Finite Group Databasebeta

← #26133026

Subgroups — #26133026

Order 12032.

396 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C2×#15689230C211yesmaximal, characteristic
2C2×#15688946C211yesmaximal, characteristic
3C2×#15688939C211yesmaximal, characteristic
4#20244142C211yesmaximal, characteristic
5#20241493C211yesmaximal, characteristic
6#20241467C211yesmaximal, characteristic
7C47×#2141C211yesmaximal, characteristic
8C2×Q8×Dic47—21no—
9C2×#13093964C2211yescharacteristic
10#15689230—22no—
11#15689230—22no—
12C2×#13093960C2211yescharacteristic
13C2×#13093956C2211yescharacteristic
14#15689150—21no—
15C2×#13093948—21no—
16C2×#13093934C2211yescharacteristic
17#15688946—22no—
18#15688939—22no—
19#15688882—21no—
20C22×Q16×C47C2211yescharacteristic
21C2×M32×C47C2211yescharacteristic
22C47×#427C2211yescharacteristic
23C47×#358C414yescomplemented
24C2×#10625329—21no—
25C22×C47⋊C8D411yescharacteristic
26Q8×Dic47—24no—
27Q8×Dic47—42no—
28C2×C4⋊Dic47—21no—
29C2×C4⋊Dic47—41no—
30C2×C4×Dic47—41no—
31C2×C4×Dic47D411yescharacteristic
32#13093964—21no—
33#13093964—21no—
34C2×C47⋊C16—21no—
35#13093960—21no—
36#13093960—21no—
37#13093956—21no—
38#13093956D412yes—
39C4⋊(C47⋊C8)—42no—
40#13093948—24no—
41#13093934—21no—
42#13093934—21no—
43C22×Q8×C47—21no—
44C2×Q16×C47—21no—
45C2×Q16×C47—21no—
46C2×Q16×C47—21no—
47C2×Q16×C47C2×C412yes—
48C2×Q16×C47D412yes—
49C2×Q16×C47Q812yes—
50C22×C8×C47C2311yescharacteristic
51#117×C47—21no—
52M32×C47—21no—
53M32×C47—21no—
54C2×C16×C47C2×C412yes—
55#107×C47C2×C412yes—
56#106×C47—21no—
57C2×Q8×C47—22no—
58C2×Q8×C47—22no—
59C2×Q8×C47—22no—
60C2×Q8×C47—41no—
61C2×Q8×C47—41no—
62C2×Q8×C47—41no—
63C22×C4×C47—41no—
64C22×C4×C47C2×D411yescharacteristic
65C22×Dic47—21no—
66C22×Dic47—41no—
67Q16×C47—22no—
68Q16×C47—22no—
69Q16×C47—22no—
70Q16×C47C4⋊C414yes—
71C2×C8×C47C22×C411yescharacteristic
72C2×C8×C47C2×D411yescharacteristic
73C2×C8×C47C2×D411yescharacteristic
74C2×C8×C47C2×Q811yescharacteristic
75C2×C8×C47#4111yescharacteristic
76C2×C8×C47#4111yescharacteristic
77C4⋊C4×C47—22no—
78C22⋊C4×C47—41no—
79C42×C47—41no—
80C4⋊Dic47—22no—
81C4⋊Dic47—44no—
82C4⋊Dic47—41no—
83C4×Dic47—21no—
84C4×Dic47—44no—
85C4×Dic47SD1612yes—
86#10625329—41no—
87#10625329—41no—
88C2×C47⋊C8—21no—
89C2×C47⋊C8—21no—
90C2×C47⋊C8—21no—
91C2×C47⋊C8—41no—
92C16×C47—22no—
93C47⋊C16—41no—
94C23×C47AGL(1,ℤ/8)11yescharacteristic
95Q8×C47—24no—
96Q8×C47—42no—
97Q8×C47—42no—
98Q8×C47—42no—
99C2×C4×C47—42no—
100C2×C4×C47—42no—
101C2×C4×C47—42no—
102C2×C4×C47—42no—
103C2×C4×C47C4×D411yescharacteristic
104C2×C4×C47#12011yescharacteristic
105C2×C4×C47C22⋊Q811yescharacteristic
106C2×C4×C47C2×SD1611yescharacteristic
107C2×C4×C47#13511yescharacteristic
108C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
109C2×Dic47—22no—
110C2×Dic47—42no—
111C2×Dic47—42no—
112C2×Dic47—82no—
113C8×C47C2×C4⋊C411yescharacteristic
114C8×C47C4×D411yescharacteristic
115C8×C47C22⋊Q811yescharacteristic
116C8×C47#12311yescharacteristic
117C47⋊C8—41no—
118C47⋊C8—81no—
119#12957—471nomaximal, Sylow
120C22×C47—22no—
121C22×C47#44211yescharacteristic
122C22×C47#45011yescharacteristic
123C22×C47#47811yescharacteristic
124C4×C47—44no—
125C4×C47—82no—
126C4×C47#38511yescharacteristic
127C4×C47C4×SD1611yescharacteristic
128C4×C47#44911yescharacteristic
129C4×C47#47611yescharacteristic
130Dic47—42no—
131Dic47—82no—
132C2×#441—471no—
133C2×#365—471no—
134C2×#362—471no—
