Finite Group Databasebeta

← #26220032

Subgroups — #26220032

Order 12032.

1858 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689685C211yesmaximal, characteristic
2C22×#13093938C214yesmaximal, complemented
3#20246002C211yesmaximal, characteristic, complemented
4#20244224C218yesmaximal, complemented
5C47×#2273C211yesmaximal, characteristic, complemented
6C23×C22⋊D47C2212yes—
7#15689685—24no—
8#15689685—24no—
9C2×#13094008C414yescomplemented
10C2×#13093998C2214yes—
11C2×#13093994C414yescomplemented
12C22×C22⋊Dic47C2212yes—
13C2×#13093938—216no—
14C2×#13093938C22116yescomplemented
15#15688998—24no—
16#15688983C2214yescomplemented
17C22×C22⋊C4×C47C2212yes—
18C22×C22⋊C4×C47C2211yescharacteristic
19C4×C22⋊C4×C47—24no—
20C4×C22⋊C4×C47C2214yescomplemented
21C24×D47C2×C412yes—
22C22×D188C2×C414yescomplemented
23C22×C22⋊D47—28no—
24C22×C22⋊D47—24no—
25C22×C22⋊D47—24no—
26C22×C22⋊D47—28no—
27C22×C22⋊D47C2×C414yes—
28C22×C22⋊D47C2314yes—
29C23×Dic47—22no—
30C23×Dic47C2×C412yes—
31#13094008—216no—
32#13093998—28no—
33#13093998—28no—
34#13093994—216no—
35C2×C4⋊Dic47—24no—
36C2×C4⋊Dic47C2×C414yescomplemented
37C2×C22⋊Dic47—28no—
38C2×C22⋊Dic47—22no—
39C2×C22⋊Dic47C2314yes—
40C2×(C2×D47)⋊C4C2×C418yescomplemented
41#13093938—232no—
42C47⋊#95—24no—
43C25×C47C2311yescharacteristic
44C23×C4×C47—22no—
45C23×C4×C47C2312yes—
46C2×C22⋊C4×C47—22no—
47C2×C22⋊C4×C47—24no—
48C2×C22⋊C4×C47C2×C414yes—
49C2×C22⋊C4×C47D414yes—
50C2×C22⋊C4×C47D414yes—
51C2×C22⋊C4×C47D418yescomplemented
52C2×C22⋊C4×C47C2314yes—
53C2×C42×C47—24no—
54#95×C47—24no—
55C24×C47—82no—
56C24×C47C22×C412yes—
57C24×C47C2×D412yes—
58C24×C47C2×D412yes—
59C24×C47C2×D414yes—
60C24×C47#4114yes—
61C24×C47C2411yescharacteristic
62C23×D47—24no—
63C23×D47—28no—
64C23×D47—84no—
65C23×D47C22×C414yes—
66C22×C4×C47—24no—
67C22×C4×C47—28no—
68C22×C4×C47—82no—
69C22×C4×C47—84no—
70C22×C4×C47C22×C414yes—
71C22×C4×C47C2×D414yes—
72C22×C4×C47C2×D414yes—
73C2×C22⋊D47—28no—
74C2×C22⋊D47—216no—
75C2×C22⋊D47—28no—
76C2×C22⋊D47—24no—
77C2×C22⋊D47—24no—
78C2×C22⋊D47—28no—
79C2×C22⋊D47—48no—
80C2×C22⋊D47—48no—
81C2×C22⋊D47—48no—
82C2×C22⋊D47—48no—
83C22×Dic47—24no—
84C22×Dic47—28no—
85C22×Dic47—24no—
86C22×Dic47—44no—
87C22×Dic47—84no—
88C22×Dic47C22×C414yes—
89C2×D188—28no—
90C2×D188—84no—
91C22⋊C4×C47—28no—
92C22⋊C4×C47—24no—
93C22⋊C4×C47C22⋊C4116yes—
94C22⋊C4×C47C2×D418yes—
95C42×C47—84no—
96C22⋊Dic47—28no—
97C22⋊Dic47—44no—
98(C2×D47)⋊C4—216no—
99C4⋊Dic47—28no—
100C4⋊Dic47—84no—
101C23×C47—44no—
102C23×C47—44no—
103C23×C47—42no—
104C23×C47—44no—
105C23×C47—84no—
106C23×C47—84no—
107C23×C47C2×C22⋊C414yes—
108C23×C47C4×D418yes—
109C23×C47#12114yes—
110C23×C47#12114yes—
111C23×C47#12414yes—
112C23×C47#12714yes—
113C23×C47C23×C411yescharacteristic
114C23×C47C22×D411yescharacteristic
