Finite Group Databasebeta

← #26107328

Subgroups — #26107328

Order 12032.

436 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252357C211yesmaximal, characteristic
2#20243075C211yesmaximal, characteristic, complemented
3#20241977C211yesmaximal, characteristic
4#20241872C211yesmaximal, characteristic
5#20241675C211yesmaximal, characteristic, complemented
6#20241286C211yesmaximal, characteristic, complemented
7C47×#1300C211yesmaximal, characteristic, complemented
8#15689657C2211yescharacteristic
9#15689250C411yescharacteristic, complemented
10#15689247C411yescharacteristic, complemented
11#15689200C411yescharacteristic, complemented
12#15689191—21no—
13#15689188C411yescharacteristic, complemented
14(C22⋊D47)⋊C8—21no—
15#15689185—21no—
16#15689176C2211yescharacteristic
17#15689080—21no—
18C2×#13093935—21no—
19C2×#13093935C2211yescharacteristic
20C2×C4×C47⋊C8—21no—
21C2×C4×C47⋊C8C2211yescharacteristic
22#15688864C412yescomplemented
23#15688862C412yescomplemented
24#15688859C412yes—
25#15688856C412yes—
26#15688845—21no—
27#15688845—21no—
28C47×#432C2211yescharacteristic
29C4×M16×C47C2211yescharacteristic
30C47×#376C2211yescharacteristic
31C2×#10625324—21no—
32C22×C47⋊C8—21no—
33C22×C47⋊C8D411yescharacteristic
34C22×C47⋊C8D411yescharacteristic
35C2×C8×D47—21no—
36#13094018—21no—
37#13094016C2×C411yescharacteristic
38#13094012C2×C411yescharacteristic
39#13094007—21no—
40#13094003C2×C411yescharacteristic
41#13093998C2×C411yescharacteristic
42C4×Dic94C2×C411yescharacteristic
43C4×D188C2×C411yescharacteristic
44C4×C22⋊D47—21no—
45#13093981—21no—
46C4⋊(C47⋊C8)—21no—
47Dic47⋊C8—21no—
48Dic47⋊C8—21no—
49C22⋊(C47⋊C8)C2×C412yes—
50(C2×D47)⋊C8—21no—
51(C2×D47)⋊C8—21no—
52#13093935—24no—
53#13093935—21no—
54#13093935C2×C412yes—
55#13093934—21no—
56C4×C47⋊C8—24no—
57C4×C47⋊C8—21no—
58C4×C47⋊C8C2×C411yescharacteristic
59C4×C47⋊C8C2×C411yescharacteristic
60C4×C47⋊C8D411yescharacteristic
61C4×C47⋊C8D411yescharacteristic
62C8×Dic47—21no—
63C2×M16×C47—21no—
64C2×C42×C47—21no—
65C2×C42×C47C2311yescharacteristic
66C4⋊C8×C47C2×C411yescharacteristic
67C4⋊C8×C47C2×C411yescharacteristic
68C22⋊C8×C47—21no—
69#97×C47—21no—
70#97×C47C2×C412yes—
71C4×C8×C47—21no—
72#95×C47—21no—
73#95×C47C2×C412yes—
74C22×C4×C47—22no—
75C22×C4×C47C22×C411yescharacteristic
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47#4111yescharacteristic
78C2×C22⋊D47—21no—
79C2×D188C4211yescharacteristic
80C2×C4×D47—21no—
81C2×Dic94C4211yescharacteristic
82M16×C47—42no—
83C2×C8×C47—21no—
84C2×C8×C47—21no—
85C2×C8×C47—22no—
86C42×C47—22no—
87C42×C47—22no—
88C42×C47C22×C411yescharacteristic
89C42×C47C22×C411yescharacteristic
90C42×C47C2×D411yescharacteristic
91C42×C47#4111yescharacteristic
92C22⋊Dic47—21no—
93(C2×D47)⋊C4—21no—
94(C2×D47)⋊C4—21no—
95(C2×D47)⋊C4—21no—
96(C2×D47)⋊C4—21no—
97C4⋊Dic47C4211yescharacteristic
