Finite Group Databasebeta

← #26109786

Subgroups — #26109786

Order 12032.

786 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 107 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#20243303C212yesmaximal
2C47⋊#1304C214yesmaximal, complemented
3C47×#2309C211yesmaximal, characteristic
4C2×#13093939—24no—
5#15688979C2214yes—
6C23×D4×C47C2211yescharacteristic
7C2×#117×C47C412yescomplemented
8C47×#390—24no—
9C47×#379C412yescomplemented
10C47×#379C2212yes—
11C2×C22⋊Dic47—24no—
12C2×C22⋊Dic47—28no—
13#13093939—48no—
14C25×C47C2×C411yescharacteristic
15C25×C47C2311yescharacteristic
16C22×D4×C47—24no—
17C22×D4×C47—44no—
18C22×D4×C47D414yes—
19C23×C4×C47C2×C411yescharacteristic
20#117×C47—44no—
21C2×C4⋊C4×C47—22no—
22C2×C22⋊C4×C47—24no—
23C2×C22⋊C4×C47—24no—
24C2×C22⋊C4×C47C2×C414yes—
25C2×C42×C47—22no—
26C24×C47—22no—
27C24×C47—42no—
28C24×C47—42no—
29C24×C47—82no—
30C24×C47—82no—
31C24×C47C22⋊C412yes—
32C24×C47C22×C411yescharacteristic
33C24×C47C2×D412yes—
34C24×C47#4114yes—
35C2×D4×C47—28no—
36C2×D4×C47—28no—
37C2×D4×C47—44no—
38C2×D4×C47—416no—
39C22×C4×C47—22no—
40C22×C4×C47—24no—
41C22×C4×C47—24no—
42C22×C4×C47C22⋊C412yes—
43C22×Dic47—44no—
44C4⋊C4×C47—44no—
45C22⋊C4×C47—44no—
46C22⋊C4×C47—84no—
47C22⋊C4×C47—84no—
48C42×C47—44no—
49C22⋊Dic47—48no—
50C22⋊Dic47—88no—
51C23×C47—22no—
52C23×C47—24no—
53C23×C47—22no—
54C23×C47—24no—
55C23×C47—42no—
56C23×C47—48no—
57C23×C47—44no—
58C23×C47—44no—
59C23×C47—44no—
60C23×C47—44no—
61C23×C47—84no—
62C23×C47—88no—
63C23×C47—88no—
64C23×C47C2×C22⋊C411yescharacteristic
65C23×C47#11712yes—
66C23×C47#12314yes—
67D4×C47—416no—
68C2×C4×C47—28no—
69C2×C4×C47—22no—
70C2×C4×C47—22no—
71C2×C4×C47—48no—
72C2×C4×C47—44no—
73C2×C4×C47#9914yes—
74C2×Dic47—84no—
75C2×Dic47—88no—
76#8749—471nomaximal, Sylow
77C22×C47—22no—
78C22×C47—24no—
79C22×C47—22no—
80C22×C47—28no—
81C22×C47—24no—
82C22×C47—44no—
83C22×C47—42no—
84C22×C47—44no—
85C22×C47—84no—
86C22×C47—84no—
87C22×C47—88no—
88C22×C47—88no—
89C22×C47#38111yescharacteristic
90C22×C47C2×#9912yes—
91C4×C47—24no—
92C4×C47—84no—
93Dic47—164no—
94#2309D4711yescharacteristic
95#1781—472no—
96#1304—474no—
97C2×C47—22no—
98C2×C47—44no—
99C2×C47—42no—
100C2×C47—82no—
101C2×C47#181911yescharacteristic
102C2×C47#182812yes—
103C23×D4C2×D4711yescharacteristic
104C2×#117Dic4712yescomplemented
105C2×#99—944no—
106#390—24no—
107#379—474no—
108#379Dic4712yescomplemented
109#379C2×D4712yes—
110C47#874911yescharacteristic, complemented, Sylow
111C25C2×Dic4711yescharacteristic
112C25C22×D4711yescharacteristic
113C22×D4—24no—
114C22×D4—44no—
115C22×D4C22⋊D4714yes—
116C23×C4C2×Dic4711yescharacteristic
117#117—44no—
118C2×C4⋊C4—22no—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—948no—
122C2×C22⋊C4—944no—
123C2×C22⋊C4C2×Dic4714yes—
124C2×C42—22no—
125#99—1888no—
126C24—22no—
127C24—42no—
128C24—42no—
129C24—82no—
130C24—82no—
131C24C22⋊Dic4712yes—
132C24#1062535014yes—
133C24C22×Dic4711yescharacteristic
134C24C2×C22⋊D4712yes—
135C2×D4—28no—
136C2×D4—28no—
137C2×D4—44no—
138C2×D4—416no—
139C22×C4—22no—
140C22×C4—24no—
141C22×C4—24no—
142C22×C4—1884no—
143C22×C4C22⋊Dic4712yes—
144C4⋊C4—44no—
145C22⋊C4—44no—
146C22⋊C4—84no—
147C22⋊C4—84no—
148C22⋊C4—3768no—
149C22⋊C4—1888no—
150C42—44no—
151C23—22no—
152C23—24no—
153C23—22no—
154C23—24no—
155C23—42no—
156C23—44no—
157C23—48no—
158C23—44no—
159C23—44no—
160C23—44no—
161C23—84no—
162C23—88no—
163C23—88no—
164C23C2×C22⋊Dic4711yescharacteristic
165C23#1309398312yes—
166C23#1309400714yes—
167D4—416no—
168C2×C4—28no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—48no—
172C2×C4—44no—
173C2×C4—3768no—
174C2×C4—3764no—
175C2×C4#1309393914yes—
176C22—22no—
177C22—22no—
178C22—24no—
179C22—28no—
180C22—24no—
181C22—44no—
182C22—42no—
183C22—44no—
184C22—84no—
185C22—84no—
186C22—88no—
187C22—88no—
188C22#1568898611yescharacteristic
189C22C2×#1309393912yes—
190C4—24no—
191C4—84no—
192C4—7524no—
193C2—22no—
194C2—44no—
195C2—42no—
196C2—82no—
197C2#2024346211yescharacteristic
198C2#2024349612yes—
subgroupG/H
centerC22#15688986
commutatorC24×C47C22×C4
FrattiniC25C22×D47
FittingC47×#2309C2
socleC22×C47#381
prime radicalC23×D4×C47C22
cosocleC25×C47C23
hypercenter#2309D47
hypocenterC47#8749
LCSG ▷ C24×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C25 ◁ #2309

Complements

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

NH ≅ G/Ntype
2. C47⋊#1304C2semidirect
7. C2×#117×C47C4semidirect
9. C47×#379C4semidirect
104. C2×#117Dic47semidirect
108. #379Dic47semidirect
110. C47#8749semidirect

Maximal subgroups

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

subgroupindex
#202433032
C47⋊#13042
C47×#23092
#874947