Finite Group Databasebeta

← #26131480

Subgroups — #26131480

Order 12032.

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

normal subgroups only

This group has 452 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#20252106C211yesmaximal, characteristic
2#20245053C211yesmaximal, characteristic
3#20243191C212yesmaximal, complemented
4#20241518C212yesmaximal, complemented
5C47×#2132C211yesmaximal, characteristic
6C2×#13094031C411yescharacteristic, complemented
7C2×#13094031C2211yescharacteristic
8#15689195—22no—
9#15689195—22no—
10#15689147C411yescharacteristic, complemented
11#15689097C2212yes—
12#15689090C411yescharacteristic, complemented
13#15689090C2211yescharacteristic
14#15689081—22no—
15C2×#13093935C411yescharacteristic, complemented
16#15688918C412yes—
17#15688916C412yes—
18#15688895—22no—
19C47×#532C2211yescharacteristic
20C47×#413C2212yes—
21C47×#356C412yescomplemented
22C47×#355C412yescomplemented
23#13094031—42no—
24#13094031—42no—
25C2×#10625329—21no—
26C2×#10625329C2×C411yescharacteristic
27C2×#10625329C2×C411yescharacteristic
28C2×#10625329D412yes—
29C22×C47⋊C8—21no—
30C22×C47⋊C8C2×C411yescharacteristic
31C22×C47⋊C8C2×C411yescharacteristic
32C22×C47⋊C8D412yes—
33#13093983—22no—
34C4⋊(C47⋊C8)—22no—
35#13093939—22no—
36#13093939C2×C414yes—
37C22⋊(C47⋊C8)—22no—
38C22⋊(C47⋊C8)—22no—
39#13093935—21no—
40#13093935—21no—
41#13093934—42no—
42C8×Dic47—42no—
43C2×#41×C47C2×C411yescharacteristic
44C2×#41×C47C2311yescharacteristic
45C2×M16×C47—22no—
46C4⋊Q8×C47C2×C411yescharacteristic
47#127×C47C2×C411yescharacteristic
48#124×C47C2×C411yescharacteristic
49#124×C47C2×C411yescharacteristic
50#121×C47—22no—
51C4×D4×C47—22no—
52C2×C42×C47C2×C411yescharacteristic
53#101×C47C2×C414yes—
54#100×C47—22no—
55#41×C47—41no—
56#41×C47—41no—
57#41×C47—41no—
58#41×C47—41no—
59C2×Q8×C47C4211yescharacteristic
60C2×Q8×C47C22×C411yescharacteristic
61C2×D4×C47—21no—
62C2×D4×C47—22no—
63C2×D4×C47C4211yescharacteristic
64C2×D4×C47C22×C411yescharacteristic
65C2×D4×C47#4112yes—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47C22×C411yescharacteristic
69C22×C4×C47C2×D412yes—
70M16×C47—24no—
71M16×C47—42no—
72C2×C8×C47—42no—
73C4⋊C4×C47—22no—
74C22⋊C4×C47—22no—
75C22⋊C4×C47—22no—
76C42×C47—21no—
77C42×C47C4211yescharacteristic
78C42×C47C4211yescharacteristic
79C22⋊Dic47—24no—
80C4⋊Dic47—42no—
81C4×Dic47—42no—
82#10625329—41no—
83#10625329—42no—
84#10625329—42no—
85#10625329—41no—
86C2×C47⋊C8—22no—
87C2×C47⋊C8—22no—
88C2×C47⋊C8—22no—
89C2×C47⋊C8—21no—
90C2×C47⋊C8—21no—
91C2×C47⋊C8—42no—
92C2×C47⋊C8—41no—
93C2×C47⋊C8C22⋊C411yescharacteristic
94C2×C47⋊C8C22⋊C412yes—
95C2×C47⋊C8C22⋊C411yescharacteristic
96C23×C47—21no—
97C23×C47#11711yescharacteristic
98C23×C47C4×D412yes—
99Q8×C47—41no—
100Q8×C47—41no—
101D4×C47—41no—
102D4×C47—42no—
103D4×C47—41no—
104D4×C47—42no—
105D4×C47—82no—
106C2×C4×C47—21no—
107C2×C4×C47—22no—
108C2×C4×C47—21no—
109C2×C4×C47—22no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47—81no—
116C2×C4×C47C2×C4211yescharacteristic
117C2×C4×C47C2×C22⋊C411yescharacteristic
118C2×C4×C47C4×D412yes—
119C2×Dic47—44no—
120C8×C47—44no—
121C47⋊C8—21no—
122C47⋊C8—22no—
123C47⋊C8—41no—
124C47⋊C8—41no—
125C47⋊C8#9912yes—
126#12634—471nomaximal, Sylow
127C22×C47—22no—
128C22×C47—21no—
129C22×C47—41no—
130C22×C47—42no—
131C22×C47—42no—
132C22×C47—41no—
