Finite Group Databasebeta

← #26129069

Subgroups — #26129069

Order 12032.

400 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 59 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#20249053C211yesmaximal, characteristic
2#20242345C212yesmaximal
3#20241408C212yesmaximal
4#20241318C211yesmaximal, characteristic
5C47×#2030C211yesmaximal, characteristic
6#15689278—22no—
7#15689059—21no—
8#15689035C2212yes—
9#15689027—22no—
10#15689010—22no—
11C4×C22⋊Dic47—21no—
12#15688860—22no—
13#15688847C2212yes—
14C47×#543C2211yescharacteristic
15C47×#381—21no—
16C47×#381C2211yescharacteristic
17C47×#342—21no—
18C47×#342C2211yescharacteristic
19C22×Dic94D412yes—
20C2×#10625337—42no—
21C2×C4⋊Dic47—21no—
22C2×Dic47⋊C4—22no—
23C2×C22⋊Dic47—22no—
24C2×C4×Dic47—21no—
25C2×C4×Dic47—21no—
26#13093947—42no—
27C22⋊(C47⋊C8)—22no—
28C47⋊#95—22no—
29C47⋊#95—21no—
30C47⋊#95—22no—
31C47⋊#95D412yes—
32C23×C4×C47D411yescharacteristic
33C22⋊Q8×C47—22no—
34C22⋊Q8×C47—42no—
35C2×C22⋊C4×C47—21no—
36C2×C22⋊C4×C47—21no—
37C2×C22⋊C4×C47D411yescharacteristic
38C2×C22⋊C4×C47C2311yescharacteristic
39#95×C47—22no—
40#95×C47—42no—
41C24×C47C2×D411yescharacteristic
42C2×Q8×C47—22no—
43C22×C4×C47—21no—
44C22×C4×C47—21no—
45C22×C4×C47—21no—
46C22×C4×C47—22no—
47C22×C4×C47—41no—
48C22×C4×C47—41no—
49C22×C4×C47—41no—
50C22×C4×C47Q1612yescomplemented
51C22×C4×C47C2×D412yes—
52C22×Dic47—22no—
53C22×Dic47—22no—
54C2×Dic94—42no—
55C2×Dic94—44no—
56C4⋊C4×C47—22no—
57C4⋊C4×C47—42no—
58C22⋊C4×C47—41no—
59C22⋊C4×C47—41no—
60C22⋊C4×C47—41no—
61C22⋊C4×C47—41no—
62C22⋊C4×C47—41no—
63C22⋊C4×C47—41no—
64C22⋊Dic47—44no—
65#10625337—84no—
66C4⋊Dic47—81no—
67Dic47⋊C4—44no—
68C4×Dic47—81no—
69C4×Dic47—81no—
70C2×C47⋊C8—42no—
71C23×C47—21no—
72C23×C47—21no—
73C23×C47—42no—
74C23×C47#12011yescharacteristic
75C23×C47C2×Q1611yescharacteristic
76C23×C47#13711yescharacteristic
77Q8×C47—82no—
78C2×C4×C47—22no—
79C2×C4×C47—22no—
80C2×C4×C47—41no—
81C2×C4×C47—41no—
82C2×C4×C47—42no—
83C2×C4×C47—42no—
84C2×C4×C47—42no—
85C2×C4×C47—41no—
86C2×C4×C47—42no—
87C2×C4×C47—81no—
88C2×C4×C47—82no—
89C2×C4×C47—81no—
90C2×C4×C47—82no—
91C2×Dic47—24no—
92C2×Dic47—42no—
93C2×Dic47—42no—
94C2×Dic47—42no—
95C2×Dic47—42no—
96C2×Dic47—42no—
97Dic94—44no—
98Dic94—82no—
99C47⋊C8—82no—
100#12215—471nomaximal, Sylow
101C22×C47—22no—
102C22×C47—21no—
103C22×C47—21no—
104C22×C47—41no—
105C22×C47—41no—
106C22×C47—42no—
107C22×C47—81no—
108C22×C47#45111yescharacteristic
109C22×C47#45511yescharacteristic
110C22×C47B(4,𝔽2)11yescharacteristic
111C4×C47—42no—
112C4×C47—42no—
113C4×C47—81no—
114C4×C47—82no—
115C4×C47—161no—
116Dic47—44no—
117Dic47—82no—
118#2607—471no—
119#2030D4711yescharacteristic
120#1602—472no—
121#1351—472no—
122#1317—471no—
