Finite Group Databasebeta

← #26109681

Subgroups — #26109681

Order 12032.

482 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 115 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#20245786C211yesmaximal, characteristic
2#20243150C211yesmaximal, characteristic
3#20241270C212yesmaximal, complemented
4#20241270C212yesmaximal, complemented
5C47×#1749C211yesmaximal, characteristic
6C2×C22⋊(C47⋊C8)—21no—
7C2×#13093935C2212yes—
8#15688991—21no—
9#15688989C2211yescharacteristic
10#15688989C2211yescharacteristic
11#15688986—21no—
12#15688983C411yescharacteristic
13#15688983C411yescharacteristic
14C4×C4⋊Dic47C411yescharacteristic
15C4×C4⋊Dic47C411yescharacteristic
16C4×C22⋊Dic47—21no—
17#15688969—21no—
18#15688878—21no—
19C22×C42×C47C2211yescharacteristic
20C2×C22⋊C8×C47—21no—
21C2×#97×C47C2212yes—
22C47×#336—21no—
23C22×C47⋊C8—22no—
24C2×C4⋊Dic47C2×C411yescharacteristic
25C2×C4⋊Dic47C2×C411yescharacteristic
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×C22⋊Dic47—21no—
29C2×C22⋊Dic47—21no—
30C2×C4×Dic47—21no—
31C2×C4×Dic47—21no—
32C22⋊(C47⋊C8)—24no—
33#13093935—24no—
34C47⋊#95—21no—
35C47⋊#95—21no—
36C47⋊#95—21no—
37C47⋊#95—21no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C23×C4×C47C2×C411yescharacteristic
43C23×C4×C47D411yescharacteristic
44C23×C4×C47D411yescharacteristic
45C22×C8×C47—22no—
46C2×C42×C47—24no—
47C2×C42×C47C2×C411yescharacteristic
48C2×C42×C47D411yescharacteristic
49C2×C42×C47D411yescharacteristic
50C2×C42×C47C2311yescharacteristic
51C22⋊C8×C47—24no—
52#97×C47—24no—
53C24×C47C22⋊C411yescharacteristic
54C22×C4×C47—22no—
55C22×C4×C47—22no—
56C22×C4×C47—22no—
57C22×C4×C47—22no—
58C22×C4×C47—22no—
59C22×C4×C47—41no—
60C22×C4×C47—41no—
61C22×C4×C47—41no—
62C22×C4×C47—41no—
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22⋊C411yescharacteristic
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22×C411yescharacteristic
67C22×C4×C47C2×D411yescharacteristic
68C22×C4×C47C2×D411yescharacteristic
69C22×Dic47—21no—
70C22×Dic47—21no—
71C22×Dic47—21no—
72C22×Dic47—21no—
73C2×C8×C47—24no—
74C2×C8×C47—42no—
75C2×C8×C47—42no—
76C42×C47—22no—
77C42×C47—22no—
78C42×C47—22no—
79C42×C47#4114yes—
80C22⋊Dic47—42no—
81C22⋊Dic47—42no—
82C4⋊Dic47—41no—
83C4⋊Dic47—41no—
84C4⋊Dic47—41no—
85C4⋊Dic47—41no—
86C4×Dic47—42no—
87C4×Dic47—42no—
88C2×C47⋊C8—24no—
89C2×C47⋊C8—42no—
90C2×C47⋊C8—42no—
91C23×C47—21no—
92C23×C47—21no—
93C23×C47—21no—
94C23×C47—21no—
95C23×C47—21no—
96C23×C47—21no—
97C23×C47#10412yes—
98C23×C47C2×C22⋊C411yescharacteristic
99C2×C4×C47—21no—
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—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—44no—
114C2×C4×C47—44no—
115C2×C4×C47—44no—
116C2×C4×C47#10412yes—
117C2×C4×C47#11712yes—
118C2×C4×C47#12312yes—
119C2×C4×C47#12312yes—
120C2×Dic47—41no—
121C2×Dic47—41no—
122C2×Dic47—41no—
123C2×Dic47—41no—
124C2×Dic47—41no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127C2×Dic47—41no—
128C2×Dic47—41no—
129C2×Dic47—41no—
130C2×Dic47—41no—
131C2×Dic47—41no—
132C8×C47—44no—
133C47⋊C8—44no—
134#8720—471nomaximal, Sylow
135C22×C47—22no—
136C22×C47—22no—
137C22×C47—22no—
138C22×C47—41no—
139C22×C47—41no—
140C22×C47—41no—
141C22×C47—41no—
142C22×C47#38111yescharacteristic
143C22×C47#41311yescharacteristic
144C22×C47#41311yescharacteristic
145C22×C47C2×#10411yescharacteristic
146C22×C47C2×#10411yescharacteristic
147C22×C47#42111yescharacteristic
148C22×C47#42111yescharacteristic
149C4×C47—22no—
150C4×C47—22no—
151C4×C47—22no—
152C4×C47—22no—
153C4×C47—41no—
154C4×C47—41no—
155Dic47—81no—
156Dic47—81no—
157Dic47—81no—
158Dic47—81no—
