Finite Group Databasebeta

← #26105548

Subgroups — #26105548

Order 12032.

418 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 83 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#20250318C211yesmaximal, characteristic
2#20244952C212yesmaximal, complemented
3#20244921C211yesmaximal, characteristic
4#20241210C212yesmaximal, complemented
5C47×#1270C211yesmaximal, characteristic
6C2×#13094005—21no—
7C2×#13093955—21no—
8C2×#13093945—21no—
9#15689053—21no—
10#15689048—22no—
11#15689021C2212yes—
12#15688998—22no—
13#15688993C2211yescharacteristic
14#15688886—22no—
15#15688886—22no—
16#15688878C2211yescharacteristic
17C47×#390C2211yescharacteristic
18C47×#336C2212yes—
19C23×Dic47—21no—
20C22×C47⋊C8—21no—
21#13094005—42no—
22#13094005—42no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47—22no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47D412yes—
27C2×C4⋊Dic47D412yes—
28C2×Dic47⋊C4—22no—
29C2×C22⋊Dic47—22no—
30C2×C22⋊Dic47—21no—
31#13093955—42no—
32#13093945—42no—
33C47⋊#95—22no—
34C47⋊#95—21no—
35C47⋊#95—22no—
36C22×D4×C47D411yescharacteristic
37C22×C8×C47—22no—
38C2×C4⋊C4×C47D411yescharacteristic
39C2×C22⋊C4×C47—22no—
40C2×C42×C47C2311yescharacteristic
41C24×C47—21no—
42C2×D4×C47—21no—
43C2×D4×C47—41no—
44C2×D4×C47—41no—
45C2×D4×C47SD1612yes—
46C22×C4×C47—21no—
47C22×C4×C47C2×D411yescharacteristic
48C22×C4×C47C2×D412yes—
49C22×Dic47—22no—
50C22×Dic47—22no—
51C22×Dic47—42no—
52C22×Dic47—41no—
53C22×Dic47—81no—
54C22×Dic47—81no—
55C2×C8×C47—42no—
56C2×C8×C47—42no—
57C2×C8×C47—42no—
58C4⋊C4×C47—41no—
59C22⋊C4×C47—42no—
60C22⋊C4×C47—42no—
61C42×C47—41no—
62C22⋊Dic47—42no—
63C22⋊Dic47—44no—
64C4⋊Dic47—22no—
65C4⋊Dic47—22no—
66C4⋊Dic47—42no—
67C4⋊Dic47—41no—
68C4⋊Dic47—41no—
69C4⋊Dic47—44no—
70C4⋊Dic47—81no—
71Dic47⋊C4—42no—
72Dic47⋊C4—42no—
73C2×C47⋊C8—22no—
74C2×C47⋊C8—41no—
75C2×C47⋊C8—41no—
76C23×C47—22no—
77C23×C47—42no—
78C23×C47—41no—
79C23×C47—42no—
80C23×C47—41no—
81C23×C47#12011yescharacteristic
82D4×C47—42no—
83D4×C47—81no—
84C2×C4×C47—21no—
85C2×C4×C47—22no—
86C2×C4×C47—21no—
87C2×C4×C47—22no—
88C2×C4×C47—42no—
89C2×C4×C47—41no—
90C2×C4×C47C2×SD1611yescharacteristic
91C2×C4×C47#13512yes—
92C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
93C2×C4×C47#13712yes—
94C2×Dic47—42no—
95C2×Dic47—42no—
96C2×Dic47—42no—
97C2×Dic47—42no—
98C2×Dic47—42no—
99C2×Dic47—42no—
100C2×Dic47—84no—
101C2×Dic47—81no—
102C2×Dic47—82no—
103C8×C47—82no—
104C47⋊C8—42no—
105#7824—471nomaximal, Sylow
106C22×C47—42no—
107C22×C47—44no—
108C22×C47—82no—
109C22×C47—81no—
110C22×C47—81no—
111C22×C47#45011yescharacteristic
112C22×C47#45212yes—
113C22×C47#45311yescharacteristic
114C22×C47#45412yes—
115C22×C47#45811yescharacteristic
116C4×C47—21no—
117C4×C47—21no—
118C4×C47—22no—
119C4×C47—22no—
120C4×C47—81no—
121Dic47—82no—
122Dic47—82no—
123Dic47—161no—
124#2724—471no—
125#2100—472no—
126#2094—471no—
127#1271—472no—
128#1270D4711yescharacteristic
