Finite Group Databasebeta

← #26143539

Subgroups — #26143539

Order 12032.

1550 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688992C213yesmaximal
2C2×#15688987C213yesmaximal
3C2×#15688968C211yesmaximal, characteristic
4#20241653C211yesmaximal, characteristic
5#20241652C213yesmaximal
6#20241651C213yesmaximal
7C47×#1444C211yesmaximal, characteristic
8C22×C4⋊Dic47C2213yes—
9C22×Dic47⋊C4C2211yescharacteristic
10C22×Dic47⋊C4C2213yes—
11C22×Dic47⋊C4C2216yes—
12C22×C4×Dic47C2213yes—
13#15688992—212no—
14#15688987—212no—
15#15688968C418yescomplemented
16C2×C47⋊#95C2213yes—
17C2×C47⋊#95C2213yes—
18C2×C47⋊#95C2213yes—
19C2×C47⋊#95C2213yes—
20C22×C4⋊C4×C47C2213yes—
21C2×#95×C47C2213yes—
22C2×#95×C47C2211yescharacteristic
23C23×Dic47C2311yescharacteristic
24C23×Dic47C2313yes—
25C23×Dic47C2313yes—
26C23×Dic47C2311yescharacteristic
27C2×C4⋊Dic47—26no—
28C2×C4⋊Dic47—46no—
29C2×Dic47⋊C4—26no—
30C2×Dic47⋊C4—26no—
31C2×Dic47⋊C4—212no—
32C2×Dic47⋊C4—212no—
33C2×Dic47⋊C4—46no—
34C2×Dic47⋊C4—46no—
35C2×C4×Dic47C2×C4112yes—
36C2×C4×Dic47D4112yes—
37C2×C4×Dic47Q8112yes—
38C47⋊#95—26no—
39C47⋊#95—26no—
40C47⋊#95—26no—
41C47⋊#95C2×C4112yes—
42C23×C4×C47C2311yescharacteristic
43C23×C4×C47C2313yes—
44C23×C4×C47C2313yes—
45C2×C4⋊C4×C47—26no—
46C2×C4⋊C4×C47—46no—
47#95×C47—26no—
48#95×C47C2×C414yes—
49C24×C47C2411yescharacteristic
50C22×C4×C47—26no—
51C22×C4×C47—26no—
52C22×C4×C47—26no—
53C22×C4×C47—23no—
54C22×C4×C47—23no—
55C22×C4×C47—23no—
56C22×C4×C47—26no—
57C22×C4×C47—23no—
58C22×C4×C47—23no—
59C22×C4×C47—21no—
60C22×C4×C47C22×C416yes—
61C22×C4×C47C2×D416yes—
62C22×C4×C47C2×Q816yes—
63C22×Dic47—23no—
64C22×Dic47—23no—
65C22×Dic47—23no—
66C22×Dic47—23no—
67C22×Dic47—23no—
68C22×Dic47—23no—
69C22×Dic47—26no—
70C22×Dic47—23no—
71C22×Dic47—26no—
72C22×Dic47—23no—
73C22×Dic47—23no—
74C22×Dic47—21no—
75C22×Dic47C22×C416yes—
76C22×Dic47C22×C412yes—
77C22×Dic47C2×D416yes—
78C22×Dic47C2×D416yes—
79C22×Dic47C2×Q816yes—
80C22×Dic47C2×Q816yes—
81C4⋊C4×C47—412no—
82C4⋊Dic47—412no—
83Dic47⋊C4—424no—
84Dic47⋊C4—44no—
85Dic47⋊C4—412no—
86C4×Dic47C4⋊C4148yescomplemented
87C23×C47C23×C411yescharacteristic
88C23×C47C22×D413yes—
89C23×C47C22×Q813yes—
90C23×C47H(5,𝔽2)13yes—
91C23×C47H(5,𝔽2)11yescharacteristic
92C23×C472−1+413yes—
93C23×C472−1+411yescharacteristic
94C2×C4×C47—212no—
95C2×C4×C47—26no—
96C2×C4×C47—26no—
97C2×C4×C47—212no—
98C2×C4×C47—43no—
99C2×C4×C47—41no—
100C2×C4×C47—46no—
101C2×C4×C47—46no—
102C2×C4×C47—43no—
103C2×C4×C47—43no—
104C2×C4×C47—43no—
105C2×C4×C47—43no—
106C2×C4×C47C2×C4⋊C4112yes—
107C2×Dic47—26no—
108C2×Dic47—26no—
109C2×Dic47—26no—
110C2×Dic47—26no—
111C2×Dic47—26no—
112C2×Dic47—26no—
113C2×Dic47—43no—
114C2×Dic47—41no—
115C2×Dic47—46no—
116C2×Dic47—46no—
117C2×Dic47—43no—
118C2×Dic47—43no—
119C2×Dic47—43no—
120C2×Dic47—43no—
121C2×Dic47C2×C4⋊C4112yes—
122C2×Dic47C2×C4⋊C4112yes—
123C2×Dic47#12714yes—
124C2×Dic47C4⋊Q8112yes—
125#14354—471nomaximal, Sylow
126C22×C47C22×C4⋊C413yes—
127C22×C47#52013yes—
128C22×C47#52011yescharacteristic
129C22×C47C2×#12711yescharacteristic
130C22×C47C2×C4⋊Q813yes—
131C22×C47#53513yes—
132C22×C47#53513yes—
133C22×C47#53613yes—
134C22×C47#53613yes—
135C22×C47#54413yes—
136C22×C47#54416yes—
137C22×C47#54413yes—
138C4×C47—212no—
139C4×C47—83no—
140C4×C47—81no—
141Dic47—212no—
142Dic47—83no—
143Dic47—81no—
144Dic47#38418yes—
145C2×#384—473no—
146C2×#382—473no—
147C2×#376—471no—
148#1444—473no—
149#1444—471no—
150#1444—473no—
151#1444D4711yescharacteristic
