Finite Group Databasebeta

← #26113877

Subgroups — #26113877

Order 12032.

504 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 107 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#20253848C211yesmaximal, characteristic
2#20242123C211yesmaximal, characteristic
3#20242072C212yesmaximal, complemented
4#20241279C212yesmaximal, complemented
5C47×#2979C211yesmaximal, characteristic
6C2×#13094054—22no—
7C2×#13094024C2211yescharacteristic
8#15689518C414yes—
9#15689502—21no—
10#15689495—21no—
11#15689147C2212yes—
12C4×#10625329C2211yescharacteristic
13#15688872—21no—
14#15688870—22no—
15#15688870—22no—
16#15688867—21no—
17C47×#532C2211yescharacteristic
18C47×#436—22no—
19C8×D4×C47—22no—
20C47×#431C2212yes—
21C47×#408—22no—
22C22×Dic94—21no—
23#13094054—44no—
24#13094054—44no—
25C2×#10625337—24no—
26C2×#10625335—24no—
27C2×#10625329—21no—
28C2×#10625329—22no—
29#13094024—22no—
30#13094024—21no—
31#13094024C2×C412yes—
32C2×C4⋊Dic47—22no—
33C4⋊(C47⋊C8)—22no—
34C4⋊(C47⋊C8)—22no—
35#13093935C2×C412yes—
36C4×C47⋊C8—21no—
37C2×#41×C47D412yes—
38C2×M16×C47—22no—
39C22×C8×C47—22no—
40C4⋊Q8×C47D411yescharacteristic
41#127×C47D411yescharacteristic
42#124×C47C2×C412yes—
43#121×C47—22no—
44C4×D4×C47—22no—
45C2×C42×C47C2311yescharacteristic
46C4⋊C8×C47D412yes—
47C4⋊C8×C47D412yes—
48C22⋊C8×C47—24no—
49C22⋊C8×C47—22no—
50C4×C8×C47—22no—
51#41×C47—42no—
52#41×C47—42no—
53C2×Q8×C47C22⋊C412yes—
54C2×D4×C47—22no—
55C2×D4×C47—22no—
56C2×D4×C47C22⋊C412yes—
57C22×C4×C47—22no—
58C22×C4×C47C2×D411yescharacteristic
59C22×C4×C47#4112yes—
60C22×Dic47—41no—
61C2×Dic94—22no—
62C2×Dic94—42no—
63C2×Dic94—42no—
64C2×Dic94—41no—
65M16×C47—82no—
66C2×C8×C47—22no—
67C2×C8×C47—22no—
68C2×C8×C47—82no—
69C4⋊C4×C47—22no—
70C22⋊C4×C47—24no—
71C42×C47C22×C411yescharacteristic
72C42×C47C2×D411yescharacteristic
73C42×C47C2×D412yes—
74#10625337—48no—
75#10625335—48no—
76C4⋊Dic47—24no—
77C4⋊Dic47—42no—
78#10625329—22no—
79#10625329—22no—
80#10625329—82no—
81C2×C47⋊C8—22no—
82C2×C47⋊C8—42no—
83C23×C47—22no—
84C23×C47#12311yescharacteristic
85Q8×C47—42no—
86D4×C47—42no—
87D4×C47—42no—
88D4×C47—42no—
89D4×C47—82no—
90C2×C4×C47—21no—
91C2×C4×C47—21no—
92C2×C4×C47—22no—
93C2×C4×C47—22no—
94C2×C4×C47—22no—
95C2×C4×C47—42no—
96C2×C4×C47—42no—
97C2×C4×C47C2×C22⋊C411yescharacteristic
98C2×C4×C47C4×D412yes—
99C2×C4×C47#12011yescharacteristic
100C2×C4×C47#12112yes—
101C2×Dic47—42no—
102C2×Dic47—82no—
103Dic94—44no—
104Dic94—81no—
105Dic94—81no—
106C8×C47—42no—
107C8×C47—42no—
108C47⋊C8—24no—
109C47⋊C8—82no—
110#9435—471nomaximal, Sylow
111C22×C47—21no—
112C22×C47—22no—
113C22×C47—42no—
114C22×C47—44no—
115C22×C47#38611yescharacteristic
116C4×C47—21no—
117C4×C47—22no—
118C4×C47—42no—
119C4×C47#45311yescharacteristic
120C4×C47#45311yescharacteristic
121C4×C47#47112yes—
122C4×C47#47312yes—
123Dic47—82no—
124#3152—471no—
125#2979D4711yescharacteristic
126#1560—471no—
127#1549—472no—