115C23×C47C22×D411yescharacteristic
116C23×C47C22×D412yes—
117C23×C47C2×#4112yes—
118C22×D47—24no—
119C22×D47—28no—
120C22×D47—48no—
121C22×D47—48no—
122C22×D47—88no—
123C22×D47—816no—
124C2×C4×C47—28no—
125C2×C4×C47—44no—
126C2×C4×C47—44no—
127C2×C4×C47—44no—
128C2×C4×C47—48no—
129C2×C4×C47—88no—
130C2×C4×C47C2×C22⋊C418yes—
131C22⋊D47—48no—
132C22⋊D47—44no—
133C22⋊D47—44no—
134C22⋊D47—416no—
135C2×Dic47—24no—
136C2×Dic47—28no—
137C2×Dic47—48no—
138C2×Dic47—48no—
139C2×Dic47—44no—
140C2×Dic47—88no—
141D188—88no—
142#21213—471nomaximal, Sylow
143C22×C47—22no—
144C22×C47—22no—
145C22×C47—22no—
146C22×C47—28no—
147C22×C47—44no—
148C22×C47—44no—
149C22×C47—48no—
150C22×C47—44no—
151C22×C47—44no—
152C22×C47—42no—
153C22×C47—42no—
154C22×C47#39018yes—
155C22×C47C22×C22⋊C411yescharacteristic
156C22×C47C2×C4×D412yes—
157C22×C47C2×#12112yes—
158C22×C47C2×#12411yescharacteristic
159C22×C47C2×#12711yescharacteristic
160C2×D47—48no—
161C2×D47—44no—
162C2×D47—816no—
163C2×D47—88no—
164C4×C47—28no—
165C4×C47—48no—
166Dic47—44no—
167Dic47—84no—
168C2×#534—471no—
169C22×#99—474no—
170#2315—471no—
171#2273D4711yescharacteristic, complemented
172#1964—478no—
173C2×C47—21no—
174C2×C47—22no—
175C2×C47—22no—
176C2×C47—28no—
177C2×C47—21no—
178C2×C47C2×#39011yescharacteristic
179C2×C47#288312yes—
180D47—84no—
181C23×D4—472no—
182#534—944no—
183#534—944no—
184C2×#123—474no—
185C2×#121—474no—
186C2×#120—474no—
187C22×C22⋊C4—472no—
188C22×C22⋊C4C2×D4712yes—
189C22×C22⋊C4C2×D4711yescharacteristic
190C2×#99—4716no—
191C2×#99—9416no—
192#385—944no—
193#380—474no—
194C4×C22⋊C4—24no—
195C4×C22⋊C4C2×D4714yescomplemented
196C47#2121311yescharacteristic, complemented, Sylow
197C25—472no—
198C25C22×D4711yescharacteristic
199C22×D4—474no—
200C22×D4—474no—
201C22×D4—474no—
202C22×D4—944no—
203C22×D4—948no—
204C22×D4—948no—
205C22×D4—944no—
206C23×C4—22no—
207C23×C4—472no—
208C23×C4—942no—
209C23×C4C22×D4712yes—
210#123—9416no—
211#121—948no—
212#121—948no—
213#120—9416no—
214C2×C4⋊C4—474no—
215C2×C4⋊C4—944no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—478no—
219C2×C22⋊C4—474no—
220C2×C22⋊C4—948no—
221C2×C22⋊C4—942no—
222C2×C22⋊C4C4×D4714yes—
223C2×C22⋊C4D18814yes—
224C2×C22⋊C4D18818yescomplemented
225C2×C22⋊C4C22⋊D4714yes—
226C2×C22⋊C4C22×D4714yes—
227C2×C42—24no—
228#99—9432no—
229#95—24no—
230#95—944no—
231C24—82no—
232C24—474no—
233C24—944no—
234C24—948no—
235C24—3764no—
236C24C2×C4×D4712yes—
237C24C2×D18812yes—
238C24C2×D18814yes—
239C24#1062535014yes—
240C24C2×C22⋊D4712yes—
241C24C23×D4711yescharacteristic
242C2×D4—948no—
243C2×D4—9416no—
244C2×D4—944no—
245C2×D4—948no—
246C2×D4—944no—
247C2×D4—948no—
248C2×D4—948no—
249C2×D4—3764no—
250C2×D4—1888no—
251C2×D4—1888no—
252C2×D4—1888no—
253C2×D4—1888no—
