Finite Group Databasebeta

← #26184855

Subgroups — #26184855

Order 12032.

950 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#533×D47C211yesmaximal, characteristic
2#20252020C212yesmaximal
3#20242126C211yesmaximal, characteristic, complemented
4#20242125C212yesmaximal, complemented
5#20242103C214yesmaximal, complemented
6#20242051C214yesmaximal, complemented
7C47×#1561C211yesmaximal, characteristic, complemented
8#15689680C2211yescharacteristic
9#15689679C412yes—
10#15689678C2212yes—
11#15689677—22no—
12#15689676C412yes—
13#15689675—22no—
14C4⋊Q8×D47C2212yes—
15#125×D47C412yes—
16C22⋊Q8×D47—22no—
17C4×Q8×D47—22no—
18#117×D47C412yes—
19C2×C42×D47C2211yescharacteristic
20#15689179—22no—
21#15689179C2214yes—
22#15689175—22no—
23#15689175C2214yes—
24#15689147C2212yes—
25#15689086—22no—
26M16×Dic47—22no—
27#15688870C2214yescomplemented
28#15688869C2214yescomplemented
29#15688868C2214yescomplemented
30C47×#533C2211yescharacteristic
31C47×#423C2212yes—
32C47×#332C2214yescomplemented
33C2×Q8×D47—22no—
34C22×C4×D47C2×C411yescharacteristic
35C22×C4×D47D412yes—
36C2×#10625329—22no—
37#13094024C2312yes—
38#13094023—22no—
39#13094023C2×C414yes—
40#13094022—22no—
41#13094016C2×C412yes—
42#13094015C2×C414yes—
43#13094014—24no—
44#13094004—22no—
45#13094003—22no—
46#13094002—24no—
47#13094001—24no—
48#13094000—22no—
49Q8×Dic47—22no—
50C4×Dic94—22no—
51C4×Dic94—24no—
52#13093989—24no—
53#13093989—22no—
54#13093983C2×C412yes—
55#13093982C2×C414yes—
56#13093981—22no—
57#13093980—22no—
58C4⋊C4×D47—24no—
59C4⋊C4×D47—22no—
60C4⋊C4×D47C2×C414yes—
61C22⋊C4×D47—22no—
62C2×C4×Dic47C2×C411yescharacteristic
63C2×C4×Dic47D412yes—
64C42×D47—22no—
65C42×D47C2×C411yescharacteristic
66C42×D47C2×C411yescharacteristic
67C42×D47D412yes—
68C42×D47D412yes—
69C42×D47C2311yescharacteristic
70C42×D47C2311yescharacteristic
71C4⋊(C47⋊C8)—22no—
72C4⋊(C47⋊C8)C2314yes—
73(C2×D47)⋊C8—24no—
74(C2×D47)⋊C8—24no—
75#13093934—24no—
76C8×Dic47—24no—
77C2×M16×C47—22no—
78C4⋊Q8×C47C2312yes—
79#125×C47C2×C412yes—
80C22⋊Q8×C47—22no—
81C4×Q8×C47—22no—
82#117×C47C2×C412yes—
83C2×C42×C47C2311yescharacteristic
84C4⋊C8×C47—22no—
85C4⋊C8×C47C2314yes—
86C23×D47C22⋊C411yescharacteristic
87C2×Q8×C47—22no—
88C22×C4×C47C22×C411yescharacteristic
89C22×C4×C47C2×D412yes—
90C22×Dic47C22⋊C411yescharacteristic
91C22×Dic47C22⋊C412yes—
92Q8×D47—44no—
93Q8×D47—44no—
94C2×C4×D47—24no—
95C2×C4×D47—24no—
96C2×C4×D47—42no—
97C2×C4×D47—42no—
98C2×C4×D47C22⋊C411yescharacteristic
99C2×C4×D47C22⋊C412yes—
100C2×C4×D47C22⋊C412yes—
101C2×C4×D47C22⋊C411yescharacteristic
102C2×C4×D47C22×C411yescharacteristic
103C2×C4×D47C22×C411yescharacteristic
104C2×C4×D47C2×D412yes—
105C2×C4×D47C2×D412yes—
106C2×Dic94—22no—
107C2×Dic94—24no—
108M16×C47—44no—
109C2×C8×C47—24no—
110C4⋊C4×C47—24no—
111C4⋊C4×C47—22no—
112C4⋊C4×C47C22×C414yes—
113C22⋊C4×C47—22no—
114C42×C47—22no—
115C42×C47C22×C411yescharacteristic
116C42×C47C2×D412yes—
117C42×C47C2411yescharacteristic
118C22⋊Dic47—22no—
119(C2×D47)⋊C4—24no—
120C4⋊Dic47—22no—