135#2141D4711yescharacteristic
136#1946—471no—
137#1386—471no—
138#1375—471no—
139C2×C47—21no—
140C2×C47—21no—
141C2×C47#186411yescharacteristic
142C2×C47#218012yes—
143C22×Q16C2×D4711yescharacteristic
144C2×C4×Q8—941no—
145C2×M32—471no—
146C2×M32C2×D4711yescharacteristic
147#441—942no—
148#441—942no—
149C2×#108—471no—
150#427C2×D4711yescharacteristic
151C2×#106—471no—
152#424—941no—
153C2×#103—941no—
154C2×#97—471no—
155#365—942no—
156#362—942no—
157#358Dic4714yescomplemented
158#339—941no—
159C47#1295711yescharacteristic, complemented, Sylow
160C22×Q8—21no—
161C2×Q16—21no—
162C2×Q16—21no—
163C2×Q16—21no—
164C2×Q16Dic9412yes—
165C2×Q16D18812yes—
166C2×Q16C2×Dic4712yes—
167C2×M16—941no—
168C22×C8—471no—
169C22×C8C22×D4711yescharacteristic
170C4×Q8—944no—
171C4×Q8—1882no—
172#117—21no—
173C2×C4⋊C4—941no—
174C2×C4⋊C4—1881no—
175C2×C42—471no—
176C2×C42—1881no—
177M32—21no—
178M32—21no—
179M32—941no—
180M32—941no—
181C2×C16—941no—
182C2×C16C2×Dic4712yes—
183#108—941no—
184#108—941no—
185#107C2×Dic4712yes—
186#106—21no—
187#106—472no—
188#106—941no—
189C4⋊C8—1882no—
190#103—944no—
191#97—941no—
192#97—941no—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—41no—
199C22×C4—41no—
200C22×C4—941no—
201C22×C4—1881no—
202C22×C4D4×D4711yescharacteristic
203Q16—22no—
204Q16—22no—
205Q16—22no—
206Q16C4⋊Dic4714yes—
207M16—1881no—
208M16—1881no—
209C2×C8—941no—
210C2×C8—941no—
211C2×C8—941no—
212C2×C8—1881no—
213C2×C8C2×Dic9411yescharacteristic
214C2×C8C2×D18811yescharacteristic
215C2×C8D4×D4711yescharacteristic
216C2×C8#1062535211yescharacteristic
217C2×C8#1062535211yescharacteristic
218C2×C8C22×Dic4711yescharacteristic
219C4⋊C4—22no—
220C4⋊C4—942no—
221C4⋊C4—1884no—
222C4⋊C4—1881no—
223C22⋊C4—41no—
224C42—41no—
225C42—472no—
226C42—941no—
227C42—1884no—
228C16—22no—
229C16—1881no—
230C23#1309404811yescharacteristic
231Q8—24no—
232Q8—42no—
233Q8—42no—
234Q8—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—942no—
240C2×C4—3762no—
241C2×C4—1882no—
242C2×C4—1882no—
243C2×C4D4×Dic4711yescharacteristic
244C2×C4#1309399411yescharacteristic
245C2×C4#1309400111yescharacteristic
246C2×C4SD16×D4711yescharacteristic
247C2×C4#1309404511yescharacteristic
248C2×C4#1309404811yescharacteristic
249C8—3761no—
250C8—1881no—
251C8C2×C4⋊Dic4711yescharacteristic
252C8D4×Dic4711yescharacteristic
253C8#1309400111yescharacteristic
254C8#1309400811yescharacteristic
255C22—22no—
256C22#1568923511yescharacteristic
257C22#1568927511yescharacteristic
258C22#1568942511yescharacteristic
259C4—44no—
260C4—82no—
261C4—3762no—
262C4—1882no—
263C4#1568899811yescharacteristic
264C4SD16×Dic4711yescharacteristic
265C4#1568926911yescharacteristic
266C4#1568941111yescharacteristic
267C2—21no—
268C2—21no—
269C2#2024365111yescharacteristic
270C2#2024528012yes—
subgroupG/H
centerC22#15689235
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC47×#2141C2
socleC22×C47#442
prime radicalC23×C47AGL(1,ℤ/8)
cosocleC22×C8×C47C23
hypercenter#2141D47
hypocenterC47#12957
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2141

Complements

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

NH ≅ G/Ntype
23. C47×#358C4semidirect
157. #358Dic47semidirect
159. C47#12957semidirect

Maximal subgroups

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

subgroupindex
C2×#156892302
C2×#156889462
C2×#156889392
#202441422
#202414932
#202414672
C47×#21412
#1295747