Finite Group Databasebeta

← #26107341

Subgroups — #26107341

Order 12032.

400 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252364C211yesmaximal, characteristic
2#20243141C211yesmaximal, characteristic, complemented
3#20241874C211yesmaximal, characteristic
4M16×C47⋊C8C211yesmaximal, characteristic, complemented
5#20241287C211yesmaximal, characteristic, complemented
6#20241282C211yesmaximal, characteristic, complemented
7C47×#1524C211yesmaximal, characteristic
8#15689254C411yescharacteristic, complemented
9Q8×C47⋊C8C411yescharacteristic, complemented
10#15689201C411yescharacteristic, complemented
11#15689195—21no—
12D4×C47⋊C8—21no—
13D4×C47⋊C8C411yescharacteristic, complemented
14#15689184—21no—
15#15689180C2211yescharacteristic
16#15689176C2211yescharacteristic
17#15689079—21no—
18C2×C4×C47⋊C8C2211yescharacteristic
19#15688969C2211yescharacteristic
20#15688877—21no—
21#15688877—21no—
22#15688860C412yes—
23#15688857C412yes—
24#15688844—21no—
25C8×C47⋊C8—21no—
26C47×#529C2211yescharacteristic
27C4×M16×C47C2211yescharacteristic
28C2×#97×C47C2211yescharacteristic
29C47×#330C412yescomplemented
30C47×#329C412yescomplemented
31C2×#10625329—21no—
32C22×C47⋊C8—21no—
33C22×C47⋊C8D411yescharacteristic
34C22×C47⋊C8D411yescharacteristic
35C2×C4×Dic47—21no—
36C4⋊(C47⋊C8)—21no—
37C4⋊(C47⋊C8)—21no—
38C4⋊(C47⋊C8)—21no—
39C4⋊(C47⋊C8)C2×C411yescharacteristic
40C4⋊(C47⋊C8)C2×C411yescharacteristic
41C22⋊(C47⋊C8)—21no—
42C22⋊(C47⋊C8)—21no—
43C22⋊(C47⋊C8)—21no—
44C22⋊(C47⋊C8)C2×C412yes—
45#13093935—21no—
46#13093935—21no—
47C4×C47⋊C8—21no—
48C4×C47⋊C8—21no—
49C4×C47⋊C8C2×C411yescharacteristic
50C4×C47⋊C8C2×C411yescharacteristic
51C4×C47⋊C8D411yescharacteristic
52C4×C47⋊C8D411yescharacteristic
53C47⋊#95—21no—
54C47⋊#95C2×C412yes—
55C2×M16×C47—21no—
56C22×C8×C47—21no—
57#126×C47—21no—
58#125×C47C2×C411yescharacteristic
59#124×C47C2×C411yescharacteristic
60#123×C47—21no—
61C22⋊Q8×C47C2×C411yescharacteristic
62#121×C47C2×C411yescharacteristic
63C4×Q8×C47C2×C411yescharacteristic
64C4×D4×C47—21no—
65C4×D4×C47C2×C411yescharacteristic
66#117×C47—21no—
67C2×C42×C47C2311yescharacteristic
68#97×C47—21no—
69#97×C47C2×C412yes—
70#97×C47C2×C412yes—
71C4×C8×C47—21no—
72C2×Q8×C47C4211yescharacteristic
73C2×D4×C47—21no—
74C2×D4×C47C4211yescharacteristic
75C22×C4×C47—21no—
76C22×C4×C47C22×C411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C22×C4×C47#4111yescharacteristic
79C22×Dic47—22no—
80M16×C47—24no—
81C2×C8×C47—22no—
82C2×C8×C47—22no—
83C2×C8×C47—41no—
84C2×C8×C47—41no—
85C4⋊C4×C47—21no—
86C4⋊C4×C47—21no—
87C4⋊C4×C47—21no—
88C4⋊C4×C47—21no—
89C4⋊C4×C47C4211yescharacteristic
90C4⋊C4×C47C4211yescharacteristic
91C22⋊C4×C47—21no—
92C22⋊C4×C47—21no—
93C22⋊C4×C47—21no—
94C22⋊C4×C47—21no—
95C22⋊C4×C47—21no—
96C42×C47—21no—
