Finite Group Databasebeta

← #26225220

Subgroups — #26225220

Order 12032.

1570 conjugacy classes of subgroups; only the 243 normal subgroups are shown.

normal subgroups only

This group has 1570 conjugacy classes of subgroups — too many to draw. Only its 243 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094008C211yesmaximal, characteristic
2#20245766C212yesmaximal, complemented
3#20243845C218yesmaximal, complemented
4#20243462C211yesmaximal, characteristic, complemented
5#20243302C212yesmaximal
6C47×#1819C211yesmaximal, characteristic, complemented
7C23×C22⋊D47C2211yescharacteristic
8C24×Dic47C2211yescharacteristic
9C2×#13094008—24no—
10C2×#13094008—24no—
11C2×#13094008C414yescomplemented
12C2×#13094008C2214yes—
13C2×#13093980C2214yescomplemented
14C22×C4⋊Dic47C412yescomplemented
15C22×(C2×D47)⋊C4C412yescomplemented
16C2×#13093938C2218yes—
17#15688984C2214yescomplemented
18#15688981—24no—
19#15688979C2212yes—
20C22⋊C4×Dic47—24no—
21#15688890C2214yescomplemented
22C22×C22⋊C4×C47C2211yescharacteristic
23C47×#379C2212yes—
24C47×#342C2214yescomplemented
25C24×D47C2×C411yescharacteristic
26C22×C22⋊D47—24no—
27C22×C22⋊D47—44no—
28C22×C22⋊D47C2×C412yes—
29C22×C22⋊D47C2312yes—
30C23×Dic47—44no—
31C23×Dic47C2×C411yescharacteristic
32C23×Dic47C2×C412yes—
33C23×Dic47C2×C412yes—
34C23×Dic47D414yes—
35C23×Dic47D414yes—
36C23×Dic47C2312yes—
37#13094008—24no—
38#13094008—28no—
39#13094008—24no—
40#13094008—28no—
41#13094008—24no—
42#13094008—24no—
43#13093980—48no—
44C2×C4⋊Dic47—22no—
45C2×C4⋊Dic47—24no—
46C2×C4⋊Dic47—22no—
47C2×C4⋊Dic47C2×C418yes—
48C2×Dic47⋊C4—24no—
49C2×C22⋊Dic47—24no—
50C2×C22⋊Dic47C2314yes—
51C2×(C2×D47)⋊C4—22no—
52C2×(C2×D47)⋊C4—24no—
53C2×(C2×D47)⋊C4—22no—
54C2×(C2×D47)⋊C4C2×C418yes—
55C2×C4×Dic47—24no—
56#13093938—48no—
57C47⋊#95—28no—
58C25×C47C2311yescharacteristic
59C23×C4×C47C2×C412yes—
60C2×C22⋊C4×C47—24no—
61C2×C22⋊C4×C47—24no—
62C2×C22⋊C4×C47C2×C412yes—
63C2×C22⋊C4×C47C2314yes—
64C2×C22⋊C4×C47C2312yes—
65C24×C47—24no—
66C24×C47—44no—
67C24×C47C22×C411yescharacteristic
68C24×C47C22×C411yescharacteristic
69C24×C47C2×D414yes—
70C24×C47C2411yescharacteristic
71C23×D47—22no—
72C23×D47—42no—
73C23×D47—44no—
74C23×D47C22×C412yes—
75C22×C4×C47—22no—
76C22×C4×C47—22no—
77C22×C4×C47—24no—
78C22×C4×C47—44no—
79C22×C4×C47C22×C414yes—
80C2×C22⋊D47—24no—
81C2×C22⋊D47—24no—
82C2×C22⋊D47—28no—
83C2×C22⋊D47—44no—
84C2×C22⋊D47—416no—
85C22×Dic47—28no—
86C22×Dic47—22no—
87C22×Dic47—24no—
88C22×Dic47—44no—
89C22×Dic47—44no—
90C22×Dic47—48no—
91C22×Dic47—48no—
92C22×Dic47—42no—
93C22×Dic47C22⋊C412yes—
94C22×Dic47C22⋊C414yes—
95C22×Dic47C22⋊C412yes—
96C22×Dic47C22⋊C412yes—
97C22×Dic47C22⋊C414yes—
98C22×Dic47C22⋊C412yes—
99C22×Dic47C22×C412yes—
100C22×Dic47C22×C414yes—
101C22×Dic47C2×D414yes—
102C22×Dic47C2×D412yes—
103C22×Dic47C2×D412yes—
104C22⋊C4×C47—24no—
105C22⋊C4×C47—24no—
106C22⋊C4×C47—28no—
107C22⋊C4×C47—48no—
108C22⋊Dic47—28no—
