Finite Group Databasebeta

← #26109870

Subgroups — #26109870

Order 12032.

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

normal subgroups only

This group has 378 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#20246353C211yesmaximal, characteristic
2#20243384C211yesmaximal, characteristic
3#20241290C212yesmaximal
4#20241247C212yesmaximal
5C47×#1461C211yesmaximal, characteristic
6#15689147C2212yes—
7C2×#13093943—21no—
8#15689030—21no—
9#15689009—22no—
10#15688991C412yes—
11#15688989C2211yescharacteristic
12C4⋊C4×Dic47—21no—
13C4×C4⋊Dic47C2211yescharacteristic
14#15688969C412yes—
15#15688892—21no—
16C2×C4×Q8×C47C2211yescharacteristic
17C47×#431C2212yes—
18C47×#324—22no—
19C2×#10625329—22no—
20C2×C4⋊Dic47—21no—
21C2×C4⋊Dic47—21no—
22C2×C4×Dic47—22no—
23C2×C4×Dic47—21no—
24C4⋊(C47⋊C8)—22no—
25C4⋊(C47⋊C8)—22no—
26#13093943—42no—
27C47⋊#95—22no—
28C47⋊#95—22no—
29C47⋊#95—22no—
30C47⋊#95—21no—
31C47⋊#95C2×C411yescharacteristic
32C47⋊#95C2×C412yes—
33C47⋊#95C2×C411yescharacteristic
34C22×Q8×C47D411yescharacteristic
35C4×Q8×C47—22no—
36C4×Q8×C47—22no—
37C2×C4⋊C4×C47D411yescharacteristic
38C2×C4⋊C4×C47D412yes—
39C2×C42×C47C2×C412yes—
40C2×C42×C47C2311yescharacteristic
41C4⋊C8×C47—22no—
42C22⋊C8×C47—22no—
43C4×C8×C47—22no—
44C2×Q8×C47—22no—
45C2×Q8×C47—22no—
46C2×Q8×C47—21no—
47C2×Q8×C47—21no—
48C22×C4×C47C22⋊C412yes—
49C22×C4×C47C22⋊C412yes—
50C22×C4×C47C22×C411yescharacteristic
51C22×C4×C47C2×D411yescharacteristic
52C22×C4×C47C2×D411yescharacteristic
53C22×Dic47—21no—
54C22×Dic47—22no—
55C22×Dic47—21no—
56C2×C8×C47—42no—
57C4⋊C4×C47—22no—
58C4⋊C4×C47—24no—
59C42×C47—22no—
60C42×C47—22no—
61C42×C47#4112yes—
62C42×C47#4112yes—
63C4⋊Dic47—42no—
64C4⋊Dic47—41no—
65C4⋊Dic47—41no—
66C4×Dic47—44no—
67C4×Dic47—42no—
68#10625329—42no—
69#10625329—42no—
70C2×C47⋊C8—42no—
71C23×C47C2×C22⋊C411yescharacteristic
72Q8×C47—42no—
73Q8×C47—42no—
74C2×C4×C47—22no—
75C2×C4×C47—22no—
76C2×C4×C47—22no—
77C2×C4×C47—22no—
78C2×C4×C47—41no—
79C2×C4×C47—41no—
80C2×C4×C47—42no—
81C2×C4×C47#11711yescharacteristic
82C2×C4×C47#11711yescharacteristic
83C2×C4×C47#12312yes—
84C2×C4×C47#12312yes—
85C2×C4×C47#13512yes—
86C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
87C2×C4×C47#13711yescharacteristic
88C2×Dic47—22no—
89C2×Dic47—41no—
90C2×Dic47—41no—
91C2×Dic47—42no—
92C2×Dic47—41no—
93C2×Dic47—41no—
94C2×Dic47—42no—
95C2×Dic47—41no—
96C2×Dic47—42no—
97C8×C47—82no—
98C47⋊C8—82no—
99#8769—471nomaximal, Sylow
100C22×C47—21no—
101C22×C47—21no—
102C22×C47#38111yescharacteristic
103C22×C47#41611yescharacteristic
104C22×C47#41711yescharacteristic
105C4×C47—22no—
106C4×C47—21no—
107C4×C47—21no—
108C4×C47—42no—
109C4×C47—42no—
110C4×C47#48212yes—
111C4×C47#48512yes—
112Dic47—42no—
113Dic47—81no—
114Dic47—82no—
115#2356—471no—
116#1801—471no—
117#1461D4711yescharacteristic
