Finite Group Databasebeta

← #26133275

Subgroups — #26133275

Order 12032.

528 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 127 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#20252969C211yesmaximal, characteristic
2C2×#15689161C211yesmaximal, characteristic, complemented
3C2×#15688934C212yesmaximal, complemented
4#20241494C212yesmaximal, complemented
5C47×#2112C211yesmaximal, characteristic, complemented
6C2×#13094046C2211yescharacteristic
7#15689634—21no—
8C2×#13093983—21no—
9#15689364C412yescomplemented
10#15689359C412yescomplemented
11#15689352—22no—
12#15689161—24no—
13#15689161C2211yescharacteristic
14C2×#13093958—21no—
15C2×#13093956C2212yes—
16#15689120—21no—
17#15688934—24no—
18C23×C8×C47C2211yescharacteristic
19C2×M32×C47C2212yes—
20C47×C22⋊C16—22no—
21C2×#10625350—21no—
22#13094046—22no—
23#13094046—22no—
24C2×Dic188C2×C411yescharacteristic
25C2×#10625325—21no—
26C2×D376C2×C411yescharacteristic
27#13094007—41no—
28#13094003—22no—
29#13093998—22no—
30#13093993—41no—
31C4×C22⋊D47—42no—
32#13093983—24no—
33#13093983—21no—
34#13093983—42no—
35C2×C4⋊Dic47—22no—
36C2×C22⋊Dic47—41no—
37C2×C4×Dic47—41no—
38#13093958—24no—
39#13093958C2×C412yes—
40#13093956—21no—
41#13093956—22no—
42#13093956D414yescomplemented
43#13093949—22no—
44#13093946—22no—
45C23×C4×C47D411yescharacteristic
46C22×C8×C47—22no—
47C22×C8×C47—22no—
48C22×C8×C47C2×C412yes—
49C22×C8×C47D411yescharacteristic
50C22×C8×C47D412yes—
51C22×C8×C47C2311yescharacteristic
52M32×C47—22no—
53M32×C47—22no—
54C2×C16×C47—22no—
55C24×C47D811yescharacteristic
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47C22⋊C412yes—
61C22×C4×C47D811yescharacteristic
62C22×C4×C47SD1612yes—
63C22×C4×C47C2×D411yescharacteristic
64C2×C22⋊D47—42no—
65C2×C22⋊D47—41no—
66C22×Dic47—42no—
67#10625350—41no—
68#10625350—41no—
69#10625350—41no—
70#10625350—41no—
71C2×D188—21no—
72C2×C4×D47—41no—
73C2×Dic94—21no—
74C2×C8×C47—21no—
75C2×C8×C47—21no—
76C2×C8×C47—24no—
77C2×C8×C47—24no—
78C2×C8×C47C22⋊C412yes—
79C2×C8×C47C22×C411yescharacteristic
80C2×C8×C47C2×D412yes—
81C2×C8×C47C2×D411yescharacteristic
82C2×C8×C47#4112yes—
83C22⋊Dic47—44no—
84C22⋊Dic47—42no—
85C22⋊Dic47—41no—
86(C2×D47)⋊C4—42no—
87C4⋊Dic47—22no—
88C4⋊Dic47—24no—
89C4⋊Dic47—42no—
90Dic47⋊C4—42no—
91C4×Dic47—44no—
92C4×Dic47—41no—
93Dic188—22no—
94D376—22no—
95#10625325—42no—
96C16×C47—42no—
97C23×C47—21no—
98C23×C47—21no—
99C23×C47—42no—
100C23×C47#10212yes—
101C23×C47C2×D811yescharacteristic
102C22×D47—41no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—22no—
106C2×C4×C47—22no—
107C2×C4×C47—22no—
108C2×C4×C47—22no—
109C2×C4×C47#10212yes—
110C2×C4×C47C2×C22⋊C411yescharacteristic
111C2×C4×C47#12011yescharacteristic
112C2×C4×C47#12311yescharacteristic
113C2×C4×C47C2×SD1611yescharacteristic
114C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
115C2×C4×C47#13711yescharacteristic
116C22⋊D47—84no—
117C22⋊D47—81no—
118C22⋊D47—81no—
119C2×Dic47—42no—
120C2×Dic47—41no—
121C2×Dic47—44no—
122C2×Dic47—84no—
123D188—41no—
124D188—41no—
125C4×D47—81no—
126C4×D47—81no—
127Dic94—41no—
128Dic94—41no—
129C8×C47—22no—
130C8×C47C4×D412yes—
131C8×C47#12112yes—
132#13002—471nomaximal, Sylow
133C22×C47—22no—
134C22×C47—22no—
135C22×C47—22no—
136C22×C47—22no—
137C22×C47—42no—
138C22×C47—42no—
139C22×C47C2×#10211yescharacteristic
140C22×C47#44711yescharacteristic
141C22×C47#48011yescharacteristic
142C2×D47—81no—
143C2×D47—82no—
144C4×C47—22no—
145C4×C47#38611yescharacteristic
146C4×C47#41711yescharacteristic
147C4×C47#44811yescharacteristic
148C4×C47#48111yescharacteristic
149Dic47—84no—
150Dic47—81no—
151Dic47—82no—
152#3050—471no—
153C2×#427—471no—
154C2×#360—472no—
155#2112D4711yescharacteristic, complemented
156#1386—472no—
157C2×C47—21no—
158C2×C47—21no—