109C22⋊Dic47—48no—
110(C2×D47)⋊C4—28no—
111(C2×D47)⋊C4—28no—
112C4⋊Dic47—28no—
113C4⋊Dic47—28no—
114Dic47⋊C4—48no—
115C4×Dic47—48no—
116C23×C47—22no—
117C23×C47—24no—
118C23×C47—22no—
119C23×C47—28no—
120C23×C47—28no—
121C23×C47—42no—
122C23×C47—44no—
123C23×C47—42no—
124C23×C47—416no—
125C23×C47C2×C22⋊C412yes—
126C23×C47C2×C22⋊C412yes—
127C23×C47C23×C411yescharacteristic
128C23×C47C22×D412yes—
129C23×C47H(5,𝔽2)14yes—
130C22×D47—22no—
131C22×D47—24no—
132C22×D47—44no—
133C22×D47—44no—
134C22×D47—44no—
135C22×D47—44no—
136C22×D47—88no—
137C2×C4×C47—24no—
138C2×C4×C47—44no—
139C2×C4×C47—44no—
140C2×C4×C47—44no—
141C2×C4×C47—48no—
142C2×C4×C47—44no—
143C22⋊D47—48no—
144C22⋊D47—48no—
145C2×Dic47—28no—
146C2×Dic47—24no—
147C2×Dic47—24no—
148C2×Dic47—28no—
149C2×Dic47—22no—
150C2×Dic47—24no—
151C2×Dic47—42no—
152C2×Dic47—44no—
153C2×Dic47—42no—
154C2×Dic47—42no—
155C2×Dic47—44no—
156C2×Dic47—42no—
157C2×Dic47—44no—
158C2×Dic47—48no—
159C2×Dic47—44no—
160C2×Dic47C2×C22⋊C414yes—
161C2×Dic47C2×C22⋊C414yes—
162#21747—471nomaximal, Sylow
163C22×C47—28no—
164C22×C47—22no—
165C22×C47—24no—
166C22×C47—22no—
167C22×C47—22no—
168C22×C47—22no—
169C22×C47—24no—
170C22×C47—44no—
171C22×C47—416no—
172C22×C47—44no—
173C22×C47#41014yes—
174C22×C47C22×C22⋊C411yescharacteristic
175C22×C47#51812yes—
176C22×C47#53414yes—
177C2×D47—42no—
178C2×D47—44no—
179C2×D47—88no—
180C2×D47—84no—
181C4×C47—44no—
182C4×C47—84no—
183Dic47—24no—
184Dic47—24no—
185Dic47—42no—
186Dic47—84no—
187C22×#123—471no—
188#2280—472no—
189#1899—478no—
190#1819—471no—
191#1819D4711yescharacteristic, complemented
192#1781—472no—
193C2×C47—24no—
194C2×C47—22no—
195C2×C47—24no—
196C2×C47—42no—
197C2×C47#230811yescharacteristic
198C2×C47C2×#41012yes—
199D47—82no—
200C23×D4—471no—
201C24×C4—471no—
202C2×#123—474no—
203C2×#123—474no—
204C2×#123—944no—
205C2×#123—944no—
206C2×#117—474no—
207C22×C4⋊C4—472no—
208C22×C22⋊C4—472no—
209C22×C22⋊C4C2×D4711yescharacteristic
210C2×#99—478no—
211#381—474no—
212#380—944no—
213#379—472no—
214#379C2×D4712yes—
215C4×C22⋊C4—944no—
216#342—474no—
217#342C2×D4714yescomplemented
218C47#2174711yescharacteristic, complemented, Sylow
219C25—471no—
220C25C22×D4711yescharacteristic
221C22×D4—472no—
222C22×D4—472no—
223C22×D4—944no—
224C22×D4—1884no—
225C23×C4—472no—
226C23×C4—472no—
227C23×C4—472no—
228C23×C4—474no—
229C23×C4—474no—
230C23×C4—471no—
231C23×C4—1884no—
232C23×C4C4×D4712yes—
233#123—944no—
234#123—948no—
235#123—944no—
236#123—948no—
237#123—944no—
238#123—944no—
239#117—1888no—
240C2×C4⋊C4—478no—
241C2×C4⋊C4—944no—
242C2×C4⋊C4—942no—
243C2×C4⋊C4—942no—
244C2×C4⋊C4—944no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—478no—
248C2×C22⋊C4—474no—
249C2×C22⋊C4—942no—
250C2×C22⋊C4—942no—
251C2×C22⋊C4—944no—
252C2×C22⋊C4—944no—
253C2×C22⋊C4C4×D4712yes—
254C2×C22⋊C4C22×D4714yes—
255C2×C22⋊C4C22×D4712yes—
256C2×C42—944no—
257#99—1888no—
258#95—948no—
259C24—24no—
