Finite Group Databasebeta

← #26104439

Subgroups — #26104439

Order 12032.

516 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20245871C211yesmaximal, characteristic
2#20243368C211yesmaximal, characteristic
3#20241212C212yesmaximal, complemented
4#20241210C212yesmaximal, complemented
5C47×#1749C211yesmaximal, characteristic
6C2×#13093983C411yescharacteristic
7C2×#13093983C411yescharacteristic
8C2×C4⋊(C47⋊C8)—21no—
9C2×C22⋊(C47⋊C8)—21no—
10#15689021C411yescharacteristic
11#15689021C411yescharacteristic
12#15689021C2211yescharacteristic
13#15689021C2211yescharacteristic
14#15688996—21no—
15#15688996—21no—
16C4×Dic47⋊C4—21no—
17C4×Dic47⋊C4—21no—
18#15688878C2212yes—
19C22×C42×C47C2211yescharacteristic
20C2×C22⋊C8×C47—21no—
21C2×#97×C47—21no—
22C47×#336C2212yes—
23C22×C47⋊C8—22no—
24#13093983—22no—
25#13093983—22no—
26#13093983—42no—
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×C4⋊Dic47C2×C411yescharacteristic
30C2×C4⋊Dic47C2×C411yescharacteristic
31C2×Dic47⋊C4—21no—
32C2×Dic47⋊C4—21no—
33C2×Dic47⋊C4—21no—
34C2×Dic47⋊C4—21no—
35C2×C22⋊Dic47—21no—
36C2×C22⋊Dic47—21no—
37C2×C4×Dic47—21no—
38C2×C4×Dic47—21no—
39C4⋊(C47⋊C8)—42no—
40C22⋊(C47⋊C8)—42no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95—21no—
45C23×C4×C47C2×C411yescharacteristic
46C23×C4×C47D411yescharacteristic
47C23×C4×C47D411yescharacteristic
48C22×C8×C47—22no—
49C2×C42×C47—42no—
50C2×C42×C47C2×C411yescharacteristic
51C2×C42×C47D411yescharacteristic
52C2×C42×C47D411yescharacteristic
53C2×C42×C47C2311yescharacteristic
54C22⋊C8×C47—24no—
55#97×C47—42no—
56C24×C47C22⋊C411yescharacteristic
57C22×C4×C47—22no—
58C22×C4×C47—22no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—22no—
62C22×C4×C47—22no—
63C22×C4×C47—21no—
64C22×C4×C47—21no—
65C22×C4×C47—41no—
66C22×C4×C47—41no—
67C22×C4×C47—41no—
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47C22×C411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47C2×D411yescharacteristic
74C22×Dic47—21no—
75C22×Dic47—21no—
76C22×Dic47—21no—
77C22×Dic47—21no—
78C2×C8×C47—24no—
79C2×C8×C47—42no—
80C2×C8×C47—42no—
81C42×C47—22no—
82C42×C47—22no—
83C42×C47—41no—
84C42×C47—41no—
85C22⋊Dic47—24no—
86C22⋊Dic47—42no—
87C4⋊Dic47—21no—
88C4⋊Dic47—21no—
89C4⋊Dic47—41no—
90C4⋊Dic47—41no—
91C4⋊Dic47M1612yes—
92C4⋊Dic47M1612yes—
93Dic47⋊C4—42no—
94Dic47⋊C4—42no—
95Dic47⋊C4—42no—
96Dic47⋊C4—41no—
97Dic47⋊C4—41no—
98C4×Dic47—24no—
99C4×Dic47—42no—
100C2×C47⋊C8—42no—
101C2×C47⋊C8—42no—
102C2×C47⋊C8—42no—
103C23×C47—21no—
104C23×C47—21no—
105C23×C47—21no—
106C23×C47—21no—
107C23×C47—21no—
108C23×C47—21no—
109C23×C47#10112yes—
110C23×C47C2×C22⋊C411yescharacteristic
111C2×C4×C47—24no—
112C2×C4×C47—24no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—44no—
127C2×C4×C47—42no—
128C2×C4×C47—42no—
129C2×C4×C47#9912yes—
130C2×C4×C47#10412yes—
131C2×C4×C47C2×M1611yescharacteristic
132C2×C4×C47C2×M1611yescharacteristic
133C2×C4×C47#13512yes—
134C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C47#13711yescharacteristic
136C2×Dic47—22no—
137C2×Dic47—22no—
138C2×Dic47—41no—
139C2×Dic47—41no—
140C2×Dic47—41no—
141C2×Dic47—41no—
142C2×Dic47—41no—
143C2×Dic47—41no—
144C2×Dic47—41no—
145C2×Dic47—41no—
146C2×Dic47—41no—
147C2×Dic47—41no—
148C8×C47—44no—
149C47⋊C8—82no—
150#7613—471nomaximal, Sylow
151C22×C47—22no—
152C22×C47—22no—
153C22×C47—22no—
154C22×C47—41no—
155C22×C47—41no—
156C22×C47—41no—
157C22×C47—41no—
158C22×C47#40711yescharacteristic
159C22×C47C2×#9911yescharacteristic
160C22×C47C2×#10111yescharacteristic
161C22×C47#41611yescharacteristic
162C22×C47#41711yescharacteristic
163C22×C47C2×#10411yescharacteristic
164C22×C47#42111yescharacteristic
165C4×C47—22no—
166C4×C47—21no—
167C4×C47—21no—
168C4×C47—22no—
169C4×C47—22no—
170C4×C47—21no—
171C4×C47—21no—
172C4×C47—41no—
173Dic47—42no—
174Dic47—42no—
175Dic47—81no—
176Dic47—81no—
