Finite Group Databasebeta

← #26131236

Subgroups — #26131236

Order 12032.

768 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689097C212yesmaximal
2#20241509C214yesmaximal, complemented
3C47×#2879C211yesmaximal, characteristic
4C2×#13093983—22no—
5#15689097—22no—
6#15689097—22no—
7#15689097C414yescomplemented
8#15689097C2214yes—
9C2×#13093939—22no—
10C2×#13093939C414yescomplemented
11C22×#41×C47C2211yescharacteristic
12C22×M16×C47C411yescharacteristic, complemented
13C22×M16×C47C411yescharacteristic, complemented
14C47×#408—22no—
15C47×#408C412yescomplemented
16C47×#408C2212yes—
17C47×#407—22no—
18#13093983—24no—
19#13093983—24no—
20#13093983—42no—
21#13093983—42no—
22C2×C4⋊Dic47—42no—
23C2×C22⋊Dic47—24no—
24C2×C4×Dic47—42no—
25#13093939—24no—
26#13093939—24no—
27#13093939—24no—
28#13093939C2×C418yescomplemented
29C2×#41×C47—42no—
30C2×#41×C47—42no—
31C2×#41×C47C2×C412yes—
32C2×#41×C47C2×C411yescharacteristic
33C2×#41×C47D411yescharacteristic
34C2×#41×C47D412yes—
35C2×#41×C47Q811yescharacteristic
36C2×#41×C47C2311yescharacteristic
37C22×Q8×C47Q811yescharacteristic
38C22×D4×C47C2×C412yes—
39C22×D4×C47D411yescharacteristic
40C23×C4×C47C2×C411yescharacteristic
41C23×C4×C47D412yes—
42C2×M16×C47—21no—
43C2×M16×C47—21no—
44C2×M16×C47—21no—
45C2×M16×C47—42no—
46C2×M16×C47—42no—
47C2×M16×C47C2×C412yescomplemented
48C22×C8×C47—21no—
49C22×C8×C47C2×C412yescomplemented
50C22⋊C8×C47—24no—
51C22⋊C8×C47—24no—
52C24×C47C4211yescharacteristic
53C24×C47C22⋊C412yes—
54#41×C47—42no—
55#41×C47—42no—
56#41×C47—41no—
57#41×C47—41no—
58#41×C47—42no—
59#41×C47—42no—
60#41×C47—42no—
61#41×C47—42no—
62#41×C47—41no—
63#41×C47—41no—
64C2×Q8×C47—42no—
65C2×Q8×C47C4⋊C412yes—
66C2×Q8×C47C4⋊C411yescharacteristic
67C2×Q8×C47C2×Q811yescharacteristic
68C2×D4×C47—42no—
69C2×D4×C47—42no—
70C2×D4×C47—42no—
71C2×D4×C47C4212yes—
72C2×D4×C47C22⋊C412yes—
73C2×D4×C47C22⋊C412yes—
74C2×D4×C47C4⋊C412yes—
75C2×D4×C47C4⋊C411yescharacteristic
76C2×D4×C47C22×C412yes—
77C2×D4×C47C2×D411yescharacteristic
78C22×C4×C47—22no—
79C22×C4×C47—22no—
80C22×C4×C47—22no—
81C22×C4×C47—22no—
82C22×C4×C47—22no—
83C22×C4×C47—22no—
84C22×C4×C47C4211yescharacteristic
85C22×C4×C47C22⋊C412yes—
86C22×C4×C47C22⋊C411yescharacteristic
87C22×C4×C47C22⋊C412yes—
88C22×C4×C47C22⋊C411yescharacteristic
89C22×C4×C47C4⋊C412yes—
90C22×C4×C47C4⋊C412yes—
91C22×C4×C47C4⋊C411yescharacteristic
92C22×C4×C47C4⋊C411yescharacteristic
93C22×C4×C47C22×C411yescharacteristic
94C22×C4×C47C2×D412yes—
95C22×Dic47—44no—
96M16×C47—42no—
97M16×C47—42no—
98M16×C47—42no—
99M16×C47—42no—
100C2×C8×C47—22no—
101C2×C8×C47—22no—
102C2×C8×C47—42no—
103C2×C8×C47—42no—
104C22⋊Dic47—28no—
105C22⋊Dic47—44no—
106C4⋊Dic47—44no—
107C4⋊Dic47—44no—
108C4×Dic47—44no—
109C4×Dic47—44no—
110C23×C47—22no—
111C23×C47—22no—
112C23×C47—22no—
113C23×C47—82no—
114C23×C47—81no—
115C23×C47#9512yes—
116C23×C47#9511yescharacteristic
117C23×C47#9511yescharacteristic
