Finite Group Databasebeta

← #26224334

Subgroups — #26224334

Order 12032.

834 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689477C211yesmaximal, characteristic
2#20245943C211yesmaximal, characteristic, complemented
3#124×Dic47C211yesmaximal, characteristic
4#20244353C211yesmaximal, characteristic, complemented
5#20244082C212yesmaximal
6#20244075C212yesmaximal, complemented
7#20243518C211yesmaximal, characteristic, complemented
8#20243220C212yesmaximal, complemented
9#20243215C212yesmaximal
10C4×#13093934C211yesmaximal, characteristic, complemented
11C47×#1958C211yesmaximal, characteristic
12C2×#13094016C2211yescharacteristic
13C2×Q8×Dic47C2211yescharacteristic
14C2×D4×Dic47C2211yescharacteristic
15#15689477C418yescomplemented
16C2×#13093948C2212yes—
17C2×#13093945C2212yes—
18C2×#13093934C2212yes—
19C2×#13093934C2212yes—
20C2×C4×C47⋊C8C2211yescharacteristic
21#15689030C2211yescharacteristic
22#15689026C2211yescharacteristic
23#15689019C2212yes—
24#15689016—22no—
25#15689005—22no—
26#15688989C2211yescharacteristic
27C4⋊C4×Dic47C2212yes—
28#15688974—22no—
29C22⋊C4×Dic47—22no—
30C42×Dic47C2211yescharacteristic
31#15688886C2212yes—
32#15688886C2212yes—
33#15688885C2212yes—
34#15688885C2212yes—
35C2×#124×C47C2211yescharacteristic
36C2×#103×C47C2212yes—
37C2×#102×C47C2212yes—
38C2×C4×C8×C47C2211yescharacteristic
39C47×#382C2211yescharacteristic
40C23×Dic47—21no—
41C22×C47⋊C8C2312yes—
42#13094016—22no—
43#13094016C2×C414yescomplemented
44Q8×Dic47—24no—
45D4×Dic47—24no—
46C2×C4⋊Dic47—21no—
47C2×C4⋊Dic47C2311yescharacteristic
48C2×C4⋊Dic47C2311yescharacteristic
49C2×C4⋊Dic47C2312yes—
50C2×Dic47⋊C4—22no—
51C2×Dic47⋊C4—22no—
52C2×C22⋊Dic47—22no—
53C2×C22⋊Dic47—21no—
54C2×C4×Dic47—21no—
55C2×C4×Dic47—22no—
56C2×C4×Dic47D412yes—
57C2×C4×Dic47D412yes—
58C2×C4×Dic47C2311yescharacteristic
59C2×C4×Dic47C2311yescharacteristic
60#13093948C2×C418yescomplemented
61#13093945C2×C418yes—
62#13093934—24no—
63#13093934—24no—
64C4×C47⋊C8C2×C414yescomplemented
65C47⋊#95—22no—
66C47⋊#95—22no—
67C47⋊#95—22no—
68C47⋊#95—22no—
69C22×Q8×C47C2311yescharacteristic
70C22×D4×C47C2311yescharacteristic
71C22×C8×C47C2312yes—
72#124×C47—22no—
73#124×C47C2×C414yes—
74C2×C4⋊C4×C47C2312yes—
75C2×C22⋊C4×C47—22no—
76C2×C42×C47C2311yescharacteristic
77#103×C47—24no—
78#102×C47—24no—
79C4×C8×C47—22no—
80#95×C47—22no—
81C24×C47—21no—
82C2×Q8×C47—21no—
83C2×Q8×C47—22no—
84C2×Q8×C47—22no—
85C2×Q8×C47C22×C412yes—
86C2×D4×C47—22no—
87C2×D4×C47—22no—
88C2×D4×C47—21no—
89C2×D4×C47C22×C412yes—
90C22×C4×C47—21no—
91C22×C4×C47—22no—
92C22×C4×C47C2×D412yes—
93C22×C4×C47C2411yescharacteristic
94C22×Dic47—22no—
95C22×Dic47—21no—
96C22×Dic47—21no—
97C22×Dic47—44no—
98C22×Dic47—42no—
99C22×Dic47C2×D412yes—
100C22×Dic47C2×D411yescharacteristic
101C22×Dic47C2×D411yescharacteristic
102C2×C8×C47—22no—
103C2×C8×C47—22no—
104C2×C8×C47#4114yes—
105C4⋊C4×C47—22no—
106C4⋊C4×C47—22no—
107C22⋊C4×C47—28no—
108C42×C47C22×C412yes—
109C42×C47#4112yes—
110C22⋊Dic47—44no—
111C22⋊Dic47—42no—
112C4⋊Dic47—22no—
113C4⋊Dic47—22no—
114C4⋊Dic47—22no—
