Finite Group Databasebeta

← #26105565

Subgroups — #26105565

Order 12032.

358 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 91 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#20244951C211yesmaximal, characteristic
2#20244951C211yesmaximal, characteristic
3#20244951C211yesmaximal, characteristic
4#20241211C211yesmaximal, characteristic
5#20241211C211yesmaximal, characteristic
6#20241211C211yesmaximal, characteristic
7C47×#2726C211yesmaximal, characteristic
8#15688993C2211yescharacteristic
9#15688885—21no—
10#15688885—21no—
11#15688885—21no—
12#15688885—21no—
13#15688885—21no—
14#15688885—21no—
15#15688878C2211yescharacteristic
16#15688878C2211yescharacteristic
17#15688878C2211yescharacteristic
18C2×#125×C47—21no—
19C47×#398—21no—
20C47×#398—21no—
21C47×#398—21no—
22C47×#389—21no—
23C47×#389C2211yescharacteristic
24C47×#389C2211yescharacteristic
25C47×#389C2211yescharacteristic
26C47×#387—21no—
27C22×C47⋊C8—21no—
28C22×C47⋊C8—21no—
29C22×C47⋊C8—21no—
30C2×C4⋊Dic47—21no—
31C2×C4⋊Dic47—21no—
32C2×C4⋊Dic47—21no—
33#125×C47—42no—
34#125×C47—42no—
35C2×C4⋊C4×C47—21no—
36C2×C4⋊C4×C47—21no—
37C2×C4⋊C4×C47—21no—
38C2×C4⋊C4×C47—21no—
39C2×C4⋊C4×C47—21no—
40C2×C4⋊C4×C47—21no—
41C2×C4⋊C4×C47—21no—
42C2×C4⋊C4×C47D411yescharacteristic
43C2×C4⋊C4×C47D411yescharacteristic
44C2×C4⋊C4×C47D411yescharacteristic
45C2×C4⋊C4×C47D411yescharacteristic
46C2×C4⋊C4×C47D411yescharacteristic
47C2×C4⋊C4×C47D411yescharacteristic
48C2×C42×C47—21no—
49C2×C42×C47C2311yescharacteristic
50#95×C47—21no—
51#95×C47—21no—
52#95×C47—21no—
53#95×C47—21no—
54#95×C47—21no—
55#95×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47C2×D411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47C2×D411yescharacteristic
65C22×Dic47—41no—
66C4⋊C4×C47—21no—
67C4⋊C4×C47—21no—
68C4⋊C4×C47—21no—
69C4⋊C4×C47—21no—
70C4⋊C4×C47—42no—
71C4⋊C4×C47—42no—
72C4⋊C4×C47—42no—
73C4⋊C4×C47—41no—
74C4⋊C4×C47—41no—
75C4⋊C4×C47—41no—
76C4⋊C4×C47—41no—
77C4⋊C4×C47—41no—
78C4⋊C4×C47—41no—
79C4⋊C4×C47—41no—
80C4⋊C4×C47—41no—
81C4⋊C4×C47—41no—
82C4⋊C4×C47—81no—
83C4⋊C4×C47SD1612yes—
84C4⋊C4×C47Q1612yes—
85C42×C47—41no—
86C42×C47—42no—
87C4⋊Dic47—42no—
88C4⋊Dic47—42no—
89C4⋊Dic47—42no—
90C2×C47⋊C8—41no—
91C2×C47⋊C8—41no—
92C2×C47⋊C8—41no—
93C2×C47⋊C8—41no—
94C2×C47⋊C8—41no—
95C2×C47⋊C8—41no—
96C2×C47⋊C8—41no—
97C2×C47⋊C8—41no—
98C2×C47⋊C8—41no—
99C23×C47#12011yescharacteristic
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—22no—
107C2×C4×C47—22no—
108C2×C4×C47—41no—
109C2×C4×C47—41no—
110C2×C4×C47—41no—
111C2×C4×C47—41no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47—41no—
121C2×C4×C47—41no—
122C2×C4×C47—41no—
123C2×C4×C47—41no—
124C2×C4×C47C2×SD1611yescharacteristic
125C2×C4×C47C2×Q1611yescharacteristic
126C2×C4×C47#13511yescharacteristic
127C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
128C2×C4×C47#13711yescharacteristic
129C2×C4×C47#13711yescharacteristic
130C2×Dic47—81no—
131C2×Dic47—81no—
132C2×Dic47—81no—
133C47⋊C8—81no—
134C47⋊C8—81no—
135C47⋊C8—81no—
136#7827—471nomaximal, Sylow
137C22×C47#45011yescharacteristic
138C22×C47#45111yescharacteristic
139C22×C47#45211yescharacteristic
140C22×C47#45411yescharacteristic
141C22×C47#45411yescharacteristic
142C22×C47#45411yescharacteristic
143C22×C47#45811yescharacteristic
144C4×C47—21no—
145C4×C47—21no—
146C4×C47—21no—
