Finite Group Databasebeta

← #26102956

Subgroups — #26102956

Order 12032.

286 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 49 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#20244398C211yesmaximal, characteristic
2#20241312C211yesmaximal, characteristic
3C47×#1289C211yesmaximal, characteristic
4C22×Dic188C411yescharacteristic, complemented
5C2×#13094003—21no—
6C2×#13093949—21no—
7C22⋊(C47⋊C16)—21no—
8#15688886C411yescharacteristic, complemented
9C2×C22⋊C8×C47C2211yescharacteristic
10C47×#380—21no—
11C22×Dic94—21no—
12C2×Dic188—41no—
13C2×Dic188—42no—
14#13094003—42no—
15#13094003—42no—
16C2×C4⋊Dic47—21no—
17C2×Dic47⋊C4—41no—
18C2×C22⋊Dic47—41no—
19C2×C47⋊C16—41no—
20#13093949—42no—
21C23×C4×C47D411yescharacteristic
22C22×C8×C47C2×C411yescharacteristic
23C22×C8×C47D411yescharacteristic
24C2×C4⋊C4×C47—41no—
25C2×C22⋊C4×C47—41no—
26C22⋊C8×C47—41no—
27#95×C47—22no—
28C24×C47D811yescharacteristic
29C22×C4×C47—21no—
30C22×C4×C47—21no—
31C22×C4×C47—41no—
32C22×C4×C47—41no—
33C22×C4×C47—42no—
34C22×C4×C47C22⋊C411yescharacteristic
35C22×C4×C47SD1611yescharacteristic
36C22×Dic47—41no—
37C22×Dic47—41no—
38C2×Dic94—22no—
39C2×Dic94—41no—
40C2×Dic94—41no—
41C2×Dic94—41no—
42C2×Dic94—41no—
43C2×Dic94—41no—
44C2×C8×C47—21no—
45C2×C8×C47—21no—
46C2×C8×C47—41no—
47C2×C8×C47—41no—
48C4⋊C4×C47—81no—
49C4⋊C4×C47—81no—
50C22⋊C4×C47—81no—
51C22⋊C4×C47—81no—
52C22⋊Dic47—44no—
53C4⋊Dic47—22no—
54C4⋊Dic47—41no—
55Dic47⋊C4—44no—
56Dic188—42no—
57Dic188—42no—
58C47⋊C16—81no—
59C23×C47—21no—
60C23×C47—21no—
61C23×C47—41no—
62C23×C47—41no—
63C23×C47#10211yescharacteristic
64C23×C47SD3211yescharacteristic
65C23×C47Q3211yescharacteristic
66C2×C4×C47—21no—
67C2×C4×C47—21no—
68C2×C4×C47—21no—
69C2×C4×C47—41no—
70C2×C4×C47—41no—
71C2×C4×C47—41no—
72C2×C4×C47—81no—
73C2×C4×C47—82no—
74C2×C4×C47—82no—
75C2×C4×C47—82no—
76C2×C4×C47#9911yescharacteristic
77C2×C4×C47#10411yescharacteristic
78C2×Dic47—42no—
79C2×Dic47—42no—
80C2×Dic47—81no—
81C2×Dic47—81no—
82C2×Dic47—81no—
83C2×Dic47—81no—
84Dic94—42no—
85Dic94—42no—
86Dic94—81no—
87Dic94—81no—
88C8×C47—41no—
89C8×C47—41no—
90#7078—471nomaximal, Sylow
91C22×C47—21no—
92C22×C47—21no—
93C22×C47—21no—
94C22×C47—21no—
95C22×C47—41no—
96C22×C47—41no—
97C22×C47—41no—
98C22×C47—41no—
99C22×C47—81no—
100C22×C47#32711yescharacteristic
101C22×C47#35811yescharacteristic
102C22×C47#36011yescharacteristic
103C4×C47—21no—
104C4×C47—41no—
105C4×C47—162no—
106C4×C47AΣL(1,𝔽16)11yescharacteristic
107C4×C47#35311yescharacteristic
108Dic47—82no—
109Dic47—82no—
110#2001—471no—
111#1314—471no—
112#1289D4711yescharacteristic
113C2×C47—21no—
114C2×C47—21no—
115C2×C47—41no—
116C2×C47—41no—
117C2×C47#134011yescharacteristic
118C2×C47#134311yescharacteristic
119C2×C47#134811yescharacteristic
