Finite Group Databasebeta

← #26126887

Subgroups — #26126887

Order 12032.

590 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C4×#13093939C211yesmaximal, characteristic
2#20241729C211yesmaximal, characteristic
3#20241409C214yesmaximal, complemented
4C47×#2409C211yesmaximal, characteristic
5#15689180—21no—
6C2×#13093939—21no—
7C4×C22⋊Dic47—21no—
8#15688890C2212yes—
9#15688847—21no—
10#15688847C2212yes—
11C23×Q8×C47C411yescharacteristic, complemented
12C2×C22⋊Q8×C47—21no—
13C2×C4×D4×C47C2211yescharacteristic
14C47×#395C412yescomplemented
15C47×#393C2212yes—
16C47×#391—21no—
17C47×#386—21no—
18C47×#385—21no—
19C47×#380C411yescharacteristic, complemented
20C2×C22⋊Dic47—22no—
21C2×C4×Dic47—41no—
22C4⋊(C47⋊C8)—41no—
23#13093939—24no—
24C22⋊(C47⋊C8)—22no—
25#13093935—41no—
26C47⋊#95—41no—
27C22×Q8×C47—21no—
28C22×Q8×C47—41no—
29C22×Q8×C47—41no—
30C22×Q8×C47—44no—
31C22×Q8×C47C2×C412yes—
32C22×D4×C47D411yescharacteristic
33C23×C4×C47—21no—
34C23×C4×C47C2×C411yescharacteristic
35C23×C4×C47D411yescharacteristic
36C22⋊Q8×C47—44no—
37C4×D4×C47—41no—
38C4×D4×C47—41no—
39C2×C4⋊C4×C47—21no—
40C2×C4⋊C4×C47—22no—
41C2×C4⋊C4×C47C2×C411yescharacteristic
42C2×C22⋊C4×C47—21no—
43C2×C22⋊C4×C47—22no—
44C2×C22⋊C4×C47D411yescharacteristic
45C2×C22⋊C4×C47C2311yescharacteristic
46C2×C42×C47D411yescharacteristic
47#95×C47—22no—
48#95×C47—22no—
49#95×C47C2×C412yes—
50C24×C47C22⋊C411yescharacteristic
51C24×C47C2×D411yescharacteristic
52C2×Q8×C47—24no—
53C2×Q8×C47—42no—
54C2×Q8×C47—44no—
55C2×Q8×C47—42no—
56C2×Q8×C47—48no—
57C2×Q8×C47—44no—
58C2×Q8×C47—44no—
59C2×Q8×C47—41no—
60C2×Q8×C47—41no—
61C2×D4×C47—22no—
62C2×D4×C47—22no—
63C2×D4×C47—41no—
64C22×C4×C47—22no—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47—22no—
70C22×C4×C47—44no—
71C22×C4×C47—42no—
72C22×C4×C47—42no—
73C22×C4×C47—42no—
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C22⋊C411yescharacteristic
76C22×C4×C47C22⋊C411yescharacteristic
77C22×C4×C47C22×C411yescharacteristic
78C22×C4×C47C2×D411yescharacteristic
79C22×Dic47—42no—
80C4⋊C4×C47—41no—
81C4⋊C4×C47—42no—
82C4⋊C4×C47—44no—
83C22⋊C4×C47—41no—
84C22⋊C4×C47—41no—
85C22⋊C4×C47—44no—
86C22⋊C4×C47—42no—
87C42×C47—21no—
88C42×C47—21no—
89C22⋊Dic47—24no—
90C22⋊Dic47—42no—
91C4×Dic47—82no—
92C2×C47⋊C8—42no—
93C23×C47—21no—
94C23×C47—21no—
95C23×C47—21no—
96C23×C47—21no—
97C23×C47—41no—
98C23×C47—41no—
99C23×C47—41no—
100C23×C47—41no—
101C23×C47#10412yes—
102C23×C47C2×C22⋊C411yescharacteristic
103C23×C47#13712yes—
104Q8×C47—44no—
105Q8×C47—44no—
106Q8×C47—44no—
107Q8×C47—44no—
108D4×C47—44no—
109C2×C4×C47—22no—
110C2×C4×C47—24no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—42no—
118C2×C4×C47—42no—
119C2×C4×C47—48no—
120C2×C4×C47—42no—
121C2×C4×C47—42no—
122C2×C4×C47—42no—
123C2×C4×C47—41no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—42no—
127C2×C4×C47—42no—
128C2×C4×C47—42no—
129C2×C4×C47—81no—
130C2×C4×C47#9911yescharacteristic
131C2×C4×C47#9911yescharacteristic
132C2×Dic47—44no—
133C2×Dic47—82no—
134C2×Dic47—82no—
135C47⋊C8—82no—
136#11785—471nomaximal, Sylow
137C22×C47—21no—
138C22×C47—21no—
139C22×C47—22no—
140C22×C47—22no—
141C22×C47—42no—
142C22×C47—42no—
143C22×C47—42no—
144C22×C47—41no—
145C22×C47—41no—
146C22×C47—41no—
147C22×C47—41no—
148C22×C47—81no—
149C22×C47C2×#9911yescharacteristic
150C22×C47#41911yescharacteristic
151C22×C47C2×#10411yescharacteristic
152C4×C47—21no—
153C4×C47—44no—
154C4×C47—41no—
155C4×C47—41no—
156C4×C47—41no—
157C4×C47—41no—
158C4×C47—82no—
159C4×C47AΣL(1,𝔽16)12yes—
160Dic47—84no—
161#2409D4711yescharacteristic
