Finite Group Databasebeta

← #26119412

Subgroups — #26119412

Order 12032.

416 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 97 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#20246681C211yesmaximal, characteristic
2#20244369C211yesmaximal, characteristic
3#20241888C212yesmaximal
4#20241290C212yesmaximal
5C47×#1274C211yesmaximal, characteristic
6#15689680C2211yescharacteristic
7C2×C4×Dic94—21no—
8#15689147C2211yescharacteristic
9#15689055—22no—
10#15689044—22no—
11C4×C4⋊Dic47—21no—
12C4×C4⋊Dic47C2212yes—
13#15688883—21no—
14#15688883—21no—
15#15688861—22no—
16#15688861—22no—
17C47×#431C2212yes—
18C47×#376C2211yescharacteristic
19C22×Dic94—21no—
20C2×#10625329—21no—
21#13094024Q812yescomplemented
22#13094016—21no—
23#13094003—22no—
24C4×Dic94—22no—
25C4×Dic94—42no—
26C4×Dic94—42no—
27#13093983—21no—
28C2×C4⋊Dic47—22no—
29C2×C4⋊Dic47—21no—
30C2×C4⋊Dic47—22no—
31C2×Dic47⋊C4—22no—
32C2×C4×Dic47—22no—
33C4⋊(C47⋊C8)—22no—
34C47⋊#95—21no—
35C47⋊#95—21no—
36C47⋊#95D412yes—
37C47⋊#95D412yes—
38C2×C42×C47D411yescharacteristic
39C2×C42×C47D411yescharacteristic
40C2×C42×C47C2311yescharacteristic
41C4⋊C8×C47—22no—
42C22⋊C8×C47—22no—
43C4×C8×C47—22no—
44#95×C47—22no—
45C22×C4×C47—21no—
46C22×C4×C47—21no—
47C22×C4×C47C2×D412yes—
48C22×C4×C47C2×D411yescharacteristic
49C22×Dic47—22no—
50C22×Dic47—22no—
51C2×Dic94—22no—
52C2×Dic94—44no—
53C2×Dic94—42no—
54C2×C8×C47—42no—
55C42×C47—21no—
56C42×C47—21no—
57C42×C47SD1612yes—
58C42×C47Q1612yes—
59C42×C47C2×Q811yescharacteristic
60C42×C47#4112yes—
61C42×C47#4111yescharacteristic
62C22⋊Dic47—41no—
63C4⋊Dic47—22no—
64C4⋊Dic47—22no—
65C4⋊Dic47—44no—
66C4⋊Dic47—44no—
67C4⋊Dic47—42no—
68Dic47⋊C4—44no—
69Dic47⋊C4—42no—
70C4×Dic47—41no—
71C4×Dic47—42no—
72C4×Dic47—42no—
73#10625329—41no—
74#10625329—41no—
75C2×C47⋊C8—41no—
76C23×C47#12011yescharacteristic
77C2×C4×C47—22no—
78C2×C4×C47—22no—
79C2×C4×C47—42no—
80C2×C4×C47—42no—
81C2×C4×C47—42no—
82C2×C4×C47C22⋊Q812yes—
83C2×C4×C47C22⋊Q811yescharacteristic
84C2×C4×C47#12412yes—
85C2×C4×C47#12411yescharacteristic
86C2×C4×C47C2×SD1611yescharacteristic
87C2×C4×C47C2×Q1611yescharacteristic
88C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
89C2×C4×C47#13711yescharacteristic
90C2×Dic47—24no—
91C2×Dic47—42no—
92C2×Dic47—42no—
93C2×Dic47—42no—
94C2×Dic47—42no—
95C2×Dic47—42no—
96C2×Dic47—42no—
97Dic94—48no—
98Dic94—82no—
99C8×C47—82no—
100C47⋊C8—81no—
101#10371—471nomaximal, Sylow
102C22×C47—22no—
103C22×C47#39311yescharacteristic
104C22×C47#45011yescharacteristic
105C22×C47#45111yescharacteristic
106C4×C47—22no—
107C4×C47—22no—
108C4×C47—42no—
109C4×C47—42no—
110C4×C47#47511yescharacteristic
111C4×C47#47711yescharacteristic
112C4×C47#48711yescharacteristic
113C4×C47#48911yescharacteristic
114Dic47—44no—
115Dic47—82no—
116Dic47—82no—
117#2399—471no—
118#1995—471no—
119#1503—472no—
120#1303—472no—
121#1274D4711yescharacteristic
122C2×C47—21no—
