Finite Group Databasebeta

← #26109673

Subgroups — #26109673

Order 12032.

588 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243456C211yesmaximal, characteristic
2#20241615C211yesmaximal, characteristic
3#20241266C214yesmaximal
4C47×#2285C211yesmaximal, characteristic
5C2×C22⋊(C47⋊C8)C2212yes—
6C2×C47⋊#95—21no—
7C2×C47⋊#95—21no—
8C2×C47⋊#95C2212yes—
9#15688878—21no—
10#15688878—21no—
11C2×#117×C47—21no—
12C22×C42×C47C2211yescharacteristic
13C47×#384C411yescharacteristic, complemented
14C47×#384C411yescharacteristic, complemented
15C47×#382—21no—
16C47×#380—21no—
17C47×#380C2212yes—
18C4×C4⋊C4×C47—21no—
19C4×C4⋊C4×C47C412yescomplemented
20C23×Dic47—22no—
21C22×C47⋊C8—22no—
22C22⋊(C47⋊C8)—24no—
23C47⋊#95—24no—
24C47⋊#95—24no—
25C47⋊#95—42no—
26C23×C4×C47D411yescharacteristic
27C23×C4×C47D411yescharacteristic
28C23×C4×C47C2311yescharacteristic
29#117×C47—44no—
30C2×C4⋊C4×C47—21no—
31C2×C4⋊C4×C47—21no—
32C2×C4⋊C4×C47—21no—
33C2×C4⋊C4×C47—21no—
34C2×C4⋊C4×C47C2×C412yes—
35C2×C4⋊C4×C47C2×C412yes—
36C2×C22⋊C4×C47—22no—
37C2×C42×C47—24no—
38C2×C42×C47—22no—
39C2×C42×C47C2×C411yescharacteristic
40C2×C42×C47C2×C411yescharacteristic
41C2×C42×C47D411yescharacteristic
42C2×C42×C47D411yescharacteristic
43#95×C47—22no—
44#95×C47—22no—
45C24×C47C2×D411yescharacteristic
46C22×C4×C47—22no—
47C22×C4×C47—22no—
48C22×C4×C47—22no—
49C22×C4×C47—22no—
50C22×C4×C47—22no—
51C22×C4×C47—22no—
52C22×C4×C47—41no—
53C22×C4×C47—41no—
54C22×C4×C47—41no—
55C22×C4×C47C22⋊C411yescharacteristic
56C22×C4×C47C22⋊C411yescharacteristic
57C22×C4×C47C22⋊C411yescharacteristic
58C22×C4×C47C22⋊C411yescharacteristic
59C22×C4×C47SD1612yes—
60C22×C4×C47Q1612yes—
61C22×C4×C47C22×C411yescharacteristic
62C22×C4×C47C2×D411yescharacteristic
63C22×C4×C47#4114yes—
64C22×Dic47—24no—
65C22×Dic47—42no—
66C22×Dic47—42no—
67C22×Dic47—42no—
68C22×Dic47—42no—
69C22×Dic47—42no—
70C22×Dic47—42no—
71C4⋊C4×C47—42no—
72C4⋊C4×C47—42no—
73C4⋊C4×C47—42no—
74C4⋊C4×C47—42no—
75C4⋊C4×C47—42no—
76C4⋊C4×C47—42no—
77C22⋊C4×C47—44no—
78C42×C47—22no—
79C42×C47—22no—
80C42×C47—22no—
81C42×C47—22no—
82C42×C47—44no—
83C2×C47⋊C8—42no—
84C2×C47⋊C8—42no—
85C2×C47⋊C8—42no—
86C23×C47—21no—
87C23×C47—21no—
88C23×C47—21no—
89C23×C47—21no—
90C23×C47C2×C22⋊C411yescharacteristic
91C23×C47#12312yes—
92C23×C47C2×SD1611yescharacteristic
93C23×C47C2×Q1611yescharacteristic
94C23×C47AGL(1,ℤ/8)12yes—
95C2×C4×C47—21no—
96C2×C4×C47—21no—
97C2×C4×C47—21no—
98C2×C4×C47—21no—
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—24no—
106C2×C4×C47—24no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—44no—
111C2×C4×C47—44no—
112C2×C4×C47—42no—
113C2×C4×C47—42no—
114C2×C4×C47—42no—
115C2×C4×C47—42no—
116C2×C4×C47—42no—
117C2×C4×C47—42no—
118C2×C4×C47#10312yes—
119C2×C4×C47#10312yes—
120C2×C4×C47#10412yes—
121C2×C4×C47#11712yes—
122C2×C4×C47#12312yes—
123C2×Dic47—44no—
124C2×Dic47—44no—
125C2×Dic47—44no—
126C2×Dic47—82no—
127C2×Dic47—82no—
128C2×Dic47—82no—
129C2×Dic47—82no—
130C47⋊C8—82no—
131#8717—471nomaximal, Sylow
132C22×C47—22no—
133C22×C47—22no—
134C22×C47—21no—
135C22×C47—21no—
136C22×C47—22no—
137C22×C47—22no—
138C22×C47—21no—
139C22×C47—21no—
140C22×C47#38111yescharacteristic
141C22×C47#41011yescharacteristic
142C22×C47#41011yescharacteristic
143C22×C47C2×#10311yescharacteristic
144C22×C47#41811yescharacteristic
145C22×C47C2×#10411yescharacteristic
146C22×C47#42111yescharacteristic
147C22×C47#48112yes—
148C22×C47#48412yes—
149C4×C47—22no—
150C4×C47—22no—
151C4×C47—22no—
152C4×C47—22no—
153C4×C47—22no—
154C4×C47—41no—
155C4×C47—82no—
156C4×C47—82no—
157Dic47—84no—
158#2285D4711yescharacteristic
159#1817—471no—
160#1434—471no—
161#1294—474no—
162C2×C47—22no—
163C2×C47—22no—