97C42×C47C22×C411yescharacteristic
98C42×C47C22×C411yescharacteristic
99C42×C47C2×D411yescharacteristic
100C42×C47#4111yescharacteristic
101C4×Dic47—41no—
102C4×Dic47—41no—
103#10625329—42no—
104C2×C47⋊C8—21no—
105C2×C47⋊C8—21no—
106C2×C47⋊C8—22no—
107C2×C47⋊C8—41no—
108C2×C47⋊C8—41no—
109C2×C47⋊C8—42no—
110C2×C47⋊C8C22⋊C411yescharacteristic
111C2×C47⋊C8C22⋊C411yescharacteristic
112C2×C47⋊C8C22⋊C411yescharacteristic
113C2×C47⋊C8C22⋊C411yescharacteristic
114C23×C47—21no—
115C23×C47C4×D411yescharacteristic
116Q8×C47—41no—
117D4×C47—41no—
118D4×C47—42no—
119C2×C4×C47—22no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47C2×C4211yescharacteristic
129C2×C4×C47C2×C22⋊C411yescharacteristic
130C2×C4×C47#11711yescharacteristic
131C2×C4×C47C4×D411yescharacteristic
132C2×C4×C47C4×D411yescharacteristic
133C2×C4×C47C4×D411yescharacteristic
134C2×Dic47—42no—
135C2×Dic47—42no—
136C2×Dic47—42no—
137C8×C47—24no—
138C8×C47—42no—
139C47⋊C8—21no—
140C47⋊C8—21no—
141C47⋊C8—21no—
142C47⋊C8—21no—
143C47⋊C8—42no—
144C47⋊C8—41no—
145C47⋊C8—41no—
146#8250—471nomaximal, Sylow
147C22×C47—21no—
148C22×C47—21no—
149C22×C47—41no—
150C22×C47—41no—
151C22×C47—41no—
152C22×C47C4×C22⋊C411yescharacteristic
153C22×C47#44312yes—
154C4×C47—21no—
155C4×C47—21no—
156C4×C47—21no—
157C4×C47—21no—
158C4×C47—21no—
159C4×C47—41no—
160C4×C47—41no—
161C4×C47—41no—
162C4×C47#42111yescharacteristic
163C4×C47#42111yescharacteristic
164Dic47—82no—
165#2966—471no—
166#1746—471no—
167#1524D4711yescharacteristic
168#1500—471no—
169C8×M16—471no—
170#1302—471no—
171#1300—471no—
172C2×C47—22no—
173C2×C47—81no—
174C2×C47#175711yescharacteristic
175C2×C47#175911yescharacteristic
176C2×C47#176411yescharacteristic
177#529C2×D4711yescharacteristic
178#446—471no—
179C8×Q8—471no—
180#436—471no—
181#435—941no—
182C8×D4—471no—
183C8×D4—941no—
184#433—941no—
185#432—471no—
186#431—471no—
187#405—941no—
188C4×M16C2×D4711yescharacteristic
189C2×#97C2×D4711yescharacteristic
190C2×C4×C8—471no—
191#376—471no—
192#336—941no—
193#336—941no—
194#330Dic4712yescomplemented
195#329Dic4712yescomplemented
196#328—472no—
197#327—472no—
198#322—941no—
199C82—941no—
200C47#825011yescharacteristic, complemented, Sylow
201C2×M16—21no—
202C2×M16—941no—
203C22×C8—21no—
204C22×C8—471no—
205C22×C8—471no—
206C22×C8—941no—
207#126—21no—
208#125C2×Dic4711yescharacteristic
209#124C2×Dic4711yescharacteristic
210#123—21no—
211C22⋊Q8C4×D4711yescharacteristic
212#121C4×D4711yescharacteristic
213C4×Q8C4×D4711yescharacteristic
214C4×D4—21no—
215C4×D4C4×D4711yescharacteristic
216#117—21no—
217C2×C42—941no—
218C2×C42C22×D4711yescharacteristic
219C4⋊C8—471no—
220C4⋊C8—471no—
221C4⋊C8—941no—
222C4⋊C8—941no—
223C4⋊C8—941no—
224C22⋊C8—472no—
225C22⋊C8—941no—
226C22⋊C8—941no—
227C22⋊C8—941no—
228#97—21no—
