Finite Group Databasebeta

← #26371518

Subgroups — #26371518

Order 12032.

520 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 135 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#20254133C211yesmaximal, characteristic, complemented
2#20245607C212yesmaximal, complemented
3#20245559C212yesmaximal, complemented
4#20245469C212yesmaximal, complemented
5#20245339C212yesmaximal
6#20245279C214yesmaximal, complemented
7#20245200C211yesmaximal, characteristic, complemented
8C47×#2956C211yesmaximal, characteristic
9#15689680—21no—
10#15689563C2214yes—
11#15689521C2211yescharacteristic, complemented
12#15689503C2211yescharacteristic, complemented
13#15689474—21no—
14#15689381—21no—
15#15689381C2212yes—
16#15689231—21no—
17#15689231C2212yescomplemented
18#15688947C2214yescomplemented
19#15688944C2212yescomplemented
20#15688938C2214yescomplemented
21#15688933C2214yes—
22#15688905C2212yes—
23#15688901C2212yescomplemented
24C47×#575C2211yescharacteristic
25C47×#568—21no—
26C47×#444—21no—
27C47×#444C2212yes—
28C47×#443—21no—
29C47×#443C2212yes—
30C47×#430C2211yescharacteristic
31C2×#104×C47—21no—
32C4×M16×C47C2211yescharacteristic
33C2×Dic188—21no—
34C2×Dic188C2312yes—
35#13094024—21no—
36#13094024—21no—
37#13094016—41no—
38#13094003—22no—
39#13094003—41no—
40C4×Dic94—22no—
41C4×Dic94—41no—
42#13093983—41no—
43#13093970—24no—
44#13093968—24no—
45#13093964C2314yes—
46#13093956—21no—
47#13093956C2312yes—
48#13093949—22no—
49#13093949—22no—
50C2×#41×C47—21no—
51#137×C47—22no—
52AGL(1,ℤ/8)×C47—22no—
53#135×C47—21no—
54#135×C47C2312yes—
55#131×C47—22no—
56#131×C47—22no—
57#131×C47—42no—
58C2×M16×C47—21no—
59C2×M16×C47—42no—
60C2×M16×C47D411yescharacteristic
61C2×M16×C47C2311yescharacteristic
62C2×C42×C47D411yescharacteristic
63#108×C47—21no—
64#108×C47C2312yes—
65#104×C47—22no—
66#104×C47—22no—
67#97×C47D411yescharacteristic
68#97×C47C2311yescharacteristic
69C4×C8×C47D411yescharacteristic
70C4×C8×C47C2311yescharacteristic
71#41×C47—22no—
72#41×C47—22no—
73#41×C47—41no—
74#41×C47—41no—
75C2×Q8×C47—21no—
76C2×D4×C47—21no—
77C2×D4×C47—41no—
78C22×C4×C47—21no—
79C22×C4×C47—41no—
80C22×C4×C47C2×D411yescharacteristic
81C2×Dic94—22no—
82C2×Dic94—41no—
83Q16×C47—21no—
84Q16×C47#4112yes—
85SD16×C47—22no—
86SD16×C47—22no—
87D8×C47—21no—
88D8×C47#4112yes—
89M16×C47—22no—
90M16×C47—22no—
91M16×C47—22no—
92M16×C47—42no—
93M16×C47—41no—
94M16×C47—42no—
95C2×C8×C47—42no—
96C2×C8×C47—41no—
97C2×C8×C47—42no—
98C2×C8×C47C2×D411yescharacteristic
99C2×C8×C47C2×D411yescharacteristic
100C2×C8×C47C2×D411yescharacteristic
101C2×C8×C47C2411yescharacteristic
102C42×C47—21no—
103C42×C47C2×D411yescharacteristic
104C42×C47C2×D411yescharacteristic
105C22⋊Dic47—42no—
106C4⋊Dic47—22no—
107C4⋊Dic47—41no—
108C4⋊Dic47—41no—
109C4⋊Dic47—41no—
110Dic47⋊C4—42no—
111Dic47⋊C4—42no—
112C4×Dic47—42no—
113Dic188—24no—
114Dic188—42no—
115C47⋊C16—24no—
116C23×C47—41no—
117C23×C47AGL(1,ℤ/8)11yescharacteristic
118Q8×C47—22no—
119Q8×C47—41no—
120D4×C47—22no—
121D4×C47—41no—
122D4×C47—42no—
123D4×C47—42no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—42no—
129C2×C4×C47—42no—
130C2×C4×C47—41no—
131C2×C4×C47—42no—
132C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C47C22×D411yescharacteristic
134C2×Dic47—42no—
135C2×Dic47—42no—
136Dic94—42no—
137Dic94—42no—
138Dic94—82no—
139C8×C47—21no—
140C8×C47—21no—
141C8×C47—42no—
142C8×C47AGL(1,ℤ/8)12yes—
143C8×C47C2×#4112yes—
144#33689—471nomaximal, Sylow
145C22×C47—21no—
146C22×C47—41no—
147C22×C47—42no—
148C22×C47—161no—
149C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
150C4×C47—22no—
151C4×C47—21no—
152C4×C47—41no—
153C4×C47—42no—
154C4×C47#52511yescharacteristic
155C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
