Finite Group Databasebeta

← #26376342

Subgroups — #26376342

Order 12032.

2390 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093982C211yesmaximal, characteristic
2#20246277C218yesmaximal, complemented
3#20246269C218yesmaximal, complemented
4#20246004C212yesmaximal, complemented
5#20245983C214yesmaximal
6#20245782C214yesmaximal, complemented
7#20245767C212yesmaximal
8#20245745C211yesmaximal, characteristic, complemented
9C47×#2274C211yesmaximal, characteristic, complemented
10C23×C4×D47C2211yescharacteristic
11C2×#13093983C2212yes—
12C2×#13093982C4116yescomplemented
13C2×#13093982C2214yes—
14C2×#13093982C2214yes—
15C22×C4⋊Dic47C2211yescharacteristic
16C22×(C2×D47)⋊C4C2212yes—
17C22×C4×Dic47C2212yes—
18#15689021C2218yescomplemented
19#15689019C2218yescomplemented
20#15689014C22116yes—
21#15689012C22116yes—
22#15688990C2218yescomplemented
23#15688987C2218yescomplemented
24#15688984C2214yes—
25#15688981C2218yes—
26#15688980C2214yes—
27#15688980C2214yes—
28#15688978C2214yes—
29C4⋊C4×Dic47C2218yescomplemented
30C4×Dic47⋊C4C2218yescomplemented
31C4×(C2×D47)⋊C4C2214yes—
32C4×(C2×D47)⋊C4C2214yes—
33C4×C22⋊Dic47C2214yes—
34#15688969C2218yescomplemented
35C2×#117×C47C2212yes—
36C22×C4⋊C4×C47C2211yescharacteristic
37C4×C4⋊C4×C47C2218yescomplemented
38C4×C22⋊C4×C47C2214yes—
39C24×D47C2311yescharacteristic
40C23×Dic47C2311yescharacteristic
41C23×Dic47C2312yes—
42C22×C4×D47—28no—
43C22×C4×D47C2×C418yes—
44C22×C4×D47C2312yes—
45C22×C4×D47C2312yes—
46#13093983—28no—
47#13093982C2×C4132yes—
48#13093982C2×C4132yes—
49C2×C4⋊Dic47C2×C418yes—
50C2×C4⋊Dic47C2314yes—
51C2×C4⋊Dic47C2314yes—
52C2×Dic47⋊C4C23116yes—
53C2×C22⋊Dic47C2314yes—
54C2×(C2×D47)⋊C4C2×C4116yes—
55C2×(C2×D47)⋊C4C2318yes—
56C2×(C2×D47)⋊C4C2318yes—
57C2×(C2×D47)⋊C4C2318yes—
58C2×C4×Dic47C2×C4116yes—
59C2×C4×Dic47C2314yes—
60C2×C4×Dic47C2318yes—
61C2×C4×Dic47C2314yes—
62C2×C4×Dic47C2314yes—
63C47⋊#95C2318yes—
64C47⋊#95C23116yes—
65C47⋊#95C2318yes—
66C47⋊#95C2318yes—
67C23×C4×C47C2312yes—
68C23×C4×C47C2311yescharacteristic
69#117×C47—28no—
70C2×C4⋊C4×C47C2×C418yes—
71C2×C4⋊C4×C47C2314yes—
72C2×C4⋊C4×C47C2314yes—
73C2×C22⋊C4×C47C2314yes—
74C2×C42×C47C2314yes—
75C2×C42×C47C2318yes—
76#95×C47C2318yes—
77C24×C47C2411yescharacteristic
78C23×D47—24no—
79C23×D47—42no—
80C23×D47—42no—
81C23×D47C22×C414yes—
82C23×D47C2412yes—
83C22×C4×C47—44no—
84C22×C4×C47C22×C418yes—
85C22×C4×C47C22×C414yes—
86C22×C4×C47#4118yes—
87C22×C4×C47C2418yes—
88C22×C4×C47C2414yes—
89C22×C4×C47C2412yes—
90C22×Dic47—24no—
91C22×Dic47—44no—
92C22×Dic47C22×C414yes—
93C22×Dic47C22×C418yes—
94C22×Dic47C2418yes—
95C22×Dic47C2411yescharacteristic
96C22×Dic47C2411yescharacteristic
97C22×Dic47C2414yes—
98C2×C4×D47—232no—
99C2×C4×D47—24no—
100C2×C4×D47—24no—
101C2×C4×D47—24no—
102C2×C4×D47—24no—
103C2×C4×D47C22×C418yes—
104C2×C4×D47C22×C418yes—
105C4⋊C4×C47—28no—
106C4⋊C4×C47C22×C4116yes—
107C22⋊C4×C47—28no—
108C42×C47—28no—
109C42×C47—48no—
110C22⋊Dic47—28no—
111(C2×D47)⋊C4—48no—
112(C2×D47)⋊C4—48no—