115C4⋊Dic47—42no—
116C4⋊Dic47C22×C414yes—
117Dic47⋊C4—28no—
118Dic47⋊C4—44no—
119C4×Dic47—22no—
120C4×Dic47—42no—
121C4×Dic47—42no—
122C4×Dic47—44no—
123C4×Dic47—42no—
124C4×Dic47#4114yes—
125C2×C47⋊C8—22no—
126C2×C47⋊C8—22no—
127C2×C47⋊C8C22×C414yes—
128C23×C47—22no—
129C23×C47—41no—
130C23×C47—41no—
131C23×C47—42no—
132C23×C47—42no—
133C23×C47C22×D411yescharacteristic
134Q8×C47—24no—
135Q8×C47—42no—
136D4×C47—24no—
137D4×C47—42no—
138C2×C4×C47—22no—
139C2×C4×C47—22no—
140C2×C4×C47—22no—
141C2×C4×C47—41no—
142C2×C4×C47—41no—
143C2×C4×C47—42no—
144C2×C4×C47—42no—
145C2×C4×C47—42no—
146C2×C4×C47C4×D414yes—
147C2×C4×C47C23×C411yescharacteristic
148C2×C4×C47C2×#4112yes—
149C2×C4×C47C2×#4112yes—
150C2×C4×C47C2×#4111yescharacteristic
151C2×Dic47—22no—
152C2×Dic47—22no—
153C2×Dic47—24no—
154C2×Dic47—22no—
155C2×Dic47—22no—
156C2×Dic47—22no—
157C2×Dic47—42no—
158C2×Dic47—42no—
159C2×Dic47—41no—
160C2×Dic47—41no—
161C2×Dic47—42no—
162C2×Dic47—42no—
163C2×Dic47—84no—
164C2×Dic47#12412yes—
165C2×Dic47#12412yes—
166C8×C47—24no—
167C47⋊C8—24no—
168#21648—471nomaximal, Sylow
169C22×C47—44no—
170C22×C47—42no—
171C22×C47—81no—
172C22×C47—81no—
173C22×C47—82no—
174C22×C47C2×C4×D411yescharacteristic
175C22×C47C2×#12411yescharacteristic
176C22×C47#53211yescharacteristic
177C22×C47#57512yes—
178C22×C47#57512yes—
179C4×C47—24no—
180C4×C47—42no—
181C4×C47—82no—
182C4×C47C4×#4112yes—
183C4×C47#52912yes—
184Dic47—22no—
185Dic47—22no—
186Dic47—42no—
187Dic47—44no—
188Dic47—81no—
189Dic47—81no—
190C2×#488—471no—
191C4×#125—471no—
192C4×#124—471no—
193#1991—471no—
194#1958D4711yescharacteristic
195#1936—472no—
196#1935—472no—
197#1835—471no—
198#1763—472no—
199#1762—472no—
200C4×#97—471no—
201C2×C47—82no—
202C2×C47C4×#12411yescharacteristic
203C2×C47#294312yes—
204C2×C47#314011yescharacteristic
205C2×C47#314211yescharacteristic
206C2×C47#318312yes—
207C2×#125—471no—
208C2×#124C2×D4711yescharacteristic
209C2×C4×Q8—471no—
210C2×C4×D4—471no—
211#488—478no—
212C2×#103—472no—
213C2×#103C2×D4712yes—
214C2×#102—472no—
215C2×#102C2×D4712yes—
216C2×#97—472no—
217C2×#97—472no—
218C2×C4×C8—471no—
219C2×C4×C8C2×D4711yescharacteristic
220#391—471no—
221#390—471no—
222#389—472no—
223#388—942no—
224#387—942no—
225#382—471no—
226#382C2×D4711yescharacteristic
227C4×C4⋊C4—472no—
228C4×C4⋊C4—942no—
229C4×C22⋊C4—942no—
230C43—471no—
231#340—472no—
232#340—472no—
233#340—472no—
234#340—472no—
235C47#2164811yescharacteristic, complemented, Sylow
236C22×Q8C22×D4711yescharacteristic
237C22×D4C22×D4711yescharacteristic
238C23×C4—941no—
239C22×C8—472no—
240C22×C8C22×D4712yes—
241#125—474no—
242#125—942no—
243#124—22no—
244#124C4×D4714yes—
245C4×Q8—944no—
246C4×D4—944no—
247C2×C4⋊C4—472no—
248C2×C4⋊C4—471no—
249C2×C4⋊C4—471no—
250C2×C4⋊C4—942no—
251C2×C4⋊C4—941no—
252C2×C4⋊C4—942no—
253C2×C4⋊C4C22×D4712yes—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—941no—
256C2×C22⋊C4—942no—
257C2×C42—471no—
258C2×C42—471no—
259C2×C42—472no—
260C2×C42—472no—
261C2×C42—941no—
262C2×C42—942no—
263C2×C42C22×D4711yescharacteristic
