Finite Group Databasebeta

← #26134157

Subgroups — #26134157

Order 12032.

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

normal subgroups only

This group has 356 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
1C2×#15688945C211yesmaximal, characteristic, complemented
2C2×#15688937C211yesmaximal, characteristic
3#20245103C211yesmaximal, characteristic
4#20244262C211yesmaximal, characteristic
5#20243274C211yesmaximal, characteristic, complemented
6#20241467C211yesmaximal, characteristic, complemented
7C47×#1386C211yesmaximal, characteristic, complemented
8C22×D376C2211yescharacteristic
9C2×#13093964C2211yescharacteristic
10C2×#13093960C2211yescharacteristic
11C2×#13093944—21no—
12#15689079—21no—
13C2×#13093934C2211yescharacteristic
14#15689024—21no—
15#15688945—22no—
16#15688937—22no—
17#15688923C414yescomplemented
18#15688882—21no—
19C2×M32×C47C2211yescharacteristic
20C47×#427C2211yescharacteristic
21C2×#106×C47C2211yescharacteristic
22C22×D188—21no—
23C2×D376—21no—
24C2×D376—21no—
25C2×D376—21no—
26C2×D376—21no—
27C2×D376—21no—
28C2×D376C2×C412yes—
29C2×#10625329—21no—
30C22×C47⋊C8D411yescharacteristic
31C2×(C2×D47)⋊C4—42no—
32C2×C4×Dic47D411yescharacteristic
33#13093964—21no—
34#13093964—21no—
35C2×C47⋊C16C2×C412yes—
36#13093960—21no—
37#13093960D412yes—
38#13093944—42no—
39#13093935—41no—
40#13093934—21no—
41#13093934—21no—
42C4×C47⋊C8—41no—
43C22×C8×C47C2311yescharacteristic
44#117×C47—21no—
45C2×C4⋊C4×C47—21no—
46M32×C47—21no—
47M32×C47—21no—
48C2×C16×C47—21no—
49#107×C47C2×C412yes—
50#106×C47—21no—
51#106×C47—21no—
52#106×C47—21no—
53C23×D47—41no—
54C22×C4×C47—41no—
55C22×C4×C47C2×D411yescharacteristic
56C22×Dic47—21no—
57C2×D188—22no—
58C2×D188—41no—
59C2×D188—41no—
60C2×D188—41no—
61C2×D188—41no—
62C2×D188—41no—
63C2×C8×C47C22×C411yescharacteristic
64C2×C8×C47C2×D411yescharacteristic
65C2×C8×C47#4111yescharacteristic
66C2×C8×C47#4111yescharacteristic
67C2×C8×C47#4111yescharacteristic
68C2×C8×C47#4111yescharacteristic
69C4⋊C4×C47—22no—
70C4⋊C4×C47—41no—
71C4⋊C4×C47—41no—
72C22⋊C4×C47—41no—
73C42×C47—41no—
74(C2×D47)⋊C4—84no—
75C4×Dic47—21no—
76C4×Dic47—21no—
77#10625329—22no—
78#10625329—41no—
79C2×C47⋊C8—21no—
80C2×C47⋊C8—21no—
81C2×C47⋊C8—41no—
82C2×C47⋊C8SD1612yes—
83D376—22no—
84D376—22no—
85D376—41no—
86D376—41no—
87C16×C47—41no—
88C47⋊C16—22no—
89C23×C47AGL(1,ℤ/8)11yescharacteristic
90C22×D47—42no—
91C22×D47—82no—
92C22×D47—82no—
93C22×D47—81no—
94C22×D47—81no—
95C2×C4×C47—42no—
96C2×C4×C47—82no—
97C2×C4×C47—81no—
98C2×C4×C47C4×D411yescharacteristic
99C2×C4×C47#12111yescharacteristic
100C2×C4×C47#12311yescharacteristic
101C2×C4×C47C2×SD1611yescharacteristic
102C2×C4×C47#13511yescharacteristic
103C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
104C2×Dic47—42no—
105C2×Dic47—41no—
106D188—42no—
107D188—42no—
108D188—81no—
109D188—81no—
110C8×C47#11711yescharacteristic
111C8×C47C4×D411yescharacteristic
112C8×C47#12411yescharacteristic
113C8×C47#12611yescharacteristic
114C47⋊C8—22no—
115C47⋊C8—42no—
116#13142—471nomaximal, Sylow
117C22×C47—22no—
118C22×C47#44211yescharacteristic
119C22×C47#46011yescharacteristic
120C22×C47#48211yescharacteristic
121C2×D47—84no—
122C2×D47—82no—
123C2×D47—162no—
124C2×D47—161no—
125C2×D47—161no—
126C4×C47—82no—
127C4×C47—161no—
128C4×C47#38811yescharacteristic
129C4×C47C4×SD1611yescharacteristic
130C4×C47#46311yescharacteristic
131C4×C47#48111yescharacteristic
132Dic47—81no—
133C2×#365—471no—
134C2×#361—471no—
135#2140—471no—
136#1972—471no—
137#1773—471no—
