Finite Group Databasebeta

← #26104889

Subgroups — #26104889

Order 12032.

464 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251727C211yesmaximal, characteristic
2#20243368C211yesmaximal, characteristic
3#20241213C212yesmaximal, complemented
4#20241210C212yesmaximal, complemented
5C47×#1797C211yesmaximal, characteristic
6C2×#13094003C411yescharacteristic
7C2×C4⋊(C47⋊C8)—21no—
8C2×C22⋊(C47⋊C8)—21no—
9#15689029C411yescharacteristic
10#15689029C2211yescharacteristic
11#15689021C411yescharacteristic
12#15689021C2211yescharacteristic
13#15689007—21no—
14#15689007—21no—
15#15688996—21no—
16#15688986—21no—
17#15688983C411yescharacteristic
18#15688878C2212yes—
19C22×C42×C47C2211yescharacteristic
20C2×C4⋊C8×C47—21no—
21C2×C22⋊C8×C47—21no—
22C47×#336C2212yes—
23C22×Dic94C2×C411yescharacteristic
24C22×C47⋊C8—22no—
25#13094003—42no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×Dic47⋊C4—21no—
32C2×C22⋊Dic47—21no—
33C2×C22⋊Dic47—21no—
34C4⋊(C47⋊C8)—42no—
35C22⋊(C47⋊C8)—42no—
36C47⋊#95—21no—
37C47⋊#95—21no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C23×C4×C47C2×C411yescharacteristic
44C23×C4×C47D411yescharacteristic
45C23×C4×C47D411yescharacteristic
46C22×C8×C47—22no—
47C2×C42×C47—42no—
48C2×C42×C47C2×C411yescharacteristic
49C2×C42×C47D411yescharacteristic
50C2×C42×C47D411yescharacteristic
51C2×C42×C47C2311yescharacteristic
52C4⋊C8×C47—42no—
53C22⋊C8×C47—42no—
54C24×C47C22⋊C411yescharacteristic
55C22×C4×C47—22no—
56C22×C4×C47—22no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—41no—
60C22×C4×C47—41no—
61C22×C4×C47—41no—
62C22×C4×C47—41no—
63C22×C4×C47—41no—
64C22×C4×C47C22⋊C411yescharacteristic
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47SD1612yes—
68C22×C4×C47Q1612yes—
69C22×C4×C47C22×C411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C22×Dic47—21no—
73C22×Dic47—21no—
74C22×Dic47—21no—
75C22×Dic47—21no—
76C2×Dic94—22no—
77C2×Dic94—41no—
78C2×Dic94—41no—
79C2×C8×C47—42no—
80C2×C8×C47—42no—
81C2×C8×C47—42no—
82C42×C47—22no—
83C42×C47—22no—
84C42×C47—41no—
85C42×C47—41no—
86C22⋊Dic47—42no—
87C22⋊Dic47—42no—
88C4⋊Dic47—22no—
89C4⋊Dic47—41no—
90C4⋊Dic47—41no—
91Dic47⋊C4—42no—
92Dic47⋊C4—42no—
93Dic47⋊C4—41no—
94Dic47⋊C4—41no—
95C2×C47⋊C8—42no—
96C2×C47⋊C8—42no—
97C2×C47⋊C8—42no—
98C23×C47—21no—
99C23×C47—21no—
100C23×C47—21no—
101C23×C47—21no—
102C23×C47—21no—
103C23×C47—21no—
104C23×C47#10312yes—
105C23×C47C2×C22⋊C411yescharacteristic
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—21no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—42no—
120C2×C4×C47—42no—
121C2×C4×C47—44no—
122C2×C4×C47—42no—
123C2×C4×C47—42no—
124C2×C4×C47#9912yes—
125C2×C4×C47#10312yes—
126C2×C4×C47#13112yes—
127C2×C4×C47C2×SD1611yescharacteristic
128C2×C4×C47C2×Q1611yescharacteristic
129C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
130C2×C4×C47#13711yescharacteristic
131C2×Dic47—41no—
132C2×Dic47—41no—
133C2×Dic47—41no—
134C2×Dic47—41no—
135C2×Dic47—41no—
136C2×Dic47—41no—
137C2×Dic47—41no—
138C2×Dic47—41no—
139C2×Dic47—41no—
140C2×Dic47—41no—
141C2×Dic47—41no—
142C2×Dic47—41no—
143Dic94—82no—
144C8×C47—82no—
145C47⋊C8—82no—
146#7715—471nomaximal, Sylow
147C22×C47—22no—
148C22×C47—22no—
149C22×C47—22no—
150C22×C47—41no—
151C22×C47—41no—
152C22×C47—41no—
153C22×C47—41no—
154C22×C47#40811yescharacteristic
155C22×C47C2×#9911yescharacteristic
156C22×C47#41311yescharacteristic
157C22×C47C2×#10311yescharacteristic
158C22×C47#41711yescharacteristic
159C22×C47#42111yescharacteristic
160C22×C47#42111yescharacteristic
161C4×C47—22no—
162C4×C47—22no—
163C4×C47—21no—
164C4×C47—21no—
165C4×C47—21no—
166C4×C47—22no—
167C4×C47—21no—
168C4×C47—41no—
169Dic47—81no—
170Dic47—81no—
171Dic47—81no—
172Dic47—81no—
173#2855—471no—
174#1797—471no—
175#1797D4711yescharacteristic
176#1272—472no—
