Finite Group Databasebeta

← #26111794

Subgroups — #26111794

Order 12032.

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

normal subgroups only

This group has 538 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#20252429C211yesmaximal, characteristic
2#20241783C212yesmaximal, complemented
3#20241709C211yesmaximal, characteristic
4#20241217C212yesmaximal, complemented
5C47×#2326C211yesmaximal, characteristic
6#15689201—24no—
7#15689176C2212yes—
8#15689176C2212yes—
9#15689087—22no—
10Q8⋊(C47⋊C8)—24no—
11#15688847—22no—
12C2×#124×C47C411yescharacteristic
13C2×C4×Q8×C47C2211yescharacteristic
14C2×C4×D4×C47C2211yescharacteristic
15C47×#391C411yescharacteristic
16C47×#390C411yescharacteristic
17C47×#388—22no—
18C47×#385—22no—
19C47×#384C411yescharacteristic
20C47×#376C2211yescharacteristic
21C2×#10625329—24no—
22C4⋊(C47⋊C8)—22no—
23C4⋊(C47⋊C8)—22no—
24C22⋊(C47⋊C8)—24no—
25C22⋊(C47⋊C8)—22no—
26C22⋊(C47⋊C8)—22no—
27C4×C47⋊C8—22no—
28C4×C47⋊C8D414yes—
29C22×Q8×C47C2×C411yescharacteristic
30C22×D4×C47C2×C411yescharacteristic
31C23×C4×C47D412yes—
32#124×C47—22no—
33#124×C47—22no—
34C4×Q8×C47—22no—
35C4×Q8×C47—22no—
36C4×D4×C47—22no—
37C4×D4×C47—22no—
38C2×C4⋊C4×C47—21no—
39C2×C4⋊C4×C47—22no—
40C2×C4⋊C4×C47C2×C411yescharacteristic
41C2×C4⋊C4×C47C2×C411yescharacteristic
42C2×C22⋊C4×C47—22no—
43C2×C22⋊C4×C47—22no—
44C2×C22⋊C4×C47D412yes—
45C2×C42×C47—21no—
46C2×C42×C47C2×C411yescharacteristic
47C2×C42×C47C2×C411yescharacteristic
48C2×C42×C47C2311yescharacteristic
49#95×C47—22no—
50#95×C47—21no—
51#95×C47—21no—
52C24×C47C22⋊C412yes—
53C2×Q8×C47—21no—
54C2×Q8×C47—22no—
55C2×Q8×C47—22no—
56C2×Q8×C47M1612yescomplemented
57C2×D4×C47—22no—
58C2×D4×C47—21no—
59C2×D4×C47—21no—
60C2×D4×C47—22no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—22no—
64C22×C4×C47—22no—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—82no—
68C22×C4×C47C22⋊C412yes—
69C22×C4×C47C22×C411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C4⋊C4×C47—22no—
73C4⋊C4×C47—22no—
74C4⋊C4×C47—41no—
75C4⋊C4×C47—41no—
76C4⋊C4×C47—82no—
77C22⋊C4×C47—28no—
78C22⋊C4×C47—24no—
79C22⋊C4×C47—82no—
80C42×C47—21no—
81C42×C47—22no—
82C42×C47—42no—
83C42×C47M1612yescomplemented
84C42×C47C2×D412yes—
85C42×C47#4112yes—
86#10625329—84no—
87C2×C47⋊C8—24no—
88C2×C47⋊C8—42no—
89C23×C47—22no—
90C23×C47—22no—
91C23×C47—22no—
92C23×C47—42no—
93C23×C47—42no—
94C23×C47C2×C22⋊C411yescharacteristic
95Q8×C47—24no—
96Q8×C47—42no—
97D4×C47—42no—
98D4×C47—42no—
99C2×C4×C47—22no—
100C2×C4×C47—22no—
101C2×C4×C47—22no—
102C2×C4×C47—41no—
103C2×C4×C47—41no—
104C2×C4×C47—44no—
105C2×C4×C47—41no—
106C2×C4×C47—41no—
107C2×C4×C47—41no—
108C2×C4×C47—41no—
109C2×C4×C47—42no—
110C2×C4×C47—41no—
111C2×C4×C47—42no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47—82no—
116C2×C4×C47C4×D412yes—
117C2×C4×C47#12011yescharacteristic
118C2×C4×C47#12112yes—
119C2×C4×C47#12311yescharacteristic
120C2×C4×C47C2×M1611yescharacteristic
121C2×C4×C47C2×M1611yescharacteristic
122C2×C4×C47#13112yes—
123C47⋊C8—44no—
124C47⋊C8—82no—
125#9103—471nomaximal, Sylow
126C22×C47—21no—
127C22×C47—21no—
128C22×C47—44no—
129C22×C47—42no—
130C22×C47—42no—
131C22×C47—42no—
132C22×C47—82no—
133C22×C47#38611yescharacteristic
134C22×C47#40711yescharacteristic
135C22×C47#40811yescharacteristic
136C4×C47—22no—
137C4×C47—21no—
138C4×C47—21no—
139C4×C47—42no—
140C4×C47—42no—
141C4×C47—42no—
142C4×C47—81no—
143C4×C47—81no—
144C4×C47#43512yes—
145C4×C47#43612yes—
146#2975—471no—
147#2326D4711yescharacteristic
148#1479—472no—
149#1460—471no—
150#1274—472no—