229#97—941no—
230#97—941no—
231#97C2×Dic4712yes—
232#97C2×Dic4712yes—
233C4×C8—21no—
234C4×C8—471no—
235C4×C8—471no—
236C4×C8—471no—
237C4×C8—471no—
238C4×C8—941no—
239C4×C8—941no—
240#95—472no—
241#95—941no—
242C2×Q8C4×Dic4711yescharacteristic
243C2×D4—21no—
244C2×D4C4×Dic4711yescharacteristic
245C22×C4—21no—
246C22×C4—942no—
247C22×C4C2×C4×D4711yescharacteristic
248C22×C4D4×D4711yescharacteristic
249C22×C4#1062535211yescharacteristic
250M16—24no—
251M16—1882no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—471no—
257C2×C8—471no—
258C2×C8—471no—
259C2×C8—471no—
260C2×C8—941no—
261C2×C8—941no—
262C2×C8—942no—
263C2×C8—1881no—
264C2×C8—1881no—
265C2×C8—1882no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4C4×Dic4711yescharacteristic
271C4⋊C4C4×Dic4711yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C42—21no—
278C42—1881no—
279C42—1881no—
280C42C2×C4×D4711yescharacteristic
281C42D4×D4711yescharacteristic
282C42#1062535211yescharacteristic
283C42C22×Dic4711yescharacteristic
284C23—21no—
285C23(C22⋊D47)⋊C411yescharacteristic
286Q8—41no—
287D4—41no—
288D4—42no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—1882no—
299C2×C4—1882no—
300C2×C4—1882no—
301C2×C4C2×C4×Dic4711yescharacteristic
302C2×C4C22⋊C4×D4711yescharacteristic
303C2×C4#1309398011yescharacteristic
304C2×C4(C22⋊D47)⋊C411yescharacteristic
305C2×C4D4×Dic4711yescharacteristic
306C2×C4D4×Dic4711yescharacteristic
307C8—24no—
308C8—42no—
309C8—941no—
310C8—941no—
311C8—941no—
312C8—941no—
313C8—1882no—
314C8—1881no—
315C8—1881no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22C22⋊C4×Dic4711yescharacteristic
322C22#1568923712yes—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—3762no—
332C4#1568913811yescharacteristic
333C4#1568913811yescharacteristic
334C2—22no—
335C2—81no—
336C2#2024319111yescharacteristic
337C2#2024320011yescharacteristic
338C2#2024322611yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D47
commutatorC2×C4×C47C2×C42
FrattiniC2×C42C22×D47
FittingC47×#1524C2
socleC22×C47C4×C22⋊C4
prime radical#121×C47C2×C4
cosocleC2×C42×C47C23
hypercenter#1524D47
hypocenterC47#8250
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1524

Complements

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

NH ≅ G/Ntype
2. #20243141C2semidirect
4. M16×C47⋊C8C2semidirect
5. #20241287C2semidirect
6. #20241282C2semidirect
8. #15689254C4semidirect
9. Q8×C47⋊C8C4semidirect
10. #15689201C4semidirect
13. D4×C47⋊C8C4semidirect
29. C47×#330C4semidirect
30. C47×#329C4semidirect
194. #330Dic47semidirect
195. #329Dic47semidirect
200. C47#8250semidirect

Maximal subgroups

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

subgroupindex
#202523642
#202431412
#202418742
M16×C47⋊C82
#202412872
#202412822
C47×#15242
#825047