Finite Group Databasebeta

← #26109854

Subgroups — #26109854

Order 12032.

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

normal subgroups only

This group has 422 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#20243384C211yesmaximal, characteristic
2#20241712C211yesmaximal, characteristic
3#20241291C212yesmaximal
4#20241246C212yesmaximal
5C47×#2351C211yesmaximal, characteristic
6#15689176C2212yes—
7#15689147C2212yes—
8C2×#13093943—21no—
9#15688892—21no—
10#15688849—22no—
11C2×C4×Q8×C47—21no—
12C2×C4×Q8×C47C2211yescharacteristic
13C47×#389—22no—
14C47×#389—21no—
15C47×#382C412yes—
16C47×#382C2211yescharacteristic
17C4×C4⋊C4×C47C412yes—
18C4×C4⋊C4×C47C2211yescharacteristic
19C2×#10625329—22no—
20C4⋊(C47⋊C8)—22no—
21C4⋊(C47⋊C8)—22no—
22C4⋊(C47⋊C8)—22no—
23#13093943—42no—
24C22⋊(C47⋊C8)—22no—
25C4×C47⋊C8—22no—
26C22×Q8×C47D411yescharacteristic
27C4×Q8×C47—22no—
28C4×Q8×C47—22no—
29C4×Q8×C47—42no—
30C4×Q8×C47—42no—
31C2×C4⋊C4×C47—22no—
32C2×C4⋊C4×C47—22no—
33C2×C4⋊C4×C47—21no—
34C2×C4⋊C4×C47—21no—
35C2×C4⋊C4×C47D412yes—
36C2×C4⋊C4×C47D411yescharacteristic
37C2×C42×C47—22no—
38C2×C42×C47—21no—
39C2×C42×C47C2×C412yes—
40C2×C42×C47C2311yescharacteristic
41#95×C47—22no—
42#95×C47—21no—
43#95×C47C2×C411yescharacteristic
44#95×C47C2×C412yes—
45#95×C47C2×C411yescharacteristic
46C2×Q8×C47—22no—
47C2×Q8×C47—22no—
48C2×Q8×C47D812yes—
49C2×Q8×C47SD1612yes—
50C22×C4×C47—21no—
51C22×C4×C47—22no—
52C22×C4×C47—21no—
53C22×C4×C47C22⋊C412yes—
54C22×C4×C47C22⋊C412yes—
55C22×C4×C47C22×C411yescharacteristic
56C22×C4×C47C2×D411yescharacteristic
57C22×C4×C47C2×D411yescharacteristic
58C4⋊C4×C47—24no—
59C4⋊C4×C47—22no—
60C4⋊C4×C47—42no—
61C4⋊C4×C47—44no—
62C4⋊C4×C47—44no—
63C4⋊C4×C47—41no—
64C4⋊C4×C47—41no—
65C42×C47—22no—
66C42×C47—22no—
67C42×C47—44no—
68C42×C47—41no—
69C42×C47—41no—
70C42×C47#4112yes—
71C42×C47#4112yes—
72#10625329—42no—
73#10625329—42no—
74C2×C47⋊C8—42no—
75C2×C47⋊C8—42no—
76C23×C47C2×C22⋊C411yescharacteristic
77Q8×C47—24no—
78Q8×C47—24no—
79C2×C4×C47—22no—
80C2×C4×C47—22no—
81C2×C4×C47—22no—
82C2×C4×C47—22no—
83C2×C4×C47—41no—
84C2×C4×C47—41no—
85C2×C4×C47—41no—
86C2×C4×C47—41no—
87C2×C4×C47—42no—
88C2×C4×C47—41no—
89C2×C4×C47—42no—
90C2×C4×C47—42no—
91C2×C4×C47—41no—
92C2×C4×C47—42no—
93C2×C4×C47—41no—
94C2×C4×C47#10214yes—
95C2×C4×C47#11711yescharacteristic
96C2×C4×C47#11711yescharacteristic
97C2×C4×C47#12312yes—
98C2×C4×C47#12312yes—
99C2×C4×C47C2×D811yescharacteristic
100C2×C4×C47C2×SD1611yescharacteristic
101C2×C4×C47#13712yes—
102C47⋊C8—82no—
103C47⋊C8—82no—
104#8766—471nomaximal, Sylow
105C22×C47—21no—
106C22×C47—21no—
107C22×C47#38111yescharacteristic
108C22×C47C2×#10211yescharacteristic
109C22×C47#41911yescharacteristic
110C4×C47—24no—
111C4×C47—22no—
112C4×C47—21no—
113C4×C47—21no—
114C4×C47—42no—
115C4×C47—42no—
116C4×C47—82no—
117C4×C47—81no—
118C4×C47#48012yes—
119C4×C47#48312yes—
120#2351D4711yescharacteristic
121#1801—471no—
122#1461—471no—
