Finite Group Databasebeta

← #26376346

Subgroups — #26376346

Order 12032.

2390 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093982C211yesmaximal, characteristic
2#20246280C218yesmaximal, complemented
3#20246272C218yesmaximal, complemented
4#20246006C211yesmaximal, characteristic, complemented
5#20245985C214yesmaximal
6#20245783C214yesmaximal, complemented
7#20245767C212yesmaximal
8#20245744C212yesmaximal, complemented
9C47×#2316C211yesmaximal, characteristic, complemented
10C23×C4×D47C2211yescharacteristic
11C2×#13093983C2212yes—
12C2×#13093982C4116yescomplemented
13C2×#13093982C2214yes—
14C2×#13093982C2214yes—
15C22×C4⋊Dic47C2211yescharacteristic
16C22×(C2×D47)⋊C4C2212yes—
17C22×C4×Dic47C2212yes—
18#15689022C2218yescomplemented
19#15689015C22116yes—
20#15689013C22116yes—
21#15688990C2218yescomplemented
22#15688988C2218yescomplemented
23#15688984C2214yes—
24#15688983C2214yes—
25#15688980C2214yes—
26#15688980C2214yes—
27#15688978C2214yes—
28C4⋊C4×Dic47C2218yescomplemented
29C4⋊C4×Dic47C2218yescomplemented
30C4×C4⋊Dic47C2218yescomplemented
31C22⋊C4×Dic47C2218yes—
32C4×(C2×D47)⋊C4C2214yes—
33C4×(C2×D47)⋊C4C2214yes—
34#15688968C2218yescomplemented
35C2×#117×C47C2212yes—
36C22×C4⋊C4×C47C2211yescharacteristic
37C47×#389C2218yescomplemented
38C47×#380C2214yes—
39C24×D47C2311yescharacteristic
40C23×Dic47C2311yescharacteristic
41C23×Dic47C2312yes—
42C22×C4×D47—24no—
43C22×C4×D47—24no—
44C22×C4×D47C2×C418yes—
45C22×C4×D47C2312yes—
46C22×C4×D47C2312yes—
47#13093983—28no—
48#13093982C2×C4132yes—
49#13093982C2×C4132yes—
50C2×C4⋊Dic47C2×C418yes—
51C2×C4⋊Dic47C2314yes—
52C2×C4⋊Dic47C2318yes—
53C2×C4⋊Dic47C2314yes—
54C2×C22⋊Dic47C2314yes—
55C2×(C2×D47)⋊C4C2×C4116yes—
56C2×(C2×D47)⋊C4C2318yes—
57C2×(C2×D47)⋊C4C2318yes—
58C2×(C2×D47)⋊C4C2318yes—
59C2×C4×Dic47C2×C4116yes—
60C2×C4×Dic47C2314yes—
61C2×C4×Dic47C23116yes—
62C2×C4×Dic47C2314yes—
63C2×C4×Dic47C2314yes—
64C47⋊#95C2318yes—
65C47⋊#95C2318yes—
66C47⋊#95C2318yes—
67C47⋊#95C23116yes—
68C23×C4×C47C2312yes—
69C23×C4×C47C2311yescharacteristic
70#117×C47—28no—
71C2×C4⋊C4×C47C2×C418yes—
72C2×C4⋊C4×C47C2314yes—
73C2×C4⋊C4×C47C2318yes—
74C2×C4⋊C4×C47C2314yes—
75C2×C22⋊C4×C47C2314yes—
76C2×C42×C47C2314yes—
77#95×C47C2318yes—
78C24×C47C2411yescharacteristic
79C23×D47—22no—
80C23×D47—22no—
81C23×D47—42no—
82C23×D47—42no—
83C23×D47C22×C414yes—
84C23×D47C2412yes—
85C22×C4×C47—44no—
86C22×C4×C47C22×C418yes—
87C22×C4×C47C22×C414yes—
88C22×C4×C47#4114yes—
89C22×C4×C47#4114yes—
90C22×C4×C47C2418yes—
91C22×C4×C47C2414yes—
92C22×C4×C47C2412yes—
93C22×Dic47—22no—
94C22×Dic47—22no—
95C22×Dic47—44no—
96C22×Dic47C22×C414yes—
97C22×Dic47C22×C418yes—
98C22×Dic47C2418yes—
99C22×Dic47C2411yescharacteristic
100C22×Dic47C2411yescharacteristic
101C22×Dic47C2414yes—
102C2×C4×D47—232no—
103C2×C4×D47—24no—
104C2×C4×D47—24no—
105C2×C4×D47—24no—
106C2×C4×D47—24no—
107C2×C4×D47C22×C418yes—
108C2×C4×D47C22×C418yes—
109C4⋊C4×C47—28no—
110C4⋊C4×C47—48no—
111C4⋊C4×C47C22×C4116yes—
112C22⋊C4×C47—28no—
113C42×C47—28no—
114C22⋊Dic47—28no—
115(C2×D47)⋊C4—48no—
116(C2×D47)⋊C4—48no—
117(C2×D47)⋊C4C22×C4132yes—
118C4⋊Dic47—28no—
119C4⋊Dic47—48no—