138#1386D4711yescharacteristic, complemented
139#1375—471no—
140C2×C47—21no—
141C2×C47—21no—
142C2×C47#192611yescharacteristic
143C2×C47AGL(1,ℤ/16)12yes—
144D47—162no—
145C22×D8—471no—
146C2×M32—471no—
147C2×M32C2×D4711yescharacteristic
148C2×#108—471no—
149#427C2×D4711yescharacteristic
150C2×#106C2×D4711yescharacteristic
151C2×#102—941no—
152#405—941no—
153C2×#97—471no—
154#390—941no—
155#365—942no—
156#361—942no—
157#357—474no—
158#339—941no—
159C47#1314211yescharacteristic, complemented, Sylow
160C22×D4—941no—
161C2×D8—472no—
162C2×D8—941no—
163C2×D8—941no—
164C2×D8—941no—
165C2×D8—941no—
166C2×D8—941no—
167C2×M16—941no—
168C22×C8—471no—
169C22×C8C22×D4711yescharacteristic
170#117—21no—
171C2×C4⋊C4—21no—
172C2×C22⋊C4—1882no—
173C2×C42—471no—
174M32—21no—
175M32—21no—
176M32—941no—
177M32—941no—
178C2×C16—21no—
179C2×C16—472no—
180#108—472no—
181#108—941no—
182#107C4×D4712yes—
183#106—21no—
184#106—21no—
185#106—21no—
186#102—1882no—
187#97—941no—
188#97—941no—
189#97—1881no—
190C4×C8—1881no—
191C24—1881no—
192C2×D4—942no—
193C2×D4—1881no—
194C2×D4—1881no—
195C2×D4—1881no—
196C2×D4—1881no—
197C2×D4—1881no—
198C22×C4—41no—
199C22×C4—941no—
200C22×C4D4×D4711yescharacteristic
201D8—942no—
202D8—942no—
203D8—1881no—
204D8—1881no—
205M16—942no—
206M16—1881no—
207C2×C8—472no—
208C2×C8—941no—
209C2×C8—941no—
210C2×C8—1881no—
211C2×C8C2×C4×D4711yescharacteristic
212C2×C8#1062535011yescharacteristic
213C2×C8#1062535011yescharacteristic
214C2×C8D4×D4711yescharacteristic
215C2×C8#1062535211yescharacteristic
216C2×C8#1062535411yescharacteristic
217C4⋊C4—22no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C22⋊C4—41no—
221C22⋊C4—3764no—
222C42—41no—
223C42—941no—
224C42—941no—
225C16—41no—
226C16—942no—
227C23—3762no—
228C23—3762no—
229C23—3761no—
230C23—3761no—
231C23—1882no—
232C23#1309404811yescharacteristic
233D4—3761no—
234D4—3761no—
235D4—1882no—
236D4—1882no—
237C2×C4—42no—
238C2×C4—82no—
239C2×C4—81no—
240C2×C4—1882no—
241C2×C4—1881no—
242C2×C4(C22⋊D47)⋊C411yescharacteristic
243C2×C4#1309399711yescharacteristic
244C2×C4#1309400911yescharacteristic
245C2×C4SD16×D4711yescharacteristic
246C2×C4#1309404711yescharacteristic
247C2×C4#1309405111yescharacteristic
248C8—942no—
249C8—1882no—
250C8#1309398111yescharacteristic
251C8#1309398611yescharacteristic
252C8#1309401011yescharacteristic
253C8#1309401911yescharacteristic
254C22—22no—
255C22—3764no—
256C22—3762no—
257C22—7521no—
258C22—7521no—
259C22—7522no—
260C22#1568923211yescharacteristic
261C22#1568932211yescharacteristic
262C22#1568945011yescharacteristic
263C4—82no—
264C4—161no—
265C4—3761no—
266C4#1568901211yescharacteristic
267C4#1568920811yescharacteristic
268C4#1568934111yescharacteristic
269C4#1568944311yescharacteristic
270C2—21no—
271C2—21no—
272C2—7522no—
273C2#2024401311yescharacteristic
274C2#2024528312yes—
subgroupG/H
centerC22#15689232
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC47×#1386C2
socleC22×C47#442
prime radicalC22×D376C22
cosocleC22×C8×C47C23
hypercenter#1386D47
hypocenterC47#13142
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1386

Complements

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

NH ≅ G/Ntype
1. C2×#15688945C2semidirect
5. #20243274C2semidirect
6. #20241467C2semidirect
7. C47×#1386C2semidirect
17. #15688923C4semidirect
138. #1386D47semidirect
159. C47#13142semidirect

Maximal subgroups

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

subgroupindex
C2×#156889452
C2×#156889372
#202451032
#202442622
#202432742
#202414672
C47×#13862
#1314247