164C2×C47#150311yescharacteristic
165C2×C47#150511yescharacteristic
166C2×C47#151411yescharacteristic
167C2×C47#151811yescharacteristic
168C2×C47#182011yescharacteristic
169C2×C47#182411yescharacteristic
170C2×C47#182611yescharacteristic
171C2×#117—21no—
172C22×C42C2×D4711yescharacteristic
173C2×C22⋊C8—472no—
174#384Dic4711yescharacteristic, complemented
175#384Dic4711yescharacteristic, complemented
176#382—21no—
177#380—21no—
178#380C2×D4712yes—
179C4×C4⋊C4—21no—
180C4×C4⋊C4Dic4712yescomplemented
181C2×#95—472no—
182C2×#95—941no—
183C2×#95—941no—
184#336—941no—
185#336—941no—
186C47#871711yescharacteristic, complemented, Sylow
187C23×C4—942no—
188C23×C4C22⋊D4711yescharacteristic
189C23×C4C22⋊D4711yescharacteristic
190C23×C4C22×D4711yescharacteristic
191C22×C8—942no—
192#117—44no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C2×Dic4712yes—
198C2×C4⋊C4C2×Dic4712yes—
199C2×C22⋊C4—22no—
200C2×C42—24no—
201C2×C42—22no—
202C2×C42C2×Dic4711yescharacteristic
203C2×C42C2×Dic4711yescharacteristic
204C2×C42C22⋊D4711yescharacteristic
205C2×C42C22⋊D4711yescharacteristic
206C22⋊C8—944no—
207#95—22no—
208#95—22no—
209#95—944no—
210#95—944no—
211#95—1882no—
212C24C2×C22⋊D4711yescharacteristic
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—944no—
223C22×C4—1882no—
224C22×C4—1882no—
225C22×C4—1882no—
226C22×C4—1882no—
227C22×C4—1882no—
228C22×C4—1882no—
229C22×C4#1062533612yes—
230C22×C4#1062533712yes—
231C22×C4C22⋊Dic4711yescharacteristic
232C22×C4C22⋊Dic4711yescharacteristic
233C22×C4C22⋊Dic4711yescharacteristic
234C22×C4C22⋊Dic4711yescharacteristic
235C22×C4#1062535014yes—
236C22×C4C22×Dic4711yescharacteristic
237C22×C4C2×C22⋊D4711yescharacteristic
238C2×C8—1882no—
239C2×C8—1882no—
240C2×C8—1882no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C22⋊C4—44no—
248C42—22no—
249C42—22no—
250C42—22no—
251C42—22no—
252C42—44no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23C2×C22⋊Dic4711yescharacteristic
258C23#1309400712yes—
259C23C2×#1062533611yescharacteristic
260C23C2×#1062533711yescharacteristic
261C23#1309405012yes—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—3762no—
286C2×C4—3762no—
287C2×C4—3762no—
288C2×C4—3762no—
289C2×C4—1884no—
290C2×C4—1884no—
291C2×C4—1884no—
292C2×C4#1309394812yes—
293C2×C4#1309394812yes—
294C2×C4#1309395112yes—
295C2×C4#1309398312yes—
296C2×C4#1309400712yes—
297C8—3762no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—22no—
306C22#1568898611yescharacteristic
307C22#1568909711yescharacteristic
308C22#1568909711yescharacteristic
309C22C2×#1309394811yescharacteristic
310C22#1568912811yescharacteristic
311C22C2×#1309395111yescharacteristic
312C22#1568914011yescharacteristic
313C22#1568944112yes—
314C22#1568945812yes—
315C4—22no—
316C4—22no—
317C4—22no—
318C4—22no—
319C4—22no—
320C4—41no—
321C4—82no—
322C4—82no—
323C4—3764no—
324C2—22no—
325C2—22no—
326C2#2024188711yescharacteristic
327C2#2024189611yescharacteristic
328C2#2024193511yescharacteristic
329C2#2024195211yescharacteristic
330C2#2024346511yescharacteristic
331C2#2024347811yescharacteristic
332C2#2024348711yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2285C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#2285D47
hypocenterC47#8717
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2285

Complements

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

NH ≅ G/Ntype
13. C47×#384C4semidirect
14. C47×#384C4semidirect
19. C4×C4⋊C4×C47C4semidirect
174. #384Dic47semidirect
175. #384Dic47semidirect
180. C4×C4⋊C4Dic47semidirect
186. C47#8717semidirect

Maximal subgroups

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

subgroupindex
#202434562
#202416152
#202412662
C47×#22852
#871747