120C4⋊Dic47C22×C4116yes—
121C4×Dic47—28no—
122C4×Dic47—416no—
123C4×Dic47C22×C4116yes—
124C4×Dic47C22×C4116yes—
125C23×C47—22no—
126C23×C47—22no—
127C23×C47C23×C412yes—
128C23×C47C2×#4112yes—
129C23×C47C2×#4112yes—
130C23×C47C2511yescharacteristic
131C22×D47—28no—
132C22×D47—22no—
133C22×D47—22no—
134C22×D47—24no—
135C22×D47—24no—
136C22×D47—44no—
137C22×D47—44no—
138C22×D47—48no—
139C22×D47—48no—
140C22×D47C23×C414yes—
141C2×C4×C47—28no—
142C2×C4×C47—28no—
143C2×C4×C47—28no—
144C2×C4×C47—28no—
145C2×C4×C47—48no—
146C2×C4×C47#117116yes—
147C2×C4×C47C23×C418yes—
148C2×C4×C47C23×C414yes—
149C2×C4×C47C2×#4114yes—
150C2×C4×C47C2×#4114yes—
151C2×Dic47—28no—
152C2×Dic47—21no—
153C2×Dic47—21no—
154C2×Dic47—21no—
155C2×Dic47—21no—
156C2×Dic47—28no—
157C2×Dic47—28no—
158C2×Dic47—28no—
159C2×Dic47—28no—
160C2×Dic47—48no—
161C2×Dic47C23×C412yes—
162C2×Dic47C23×C412yes—
163C2×Dic47C23×C418yes—
164C4×D47—216no—
165C4×D47—216no—
166#34179—471nomaximal, Sylow
167C22×C47—22no—
168C22×C47—22no—
169C22×C47—24no—
170C22×C47—24no—
171C22×C47C2×#11714yes—
172C22×C47C24×C411yescharacteristic
173C22×C47C22×#4111yescharacteristic
174C22×C47C22×#4111yescharacteristic
175C22×C47#58512yes—
176C22×C47#58512yes—
177C2×D47—24no—
178C2×D47—24no—
179C2×D47—24no—
180C2×D47—416no—
181C2×D47—44no—
182C2×D47—44no—
183C4×C47—28no—
184C4×C47—48no—
185C4×C47C2×#11718yes—
186Dic47—22no—
187Dic47—22no—
188Dic47—28no—
189Dic47—48no—
190C22×#117—471no—
191#2347—478no—
192#2346—478no—
193#2316—471no—
194#2316D4711yescharacteristic, complemented
195#2312—474no—
196#2283—474no—
197#2280—472no—
198#2274—472no—
199C2×C47—24no—
200C2×C47C22×#11711yescharacteristic
201C2×C47#342811yescharacteristic
202C2×C47#342811yescharacteristic
203C2×C47#349312yes—
204C2×C47#349312yes—
205D47—44no—
206C24×C4—471no—
207C2×#117—472no—
208C2×#117—4716no—
209C2×#117—474no—
210C2×#117—474no—
211C2×#117C2×D4712yes—
212C22×C4⋊C4—471no—
213C22×C4⋊C4C2×D4711yescharacteristic
214C22×C22⋊C4—472no—
215C22×C42—472no—
216#389—478no—
217#389C2×D4718yescomplemented
218#388—4716no—
219#388—4716no—
220#383—478no—
221#382—478no—
222#381—474no—
223#380—474no—
224#380—474no—
225#380—474no—
226#380C2×D4714yes—
227#379—474no—
228C4×C4⋊C4—478no—
229C4×C4⋊C4—478no—
230C4×C4⋊C4—478no—
231C4×C22⋊C4—478no—
232C4×C22⋊C4—474no—
233C4×C22⋊C4—474no—
234#376—478no—
235C47#3417911yescharacteristic, complemented, Sylow
236C25—471no—
237C23×C4—471no—
238C23×C4—472no—
239C23×C4—478no—
240C23×C4—472no—
241C23×C4—472no—
242C23×C4—944no—
243C23×C4—944no—
244C23×C4C22×D4712yes—
245C23×C4C22×D4711yescharacteristic
246#117—28no—
247#117—4732no—
248#117—4732no—
249#117—948no—
250C2×C4⋊C4—478no—
251C2×C4⋊C4—474no—
252C2×C4⋊C4—474no—
253C2×C4⋊C4—478no—
254C2×C4⋊C4C4×D4718yes—
255C2×C4⋊C4C22×D4714yes—
256C2×C4⋊C4C22×D4718yes—
257C2×C4⋊C4C22×D4714yes—
258C2×C22⋊C4—4716no—
259C2×C22⋊C4—478no—
260C2×C22⋊C4—474no—
261C2×C22⋊C4—478no—
262C2×C22⋊C4—478no—
263C2×C22⋊C4C22×D4714yes—
264C2×C42—4716no—
265C2×C42—474no—
266C2×C42—474no—
267C2×C42—474no—
268C2×C42—4716no—
269C2×C42C22×D4714yes—
270#95—478no—
271#95—478no—
272#95—478no—
273#95—4716no—
274#95C22×D4718yes—
275C24—472no—
276C24—474no—
