Finite Group Databasebeta

← #26131475

Subgroups — #26131475

Order 12032.

460 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 121 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#20252100C211yesmaximal, characteristic
2#20245053C211yesmaximal, characteristic
3#20243191C212yesmaximal, complemented
4#20241518C212yesmaximal, complemented
5C47×#2132C211yesmaximal, characteristic
6C2×#13094031C411yescharacteristic, complemented
7C2×#13094031C2211yescharacteristic
8#15689201—22no—
9D4×C47⋊C8—22no—
10#15689147C411yescharacteristic, complemented
11#15689097C2212yes—
12#15689090C411yescharacteristic, complemented
13#15689090C2211yescharacteristic
14#15689081—22no—
15C2×C4×C47⋊C8C411yescharacteristic, complemented
16#15688918C412yes—
17#15688916C412yes—
18#15688895—22no—
19C47×#532C2211yescharacteristic
20C47×#413C2212yes—
21C47×#356C412yescomplemented
22C47×#355C412yescomplemented
23#13094031—41no—
24#13094031—41no—
25#13094031—41no—
26#13094031—41no—
27C2×#10625329—21no—
28C2×#10625329C2×C411yescharacteristic
29C2×#10625329C2×C411yescharacteristic
30C2×#10625329D412yes—
31C22×C47⋊C8—21no—
32C22×C47⋊C8C2×C411yescharacteristic
33C22×C47⋊C8C2×C411yescharacteristic
34C22×C47⋊C8D412yes—
35#13093983—22no—
36C4⋊(C47⋊C8)—22no—
37#13093939—22no—
38#13093939C2×C414yes—
39C22⋊(C47⋊C8)—22no—
40C22⋊(C47⋊C8)—22no—
41#13093934—42no—
42C4×C47⋊C8—21no—
43C4×C47⋊C8—21no—
44C8×Dic47—42no—
45C2×#41×C47C2×C411yescharacteristic
46C2×#41×C47C2311yescharacteristic
47C2×M16×C47—22no—
48C4⋊Q8×C47C2×C411yescharacteristic
49#127×C47C2×C411yescharacteristic
50#124×C47C2×C411yescharacteristic
51#124×C47C2×C411yescharacteristic
52#121×C47—22no—
53C4×D4×C47—22no—
54C2×C42×C47C2×C411yescharacteristic
55#101×C47C2×C414yes—
56#100×C47—22no—
57#41×C47—41no—
58#41×C47—41no—
59#41×C47—41no—
60#41×C47—41no—
61C2×Q8×C47C4211yescharacteristic
62C2×Q8×C47C22×C411yescharacteristic
63C2×D4×C47—21no—
64C2×D4×C47—22no—
65C2×D4×C47C4211yescharacteristic
66C2×D4×C47C22×C411yescharacteristic
67C2×D4×C47#4112yes—
68C22×C4×C47—21no—
69C22×C4×C47—21no—
70C22×C4×C47C22×C411yescharacteristic
71C22×C4×C47C2×D412yes—
72M16×C47—24no—
73M16×C47—42no—
74C2×C8×C47—42no—
75C4⋊C4×C47—22no—
76C22⋊C4×C47—22no—
77C22⋊C4×C47—22no—
78C42×C47—21no—
79C42×C47C4211yescharacteristic
80C42×C47C4211yescharacteristic
81C22⋊Dic47—24no—
82C4⋊Dic47—42no—
83C4×Dic47—42no—
84#10625329—41no—
85#10625329—42no—
86#10625329—42no—
87#10625329—41no—
88C2×C47⋊C8—22no—
89C2×C47⋊C8—22no—
90C2×C47⋊C8—21no—
91C2×C47⋊C8—21no—
92C2×C47⋊C8—21no—
93C2×C47⋊C8—21no—
94C2×C47⋊C8—41no—
95C2×C47⋊C8—41no—
96C2×C47⋊C8—41no—
97C2×C47⋊C8C22⋊C411yescharacteristic
98C2×C47⋊C8C22⋊C412yes—
99C2×C47⋊C8C22⋊C411yescharacteristic
100C23×C47—21no—
101C23×C47#11711yescharacteristic
102C23×C47C4×D412yes—
103Q8×C47—41no—
104Q8×C47—41no—
105D4×C47—41no—
106D4×C47—42no—
107D4×C47—41no—
108D4×C47—44no—
109D4×C47—42no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—22no—
113C2×C4×C47—21no—
114C2×C4×C47—22no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—42no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47C2×C4211yescharacteristic
121C2×C4×C47C2×C22⋊C411yescharacteristic
122C2×C4×C47C4×D412yes—
123C2×Dic47—44no—
124C8×C47—44no—
125C47⋊C8—22no—
126C47⋊C8—21no—
127C47⋊C8—41no—
128C47⋊C8—41no—
129C47⋊C8#9911yescharacteristic
130C47⋊C8#9911yescharacteristic
131#12633—471nomaximal, Sylow
132C22×C47—22no—
133C22×C47—21no—
134C22×C47—41no—
135C22×C47—42no—
136C22×C47—42no—
137C22×C47—41no—
138C22×C47C4×C22⋊C411yescharacteristic
139C4×C47—22no—
140C4×C47—22no—
141C4×C47—21no—
142C4×C47—41no—
143C4×C47—41no—
144C4×C47C2×#9911yescharacteristic
145C4×C47#41011yescharacteristic