98C4⋊Dic47C4211yescharacteristic
99Dic47⋊C4—21no—
100Dic47⋊C4—21no—
101Dic47⋊C4—21no—
102Dic47⋊C4—21no—
103C4×Dic47—21no—
104C2×C47⋊C8—22no—
105C2×C47⋊C8—22no—
106C2×C47⋊C8—22no—
107C2×C47⋊C8—22no—
108C2×C47⋊C8—22no—
109C2×C47⋊C8—22no—
110C2×C47⋊C8C22⋊C411yescharacteristic
111C2×C47⋊C8C22⋊C411yescharacteristic
112C2×C47⋊C8C22⋊C411yescharacteristic
113C2×C47⋊C8C22⋊C411yescharacteristic
114#10625324—42no—
115C8×D47—42no—
116C23×C47C4×D411yescharacteristic
117C22×D47—21no—
118C2×C4×C47—22no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—24no—
122C2×C4×C47—41no—
123C2×C4×C47—42no—
124C2×C4×C47—42no—
125C2×C4×C47C2×C4211yescharacteristic
126C2×C4×C47C2×C22⋊C411yescharacteristic
127C2×C4×C47#11711yescharacteristic
128C2×C4×C47C4×D411yescharacteristic
129C2×C4×C47C4×D411yescharacteristic
130C2×C4×C47C4×D411yescharacteristic
131C22⋊D47—42no—
132C2×Dic47—21no—
133C2×Dic47—21no—
134C2×Dic47—21no—
135D188—41no—
136C4×D47—41no—
137C4×D47—41no—
138Dic94—41no—
139C8×C47—42no—
140C8×C47—41no—
141C8×C47—41no—
142C47⋊C8—24no—
143C47⋊C8—21no—
144C47⋊C8—21no—
145C47⋊C8—42no—
146C47⋊C8#10412yes—
147C47⋊C8#10412yes—
148#8249—471nomaximal, Sylow
149C22×C47—21no—
150C22×C47—21no—
151C22×C47C4×C22⋊C411yescharacteristic
152C22×C47#44412yes—
153C2×D47—41no—
154C2×D47—41no—
155C2×D47—41no—
156C4×C47—21no—
157C4×C47—21no—
158C4×C47—21no—
159C4×C47—21no—
160C4×C47—21no—
161C4×C47—44no—
162C4×C47C2×#10411yescharacteristic
163C4×C47C2×#10411yescharacteristic
164Dic47—41no—
165Dic47—41no—
166Dic47—41no—
167#2966—471no—
168#1726—471no—
169#1524—471no—
170#1500—471no—
171#1451—471no—
172#1302—471no—
173#1300D4711yescharacteristic, complemented
174C2×C47—22no—
175C2×C47#175711yescharacteristic
176C2×C47C4×#10411yescharacteristic
177C2×C47#176511yescharacteristic
178D47—81no—
179#529—471no—
180#446—471no—
181C8×Q8—471no—
182#436—471no—
183#435—941no—
184C8×D4—471no—
185C8×D4—941no—
186#433—941no—
187#432C2×D4711yescharacteristic
188#431—471no—
189#405—941no—
190C4×M16C2×D4711yescharacteristic
191C2×#97—471no—
192C2×#97—941no—
193C2×C4×C8—471no—
194C2×C4×C8—941no—
195#376C2×D4711yescharacteristic
196#330—472no—
197#329—472no—
198#328—472no—
199#327—472no—
200#322—941no—
201#322—941no—
202C47#824911yescharacteristic, complemented, Sylow
203C2×M16—21no—
204C2×M16—941no—
205C22×C8—471no—
206C22×C8—471no—
207C22×C8—941no—
208C22×C8—941no—
209#126—941no—
210#125—471no—
211#124—471no—
212#123—941no—
213C22⋊Q8—471no—
214#121—471no—
215C4×Q8—471no—
216C4×D4—471no—
217C4×D4—941no—
218#117—941no—
219C2×C42—21no—
220C2×C42C22×D4711yescharacteristic
221C4⋊C8—941no—
222C4⋊C8—941no—
223C4⋊C8—941no—
224C4⋊C8C4×D4711yescharacteristic
225C4⋊C8C4×D4711yescharacteristic
226C22⋊C8—21no—
227C22⋊C8—472no—
228C22⋊C8—941no—
229C22⋊C8—941no—
230#97—21no—
231#97—472no—
232#97—944no—
233#97—941no—
234#97—941no—
235#97C4×D4712yes—
236C4×C8—21no—