254C22×C4—24no—
255C22×C4—28no—
256C22×C4—82no—
257C22×C4—84no—
258C22×C4—474no—
259C22×C4—944no—
260C22×C4—948no—
261C22×C4—944no—
262C22×C4—3764no—
263C22×C4—1884no—
264C22×C4C2×C4×D4714yes—
265C22×C4C2×D18814yes—
266C22×C4C2×C22⋊D4714yes—
267C4⋊C4—948no—
268C4⋊C4—3764no—
269C22⋊C4—28no—
270C22⋊C4—24no—
271C22⋊C4—9416no—
272C22⋊C4—948no—
273C22⋊C4—1884no—
274C22⋊C4(C2×D47)⋊C4116yes—
275C22⋊C4C2×D18818yes—
276C42—84no—
277C23—44no—
278C23—42no—
279C23—44no—
280C23—44no—
281C23—84no—
282C23—84no—
283C23—948no—
284C23—944no—
285C23—3768no—
286C23—37616no—
287C23—1888no—
288C23—1888no—
289C23C2×(C2×D47)⋊C414yes—
290C23C4×D18818yes—
291C23#1309399814yes—
292C23#1309399814yes—
293C23#1309401214yes—
294C23#1309402114yes—
295C23C22×C4×D4711yescharacteristic
296C23C22×C22⋊D4711yescharacteristic
297C23C22×D18811yescharacteristic
298C23C22×D18812yes—
299C23C2×#1062535012yes—
300D4—3768no—
301D4—18816no—
302D4—1888no—
303D4—1884no—
304D4—1884no—
305C2×C4—28no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—48no—
310C2×C4—88no—
311C2×C4—944no—
312C2×C4—948no—
313C2×C4—3768no—
314C2×C4—1888no—
315C2×C4—1884no—
316C2×C4—1888no—
317C2×C4C2×(C2×D47)⋊C418yes—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—28no—
322C22—44no—
323C22—44no—
324C22—48no—
325C22—44no—
326C22—44no—
327C22—42no—
328C22—42no—
329C22—3768no—
330C22—37616no—
331C22—1884no—
332C22—1888no—
333C22#1568902518yes—
334C22C22×(C2×D47)⋊C411yescharacteristic
335C22C2×C4×D18812yes—
336C22C2×#1309399812yes—
337C22C2×#1309401211yescharacteristic
338C22C2×#1309402111yescharacteristic
339C4—28no—
340C4—48no—
341C4—3764no—
342C4—1884no—
343C2—22no—
344C2—21no—
345C2—22no—
346C2—28no—
347C2—21no—
348C2—3764no—
349C2C2×#1568902511yescharacteristic
350C2#2025184812yes—
subgroupG/H
centerC22C22×(C2×D47)⋊C4
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC47×#2273C2
socleC22×C47C22×C22⋊C4
prime radicalC2×#15689685C2
cosocleC24×C47C24
hypercenter#2273D47
hypocenterC47#21213
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C22⋊C4 ◁ #2273

Complements

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

NH ≅ G/Ntype
2. C22×#13093938C2semidirect
3. #20246002C2semidirect
4. #20244224C2semidirect
5. C47×#2273C2semidirect
9. C2×#13094008C4semidirect
11. C2×#13093994C4semidirect
14. C2×#13093938C22semidirect
16. #15688983C22semidirect
20. C4×C22⋊C4×C47C22semidirect
22. C22×D188C2×C4semidirect
36. C2×C4⋊Dic47C2×C4semidirect
40. C2×(C2×D47)⋊C4C2×C4semidirect
51. C2×C22⋊C4×C47D4semidirect
171. #2273D47semidirect
195. C4×C22⋊C4C2×D47semidirect
196. C47#21213semidirect
224. C2×C22⋊C4D188semidirect

Maximal subgroups

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

subgroupindex
C2×#156896852
C22×#130939382
#202460022
#202442242
C47×#22732
#2121347