129C2×C47—82no—
130C2×C47#161311yescharacteristic
131C2×C47#161512yes—
132C2×C47#163011yescharacteristic
133C2×C47#163512yes—
134C2×C47#165711yescharacteristic
135C2×#123—941no—
136C2×#106—941no—
137C2×#102—941no—
138#398—941no—
139#397—942no—
140#390C2×D4711yescharacteristic
141#389—472no—
142#385—942no—
143#384—471no—
144#340—942no—
145#340—942no—
146#336—471no—
147#336C2×D4712yes—
148C47#782411yescharacteristic, complemented, Sylow
149C22×D4D18811yescharacteristic
150C23×C4—941no—
151C22×C8—22no—
152C22×C8—941no—
153#123—1882no—
154#123—1882no—
155C2×C4⋊C4—472no—
156C2×C4⋊C4—472no—
157C2×C4⋊C4—941no—
158C2×C4⋊C4—942no—
159C2×C4⋊C4—941no—
160C2×C4⋊C4—942no—
161C2×C4⋊C4D18811yescharacteristic
162C2×C22⋊C4—22no—
163C2×C22⋊C4—942no—
164C2×C22⋊C4—941no—
165C2×C42C22×D4711yescharacteristic
166#106—1882no—
167#102—1882no—
168#95—942no—
169#95—941no—
170#95—942no—
171C24—21no—
172C2×D4—21no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4#1062532512yes—
176C22×C4—21no—
177C22×C4—942no—
178C22×C4—942no—
179C22×C4—3761no—
180C22×C4—3761no—
181C22×C4—1881no—
182C22×C4—1882no—
183C22×C4C2×D18811yescharacteristic
184C22×C4D4×D4712yes—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—942no—
189C2×C8—1881no—
190C2×C8—1881no—
191C4⋊C4—41no—
192C4⋊C4—942no—
193C4⋊C4—942no—
194C4⋊C4—3761no—
195C4⋊C4—1881no—
196C4⋊C4—1881no—
197C4⋊C4—1884no—
198C4⋊C4—1882no—
199C4⋊C4—1882no—
200C4⋊C4—1882no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—1882no—
204C22⋊C4—1884no—
205C42—41no—
206C23—22no—
207C23—42no—
208C23—41no—
209C23—42no—
210C23—41no—
211C23#1309399411yescharacteristic
212D4—42no—
213D4—81no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—3764no—
221C2×C4—3761no—
222C2×C4—3762no—
223C2×C4—1882no—
224C2×C4—1882no—
225C2×C4—1882no—
226C2×C4—1882no—
227C2×C4—1882no—
228C2×C4—1882no—
229C2×C4C2×#1062532511yescharacteristic
230C2×C4#1309404312yes—
231C2×C4#1309405211yescharacteristic
232C2×C4#1309405312yes—
233C8—82no—
234C8—1882no—
235C22—44no—
236C22—42no—
237C22—81no—
238C22—82no—
239C22—81no—
240C22#1568927611yescharacteristic
241C22#1568928612yes—
242C22#1568929311yescharacteristic
243C22#1568929712yes—
244C22#1568931411yescharacteristic
245C4—21no—
246C4—21no—
247C4—22no—
248C4—22no—
249C4—81no—
250C4—3762no—
251C4—3762no—
252C4—7521no—
253C2—82no—
254C2#2024241011yescharacteristic
255C2#2024242012yes—
256C2#2024251011yescharacteristic
257C2#2024254112yes—
258C2#2024268711yescharacteristic
subgroupG/H
centerC23#13093994
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1270C2
socleC23×C47#120
prime radicalC22×D4×C47D4
hypercenter#1270D47
hypocenterC47#7824
LCSG ▷ C2×C42×C47 ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #20244952C2semidirect
4. #20241210C2semidirect
148. C47#7824semidirect

Maximal subgroups

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

subgroupindex
#202503182
#202449522
#202449212
#202412102
C47×#12702
#782447