Finite Group Databasebeta

← #26115859

Subgroups — #26115859

Order 12032.

426 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 121 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×#15689519C211yesmaximal, characteristic
2#20244376C211yesmaximal, characteristic
3#20244233C211yesmaximal, characteristic
4#20242143C211yesmaximal, characteristic
5#20242072C211yesmaximal, characteristic
6#20241669C211yesmaximal, characteristic
7C47×#1546C211yesmaximal, characteristic
8C2×#13094024C2211yescharacteristic
9#15689519—21no—
10#15689519—21no—
11#15689519—22no—
12C2×#13093955C414yes—
13#15689147C2211yescharacteristic
14C2×#13093943—22no—
15C4×#10625329C2211yescharacteristic
16C2×C4×C47⋊C8C2211yescharacteristic
17#15688891—22no—
18#15688873—22no—
19#15688870—22no—
20#15688845—21no—
21C8×C47⋊C8—21no—
22C2×C4⋊Q8×C47C2211yescharacteristic
23C47×#431C2211yescharacteristic
24C47×#423C2211yescharacteristic
25C47×#332—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—
36#13093955—24no—
37#13093955—24no—
38C4⋊(C47⋊C8)—21no—
39C4⋊(C47⋊C8)—21no—
40#13093943—44no—
41#13093935—21no—
42C4×C47⋊C8—21no—
43C4×C47⋊C8D411yescharacteristic
44C4×C47⋊C8D411yescharacteristic
45C4×C47⋊C8D412yes—
46C22×Q8×C47—21no—
47C2×M16×C47—21no—
48C4⋊Q8×C47—21no—
49C4⋊Q8×C47—21no—
50C4⋊Q8×C47—21no—
51C4⋊Q8×C47—21no—
52C2×C4⋊C4×C47—21no—
53C2×C4⋊C4×C47C2×C412yes—
54C2×C42×C47C2311yescharacteristic
55C4⋊C8×C47—21no—
56C4⋊C8×C47—21no—
57C4⋊C8×C47D412yes—
58C22⋊C8×C47—21no—
59C4×C8×C47—21no—
60C2×Q8×C47—41no—
61C2×Q8×C47—41no—
62C2×Q8×C47—42no—
63C2×Q8×C47—42no—
64C22×C4×C47—22no—
65C22×C4×C47C22×C411yescharacteristic
66C22×C4×C47C2×D411yescharacteristic
67C22×C4×C47#4111yescharacteristic
68C22×Dic47—22no—
69C2×Dic94—41no—
70C2×Dic94—41no—
71C2×Dic94—82no—
72M16×C47—42no—
73C2×C8×C47—22no—
74C2×C8×C47—41no—
75C4⋊C4×C47—22no—
76C4⋊C4×C47—22no—
77C4⋊C4×C47—41no—
78C4⋊C4×C47—41no—
79C42×C47C2×D411yescharacteristic
80C42×C47C2×D411yescharacteristic
81C42×C47C2×D411yescharacteristic
82C42×C47#4111yescharacteristic
83C4⋊Dic47—22no—
84C4⋊Dic47—22no—
85C4⋊Dic47—41no—
86C4⋊Dic47—41no—
87#10625329—22no—
88#10625329—42no—
89#10625329SD1614yes—
90C2×C47⋊C8—22no—
91C2×C47⋊C8—24no—
92C2×C47⋊C8—41no—
93C2×C47⋊C8C22⋊C412yes—
94C2×C47⋊C8C22⋊C412yes—
95C23×C47C4×D411yescharacteristic
96Q8×C47—82no—
97Q8×C47—81no—
98Q8×C47—81no—
99C2×C4×C47—22no—
100C2×C4×C47—22no—
101C2×C4×C47—42no—
102C2×C4×C47—44no—
103C2×C4×C47C2×C22⋊C411yescharacteristic
104C2×C4×C47C4×D411yescharacteristic
105C2×C4×C47#12111yescharacteristic
106C2×C4×C47#12111yescharacteristic
107C2×C4×C47#12411yescharacteristic
108C2×C4×C47#12711yescharacteristic
109C2×C4×C47C2×SD1612yes—
110C2×C4×C47#13512yes—
111C2×Dic47—42no—
112C2×Dic47—82no—
113Dic94—82no—
114Dic94—82no—
115C8×C47—42no—
116C8×C47—81no—
117C47⋊C8—22no—
118C47⋊C8—24no—
119C47⋊C8—81no—
120C47⋊C8#10114yescomplemented
121#9794—471nomaximal, Sylow
122C22×C47—21no—
123C22×C47—21no—
124C22×C47#39011yescharacteristic
125C22×C47C4×SD1612yes—
126C4×C47—21no—
127C4×C47—21no—
128C4×C47—82no—
129C4×C47C2×#10112yes—
130C4×C47#46512yes—
131C4×C47#47112yes—
132C4×C47#49211yescharacteristic
133C4×C47#49511yescharacteristic
134Dic47—82no—
135C2×#498—471no—
136#1997—471no—
137#1966—471no—
