Finite Group Databasebeta

← #26129126

Subgroups — #26129126

Order 12032.

336 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 63 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#20244549C211yesmaximal, characteristic
2#20242429C212yesmaximal
3#20241415C212yesmaximal
4#20241322C211yesmaximal, characteristic
5C47×#2809C211yesmaximal, characteristic
6#15689754C2211yescharacteristic
7#15689458—22no—
8#15689132—21no—
9#15688892C2211yescharacteristic
10#15688883—21no—
11#15688859—22no—
12#15688849C2212yes—
13C47×#400C2212yes—
14C47×#398—22no—
15C47×#391—21no—
16C47×#391C2211yescharacteristic
17C47×#389—21no—
18C47×#387—22no—
19C2×#10625329—21no—
20#13094023—41no—
21#13094015—41no—
22#13094003—22no—
23#13093983—21no—
24#13093957—42no—
25#13093948—41no—
26#13093948—41no—
27#13093947—42no—
28C22⋊(C47⋊C8)—22no—
29C22×Q8×C47D411yescharacteristic
30C2×C4⋊C4×C47—22no—
31C2×C4⋊C4×C47—21no—
32C2×C4⋊C4×C47—21no—
33C2×C4⋊C4×C47—21no—
34C2×C4⋊C4×C47D412yes—
35C2×C4⋊C4×C47C2311yescharacteristic
36C2×C42×C47—21no—
37C2×C42×C47D411yescharacteristic
38#95×C47—22no—
39#95×C47—22no—
40#95×C47—22no—
41#95×C47—22no—
42#95×C47D412yes—
43C2×Q8×C47—21no—
44C2×Q8×C47—41no—
45C2×Q8×C47—41no—
46C2×Q8×C47Q1612yes—
47C22×C4×C47—21no—
48C22×C4×C47—21no—
49C22×C4×C47—22no—
50C22×C4×C47—22no—
51C22×C4×C47C2×D411yescharacteristic
52C22×C4×C47C2×D412yes—
53C2×Dic94—22no—
54C4⋊C4×C47—41no—
55C4⋊C4×C47—44no—
56C4⋊C4×C47—42no—
57C4⋊C4×C47—42no—
58C4⋊C4×C47—81no—
59C4⋊C4×C47—81no—
60C42×C47—41no—
61C42×C47—81no—
62C22⋊Dic47—41no—
63C4⋊Dic47—21no—
64C4⋊Dic47—21no—
65C4⋊Dic47—22no—
66Dic47⋊C4—42no—
67C4×Dic47—41no—
68#10625329—41no—
69#10625329—41no—
70C2×C47⋊C8—42no—
71C2×C47⋊C8—41no—
72C23×C47#12011yescharacteristic
73Q8×C47—42no—
74Q8×C47—81no—
75C2×C4×C47—21no—
76C2×C4×C47—21no—
77C2×C4×C47—22no—
78C2×C4×C47—22no—
79C2×C4×C47—22no—
80C2×C4×C47—41no—
81C2×C4×C47—41no—
82C2×C4×C47—41no—
83C2×C4×C47—41no—
84C2×C4×C47—42no—
85C2×C4×C47—42no—
86C2×C4×C47—42no—
87C2×C4×C47—42no—
88C2×C4×C47—42no—
89C2×C4×C47—42no—
90C2×C4×C47—42no—
91C2×C4×C47C2×Q1611yescharacteristic
92C2×C4×C47#13711yescharacteristic
93C2×Dic47—41no—
94C2×Dic47—41no—
95Dic94—82no—
96C47⋊C8—81no—
97C47⋊C8—82no—
98#12225—471nomaximal, Sylow
99C22×C47—21no—
100C22×C47—21no—
101C22×C47#45111yescharacteristic
102C22×C47#45511yescharacteristic
103C22×C47B(4,𝔽2)11yescharacteristic
104C4×C47—21no—
105C4×C47—21no—
106C4×C47—42no—
107C4×C47—42no—
108C4×C47—42no—
109C4×C47—82no—
110C4×C47—82no—
111Dic47—81no—
112Dic47—81no—
113#2809D4711yescharacteristic
114#2028—471no—
115#1617—472no—
116#1353—472no—
117#1319—471no—
118C2×C47—21no—
119C2×C47—21no—
120C2×C47#160211yescharacteristic
121C2×C47#220111yescharacteristic
122C2×C47#220411yescharacteristic
123#544—471no—
124#484—942no—
125#419—941no—
126#400C2×D4712yes—
127#398—22no—
128#391—21no—
129#391C2×D4711yescharacteristic
130#389—21no—
131#387—22no—
132#343—471no—
133#339—941no—
134#328—942no—
135#324—472no—
136C47#1222511yescharacteristic, complemented, Sylow
137C22×Q8C22⋊D4711yescharacteristic
138C2×M16—941no—
139C4⋊Q8—1881no—
140#125—1881no—
141C22⋊Q8—942no—
142#117—941no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4C22⋊D4712yes—
148C2×C4⋊C4C22×D4711yescharacteristic
149C2×C42—21no—
150C2×C42C22⋊D4711yescharacteristic
151#107—1882no—
152#103—1882no—
153#103—1881no—
154#103—1881no—
155C22⋊C8—942no—
156#95—22no—
157#95—22no—
158#95—22no—
159#95—22no—
160#95C22⋊D4712yes—
161C2×Q8—21no—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—942no—
165C2×Q8#1062533712yes—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4C2×C22⋊D4711yescharacteristic
171C22×C4C2×C22⋊D4712yes—
172M16—1881no—
173M16—1881no—
174C2×C8—1882no—
175C2×C8—1881no—
176C4⋊C4—41no—
177C4⋊C4—42no—
178C4⋊C4—44no—
179C4⋊C4—42no—
180C4⋊C4—81no—
181C4⋊C4—81no—
182C4⋊C4—941no—
183C4⋊C4—941no—
184C4⋊C4—942no—
185C4⋊C4—1882no—
186C22⋊C4—1881no—
187C42—41no—
188C42—81no—
189C42—1881no—
190C23#1309399311yescharacteristic
191Q8—42no—
192Q8—81no—
193Q8—3762no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—1881no—
211C2×C4—1881no—
212C2×C4C2×#1062533711yescharacteristic
213C2×C4#1309405511yescharacteristic
214C8—3761no—
215C8—3762no—
216C22—21no—
217C22—21no—
218C22#1568927911yescharacteristic
219C22#1568930111yescharacteristic
220C22#1568931011yescharacteristic
221C4—21no—
222C4—21no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—82no—
227C4—82no—
228C4—3761no—
229C4—3761no—
230C2—21no—
231C2—21no—
232C2#2024234511yescharacteristic
233C2#2024540111yescharacteristic
234C2#2024541411yescharacteristic
subgroupG/H
centerC22#15689310
commutator, cosocleC2×C4⋊C4×C47C23
FrattiniC2×C4⋊C4C22×D47
FittingC47×#2809C2
socleC22×C47B(4,𝔽2)
prime radicalC23×C47#120
hypercenter#2809D47
hypocenterC47#12225
LCSG ▷ C2×C4⋊C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2809

Complements

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

NH ≅ G/Ntype
136. C47#12225semidirect

Maximal subgroups

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

subgroupindex
#202445492
#202424292
#202414152
#202413222
C47×#28092
#1222547