133C22×C47C4×C22⋊C411yescharacteristic
134C4×C47—22no—
135C4×C47—21no—
136C4×C47—41no—
137C4×C47—41no—
138C4×C47—41no—
139C4×C47C2×#9911yescharacteristic
140C4×C47#41011yescharacteristic
141Dic47—84no—
142#2932—471no—
143#2132D4711yescharacteristic
144#2126—471no—
145#1757—472no—
146#1395—472no—
147C2×C47—21no—
148C2×C47—42no—
149C2×C47—41no—
150C2×C47—81no—
151C2×C47C4×#9911yescharacteristic
152C2×#131—471no—
153C2×#131—471no—
154#532C2×D4711yescharacteristic
155#435—942no—
156#435—942no—
157#423—471no—
158#413C2×D4712yes—
159#410—472no—
160#408—471no—
161#408—471no—
162#405—942no—
163C2×#97—471no—
164#356Dic4712yescomplemented
165#355Dic4712yescomplemented
166#354—472no—
167#353—472no—
168#344—942no—
169C47#1263411yescharacteristic, complemented, Sylow
170C2×#41C4×D4711yescharacteristic
171C2×#41C22×D4711yescharacteristic
172#131—1882no—
173#131—1882no—
174C2×M16—22no—
175C2×M16—471no—
176C2×M16—472no—
177C2×M16—471no—
178C2×M16—941no—
179C22×C8—471no—
180C22×C8—472no—
181C22×C8—471no—
182C22×C8—941no—
183C4⋊Q8C2×Dic4711yescharacteristic
184#127C4×D4711yescharacteristic
185#124C4×D4711yescharacteristic
186#124C2×Dic4711yescharacteristic
187#121—22no—
188C4×D4—22no—
189#117—942no—
190C2×C42C4×D4711yescharacteristic
191C4⋊C8—942no—
192#101C2×Dic4714yes—
193#100—22no—
194#99—474no—
195#99—942no—
196C22⋊C8—942no—
197C22⋊C8—942no—
198#97—941no—
199#97—941no—
200#97—1882no—
201C4×C8—1882no—
202#41—41no—
203#41—41no—
204#41—41no—
205#41—41no—
206C2×Q8C4×Dic4711yescharacteristic
207C2×Q8C22×Dic4711yescharacteristic
208C2×D4—21no—
209C2×D4—22no—
210C2×D4C4×Dic4711yescharacteristic
211C2×D4C2×C4×D4711yescharacteristic
212C2×D4#1062535212yes—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4C2×C4×D4711yescharacteristic
216C22×C4D4×D4712yes—
217M16—24no—
218M16—42no—
219M16—1882no—
220M16—1882no—
221M16—1881no—
222M16—1881no—
223C2×C8—42no—
224C2×C8—471no—
225C2×C8—471no—
226C2×C8—472no—
227C2×C8—942no—
228C2×C8—942no—
229C2×C8—942no—
230C2×C8—941no—
231C2×C8—941no—
232C2×C8—1882no—
233C2×C8—1881no—
234C4⋊C4—22no—
235C4⋊C4—1882no—
236C22⋊C4—22no—
237C22⋊C4—22no—
238C22⋊C4—944no—
239C42—21no—
240C42—1882no—
241C42C4×Dic4711yescharacteristic
242C42C4×Dic4711yescharacteristic
243C23—21no—
244C23#1309398011yescharacteristic
245C23(C22⋊D47)⋊C412yes—
246Q8—41no—
247Q8—41no—
248D4—41no—
249D4—42no—
250D4—41no—
251D4—42no—
252D4—82no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—81no—
263C2×C4—1884no—
264C2×C4C2×C4×Dic4711yescharacteristic
265C2×C4C22⋊C4×D4711yescharacteristic
266C2×C4D4×Dic4712yes—
267C8—44no—
268C8—472no—
269C8—941no—
270C8—942no—
271C8—1881no—
272C8—1881no—
273C22—22no—
274C22—21no—
275C22—41no—
276C22—42no—
277C22—42no—
278C22—41no—
279C22C22⋊C4×Dic4711yescharacteristic
280C4—22no—
281C4—21no—
282C4—41no—
283C4—41no—
284C4—41no—
285C4—3764no—
286C4#99×D4711yescharacteristic
287C4#1568909511yescharacteristic
288C2—21no—
289C2—42no—
290C2—41no—
291C2—81no—
292C2#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#12634
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×#13093935C4semidirect
21. C47×#356C4semidirect
22. C47×#355C4semidirect
164. #356Dic47semidirect
165. #355Dic47semidirect
169. C47#12634semidirect

Maximal subgroups

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

subgroupindex
#202521062
#202450532
#202431912
#202415182
C47×#21322
#1263447