113(C2×D47)⋊C4C22×C4132yes—
114C4⋊Dic47—28no—
115C4⋊Dic47C22×C4116yes—
116Dic47⋊C4—416no—
117C4×Dic47—28no—
118C4×Dic47—48no—
119C4×Dic47C22×C4116yes—
120C4×Dic47C22×C4116yes—
121C23×C47—22no—
122C23×C47—22no—
123C23×C47C23×C412yes—
124C23×C47C2×#4114yes—
125C23×C47C2511yescharacteristic
126C22×D47—28no—
127C22×D47—22no—
128C22×D47—22no—
129C22×D47—24no—
130C22×D47—24no—
131C22×D47—44no—
132C22×D47—44no—
133C22×D47—416no—
134C22×D47C23×C414yes—
135C2×C4×C47—28no—
136C2×C4×C47—28no—
137C2×C4×C47—28no—
138C2×C4×C47—28no—
139C2×C4×C47—48no—
140C2×C4×C47#117116yes—
141C2×C4×C47C23×C418yes—
142C2×C4×C47C23×C414yes—
143C2×C4×C47C2×#4114yes—
144C2×C4×C47C2×#4114yes—
145C2×Dic47—28no—
146C2×Dic47—22no—
147C2×Dic47—22no—
148C2×Dic47—28no—
149C2×Dic47—28no—
150C2×Dic47—28no—
151C2×Dic47—28no—
152C2×Dic47—48no—
153C2×Dic47C23×C412yes—
154C2×Dic47C23×C412yes—
155C2×Dic47C23×C418yes—
156C4×D47—216no—
157C4×D47—216no—
158#34179—471nomaximal, Sylow
159C22×C47—22no—
160C22×C47—22no—
161C22×C47—28no—
162C22×C47C2×#11714yes—
163C22×C47C24×C411yescharacteristic
164C22×C47C22×#4112yes—
165C22×C47#58512yes—
166C22×C47#58512yes—
167C2×D47—24no—
168C2×D47—24no—
169C2×D47—24no—
170C2×D47—416no—
171C2×D47—44no—
172C2×D47—44no—
173C4×C47—28no—
174C4×C47—48no—
175C4×C47C2×#11718yes—
176Dic47—22no—
177Dic47—22no—
178Dic47—28no—
179Dic47—48no—
180C22×#117—471no—
181#2347—478no—
182#2346—478no—
183#2316—472no—
184#2312—474no—
185#2283—474no—
186#2280—472no—
187#2274—471no—
188#2274D4711yescharacteristic, complemented
189C2×C47—24no—
190C2×C47C22×#11711yescharacteristic
191C2×C47#342811yescharacteristic
192C2×C47#342811yescharacteristic
193C2×C47#349314yes—
194D47—44no—
195C24×C4—471no—
196C2×#117—472no—
197C2×#117—4716no—
198C2×#117—474no—
199C2×#117—474no—
200C2×#117C2×D4712yes—
201C22×C4⋊C4—471no—
202C22×C4⋊C4C2×D4711yescharacteristic
203C22×C22⋊C4—472no—
204C22×C42—472no—
205#389—478no—
206#389—478no—
207#388—4716no—
208#388—4716no—
209#383—478no—
210#382—478no—
211#381—474no—
212#380—474no—
213#380—474no—
214#380—478no—
215#379—474no—
216C4×C4⋊C4—478no—
217C4×C4⋊C4—478no—
218C4×C4⋊C4C2×D4718yescomplemented
219C4×C22⋊C4—474no—
220C4×C22⋊C4—474no—
221C4×C22⋊C4—474no—
222C4×C22⋊C4C2×D4714yes—
223#376—478no—
224C47#3417911yescharacteristic, complemented, Sylow
225C25—471no—
226C23×C4—471no—
227C23×C4—472no—
228C23×C4—478no—
229C23×C4—472no—
230C23×C4—472no—
231C23×C4—948no—
232C23×C4C22×D4712yes—
233C23×C4C22×D4711yescharacteristic
234#117—28no—
235#117—4732no—
236#117—4732no—
237#117—948no—
238C2×C4⋊C4—478no—
239C2×C4⋊C4—474no—
240C2×C4⋊C4—474no—
241C2×C4⋊C4—4716no—
242C2×C4⋊C4C4×D4718yes—
243C2×C4⋊C4C22×D4714yes—
244C2×C4⋊C4C22×D4714yes—
245C2×C22⋊C4—4716no—
246C2×C22⋊C4—478no—
247C2×C22⋊C4—474no—
248C2×C22⋊C4—478no—
249C2×C22⋊C4—478no—
250C2×C22⋊C4C22×D4714yes—
251C2×C42—4716no—
252C2×C42—474no—
253C2×C42—474no—
254C2×C42—474no—
255C2×C42—478no—
256C2×C42C22×D4714yes—
257C2×C42C22×D4718yes—
258#95—478no—
259#95—4716no—
260#95—478no—
