Finite Group Databasebeta

← #26103788

Subgroups — #26103788

Order 12032.

282 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 43 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#20241513C211yesmaximal, characteristic
2#20241320C211yesmaximal, characteristic
3C47×#2013C211yesmaximal, characteristic
4C2×#13093939—21no—
5#15688849—21no—
6C22×SD16×C47—21no—
7C2×C4⋊Q8×C47C411yescharacteristic, complemented
8C2×#103×C47—21no—
9C47×#390C2211yescharacteristic
10C47×#336C411yescharacteristic, complemented
11C2×C22⋊Dic47—41no—
12#13093939—22no—
13#13093939—41no—
14C22⋊(C47⋊C8)—41no—
15C22×Q8×C47—21no—
16C22×D4×C47D411yescharacteristic
17C2×SD16×C47—42no—
18C2×SD16×C47—42no—
19C2×SD16×C47—41no—
20C2×SD16×C47—41no—
21C22×C8×C47—21no—
22C4⋊Q8×C47—42no—
23C2×C4⋊C4×C47—41no—
24C2×C4⋊C4×C47D411yescharacteristic
25C2×C22⋊C4×C47—41no—
26C2×C42×C47C2×C411yescharacteristic
27#103×C47—42no—
28C24×C47—21no—
29C2×Q8×C47—22no—
30C2×Q8×C47—41no—
31C2×Q8×C47—41no—
32C2×Q8×C47—41no—
33C2×Q8×C47—41no—
34C2×Q8×C47—41no—
35C2×D4×C47—21no—
36C2×D4×C47—41no—
37C2×D4×C47—41no—
38C2×D4×C47D811yescharacteristic
39C2×D4×C47SD1611yescharacteristic
40C22×C4×C47—21no—
41C22×C4×C47—21no—
42C22×C4×C47—41no—
43C22×C4×C47C22⋊C411yescharacteristic
44C22×Dic47—81no—
45SD16×C47—44no—
46SD16×C47—44no—
47C2×C8×C47—22no—
48C2×C8×C47—41no—
49C2×C8×C47—41no—
50C4⋊C4×C47—41no—
51C4⋊C4×C47—42no—
52C4⋊C4×C47—42no—
53C22⋊C4×C47—82no—
54C42×C47—41no—
55C22⋊Dic47—42no—
56C22⋊Dic47—81no—
57C2×C47⋊C8—81no—
58C23×C47—21no—
59C23×C47—21no—
60C23×C47—41no—
61C23×C47—41no—
62C23×C47—81no—
63C23×C47—81no—
64C23×C47#9911yescharacteristic
65Q8×C47—42no—
66Q8×C47—42no—
67Q8×C47—81no—
68Q8×C47—81no—
69D4×C47—41no—
70D4×C47—41no—
71D4×C47—81no—
72C2×C4×C47—21no—
73C2×C4×C47—21no—
74C2×C4×C47—22no—
75C2×C4×C47—41no—
76C2×C4×C47—41no—
77C2×C4×C47—42no—
78C2×C4×C47—41no—
79C2×C4×C47—81no—
80C2×C4×C47—81no—
81C2×C4×C47—81no—
82C2×C4×C47#10211yescharacteristic
83C2×C4×C47#10411yescharacteristic
84C2×Dic47—82no—
85C2×Dic47—161no—
86C2×Dic47—161no—
87C8×C47—42no—
88C47⋊C8—161no—
89#7403—471nomaximal, Sylow
90C22×C47—21no—
91C22×C47—21no—
92C22×C47—41no—
93C22×C47—41no—
94C22×C47—81no—
95C22×C47—81no—
96C22×C47—81no—
97C22×C47—81no—
98C22×C47—161no—
99C22×C47#32711yescharacteristic
100C22×C47AΣL(1,𝔽16)11yescharacteristic
101C22×C47#35311yescharacteristic
102C4×C47—21no—
103C4×C47—21no—
104C4×C47—42no—
105C4×C47—82no—
106C4×C47—81no—
107Dic47—162no—
108#2013D4711yescharacteristic
109#1393—471no—
110#1318—471no—
111C2×C47—21no—
112C2×C47—21no—
113C2×C47—81no—
114C2×C47—81no—
115C2×C47#134311yescharacteristic
116C2×C47#140711yescharacteristic
117C2×C47#140911yescharacteristic
118C22×SD16—21no—
