Finite Group Databasebeta

← #26117148

Subgroups — #26117148

Order 12032.

528 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 105 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#20253427C212yesmaximal
2#20244369C211yesmaximal, characteristic
3#20241817C212yesmaximal
4#20241217C211yesmaximal, characteristic
5C47×#2399C211yesmaximal, characteristic
6#15689680C2211yescharacteristic
7#15689458—22no—
8#15689418—22no—
9#15689333—22no—
10#15689279—22no—
11C4×#10625337—24no—
12#15689176C2212yes—
13#15689147C2211yescharacteristic
14#15688883—22no—
15Q8⋊(C47⋊C8)—24no—
16C2×C4×Q8×C47C2212yes—
17C47×#398—22no—
18C47×#395—22no—
19C4×C4⋊C4×C47—22no—
20C47×#376C2211yescharacteristic
21C2×#10625337—24no—
22C2×#10625329—21no—
23#13094024D412yes—
24#13094016—21no—
25#13094003—22no—
26C4×Dic94—22no—
27#13093983—21no—
28#13093957—24no—
29C4⋊(C47⋊C8)—22no—
30C4⋊(C47⋊C8)—22no—
31#13093948—24no—
32#13093947—24no—
33C22⋊(C47⋊C8)—22no—
34C4×C47⋊C8D414yes—
35C22×Q8×C47D412yes—
36C4×Q8×C47—22no—
37C4×Q8×C47—22no—
38C4×Q8×C47—22no—
39C4×Q8×C47—22no—
40C2×C4⋊C4×C47—24no—
41C2×C4⋊C4×C47—22no—
42C2×C4⋊C4×C47D412yes—
43C2×C42×C47—22no—
44C2×C42×C47D412yes—
45C2×C42×C47C2311yescharacteristic
46#95×C47—22no—
47#95×C47—22no—
48#95×C47—22no—
49C2×Q8×C47—22no—
50C2×Q8×C47—22no—
51C2×Q8×C47—44no—
52C22×C4×C47—22no—
53C22×C4×C47—22no—
54C22×C4×C47—22no—
55C22×C4×C47C2×D412yes—
56C22×C4×C47C2×D411yescharacteristic
57C2×Dic94—22no—
58C4⋊C4×C47—22no—
59C4⋊C4×C47—22no—
60C4⋊C4×C47—48no—
61C4⋊C4×C47—42no—
62C4⋊C4×C47—42no—
63C42×C47—22no—
64C42×C47—42no—
65C42×C47—42no—
66C42×C47Q1614yescomplemented
67C42×C47C2×D411yescharacteristic
68C42×C47C2×D412yes—
69C42×C47#4111yescharacteristic
70C22⋊Dic47—41no—
71#10625337—48no—
72C4⋊Dic47—22no—
73C4⋊Dic47—22no—
74Dic47⋊C4—42no—
75C4×Dic47—41no—
76#10625329—42no—
77C2×C47⋊C8—24no—
78C2×C47⋊C8—41no—
79C23×C47#12011yescharacteristic
80Q8×C47—24no—
81Q8×C47—24no—
82Q8×C47—42no—
83Q8×C47—42no—
84C2×C4×C47—24no—
85C2×C4×C47—24no—
86C2×C4×C47—42no—
87C2×C4×C47—44no—
88C2×C4×C47—42no—
89C2×C4×C47—42no—
90C2×C4×C47—44no—
91C2×C4×C47—42no—
92C2×C4×C47#12012yes—
93C2×C4×C47#12111yescharacteristic
94C2×C4×C47#12112yes—
95C2×C4×C47#12711yescharacteristic
96C2×C4×C47C2×Q1612yes—
97C2×C4×C47#13712yes—
98C2×Dic47—42no—
99Dic94—44no—
100C47⋊C8—44no—
101C47⋊C8—81no—
102#10017—471nomaximal, Sylow
103C22×C47—22no—
104C22×C47#39211yescharacteristic
105C22×C47#45112yes—
106C4×C47—22no—
107C4×C47—22no—
108C4×C47—44no—
109C4×C47—44no—
110C4×C47—82no—
111C4×C47#46212yes—
112C4×C47#49411yescharacteristic
113C4×C47#49711yescharacteristic
114Dic47—82no—
115#3104—472no—
116#2399D4711yescharacteristic
117#1995—471no—
118#1487—472no—
119#1274—471no—
120C2×C47—22no—
121C2×C47#201012yes—
122C2×C47#201611yescharacteristic
123#533—471no—
124C2×C4×Q8C2×D4712yes—
125#484—942no—
126#477—942no—
127#462—942no—
128#451—942no—
129C4×Q16—944no—
130#431—472no—
131#423—471no—
132#398—22no—
133#395—22no—
134C4×C4⋊C4—22no—
135#376C2×D4711yescharacteristic
136#339—942no—
137Q8⋊C8—944no—
138C47#1001711yescharacteristic, complemented, Sylow
139C22×Q8C22⋊D4712yes—
140C2×Q16—944no—
141C2×M16—941no—
142C4⋊Q8—472no—
143#125—941no—
144C22⋊Q8—942no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—942no—
150#117—941no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4C22⋊D4712yes—
154C2×C42—22no—
155C2×C42C22⋊D4712yes—
156C2×C42C22×D4711yescharacteristic
157#107—944no—
158C4⋊C8—942no—
159C4⋊C8—942no—
160#103—944no—
161#103—944no—
162C22⋊C8—942no—
163C4×C8—474no—
164#95—22no—
165#95—22no—
166#95—22no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—44no—
170C2×Q8—942no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4C2×C22⋊D4712yes—
175C22×C4C2×C22⋊D4711yescharacteristic
176Q16—1888no—
177M16—1882no—
178C2×C8—944no—
179C2×C8—1881no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—48no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—942no—
186C4⋊C4—942no—
187C4⋊C4—1882no—
188C22⋊C4—1881no—
189C42—22no—
190C42—42no—
191C42—42no—
192C42—1881no—
193C42#1062533714yescomplemented
194C42D4×D4711yescharacteristic
195C42D4×D4712yes—
196C42#1062535211yescharacteristic
197C23#1309399311yescharacteristic
198Q8—24no—
199Q8—24no—
200Q8—42no—
201Q8—42no—
202Q8—1884no—
203C2×C4—24no—
204C2×C4—24no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—1882no—
212C2×C4#1309399212yes—
213C2×C4#1309399511yescharacteristic
214C2×C4#1309399612yes—
215C2×C4#1309402011yescharacteristic
216C2×C4C2×#1062533712yes—
217C2×C4#1309405512yes—
218C8—3761no—
219C8—1884no—
220C22—22no—
221C22#1568903111yescharacteristic
222C22#1568927912yes—
223C4—22no—
224C4—22no—
225C4—44no—
226C4—44no—
227C4—82no—
228C4—3762no—
229C4#1568933012yes—
230C4#1568950211yescharacteristic
231C4#1568951511yescharacteristic
232C2—22no—
233C2#2024445312yes—
234C2#2024448811yescharacteristic
subgroupG/H
centerC2×C4#13094020
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2399C2
socleC22×C47#392
prime radicalC23×C47#120
hypercenter#2399D47
hypocenterC47#10017
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2399

Complements

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

NH ≅ G/Ntype
66. C42×C47Q16semidirect
138. C47#10017semidirect
193. C42#10625337semidirect

Maximal subgroups

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

subgroupindex
#202534272
#202443692
#202418172
#202412172
C47×#23992
#1001747