177#1271—472no—
178C2×C47—42no—
179C2×C47#146811yescharacteristic
180C2×C47#148111yescharacteristic
181C2×C47#149511yescharacteristic
182C2×C47#151311yescharacteristic
183C2×C47#152011yescharacteristic
184C2×C47#153511yescharacteristic
185C2×C47#155011yescharacteristic
186C2×C22⋊Q8—471no—
187C22×C42C2×D4711yescharacteristic
188C2×C4⋊C8—21no—
189C2×C4⋊C8—941no—
190C2×C22⋊C8—21no—
191C2×C22⋊C8—941no—
192#391—471no—
193#391—471no—
194#389—471no—
195#389—471no—
196#387—941no—
197#387—941no—
198#385—941no—
199#381—941no—
200#380—471no—
201#336—472no—
202#336C2×D4712yes—
203C47#771511yescharacteristic, complemented, Sylow
204C22×Q8—471no—
205C23×C4C4×D4711yescharacteristic
206C23×C4D18811yescharacteristic
207C23×C4C22⋊D4711yescharacteristic
208C22×C8—22no—
209C22×C8—942no—
210C22⋊Q8—1882no—
211C2×C4⋊C4—471no—
212C2×C4⋊C4—471no—
213C2×C4⋊C4—471no—
214C2×C4⋊C4—941no—
215C2×C4⋊C4—941no—
216C2×C4⋊C4—941no—
217C2×C22⋊C4—941no—
218C2×C22⋊C4—941no—
219C2×C42—42no—
220C2×C42C4×D4711yescharacteristic
221C2×C42D18811yescharacteristic
222C2×C42C22⋊D4711yescharacteristic
223C2×C42C22×D4711yescharacteristic
224C4⋊C8—42no—
225C4⋊C8—1882no—
226C22⋊C8—42no—
227C22⋊C8—1882no—
228#95—941no—
229#95—941no—
230#95—941no—
231#95—941no—
232#95—941no—
233#95—941no—
234#95—941no—
235C24(C2×D47)⋊C411yescharacteristic
236C2×Q8—942no—
237C2×Q8—1881no—
238C2×Q8—1881no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—941no—
249C22×C4—941no—
250C22×C4—941no—
251C22×C4—941no—
252C22×C4(C2×D47)⋊C411yescharacteristic
253C22×C4(C2×D47)⋊C411yescharacteristic
254C22×C4(C2×D47)⋊C411yescharacteristic
255C22×C4#1062533512yes—
256C22×C4#1062533712yes—
257C22×C4C2×C4×D4711yescharacteristic
258C22×C4C2×D18811yescharacteristic
259C22×C4C2×C22⋊D4711yescharacteristic
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—1882no—
264C2×C8—1882no—
265C2×C8—1882no—
266C4⋊C4—942no—
267C4⋊C4—1881no—
268C4⋊C4—1881no—
269C4⋊C4—1882no—
270C4⋊C4—1882no—
271C4⋊C4—1881no—
272C4⋊C4—1881no—
273C22⋊C4—1882no—
274C22⋊C4—1882no—
275C42—22no—
276C42—22no—
277C42—41no—
278C42—41no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23#1309394712yes—
286C23C2×(C2×D47)⋊C411yescharacteristic
287Q8—3762no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—1881no—
307C2×C4—1881no—
308C2×C4—1881no—
309C2×C4—1881no—
310C2×C4—1881no—
311C2×C4—1881no—
312C2×C4—1881no—
313C2×C4—1881no—
314C2×C4—1881no—
315C2×C4—1881no—
316C2×C4—1881no—
317C2×C4—1881no—
318C2×C4#1309393812yes—
319C2×C4#1309394712yes—
320C2×C4#1309403012yes—
321C2×C4C2×#1062533511yescharacteristic
322C2×C4C2×#1062533711yescharacteristic
323C2×C4#1309405211yescharacteristic
324C2×C4#1309405711yescharacteristic
325C8—82no—
326C8—3762no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22#1568908911yescharacteristic
335C22C2×#1309393811yescharacteristic
336C22#1568910511yescharacteristic
337C22C2×#1309394711yescharacteristic
338C22#1568912511yescharacteristic
339C22#1568913911yescharacteristic
340C22#1568914111yescharacteristic
341C4—21no—
342C4—21no—
343C4—21no—
344C4—22no—
345C4—21no—
346C4—22no—
347C4—22no—
348C4—41no—
349C4—3761no—
350C4—3761no—
351C4—3761no—
352C4—3761no—
353C2—42no—
354C2#2024173411yescharacteristic
355C2#2024179011yescharacteristic
356C2#2024185411yescharacteristic
357C2#2024193211yescharacteristic
358C2#2024195911yescharacteristic
359C2#2024202011yescharacteristic
360C2#2024207911yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1797C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1797D47
hypocenterC47#7715
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
3. #20241213C2semidirect
4. #20241210C2semidirect
203. C47#7715semidirect

Maximal subgroups

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

subgroupindex
#202517272
#202433682
#202412132
#202412102
C47×#17972
#771547