22×C47—81no—
120C22×C47#44711yescharacteristic
121C2×D47—161no—
122C2×D47—161no—
123C2×D47—161no—
124C4×C47—21no—
125C4×C47—41no—
126C4×C47—41no—
127C4×C47—82no—
128C4×C47—81no—
129C4×C47#44911yescharacteristic
130C4×C47#45311yescharacteristic
131C4×C47#50811yescharacteristic
132C4×C47#50911yescharacteristic
133#3393D4711yescharacteristic, complemented
134#2248—471no—
135#2206—471no—
136#1668—471no—
137#1362—471no—
138Q8⋊C16—471no—
139#1331—471no—
140C2×C47—161no—
141C2×C47#162111yescharacteristic
142C2×C47#218611yescharacteristic
143C2×C47#219311yescharacteristic
144D47—321no—
145#556—21no—
146C2×SD32—942no—
147C2×D16—941no—
148#495—21no—
149#494C2×D4711yescharacteristic
150#493—471no—
151#464—22no—
152#461—21no—
153#460—941no—
154C8×Q8C2×D4711yescharacteristic
155C4×Q16—21no—
156C4×D8C2×D4711yescharacteristic
157C4⋊C16—471no—
158#357—941no—
159C4×C16—471no—
160#335—471no—
161#331—941no—
162C47#1233111yescharacteristic, complemented, Sylow
163C2×Q16—21no—
164C2×Q16C22⋊D4712yes—
165C2×SD16—22no—
166C2×SD16—1881no—
167C2×D8—941no—
168C2×D8—941no—
169C2×D8C22⋊D4711yescharacteristic
170C4⋊Q8—21no—
171#127—941no—
172#126—42no—
173#124—22no—
174C22⋊Q8—41no—
175C4×Q8—22no—
176C4×Q8C22⋊D4711yescharacteristic
177C4×D4—21no—
178#117—41no—
179SD32—1884no—
180D16—1882no—
181C2×C16—941no—
182C2×C16—941no—
183#107C22⋊D4711yescharacteristic
184C4⋊C8—21no—
185C4⋊C8—941no—
186C4⋊C8C22⋊D4711yescharacteristic
187#103—22no—
188#102—21no—
189#102—1881no—
190C4×C8—21no—
191C4×C8C22×D4711yescharacteristic
192C2×Q8—22no—
193C2×Q8#1062533411yescharacteristic
194C2×D4—21no—
195C2×D4—1881no—
196C2×D4—1881no—
197C2×D4—1881no—
198C22×C4—41no—
199Q16—42no—
200Q16—41no—
201Q16—41no—
202SD16—82no—
203SD16—3762no—
204D8—21no—
205D8—21no—
206D8—1882no—
207D8—1881no—
208D8—1881no—
209C2×C8—21no—
210C2×C8—1881no—
211C2×C8C2×C22⋊D4711yescharacteristic
212C2×C8C2×C22⋊D4711yescharacteristic
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—42no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4#1062533411yescharacteristic
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—41no—
222C42—21no—
223C42—42no—
224C42C2×C22⋊D4711yescharacteristic
225C16—942no—
226C16—1881no—
227C23—41no—
228C23—3761no—
229Q8—21no—
230Q8—21no—
231Q8—42no—
232Q8—42no—
233D4—41no—
234D4—41no—
235D4—3761no—
236D4—3761no—
237D4—3761no—
238D4—3761no—
239D4—7521no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4#1309399311yescharacteristic
247C2×C4C2×#1062533411yescharacteristic
248C2×C4#1309405011yescharacteristic
249C8—21no—
250C8—41no—
251C8—3761no—
252C8#1309404411yescharacteristic
253C8#1309404911yescharacteristic
254C22—81no—
255C22—81no—
256C22—81no—
257C22—7521no—
258C22—7521no—
259C22—7521no—
260C22#1568925711yescharacteristic
261C4—21no—
262C4—41no—
263C4—41no—
264C4—82no—
265C4—81no—
266C4#1568926811yescharacteristic
267C4#1568929111yescharacteristic
268C4#1568955311yescharacteristic
269C4#1568955811yescharacteristic
270C2—161no—
271C2—15041no—
272C2#2024245311yescharacteristic
273C2#2024531111yescharacteristic
274C2#2024535211yescharacteristic
subgroupG/H
centerC22#15689257
commutator, cosocleC4×C8×C47C23
FrattiniC4×C8C22×D47
FittingC47×#3393C2
socleC22×C47#447
prime radical#20245423C2
hypercenter#3393D47
hypocenterC47#12331
LCSG ▷ C4×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #3393

Complements

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

NH ≅ G/Ntype
1. #20245645C2semidirect
3. #20242748C2semidirect
4. #20241437C2semidirect
5. Q8⋊(C47⋊C16)C2semidirect
6. #20241349C2semidirect
7. C47×#3393C2semidirect
133. #3393D47semidirect
162. C47#12331semidirect

Maximal subgroups

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

subgroupindex
#202456452
#202454232
#202427482
#202414372
Q8⋊(C47⋊C16)2
#202413492
C47×#33932
#1233147