162C4×#99—471no—
163#1466—471no—
164#1351—474no—
165C2×C47—21no—
166C2×C47—21no—
167C2×C47—42no—
168C2×C47—42no—
169C2×C47#150811yescharacteristic
170C2×C47C2×AΣL(1,𝔽16)11yescharacteristic
171C2×C47#212911yescharacteristic
172C23×Q8Dic4711yescharacteristic, complemented
173C2×C22⋊Q8—21no—
174C2×C4×D4C2×D4711yescharacteristic
175#432—941no—
176C2×#99—941no—
177#395Dic4712yescomplemented
178#393C2×D4712yes—
179#391—21no—
180#386—21no—
181#385—21no—
182#380Dic4711yescharacteristic, complemented
183C4×C22⋊C4—941no—
184#342—472no—
185#323—472no—
186#323—941no—
187C47#1178511yescharacteristic, complemented, Sylow
188C22×Q8—21no—
189C22×Q8—41no—
190C22×Q8—41no—
191C22×Q8—44no—
192C22×Q8C2×Dic4712yes—
193C22×D4C22⋊D4711yescharacteristic
194C23×C4—21no—
195C23×C4C2×Dic4711yescharacteristic
196C23×C4C22⋊D4711yescharacteristic
197C22⋊Q8—44no—
198C4×D4—41no—
199C4×D4—41no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4C2×Dic4711yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—942no—
206C2×C22⋊C4C22⋊D4711yescharacteristic
207C2×C22⋊C4C22×D4711yescharacteristic
208C2×C42—1881no—
209C2×C42C22⋊D4711yescharacteristic
210C4⋊C8—1881no—
211#99—944no—
212C22⋊C8—942no—
213#97—1881no—
214#95—22no—
215#95—22no—
216#95—1881no—
217#95C2×Dic4712yes—
218C24C22⋊Dic4711yescharacteristic
219C24C2×C22⋊D4711yescharacteristic
220C2×Q8—24no—
221C2×Q8—44no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—44no—
225C2×Q8—48no—
226C2×Q8—44no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—41no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—44no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—1882no—
243C22×C4C22⋊Dic4711yescharacteristic
244C22×C4C22⋊Dic4711yescharacteristic
245C22×C4C22⋊Dic4711yescharacteristic
246C22×C4C22×Dic4711yescharacteristic
247C22×C4C2×C22⋊D4711yescharacteristic
248C2×C8—1882no—
249C4⋊C4—41no—
250C4⋊C4—42no—
251C4⋊C4—44no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—44no—
255C22⋊C4—42no—
256C22⋊C4—944no—
257C22⋊C4—1882no—
258C42—21no—
259C42—21no—
260C42—3762no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23#1309395112yes—
270C23C2×C22⋊Dic4711yescharacteristic
271C23#1309405512yes—
272Q8—44no—
273Q8—44no—
274Q8—44no—
275Q8—44no—
276D4—44no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—48no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—81no—
298C2×C4—3762no—
299C2×C4—3762no—
300C2×C4—1884no—
301C2×C4#1309393911yescharacteristic
302C2×C4#1309393911yescharacteristic
303C8—3762no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—22no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22C2×#1309393911yescharacteristic
317C22#1568913211yescharacteristic
318C22C2×#1309395111yescharacteristic
319C4—21no—
320C4—44no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—82no—
326C4—3764no—
327C4#1568891012yes—
328C2—21no—
329C2—21no—
330C2—42no—
331C2—42no—
332C2#2024190811yescharacteristic
333C2C2×#1568891011yescharacteristic
334C2#2024506311yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#2409C2
socleC22×C47C2×#99
prime radicalC22×D4×C47D4
cosocleC2×C22⋊C4×C47C23
hypercenter#2409D47
hypocenterC47#11785
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
3. #20241409C2semidirect
11. C23×Q8×C47C4semidirect
14. C47×#395C4semidirect
19. C47×#380C4semidirect
172. C23×Q8Dic47semidirect
177. #395Dic47semidirect
182. #380Dic47semidirect
187. C47#11785semidirect

Maximal subgroups

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

subgroupindex
C4×#130939392
#202417292
#202414092
C47×#24092
#1178547