Finite Group Databasebeta

← #26191177

Subgroups — #26191177

Order 12032.

906 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 906 conjugacy classes of subgroups — too many to draw. Only its 191 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#20257198C211yesmaximal, characteristic
2#20254532C212yesmaximal
3#20242395C218yesmaximal
4#20241886C212yesmaximal
5#20241712C211yesmaximal, characteristic
6C47×#2546C211yesmaximal, characteristic
7Q8×Dic94—22no—
8#15689852—22no—
9#15689849—24no—
10#15689754—21no—
11#15689754C2214yes—
12#15689680—22no—
13#15689680C2212yes—
14#15689601—21no—
15#15689519—22no—
16#15689475—22no—
17#15689453C2218yes—
18#15689405—24no—
19#15689176C2212yes—
20#15688859C2218yes—
21#15688849C2214yes—
22C2×C4⋊Q8×C47—21no—
23C2×C4⋊Q8×C47C2211yescharacteristic
24C2×C4×Q8×C47C2211yescharacteristic
25C47×#398C2214yes—
26C47×#391—21no—
27C47×#389—22no—
28C4×C4⋊C4×C47C2211yescharacteristic
29#13094024—22no—
30#13094024—22no—
31#13094024D412yes—
32#13094024D412yes—
33#13094023—24no—
34#13094023—28no—
35#13094016—24no—
36#13094016—22no—
37#13094015—24no—
38#13094015—28no—
39#13094003—24no—
40#13094003—22no—
41#13094003C2314yes—
42Q8×Dic47—24no—
43Q8×Dic47—24no—
44C4×Dic94—24no—
45C4×Dic94—24no—
46C4×Dic94—22no—
47C4×Dic94D414yes—
48#13093983—22no—
49#13093983—24no—
50#13093955—28no—
51C4⋊(C47⋊C8)—22no—
52#13093947—28no—
53C22⋊(C47⋊C8)C2314yes—
54C4×C47⋊C8—22no—
55C22×Q8×C47—21no—
56C22×Q8×C47D411yescharacteristic
57C4⋊Q8×C47—22no—
58C4⋊Q8×C47—22no—
59C4⋊Q8×C47—44no—
60C4×Q8×C47—22no—
61C4×Q8×C47—22no—
62C2×C4⋊C4×C47—21no—
63C2×C4⋊C4×C47—24no—
64C2×C4⋊C4×C47—21no—
65C2×C4⋊C4×C47D411yescharacteristic
66C2×C4⋊C4×C47C2312yes—
67C2×C4⋊C4×C47C2312yes—
68C2×C42×C47—21no—
69C2×C42×C47D412yes—
70C2×C42×C47C2311yescharacteristic
71#95×C47—24no—
72#95×C47C2312yes—
73C2×Q8×C47—22no—
74C2×Q8×C47—22no—
75C2×Q8×C47—42no—
76C2×Q8×C47—42no—
77C2×Q8×C47—42no—
78C2×Q8×C47SD1614yes—
79C22×C4×C47—22no—
80C22×C4×C47—22no—
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47C2×D412yes—
83C22×C4×C47C2×D412yes—
84C22×C4×C47C2×D411yescharacteristic
85C22×C4×C47C2411yescharacteristic
86C2×Dic94—22no—
87C2×Dic94—24no—
88C2×Dic94C2×D414yes—
89C4⋊C4×C47—24no—
90C4⋊C4×C47—22no—
91C4⋊C4×C47—24no—
92C4⋊C4×C47—24no—
93C4⋊C4×C47—41no—
94C4⋊C4×C47—41no—
95C4⋊C4×C47—48no—
96C42×C47—22no—
97C42×C47—22no—
98C42×C47—42no—
99C42×C47C2×D412yes—
100C42×C47C2×D412yes—
101C22⋊Dic47—24no—
102C4⋊Dic47—24no—
103C4⋊Dic47—24no—
104C4⋊Dic47—24no—
105C4⋊Dic47—22no—
106C4⋊Dic47—22no—
107C4⋊Dic47—22no—
108C4⋊Dic47C2×D414yes—
109Dic47⋊C4—28no—
110Dic47⋊C4—24no—
111Dic47⋊C4—24no—
112C4×Dic47—28no—
113C4×Dic47—24no—
114C2×C47⋊C8—24no—
115C23×C47C22×D411yescharacteristic
116Q8×C47—28no—
117Q8×C47—84no—
118C2×C4×C47—22no—
119C2×C4×C47—22no—
120C2×C4×C47—22no—
121C2×C4×C47—24no—
122C2×C4×C47—22no—
123C2×C4×C47—41no—
124C2×C4×C47—42no—
125C2×C4×C47—42no—
126C2×C4×C47—42no—
127C2×C4×C47—42no—
128C2×C4×C47—42no—
129C2×C4×C47—42no—
130C2×C4×C47#12011yescharacteristic
131C2×C4×C47#12012yes—
132C2×C4×C47#12011yescharacteristic
133C2×C4×C47C2×SD1614yes—
134C2×C4×C47C2×SD1612yes—
135C2×C4×C47#13712yes—
136C2×C4×C47C22×D412yes—
137C2×Dic47—24no—
138C2×Dic47—24no—
139Dic94—44no—
140Dic94—44no—
141Dic94—84no—
142C47⋊C8—44no—
143#18410—471nomaximal, Sylow
144C22×C47—21no—
145C22×C47—21no—
146C22×C47C2×#12011yescharacteristic
147C22×C47C22×SD1611yescharacteristic
148C22×C47C2×#13711yescharacteristic
149C4×C47—24no—
150C4×C47—21no—
151C4×C47—22no—
152C4×C47—44no—
153C4×C47—42no—
154C4×C47—82no—
155C4×C47#44814yes—
156C4×C47#45612yes—