123C2×C47—21no—
124C2×C47—21no—
125C2×C47—41no—
126C2×C47—41no—
127C2×C47#160211yescharacteristic
128C2×C47#219712yes—
129#543C2×D4711yescharacteristic
130#451—942no—
131#399—941no—
132#393—472no—
133#391—942no—
134#387—942no—
135#381—21no—
136#381C2×D4711yescharacteristic
137C4×C22⋊C4—941no—
138#342—21no—
139#342C2×D4711yescharacteristic
140#328—942no—
141#323—472no—
142C47#1221511yescharacteristic, complemented, Sylow
143C22×Q8—472no—
144C23×C4C22⋊D4711yescharacteristic
145C2×Q16—1882no—
146C22⋊Q8—22no—
147C22⋊Q8—42no—
148C2×C4⋊C4—941no—
149C2×C4⋊C4—942no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—942no—
153C2×C22⋊C4C22⋊D4711yescharacteristic
154C2×C22⋊C4C22×D4711yescharacteristic
155C2×C42—941no—
156C2×C42—941no—
157#103—1882no—
158C22⋊C8—942no—
159#95—22no—
160#95—42no—
161#95—472no—
162#95—942no—
163#95—941no—
164#95—942no—
165C24C2×C22⋊D4711yescharacteristic
166C2×Q8—22no—
167C2×Q8—1882no—
168C2×Q8—1884no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—942no—
177C22×C4—942no—
178C22×C4#1062533712yescomplemented
179C22×C4D4×D4712yes—
180Q16—3764no—
181C2×C8—1882no—
182C4⋊C4—22no—
183C4⋊C4—42no—
184C4⋊C4—3761no—
185C4⋊C4—1884no—
186C22⋊C4—41no—
187C22⋊C4—41no—
188C22⋊C4—41no—
189C22⋊C4—41no—
190C22⋊C4—41no—
191C22⋊C4—41no—
192C22⋊C4—1884no—
193C42—3761no—
194C42—3761no—
195C23—21no—
196C23—21no—
197C23—42no—
198C23#1309399211yescharacteristic
199C23C2×#1062533711yescharacteristic
200C23#1309405411yescharacteristic
201Q8—82no—
202Q8—3762no—
203Q8—1884no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—81no—
214C2×C4—82no—
215C2×C4—81no—
216C2×C4—82no—
217C2×C4—944no—
218C2×C4—1882no—
219C2×C4—1882no—
220C2×C4—1882no—
221C2×C4—1882no—
222C2×C4—1882no—
223C8—3762no—
224C22—22no—
225C22—21no—
226C22—21no—
227C22—41no—
228C22—41no—
229C22—42no—
230C22—81no—
231C22#1568927811yescharacteristic
232C22#1568930011yescharacteristic
233C22#1568930911yescharacteristic
234C4—42no—
235C4—42no—
236C4—81no—
237C4—82no—
238C4—161no—
239C4—3762no—
240C4—1884no—
241C2—21no—
242C2—21no—
243C2—41no—
244C2—41no—
245C2#2024234411yescharacteristic
246C2#2024537512yes—
subgroupG/H
centerC22#15689309
commutator, cosocleC2×C22⋊C4×C47C23
FrattiniC2×C22⋊C4C22×D47
FittingC47×#2030C2
socleC22×C47B(4,𝔽2)
prime radicalC24×C47C2×D4
hypercenter#2030D47
hypocenterC47#12215
LCSG ▷ C2×C22⋊C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2030

Complements

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

NH ≅ G/Ntype
50. C22×C4×C47Q16semidirect
142. C47#12215semidirect
178. C22×C4#10625337semidirect

Maximal subgroups

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

subgroupindex
#202490532
#202423452
#202414082
#202413182
C47×#20302
#1221547