Finite Group Databasebeta

← #26114715

Subgroups — #26114715

Order 12032.

338 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 101 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#20244364C211yesmaximal, characteristic
2C4⋊C8×Dic47C211yesmaximal, characteristic
3#20242160C211yesmaximal, characteristic
4#20242046C211yesmaximal, characteristic
5#20241792C211yesmaximal, characteristic
6#20241220C211yesmaximal, characteristic
7C47×#1301C211yesmaximal, characteristic
8#15689680C2211yescharacteristic
9#15689176C2211yescharacteristic
10C2×C4⋊(C47⋊C8)C2211yescharacteristic
11C2×C8×Dic47—21no—
12C42×Dic47C2211yescharacteristic
13#15688884—21no—
14#15688884—21no—
15#15688877—21no—
16#15688873—22no—
17#15688871C412yescomplemented
18#15688868C412yescomplemented
19#15688855—22no—
20C47×#431C2211yescharacteristic
21C47×#423C2211yescharacteristic
22C2×C4⋊C8×C47C2211yescharacteristic
23C22×C47⋊C8—21no—
24#13094024C2×C411yescharacteristic
25#13094024C2×C411yescharacteristic
26#13094016—21no—
27#13094003—21no—
28#13094003—21no—
29C4×Dic94—21no—
30C4×Dic94—21no—
31#13093983—21no—
32C2×C4×Dic47—21no—
33C2×C4×Dic47—21no—
34C2×C4×Dic47D411yescharacteristic
35C2×C4×Dic47D411yescharacteristic
36C4⋊(C47⋊C8)—21no—
37C4⋊(C47⋊C8)C2×C412yes—
38C4⋊(C47⋊C8)D412yes—
39C22⋊(C47⋊C8)—21no—
40C4×C47⋊C8—21no—
41C8×Dic47—42no—
42C2×M16×C47—21no—
43C22×C8×C47—21no—
44C2×C42×C47C2311yescharacteristic
45C4⋊C8×C47—21no—
46C4⋊C8×C47—21no—
47C4⋊C8×C47—21no—
48C4⋊C8×C47—21no—
49C4⋊C8×C47C2×C412yes—
50C22⋊C8×C47—21no—
51C4×C8×C47—21no—
52C22×C4×C47C2×D411yescharacteristic
53C22×C4×C47#4111yescharacteristic
54C22×C4×C47#4111yescharacteristic
55C22×Dic47—21no—
56C22×Dic47—21no—
57C2×Dic94—21no—
58C2×Dic94—21no—
59M16×C47—41no—
60M16×C47—41no—
61C2×C8×C47—22no—
62C2×C8×C47—41no—
63C2×C8×C47—41no—
64C2×C8×C47—41no—
65C2×C8×C47—41no—
66C42×C47C22×C411yescharacteristic
67C42×C47C2×D411yescharacteristic
68C42×C47#4111yescharacteristic
69C42×C47#4111yescharacteristic
70C22⋊Dic47—41no—
71C4⋊Dic47—21no—
72C4⋊Dic47—21no—
73C4⋊Dic47—21no—
74C4⋊Dic47—21no—
75Dic47⋊C4—41no—
76Dic47⋊C4—41no—
77C4×Dic47—24no—
78C4×Dic47—21no—
79C4×Dic47—21no—
80C4×Dic47—21no—
81C4×Dic47—42no—
82C4×Dic47—41no—
83C4×Dic47SD1612yes—
84C2×C47⋊C8—22no—
85C2×C47⋊C8—22no—
86C2×C47⋊C8—41no—
87C2×C47⋊C8—41no—
88C23×C47#12311yescharacteristic
89C2×C4×C47—22no—
90C2×C4×C47—41no—
91C2×C4×C47#11711yescharacteristic
92C2×C4×C47C4×D411yescharacteristic
93C2×C4×C47C4×D411yescharacteristic
94C2×C4×C47#12111yescharacteristic
95C2×C4×C47#12411yescharacteristic
96C2×C4×C47#12611yescharacteristic
97C2×C4×C47C2×SD1611yescharacteristic
98C2×C4×C47#13511yescharacteristic
99C2×C4×C47#13711yescharacteristic
100C2×C4×C47#13711yescharacteristic
101C2×Dic47—22no—
102C2×Dic47—22no—
103C2×Dic47—22no—
104C2×Dic47—22no—
105C2×Dic47—41no—
106C2×Dic47—41no—
107C2×Dic47—41no—
108C2×Dic47—41no—
109Dic94—42no—
110Dic94—81no—
111C8×C47—42no—
112C8×C47—81no—
113C8×C47—81no—
114C47⋊C8—42no—
115C47⋊C8—42no—
116#9579—471nomaximal, Sylow
117C22×C47—21no—
118C22×C47—21no—
119C22×C47#38811yescharacteristic
120C22×C47#48311yescharacteristic
121C22×C47#48511yescharacteristic
122C4×C47—21no—
123C4×C47—21no—
124C4×C47—21no—
125C4×C47C4×SD1611yescharacteristic
126C4×C47#44111yescharacteristic
127C4×C47#44312yes—
128C4×C47#46111yescharacteristic
129C4×C47#46411yescharacteristic
130Dic47—24no—
131Dic47—42no—
132Dic47—81no—
133Dic47—81no—
134#1994—471no—