121C4⋊Dic47—24no—
122C4⋊Dic47C22×C414yes—
123Dic47⋊C4—24no—
124Dic47⋊C4—24no—
125Dic47⋊C4—24no—
126Dic47⋊C4—24no—
127C4×Dic47—22no—
128C4×Dic47—24no—
129C4×Dic47C22⋊C412yes—
130C4×Dic47C22⋊C412yes—
131C4×Dic47C22⋊C412yes—
132C4×Dic47C22×C411yescharacteristic
133C4×Dic47C22×C411yescharacteristic
134C4×Dic47C2×D412yes—
135C4×Dic47C2×D412yes—
136#10625329—44no—
137C2×C47⋊C8—24no—
138C23×C47C2×C22⋊C411yescharacteristic
139C22×D47—22no—
140C22×D47—42no—
141C22×D47C2×C22⋊C411yescharacteristic
142C22×D47C2×C22⋊C411yescharacteristic
143Q8×C47—42no—
144Q8×C47—42no—
145C2×C4×C47—22no—
146C2×C4×C47—22no—
147C2×C4×C47—22no—
148C2×C4×C47—24no—
149C2×C4×C47C2×C22⋊C412yes—
150C2×C4×C47C2×C22⋊C411yescharacteristic
151C2×C4×C47C23×C411yescharacteristic
152C2×C4×C47C22×D412yes—
153C2×Dic47—22no—
154C2×Dic47—24no—
155C2×Dic47—42no—
156C2×Dic47C2×C22⋊C412yes—
157C2×Dic47C2×C22⋊C412yes—
158C2×Dic47C2×C22⋊C411yescharacteristic
159C2×Dic47C2×C22⋊C411yescharacteristic
160C4×D47—22no—
161C4×D47—22no—
162C4×D47—22no—
163C4×D47—22no—
164C4×D47—22no—
165C4×D47—22no—
166C4×D47—22no—
167C4×D47—42no—
168C4×D47—42no—
169C4×D47—48no—
170C4×D47#10114yes—
171Dic94—48no—
172Dic94—42no—
173Dic94—42no—
174C8×C47—44no—
175C47⋊C8—44no—
176#17739—471nomaximal, Sylow
177C22×C47—21no—
178C22×C47—22no—
179C22×C47C22×C22⋊C411yescharacteristic
180C2×D47—22no—
181C2×D47—21no—
182C2×D47—22no—
183C2×D47—44no—
184C2×D47C2×#10112yes—
185C4×C47—21no—
186C4×C47—22no—
187C4×C47—44no—
188C4×C47C2×#10112yes—
189C4×C47#57512yes—
190C4×C47#57512yes—
191Dic47—21no—
192Dic47—22no—
193Dic47—22no—
194Dic47—44no—
195Dic47C2×#10112yes—
196C2×#533—471no—
197#2920—472no—
198#1561—472no—
199#1561—471no—
200#1561D4711yescharacteristic, complemented
201#1556—474no—
202#1544—474no—
203C2×C47—41no—
204C2×C47C22×#10111yescharacteristic
205C2×C47#289612yes—
206D47—22no—
207D47—41no—
208#533—472no—
209#533—472no—
210#533—472no—
211#533—471no—
212#533—942no—
213#533—942no—
214#533C2×D4711yescharacteristic
215C2×C4⋊Q8—472no—
216C2×#125—472no—
217C2×C22⋊Q8—942no—
218C2×C4×Q8—942no—
219C2×#117—472no—
220C22×C42—471no—
221#432—474no—
222#432—942no—
223#431—474no—
224#431—942no—
225#423—472no—
226#423C2×D4712yes—
227#407—942no—
228C4×M16—942no—
229#332—474no—
230#332—474no—
231#332—474no—
232#332C2×D4714yescomplemented
233C47#1773911yescharacteristic, complemented, Sylow
234C22×Q8—942no—
235C23×C4—471no—
236C23×C4—472no—
237C2×M16—22no—
238C2×M16—942no—
239C4⋊Q8—474no—
240C4⋊Q8—472no—
241C4⋊Q8—942no—
242C4⋊Q8—942no—
243C4⋊Q8C22×D4712yes—
244#125—474no—
245#125—472no—
246#125—944no—
247#125C4×D4712yes—
248C22⋊Q8—22no—
249C22⋊Q8—944no—
250C22⋊Q8—944no—
251C22⋊Q8—942no—
252C22⋊Q8—942no—
253C22⋊Q8—942no—
254C4×Q8—22no—
255C4×Q8—944no—
256C4×Q8—944no—
257C4×Q8—942no—
258C4×Q8—942no—
259C4×Q8—942no—
260#117—472no—
261#117—474no—
262#117—942no—
263#117—942no—
264#117C4×D4712yes—
265C2×C4⋊C4—474no—
266C2×C4⋊C4—944no—
267C2×C4⋊C4—942no—
268C2×C22⋊C4—942no—
269C2×C42—471no—
270C2×C42—472no—
271C2×C42—471no—
272C2×C42—471no—
