Finite Group Databasebeta

← #26115866

Subgroups — #26115866

Order 12032.

410 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C2×#15689473C211yesmaximal, characteristic
2#20244378C211yesmaximal, characteristic
3#20244376C211yesmaximal, characteristic
4#20242146C211yesmaximal, characteristic
5#20242072C211yesmaximal, characteristic
6#20241669C211yesmaximal, characteristic
7C47×#1548C211yesmaximal, characteristic
8C2×#13094024C2211yescharacteristic
9#15689473—21no—
10#15689473—21no—
11#15689473—22no—
12#15689147C2211yescharacteristic
13C2×#13093948C414yes—
14C4×#10625329C2211yescharacteristic
15C2×C4×C47⋊C8C2211yescharacteristic
16#15688892—22no—
17#15688891—22no—
18#15688875—22no—
19#15688870—22no—
20#15688845—21no—
21C8×C47⋊C8—21no—
22C2×C4⋊Q8×C47C2211yescharacteristic
23C47×#431C2211yescharacteristic
24C47×#423C2211yescharacteristic
25C47×#333—22no—
26C22×Dic94—21no—
27C2×#10625329—21no—
28C2×#10625329D412yes—
29C22×C47⋊C8C2×C412yes—
30#13094024—21no—
31#13094024—21no—
32#13094024—21no—
33#13094024—21no—
34C2×C4⋊Dic47—21no—
35C2×C4⋊Dic47C2×C412yes—
36C4⋊(C47⋊C8)—21no—
37C4⋊(C47⋊C8)—21no—
38#13093948—24no—
39#13093948—24no—
40#13093935—21no—
41C4×C47⋊C8—21no—
42C4×C47⋊C8D411yescharacteristic
43C4×C47⋊C8D411yescharacteristic
44C4×C47⋊C8D412yes—
45C22×Q8×C47C2×C412yes—
46C2×M16×C47—21no—
47C4⋊Q8×C47—21no—
48C4⋊Q8×C47—21no—
49C4⋊Q8×C47—21no—
50C4⋊Q8×C47—21no—
51C2×C4⋊C4×C47—21no—
52C2×C4⋊C4×C47—21no—
53C2×C42×C47C2311yescharacteristic
54C4⋊C8×C47—21no—
55C4⋊C8×C47—21no—
56C4⋊C8×C47D412yes—
57C22⋊C8×C47—21no—
58C4×C8×C47—21no—
59C2×Q8×C47—22no—
60C2×Q8×C47—22no—
61C2×Q8×C47—44no—
62C22×C4×C47—22no—
63C22×C4×C47C22×C411yescharacteristic
64C22×C4×C47C2×D411yescharacteristic
65C22×C4×C47#4111yescharacteristic
66C22×Dic47—22no—
67C2×Dic94—41no—
68C2×Dic94—41no—
69C2×Dic94—82no—
70M16×C47—42no—
71C2×C8×C47—22no—
72C2×C8×C47—41no—
73C4⋊C4×C47—41no—
74C4⋊C4×C47—41no—
75C4⋊C4×C47—41no—
76C4⋊C4×C47—41no—
77C42×C47C2×D411yescharacteristic
78C42×C47C2×D411yescharacteristic
79C42×C47C2×D411yescharacteristic
80C42×C47#4111yescharacteristic
81C4⋊Dic47—22no—
82C4⋊Dic47—22no—
83C4⋊Dic47—41no—
84C4⋊Dic47—41no—
85#10625329—22no—
86#10625329—42no—
87#10625329Q1614yes—
88C2×C47⋊C8—22no—
89C2×C47⋊C8—42no—
90C2×C47⋊C8—41no—
91C2×C47⋊C8C22⋊C412yes—
92C2×C47⋊C8C22⋊C412yes—
93C23×C47C4×D411yescharacteristic
94Q8×C47—44no—
95Q8×C47—82no—
96C2×C4×C47—22no—
97C2×C4×C47—22no—
98C2×C4×C47—42no—
99C2×C4×C47—82no—
100C2×C4×C47C2×C22⋊C411yescharacteristic
101C2×C4×C47C4×D411yescharacteristic
102C2×C4×C47#12111yescharacteristic
103C2×C4×C47#12111yescharacteristic
104C2×C4×C47#12411yescharacteristic
105C2×C4×C47#12711yescharacteristic
106C2×C4×C47C2×Q1612yes—
107C2×C4×C47#13512yes—
108C2×Dic47—42no—
109C2×Dic47—82no—
110Dic94—82no—
111Dic94—82no—
112C8×C47—42no—
113C8×C47—81no—
114C47⋊C8—22no—
115C47⋊C8—22no—
116C47⋊C8—24no—
117C47⋊C8—81no—
118#9795—471nomaximal, Sylow
119C22×C47—21no—
120C22×C47—21no—
121C22×C47#39011yescharacteristic
122C22×C47C4×Q1612yes—
123C4×C47—21no—
124C4×C47—21no—
125C4×C47—82no—
126C4×C47#41312yes—
127C4×C47#46712yes—
128C4×C47#47112yes—
129C4×C47#49411yescharacteristic
130C4×C47#49511yescharacteristic
131Dic47—82no—
132C2×#487—471no—
133#1997—471no—