152C2×C47C2×#38411yescharacteristic
153C2×C47#231713yes—
154C2×C47#231713yes—
155C2×C47#265711yescharacteristic
156C2×C47#265811yescharacteristic
157C2×C47#266013yes—
158C2×C47#266013yes—
159C22×C4⋊C4—471no—
160C22×C4⋊C4—473no—
161C22×C4⋊C4—476no—
162C22×C4⋊C4—473no—
163C22×C4⋊C4C2×D4713yes—
164C22×C42—473no—
165#384—9412no—
166#382—9412no—
167#376—478no—
168C2×#95—473no—
169C2×#95—473no—
170C2×#95—473no—
171C2×#95—473no—
172C2×#95C2×D4713yes—
173C2×#95C2×D4711yescharacteristic
174C47#1435411yescharacteristic, complemented, Sylow
175C23×C4—471no—
176C23×C4—473no—
177C23×C4—473no—
178C23×C4—471no—
179C23×C4C22×D4713yes—
180C23×C4C22×D4711yescharacteristic
181C23×C4C22×D4713yes—
182C2×C4⋊C4—26no—
183C2×C4⋊C4—46no—
184C2×C4⋊C4—946no—
185C2×C4⋊C4—946no—
186C2×C4⋊C4—9412no—
187C2×C4⋊C4—9412no—
188C2×C4⋊C4—946no—
189C2×C4⋊C4—1886no—
190C2×C4⋊C4—1886no—
191C2×C4⋊C4—1886no—
192C2×C42—4712no—
193C2×C42—4712no—
194C2×C42—4712no—
195#95—26no—
196#95—4712no—
197#95—946no—
198#95—946no—
199#95—946no—
200#95C4×D4714yes—
201C24C23×D4711yescharacteristic
202C22×C4—26no—
203C22×C4—26no—
204C22×C4—23no—
205C22×C4—23no—
206C22×C4—26no—
207C22×C4—23no—
208C22×C4—26no—
209C22×C4—23no—
210C22×C4—23no—
211C22×C4—21no—
212C22×C4—472no—
213C22×C4—476no—
214C22×C4—476no—
215C22×C4—476no—
216C22×C4—476no—
217C22×C4—476no—
218C22×C4—943no—
219C22×C4—943no—
220C22×C4—943no—
221C22×C4—943no—
222C22×C4—943no—
223C22×C4—943no—
224C22×C4—946no—
225C22×C4—943no—
226C22×C4—946no—
227C22×C4—943no—
228C22×C4—943no—
229C22×C4—941no—
230C22×C4C2×C4×D4716yes—
231C22×C4D4×D4716yes—
232C22×C4Q8×D4716yes—
233C4⋊C4—412no—
234C4⋊C4—18812no—
235C4⋊C4—1884no—
236C4⋊C4—18812no—
237C4⋊C4—18824no—
238C42—4748no—
239C23C22×C4×D4711yescharacteristic
240C23C2×D4×D4713yes—
241C23C2×Q8×D4713yes—
242C23#1309406913yes—
243C23#1309407011yescharacteristic
244C23#1309407113yes—
245C23#1309407211yescharacteristic
246C2×C4—212no—
247C2×C4—212no—
248C2×C4—26no—
249C2×C4—26no—
250C2×C4—43no—
251C2×C4—41no—
252C2×C4—46no—
253C2×C4—43no—
254C2×C4—46no—
255C2×C4—43no—
256C2×C4—43no—
257C2×C4—43no—
258C2×C4—4712no—
259C2×C4—474no—
260C2×C4—4712no—
261C2×C4—4712no—
262C2×C4—946no—
263C2×C4—946no—
264C2×C4—946no—
265C2×C4—946no—
266C2×C4—946no—
267C2×C4—946no—
268C2×C4—1883no—
269C2×C4—1881no—
270C2×C4—1886no—
271C2×C4—1883no—
272C2×C4—1886no—
273C2×C4—1883no—
274C2×C4—1883no—
275C2×C4—1883no—
276C2×C4C4⋊C4×D47112yes—
277C22C2×C4⋊C4×D4713yes—
278C22#1568960213yes—
279C22#1568960511yescharacteristic
280C22#127×D4711yescharacteristic
281C22C4⋊Q8×D4713yes—
282C22#1568968813yes—
283C22#1568968913yes—
284C22#1568969513yes—
285C22#1568969613yes—
286C22#1568975013yes—
287C22#1568975216yes—
288C22#1568975313yes—
289C4—212no—
290C4—83no—
291C4—81no—
292C4—478no—
293C4—9412no—
294C4—3763no—
295C4—3761no—
296C2#384×D4711yescharacteristic
297C2#2024600713yes—
298C2#2024601013yes—
299C2#2024949311yescharacteristic
300C2#2024949711yescharacteristic
301C2#2024950613yes—
302C2#2024951113yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×#1444C2
socle, prime radical, cosocleC24×C47C24
hypercenter#1444D47
hypocenterC47#14354
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
15. #15688968C4semidirect
86. C4×Dic47C4⋊C4semidirect
174. C47#14354semidirect

Maximal subgroups

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

subgroupindex
C2×#156889922
C2×#156889872
C2×#156889682
#202416532
#202416522
#202416512
C47×#14442
#1435447