277C24—942no—
278C24—942no—
279C24—1882no—
280C24—1882no—
281C24C23×D4711yescharacteristic
282C22×C4—44no—
283C22×C4—474no—
284C22×C4—471no—
285C22×C4—471no—
286C22×C4—474no—
287C22×C4—478no—
288C22×C4—478no—
289C22×C4—478no—
290C22×C4—478no—
291C22×C4—942no—
292C22×C4—942no—
293C22×C4—9432no—
294C22×C4—944no—
295C22×C4—944no—
296C22×C4—944no—
297C22×C4—944no—
298C22×C4—1884no—
299C22×C4C2×C4×D4718yes—
300C22×C4C2×C4×D4714yes—
301C22×C4#1062535214yes—
302C22×C4#1062535414yes—
303C22×C4C23×D4718yes—
304C22×C4C23×D4714yes—
305C22×C4C23×D4712yes—
306C4⋊C4—28no—
307C4⋊C4—48no—
308C4⋊C4—4716no—
309C4⋊C4—948no—
310C4⋊C4—1888no—
311C4⋊C4C2×C4×D47116yes—
312C22⋊C4—28no—
313C22⋊C4—4732no—
314C22⋊C4—948no—
315C22⋊C4—1888no—
316C22⋊C4—1888no—
317C42—28no—
318C42—4716no—
319C42—4716no—
320C42—948no—
321C42—18816no—
322C23—22no—
323C23—22no—
324C23—474no—
325C23—948no—
326C23—942no—
327C23—942no—
328C23—944no—
329C23—944no—
330C23—1884no—
331C23—1884no—
332C23—1888no—
333C23—1888no—
334C23C22×C4×D4712yes—
335C23C2×#1062535212yes—
336C23C2×#1062535412yes—
337C23C24×D4711yescharacteristic
338C2×C4—28no—
339C2×C4—28no—
340C2×C4—28no—
341C2×C4—28no—
342C2×C4—48no—
343C2×C4—472no—
344C2×C4—472no—
345C2×C4—478no—
346C2×C4—948no—
347C2×C4—941no—
348C2×C4—941no—
349C2×C4—941no—
350C2×C4—941no—
351C2×C4—948no—
352C2×C4—9416no—
353C2×C4—9416no—
354C2×C4—948no—
355C2×C4—948no—
356C2×C4—948no—
357C2×C4—1888no—
358C2×C4#13093982116yes—
359C2×C4C22×C4×D4718yes—
360C2×C4C22×C4×D4714yes—
361C2×C4C2×#1062535214yes—
362C2×C4C2×#1062535414yes—
363C22—22no—
364C22—22no—
365C22—24no—
366C22—24no—
367C22—944no—
368C22—944no—
369C22—944no—
370C22—18816no—
371C22—1884no—
372C22—1884no—
373C22C2×#1309398214yes—
374C22C23×C4×D4711yescharacteristic
375C22C22×#1062535211yescharacteristic
376C22C22×#1062535411yescharacteristic
377C22#1568995812yes—
378C22#1568996012yes—
379C4—28no—
380C4—48no—
381C4—942no—
382C4—942no—
383C4—948no—
384C4—1888no—
385C4C2×#1309398218yes—
386C2—24no—
387C2—1884no—
388C2C22×#1309398211yescharacteristic
389C2#2025658611yescharacteristic
390C2#2025659011yescharacteristic
391C2#2025710712yes—
392C2#2025710912yes—
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingC47×#2316C2
socle, cosocleC23×C47C25
prime radicalC24×D47C23
hypercenter#2316D47
hypocenterC47#34179
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ #2316

Complements

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

NH ≅ G/Ntype
2. #20246280C2semidirect
3. #20246272C2semidirect
4. #20246006C2semidirect
6. #20245783C2semidirect
8. #20245744C2semidirect
9. C47×#2316C2semidirect
12. C2×#13093982C4semidirect
18. #15689022C22semidirect
21. #15688990C22semidirect
22. #15688988C22semidirect
28. C4⋊C4×Dic47C22semidirect
29. C4⋊C4×Dic47C22semidirect
30. C4×C4⋊Dic47C22semidirect
34. #15688968C22semidirect
37. C47×#389C22semidirect
194. #2316D47semidirect
217. #389C2×D47semidirect
235. C47#34179semidirect

Maximal subgroups

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

subgroupindex
C22×#130939822
#202462802
#202462722
#202460062
#202459852
#202457832
#202457672
#202457442
C47×#23162
#3417947