118C23×C47C2×C4211yescharacteristic
119C23×C47C2×C22⋊C412yes—
120Q8×C47—42no—
121Q8×C47—41no—
122Q8×C47—41no—
123D4×C47—42no—
124D4×C47—42no—
125D4×C47—42no—
126D4×C47—42no—
127D4×C47—41no—
128D4×C47—42no—
129D4×C47—41no—
130C2×C4×C47—22no—
131C2×C4×C47—22no—
132C2×C4×C47—22no—
133C2×C4×C47—22no—
134C2×C4×C47—22no—
135C2×C4×C47—22no—
136C2×C4×C47—22no—
137C2×C4×C47—22no—
138C2×C4×C47—22no—
139C2×C4×C47—22no—
140C2×C4×C47—22no—
141C2×C4×C47—21no—
142C2×C4×C47—21no—
143C2×C4×C47—21no—
144C2×C4×C47—21no—
145C2×C4×C47—22no—
146C2×C4×C47—22no—
147C2×C4×C47#9511yescharacteristic
148C2×C4×C47#9512yes—
149C2×C4×C47#9511yescharacteristic
150C2×C4×C47#9912yes—
151C2×C4×C47#9912yes—
152C2×C4×C47C2×C22⋊C411yescharacteristic
153C2×C4×C47C2×C4⋊C412yes—
154C2×C4×C47C2×C4⋊C411yescharacteristic
155C2×Dic47—48no—
156C2×Dic47—88no—
157C8×C47—42no—
158C8×C47—42no—
159#12577—471nomaximal, Sylow
160C22×C47—22no—
161C22×C47—22no—
162C22×C47—21no—
163C22×C47—21no—
164C22×C47—41no—
165C22×C47—42no—
166C22×C47—41no—
167C22×C47—41no—
168C22×C47—42no—
169C22×C47—44no—
170C22×C47—82no—
171C22×C47—82no—
172C22×C47#34212yes—
173C22×C47C2×#9511yescharacteristic
174C4×C47—22no—
175C4×C47—21no—
176C4×C47—22no—
177C4×C47—21no—
178C4×C47#34212yes—
179C4×C47C2×#9911yescharacteristic
180C4×C47C2×#9911yescharacteristic
181Dic47—88no—
182C2×#410—472no—
183#2879D4711yescharacteristic
184#1391—474no—
185C2×C47—21no—
186C2×C47—22no—
187C2×C47—42no—
188C2×C47—42no—
189C2×C47—41no—
190C2×C47—41no—
191C2×C47C2×#34211yescharacteristic
192C22×#41C2×D4711yescharacteristic
193C22×M16Dic4711yescharacteristic, complemented
194C22×M16Dic4711yescharacteristic, complemented
195C2×#117—942no—
196#410—474no—
197#410—474no—
198#410—942no—
199#410—942no—
200C2×#99—474no—
201C2×#99—942no—
202#408—22no—
203#408Dic4712yescomplemented
204#408C2×D4712yes—
205#407—22no—
206C47#1257711yescharacteristic, complemented, Sylow
207C2×#41—42no—
208C2×#41—42no—
209C2×#41Dic9411yescharacteristic
210C2×#41C4×D4712yes—
211C2×#41D18811yescharacteristic
212C2×#41C2×Dic4711yescharacteristic
213C2×#41C22⋊D4712yes—
214C2×#41C22×D4711yescharacteristic
215C22×Q8Dic9411yescharacteristic
216C22×D4C4×D4712yes—
217C22×D4D18811yescharacteristic
218C23×C4C2×Dic4711yescharacteristic
219C23×C4C22⋊D4712yes—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—42no—
224C2×M16—42no—
225C2×M16C2×Dic4712yescomplemented
226C22×C8—21no—
227C22×C8C2×Dic4712yescomplemented
228#117—944no—
229#117—944no—
230#117—1882no—
231#117—1882no—
232C2×C4⋊C4—1882no—
233C2×C22⋊C4—944no—
234C2×C42—1882no—
235#99—478no—
236#99—944no—
237#99—944no—
238#99—944no—
239C22⋊C8—24no—
240C22⋊C8—24no—
241C24C4×Dic4711yescharacteristic
242C24(C2×D47)⋊C412yes—
243#41—42no—
244#41—42no—
245#41—42no—
246#41—42no—
247#41—41no—
248#41—41no—
249#41—42no—
250#41—42no—
251#41—41no—
252#41—41no—
253C2×Q8—42no—
254C2×Q8Dic47⋊C412yes—
255C2×Q8C4⋊Dic4711yescharacteristic
256C2×Q8C2×Dic9411yescharacteristic