261#95—478no—
262#95C22×D4718yes—
263C24—472no—
264C24—474no—
265C24—944no—
266C24—1882no—
267C24—1882no—
268C24C23×D4711yescharacteristic
269C22×C4—44no—
270C22×C4—474no—
271C22×C4—471no—
272C22×C4—471no—
273C22×C4—474no—
274C22×C4—478no—
275C22×C4—478no—
276C22×C4—478no—
277C22×C4—478no—
278C22×C4—944no—
279C22×C4—9432no—
280C22×C4—944no—
281C22×C4—944no—
282C22×C4—944no—
283C22×C4—944no—
284C22×C4—1884no—
285C22×C4C2×C4×D4718yes—
286C22×C4C2×C4×D4714yes—
287C22×C4#1062535018yes—
288C22×C4C23×D4718yes—
289C22×C4C23×D4714yes—
290C22×C4C23×D4712yes—
291C4⋊C4—28no—
292C4⋊C4—4716no—
293C4⋊C4—948no—
294C4⋊C4—18816no—
295C4⋊C4C2×C4×D47116yes—
296C22⋊C4—28no—
297C22⋊C4—4732no—
298C22⋊C4—948no—
299C22⋊C4—1888no—
300C22⋊C4—1888no—
301C42—28no—
302C42—48no—
303C42—4716no—
304C42—4716no—
305C42—948no—
306C42—1888no—
307C23—22no—
308C23—22no—
309C23—474no—
310C23—948no—
311C23—942no—
312C23—942no—
313C23—944no—
314C23—944no—
315C23—1884no—
316C23—1884no—
317C23—18816no—
318C23C22×C4×D4712yes—
319C23C2×#1062535014yes—
320C23C24×D4711yescharacteristic
321C2×C4—28no—
322C2×C4—28no—
323C2×C4—28no—
324C2×C4—28no—
325C2×C4—48no—
326C2×C4—472no—
327C2×C4—472no—
328C2×C4—478no—
329C2×C4—948no—
330C2×C4—942no—
331C2×C4—942no—
332C2×C4—948no—
333C2×C4—9416no—
334C2×C4—9416no—
335C2×C4—948no—
336C2×C4—948no—
337C2×C4—948no—
338C2×C4—1888no—
339C2×C4#13093981116yes—
340C2×C4C22×C4×D4718yes—
341C2×C4C22×C4×D4714yes—
342C2×C4C2×#1062535014yes—
343C2×C4C2×#1062535014yes—
344C22—22no—
345C22—28no—
346C22—22no—
347C22—944no—
348C22—944no—
349C22—944no—
350C22—18816no—
351C22—1884no—
352C22—1884no—
353C22C2×#1309398114yes—
354C22C23×C4×D4711yescharacteristic
355C22C22×#1062535012yes—
356C22#1568995812yes—
357C22#1568996012yes—
358C4—28no—
359C4—48no—
360C4—942no—
361C4—942no—
362C4—948no—
363C4—1888no—
364C4C2×#1309398118yes—
365C2—24no—
366C2—1884no—
367C2C22×#1309398111yescharacteristic
368C2#2025658611yescharacteristic
369C2#2025659011yescharacteristic
370C2#2025710114yes—
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingC47×#2274C2
socle, cosocleC23×C47C25
prime radicalC24×D47C23
hypercenter#2274D47
hypocenterC47#34179
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ #2274

Complements

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

NH ≅ G/Ntype
2. #20246277C2semidirect
3. #20246269C2semidirect
4. #20246004C2semidirect
6. #20245782C2semidirect
8. #20245745C2semidirect
9. C47×#2274C2semidirect
12. C2×#13093982C4semidirect
18. #15689021C22semidirect
19. #15689019C22semidirect
22. #15688990C22semidirect
23. #15688987C22semidirect
29. C4⋊C4×Dic47C22semidirect
30. C4×Dic47⋊C4C22semidirect
34. #15688969C22semidirect
37. C4×C4⋊C4×C47C22semidirect
188. #2274D47semidirect
218. C4×C4⋊C4C2×D47semidirect
224. C47#34179semidirect

Maximal subgroups

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

subgroupindex
C22×#130939822
#202462772
#202462692
#202460042
#202459832
#202457822
#202457672
#202457452
C47×#22742
#3417947