156Dic47—82no—
157Dic47—82no—
158#3185—471no—
159#2956D4711yescharacteristic
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—82no—
169C2×C47#310911yescharacteristic
170#575C2×D4711yescharacteristic
171#568—21no—
172#533—941no—
173#510—474no—
174#498—471no—
175#494—471no—
176#487—941no—
177#470—472no—
178#470—941no—
179#444—21no—
180#444C2×D4712yes—
181#443—21no—
182#443C2×D4712yes—
183#441—472no—
184#441—941no—
185#430C2×D4711yescharacteristic
186C2×#104—21no—
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—21no—
196#137—22no—
197AGL(1,ℤ/8)—22no—
198#135—21no—
199#135C22×D4712yes—
200C2×Q16—472no—
201C2×Q16—941no—
202#131—22no—
203#131—22no—
204#131—42no—
205C2×M16—21no—
206C2×M16—42no—
207C2×M16C22⋊D4711yescharacteristic
208C2×M16C22×D4711yescharacteristic
209C4⋊Q8—941no—
210C4⋊Q8—941no—
211#125—1881no—
212C22⋊Q8—942no—
213C22⋊Q8—1881no—
214C4×Q8—942no—
215C4×Q8—1881no—
216#117—1881no—
217C2×C42C22⋊D4711yescharacteristic
218Q32—944no—
219SD32—944no—
220M32—474no—
221#108—21no—
222#108C22×D4712yes—
223#106—472no—
224#106—941no—
225#104—22no—
226#104—22no—
227#103—942no—
228#103—942no—
229#97C22⋊D4711yescharacteristic
230#97C22×D4711yescharacteristic
231C4×C8C22⋊D4711yescharacteristic
232C4×C8C22×D4711yescharacteristic
233#41—22no—
234#41—22no—
235#41—41no—
236#41—41no—
237C2×Q8—21no—
238C2×Q8—942no—
239C2×Q8—1881no—
240C2×D4—21no—
241C2×D4—41no—
242C22×C4—21no—
243C22×C4—41no—
244C22×C4C2×C22⋊D4711yescharacteristic
245Q16—21no—
246Q16—944no—
247Q16—1882no—
248Q16#1062535012yes—
249SD16—22no—
250SD16—22no—
251D8—21no—
252D8#1062535012yes—
253M16—22no—
254M16—22no—
255M16—22no—
256M16—42no—
257M16—41no—
258M16—42no—
259C2×C8—42no—
260C2×C8—41no—
261C2×C8—42no—
262C2×C8C2×C22⋊D4711yescharacteristic
263C2×C8C2×C22⋊D4711yescharacteristic
264C2×C8C2×C22⋊D4711yescharacteristic
265C2×C8C23×D4711yescharacteristic
266C4⋊C4—942no—
267C4⋊C4—1882no—
268C4⋊C4—1881no—
269C4⋊C4—1881no—
270C4⋊C4—1881no—
271C4⋊C4—1882no—
272C22⋊C4—1882no—
273C42—21no—
274C42—1882no—
275C42C2×C22⋊D4711yescharacteristic
276C42C2×C22⋊D4711yescharacteristic
277C16—944no—
278C23—41no—
279C23#1309404911yescharacteristic
280Q8—22no—
281Q8—41no—
282Q8—3762no—
283Q8—1882no—
284Q8—1882no—
285D4—22no—
286D4—41no—
287D4—42no—
288D4—42no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—1882no—
298C2×C4—1882no—
299C2×C4#1309404911yescharacteristic
300C2×C4C22×C22⋊D4711yescharacteristic
301C8—21no—
302C8—21no—
303C8—42no—
304C8#1309405012yes—
305C8C2×#1062535012yes—
306C22—21no—
307C22—41no—
308C22—42no—
309C22—161no—
310C22C2×#1309404911yescharacteristic
311C4—22no—
312C4—21no—
313C4—41no—
314C4—42no—
315C4—3762no—
316C4—3762no—
317C4#1568963411yescharacteristic
318C4C2×#1309405011yescharacteristic
319C2—21no—
320C2—41no—
321C2—82no—
322C2#2025347911yescharacteristic
subgroupG/H
centerC2#20253479
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC47×#2956C2
socleC2×C47#3109
prime radicalC47×#575C22
hypercenter#2956D47
hypocenterC47#33689
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #2956

Complements

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

NH ≅ G/Ntype
1. #20254133C2semidirect
2. #20245607C2semidirect
3. #20245559C2semidirect
4. #20245469C2semidirect
6. #20245279C2semidirect
7. #20245200C2semidirect
11. #15689521C22semidirect
12. #15689503C22semidirect
17. #15689231C22semidirect
18. #15688947C22semidirect
19. #15688944C22semidirect
20. #15688938C22semidirect
23. #15688901C22semidirect
194. C47#33689semidirect

Maximal subgroups

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

subgroupindex
#202541332
#202456072
#202455592
#202454692
#202453392
#202452792
#202452002
C47×#29562
#3368947