Finite Group Databasebeta

← #26126174

Subgroups — #26126174

Order 12032.

328 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 93 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#20252353C211yesmaximal, characteristic
2#20245189C211yesmaximal, characteristic, complemented
3C4⋊C16×D47C211yesmaximal, characteristic, complemented
4#20241370C211yesmaximal, characteristic, complemented
5#20241347C211yesmaximal, characteristic, complemented
6#20241345C211yesmaximal, characteristic, complemented
7C47×#1338C211yesmaximal, characteristic, complemented
8C4×#10625354C411yescharacteristic
9C2×C16×D47—21no—
10#15689254C411yescharacteristic
11#15689202C2211yescharacteristic
12#15689200C411yescharacteristic
13#15689196—21no—
14#15689193—21no—
15#15689175—21no—
16#15689175C411yescharacteristic
17#15689075—21no—
18C4×C8×D47C2211yescharacteristic
19C4⋊(C47⋊C16)C2211yescharacteristic
20Dic47⋊C16—21no—
21Dic47⋊C16—21no—
22(C2×D47)⋊C16—21no—
23#15688899C2211yescharacteristic
24C47×#446C2211yescharacteristic
25C47×C4⋊C16C2211yescharacteristic
26C47×#346C2211yescharacteristic
27C2×#10625354C811yescharacteristic
28C2×#10625324—21no—
29C2×#10625324—21no—
30C2×C8×D47D411yescharacteristic
31C2×C8×D47D411yescharacteristic
32Q8×Dic47C811yescharacteristic
33C4×D188—21no—
34C4×D188C2×C411yescharacteristic
35#13093986—21no—
36#13093986C811yescharacteristic
37#13093982—21no—
38#13093982C811yescharacteristic
39C42×D47—21no—
40C42×D47C2×C411yescharacteristic
41C2×C47⋊C16—21no—
42C2×C47⋊C16—21no—
43C16×D47—42no—
44C4⋊(C47⋊C8)—21no—
45C4⋊(C47⋊C8)C2×C411yescharacteristic
46(C2×D47)⋊C8—21no—
47(C2×D47)⋊C8—21no—
48(C2×D47)⋊C8—21no—
49#13093935—21no—
50C4×C47⋊C8C2×C411yescharacteristic
51C8×Dic47—21no—
52C8×Dic47D411yescharacteristic
53C8×Dic47D411yescharacteristic
54C4×Q8×C47C2×C411yescharacteristic
55C2×C16×C47—21no—
56C2×C16×C47—21no—
57C4⋊C8×C47—21no—
58C4⋊C8×C47C2×C411yescharacteristic
59#97×C47—21no—
60C4×C8×C47C2311yescharacteristic
61C2×Q8×C47C2×C811yescharacteristic
62#10625354—42no—
63C2×D188—21no—
64C2×D188C2×C811yescharacteristic
65C2×C4×D47—21no—
66C2×C4×D47—21no—
67C2×C4×D47—21no—
68C2×C4×D47C22⋊C411yescharacteristic
69C2×C4×D47C2×C811yescharacteristic
70C2×C4×D47M1611yescharacteristic
71C2×C8×C47—21no—
72C2×C8×C47C2×D411yescharacteristic
73C2×C8×C47C2×D411yescharacteristic
74C4⋊C4×C47—21no—
75C4⋊C4×C47C2×C811yescharacteristic
76C42×C47—21no—
77C42×C47C22×C411yescharacteristic
78(C2×D47)⋊C4—21no—
79(C2×D47)⋊C4—21no—
80(C2×D47)⋊C4—21no—
81C4⋊Dic47—21no—
82C4⋊Dic47C2×C811yescharacteristic
83C4×Dic47—21no—
84C4×Dic47—21no—
85C4×Dic47C22⋊C411yescharacteristic
86C4×Dic47C2×C811yescharacteristic
87C4×Dic47M1611yescharacteristic
88C2×C47⋊C8—21no—
89C2×C47⋊C8C22⋊C411yescharacteristic
90C2×C47⋊C8C22⋊C411yescharacteristic
91#10625324—42no—
92#10625324—41no—
93#10625324—41no—
94C8×D47—21no—
95C8×D47—21no—
96C8×D47—41no—
97C16×C47—41no—
98C16×C47—41no—
99C47⋊C16—41no—
100C47⋊C16—41no—
101C22×D47—21no—
102C22×D47C22⋊C811yescharacteristic
103Q8×C47—41no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47C2×C22⋊C411yescharacteristic
107C2×C4×C47C22×C811yescharacteristic
108C2×C4×C47C2×M1611yescharacteristic
109C2×Dic47—21no—
110C2×Dic47C22⋊C811yescharacteristic
111C2×Dic47C22⋊C811yescharacteristic
112C2×Dic47C22⋊C811yescharacteristic
113D188—41no—
114D188—42no—
115C4×D47—22no—
116C4×D47—21no—
117C4×D47—21no—
118C4×D47—22no—
119C4×D47—42no—
120C4×D47—41no—
121C4×D47—41no—
122C4×D47—42no—
123C8×C47—21no—
124C8×C47—41no—
125C8×C47AGL(1,ℤ/8)11yescharacteristic
126C8×C47#13711yescharacteristic
127C47⋊C8—21no—
128C47⋊C8—21no—
129C47⋊C8—41no—
130#11626—471nomaximal, Sylow
131C22×C47C2×C22⋊C811yescharacteristic
132C2×D47—21no—
133C2×D47—21no—
134C2×D47—41no—
135C2×D47—41no—
136C2×D47—41no—
137C2×D47#34912yes—
138C4×C47—21no—
139C4×C47—21no—
140C4×C47—41no—
141C4×C47—41no—
142C4×C47#41711yescharacteristic