138#1566—471no—
139#1549—471no—
140#1546D4711yescharacteristic
141C8×M16—471no—
142C2×C47—22no—
143C2×C47#196611yescharacteristic
144C2×C47#196811yescharacteristic
145C2×C47#197711yescharacteristic
146C2×C4⋊Q8—471no—
147C2×C4⋊Q8C2×D4711yescharacteristic
148#498—941no—
149#498—941no—
150#498—942no—
151#431C2×D4711yescharacteristic
152C2×#106—474no—
153#423—471no—
154#423C2×D4711yescharacteristic
155C2×#101—942no—
156C4×M16—471no—
157C2×C4×C8—471no—
158#343—942no—
159#334—942no—
160#332—22no—
161#332—942no—
162#322—941no—
163C82—941no—
164C47#979411yescharacteristic, complemented, Sylow
165C22×Q8—21no—
166C22×Q8—941no—
167C2×M16—21no—
168C2×M16—472no—
169C2×M16—941no—
170C22×C8—472no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—941no—
176C4⋊Q8—941no—
177C4⋊Q8—941no—
178C4⋊Q8—941no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—472no—
181C2×C4⋊C4—941no—
182C2×C4⋊C4C4×D4712yes—
183C2×C42C22×D4711yescharacteristic
184#106—944no—
185#106—944no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—941no—
189C4⋊C8—941no—
190C4⋊C8C22⋊D4712yes—
191#101—1884no—
192C22⋊C8—21no—
193#97—941no—
194C4×C8—21no—
195C4×C8—471no—
196C4×C8—471no—
197C4×C8—472no—
198C4×C8—941no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—3762no—
204C2×Q8—1881no—
205C2×Q8—1881no—
206C22×C4—22no—
207C22×C4—942no—
208C22×C4C2×C4×D4711yescharacteristic
209C22×C4D4×D4711yescharacteristic
210C22×C4#1062535211yescharacteristic
211M16—42no—
212M16—474no—
213M16—942no—
214M16—1882no—
215C2×C8—22no—
216C2×C8—41no—
217C2×C8—472no—
218C2×C8—472no—
219C2×C8—944no—
220C2×C8—942no—
221C2×C8—1881no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—942no—
227C4⋊C4—942no—
228C4⋊C4—1881no—
229C4⋊C4—1881no—
230C42#1062535011yescharacteristic
231C42D4×D4711yescharacteristic
232C42D4×D4711yescharacteristic
233C42C2×C22⋊D4711yescharacteristic
234C23(C22⋊D47)⋊C411yescharacteristic
235Q8—81no—
236Q8—81no—
237Q8—82no—
238Q8—3762no—
239Q8—3762no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—3762no—
245C2×C4—1882no—
246C2×C4C22⋊C4×D4711yescharacteristic
247C2×C4C4×C22⋊D4711yescharacteristic
248C2×C4#1309399611yescharacteristic
249C2×C4#1309399711yescharacteristic
250C2×C4#1309401011yescharacteristic
251C2×C4#1309402011yescharacteristic
252C2×C4SD16×D4712yes—
253C2×C4#1309404312yes—
254C8—42no—
255C8—81no—
256C8—474no—
257C8—944no—
258C8—942no—
259C8—3761no—
260C22—21no—
261C22—21no—
262C22#1568902311yescharacteristic
263C22#1568921412yes—
264C4—21no—
265C4—21no—
266C4—82no—
267C4—3762no—
268C4#101×D4712yes—
269C4#1568935112yes—
270C4#1568938812yes—
271C4#1568949411yescharacteristic
272C4#1568950611yescharacteristic
273C2—22no—
274C2#2024423011yescharacteristic
275C2#2024424311yescharacteristic
276C2#2024429011yescharacteristic
subgroupG/H
centerC22#15689023
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1546C2
socleC22×C47#390
prime radicalC23×C47C4×D4
cosocleC2×C42×C47C23
hypercenter#1546D47
hypocenterC47#9794
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
120. C47⋊C8#101semidirect
164. C47#9794semidirect

Maximal subgroups

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

subgroupindex
C2×#156895192
#202443762
#202442332
#202421432
#202420722
#202416692
C47×#15462
#979447