Finite Group Databasebeta

← #26109820

Subgroups — #26109820

Order 12032.

474 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 99 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#20241715C212yesmaximal
2#20241294C214yesmaximal, complemented
3C47×#2312C211yesmaximal, characteristic
4#15689087C2214yes—
5#15688847—24no—
6C22×C22⋊C4×C47C2211yescharacteristic
7C47×#388—24no—
8C47×#380C2212yes—
9C47×#379C412yes—
10C4×C22⋊C4×C47C412yes—
11C2×#10625329—24no—
12C22⋊(C47⋊C8)—24no—
13C22⋊(C47⋊C8)—24no—
14C25×C47C2×C411yescharacteristic
15C23×C4×C47C2×C411yescharacteristic
16C23×C4×C47C2311yescharacteristic
17C2×C4⋊C4×C47—22no—
18C2×C22⋊C4×C47—24no—
19C2×C22⋊C4×C47—22no—
20C2×C22⋊C4×C47—22no—
21C2×C22⋊C4×C47C2×C414yes—
22C2×C22⋊C4×C47D414yes—
23C2×C42×C47—22no—
24#95×C47—22no—
25#95×C47—22no—
26C24×C47—22no—
27C24×C47—42no—
28C24×C47—81no—
29C24×C47—81no—
30C24×C47C22⋊C412yes—
31C24×C47C22×C411yescharacteristic
32C22×C4×C47—22no—
33C22×C4×C47—24no—
34C22×C4×C47—81no—
35C22×C4×C47—81no—
36C22×C4×C47C22⋊C412yes—
37C22×C4×C47C2×D412yes—
38C22×C4×C47#4114yes—
39C4⋊C4×C47—82no—
40C22⋊C4×C47—28no—
41C22⋊C4×C47—44no—
42C22⋊C4×C47—82no—
43C22⋊C4×C47—82no—
44C42×C47—82no—
45#10625329—84no—
46C2×C47⋊C8—44no—
47C23×C47—22no—
48C23×C47—22no—
49C23×C47—24no—
50C23×C47—41no—
51C23×C47—41no—
52C23×C47—44no—
53C23×C47—42no—
54C23×C47—42no—
55C23×C47—82no—
56C23×C47—82no—
57C23×C47—88no—
58C23×C47C2×C22⋊C411yescharacteristic
59C23×C47#11712yes—
60C2×C4×C47—22no—
61C2×C4×C47—44no—
62C2×C4×C47—42no—
63C2×C4×C47—42no—
64C2×C4×C47—44no—
65C2×C4×C47—44no—
66C2×C4×C47—44no—
67C2×C4×C47—82no—
68C2×C4×C47#12314yes—
69C47⋊C8—84no—
70#8759—471nomaximal, Sylow
71C22×C47—21no—
72C22×C47—21no—
73C22×C47—22no—
74C22×C47—22no—
75C22×C47—22no—
76C22×C47—44no—
77C22×C47—42no—
78C22×C47—42no—
79C22×C47—42no—
80C22×C47—84no—
81C22×C47—88no—
82C22×C47#38111yescharacteristic
83C22×C47#41312yes—
84C4×C47—24no—
85C4×C47—42no—
86C4×C47—84no—
87#2312D4711yescharacteristic
88#1462—472no—
89#1305—474no—
90C2×C47—22no—
91C2×C47—42no—
92C2×C47—82no—
93C2×C47#182211yescharacteristic
94C2×C47#182811yescharacteristic
95C2×C47#182911yescharacteristic
96C22×C22⋊C4C2×D4711yescharacteristic
97#407—474no—
98#388—24no—
99#380C2×D4712yes—
100#379Dic4712yes—
101C4×C22⋊C4Dic4712yes—
102#323—944no—
103C47#875911yescharacteristic, complemented, Sylow
104C25C2×Dic4711yescharacteristic
105C23×C4C2×Dic4711yescharacteristic
106C23×C4C22×D4711yescharacteristic
107C2×M16—944no—
108C2×C4⋊C4—22no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4C2×Dic4714yes—
113C2×C22⋊C4C22⋊D4714yes—
114C2×C42—22no—
115C22⋊C8—944no—
116C22⋊C8—944no—
117#95—22no—
118#95—22no—
119C24—22no—
120C24—42no—
121C24—81no—
122C24—81no—
123C24C22⋊Dic4712yes—
124C24C22×Dic4711yescharacteristic
125C22×C4—22no—
126C22×C4—24no—
127C22×C4—81no—
128C22×C4—81no—
129C22×C4C22⋊Dic4712yes—
130C22×C4#1062535014yes—
131C22×C4C2×C22⋊D4712yes—
132M16—3764no—
133C2×C8—1884no—
134C4⋊C4—82no—
135C22⋊C4—28no—
136C22⋊C4—44no—
137C22⋊C4—82no—
138C22⋊C4—82no—
139C42—82no—
140C23—22no—
141C23—22no—
142C23—24no—
143C23—41no—
144C23—41no—
145C23—44no—
146C23—42no—
147C23—42no—
148C23—82no—
149C23—82no—
150C23—88no—
151C23C2×C22⋊Dic4711yescharacteristic
152C23#1309398312yes—
153C2×C4—22no—
154C2×C4—42no—
155C2×C4—44no—
156C2×C4—42no—
157C2×C4—44no—
158C2×C4—44no—
159C2×C4—44no—
160C2×C4—82no—
161C2×C4#1309400714yes—
162C8—3764no—
163C22—21no—
164C22—21no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—44no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22—84no—
173C22—88no—
174C22#1568898611yescharacteristic
175C22#1568910712yes—
176C4—24no—
177C4—42no—
178C4—84no—
179C2—22no—
180C2—42no—
181C2—82no—
182C2#2024347111yescharacteristic
183C2#2024349611yescharacteristic
184C2#2024349911yescharacteristic
subgroupG/H
centerC22#15688986
commutatorC24×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2312C2
socleC22×C47#381
prime radicalC25×C47C2×C4
cosocleC23×C4×C47C23
hypercenter#2312D47
hypocenterC47#8759
LCSG ▷ C24×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. #20241294C2semidirect
103. C47#8759semidirect

Maximal subgroups

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

subgroupindex
#202417152
#202412942
C47×#23122
#875947