Finite Group Databasebeta

← #26371544

Subgroups — #26371544

Order 12032.

536 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 131 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
1#20252292C211yesmaximal, characteristic, complemented
2#20245600C212yesmaximal, complemented
3#20245561C212yesmaximal, complemented
4#20245461C212yesmaximal, complemented
5#20245333C212yesmaximal, complemented
6#20245284C214yesmaximal, complemented
7#20245204C211yesmaximal, characteristic, complemented
8C47×#3176C211yesmaximal, characteristic, complemented
9#15689923C2211yescharacteristic
10C2×#13094030—21no—
11#15689559C2214yes—
12#15689246—21no—
13#15689246C2212yes—
14#15689241—21no—
15#15689241C2212yes—
16#15689172C2211yescharacteristic
17#15689141—21no—
18#15688945C2214yescomplemented
19#15688944C2212yescomplemented
20#15688936C2214yescomplemented
21#15688932C2214yes—
22#15688905C2212yes—
23#15688901C2212yescomplemented
24C47×#540—21no—
25C47×#499C2211yescharacteristic, complemented
26C47×#493C2211yescharacteristic, complemented
27C47×#490—21no—
28C47×#469C2212yes—
29C47×#466—21no—
30C47×#442C2212yescomplemented
31C4×D8×C47—21no—
32C47×#405C2211yescharacteristic
33C2×#10625350—21no—
34#13094057—22no—
35#13094052—22no—
36#13094046—21no—
37#13094046C2312yes—
38#13094030—22no—
39#13094030—22no—
40#13094030—41no—
41#13094030—41no—
42C2×#10625329—21no—
43C2×#10625324—41no—
44C2×C8×D47—41no—
45#13093968—24no—
46#13093965—24no—
47#13093964C2314yes—
48#13093960—21no—
49#13093960C2312yes—
50#13093952—22no—
51#13093952—22no—
52C2×D8×C47—21no—
53C2×D8×C47C2312yes—
54C2×M16×C47C2311yescharacteristic
55C22×C8×C47D411yescharacteristic
56#127×C47—21no—
57#125×C47—21no—
58#123×C47—42no—
59#121×C47—22no—
60#121×C47—42no—
61C4×D4×C47—22no—
62#117×C47D411yescharacteristic
63#107×C47—21no—
64#106×C47C2312yes—
65#102×C47—22no—
66#102×C47—22no—
67#97×C47D411yescharacteristic
68#97×C47C2311yescharacteristic
69C4×C8×C47D411yescharacteristic
70C4×C8×C47C2311yescharacteristic
71C2×D4×C47—22no—
72C2×D4×C47—42no—
73C2×D4×C47—42no—
74C22×C4×C47—42no—
75C22×C4×C47C2×D411yescharacteristic
76C2×C22⋊D47—41no—
77#10625350—22no—
78#10625350—22no—
79#10625350—41no—
80#10625350—41no—
81C2×D188—21no—
82C2×C4×D47—41no—
83C2×Dic94—21no—
84D8×C47—24no—
85D8×C47—41no—
86D8×C47—41no—
87M16×C47—22no—
88C2×C8×C47—21no—
89C2×C8×C47—21no—
90C2×C8×C47C2×D411yescharacteristic
91C2×C8×C47C2×D411yescharacteristic
92C2×C8×C47C2×D411yescharacteristic
93C2×C8×C47C2411yescharacteristic
94C4⋊C4×C47—21no—
95C4⋊C4×C47—22no—
96C4⋊C4×C47—42no—
97C22⋊C4×C47—21no—
98C22⋊C4×C47—42no—
99C22⋊C4×C47—42no—
100C42×C47C2×D411yescharacteristic
101C42×C47C2×D411yescharacteristic
102#10625329—22no—
103#10625329—41no—
104C2×C47⋊C8—41no—
105Dic188—21no—
106Dic188#4112yes—
107D376—21no—
108D376#4112yes—
109#10625325—22no—
110#10625325—22no—
111#10625324—42no—
112#10625324—42no—
113C8×D47—42no—
114C8×D47—42no—
115C47⋊C16—24no—
116C23×C47—42no—
117C23×C47AGL(1,ℤ/8)11yescharacteristic
118C22×D47—41no—
119D4×C47—42no—
120D4×C47—42no—
121D4×C47—162no—
122D4×C47—162no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—42no—
128C2×C4×C47—162no—
129C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
130C2×C4×C47C22×D411yescharacteristic
131C22⋊D47—42no—
132C22⋊D47—42no—
133C2×Dic47—41no—
134D188—22no—
135D188—41no—
136C4×D47—42no—
137C4×D47—42no—
138Dic94—22no—
139Dic94—41no—
140C8×C47—21no—
141C8×C47—21no—
142C8×C47AGL(1,ℤ/8)11yescharacteristic
143C8×C47AGL(1,ℤ/8)11yescharacteristic
144C8×C47C2×#4112yes—
145C47⋊C8—42no—
146#33692—471nomaximal, Sylow
147C22×C47—21no—
148C22×C47—41no—
149C22×C47—82no—
150C22×C47—82no—
151C22×C47—82no—
152C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
153C2×D47—42no—
154C2×D47—82no—
155C4×C47—21no—
156C4×C47—41no—
157C4×C47—41no—
158C4×C47—82no—
159C4×C47#52511yescharacteristic
160C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
161Dic47—42no—
162#3176D4711yescharacteristic, complemented
163#2955—471no—
164#2238—472no—
165#2231—472no—
166#2213—472no—
167#2190—472no—
168AGL(1,ℤ/16)—474no—
169#2166—471no—
170C2×C47—21no—