177#2296—471no—
178#1797—471no—
179#1749D4711yescharacteristic
180#1271—472no—
181#1271—472no—
182C2×C47—42no—
183C2×C47#146311yescharacteristic
184C2×C47#148811yescharacteristic
185C2×C47#148911yescharacteristic
186C2×C47#150111yescharacteristic
187C2×C47#151311yescharacteristic
188C2×C47#153111yescharacteristic
189C2×C47#155211yescharacteristic
190C2×#117—471no—
191C2×#117—471no—
192C22×C42C2×D4711yescharacteristic
193C2×C4⋊C8—941no—
194C2×C22⋊C8—21no—
195C2×C22⋊C8—941no—
196C2×#97—21no—
197#389—471no—
198#389—471no—
199#389—471no—
200#389—471no—
201#385—941no—
202#385—941no—
203C4×C4⋊C4—941no—
204C4×C4⋊C4—941no—
205#336—472no—
206#336C2×D4712yes—
207C47#761311yescharacteristic, complemented, Sylow
208C23×C4C4×D4711yescharacteristic
209C23×C4D18811yescharacteristic
210C23×C4C22⋊D4711yescharacteristic
211C22×C8—22no—
212C22×C8—942no—
213#117—942no—
214#117—942no—
215#117—1882no—
216C2×C4⋊C4—471no—
217C2×C4⋊C4—471no—
218C2×C4⋊C4—471no—
219C2×C4⋊C4—471no—
220C2×C4⋊C4—941no—
221C2×C4⋊C4—941no—
222C2×C4⋊C4—941no—
223C2×C4⋊C4—941no—
224C2×C22⋊C4—941no—
225C2×C22⋊C4—941no—
226C2×C42—42no—
227C2×C42—941no—
228C2×C42—941no—
229C2×C42C4×D4711yescharacteristic
230C2×C42D18811yescharacteristic
231C2×C42C22⋊D4711yescharacteristic
232C2×C42C22×D4711yescharacteristic
233C4⋊C8—1882no—
234C22⋊C8—24no—
235C22⋊C8—1882no—
236#97—42no—
237#95—941no—
238#95—941no—
239#95—941no—
240#95—941no—
241C24(C2×D47)⋊C411yescharacteristic
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—941no—
254C22×C4—941no—
255C22×C4—941no—
256C22×C4—941no—
257C22×C4(C2×D47)⋊C411yescharacteristic
258C22×C4(C2×D47)⋊C411yescharacteristic
259C22×C4(C2×D47)⋊C411yescharacteristic
260C22×C4C2×C4×D4711yescharacteristic
261C22×C4C2×D18811yescharacteristic
262C22×C4C2×C22⋊D4711yescharacteristic
263C2×C8—24no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—1882no—
267C2×C8—1882no—
268C2×C8—1882no—
269C4⋊C4—472no—
270C4⋊C4—472no—
271C4⋊C4—941no—
272C4⋊C4—941no—
273C4⋊C4—1881no—
274C4⋊C4—1881no—
275C4⋊C4—1882no—
276C4⋊C4—1882no—
277C4⋊C4—1882no—
278C4⋊C4—1881no—
279C4⋊C4—1881no—
280C22⋊C4—944no—
281C22⋊C4—1882no—
282C42—22no—
283C42—22no—
284C42—41no—
285C42—41no—
286C42—944no—
287C42—1882no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23#1309394212yes—
295C23C2×(C2×D47)⋊C411yescharacteristic
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—942no—
315C2×C4—942no—
316C2×C4—1881no—
317C2×C4—1881no—
318C2×C4—1881no—
319C2×C4—1881no—
320C2×C4—1881no—
321C2×C4—1881no—
322C2×C4—1881no—
323C2×C4—1881no—
324C2×C4—1881no—
325C2×C4—1881no—
326C2×C4#1309393812yes—
327C2×C4#1309395212yes—
328C2×C4M16×D4711yescharacteristic
329C2×C4M16×D4711yescharacteristic
330C2×C4#1309404412yes—
331C2×C4#1309405211yescharacteristic
332C2×C4#1309405711yescharacteristic
333C8—44no—
334C8—3762no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22#1568908611yescharacteristic
343C22C2×#1309393811yescharacteristic
344C22C2×#1309394211yescharacteristic
345C22#1568911711yescharacteristic
346C22#1568912511yescharacteristic
347C22C2×#1309395211yescharacteristic
348C22#1568913911yescharacteristic
349C4—21no—
350C4—21no—
351C4—22no—
352C4—21no—
353C4—21no—
354C4—22no—
355C4—22no—
356C4—41no—
357C4—3761no—
358C4—3761no—
359C4—1882no—
360C4—1882no—
361C2—42no—
362C2#2024171711yescharacteristic
363C2#2024182011yescharacteristic
364C2#2024182811yescharacteristic
365C2#2024187711yescharacteristic
366C2#2024193211yescharacteristic
367C2#2024200211yescharacteristic
368C2#2024208911yescharacteristic
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#7613
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. #20241212C2semidirect
4. #20241210C2semidirect
207. C47#7613semidirect

Maximal subgroups

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

subgroupindex
#202458712
#202433682
#202412122
#202412102
C47×#17492
#761347