Finite Group Databasebeta

← #26104079

Subgroups — #26104079

Order 12032.

290 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 63 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×#15688918C211yesmaximal, characteristic
2#20244205C211yesmaximal, characteristic
3C2×C47×#355C211yesmaximal, characteristic
4C2×#13093948—21no—
5C2×#13093945—21no—
6C2×#13093939—21no—
7C2×#13093935C411yescharacteristic, complemented
8#15688993C411yescharacteristic, complemented
9#15688918—21no—
10#15688918C412yescomplemented
11C2×#124×C47C2211yescharacteristic
12C2×#101×C47—21no—
13C47×#355—21no—
14C47×#355C412yescomplemented
15C22×C47⋊C8—21no—
16C2×C4⋊Dic47—21no—
17C2×C4⋊Dic47—41no—
18C2×C22⋊Dic47—41no—
19#13093948—42no—
20#13093945—42no—
21#13093939—22no—
22#13093939—41no—
23#13093935—41no—
24C22×Q8×C47D411yescharacteristic
25C22×D4×C47D411yescharacteristic
26C2×M16×C47—41no—
27#124×C47—41no—
28#124×C47C2×C411yescharacteristic
29#124×C47C2×C411yescharacteristic
30#124×C47D411yescharacteristic
31#124×C47Q811yescharacteristic
32C2×C22⋊C4×C47—41no—
33C2×C42×C47C2×C411yescharacteristic
34#101×C47—22no—
35#101×C47—41no—
36C24×C47—21no—
37C2×Q8×C47—21no—
38C2×Q8×C47—41no—
39C2×Q8×C47—41no—
40C2×Q8×C47C22⋊C411yescharacteristic
41C2×Q8×C47C4⋊C411yescharacteristic
42C2×D4×C47—21no—
43C2×D4×C47—41no—
44C2×D4×C47—41no—
45C2×D4×C47C22⋊C411yescharacteristic
46C2×D4×C47C4⋊C411yescharacteristic
47C22×C4×C47—21no—
48C22×C4×C47—21no—
49C22×C4×C47C22⋊C411yescharacteristic
50C22×Dic47—41no—
51C22×Dic47—81no—
52M16×C47—42no—
53M16×C47—81no—
54C2×C8×C47—81no—
55C22⋊C4×C47—41no—
56C22⋊C4×C47—41no—
57C22⋊C4×C47—41no—
58C22⋊C4×C47—41no—
59C42×C47—21no—
60C42×C47C4211yescharacteristic
61C42×C47C4⋊C411yescharacteristic
62C22⋊Dic47—42no—
63C22⋊Dic47—81no—
64C4⋊Dic47—41no—
65C4⋊Dic47—41no—
66C4⋊Dic47—82no—
67C2×C47⋊C8—22no—
68C2×C47⋊C8—41no—
69C2×C47⋊C8—41no—
70C23×C47—21no—
71C23×C47—21no—
72C23×C47—41no—
73C23×C47—41no—
74C23×C47—81no—
75C23×C47—81no—
76C23×C47#9911yescharacteristic
77Q8×C47—41no—
78Q8×C47—41no—
79Q8×C47—81no—
80D4×C47—41no—
81D4×C47—41no—
82D4×C47—81no—
83C2×C4×C47—21no—
84C2×C4×C47—21no—
85C2×C4×C47—21no—
86C2×C4×C47—21no—
87C2×C4×C47—21no—
88C2×C4×C47—21no—
89C2×C4×C47—41no—
90C2×C4×C47—41no—
91C2×C4×C47—41no—
92C2×C4×C47—41no—
93C2×C4×C47#9511yescharacteristic
94C2×C4×C47#9911yescharacteristic
95C2×Dic47—82no—
96C2×Dic47—82no—
97C2×Dic47—81no—
98C2×Dic47—161no—
99C2×Dic47—161no—
100C8×C47—82no—
101C47⋊C8—42no—
102#7508—471nomaximal, Sylow
103C22×C47—21no—
104C22×C47—21no—
105C22×C47—41no—
106C22×C47—41no—
107C22×C47—81no—
108C22×C47—81no—
109C22×C47—81no—
110C22×C47—81no—
111C22×C47—161no—
112C22×C47#34211yescharacteristic
113C22×C47#35311yescharacteristic
114C22×C47#35411yescharacteristic
115C4×C47—21no—
116C4×C47—21no—
117C4×C47—41no—
118C4×C47—41no—
119C4×C47—41no—
120C4×C47—41no—
121Dic47—162no—
122Dic47—161no—
123C2×#355D4711yescharacteristic
124C2×#354—471no—
125#1959—471no—
126C2×C47—21no—
127C2×C47—21no—
128C2×C47—81no—
129C2×C47—81no—
130C2×C47#139311yescharacteristic
131C2×C47#140611yescharacteristic