118#1303—472no—
119#1286—472no—
120C2×C47—21no—
121C2×C47—21no—
122C2×C47#182511yescharacteristic
123C2×C47#182711yescharacteristic
124C2×C47#183011yescharacteristic
125C2×C4×Q8C2×D4711yescharacteristic
126#431C2×D4712yes—
127#423—472no—
128C2×#101—941no—
129#391—941no—
130#387—942no—
131#383—472no—
132#382—471no—
133C4×C4⋊C4—471no—
134C4×C4⋊C4—941no—
135#376—472no—
136#343—941no—
137#324—22no—
138C47#876911yescharacteristic, complemented, Sylow
139C22×Q8D18811yescharacteristic
140C2×M16—942no—
141C4×Q8—22no—
142C4×Q8—22no—
143C2×C4⋊C4—941no—
144C2×C4⋊C4—941no—
145C2×C4⋊C4D18811yescharacteristic
146C2×C4⋊C4C22⋊D4712yes—
147C2×C42—942no—
148C2×C42—941no—
149C2×C42C4×D4712yes—
150C2×C42C22×D4711yescharacteristic
151C4⋊C8—22no—
152C4⋊C8—942no—
153C4⋊C8—942no—
154#101—1882no—
155C22⋊C8—22no—
156C4×C8—22no—
157#95—471no—
158#95—472no—
159#95—471no—
160#95—942no—
161#95—942no—
162#95—942no—
163#95—941no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—21no—
167C2×Q8—21no—
168C22×C4—941no—
169C22×C4—942no—
170C22×C4—941no—
171C22×C4(C2×D47)⋊C412yes—
172C22×C4(C2×D47)⋊C412yes—
173C22×C4C2×C4×D4711yescharacteristic
174C22×C4C2×D18811yescharacteristic
175C22×C4C2×C22⋊D4711yescharacteristic
176M16—1882no—
177M16—1882no—
178C2×C8—42no—
179C2×C8—1882no—
180C4⋊C4—24no—
181C4⋊C4—22no—
182C4⋊C4—1882no—
183C4⋊C4—1881no—
184C4⋊C4—1881no—
185C42—22no—
186C42—22no—
187C42—1884no—
188C42—1882no—
189C42#1062535212yes—
190C42#1062535212yes—
191C23C2×(C2×D47)⋊C411yescharacteristic
192Q8—42no—
193Q8—42no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—942no—
202C2×C4—1882no—
203C2×C4—1881no—
204C2×C4—1882no—
205C2×C4—1881no—
206C2×C4—1881no—
207C2×C4—1882no—
208C2×C4—1881no—
209C2×C4—1881no—
210C2×C4#1309398011yescharacteristic
211C2×C4#1309398011yescharacteristic
212C2×C4#1309400512yes—
213C2×C4#1309400812yes—
214C2×C4#1309404612yes—
215C2×C4#1309405211yescharacteristic
216C2×C4#1309405711yescharacteristic
217C8—82no—
218C8—3762no—
219C22—21no—
220C22—21no—
221C22#1568898411yescharacteristic
222C22#1568912011yescharacteristic
223C22#1568912511yescharacteristic
224C4—22no—
225C4—21no—
226C4—21no—
227C4—42no—
228C4—42no—
229C4—3762no—
230C4—3761no—
231C4—1882no—
232C4#1568944812yes—
233C4#1568946512yes—
234C2—21no—
235C2—21no—
236C2#2024348411yescharacteristic
237C2#2024348911yescharacteristic
238C2#2024350011yescharacteristic
subgroupG/H
centerC22#15688984
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1461C2
socleC22×C47#381
prime radicalC23×C47C2×C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1461D47
hypocenterC47#8769
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
138. C47#8769semidirect

Maximal subgroups

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

subgroupindex
#202463532
#202433842
#202412902
#202412472
C47×#14612
#876947