135C4×C4⋊C8—471no—
136#1570—471no—
137#1542—471no—
138#1481—471no—
139#1301D4711yescharacteristic
140#1276—471no—
141C2×C47—21no—
142C2×C47—21no—
143C2×C47#192711yescharacteristic
144C2×C47#193211yescharacteristic
145C2×C47#193711yescharacteristic
146#533—471no—
147#431—471no—
148#431C2×D4711yescharacteristic
149#423C2×D4711yescharacteristic
150C2×C4⋊C8—471no—
151C2×C4⋊C8C2×D4711yescharacteristic
152C2×C4×C8—941no—
153C43—471no—
154#339—941no—
155#339—941no—
156#336—941no—
157#334—942no—
158#333—472no—
159#332—472no—
160Q8⋊C8—942no—
161C47#957911yescharacteristic, complemented, Sylow
162C2×M16—21no—
163C22×C8—21no—
164C22×C8—941no—
165C4⋊Q8—471no—
166C4⋊Q8—471no—
167#125—941no—
168C22⋊Q8—941no—
169C22⋊Q8—941no—
170C4×Q8—941no—
171C4×Q8—941no—
172#117—941no—
173C2×C42—471no—
174C2×C42—471no—
175C2×C42—941no—
176C2×C42—941no—
177C2×C42C22×D4711yescharacteristic
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—472no—
183C4⋊C8—472no—
184C4⋊C8—941no—
185C4⋊C8C4×D4712yes—
186C22⋊C8—21no—
187C22⋊C8—941no—
188C4×C8—21no—
189C4×C8—941no—
190C4×C8—1882no—
191C2×Q8—941no—
192C2×Q8—941no—
193C22×C4—941no—
194C22×C4—941no—
195C22×C4#1062535011yescharacteristic
196C22×C4D4×D4711yescharacteristic
197C22×C4#1062535411yescharacteristic
198M16—41no—
199M16—41no—
200C2×C8—22no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—942no—
206C2×C8—942no—
207C2×C8—1881no—
208C2×C8—1881no—
209C4⋊C4—941no—
210C4⋊C4—941no—
211C4⋊C4—941no—
212C4⋊C4—941no—
213C4⋊C4—1881no—
214C4⋊C4—1881no—
215C22⋊C4—1881no—
216C42—472no—
217C42—944no—
218C42—941no—
219C42—941no—
220C42—941no—
221C42—1882no—
222C42—1881no—
223C42C2×C4×D4711yescharacteristic
224C42#1062535011yescharacteristic
225C42D4×D4711yescharacteristic
226C42#1062535211yescharacteristic
227C23#1309400911yescharacteristic
228Q8—3761no—
229Q8—1882no—
230C2×C4—22no—
231C2×C4—41no—
232C2×C4—942no—
233C2×C4—942no—
234C2×C4—942no—
235C2×C4—942no—
236C2×C4—1881no—
237C2×C4—1881no—
238C2×C4—1881no—
239C2×C4—1881no—
240C2×C4#1309398111yescharacteristic
241C2×C4(C22⋊D47)⋊C411yescharacteristic
242C2×C4#1309398611yescharacteristic
243C2×C4#1309399711yescharacteristic
244C2×C4#1309401011yescharacteristic
245C2×C4#1309401911yescharacteristic
246C2×C4SD16×D4711yescharacteristic
247C2×C4#1309404311yescharacteristic
248C2×C4#1309405311yescharacteristic
249C2×C4#1309405611yescharacteristic
250C8—42no—
251C8—81no—
252C8—81no—
253C8—1882no—
254C8—1882no—
255C22—21no—
256C22—21no—
257C22#1568901211yescharacteristic
258C22#1568945511yescharacteristic
259C22#1568946611yescharacteristic
260C4—21no—
261C4—21no—
262C4—21no—
263C4—944no—
264C4—3761no—
265C4—3761no—
266C4—1882no—
267C4#1568921411yescharacteristic
268C4#1568922711yescharacteristic
269C4#1568923912yes—
270C4#1568932711yescharacteristic
271C4#1568934411yescharacteristic
272C2—21no—
273C2—21no—
274C2#2024402011yescharacteristic
275C2#2024405511yescharacteristic
276C2#2024409011yescharacteristic
subgroupG/H
centerC2×C4#13093997
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1301C2
socleC22×C47#388
prime radicalC23×C47#123
cosocleC2×C42×C47C23
hypercenter#1301D47
hypocenterC47#9579
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
17. #15688871C4semidirect
18. #15688868C4semidirect
161. C47#9579semidirect

Maximal subgroups

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

subgroupindex
#202443642
C4⋊C8×Dic472
#202421602
#202420462
#202417922
#202412202
C47×#13012
#957947