237C4×C8—471no—
238C4×C8—471no—
239C4×C8—471no—
240C4×C8—471no—
241C4×C8—944no—
242C4×C8—941no—
243C4×C8—941no—
244#95—21no—
245#95C4×D4712yes—
246C2×Q8—471no—
247C2×D4—471no—
248C2×D4—941no—
249C22×C4—22no—
250C22×C4—941no—
251C22×C4C2×C4×D4711yescharacteristic
252C22×C4D4×D4711yescharacteristic
253C22×C4#1062535211yescharacteristic
254M16—42no—
255M16—1882no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—22no—
259C2×C8—471no—
260C2×C8—471no—
261C2×C8—471no—
262C2×C8—471no—
263C2×C8—942no—
264C2×C8—942no—
265C2×C8—942no—
266C2×C8—942no—
267C2×C8—942no—
268C2×C8—942no—
269C2×C8—1882no—
270C4⋊C4—471no—
271C4⋊C4—471no—
272C4⋊C4—941no—
273C4⋊C4—941no—
274C4⋊C4—941no—
275C4⋊C4—941no—
276C22⋊C4—941no—
277C22⋊C4—941no—
278C22⋊C4—941no—
279C22⋊C4—941no—
280C22⋊C4—941no—
281C42—22no—
282C42—22no—
283C42—941no—
284C42C2×C4×D4711yescharacteristic
285C42C2×C4×D4711yescharacteristic
286C42D4×D4711yescharacteristic
287C42#1062535411yescharacteristic
288C23—941no—
289C23(C22⋊D47)⋊C411yescharacteristic
290Q8—1881no—
291D4—1881no—
292D4—1882no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—24no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—941no—
301C2×C4—941no—
302C2×C4—941no—
303C2×C4—1881no—
304C2×C4—1881no—
305C2×C4C42×D4711yescharacteristic
306C2×C4C22⋊C4×D4711yescharacteristic
307C2×C4#1309398211yescharacteristic
308C2×C4(C22⋊D47)⋊C411yescharacteristic
309C2×C4#1309398611yescharacteristic
310C2×C4#1309398611yescharacteristic
311C8—42no—
312C8—41no—
313C8—41no—
314C8—472no—
315C8—472no—
316C8—944no—
317C8—941no—
318C8—941no—
319C8—1882no—
320C22—21no—
321C22—21no—
322C22—1881no—
323C22—1881no—
324C22—1881no—
325C22#1568897011yescharacteristic
326C22#1568924212yes—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—44no—
333C4—1881no—
334C4—1881no—
335C4—1881no—
336C4#104×D4711yescharacteristic
337C4#104×D4711yescharacteristic
338C2—22no—
339C2—3761no—
340C2#2024318711yescharacteristic
341C2#2024319211yescharacteristic
342C2#2024322911yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D47
commutatorC2×C4×C47C2×C42
FrattiniC2×C42C22×D47
FittingC47×#1300C2
socleC22×C47C4×C22⋊C4
prime radical#13093998C2×C4
cosocleC2×C42×C47C23
hypercenter#1300D47
hypocenterC47#8249
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1300

Complements

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

NH ≅ G/Ntype
2. #20243075C2semidirect
5. #20241675C2semidirect
6. #20241286C2semidirect
7. C47×#1300C2semidirect
9. #15689250C4semidirect
10. #15689247C4semidirect
11. #15689200C4semidirect
13. #15689188C4semidirect
22. #15688864C4semidirect
23. #15688862C4semidirect
173. #1300D47semidirect
202. C47#8249semidirect

Maximal subgroups

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

subgroupindex
#202523572
#202430752
#202419772
#202418722
#202416752
#202412862
C47×#13002
#824947