Finite Group Databasebeta

← #26109192

Subgroups — #26109192

Order 12032.

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

normal subgroups only

This group has 460 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×#15689191C211yesmaximal, characteristic
2#20243424C212yesmaximal, complemented
3#20241724C212yesmaximal
4#20241265C211yesmaximal, characteristic, complemented
5C47×#1293C211yesmaximal, characteristic, complemented
6C22×#10625324C411yescharacteristic
7C23×C47⋊C8C2211yescharacteristic
8C2×C4×C22⋊D47C2211yescharacteristic
9#15689191—22no—
10#15689191—22no—
11C2×#13093959—22no—
12C2×Dic47⋊C8C411yescharacteristic
13#15689087C2212yes—
14C2×(C2×D47)⋊C8C411yescharacteristic
15C2×#13093934C411yescharacteristic
16#15688887—22no—
17#15688848—22no—
18#15688846—22no—
19C47×#407C2212yes—
20C2×C22⋊C8×C47C2211yescharacteristic
21C22×C22⋊D47Q811yescharacteristic
22C22×C4×D47C2×C411yescharacteristic
23C2×#10625329—22no—
24C2×#10625324—21no—
25C2×#10625324—21no—
26C2×#10625324—42no—
27C22×C47⋊C8—42no—
28C22×C47⋊C8C2×C411yescharacteristic
29C22×C47⋊C8C2×C411yescharacteristic
30C22×C47⋊C8D412yes—
31C22×C47⋊C8D412yes—
32C4×C22⋊D47—22no—
33C4×C22⋊D47—22no—
34C2×Dic47⋊C4D411yescharacteristic
35C2×C22⋊Dic47Q811yescharacteristic
36C2×(C2×D47)⋊C4D411yescharacteristic
37C2×C4×Dic47C2×C411yescharacteristic
38#13093959—44no—
39Dic47⋊C8—22no—
40C22⋊(C47⋊C8)—22no—
41C22⋊(C47⋊C8)—22no—
42(C2×D47)⋊C8—22no—
43#13093934—21no—
44#13093934—21no—
45C23×C4×C47C2311yescharacteristic
46C2×M16×C47—22no—
47C22×C8×C47C2×C411yescharacteristic
48C22×C8×C47C2×C411yescharacteristic
49C22⋊C8×C47—22no—
50C22⋊C8×C47—22no—
51C22⋊C8×C47—22no—
52C24×C47C2×Q811yescharacteristic
53C23×D47C4⋊C411yescharacteristic
54C22×C4×C47—41no—
55C22×C4×C47—41no—
56C22×C4×C47C22×C411yescharacteristic
57C22×C4×C47C2×D412yes—
58C22×C4×C47C2×D411yescharacteristic
59C22×C4×C47#4112yes—
60C2×C22⋊D47—22no—
61C2×C22⋊D47—42no—
62C22×Dic47C4⋊C411yescharacteristic
63C22×Dic47C4⋊C411yescharacteristic
64C22×Dic47C4⋊C411yescharacteristic
65C2×C4×D47—21no—
66C2×C4×D47—21no—
67C2×C4×D47—41no—
68C2×C4×D47—41no—
69M16×C47—42no—
70M16×C47—42no—
71C2×C8×C47—21no—
72C2×C8×C47—21no—
73C2×C8×C47—41no—
74C2×C8×C47—41no—
75C2×C8×C47—42no—
76C22⋊Dic47—22no—
77(C2×D47)⋊C4—22no—
78Dic47⋊C4—22no—
79C4×Dic47—21no—
80C4×Dic47—21no—
81#10625329—42no—
82#10625329—42no—
83C2×C47⋊C8—24no—
84C2×C47⋊C8—22no—
85C2×C47⋊C8—41no—
86C2×C47⋊C8—42no—
87C2×C47⋊C8—42no—
88C2×C47⋊C8C22⋊C411yescharacteristic
89C2×C47⋊C8C22⋊C412yes—
90C2×C47⋊C8C22⋊C411yescharacteristic
91#10625324—42no—
92#10625324—42no—
93C23×C47—22no—
94C23×C47—41no—
95C23×C47—41no—
96C23×C47C2×C4⋊C411yescharacteristic
97C23×C47C22⋊Q812yes—
98C22×D47—21no—
99C22×D47—22no—
100C22×D47—41no—
101C22×D47—41no—
102C2×C4×C47—22no—
103C2×C4×C47—21no—
104C2×C4×C47—22no—
105C2×C4×C47—21no—
106C2×C4×C47—41no—
107C2×C4×C47—42no—
108C2×C4×C47C2×C22⋊C411yescharacteristic
109C2×C4×C47#11711yescharacteristic
110C2×C4×C47#12112yes—
111C22⋊D47—44no—
112C2×Dic47—21no—
113C2×Dic47—21no—
114C2×Dic47—21no—
115C2×Dic47—22no—
116C2×Dic47—41no—
117C2×Dic47—41no—
118C4×D47—42no—
119C4×D47—41no—
120C4×D47—41no—
121C8×C47—41no—
122C8×C47—41no—
123C8×C47—82no—
124C47⋊C8—21no—
125C47⋊C8—22no—
126C47⋊C8—82no—
127C47⋊C8#9912yes—
128#8613—471nomaximal, Sylow
129C22×C47—21no—
130C22×C47—21no—
131C22×C47—22no—
132C22×C47—22no—
133C22×C47—22no—
134C22×C47—44no—
135C22×C47#38011yescharacteristic
136C22×C47#43012yes—
137C2×D47—41no—
138C2×D47—41no—
139C2×D47—41no—
140C2×D47—82no—
141C4×C47—22no—
142C4×C47—21no—
143C4×C47C2×#9911yescharacteristic