264#103—24no—
265#103—478no—
266#102—24no—
267#102—478no—
268#97—944no—
269#97—944no—
270C4×C8—22no—
271C4×C8—474no—
272#95—22no—
273#95—942no—
274#95—942no—
275#95—942no—
276#95—942no—
277C24—21no—
278C2×Q8—21no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8C2×C4×D4712yes—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—21no—
285C2×D4C2×C4×D4712yes—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—471no—
289C22×C4—471no—
290C22×C4—472no—
291C22×C4—942no—
292C22×C4—941no—
293C22×C4—941no—
294C22×C4—1882no—
295C22×C4—1884no—
296C22×C4D4×D4712yes—
297C22×C4C23×D4711yescharacteristic
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—474no—
301C2×C8—942no—
302C2×C8—942no—
303C2×C8#1062535214yes—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—474no—
307C4⋊C4—948no—
308C4⋊C4—942no—
309C4⋊C4—942no—
310C4⋊C4—942no—
311C4⋊C4—1882no—
312C4⋊C4—1884no—
313C22⋊C4—28no—
314C22⋊C4—1882no—
315C22⋊C4—1884no—
316C42—474no—
317C42—942no—
318C42—1882no—
319C42—1882no—
320C42—1882no—
321C42—1884no—
322C42C2×C4×D4712yes—
323C42#1062535212yes—
324C23—22no—
325C23—41no—
326C23—42no—
327C23—42no—
328C23—41no—
329C23C2×D4×D4711yescharacteristic
330Q8—24no—
331Q8—42no—
332D4—24no—
333D4—42no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—472no—
343C2×C4—472no—
344C2×C4—942no—
345C2×C4—942no—
346C2×C4—944no—
347C2×C4—942no—
348C2×C4—942no—
349C2×C4—942no—
350C2×C4—3764no—
351C2×C4—1881no—
352C2×C4—1881no—
353C2×C4—1882no—
354C2×C4—1882no—
355C2×C4—1882no—
356C2×C4—1882no—
357C2×C4(C22⋊D47)⋊C414yes—
358C2×C4C22×C4×D4711yescharacteristic
359C2×C4#41×D4712yes—
360C2×C4C2×#1062535212yes—
361C2×C4C2×#1062535211yescharacteristic
362C8—24no—
363C8—944no—
364C22—44no—
365C22—42no—
366C22—81no—
367C22—81no—
368C22—82no—
369C22C2×(C22⋊D47)⋊C411yescharacteristic
370C22#124×D4711yescharacteristic
371C22#1568967111yescharacteristic
372C22#1568992012yes—
373C22#1568992012yes—
374C4—24no—
375C4—42no—
376C4—82no—
377C4—942no—
378C4—942no—
379C4—3761no—
380C4—3761no—
381C4—1884no—
382C4—1882no—
383C4C4×#1062535212yes—
384C4#1568965412yes—
385C2—82no—
386C2#2024593511yescharacteristic
387C2#2025218412yes—
388C2#2025377111yescharacteristic
389C2#2025378511yescharacteristic
390C2#2025411412yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1958C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenter#1958D47
hypocenterC47#21648
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1958

Complements

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

NH ≅ G/Ntype
2. #20245943C2semidirect
4. #20244353C2semidirect
6. #20244075C2semidirect
7. #20243518C2semidirect
8. #20243220C2semidirect
10. C4×#13093934C2semidirect
15. #15689477C4semidirect
43. #13094016C2×C4semidirect
60. #13093948C2×C4semidirect
64. C4×C47⋊C8C2×C4semidirect
235. C47#21648semidirect

Maximal subgroups

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

subgroupindex
C2×#156894772
#202459432
#124×Dic472
#202443532
#202440822
#202440752
#202435182
#202432202
#202432152
C4×#130939342
C47×#19582
#2164847