120C22×Q16—471no—
121C2×C22⋊Q8—941no—
122C2×#103—941no—
123C2×C22⋊C8C2×D4711yescharacteristic
124#380—21no—
125C22⋊C16—941no—
126#340—471no—
127C47#707811yescharacteristic, complemented, Sylow
128C22×Q8—941no—
129C23×C4C22⋊D4711yescharacteristic
130C2×Q16—1881no—
131C2×Q16—1882no—
132C22×C8C4×D4711yescharacteristic
133C22×C8D18811yescharacteristic
134C22⋊Q8—1882no—
135C22⋊Q8—1882no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—941no—
138C2×C4⋊C4—1881no—
139C2×C22⋊C4—41no—
140C2×C22⋊C4—1881no—
141C2×C16—1881no—
142#103—1882no—
143C22⋊C8—41no—
144#95—22no—
145C24#1062533411yescharacteristic
146C2×Q8—942no—
147C2×Q8—1881no—
148C2×Q8—1881no—
149C2×Q8—1881no—
150C2×Q8—1881no—
151C2×Q8—1881no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—42no—
157C22×C4—1881no—
158C22×C4—1881no—
159C22×C4(C2×D47)⋊C411yescharacteristic
160C22×C4#1062533611yescharacteristic
161Q16—1882no—
162Q16—1882no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—41no—
166C2×C8—41no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C4⋊C4—942no—
170C4⋊C4—1881no—
171C4⋊C4—1884no—
172C22⋊C4—81no—
173C22⋊C4—81no—
174C22⋊C4—1884no—
175C16—3761no—
176C23—21no—
177C23—21no—
178C23—41no—
179C23—41no—
180C23#1309394411yescharacteristic
181C23#1309396811yescharacteristic
182C23#1309397011yescharacteristic
183Q8—3761no—
184Q8—3761no—
185Q8—1882no—
186Q8—1882no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—81no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—3761no—
198C2×C4—3761no—
199C2×C4—3761no—
200C2×C4—3761no—
201C2×C4—1882no—
202C2×C4—1882no—
203C2×C4#1309393811yescharacteristic
204C2×C4#1309395211yescharacteristic
205C8—41no—
206C8—41no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—41no—
212C22—41no—
213C22—41no—
214C22—41no—
215C22—81no—
216C22#1568885611yescharacteristic
217C22#1568892611yescharacteristic
218C22#1568893211yescharacteristic
219C4—21no—
220C4—41no—
221C4—162no—
222C4—3762no—
223C4—3762no—
224C4#1568890911yescharacteristic
225C4#1568891511yescharacteristic
226C2—21no—
227C2—21no—
228C2—41no—
229C2—41no—
230C2#2024137511yescharacteristic
231C2#2024138411yescharacteristic
232C2#2024139911yescharacteristic
subgroupG/H
centerC22#15688856
commutatorC22×C8×C47C2×C4
FrattiniC2×C22⋊C8C2×D47
FittingC47×#1289C2
socleC22×C47#327
prime radicalC24×C47D8
cosocleC2×C22⋊C8×C47C22
hypercenter#1289D47
hypocenterC47#7078
LCSG ▷ C22×C8×C47 ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #1289

Complements

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

NH ≅ G/Ntype
4. C22×Dic188C4semidirect
8. #15688886C4semidirect
127. C47#7078semidirect

Maximal subgroups

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

subgroupindex
#202443982
#202413122
C47×#12892
#707847