146Dic47—84no—
147#2931—471no—
148#2132D4711yescharacteristic
149#2126—471no—
150#1757—472no—
151#1395—472no—
152C2×C47—21no—
153C2×C47—42no—
154C2×C47—41no—
155C2×C47—81no—
156C2×C47C4×#9911yescharacteristic
157C2×#131—471no—
158C2×#131—471no—
159#532C2×D4711yescharacteristic
160#436—942no—
161C8×D4—942no—
162#423—471no—
163#413C2×D4712yes—
164#410—472no—
165#408—471no—
166#408—471no—
167#405—942no—
168C2×C4×C8—471no—
169#356Dic4712yescomplemented
170#355Dic4712yescomplemented
171#354—472no—
172#353—472no—
173#344—942no—
174C47#1263311yescharacteristic, complemented, Sylow
175C2×#41C4×D4711yescharacteristic
176C2×#41C22×D4711yescharacteristic
177#131—1881no—
178#131—1881no—
179#131—1881no—
180#131—1881no—
181C2×M16—22no—
182C2×M16—471no—
183C2×M16—472no—
184C2×M16—471no—
185C2×M16—941no—
186C22×C8—471no—
187C22×C8—472no—
188C22×C8—471no—
189C22×C8—941no—
190C4⋊Q8C2×Dic4711yescharacteristic
191#127C4×D4711yescharacteristic
192#124C4×D4711yescharacteristic
193#124C2×Dic4711yescharacteristic
194#121—22no—
195C4×D4—22no—
196#117—942no—
197C2×C42C4×D4711yescharacteristic
198C4⋊C8—942no—
199#101C2×Dic4714yes—
200#100—22no—
201#99—474no—
202#99—942no—
203C22⋊C8—942no—
204C22⋊C8—942no—
205#97—1882no—
206C4×C8—941no—
207C4×C8—941no—
208C4×C8—1882no—
209#41—41no—
210#41—41no—
211#41—41no—
212#41—41no—
213C2×Q8C4×Dic4711yescharacteristic
214C2×Q8C22×Dic4711yescharacteristic
215C2×D4—21no—
216C2×D4—22no—
217C2×D4C4×Dic4711yescharacteristic
218C2×D4C2×C4×D4711yescharacteristic
219C2×D4#1062535212yes—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4C2×C4×D4711yescharacteristic
223C22×C4D4×D4712yes—
224M16—24no—
225M16—42no—
226M16—1881no—
227M16—1882no—
228M16—1882no—
229M16—1881no—
230C2×C8—42no—
231C2×C8—471no—
232C2×C8—471no—
233C2×C8—472no—
234C2×C8—942no—
235C2×C8—942no—
236C2×C8—941no—
237C2×C8—941no—
238C2×C8—941no—
239C2×C8—941no—
240C2×C8—1881no—
241C2×C8—1881no—
242C2×C8—1881no—
243C4⋊C4—22no—
244C4⋊C4—1882no—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—944no—
248C42—21no—
249C42—1882no—
250C42C4×Dic4711yescharacteristic
251C42C4×Dic4711yescharacteristic
252C23—21no—
253C23#1309398011yescharacteristic
254C23(C22⋊D47)⋊C412yes—
255Q8—41no—
256Q8—41no—
257D4—41no—
258D4—42no—
259D4—41no—
260D4—44no—
261D4—42no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—1884no—
273C2×C4C2×C4×Dic4711yescharacteristic
274C2×C4C22⋊C4×D4711yescharacteristic
275C2×C4D4×Dic4712yes—
276C8—44no—
277C8—471no—
278C8—471no—
279C8—942no—
280C8—941no—
281C8—1881no—
282C8—1881no—
283C22—22no—
284C22—21no—
285C22—41no—
286C22—42no—
287C22—42no—
288C22—41no—
289C22C22⋊C4×Dic4711yescharacteristic
290C4—22no—
291C4—22no—
292C4—21no—
293C4—41no—
294C4—41no—
295C4—3764no—
296C4#99×D4711yescharacteristic
297C4#1568909511yescharacteristic
298C2—21no—
299C2—42no—
300C2—41no—
301C2—81no—
302C2#99×Dic4711yescharacteristic
subgroupG/H
centerC4#99×D47
commutatorC2×C4×C47C2×C42
FrattiniC2×#41C22×D47
FittingC47×#2132C2
socleC2×C47C4×#99
prime radicalC47×#532C22
cosocleC2×#41×C47C23
hypercenter#2132D47
hypocenterC47#12633
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #2132

Complements

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

NH ≅ G/Ntype
3. #20243191C2semidirect
4. #20241518C2semidirect
6. C2×#13094031C4semidirect
10. #15689147C4semidirect
12. #15689090C4semidirect
15. C2×C4×C47⋊C8C4semidirect
21. C47×#356C4semidirect
22. C47×#355C4semidirect
169. #356Dic47semidirect
170. #355Dic47semidirect
174. C47#12633semidirect

Maximal subgroups

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

subgroupindex
#202521002
#202450532
#202431912
#202415182
C47×#21322
#1263347