123#1303—472no—
124#1286—472no—
125C2×C47—21no—
126C2×C47—21no—
127C2×C47#182311yescharacteristic
128C2×C47#182711yescharacteristic
129C2×C47#183011yescharacteristic
130C2×C4×Q8—21no—
131C2×C4×Q8C2×D4711yescharacteristic
132#431—472no—
133#423—472no—
134C2×#101—941no—
135#389—21no—
136#389—22no—
137#382Dic4712yes—
138#382C2×D4711yescharacteristic
139C4×C4⋊C4Dic4712yes—
140C4×C4⋊C4C2×D4711yescharacteristic
141#343—941no—
142#324—942no—
143C47#876611yescharacteristic, complemented, Sylow
144C22×Q8C22⋊D4711yescharacteristic
145C2×M16—942no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—42no—
149C4×Q8—42no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4C22⋊D4712yes—
155C2×C4⋊C4C22⋊D4711yescharacteristic
156C2×C42—22no—
157C2×C42—21no—
158C2×C42C2×Dic4712yes—
159C2×C42C22×D4711yescharacteristic
160C4⋊C8—942no—
161C4⋊C8—942no—
162C4⋊C8—942no—
163#101—1882no—
164C22⋊C8—942no—
165C4×C8—942no—
166#95—22no—
167#95—21no—
168#95C2×Dic4711yescharacteristic
169#95C2×Dic4712yes—
170#95C2×Dic4711yescharacteristic
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8#1062533412yes—
174C2×Q8#1062533512yes—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4C22⋊Dic4712yes—
179C22×C4C22⋊Dic4712yes—
180C22×C4C22×Dic4711yescharacteristic
181C22×C4C2×C22⋊D4711yescharacteristic
182C22×C4C2×C22⋊D4711yescharacteristic
183M16—1882no—
184M16—1882no—
185C2×C8—1882no—
186C2×C8—1882no—
187C4⋊C4—24no—
188C4⋊C4—22no—
189C4⋊C4—42no—
190C4⋊C4—44no—
191C4⋊C4—44no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C42—22no—
195C42—22no—
196C42—44no—
197C42—41no—
198C42—41no—
199C42#1062535012yes—
200C42#1062535012yes—
201C23C2×C22⋊Dic4711yescharacteristic
202Q8—24no—
203Q8—24no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4#1309394514yes—
220C2×C4#1309398311yescharacteristic
221C2×C4#1309398311yescharacteristic
222C2×C4#1309400712yes—
223C2×C4#1309400712yes—
224C2×C4C2×#1062533411yescharacteristic
225C2×C4C2×#1062533511yescharacteristic
226C2×C4#1309405512yes—
227C8—3762no—
228C8—3762no—
229C22—21no—
230C22—21no—
231C22#1568898611yescharacteristic
232C22C2×#1309394511yescharacteristic
233C22#1568913211yescharacteristic
234C4—24no—
235C4—22no—
236C4—21no—
237C4—21no—
238C4—42no—
239C4—42no—
240C4—82no—
241C4—81no—
242C4#1568943612yes—
243C4#1568945312yes—
244C2—21no—
245C2—21no—
246C2#2024347411yescharacteristic
247C2#2024349211yescharacteristic
248C2#2024350211yescharacteristic
subgroupG/H
centerC22#15688986
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2351C2
socleC22×C47#381
prime radicalC23×C47C2×C22⋊C4
cosocleC2×C42×C47C23
hypercenter#2351D47
hypocenterC47#8766
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2351

Complements

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

NH ≅ G/Ntype
143. C47#8766semidirect

Maximal subgroups

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

subgroupindex
#202433842
#202417122
#202412912
#202412462
C47×#23512
#876647