171C2×C47—41no—
172C2×C47—162no—
173C2×C47#310911yescharacteristic
174D47—82no—
175#575—471no—
176C2×#131—941no—
177#540—21no—
178#509—474no—
179#499C2×D4711yescharacteristic, complemented
180#493C2×D4711yescharacteristic, complemented
181#490—21no—
182#469C2×D4712yes—
183#466—21no—
184#444—472no—
185#444—941no—
186#443—472no—
187#443—941no—
188#442C2×D4712yescomplemented
189C4×D8—21no—
190#430—471no—
191#421—941no—
192#405C2×D4711yescharacteristic
193#365—474no—
194#364—472no—
195#361—474no—
196#360—474no—
197#349—472no—
198#347—472no—
199C47#3369211yescharacteristic, complemented, Sylow
200C2×#41—941no—
201#137—942no—
202AGL(1,ℤ/8)—942no—
203#135—472no—
204#135—941no—
205C2×D8—21no—
206C2×D8C22×D4712yes—
207#131—942no—
208#131—942no—
209#131—1881no—
210#131—1881no—
211C2×M16—941no—
212C2×M16—1881no—
213C2×M16C22×D4711yescharacteristic
214C22×C8—1881no—
215C22×C8C22⋊D4711yescharacteristic
216#127—21no—
217#125—21no—
218#123—42no—
219#121—22no—
220#121—42no—
221C4×D4—22no—
222#117C22⋊D4711yescharacteristic
223SD32—944no—
224D16—944no—
225M32—474no—
226#108—472no—
227#108—941no—
228#107—21no—
229#106C22×D4712yes—
230#104—942no—
231#104—942no—
232#102—22no—
233#102—22no—
234#97C22⋊D4711yescharacteristic
235#97C22×D4711yescharacteristic
236C4×C8C22⋊D4711yescharacteristic
237C4×C8C22×D4711yescharacteristic
238#41—942no—
239#41—942no—
240#41—1881no—
241#41—1881no—
242C2×Q8—941no—
243C2×D4—22no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—941no—
247C2×D4—1881no—
248C22×C4—42no—
249C22×C4—1881no—
250C22×C4C2×C22⋊D4711yescharacteristic
251Q16—472no—
252Q16—941no—
253SD16—942no—
254SD16—942no—
255D8—24no—
256D8—41no—
257D8—41no—
258D8—472no—
259D8—941no—
260M16—22no—
261M16—942no—
262M16—1882no—
263M16—1881no—
264M16—1882no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—1882no—
268C2×C8—1881no—
269C2×C8—1882no—
270C2×C8C2×C22⋊D4711yescharacteristic
271C2×C8C2×C22⋊D4711yescharacteristic
272C2×C8C2×C22⋊D4711yescharacteristic
273C2×C8C23×D4711yescharacteristic
274C4⋊C4—21no—
275C4⋊C4—22no—
276C4⋊C4—42no—
277C22⋊C4—21no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C42C2×C22⋊D4711yescharacteristic
281C42C2×C22⋊D4711yescharacteristic
282C16—944no—
283C23—42no—
284C23—1881no—
285C23#1309405011yescharacteristic
286Q8—942no—
287Q8—1881no—
288D4—42no—
289D4—42no—
290D4—162no—
291D4—162no—
292D4—942no—
293D4—1881no—
294D4—1882no—
295D4—1882no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—42no—
301C2×C4—162no—
302C2×C4—1882no—
303C2×C4—1881no—
304C2×C4—1882no—
305C2×C4#1309405011yescharacteristic
306C2×C4C22×C22⋊D4711yescharacteristic
307C8—21no—
308C8—21no—
309C8—1882no—
310C8#1309404911yescharacteristic
311C8#1309404911yescharacteristic
312C8#41×D4712yes—
313C22—21no—
314C22—41no—
315C22—82no—
316C22—82no—
317C22—82no—
318C22—3762no—
319C22—1882no—
320C22C2×#1309405011yescharacteristic
321C4—21no—
322C4—41no—
323C4—41no—
324C4—82no—
325C4—1882no—
326C4#1568963011yescharacteristic
327C4C2×#1309404911yescharacteristic
328C2—21no—
329C2—41no—
330C2—162no—
331C2—3762no—
332C2#2025347511yescharacteristic
subgroupG/H
centerC2#20253475
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC47×#3176C2
socleC2×C47#3109
hypercenter#3176D47
hypocenterC47#33692
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #3176

Complements

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

NH ≅ G/Ntype
1. #20252292C2semidirect
2. #20245600C2semidirect
3. #20245561C2semidirect
4. #20245461C2semidirect
5. #20245333C2semidirect
6. #20245284C2semidirect
7. #20245204C2semidirect
8. C47×#3176C2semidirect
18. #15688945C22semidirect
19. #15688944C22semidirect
20. #15688936C22semidirect
23. #15688901C22semidirect
25. C47×#499C22semidirect
26. C47×#493C22semidirect
30. C47×#442C22semidirect
162. #3176D47semidirect
179. #499C2×D47semidirect
180. #493C2×D47semidirect
188. #442C2×D47semidirect
199. C47#33692semidirect

Maximal subgroups

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

subgroupindex
#202522922
#202456002
#202455612
#202454612
#202453332
#202452842
#202452042
C47×#31762
#3369247