Finite Group Databasebeta

← #26108649

Subgroups — #26108649

Order 12032.

732 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 131 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#20244370C211yesmaximal, characteristic, complemented
2#20242110C212yesmaximal, complemented
3#20242026C211yesmaximal, characteristic, complemented
4#20241896C212yesmaximal, complemented
5C47×#3548C211yesmaximal, characteristic
6#15689180C2212yes—
7#15689147C2211yescharacteristic
8#15688969C2211yescharacteristic
9#15688882—22no—
10#15688869—22no—
11#15688867—22no—
12#15688865—22no—
13#15688857—24no—
14C47×#562—22no—
15C47×#554—22no—
16C47×#550—22no—
17C47×#547C414yescomplemented
18C47×#540—22no—
19C47×#537—24no—
20C47×#533C2211yescharacteristic
21C47×#532C2212yes—
22C2×#10625329—21no—
23C2×C4×Dic47—21no—
24C4⋊(C47⋊C8)—22no—
25C4⋊(C47⋊C8)—22no—
26C22⋊(C47⋊C8)—22no—
27#13093935—22no—
28C47⋊#95—22no—
29C2×#41×C47—24no—
30C2×#41×C47—22no—
31C4⋊Q8×C47C2×C412yes—
32C4⋊Q8×C47D412yes—
33#127×C47—22no—
34#127×C47D412yes—
35#125×C47—22no—
36#125×C47D412yes—
37#124×C47—22no—
38#123×C47—28no—
39C22⋊Q8×C47—28no—
40C22⋊Q8×C47—22no—
41C22⋊Q8×C47—24no—
42#121×C47—28no—
43#121×C47—28no—
44#121×C47—24no—
45#121×C47—22no—
46#121×C47D414yes—
47C4×Q8×C47—22no—
48C4×D4×C47—24no—
49C4×D4×C47—22no—
50C4×D4×C47C2×C414yes—
51#117×C47D412yes—
52C2×C4⋊C4×C47—24no—
53C2×C42×C47C2311yescharacteristic
54#41×C47—84no—
55#41×C47—88no—
56C2×Q8×C47—22no—
57C2×Q8×C47—22no—
58C2×D4×C47—28no—
59C2×D4×C47—24no—
60C2×D4×C47—22no—
61C2×D4×C47—22no—
62C2×D4×C47C22⋊C414yes—
63C22×C4×C47—28no—
64C22×C4×C47—24no—
65C22×C4×C47C2×D411yescharacteristic
66C22×C4×C47C2×D412yes—
67C22×Dic47—41no—
68C4⋊C4×C47—28no—
69C4⋊C4×C47—22no—
70C4⋊C4×C47—22no—
71C4⋊C4×C47—24no—
72C4⋊C4×C47C22⋊C414yes—
73C4⋊C4×C47C22⋊C414yes—
74C22⋊C4×C47—28no—
75C22⋊C4×C47—22no—
76C22⋊C4×C47—24no—
77C22⋊C4×C47—24no—
78C42×C47—22no—
79C42×C47C22×C411yescharacteristic
80C42×C47C2×D412yes—
81C42×C47C2×D411yescharacteristic
82C4×Dic47—22no—
83C4×Dic47—41no—
84#10625329—42no—
85C2×C47⋊C8—42no—
86C2×C47⋊C8—41no—
87C23×C47—24no—
88C23×C47#12011yescharacteristic
89Q8×C47—44no—
90Q8×C47—82no—
91D4×C47—44no—
92D4×C47—42no—
93D4×C47—42no—
94D4×C47—82no—
95D4×C47—84no—
96D4×C47—88no—
97C2×C4×C47—21no—
98C2×C4×C47—21no—
99C2×C4×C47—24no—
100C2×C4×C47—24no—
101C2×C4×C47—42no—
102C2×C4×C47—48no—
103C2×C4×C47—88no—
104C2×C4×C47C2×C22⋊C411yescharacteristic
105C2×C4×C47C2×C22⋊C412yes—
106C2×C4×C47#12012yes—
107C2×C4×C47#12011yescharacteristic
108C2×Dic47—42no—
109C2×Dic47—82no—
110C47⋊C8—82no—
111C47⋊C8—81no—
112#8494—471nomaximal, Sylow
113C22×C47—21no—
114C22×C47—22no—
115C22×C47—44no—
116C22×C47—44no—
117C22×C47—44no—
118C22×C47#37911yescharacteristic
119C4×C47—21no—
120C4×C47—22no—
121C4×C47—22no—
122C4×C47—44no—
123C4×C47—44no—
124C4×C47#45411yescharacteristic
125C4×C47#45611yescharacteristic
126Dic47—82no—
127#3548D4711yescharacteristic
128#1995—471no—
129#1557—472no—
130#1537—471no—
131#1505—472no—
132C2×C47—41no—
133C2×C47—84no—