144C4×C47#41011yescharacteristic
145Dic47—41no—
146Dic47—41no—
147Dic47—41no—
148C2×#435—471no—
149#1810—472no—
150#1465—472no—
151#1293—471no—
152#1293D4711yescharacteristic, complemented
153C2×C47—22no—
154C2×C47—42no—
155C2×C47#179911yescharacteristic
156C2×C47#181111yescharacteristic
157C2×C47#181311yescharacteristic
158D47—81no—
159C22×M16—471no—
160C23×C8—471no—
161C2×C4×D4—471no—
162#435—942no—
163#435—942no—
164C2×#108—942no—
165C2×C4⋊C8—471no—
166#407—472no—
167#407C2×D4712yes—
168C2×C22⋊C8—471no—
169C2×C22⋊C8C2×D4711yescharacteristic
170C2×#97—471no—
171#341—942no—
172#324—942no—
173#323—942no—
174C47#861311yescharacteristic, complemented, Sylow
175C22×D4—471no—
176C23×C4—471no—
177C23×C4C22×D4711yescharacteristic
178C2×M16—22no—
179C2×M16—941no—
180C2×M16—941no—
181C2×M16—942no—
182C2×M16—1882no—
183C22×C8—471no—
184C22×C8—472no—
185C22×C8—471no—
186C22×C8—472no—
187C22×C8—1882no—
188C22×C8C4×D4711yescharacteristic
189C22×C8C4×D4711yescharacteristic
190C4×D4—942no—
191C4×D4—942no—
192C2×C4⋊C4—471no—
193C2×C22⋊C4—471no—
194C2×C22⋊C4—471no—
195C2×C42—471no—
196#108—1884no—
197C4⋊C8—942no—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—942no—
202C22⋊C8—942no—
203C22⋊C8—942no—
204#97—941no—
205#97—941no—
206C24—471no—
207C24Q8×D4711yescharacteristic
208C2×D4—942no—
209C2×D4—1882no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—471no—
213C22×C4—471no—
214C22×C4—471no—
215C22×C4—941no—
216C22×C4—941no—
217C22×C4—1881no—
218C22×C4—1881no—
219C22×C4C2×C4×D4711yescharacteristic
220C22×C4#1062535012yes—
221C22×C4D4×D4712yes—
222C22×C4D4×D4711yescharacteristic
223M16—42no—
224M16—42no—
225M16—1882no—
226M16—1882no—
227M16—1882no—
228M16—1882no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—42no—
234C2×C8—471no—
235C2×C8—471no—
236C2×C8—472no—
237C2×C8—944no—
238C2×C8—942no—
239C2×C8—1881no—
240C2×C8—1882no—
241C2×C8—1882no—
242C4⋊C4—942no—
243C22⋊C4—942no—
244C22⋊C4—942no—
245C42—941no—
246C42—941no—
247C23—22no—
248C23—41no—
249C23—41no—
250C23—942no—
251C23—941no—
252C23—1881no—
253C23—1881no—
254C23C4⋊C4×D4711yescharacteristic
255C23#1309400212yes—
256D4—1884no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—941no—
264C2×C4—941no—
265C2×C4—941no—
266C2×C4—942no—
267C2×C4—1882no—
268C2×C4—1881no—
269C2×C4—1881no—
270C2×C4—1881no—
271C2×C4—1881no—
272C2×C4C22⋊C4×D4711yescharacteristic
273C2×C4#1309398111yescharacteristic
274C2×C4#1309399712yes—
275C8—41no—
276C8—41no—
277C8—82no—
278C8—472no—
279C8—942no—
280C8—941no—
281C8—3762no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—44no—
288C22—3762no—
289C22—1881no—
290C22—1881no—
291C22—1881no—
292C22#1568898211yescharacteristic
293C22#1568917012yes—
294C4—22no—
295C4—21no—
296C4—1881no—
297C4—1881no—
298C4—1881no—
299C4#99×D4711yescharacteristic
300C4#1568909511yescharacteristic
301C2—22no—
302C2—42no—
303C2—3761no—
304C2#2024337511yescharacteristic
305C2#2024342911yescharacteristic
306C2#2024343711yescharacteristic
subgroupG/H
centerC22#15688982
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1293C2
socleC22×C47#380
prime radicalC22×C22⋊D47Q8
cosocleC23×C4×C47C23
hypercenter#1293D47
hypocenterC47#8613
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1293

Complements

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

NH ≅ G/Ntype
2. #20243424C2semidirect
4. #20241265C2semidirect
5. C47×#1293C2semidirect
152. #1293D47semidirect
174. C47#8613semidirect

Maximal subgroups

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

subgroupindex
C2×#156891912
#202434242
#202417242
#202412652
C47×#12932
#861347