159#2283—471no—
160#1749—471no—
161#1749D4711yescharacteristic
162#1295—472no—
163#1295—472no—
164C2×C47—42no—
165C2×C47#152411yescharacteristic
166C2×C47#152911yescharacteristic
167C2×C47#153211yescharacteristic
168C2×C47#153511yescharacteristic
169C2×C47#182211yescharacteristic
170C2×C47#182611yescharacteristic
171C2×C47#182611yescharacteristic
172C22×C42C2×D4711yescharacteristic
173C2×C22⋊C8—21no—
174C2×C22⋊C8—941no—
175C2×#97—472no—
176C2×#97C2×D4712yes—
177#383—941no—
178#382—471no—
179#382—471no—
180#381—941no—
181#380—471no—
182#380—471no—
183C4×C4⋊C4—471no—
184C4×C4⋊C4—471no—
185C4×C22⋊C4—941no—
186#376—941no—
187#336—21no—
188#336—941no—
189C47#872011yescharacteristic, complemented, Sylow
190C23×C4C4×D4711yescharacteristic
191C23×C4D18811yescharacteristic
192C23×C4C22⋊D4711yescharacteristic
193C22×C8—22no—
194C22×C8—942no—
195C2×C4⋊C4—471no—
196C2×C4⋊C4—471no—
197C2×C4⋊C4—471no—
198C2×C4⋊C4—471no—
199C2×C22⋊C4—941no—
200C2×C22⋊C4—941no—
201C2×C42—24no—
202C2×C42—941no—
203C2×C42—941no—
204C2×C42C4×D4711yescharacteristic
205C2×C42D18811yescharacteristic
206C2×C42C22⋊D4711yescharacteristic
207C2×C42C22×D4711yescharacteristic
208C22⋊C8—24no—
209C22⋊C8—944no—
210#97—24no—
211#97—944no—
212#95—941no—
213#95—941no—
214#95—941no—
215#95—941no—
216#95—941no—
217#95—941no—
218#95—941no—
219#95—941no—
220C24(C2×D47)⋊C411yescharacteristic
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—941no—
231C22×C4—941no—
232C22×C4—941no—
233C22×C4—941no—
234C22×C4(C2×D47)⋊C411yescharacteristic
235C22×C4(C2×D47)⋊C411yescharacteristic
236C22×C4(C2×D47)⋊C411yescharacteristic
237C22×C4C2×C4×D4711yescharacteristic
238C22×C4C2×D18811yescharacteristic
239C22×C4C2×C22⋊D4711yescharacteristic
240C2×C8—24no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—944no—
244C2×C8—1882no—
245C2×C8—1882no—
246C4⋊C4—1881no—
247C4⋊C4—1881no—
248C4⋊C4—1881no—
249C4⋊C4—1881no—
250C22⋊C4—1882no—
251C22⋊C4—1882no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42—1882no—
256C42—1882no—
257C42#1062535214yes—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23#1309395012yes—
265C23C2×(C2×D47)⋊C411yescharacteristic
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—1881no—
284C2×C4—1881no—
285C2×C4—1881no—
286C2×C4—1881no—
287C2×C4—1881no—
288C2×C4—1881no—
289C2×C4—1881no—
290C2×C4—1881no—
291C2×C4—1881no—
292C2×C4—1881no—
293C2×C4—1881no—
294C2×C4—1881no—
295C2×C4#1309395212yes—
296C2×C4#1309398012yes—
297C2×C4#1309400512yes—
298C2×C4#1309400812yes—
299C8—44no—
300C8—1884no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22#1568898411yescharacteristic
309C22#1568910511yescharacteristic
310C22#1568910511yescharacteristic
311C22C2×#1309395011yescharacteristic
312C22C2×#1309395211yescharacteristic
313C22#1568913911yescharacteristic
314C22#1568914111yescharacteristic
315C4—22no—
316C4—22no—
317C4—22no—
318C4—22no—
319C4—41no—
320C4—41no—
321C4—3761no—
322C4—3761no—
323C4—3761no—
324C4—3761no—
325C2—42no—
326C2#2024197411yescharacteristic
327C2#2024199311yescharacteristic
328C2#2024200911yescharacteristic
329C2#2024201711yescharacteristic
330C2#2024346911yescharacteristic
331C2#2024348511yescharacteristic
332C2#2024348811yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1749C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1749D47
hypocenterC47#8720
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
3. #20241270C2semidirect
4. #20241270C2semidirect
189. C47#8720semidirect

Maximal subgroups

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

subgroupindex
#202457862
#202431502
#202412702
C47×#17492
#872047