134#1997—471no—
135#1567—471no—
136#1549—471no—
137#1548D4711yescharacteristic
138C8×M16—471no—
139C2×C47—22no—
140C2×C47#196711yescharacteristic
141C2×C47#196911yescharacteristic
142C2×C47#197711yescharacteristic
143C2×C4⋊Q8—471no—
144C2×C4⋊Q8C2×D4711yescharacteristic
145#487—941no—
146#487—941no—
147#487—942no—
148#431C2×D4711yescharacteristic
149#423—471no—
150#423C2×D4711yescharacteristic
151C2×#103—474no—
152C4×M16—471no—
153C2×C4×C8—471no—
154#343—942no—
155#343—942no—
156#335—942no—
157#333—22no—
158#332—942no—
159#322—941no—
160C82—941no—
161C47#979511yescharacteristic, complemented, Sylow
162C22×Q8—941no—
163C22×Q8C4×D4712yes—
164C2×M16—21no—
165C2×M16—472no—
166C2×M16—941no—
167C22×C8—472no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—941no—
173C4⋊Q8—941no—
174C4⋊Q8—941no—
175C4⋊Q8—941no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—472no—
179C2×C4⋊C4—941no—
180C2×C42C22×D4711yescharacteristic
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—941no—
184C4⋊C8—941no—
185C4⋊C8C22⋊D4712yes—
186#103—944no—
187#103—944no—
188C22⋊C8—21no—
189#97—941no—
190C4×C8—21no—
191C4×C8—471no—
192C4×C8—471no—
193C4×C8—472no—
194C4×C8—941no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—44no—
198C2×Q8—3762no—
199C2×Q8—1881no—
200C2×Q8—1881no—
201C22×C4—22no—
202C22×C4—942no—
203C22×C4C2×C4×D4711yescharacteristic
204C22×C4D4×D4711yescharacteristic
205C22×C4#1062535211yescharacteristic
206M16—42no—
207M16—474no—
208M16—942no—
209M16—1882no—
210C2×C8—22no—
211C2×C8—41no—
212C2×C8—472no—
213C2×C8—472no—
214C2×C8—942no—
215C2×C8—1882no—
216C2×C8—1881no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—942no—
222C4⋊C4—942no—
223C4⋊C4—1881no—
224C4⋊C4—1881no—
225C42#1062535011yescharacteristic
226C42D4×D4711yescharacteristic
227C42D4×D4711yescharacteristic
228C42C2×C22⋊D4711yescharacteristic
229C23(C22⋊D47)⋊C411yescharacteristic
230Q8—44no—
231Q8—82no—
232Q8—3762no—
233Q8—3762no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—42no—
237C2×C4—82no—
238C2×C4—3762no—
239C2×C4—1882no—
240C2×C4C22⋊C4×D4711yescharacteristic
241C2×C4C4×C22⋊D4711yescharacteristic
242C2×C4#1309399611yescharacteristic
243C2×C4#1309399711yescharacteristic
244C2×C4#1309401011yescharacteristic
245C2×C4#1309402011yescharacteristic
246C2×C4Q16×D4712yes—
247C2×C4#1309404512yes—
248C8—42no—
249C8—81no—
250C8—942no—
251C8—942no—
252C8—944no—
253C8—3761no—
254C22—21no—
255C22—21no—
256C22#1568902311yescharacteristic
257C22#1568921512yes—
258C4—21no—
259C4—21no—
260C4—82no—
261C4—3762no—
262C4#1568910412yes—
263C4#1568936312yes—
264C4#1568938812yes—
265C4#1568950111yescharacteristic
266C4#1568950411yescharacteristic
267C2—22no—
268C2#2024423411yescharacteristic
269C2#2024424611yescharacteristic
270C2#2024429011yescharacteristic
subgroupG/H
centerC22#15689023
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1548C2
socleC22×C47#390
prime radicalC23×C47C4×D4
cosocleC2×C42×C47C23
hypercenter#1548D47
hypocenterC47#9795
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
161. C47#9795semidirect

Maximal subgroups

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

subgroupindex
C2×#156894732
#202443782
#202443762
#202421462
#202420722
#202416692
C47×#15482
#979547