Finite Group Databasebeta

← #26143439

Subgroups — #26143439

Order 12032.

3998 conjugacy classes of subgroups; only the 467 normal subgroups are shown.

normal subgroups only

This group has 3998 conjugacy classes of subgroups — too many to draw. Only its 467 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
1C23×Q8×D47C211yesmaximal, characteristic
2C2×#15688985C216yesmaximal
3#20241659C211yesmaximal, characteristic, complemented
4#20241658C216yesmaximal, complemented
5C47×#1446C211yesmaximal, characteristic, complemented
6C23×Dic94C2213yes—
7C22×Q8×D47—224no—
8C22×Q8×D47C418yescomplemented
9C23×C4×D47C2213yes—
10C22×(C2×D47)⋊C4C2214yes—
11#15688985—224no—
12C2×C47⋊#95C22112yes—
13C2×C47⋊#95C2216yes—
14C23×Q8×C47C2211yescharacteristic
15C2×#95×C47C2216yes—
16C24×D47C2311yescharacteristic
17C22×Dic94—212no—
18C22×Dic94—224no—
19C22×Dic94C2×C4112yes—
20C2×Q8×D47—2128no—
21C2×Q8×D47—296no—
22C23×Dic47C2314yes—
23C23×Dic47C2313yes—
24C22×C4×D47—224no—
25C22×C4×D47C2×C4112yes—
26C22×C4×D47D4124yes—
27C2×(C2×D47)⋊C4—224no—
28C47⋊#95—212no—
29C47⋊#95—224no—
30C22×Q8×C47—212no—
31C22×Q8×C47C2×C414yes—
32C23×C4×C47C2314yes—
33C23×C4×C47C2313yes—
34#95×C47—212no—
35C24×C47C2411yescharacteristic
36C23×D47—24no—
37C23×D47—24no—
38C23×D47C22×C412yes—
39C23×D47C2×D4112yes—
40C2×Q8×C47—232no—
41C2×Q8×C47—224no—
42C22×C4×C47—212no—
43C22×C4×C47—212no—
44C22×C4×C47—212no—
45C22×C4×C47—24no—
46C22×C4×C47C22×C416yes—
47C22×C4×C47C2×D4112yes—
48C22×Dic47—212no—
49C22×Dic47—212no—
50C22×Dic47—24no—
51C22×Dic47—212no—
52C22×Dic47C22×C416yes—
53C22×Dic47C2×D4112yes—
54Q8×D47—2256no—
55C2×C4×D47—248no—
56C2×C4×D47—296no—
57C2×C4×D47C22⋊C4148yes—
58C2×Dic94—296no—
59C2×Dic94—224no—
60C2×Dic94—248no—
61(C2×D47)⋊C4—416no—
62C23×C47C23×C411yescharacteristic
63C23×C47C22×D416yes—
64C23×C472−1+414yes—
65C23×C472−1+414yes—
66C22×D47—28no—
67C22×D47—212no—
68C22×D47—224no—
69C22×D47—212no—
70C22×D47C2×C22⋊C4112yes—
71C22×D47#120116yes—
72Q8×C47—232no—
73C2×C4×C47—212no—
74C2×C4×C47—224no—
75C2×C4×C47—44no—
76C2×C4×C47—412no—
77C2×C4×C47—412no—
78C2×C4×C47C2×C22⋊C4112yes—
79C2×Dic47—212no—
80C2×Dic47—224no—
81C2×Dic47—44no—
82C2×Dic47—412no—
83C2×Dic47—412no—
84C2×Dic47C2×C22⋊C4112yes—
85C4×D47—296no—
86Dic94—296no—
87#14336—471nomaximal, Sylow
88C22×C47C22×C22⋊C413yes—
89C22×C47#51914yes—
90C22×C47C2×#12014yes—
91C22×C47#53616yes—
92C22×C47#536112yes—
93C22×C47#53616yes—
94C2×D47—224no—
95C2×D47—216no—
96C2×D47—216no—
97C2×D47#37918yes—
98C4×C47—212no—
99C4×C47—84no—
100Dic47—212no—
101Dic47—84no—
102C24×Q8—471no—
103C2×#381—476no—
104#1446—476no—
105#1446—471no—