123C2×C47—21no—
124C2×C47#202411yescharacteristic
125C2×C47#203112yes—
126#533—471no—
127C2×C4×Q8—941no—
128#431C2×D4712yes—
129#423—471no—
130#398—942no—
131#395—942no—
132C4×C4⋊C4—472no—
133C4×C4⋊C4—941no—
134#376C2×D4711yescharacteristic
135#339—941no—
136#339—941no—
137#328—942no—
138#328—942no—
139C47#1037111yescharacteristic, complemented, Sylow
140C22×Q8—941no—
141C2×M16—941no—
142C4⋊Q8—472no—
143#125—941no—
144C22⋊Q8—942no—
145C4×Q8—942no—
146C4×Q8—1882no—
147C4×Q8—1882no—
148#117—941no—
149C2×C4⋊C4—942no—
150C2×C4⋊C4—941no—
151C2×C4⋊C4—942no—
152C2×C4⋊C4—942no—
153C2×C42—942no—
154C2×C42D18811yescharacteristic
155C2×C42D18811yescharacteristic
156C2×C42C22×D4711yescharacteristic
157C4⋊C8—22no—
158C4⋊C8—942no—
159C22⋊C8—22no—
160C4×C8—22no—
161#95—22no—
162#95—472no—
163#95—472no—
164#95—941no—
165#95—941no—
166C2×Q8—942no—
167C2×Q8—1882no—
168C2×Q8—1884no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—942no—
172C22×C4—942no—
173C22×C4C2×D18811yescharacteristic
174C22×C4D4×D4712yes—
175M16—1881no—
176M16—1881no—
177C2×C8—42no—
178C2×C8—1881no—
179C4⋊C4—942no—
180C4⋊C4—942no—
181C4⋊C4—1884no—
182C4⋊C4—1882no—
183C4⋊C4—1884no—
184C4⋊C4—1884no—
185C4⋊C4—1882no—
186C22⋊C4—1881no—
187C42—21no—
188C42—21no—
189C42—1881no—
190C42—1882no—
191C42—1882no—
192C42#1062532512yes—
193C42Dic18812yes—
194C42#1062535012yes—
195C42#1062535211yescharacteristic
196C42Q8×D4711yescharacteristic
197C23#1309399411yescharacteristic
198Q8—3762no—
199Q8—1888no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—944no—
206C2×C4—1882no—
207C2×C4—1882no—
208C2×C4—1882no—
209C2×C4—1882no—
210C2×C4—1882no—
211C2×C4—1882no—
212C2×C4#1309400212yes—
213C2×C4#1309400411yescharacteristic
214C2×C4#1309401012yes—
215C2×C4#1309401211yescharacteristic
216C2×C4C2×#1062532511yescharacteristic
217C2×C4C2×Dic18811yescharacteristic
218C2×C4#1309405211yescharacteristic
219C2×C4#1309405711yescharacteristic
220C8—82no—
221C8—3761no—
222C22—22no—
223C22#1568903611yescharacteristic
224C22#1568927611yescharacteristic
225C22#1568928111yescharacteristic
226C4—22no—
227C4—22no—
228C4—42no—
229C4—42no—
230C4—3762no—
231C4—3762no—
232C4—1884no—
233C4#1568940911yescharacteristic
234C4#1568941911yescharacteristic
235C4#1568947411yescharacteristic
236C4#1568948311yescharacteristic
237C2—21no—
238C2—21no—
239C2#2024453011yescharacteristic
240C2#2024455912yes—
subgroupG/H
centerC2×C4#13094012
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1274C2
socleC22×C47#393
prime radicalC23×C47#120
hypercenter#1274D47
hypocenterC47#10371
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1274

Complements

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

NH ≅ G/Ntype
21. #13094024Q8semidirect
139. C47#10371semidirect

Maximal subgroups

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

subgroupindex
#202466812
#202443692
#202418882
#202412902
C47×#12742
#1037147