273C2×C42—472no—
274C2×C42—472no—
275C2×C42—471no—
276C2×C42—471no—
277C2×C42—942no—
278C2×C42C22×D4711yescharacteristic
279C4⋊C8—22no—
280C4⋊C8—474no—
281C4⋊C8—942no—
282C4⋊C8C22×D4714yes—
283C22⋊C8—944no—
284C22⋊C8—944no—
285#97—944no—
286C4×C8—944no—
287C24—471no—
288C2×Q8—22no—
289C2×Q8—944no—
290C2×Q8—942no—
291C2×Q8—1884no—
292C2×Q8—1884no—
293C22×C4—471no—
294C22×C4—471no—
295C22×C4—472no—
296C22×C4—471no—
297C22×C4—471no—
298C22×C4—471no—
299C22×C4—472no—
300C22×C4—472no—
301C22×C4—472no—
302C22×C4—472no—
303C22×C4—944no—
304C22×C4—944no—
305C22×C4—1882no—
306C22×C4—1882no—
307C22×C4C2×C4×D4711yescharacteristic
308C22×C4D4×D4712yes—
309M16—44no—
310M16—1884no—
311C2×C8—24no—
312C2×C8—944no—
313C4⋊C4—24no—
314C4⋊C4—22no—
315C4⋊C4—474no—
316C4⋊C4—944no—
317C4⋊C4—944no—
318C4⋊C4—944no—
319C4⋊C4—944no—
320C4⋊C4—944no—
321C4⋊C4—942no—
322C4⋊C4C2×C4×D4714yes—
323C22⋊C4—22no—
324C22⋊C4—944no—
325C22⋊C4—942no—
326C42—22no—
327C42—472no—
328C42—472no—
329C42—472no—
330C42—471no—
331C42—471no—
332C42—472no—
333C42—472no—
334C42—944no—
335C42—942no—
336C42C2×C4×D4711yescharacteristic
337C42D4×D4712yes—
338C42C23×D4711yescharacteristic
339C23—471no—
340C23—471no—
341C23—942no—
342C23—1882no—
343C23C22⋊C4×D4711yescharacteristic
344Q8—42no—
345Q8—42no—
346Q8—1888no—
347Q8—1882no—
348Q8—1882no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—471no—
354C2×C4—474no—
355C2×C4—472no—
356C2×C4—471no—
357C2×C4—472no—
358C2×C4—942no—
359C2×C4—942no—
360C2×C4—942no—
361C2×C4—942no—
362C2×C4—942no—
363C2×C4—942no—
364C2×C4—942no—
365C2×C4—942no—
366C2×C4—944no—
367C2×C4—1882no—
368C2×C4—1882no—
369C2×C4—1888no—
370C2×C4—1882no—
371C2×C4C22⋊C4×D4712yes—
372C2×C4C22⋊C4×D4711yescharacteristic
373C2×C4C22×C4×D4711yescharacteristic
374C2×C4C2×D4×D4712yes—
375C8—44no—
376C8—1884no—
377C22—22no—
378C22—21no—
379C22—472no—
380C22—942no—
381C22—942no—
382C22—941no—
383C22—1884no—
384C22C2×C22⋊C4×D4711yescharacteristic
385C4—22no—
386C4—21no—
387C4—44no—
388C4—472no—
389C4—942no—
390C4—942no—
391C4—941no—
392C4—1884no—
393C4#101×D4712yes—
394C4#1568991912yes—
395C4#1568992212yes—
396C2—41no—
397C2—942no—
398C2—1881no—
399C2C2×#101×D4711yescharacteristic
400C2#2025190412yes—
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#1561C2
socleC22×C47C22×C22⋊C4
prime radicalC23×D47C22⋊C4
cosocleC42×C47C24
hypercenter#1561D47
hypocenterC47#17739
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1561

Complements

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

NH ≅ G/Ntype
3. #20242126C2semidirect
4. #20242125C2semidirect
5. #20242103C2semidirect
6. #20242051C2semidirect
7. C47×#1561C2semidirect
27. #15688870C22semidirect
28. #15688869C22semidirect
29. #15688868C22semidirect
32. C47×#332C22semidirect
200. #1561D47semidirect
232. #332C2×D47semidirect
233. C47#17739semidirect

Maximal subgroups

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

subgroupindex
#533×D472
#202520202
#202421262
#202421252
#202421032
#202420512
C47×#15612
#1773947