106#1446D4711yescharacteristic, complemented
107C2×C47C2×#37911yescharacteristic
108C2×C47#231016yes—
109C2×C47#261414yes—
110C2×C47#261414yes—
111D47—28no—
112C23×Q8—478no—
113C23×Q8—473no—
114C23×Q8—9424no—
115C23×Q8C2×D4711yescharacteristic
116C24×C4—473no—
117C22×C22⋊C4—474no—
118#381—9424no—
119C2×#95—4712no—
120C2×#95—476no—
121C2×#95C2×D4716yes—
122C47#1433611yescharacteristic, complemented, Sylow
123C25—471no—
124C22×Q8—212no—
125C22×Q8—4712no—
126C22×Q8—9424no—
127C22×Q8—9496no—
128C22×Q8—94128no—
129C22×Q8—9412no—
130C22×Q8C4×D4714yes—
131C23×C4—473no—
132C23×C4—4712no—
133C23×C4—4724no—
134C23×C4—474no—
135C23×C4—9424no—
136C23×C4C22×D4714yes—
137C23×C4C22×D4713yes—
138C2×C22⋊C4—9424no—
139#95—212no—
140#95—9424no—
141#95—9412no—
142C24—472no—
143C24—4712no—
144C24—944no—
145C24—944no—
146C24C23×D4711yescharacteristic
147C2×Q8—232no—
148C2×Q8—224no—
149C2×Q8—9448no—
150C2×Q8—9496no—
151C2×Q8—94256no—
152C2×Q8—9424no—
153C22×C4—212no—
154C22×C4—212no—
155C22×C4—212no—
156C22×C4—24no—
157C22×C4—476no—
158C22×C4—4712no—
159C22×C4—4748no—
160C22×C4—9448no—
161C22×C4—9496no—
162C22×C4—9412no—
163C22×C4—9412no—
164C22×C4—944no—
165C22×C4—9412no—
166C22×C4C2×C4×D4716yes—
167C22×C4D4×D47112yes—
168C22⋊C4—18816no—
169C23—4712no—
170C23—4716no—
171C23—948no—
172C23—9412no—
173C23—9424no—
174C23—9412no—
175C23C22×C4×D4711yescharacteristic
176C23C2×D4×D4716yes—
177C23#1309407114yes—
178C23#1309407214yes—
179Q8—232no—
180Q8—9496no—
181C2×C4—212no—
182C2×C4—224no—
183C2×C4—44no—
184C2×C4—412no—
185C2×C4—412no—
186C2×C4—4712no—
187C2×C4—9496no—
188C2×C4—9412no—
189C2×C4—9424no—
190C2×C4—1884no—
191C2×C4—18812no—
192C2×C4—18812no—
193C2×C4C22⋊C4×D47112yes—
194C22—478no—
195C22—9416no—
196C22—9416no—
197C22—9424no—
198C22C2×C22⋊C4×D4713yes—
199C22#1568959914yes—
200C22#120×D4714yes—
201C22#1568969316yes—
202C22#15689696112yes—
203C22#1568969816yes—
204C4—212no—
205C4—84no—
206C4—9412no—
207C4—3764no—
208C2—948no—
209C2#379×D4711yescharacteristic
210C2#2024597116yes—
211C2#2024910614yes—
212C2#2024910814yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×#1446C2
socle, cosocleC24×C47C24
prime radicalC24×D47C23
hypercenter#1446D47
hypocenterC47#14336
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1446

Complements

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

NH ≅ G/Ntype
3. #20241659C2semidirect
4. #20241658C2semidirect
5. C47×#1446C2semidirect
8. C22×Q8×D47C4semidirect
106. #1446D47semidirect
122. C47#14336semidirect

Maximal subgroups

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

subgroupindex
C23×Q8×D472
C2×#156889852
#202416592
#202416582
C47×#14462
#1433647