159C2×C47—42no—
160C2×C47#187411yescharacteristic
161C2×C47#214512yes—
162D47—161no—
163C2×#135—471no—
164C23×C8C2×D4711yescharacteristic
165#525—941no—
166C2×#117—941no—
167C2×M32C2×D4712yes—
168#467—472no—
169#466—472no—
170#465—942no—
171#427—471no—
172#427—944no—
173C2×#107—941no—
174C2×#106—472no—
175#416—941no—
176#360—944no—
177C22⋊C16—22no—
178C47#1300211yescharacteristic, complemented, Sylow
179C2×#41—941no—
180C23×C4D18811yescharacteristic
181#135—942no—
182#135—942no—
183C2×Q16—471no—
184C2×SD16—941no—
185C2×D8—471no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8C4×D4712yes—
189C22×C8D18811yescharacteristic
190C22×C8C22⋊D4712yes—
191C22×C8C22×D4711yescharacteristic
192#123—1881no—
193C22⋊Q8—942no—
194#121—942no—
195#120—1881no—
196C4×D4—1882no—
197#117—944no—
198#117—941no—
199#117—1882no—
200C2×C4⋊C4—942no—
201C2×C22⋊C4—1881no—
202C2×C42—1881no—
203M32—22no—
204M32—22no—
205C2×C16—22no—
206#107—472no—
207#107—944no—
208#106—474no—
209#106—942no—
210#106—941no—
211#103—942no—
212#102—942no—
213C24D37611yescharacteristic
214#41—1881no—
215#41—1881no—
216#41—1881no—
217#41—1881no—
218C2×Q8—941no—
219C2×D4—941no—
220C2×D4—1882no—
221C2×D4—1881no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—1882no—
227C22×C4—1881no—
228C22×C4#1062532512yes—
229C22×C4D37611yescharacteristic
230C22×C4(C2×D47)⋊C412yes—
231C22×C4C2×D18811yescharacteristic
232Q16—942no—
233SD16—1882no—
234D8—942no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—24no—
238C2×C8—24no—
239C2×C8(C2×D47)⋊C412yes—
240C2×C8C2×C4×D4711yescharacteristic
241C2×C8D4×D4712yes—
242C2×C8#1062535212yes—
243C2×C8C2×C22⋊D4711yescharacteristic
244C4⋊C4—942no—
245C4⋊C4—944no—
246C4⋊C4—1882no—
247C4⋊C4—1882no—
248C22⋊C4—1884no—
249C22⋊C4—1882no—
250C22⋊C4—1881no—
251C22⋊C4—1882no—
252C42—1884no—
253C42—1881no—
254C16—42no—
255C23—21no—
256C23—21no—
257C23—42no—
258C23—1881no—
259C23#1309394612yes—
260C23C2×D37611yescharacteristic
261Q8—1881no—
262Q8—1881no—
263D4—3764no—
264D4—3761no—
265D4—3761no—
266D4—1881no—
267D4—1881no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—3764no—
275C2×C4—3761no—
276C2×C4—3761no—
277C2×C4—1884no—
278C2×C4—1882no—
279C2×C4—1881no—
280C2×C4#1309394612yes—
281C2×C4C2×(C2×D47)⋊C411yescharacteristic
282C2×C4#1309399411yescharacteristic
283C2×C4#1309400811yescharacteristic
284C2×C4C2×#1062532511yescharacteristic
285C2×C4#1309405211yescharacteristic
286C2×C4#1309405711yescharacteristic
287C8—22no—
288C8(C22⋊D47)⋊C412yes—
289C8#1309399612yes—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—42no—
295C22—42no—
296C22—3762no—
297C22—3761no—
298C22C2×#1309394611yescharacteristic
299C22#1568925911yescharacteristic
300C22#1568943711yescharacteristic
301C4—22no—
302C4—3764no—
303C4—3762no—
304C4—3761no—
305C4#1568900011yescharacteristic
306C4#1568912511yescharacteristic
307C4#1568926411yescharacteristic
308C4#1568944211yescharacteristic
309C2—21no—
310C2—21no—
311C2—42no—
312C2—7521no—
313C2#2024370311yescharacteristic
314C2#2024512512yes—
subgroupG/H
centerC22C2×#13093946
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC47×#2112C2
socleC22×C47C2×#102
prime radical#20252969C2
cosocleC22×C8×C47C23
hypercenter#2112D47
hypocenterC47#13002
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #2112

Complements

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

NH ≅ G/Ntype
2. C2×#15689161C2semidirect
3. C2×#15688934C2semidirect
4. #20241494C2semidirect
5. C47×#2112C2semidirect
9. #15689364C4semidirect
10. #15689359C4semidirect
42. #13093956D4semidirect
155. #2112D47semidirect
178. C47#13002semidirect

Maximal subgroups

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

subgroupindex
#202529692
C2×#156891612
C2×#156889342
#202414942
C47×#21122
#1300247