Finite Group Databasebeta

← #26104055

Subgroups — #26104055

Order 12032.

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

normal subgroups only

This group has 348 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×#15688916C211yesmaximal, characteristic
2#20244195C211yesmaximal, characteristic
3C2×C47×#353C211yesmaximal, characteristic
4C2×#13093945—21no—
5C2×#13093945—21no—
6C2×#13093939—21no—
7C2×C4×C47⋊C8C411yescharacteristic, complemented
8#15688993C411yescharacteristic, complemented
9#15688916—21no—
10#15688916C412yescomplemented
11C2×#127×C47C2211yescharacteristic
12C2×#99×C47—21no—
13C47×#353—21no—
14C47×#353C412yescomplemented
15C22×C47⋊C8—21no—
16C2×C4⋊Dic47—21no—
17C2×C4⋊Dic47—41no—
18C2×C22⋊Dic47—41no—
19#13093945—42no—
20#13093945—42no—
21#13093939—22no—
22#13093939—41no—
23C4×C47⋊C8—41no—
24C22×D4×C47—41no—
25C22×D4×C47D411yescharacteristic
26C22×D4×C47D411yescharacteristic
27#127×C47—41no—
28#127×C47C2×C411yescharacteristic
29#127×C47C2×C411yescharacteristic
30#127×C47D411yescharacteristic
31#127×C47Q811yescharacteristic
32C2×C22⋊C4×C47—41no—
33C2×C42×C47C2×C411yescharacteristic
34#99×C47—22no—
35#99×C47—41no—
36C24×C47—21no—
37C24×C47—21no—
38C2×D4×C47—21no—
39C2×D4×C47—21no—
40C2×D4×C47—41no—
41C2×D4×C47—41no—
42C2×D4×C47—41no—
43C2×D4×C47—41no—
44C2×D4×C47—41no—
45C2×D4×C47—41no—
46C2×D4×C47—41no—
47C2×D4×C47—41no—
48C2×D4×C47—82no—
49C2×D4×C47—82no—
50C2×D4×C47C22⋊C411yescharacteristic
51C2×D4×C47C22⋊C411yescharacteristic
52C2×D4×C47C4⋊C411yescharacteristic
53C2×D4×C47C4⋊C411yescharacteristic
54C22×C4×C47—21no—
55C22×C4×C47—81no—
56C22×C4×C47C22⋊C411yescharacteristic
57C22×Dic47—41no—
58C22×Dic47—81no—
59C22⋊C4×C47—42no—
60C22⋊C4×C47—81no—
61C42×C47—21no—
62C42×C47C4211yescharacteristic
63C42×C47C4⋊C411yescharacteristic
64C22⋊Dic47—42no—
65C22⋊Dic47—81no—
66C4⋊Dic47—41no—
67C4⋊Dic47—41no—
68C4⋊Dic47—82no—
69C2×C47⋊C8—22no—
70C2×C47⋊C8—41no—
71C2×C47⋊C8—41no—
72C23×C47—21no—
73C23×C47—21no—
74C23×C47—21no—
75C23×C47—21no—
76C23×C47—41no—
77C23×C47—41no—
78C23×C47—41no—
79C23×C47—41no—
80C23×C47—81no—
81C23×C47—81no—
82C23×C47—81no—
83C23×C47—81no—
84C23×C47#9911yescharacteristic
85D4×C47—41no—
86D4×C47—41no—
87D4×C47—41no—
88D4×C47—41no—
89D4×C47—81no—
90D4×C47—81no—
91D4×C47—82no—
92D4×C47—82no—
93D4×C47—82no—
94D4×C47—82no—
95C2×C4×C47—22no—
96C2×C4×C47—22no—
97C2×C4×C47—21no—
98C2×C4×C47—21no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—82no—
102C2×C4×C47—161no—
103C2×C4×C47—161no—
104C2×C4×C47#9511yescharacteristic
105C2×C4×C47#9911yescharacteristic
106C2×Dic47—82no—
107C2×Dic47—82no—
108C2×Dic47—81no—
109C2×Dic47—161no—
110C2×Dic47—161no—
111C47⋊C8—42no—
112#7498—471nomaximal, Sylow
113C22×C47—21no—
114C22×C47—21no—
115C22×C47—41no—
116C22×C47—41no—
117C22×C47—41no—
118C22×C47—41no—
119C22×C47—81no—
120C22×C47—81no—
121C22×C47—81no—
122C22×C47—81no—
123C22×C47—81no—
124C22×C47—81no—
125C22×C47—81no—
126C22×C47—81no—
127C22×C47—161no—
128C22×C47—161no—
129C22×C47#34211yescharacteristic
130C22×C47#35311yescharacteristic
131C22×C47#35411yescharacteristic
132C4×C47—22no—
133C4×C47—22no—
134C4×C47—21no—
135C4×C47—21no—
136C4×C47—162no—
137Dic47—162no—
138Dic47—161no—
139C2×#353—471no—
140C2×#353D4711yescharacteristic
141#1957—471no—
142C2×C47—21no—
143C2×C47—21no—