132C2×C47#140711yescharacteristic
133C2×#124C2×D4711yescharacteristic
134C2×#103—941no—
135C2×#102—941no—
136C2×#101—21no—
137C2×#99—941no—
138C2×#97—471no—
139#384—471no—
140#355—21no—
141#355Dic4712yescomplemented
142#354—472no—
143#354—941no—
144C47#750811yescharacteristic, complemented, Sylow
145C22×Q8C22⋊D4711yescharacteristic
146C22×D4D18811yescharacteristic
147C2×M16—41no—
148C22×C8—941no—
149#124—41no—
150#124Dic9411yescharacteristic
151#124C4×D4711yescharacteristic
152#124C2×Dic4711yescharacteristic
153#124C22⋊D4711yescharacteristic
154C2×C4⋊C4—941no—
155C2×C4⋊C4—1881no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—1881no—
158C2×C42C4×D4711yescharacteristic
159#103—1882no—
160#102—1882no—
161#101—22no—
162#101—41no—
163#99—942no—
164#99—1881no—
165#97—1881no—
166C24—21no—
167C2×Q8—21no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8Dic47⋊C411yescharacteristic
171C2×Q8C22⋊Dic4711yescharacteristic
172C2×D4—21no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4C4⋊Dic4711yescharacteristic
176C2×D4(C2×D47)⋊C411yescharacteristic
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—3761no—
180C22×C4—1881no—
181C22×C4(C2×D47)⋊C411yescharacteristic
182M16—42no—
183M16—81no—
184C2×C8—81no—
185C2×C8—942no—
186C2×C8—1881no—
187C2×C8—1881no—
188C4⋊C4—3762no—
189C4⋊C4—1881no—
190C4⋊C4—1881no—
191C22⋊C4—41no—
192C22⋊C4—41no—
193C22⋊C4—41no—
194C22⋊C4—41no—
195C22⋊C4—3761no—
196C22⋊C4—1882no—
197C42—21no—
198C42C4×Dic4711yescharacteristic
199C42Dic47⋊C411yescharacteristic
200C23—21no—
201C23—21no—
202C23—41no—
203C23—41no—
204C23—81no—
205C23—81no—
206C23#1309393811yescharacteristic
207Q8—41no—
208Q8—41no—
209Q8—81no—
210D4—41no—
211D4—41no—
212D4—81no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—3762no—
224C2×C4—3762no—
225C2×C4—3761no—
226C2×C4—7521no—
227C2×C4—7521no—
228C2×C4C47⋊#9511yescharacteristic
229C2×C4#1309393811yescharacteristic
230C8—82no—
231C8—1882no—
232C22—21no—
233C22—21no—
234C22—41no—
235C22—41no—
236C22—81no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—161no—
241C22#1568888911yescharacteristic
242C22#1568891511yescharacteristic
243C22#1568891711yescharacteristic
244C4—21no—
245C4—21no—
246C4—41no—
247C4—41no—
248C4—41no—
249C4—41no—
250C4—7522no—
251C4—7521no—
252C2—21no—
253C2—21no—
254C2—81no—
255C2—81no—
256C2#2024151211yescharacteristic
257C2#2024154611yescharacteristic
258C2#2024154911yescharacteristic
subgroupG/H
centerC22#15688915
commutatorC42×C47C42
FrattiniC2×#124C2×D47
FittingC2×C47×#355C2
socleC22×C47#353
prime radicalC22×D4×C47D4
cosocleC2×#124×C47C22
hypercenterC2×#355D47
hypocenterC47#7508
LCSG ▷ C42×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ C2×#355

Complements

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

NH ≅ G/Ntype
7. C2×#13093935C4semidirect
8. #15688993C4semidirect
10. #15688918C4semidirect
14. C47×#355C4semidirect
141. #355Dic47semidirect
144. C47#7508semidirect

Maximal subgroups

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

subgroupindex
C2×#156889182
#202442052
C2×C47×#3552
#750847