Finite Group Databasebeta

← #26371514

Subgroups — #26371514

Order 12032.

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

normal subgroups only

This group has 520 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#20252299C211yesmaximal, characteristic
2#20245605C212yesmaximal, complemented
3#20245556C212yesmaximal, complemented
4#20245466C212yesmaximal, complemented
5#20245338C212yesmaximal
6#20245277C214yesmaximal, complemented
7#20245200C211yesmaximal, characteristic, complemented
8C47×#3185C211yesmaximal, characteristic, complemented
9#15689923C2211yescharacteristic
10#15689889—21no—
11#15689564C2214yes—
12#15689246—21no—
13#15689246C2212yes—
14#15689241—21no—
15#15689241C2212yes—
16#15689172C2211yescharacteristic
17C2×#13093952—21no—
18#15688945C2214yescomplemented
19#15688944C2212yescomplemented
20#15688939C2214yescomplemented
21#15688932C2214yes—
22#15688905C2212yes—
23#15688901C2212yescomplemented
24C47×#533—21no—
25C47×#498C2211yescharacteristic, complemented
26C47×#494C2211yescharacteristic, complemented
27C47×#487—21no—
28C47×#470—21no—
29C47×#470C2212yes—
30C47×#441—21no—
31C47×#441C2212yescomplemented
32C4×M16×C47C2211yescharacteristic
33C2×#10625350—21no—
34#13094057—22no—
35#13094052—22no—
36#13094046—21no—
37#13094046C2312yes—
38#13094032—42no—
39#13094030—22no—
40#13094030—22no—
41C2×#10625329—21no—
42M16×D47—42no—
43#13093970—24no—
44#13093967—24no—
45#13093964C2314yes—
46#13093960—21no—
47#13093960C2312yes—
48#13093952—22no—
49#13093952—22no—
50C2×Q16×C47—21no—
51C2×Q16×C47C2312yes—
52C2×M16×C47D411yescharacteristic
53C2×M16×C47C2311yescharacteristic
54C4⋊Q8×C47—21no—
55C4⋊Q8×C47—21no—
56#125×C47—41no—
57C22⋊Q8×C47—22no—
58C22⋊Q8×C47—41no—
59C4×Q8×C47—22no—
60C4×Q8×C47—41no—
61#117×C47—41no—
62C2×C42×C47D411yescharacteristic
63#106×C47—21no—
64#106×C47C2312yes—
65#103×C47—22no—
66#103×C47—22no—
67#97×C47D411yescharacteristic
68#97×C47C2311yescharacteristic
69C4×C8×C47D411yescharacteristic
70C4×C8×C47C2311yescharacteristic
71C2×Q8×C47—22no—
72C2×Q8×C47—41no—
73C22×C4×C47—21no—
74C22×C4×C47C2×D411yescharacteristic
75C2×C22⋊D47—41no—
76#10625350—22no—
77#10625350—22no—
78#10625350—41no—
79#10625350—41no—
80C2×D188—21no—
81C2×C4×D47—41no—
82C2×Dic94—21no—
83Q16×C47—24no—
84Q16×C47—42no—
85M16×C47—22no—
86M16×C47—22no—
87C2×C8×C47C2×D411yescharacteristic
88C2×C8×C47C2×D411yescharacteristic
89C2×C8×C47C2×D411yescharacteristic
90C2×C8×C47C2411yescharacteristic
91C4⋊C4×C47—22no—
92C4⋊C4×C47—41no—
93C4⋊C4×C47—41no—
94C4⋊C4×C47—42no—
95C4⋊C4×C47—41no—
96C4⋊C4×C47—42no—
97C22⋊C4×C47—42no—
98C42×C47—21no—
99C42×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×C47AGL(1,ℤ/8)11yescharacteristic
117C22×D47—41no—
118Q8×C47—42no—
119Q8×C47—42no—
120Q8×C47—82no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—42no—
126C2×C4×C47—42no—
127C2×C4×C47—42no—
128C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C47C22×D411yescharacteristic
130C22⋊D47—42no—
131C22⋊D47—42no—
132C2×Dic47—41no—
133D188—22no—
134D188—41no—
135C4×D47—42no—
136C4×D47—42no—
137Dic94—22no—
138Dic94—41no—
139C8×C47—21no—
140C8×C47—21no—
141C8×C47AGL(1,ℤ/8)12yes—
142C8×C47C2×#4112yes—
143C47⋊C8—42no—
144#33689—471nomaximal, Sylow
145C22×C47—21no—
146C22×C47—41no—
147C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
148C2×D47—42no—
149C2×D47—161no—
150C4×C47—22no—
151C4×C47—21no—
152C4×C47—41no—
153C4×C47—82no—
154C4×C47—82no—
155C4×C47#52511yescharacteristic
156C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
157Dic47—42no—
158#3185D4711yescharacteristic, complemented
159#2956—471no—
160#2239—472no—
161#2230—472no—
162#2214—472no—
163#2191—472no—
164#2180—474no—
165#2165—471no—
166C2×C47—21no—
167C2×C47—41no—
168C2×C47#310911yescharacteristic
169D47—82no—
170#575—471no—
171#568—941no—