144C2×C47—81no—
145C2×C47—81no—
146C2×C47—81no—
147C2×C47—81no—
148C2×C47#139311yescharacteristic
149C2×C47#140811yescharacteristic
150C2×C47#140911yescharacteristic
151C2×#127C2×D4711yescharacteristic
152C2×#102—941no—
153C2×#102—941no—
154C2×#99—21no—
155C2×#99—941no—
156C2×C4×C8—471no—
157#384—471no—
158#353—21no—
159#353—472no—
160#353—941no—
161#353Dic4712yescomplemented
162C47#749811yescharacteristic, complemented, Sylow
163C22×D4—41no—
164C22×D4D18811yescharacteristic
165C22×D4C22⋊D4711yescharacteristic
166C22×C8—941no—
167#127—41no—
168#127Dic9411yescharacteristic
169#127C4×D4711yescharacteristic
170#127C2×Dic4711yescharacteristic
171#127C22⋊D4711yescharacteristic
172C2×C4⋊C4—941no—
173C2×C4⋊C4—1881no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—1881no—
176C2×C42C4×D4711yescharacteristic
177#102—1882no—
178#102—1882no—
179#99—22no—
180#99—41no—
181#99—942no—
182#99—1881no—
183C4×C8—1881no—
184C24—21no—
185C24—21no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4—82no—
197C2×D4—82no—
198C2×D4Dic47⋊C411yescharacteristic
199C2×D4C4⋊Dic4711yescharacteristic
200C2×D4(C2×D47)⋊C411yescharacteristic
201C2×D4C22⋊Dic4711yescharacteristic
202C22×C4—21no—
203C22×C4—81no—
204C22×C4—3761no—
205C22×C4—1881no—
206C22×C4(C2×D47)⋊C411yescharacteristic
207C2×C8—942no—
208C2×C8—1881no—
209C2×C8—1881no—
210C4⋊C4—3762no—
211C4⋊C4—1881no—
212C4⋊C4—1881no—
213C22⋊C4—42no—
214C22⋊C4—81no—
215C22⋊C4—3761no—
216C22⋊C4—1882no—
217C42—21no—
218C42C4×Dic4711yescharacteristic
219C42Dic47⋊C411yescharacteristic
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—81no—
229C23—81no—
230C23—81no—
231C23—81no—
232C23#1309393811yescharacteristic
233D4—41no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—81no—
238D4—81no—
239D4—82no—
240D4—82no—
241D4—82no—
242D4—82no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—82no—
250C2×C4—161no—
251C2×C4—161no—
252C2×C4—3762no—
253C2×C4—3762no—
254C2×C4—3761no—
255C2×C4—7521no—
256C2×C4—7521no—
257C2×C4C47⋊#9511yescharacteristic
258C2×C4#1309393811yescharacteristic
259C8—1882no—
260C22—21no—
261C22—21no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—41no—
266C22—81no—
267C22—81no—
268C22—81no—
269C22—81no—
270C22—81no—
271C22—81no—
272C22—81no—
273C22—81no—
274C22—161no—
275C22—161no—
276C22#1568888911yescharacteristic
277C22#1568891511yescharacteristic
278C22#1568891711yescharacteristic
279C4—22no—
280C4—22no—
281C4—21no—
282C4—21no—
283C4—162no—
284C4—7522no—
285C4—7521no—
286C2—21no—
287C2—21no—
288C2—81no—
289C2—81no—
290C2—81no—
291C2—81no—
292C2#2024151211yescharacteristic
293C2#2024155011yescharacteristic
294C2#2024155211yescharacteristic
subgroupG/H
centerC22#15688915
commutatorC42×C47C42
FrattiniC2×#127C2×D47
FittingC2×C47×#353C2
socleC22×C47#353
prime radical, cosocleC2×#127×C47C22
hypercenterC2×#353D47
hypocenterC47#7498
LCSG ▷ C42×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ C2×#353

Complements

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

NH ≅ G/Ntype
7. C2×C4×C47⋊C8C4semidirect
8. #15688993C4semidirect
10. #15688916C4semidirect
14. C47×#353C4semidirect
161. #353Dic47semidirect
162. C47#7498semidirect

Maximal subgroups

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

subgroupindex
C2×#156889162
#202441952
C2×C47×#3532
#749847