128#1299—472no—
129C2×C47—41no—
130C2×C47—82no—
131C2×C47#188711yescharacteristic
132C2×C47#190812yes—
133C2×#137—942no—
134#532C2×D4711yescharacteristic
135C2×C4⋊Q8—471no—
136#497—474no—
137#494—941no—
138#492—941no—
139#436—22no—
140C8×D4—22no—
141#431C2×D4712yes—
142#423—472no—
143#408—22no—
144C4×M16—471no—
145#333—941no—
146#332—942no—
147#332—942no—
148#331—941no—
149C47#943511yescharacteristic, complemented, Sylow
150C2×#41C22⋊D4712yes—
151C22×Q8—941no—
152#137—1884no—
153#137—1884no—
154C2×Q16—944no—
155C2×SD16—944no—
156C2×M16—22no—
157C2×M16—941no—
158C2×M16—942no—
159C22×C8—22no—
160C4⋊Q8—472no—
161C4⋊Q8—941no—
162C4⋊Q8—942no—
163C4⋊Q8D18811yescharacteristic
164#127D18811yescharacteristic
165#124C4×D4712yes—
166#121—22no—
167C4×D4—22no—
168C2×C4⋊C4—942no—
169C2×C42C22×D4711yescharacteristic
170C4⋊C8—942no—
171C4⋊C8—942no—
172C4⋊C8C22⋊D4712yes—
173C4⋊C8C22⋊D4712yes—
174C22⋊C8—24no—
175C22⋊C8—22no—
176#97—472no—
177C4×C8—22no—
178C4×C8—941no—
179#41—42no—
180#41—42no—
181C2×Q8—942no—
182C2×Q8—1882no—
183C2×Q8—1882no—
184C2×Q8—1881no—
185C2×Q8(C2×D47)⋊C412yes—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4(C2×D47)⋊C412yes—
189C22×C4—22no—
190C22×C4—1881no—
191C22×C4#1062535012yes—
192C22×C4C2×C22⋊D4711yescharacteristic
193Q16—1888no—
194SD16—1888no—
195M16—82no—
196M16—942no—
197M16—942no—
198M16—3762no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—82no—
202C2×C8—942no—
203C2×C8—1882no—
204C4⋊C4—22no—
205C4⋊C4—944no—
206C4⋊C4—1882no—
207C22⋊C4—24no—
208C42C2×C4×D4711yescharacteristic
209C42C2×D18811yescharacteristic
210C42C2×C22⋊D4712yes—
211C23—22no—
212C23#1309400711yescharacteristic
213Q8—42no—
214Q8—3761no—
215Q8—3761no—
216Q8—1884no—
217D4—42no—
218D4—42no—
219D4—42no—
220D4—82no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—3762no—
229C2×C4—1882no—
230C2×C4C2×(C2×D47)⋊C411yescharacteristic
231C2×C4C4×C22⋊D4712yes—
232C2×C4#1309399311yescharacteristic
233C2×C4#1309399812yes—
234C8—42no—
235C8—42no—
236C8—944no—
237C8—3762no—
238C22—21no—
239C22—22no—
240C22—42no—
241C22—44no—
242C22#1568900211yescharacteristic
243C4—21no—
244C4—22no—
245C4—42no—
246C4—3762no—
247C4#1568929111yescharacteristic
248C4#1568929111yescharacteristic
249C4#1568938612yes—
250C4#1568939812yes—
251C2—41no—
252C2—82no—
253C2#2024378611yescharacteristic
254C2#2024390312yes—
subgroupG/H
centerC22#15689002
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2979C2
socleC22×C47#386
prime radicalC47×#532C22
cosocleC2×C42×C47C23
hypercenter#2979D47
hypocenterC47#9435
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2979

Complements

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

NH ≅ G/Ntype
3. #20242072C2semidirect
4. #20241279C2semidirect
149. C47#9435semidirect

Maximal subgroups

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

subgroupindex
#202538482
#202421232
#202420722
#202412792
C47×#29792
#943547