143C4×C47#42111yescharacteristic
144Dic47—21no—
145Dic47—21no—
146Dic47—41no—
147Dic47#35012yes—
148#2965—471no—
149#2163—471no—
150C2×C4⋊C16—471no—
151#1338—471no—
152#1338D4711yescharacteristic, complemented
153#1330—471no—
154#1330—471no—
155C2×C47#147711yescharacteristic
156C2×C47C2×#34911yescharacteristic
157C2×C47C2×#35011yescharacteristic
158D47—22no—
159D47—81no—
160C4×#41—471no—
161C22×C16—941no—
162#446—471no—
163#446C2×D4711yescharacteristic
164#436—471no—
165#436—471no—
166#435—941no—
167#435—941no—
168#431—471no—
169#431—941no—
170C4×M16—941no—
171C2×C4×C8—471no—
172C4⋊C16—471no—
173C4⋊C16—941no—
174C4⋊C16—941no—
175C4⋊C16C2×D4711yescharacteristic
176C22⋊C16—941no—
177#346—471no—
178#346C2×D4711yescharacteristic
179C47#1162611yescharacteristic, complemented, Sylow
180C2×#41—471no—
181C2×M16—941no—
182C2×M16—941no—
183C22×C8—471no—
184C22×C8—471no—
185C4×Q8—471no—
186C4×Q8C4×D4711yescharacteristic
187C4×D4—471no—
188C4×D4—471no—
189C4×D4—941no—
190C4×D4—941no—
191#117—471no—
192#117—941no—
193C2×C42—471no—
194C2×C42—941no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—941no—
198C2×C16—941no—
199C2×C16—1882no—
200C4⋊C8—21no—
201C4⋊C8—471no—
202C4⋊C8—941no—
203C4⋊C8C4×D4711yescharacteristic
204C22⋊C8—941no—
205C22⋊C8—941no—
206C22⋊C8—941no—
207#97—21no—
208#97—941no—
209C4×C8—471no—
210C4×C8—471no—
211C4×C8—471no—
212C4×C8—941no—
213C4×C8C22×D4711yescharacteristic
214#41—1882no—
215C2×Q8C8×D4711yescharacteristic
216C2×D4—471no—
217C2×D4—941no—
218C22×C4—471no—
219C22×C4—471no—
220C22×C4—471no—
221C22×C4—941no—
222C22×C4—941no—
223C22×C4—941no—
224M16—1882no—
225M16—1881no—
226M16—1881no—
227C2×C8—21no—
228C2×C8—471no—
229C2×C8—471no—
230C2×C8—941no—
231C2×C8—941no—
232C2×C8—941no—
233C2×C8—1881no—
234C2×C8D4×D4711yescharacteristic
235C2×C8D4×D4711yescharacteristic
236C4⋊C4—21no—
237C4⋊C4—471no—
238C4⋊C4—941no—
239C4⋊C4C8×D4711yescharacteristic
240C22⋊C4—941no—
241C22⋊C4—941no—
242C22⋊C4—941no—
243C42—21no—
244C42—471no—
245C42—471no—
246C42—471no—
247C42—941no—
248C42—941no—
249C42C2×C4×D4711yescharacteristic
250C16—41no—
251C16—41no—
252C16—1881no—
253C16—1881no—
254C23—471no—
255C23—941no—
256Q8—41no—
257D4—1881no—
258D4—1882no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—471no—
262C2×C4—471no—
263C2×C4—471no—
264C2×C4—942no—
265C2×C4—942no—
266C2×C4—941no—
267C2×C4—941no—
268C2×C4—941no—
269C2×C4—1882no—
270C2×C4—1882no—
271C2×C4—1881no—
272C2×C4—1881no—
273C2×C4C22⋊C4×D4711yescharacteristic
274C2×C4C2×C8×D4711yescharacteristic
275C2×C4M16×D4711yescharacteristic
276C8—21no—
277C8—41no—
278C8—941no—
279C8—941no—
280C8—1881no—
281C8#1309405111yescharacteristic
282C8#1309405611yescharacteristic
283C22—472no—
284C22—941no—
285C22—941no—
286C22—1881no—
287C22—1881no—
288C22—1881no—
289C22C22⋊C8×D4711yescharacteristic
290C4—21no—
291C4—21no—
292C4—41no—
293C4—41no—
294C4—472no—
295C4—941no—
296C4—941no—
297C4—1881no—
298C4#1568912311yescharacteristic
299C4#1568913811yescharacteristic
300C2—942no—
301C2—3761no—
302C2#2024177411yescharacteristic
303C2#349×D4711yescharacteristic
304C2#350×D4711yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D47
commutatorC2×C4×C47C22×C8
FrattiniC4×C8C22×D47
FittingC47×#1338C2
socleC22×C47C2×C22⋊C8
prime radicalC2×#10625354C8
cosocleC4×C8×C47C23
hypercenter#1338D47
hypocenterC47#11626
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1338

Complements

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

NH ≅ G/Ntype
2. #20245189C2semidirect
3. C4⋊C16×D47C2semidirect
4. #20241370C2semidirect
5. #20241347C2semidirect
6. #20241345C2semidirect
7. C47×#1338C2semidirect
152. #1338D47semidirect
179. C47#11626semidirect

Maximal subgroups

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

subgroupindex
#202523532
#202451892
C4⋊C16×D472
#202413702
#202413472
#202413452
C47×#13382
#1162647