260C24—44no—
261C24—472no—
262C24—942no—
263C24—1882no—
264C24—1884no—
265C24C2×C4×D4711yescharacteristic
266C24C2×C4×D4711yescharacteristic
267C24D4×D4714yes—
268C24C23×D4711yescharacteristic
269C2×D4—944no—
270C2×D4—944no—
271C2×D4—948no—
272C2×D4—1884no—
273C2×D4—18816no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—44no—
278C22×C4—474no—
279C22×C4—474no—
280C22×C4—472no—
281C22×C4—472no—
282C22×C4—472no—
283C22×C4—474no—
284C22×C4—472no—
285C22×C4—472no—
286C22×C4—474no—
287C22×C4—472no—
288C22×C4—472no—
289C22×C4—948no—
290C22×C4—942no—
291C22×C4—944no—
292C22×C4—1884no—
293C22×C4—1884no—
294C22×C4—1888no—
295C22×C4—1888no—
296C22×C4—1882no—
297C22×C4C2×C4×D4714yes—
298C4⋊C4—948no—
299C4⋊C4—948no—
300C4⋊C4—1888no—
301C22⋊C4—24no—
302C22⋊C4—24no—
303C22⋊C4—28no—
304C22⋊C4—48no—
305C22⋊C4—948no—
306C22⋊C4—948no—
307C22⋊C4—948no—
308C22⋊C4—1888no—
309C42—1888no—
310C23—22no—
311C23—24no—
312C23—22no—
313C23—28no—
314C23—28no—
315C23—42no—
316C23—44no—
317C23—42no—
318C23—416no—
319C23—944no—
320C23—942no—
321C23—3768no—
322C23—1884no—
323C23—1884no—
324C23—1884no—
325C23—1884no—
326C23C22⋊C4×D4712yes—
327C23C22⋊C4×D4712yes—
328C23C22×C4×D4711yescharacteristic
329C23C2×D4×D4712yes—
330C23#1309407014yes—
331D4—1888no—
332D4—1888no—
333C2×C4—24no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—48no—
338C2×C4—44no—
339C2×C4—474no—
340C2×C4—474no—
341C2×C4—948no—
342C2×C4—944no—
343C2×C4—944no—
344C2×C4—948no—
345C2×C4—942no—
346C2×C4—944no—
347C2×C4—1882no—
348C2×C4—1884no—
349C2×C4—1882no—
350C2×C4—1882no—
351C2×C4—1884no—
352C2×C4—1882no—
353C2×C4—1888no—
354C2×C4—1884no—
355C2×C4—1884no—
356C22—28no—
357C22—24no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—24no—
362C22—22no—
363C22—44no—
364C22—416no—
365C22—44no—
366C22—3764no—
367C22—3768no—
368C22—1884no—
369C22—1882no—
370C22#1568909514yes—
371C22C2×C22⋊C4×D4711yescharacteristic
372C22#1568959712yes—
373C22#1568968414yes—
374C4—44no—
375C4—84no—
376C4—944no—
377C4—944no—
378C4—3764no—
379C4—1882no—
380C2—24no—
381C2—22no—
382C2—24no—
383C2—42no—
384C2—3762no—
385C2#2024596511yescharacteristic
386C2C2×#1568909512yes—
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC47×#1819C2
socleC22×C47C22×C22⋊C4
prime radicalC23×C22⋊D47C22
cosocleC24×C47C24
hypercenter#1819D47
hypocenterC47#21747
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
2. #20245766C2semidirect
3. #20243845C2semidirect
4. #20243462C2semidirect
6. C47×#1819C2semidirect
11. C2×#13094008C4semidirect
13. C2×#13093980C22semidirect
14. C22×C4⋊Dic47C4semidirect
15. C22×(C2×D47)⋊C4C4semidirect
17. #15688984C22semidirect
21. #15688890C22semidirect
24. C47×#342C22semidirect
191. #1819D47semidirect
217. #342C2×D47semidirect
218. C47#21747semidirect

Maximal subgroups

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

subgroupindex
C22×#130940082
#202457662
#202438452
#202434622
#202433022
C47×#18192
#2174747