147C4×C47—21no—
148C4×C47—21no—
149C4×C47—21no—
150C4×C47—42no—
151C4×C47—42no—
152C4×C47—81no—
153C4×C47—81no—
154C4×C47—81no—
155C4×C47—81no—
156Dic47—161no—
157#2726D4711yescharacteristic
158#2100—471no—
159#2100—471no—
160#2100—471no—
161#1271—471no—
162#1271—471no—
163#1271—471no—
164C2×C47#161511yescharacteristic
165C2×C47#161611yescharacteristic
166C2×C47#161711yescharacteristic
167C2×C47#163511yescharacteristic
168C2×C47#163611yescharacteristic
169C2×C47#163711yescharacteristic
170C2×C47#165811yescharacteristic
171C2×#125—21no—
172#398—21no—
173#398—21no—
174#398—21no—
175#389—21no—
176#389C2×D4711yescharacteristic
177#389C2×D4711yescharacteristic
178#389C2×D4711yescharacteristic
179#387—21no—
180#384—471no—
181#340—941no—
182#340—941no—
183#340—941no—
184#340—941no—
185#340—941no—
186#340—941no—
187#336—471no—
188#336—471no—
189#336—471no—
190C47#782711yescharacteristic, complemented, Sylow
191C22×C8—941no—
192C22×C8—941no—
193C22×C8—941no—
194#125—42no—
195#125—42no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—941no—
204C2×C4⋊C4—941no—
205C2×C4⋊C4—941no—
206C2×C4⋊C4C22⋊D4711yescharacteristic
207C2×C4⋊C4C22⋊D4711yescharacteristic
208C2×C4⋊C4C22⋊D4711yescharacteristic
209C2×C4⋊C4C22⋊D4711yescharacteristic
210C2×C4⋊C4C22⋊D4711yescharacteristic
211C2×C4⋊C4C22⋊D4711yescharacteristic
212C2×C42—21no—
213C2×C42C22×D4711yescharacteristic
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—1881no—
227C22×C4C2×C22⋊D4711yescharacteristic
228C22×C4C2×C22⋊D4711yescharacteristic
229C22×C4C2×C22⋊D4711yescharacteristic
230C2×C8—1881no—
231C2×C8—1881no—
232C2×C8—1881no—
233C2×C8—1881no—
234C2×C8—1881no—
235C2×C8—1881no—
236C2×C8—1881no—
237C2×C8—1881no—
238C2×C8—1881no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—81no—
256C4⋊C4—1882no—
257C4⋊C4—1882no—
258C4⋊C4—1882no—
259C4⋊C4#1062533512yes—
260C4⋊C4#1062533712yes—
261C42—41no—
262C42—42no—
263C23#1309399311yescharacteristic
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—3761no—
289C2×C4—3761no—
290C2×C4—3761no—
291C2×C4C2×#1062533511yescharacteristic
292C2×C4C2×#1062533711yescharacteristic
293C2×C4#1309404411yescharacteristic
294C2×C4#1309405011yescharacteristic
295C2×C4#1309405411yescharacteristic
296C2×C4#1309405511yescharacteristic
297C8—3761no—
298C8—3761no—
299C8—3761no—
300C22#1568927411yescharacteristic
301C22#1568927911yescharacteristic
302C22#1568928511yescharacteristic
303C22#1568929611yescharacteristic
304C22#1568929611yescharacteristic
305C22#1568929611yescharacteristic
306C22#1568931311yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—42no—
314C4—42no—
315C4—81no—
316C4—81no—
317C4—81no—
318C4—81no—
319C4—7521no—
320C2#2024241911yescharacteristic
321C2#2024242511yescharacteristic
322C2#2024243011yescharacteristic
323C2#2024253911yescharacteristic
324C2#2024254611yescharacteristic
325C2#2024255311yescharacteristic
326C2#2024269011yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2726C2
socle, prime radicalC23×C47#120
hypercenter#2726D47
hypocenterC47#7827
LCSG ▷ C2×C42×C47 ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2726

Complements

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

NH ≅ G/Ntype
190. C47#7827semidirect

Maximal subgroups

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

subgroupindex
#202449512
#202412112
C47×#27262
#782747