134C2×C47#179311yescharacteristic
135C2×C47#179512yes—
136#562—22no—
137#554—22no—
138#550—22no—
139#547Dic4714yescomplemented
140#540—22no—
141#537—24no—
142#533C2×D4711yescharacteristic
143#532C2×D4712yes—
144#432—472no—
145#423—471no—
146#376—471no—
147#339—942no—
148#332—942no—
149#331—942no—
150#330—942no—
151#327—944no—
152C47#849411yescharacteristic, complemented, Sylow
153C2×#41—24no—
154C2×#41—22no—
155C2×M16—941no—
156C4⋊Q8C2×Dic4712yes—
157C4⋊Q8C22⋊D4712yes—
158#127—22no—
159#127C22⋊D4712yes—
160#125—22no—
161#125C22⋊D4712yes—
162#124—22no—
163#123—28no—
164C22⋊Q8—28no—
165C22⋊Q8—22no—
166C22⋊Q8—24no—
167#121—28no—
168#121—28no—
169#121—24no—
170#121—22no—
171#121C22⋊D4714yes—
172C4×Q8—22no—
173C4×D4—24no—
174C4×D4—22no—
175C4×D4C2×Dic4714yes—
176#117C22⋊D4712yes—
177C2×C4⋊C4—24no—
178C2×C42—941no—
179C2×C42C22×D4711yescharacteristic
180C4⋊C8—942no—
181C4⋊C8—942no—
182C22⋊C8—942no—
183#97—942no—
184#95—942no—
185#41—84no—
186#41—88no—
187C2×Q8—22no—
188C2×Q8—22no—
189C2×D4—28no—
190C2×D4—24no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4C22⋊Dic4714yes—
194C22×C4—28no—
195C22×C4—24no—
196C22×C4—1881no—
197C22×C4C2×C22⋊D4711yescharacteristic
198C22×C4C2×C22⋊D4712yes—
199M16—1882no—
200C2×C8—1882no—
201C2×C8—1881no—
202C4⋊C4—28no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—24no—
206C4⋊C4C22⋊Dic4714yes—
207C4⋊C4C22⋊Dic4714yes—
208C22⋊C4—28no—
209C22⋊C4—22no—
210C22⋊C4—24no—
211C22⋊C4—24no—
212C42—22no—
213C42—942no—
214C42—1881no—
215C42C22×Dic4711yescharacteristic
216C42C2×C22⋊D4711yescharacteristic
217C42C2×C22⋊D4712yes—
218C23—24no—
219C23#1309399311yescharacteristic
220Q8—44no—
221Q8—82no—
222D4—44no—
223D4—42no—
224D4—42no—
225D4—82no—
226D4—84no—
227D4—88no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—42no—
233C2×C4—48no—
234C2×C4—88no—
235C2×C4—3762no—
236C2×C4—1882no—
237C2×C4C2×C22⋊Dic4711yescharacteristic
238C2×C4C2×C22⋊Dic4712yes—
239C2×C4#1309399312yes—
240C2×C4#1309399311yescharacteristic
241C8—3762no—
242C8—3761no—
243C22—21no—
244C22—22no—
245C22—44no—
246C22—44no—
247C22—44no—
248C22#1568897911yescharacteristic
249C4—21no—
250C4—22no—
251C4—22no—
252C4—44no—
253C4—44no—
254C4—3762no—
255C4#1568929611yescharacteristic
256C4#1568930611yescharacteristic
257C2—41no—
258C2—84no—
259C2#2024334711yescharacteristic
260C2#2024335812yes—
subgroupG/H
centerC22#15688979
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
Fitting, prime radicalC47×#3548C2
socleC22×C47#379
cosocleC2×C42×C47C23
hypercenter#3548D47
hypocenterC47#8494
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3548

Complements

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

NH ≅ G/Ntype
1. #20244370C2semidirect
2. #20242110C2semidirect
3. #20242026C2semidirect
4. #20241896C2semidirect
17. C47×#547C4semidirect
139. #547Dic47semidirect
152. C47#8494semidirect

Maximal subgroups

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

subgroupindex
#202443702
#202421102
#202420262
#202418962
C47×#35482
#849447