Finite Group Databasebeta

← #26102957

Subgroups — #26102957

Order 12032.

286 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 51 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#20241312C211yesmaximal, characteristic
2#20241256C211yesmaximal, characteristic
3C47×#2001C211yesmaximal, characteristic
4#15688983—21no—
5C22⋊(C47⋊C16)—21no—
6C22×Q16×C47C411yescharacteristic, complemented
7C2×C22⋊Q8×C47—21no—
8C2×#103×C47—21no—
9C2×C22⋊C8×C47C2211yescharacteristic
10C47×#340C411yescharacteristic, complemented
11C2×C4⋊Dic47—41no—
12C2×C22⋊Dic47—41no—
13C2×C47⋊C16—41no—
14C47⋊#95—22no—
15C22×Q8×C47—21no—
16C23×C4×C47D411yescharacteristic
17C2×Q16×C47—41no—
18C2×Q16×C47—42no—
19C22×C8×C47C2×C411yescharacteristic
20C22×C8×C47D411yescharacteristic
21C22⋊Q8×C47—42no—
22C22⋊Q8×C47—42no—
23C2×C4⋊C4×C47—21no—
24C2×C4⋊C4×C47—41no—
25C2×C22⋊C4×C47—41no—
26#103×C47—42no—
27C22⋊C8×C47—41no—
28C24×C47D811yescharacteristic
29C2×Q8×C47—22no—
30C2×Q8×C47—41no—
31C2×Q8×C47—41no—
32C2×Q8×C47—41no—
33C2×Q8×C47—41no—
34C2×Q8×C47—41no—
35C22×C4×C47—21no—
36C22×C4×C47—21no—
37C22×C4×C47—41no—
38C22×C4×C47—41no—
39C22×C4×C47—41no—
40C22×C4×C47—41no—
41C22×C4×C47C22⋊C411yescharacteristic
42C22×C4×C47SD1611yescharacteristic
43C22×Dic47—42no—
44Q16×C47—42no—
45Q16×C47—42no—
46C2×C8×C47—21no—
47C2×C8×C47—21no—
48C2×C8×C47—41no—
49C2×C8×C47—41no—
50C4⋊C4×C47—22no—
51C4⋊C4×C47—41no—
52C4⋊C4×C47—44no—
53C22⋊C4×C47—44no—
54C22⋊Dic47—81no—
55C22⋊Dic47—81no—
56C4⋊Dic47—81no—
57C4⋊Dic47—81no—
58C47⋊C16—81no—
59C23×C47—21no—
60C23×C47—21no—
61C23×C47—41no—
62C23×C47—41no—
63C23×C47#10211yescharacteristic
64C23×C47SD3211yescharacteristic
65C23×C47Q3211yescharacteristic
66Q8×C47—42no—
67Q8×C47—42no—
68Q8×C47—81no—
69Q8×C47—81no—
70C2×C4×C47—21no—
71C2×C4×C47—21no—
72C2×C4×C47—21no—
73C2×C4×C47—41no—
74C2×C4×C47—41no—
75C2×C4×C47—41no—
76C2×C4×C47—42no—
77C2×C4×C47—42no—
78C2×C4×C47—81no—
79C2×C4×C47—81no—
80C2×C4×C47—81no—
81C2×C4×C47—81no—
82C2×C4×C47—81no—
83C2×C4×C47#9911yescharacteristic
84C2×C4×C47#10411yescharacteristic
85C2×Dic47—82no—
86C2×Dic47—82no—
87C2×Dic47—82no—
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—82no—
106C4×C47—82no—
107C4×C47AΣL(1,𝔽16)11yescharacteristic
108C4×C47#35311yescharacteristic
109Dic47—162no—
110#2001D4711yescharacteristic
111#1314—471no—
112#1289—471no—
113C2×C47—21no—
114C2×C47—21no—
115C2×C47—41no—
116C2×C47—41no—
117C2×C47#134011yescharacteristic
118C2×C47#134311yescharacteristic
119C2×C47#134811yescharacteristic
120C22×Q16Dic4711yescharacteristic, complemented
121C2×C22⋊Q8—21no—
122C2×#103—21no—
123C2×C22⋊C8C2×D4711yescharacteristic
124#380—941no—
125C22⋊C16—941no—
126#340Dic4711yescharacteristic, complemented
127C47#707811yescharacteristic, complemented, Sylow
128C22×Q8—21no—
129C23×C4C22⋊D4711yescharacteristic
130C2×Q16—41no—
131C2×Q16—42no—
132C22×C8C2×Dic4711yescharacteristic
133C22×C8C22⋊D4711yescharacteristic
134C22⋊Q8—42no—
135C22⋊Q8—42no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—1881no—
139C2×C22⋊C4—41no—
140C2×C22⋊C4—1881no—
141C2×C16—1881no—
142#103—42no—
143C22⋊C8—41no—
144#95—942no—
145C24#1062533411yescharacteristic
146C2×Q8—22no—
147C2×Q8—41no—
148C2×Q8—41no—
149C2×Q8—41no—
150C2×Q8—41no—
151C2×Q8—41no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—1882no—
159C22×C4#1062533511yescharacteristic
160C22×C4C22⋊Dic4711yescharacteristic
161Q16—42no—
162Q16—42no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—41no—
166C2×C8—41no—
167C4⋊C4—22no—
168C4⋊C4—41no—
169C4⋊C4—44no—
170C4⋊C4—3761no—
171C4⋊C4—3761no—
172C22⋊C4—44no—
173C22⋊C4—3761no—
174C22⋊C4—3761no—
175C16—3761no—
176C23—21no—
177C23—21no—
178C23—41no—
179C23—41no—
180C23#1309394511yescharacteristic
181C23#1309396711yescharacteristic
182C23#1309397011yescharacteristic
183Q8—42no—
184Q8—42no—
185Q8—81no—
186Q8—81no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—3762no—
201C2×C4—3762no—
202C2×C4—3762no—
203C2×C4#1309393911yescharacteristic
204C2×C4#1309395111yescharacteristic
205C8—41no—
206C8—41no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—41no—
212C22—41no—
213C22—41no—
214C22—41no—
215C22—81no—
216C22#1568885711yescharacteristic
217C22#1568892711yescharacteristic
218C22#1568893311yescharacteristic
219C4—21no—
220C4—41no—
221C4—82no—
222C4—82no—
223C4—7522no—
224C4#1568891011yescharacteristic
225C4#1568891611yescharacteristic
226C2—21no—
227C2—21no—
228C2—41no—
229C2—41no—
230C2#2024137611yescharacteristic
231C2#2024138511yescharacteristic
232C2#2024140011yescharacteristic
subgroupG/H
centerC22#15688857
commutatorC22×C8×C47C2×C4
FrattiniC2×C22⋊C8C2×D47
FittingC47×#2001C2
socleC22×C47#327
prime radicalC24×C47D8
cosocleC2×C22⋊C8×C47C22
hypercenter#2001D47
hypocenterC47#7078
LCSG ▷ C22×C8×C47 ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #2001

Complements

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

NH ≅ G/Ntype
6. C22×Q16×C47C4semidirect
10. C47×#340C4semidirect
120. C22×Q16Dic47semidirect
126. #340Dic47semidirect
127. C47#7078semidirect

Maximal subgroups

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

subgroupindex
#202413122
#202412562
C47×#20012
#707847