119C2×C4⋊Q8Dic4711yescharacteristic, complemented
120C2×#103—21no—
121C2×#99—941no—
122#390C2×D4711yescharacteristic
123#336Dic4711yescharacteristic, complemented
124#324—941no—
125C47#740311yescharacteristic, complemented, Sylow
126C22×Q8—21no—
127C22×D4C22⋊D4711yescharacteristic
128C2×SD16—42no—
129C2×SD16—42no—
130C2×SD16—41no—
131C2×SD16—41no—
132C22×C8—21no—
133C4⋊Q8—42no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4C22⋊D4711yescharacteristic
136C2×C22⋊C4—41no—
137C2×C22⋊C4—1881no—
138C2×C42C2×Dic4711yescharacteristic
139#103—42no—
140#99—942no—
141#99—1881no—
142C22⋊C8—1881no—
143C24—21no—
144C2×Q8—22no—
145C2×Q8—41no—
146C2×Q8—41no—
147C2×Q8—41no—
148C2×Q8—41no—
149C2×Q8—41no—
150C2×D4—21no—
151C2×D4—41no—
152C2×D4—41no—
153C2×D4#1062533411yescharacteristic
154C2×D4#1062533511yescharacteristic
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—41no—
158C22×C4—3761no—
159C22×C4C22⋊Dic4711yescharacteristic
160SD16—44no—
161SD16—44no—
162C2×C8—22no—
163C2×C8—41no—
164C2×C8—41no—
165C2×C8—3761no—
166C4⋊C4—41no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C22⋊C4—82no—
170C22⋊C4—3761no—
171C22⋊C4—1882no—
172C42—41no—
173C23—21no—
174C23—21no—
175C23—41no—
176C23—41no—
177C23—81no—
178C23—81no—
179C23#1309393911yescharacteristic
180Q8—42no—
181Q8—42no—
182Q8—81no—
183Q8—81no—
184D4—41no—
185D4—41no—
186D4—81no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—81no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—3762no—
198C2×C4—7521no—
199C2×C4—7521no—
200C2×C4#1309394511yescharacteristic
201C2×C4#1309395111yescharacteristic
202C8—42no—
203C8—7521no—
204C22—21no—
205C22—21no—
206C22—41no—
207C22—41no—
208C22—81no—
209C22—81no—
210C22—81no—
211C22—81no—
212C22—161no—
213C22#1568885711yescharacteristic
214C22#1568891011yescharacteristic
215C22#1568891611yescharacteristic
216C4—21no—
217C4—21no—
218C4—42no—
219C4—81no—
220C4—82no—
221C4—7522no—
222C2—21no—
223C2—21no—
224C2—81no—
225C2—81no—
226C2#2024138511yescharacteristic
227C2#2024154811yescharacteristic
228C2#2024155311yescharacteristic
subgroupG/H
centerC22#15688916
commutatorC2×C42×C47C2×C4
Frattini#390C2×D47
FittingC47×#2013C2
socleC22×C47#353
prime radicalC22×D4×C47D4
cosocleC47×#390C22
hypercenter#2013D47
hypocenterC47#7403
LCSG ▷ C2×C42×C47 ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #390 ◁ #2013

Complements

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

NH ≅ G/Ntype
7. C2×C4⋊Q8×C47C4semidirect
10. C47×#336C4semidirect
119. C2×C4⋊Q8Dic47semidirect
123. #336Dic47semidirect
125. C47#7403semidirect

Maximal subgroups

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

subgroupindex
#202415132
#202413202
C47×#20132
#740347