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4C4×Dic4712yes—
261C2×D4Dic47⋊C412yes—
262C2×D4C4⋊Dic4711yescharacteristic
263C2×D4(C2×D47)⋊C412yes—
264C2×D4(C2×D47)⋊C412yes—
265C2×D4C2×C4×D4712yes—
266C2×D4C2×D18811yescharacteristic
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—1884no—
274C22×C4C4×Dic4711yescharacteristic
275C22×C4Dic47⋊C412yes—
276C22×C4Dic47⋊C412yes—
277C22×C4C4⋊Dic4711yescharacteristic
278C22×C4C4⋊Dic4711yescharacteristic
279C22×C4(C2×D47)⋊C412yes—
280C22×C4C22⋊Dic4712yes—
281C22×C4C22⋊Dic4711yescharacteristic
282C22×C4C22⋊Dic4711yescharacteristic
283C22×C4C22×Dic4711yescharacteristic
284C22×C4C2×C22⋊D4712yes—
285M16—42no—
286M16—42no—
287M16—42no—
288M16—42no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—42no—
292C2×C8—42no—
293C4⋊C4—1884no—
294C4⋊C4—1884no—
295C22⋊C4—948no—
296C22⋊C4—1884no—
297C42—1884no—
298C42—1884no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—82no—
303C23—81no—
304C23C47⋊#9512yes—
305C23C47⋊#9511yescharacteristic
306C23C47⋊#9511yescharacteristic
307C23C2×C4×Dic4711yescharacteristic
308C23C2×(C2×D47)⋊C412yes—
309Q8—42no—
310Q8—41no—
311Q8—41no—
312D4—42no—
313D4—42no—
314D4—41no—
315D4—42no—
316D4—42no—
317D4—42no—
318D4—41no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—3768no—
337C2×C4—1888no—
338C2×C4C47⋊#9511yescharacteristic
339C2×C4C47⋊#9512yes—
340C2×C4C47⋊#9511yescharacteristic
341C2×C4#1309393912yes—
342C2×C4#1309393912yes—
343C2×C4C2×C22⋊Dic4711yescharacteristic
344C2×C4C2×Dic47⋊C412yes—
345C2×C4C2×C4⋊Dic4711yescharacteristic
346C8—42no—
347C8—42no—
348C22—22no—
349C22—22no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—42no—
354C22—41no—
355C22—41no—
356C22—42no—
357C22—44no—
358C22—82no—
359C22—82no—
360C22#1568889012yes—
361C22C2×C47⋊#9511yescharacteristic
362C4—22no—
363C4—21no—
364C4—22no—
365C4—21no—
366C4—3768no—
367C4#1568889012yes—
368C4C2×#1309393911yescharacteristic
369C4C2×#1309393911yescharacteristic
370C2—21no—
371C2—22no—
372C2—42no—
373C2—42no—
374C2—41no—
375C2—41no—
376C2C2×#1568889011yescharacteristic
subgroupG/H
centerC4C2×#13093939
commutatorC23×C47C2×C42
FrattiniC2×#41C22×D47
FittingC47×#2879C2
socleC2×C47C2×#342
prime radicalC22×#41×C47C22
cosocleC2×#41×C47C23
hypercenter#2879D47
hypocenterC47#12577
LCSG ▷ C23×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ #2879

Complements

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

NH ≅ G/Ntype
2. #20241509C2semidirect
7. #15689097C4semidirect
10. C2×#13093939C4semidirect
12. C22×M16×C47C4semidirect
13. C22×M16×C47C4semidirect
15. C47×#408C4semidirect
28. #13093939C2×C4semidirect
47. C2×M16×C47C2×C4semidirect
49. C22×C8×C47C2×C4semidirect
193. C22×M16Dic47semidirect
194. C22×M16Dic47semidirect
203. #408Dic47semidirect
206. C47#12577semidirect
225. C2×M16C2×Dic47semidirect
227. C22×C8C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#156890972
#202415092
C47×#28792
#1257747