151C2×C47—21no—
152C2×C47—21no—
153C2×C47—82no—
154C2×C47#187011yescharacteristic
155C2×C47#190911yescharacteristic
156C2×C47#191111yescharacteristic
157C2×#124Dic4711yescharacteristic
158C2×C4×Q8C2×D4711yescharacteristic
159C2×C4×D4C2×D4711yescharacteristic
160#436—944no—
161#431—472no—
162#431—472no—
163#407—942no—
164#391Dic4711yescharacteristic
165#390Dic4711yescharacteristic
166#388—22no—
167#385—22no—
168#384Dic4711yescharacteristic
169#376C2×D4711yescharacteristic
170Q8⋊C8—944no—
171#323—942no—
172C47#910311yescharacteristic, complemented, Sylow
173C22×Q8C2×Dic4711yescharacteristic
174C22×D4C2×Dic4711yescharacteristic
175C23×C4C22⋊D4712yes—
176C2×M16—944no—
177#124—22no—
178#124—22no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×D4—22no—
182C4×D4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4C2×Dic4711yescharacteristic
186C2×C4⋊C4C2×Dic4711yescharacteristic
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4C22⋊D4712yes—
190C2×C42—21no—
191C2×C42C2×Dic4711yescharacteristic
192C2×C42C2×Dic4711yescharacteristic
193C2×C42C22×D4711yescharacteristic
194C4⋊C8—942no—
195C4⋊C8—942no—
196C22⋊C8—944no—
197C22⋊C8—942no—
198C22⋊C8—942no—
199C4×C8—474no—
200C4×C8—942no—
201#95—22no—
202#95—21no—
203#95—21no—
204C24C22⋊Dic4712yes—
205C2×Q8—21no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8#1062532912yescomplemented
209C2×D4—22no—
210C2×D4—21no—
211C2×D4—21no—
212C2×D4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—82no—
220C22×C4C22⋊Dic4712yes—
221C22×C4C22×Dic4711yescharacteristic
222C22×C4C2×C22⋊D4711yescharacteristic
223C22×C4C2×C22⋊D4711yescharacteristic
224M16—3764no—
225C2×C8—944no—
226C2×C8—1882no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—82no—
232C22⋊C4—28no—
233C22⋊C4—24no—
234C22⋊C4—82no—
235C42—21no—
236C42—22no—
237C42—42no—
238C42#1062532912yescomplemented
239C42D4×D4712yes—
240C42#1062535212yes—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—42no—
245C23—42no—
246C23C2×C22⋊Dic4711yescharacteristic
247Q8—24no—
248Q8—42no—
249D4—42no—
250D4—42no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—44no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—82no—
268C2×C4D4×Dic4712yes—
269C2×C4#1309399211yescharacteristic
270C2×C4#1309399612yes—
271C2×C4#1309400511yescharacteristic
272C2×C4C2×#1062532911yescharacteristic
273C2×C4C2×#1062532911yescharacteristic
274C2×C4#1309403112yes—
275C8—3762no—
276C8—1884no—
277C22—21no—
278C22—21no—
279C22—44no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—82no—
284C22#1568900311yescharacteristic
285C22#1568908711yescharacteristic
286C22#1568909011yescharacteristic
287C4—22no—
288C4—21no—
289C4—21no—
290C4—42no—
291C4—42no—
292C4—42no—
293C4—81no—
294C4—81no—
295C4#1568919512yes—
296C4#1568920112yes—
297C2—21no—
298C2—21no—
299C2—82no—
300C2#2024368611yescharacteristic
301C2#2024391111yescharacteristic
302C2#2024392111yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2326C2
socleC22×C47#386
prime radicalC22×D4×C47C2×C4
cosocleC2×C42×C47C23
hypercenter#2326D47
hypocenterC47#9103
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2326

Complements

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

NH ≅ G/Ntype
2. #20241783C2semidirect
4. #20241217C2semidirect
56. C2×Q8×C47M16semidirect
83. C42×C47M16semidirect
172. C47#9103semidirect
208. C2×Q8#10625329semidirect
238. C42#10625329semidirect

Maximal subgroups

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

subgroupindex
#202524292
#202417832
#202417092
#202412172
C47×#23262
#910347