157Dic47—44no—
158Dic47—44no—
159#3502—471no—
160#3229—472no—
161#2546D4711yescharacteristic
162#1611—478no—
163#1503—472no—
164#1461—471no—
165C2×C47—21no—
166C2×C47—21no—
167C2×C47C2×#44811yescharacteristic
168C2×C47C2×#45611yescharacteristic
169C2×C47#302811yescharacteristic
170Q82—942no—
171#557—944no—
172#557—942no—
173#544—474no—
174#544—941no—
175#533—472no—
176#533—942no—
177C2×C4⋊Q8—21no—
178C2×C4⋊Q8C2×D4711yescharacteristic
179#519—941no—
180C2×C4×Q8C2×D4711yescharacteristic
181#498—942no—
182#487—942no—
183#483—478no—
184#475—944no—
185#431—472no—
186#398C2×D4714yes—
187#391—21no—
188#389—22no—
189C4×C4⋊C4C2×D4711yescharacteristic
190#328—478no—
191#324—474no—
192C47#1841011yescharacteristic, complemented, Sylow
193C22×Q8—21no—
194C22×Q8C22⋊D4711yescharacteristic
195C4⋊Q8—22no—
196C4⋊Q8—22no—
197C4⋊Q8—44no—
198C4⋊Q8—472no—
199C4⋊Q8—472no—
200C4⋊Q8—942no—
201C4⋊Q8—942no—
202C4⋊Q8—948no—
203C4⋊Q8—944no—
204#125—944no—
205#125—948no—
206#125—944no—
207#125—942no—
208C22⋊Q8—474no—
209C22⋊Q8—942no—
210C22⋊Q8—944no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×Q8—474no—
214C4×Q8—944no—
215C4×Q8—944no—
216C4×Q8—944no—
217C4×Q8—944no—
218C4×Q8—942no—
219#117—942no—
220#117—944no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4C22⋊D4711yescharacteristic
225C2×C4⋊C4C22×D4712yes—
226C2×C4⋊C4C22×D4712yes—
227C2×C42—21no—
228C2×C42C22⋊D4712yes—
229C2×C42C22×D4711yescharacteristic
230#106—948no—
231C4⋊C8—942no—
232#103—948no—
233C22⋊C8—474no—
234C4×C8—942no—
235#95—24no—
236#95C22×D4712yes—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—474no—
243C2×Q8—944no—
244C2×Q8—942no—
245C2×Q8#1062533514yes—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4C2×C22⋊D4711yescharacteristic
249C22×C4C2×C22⋊D4712yes—
250C22×C4C2×C22⋊D4712yes—
251C22×C4C2×C22⋊D4711yescharacteristic
252C22×C4C23×D4711yescharacteristic
253C2×C8—944no—
254C4⋊C4—24no—
255C4⋊C4—22no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—48no—
261C4⋊C4—474no—
262C4⋊C4—948no—
263C4⋊C4—944no—
264C4⋊C4—944no—
265C4⋊C4—944no—
266C4⋊C4—944no—
267C4⋊C4—942no—
268C4⋊C4—942no—
269C4⋊C4—942no—
270C4⋊C4—944no—
271C22⋊C4—944no—
272C42—22no—
273C42—22no—
274C42—42no—
275C42—948no—
276C42—944no—
277C42D4×D4712yes—
278C42D4×D4712yes—
279C23C22×C22⋊D4711yescharacteristic
280Q8—28no—
281Q8—84no—
282Q8—3764no—
283Q8—1884no—
284Q8—1884no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—944no—
298C2×C4—944no—
299C2×C4#1309399211yescharacteristic
300C2×C4#1309399212yes—
301C2×C4#1309399211yescharacteristic
302C2×C4C2×#1062533514yes—
303C2×C4C2×#1062533512yes—
304C2×C4#1309405412yes—
305C2×C4C2×D4×D4712yes—
306C8—1884no—
307C22—21no—
308C22—21no—
309C22C2×#1309399211yescharacteristic
310C22C22×#1062533511yescharacteristic
311C22C2×#1309405411yescharacteristic
312C4—21no—
313C4—22no—
314C4—24no—
315C4—44no—
316C4—42no—
317C4—82no—
318C4—1884no—
319C4—1884no—
320C4#1568926114yes—
321C4#1568930512yes—
322C2—21no—
323C2—21no—
324C2C2×#1568926111yescharacteristic
325C2C2×#1568930511yescharacteristic
326C2#2025279311yescharacteristic
subgroupG/H
centerC22C2×#13093992
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2546C2
socleC22×C47C2×#120
prime radicalC23×C47C22×D4
hypercenter#2546D47
hypocenterC47#18410
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2546

Complements

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

NH ≅ G/Ntype
192. C47#18410semidirect

Maximal subgroups

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

subgroupindex
#202571982
#202545322
#202423952
#202418862
#202417122
C47×#25462
#1841047