172#533—21no—
173#510—474no—
174#498C2×D4711yescharacteristic, complemented
175#494C2×D4711yescharacteristic, complemented
176#487—21no—
177#470—21no—
178#470C2×D4712yes—
179#444—472no—
180#444—941no—
181#443—472no—
182#443—941no—
183#441—21no—
184#441C2×D4712yescomplemented
185#430—471no—
186C2×#104—941no—
187C4×M16C2×D4711yescharacteristic
188#365—474no—
189#364—472no—
190#362—474no—
191#360—474no—
192#349—472no—
193#347—472no—
194C47#3368911yescharacteristic, complemented, Sylow
195C2×#41—941no—
196#137—942no—
197AGL(1,ℤ/8)—942no—
198#135—472no—
199#135—941no—
200C2×Q16—21no—
201C2×Q16C22×D4712yes—
202#131—942no—
203#131—942no—
204#131—1882no—
205C2×M16—941no—
206C2×M16—1882no—
207C2×M16C22⋊D4711yescharacteristic
208C2×M16C22×D4711yescharacteristic
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211#125—41no—
212C22⋊Q8—22no—
213C22⋊Q8—41no—
214C4×Q8—22no—
215C4×Q8—41no—
216#117—41no—
217C2×C42C22⋊D4711yescharacteristic
218Q32—944no—
219SD32—944no—
220M32—474no—
221#108—472no—
222#108—941no—
223#106—21no—
224#106C22×D4712yes—
225#104—942no—
226#104—942no—
227#103—22no—
228#103—22no—
229#97C22⋊D4711yescharacteristic
230#97C22×D4711yescharacteristic
231C4×C8C22⋊D4711yescharacteristic
232C4×C8C22×D4711yescharacteristic
233#41—942no—
234#41—942no—
235#41—1881no—
236#41—1881no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—941no—
240C2×D4—941no—
241C2×D4—1881no—
242C22×C4—21no—
243C22×C4—1881no—
244C22×C4C2×C22⋊D4711yescharacteristic
245Q16—24no—
246Q16—42no—
247Q16—472no—
248Q16—941no—
249SD16—942no—
250SD16—942no—
251D8—472no—
252D8—941no—
253M16—22no—
254M16—22no—
255M16—942no—
256M16—1882no—
257M16—1881no—
258M16—1882no—
259C2×C8—1882no—
260C2×C8—1881no—
261C2×C8—1882no—
262C2×C8C2×C22⋊D4711yescharacteristic
263C2×C8C2×C22⋊D4711yescharacteristic
264C2×C8C2×C22⋊D4711yescharacteristic
265C2×C8C23×D4711yescharacteristic
266C4⋊C4—22no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C22⋊C4—42no—
273C42—21no—
274C42—42no—
275C42C2×C22⋊D4711yescharacteristic
276C42C2×C22⋊D4711yescharacteristic
277C16—944no—
278C23—1881no—
279C23#1309405011yescharacteristic
280Q8—42no—
281Q8—42no—
282Q8—82no—
283Q8—942no—
284Q8—1881no—
285D4—942no—
286D4—1881no—
287D4—1882no—
288D4—1882no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—1882no—
297C2×C4—1881no—
298C2×C4—1882no—
299C2×C4#1309405011yescharacteristic
300C2×C4C22×C22⋊D4711yescharacteristic
301C8—21no—
302C8—21no—
303C8—1882no—
304C8#1309404912yes—
305C8#41×D4712yes—
306C22—21no—
307C22—41no—
308C22—1882no—
309C22—7521no—
310C22C2×#1309405011yescharacteristic
311C4—22no—
312C4—21no—
313C4—41no—
314C4—82no—
315C4—82no—
316C4—1882no—
317C4#1568963011yescharacteristic
318C4C2×#1309404911yescharacteristic
319C2—21no—
320C2—41no—
321C2—3762no—
322C2#2025347511yescharacteristic
subgroupG/H
centerC2#20253475
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC47×#3185C2
socleC2×C47#3109
prime radical#15689923C22
hypercenter#3185D47
hypocenterC47#33689
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3185

Complements

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

NH ≅ G/Ntype
2. #20245605C2semidirect
3. #20245556C2semidirect
4. #20245466C2semidirect
6. #20245277C2semidirect
7. #20245200C2semidirect
8. C47×#3185C2semidirect
18. #15688945C22semidirect
19. #15688944C22semidirect
20. #15688939C22semidirect
23. #15688901C22semidirect
25. C47×#498C22semidirect
26. C47×#494C22semidirect
31. C47×#441C22semidirect
158. #3185D47semidirect
174. #498C2×D47semidirect
175. #494C2×D47semidirect
184. #441C2×D47semidirect
194. C47#33689semidirect

Maximal subgroups

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

subgroupindex
#202522992
#202456052
#202455562
#202454662
#202453382
#202452772
#202452002
C47×#31852
#3368947