Finite Group Databasebeta

← #26371556

Subgroups — #26371556

Order 12032.

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

normal subgroups only

This group has 512 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#20254047C211yesmaximal, characteristic, complemented
2#20245607C212yesmaximal, complemented
3#20245564C212yesmaximal, complemented
4#20245469C212yesmaximal, complemented
5#20245339C212yesmaximal
6#20245279C214yesmaximal, complemented
7#20245204C211yesmaximal, characteristic, complemented
8C47×#2955C211yesmaximal, characteristic
9#15689735—21no—
10#15689563C2214yes—
11#15689525C2211yescharacteristic, complemented
12#15689503C2211yescharacteristic, complemented
13#15689492—21no—
14#15689381C2212yes—
15#15689364—21no—
16#15689231C2212yescomplemented
17C4×Dic188—21no—
18#15688947C2214yescomplemented
19#15688944C2212yescomplemented
20#15688938C2214yescomplemented
21#15688933C2214yes—
22#15688905C2212yes—
23#15688901C2212yescomplemented
24C47×#575C2211yescharacteristic
25C2×#131×C47—21no—
26C47×#444—21no—
27C47×#444C2212yes—
28C47×#443—21no—
29C47×#443C2212yes—
30C47×#430C2211yescharacteristic
31C47×#421—21no—
32C47×#405C2211yescharacteristic
33C2×Dic188—21no—
34C2×Dic188C2312yes—
35#13094024—21no—
36#13094017—42no—
37#13094016—21no—
38#13094014—42no—
39#13094003—22no—
40C4×Dic94—22no—
41#13093970—24no—
42#13093968—24no—
43#13093964C2314yes—
44#13093958—21no—
45#13093956C2312yes—
46#13093949—22no—
47#13093949—22no—
48C2×#41×C47—21no—
49#137×C47—22no—
50AGL(1,ℤ/8)×C47—22no—
51#135×C47—21no—
52#135×C47C2312yes—
53#131×C47—22no—
54#131×C47—22no—
55#131×C47—41no—
56#131×C47—41no—
57C2×M16×C47—21no—
58C2×M16×C47—41no—
59C2×M16×C47C2311yescharacteristic
60C22×C8×C47—41no—
61C22×C8×C47D411yescharacteristic
62#117×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—41no—
79C22×C4×C47C2×D411yescharacteristic
80C2×Dic94—22no—
81Q16×C47—21no—
82Q16×C47#4112yes—
83SD16×C47—22no—
84SD16×C47—22no—
85D8×C47—21no—
86D8×C47#4112yes—
87M16×C47—22no—
88M16×C47—22no—
89M16×C47—42no—
90M16×C47—41no—
91M16×C47—42no—
92C2×C8×C47—21no—
93C2×C8×C47—21no—
94C2×C8×C47—42no—
95C2×C8×C47—41no—
96C2×C8×C47—42no—
97C2×C8×C47C2×D411yescharacteristic
98C2×C8×C47C2×D411yescharacteristic
99C2×C8×C47C2×D411yescharacteristic
100C2×C8×C47C2411yescharacteristic
101C4⋊C4×C47—21no—
102C22⋊C4×C47—21no—
103C42×C47C2×D411yescharacteristic
104C42×C47C2×D411yescharacteristic
105C22⋊Dic47—42no—
106C4⋊Dic47—22no—
107C4⋊Dic47—42no—
108Dic47⋊C4—42no—
109Dic47⋊C4—42no—
110Dic47⋊C4—42no—
111C4×Dic47—42no—
112Dic188—24no—
113Dic188—41no—
114Dic188—41no—
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—41no—
130C2×C4×C47—42no—
131C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C47C22×D411yescharacteristic
133C2×Dic47—42no—
134C2×Dic47—42no—
135Dic94—42no—
136Dic94—42no—
137C8×C47—21no—
138C8×C47—21no—
139C8×C47—42no—
140C8×C47AGL(1,ℤ/8)11yescharacteristic
141C8×C47AGL(1,ℤ/8)11yescharacteristic
142C8×C47C2×#4112yes—
143#33693—471nomaximal, Sylow
144C22×C47—21no—
145C22×C47—41no—
146C22×C47—42no—
147C22×C47—82no—
148C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
149C4×C47—21no—
150C4×C47—41no—
151C4×C47—41no—
152C4×C47—42no—
153C4×C47#52511yescharacteristic
154C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
155Dic47—82no—
156Dic47—82no—
157#3177—471no—
158#2955D4711yescharacteristic
159#2239—472no—
160#2231—472no—
161#2214—472no—
162#2191—472no—
163#2180—474no—
164#2166—471no—
165C2×C47—21no—
166C2×C47—41no—
167C2×C47—82no—
168C2×C47#310911yescharacteristic
169#575C2×D4711yescharacteristic
170C2×#131—21no—
171#541—941no—
172#510—474no—
173#499—471no—
174#494—471no—
175#491—941no—
176#470—472no—
177#467—941no—
178#444—21no—
179#444C2×D4712yes—
180#443—21no—
181#443C2×D4712yes—
182#441—472no—
183C4×Q16—941no—
184#430C2×D4711yescharacteristic
185#421—21no—
186#405C2×D4711yescharacteristic
187#365—474no—
188#364—472no—
189#362—474no—
190#360—474no—
191#349—472no—
192#347—472no—
193C47#3369311yescharacteristic, complemented, Sylow
194C2×#41—21no—
195#137—22no—
196AGL(1,ℤ/8)—22no—
197#135—21no—
198#135C22×D4712yes—
199C2×Q16—472no—
200C2×Q16—941no—
201#131—22no—
202#131—22no—
203#131—41no—
204#131—41no—
205C2×M16—21no—
206C2×M16—41no—
207C2×M16C22×D4711yescharacteristic
208C22×C8—41no—
209C22×C8C22⋊D4711yescharacteristic
210C4⋊Q8—941no—
211#126—1882no—
212#125—941no—
213#125—1882no—
214C22⋊Q8—942no—
215C4×Q8—942no—
216#117C22⋊D4711yescharacteristic
217Q32—944no—
218SD32—944no—
219M32—474no—
220#108—21no—
221#108C22×D4712yes—
222#107—941no—
223#106—472no—
224#104—22no—
225#104—22no—
226#103—942no—
227#103—942no—
228#97C22⋊D4711yescharacteristic
229#97C22×D4711yescharacteristic
230C4×C8C22⋊D4711yescharacteristic
231C4×C8C22×D4711yescharacteristic
232#41—22no—
233#41—22no—
234#41—41no—
235#41—41no—
236C2×Q8—21no—
237C2×Q8—942no—
238C2×D4—21no—
239C2×D4—41no—
240C22×C4—41no—
241C22×C4C2×C22⋊D4711yescharacteristic
242Q16—21no—
243Q16—944no—
244Q16—1881no—
245Q16—1881no—
246Q16#1062535012yes—
247SD16—22no—
248SD16—22no—
249D8—21no—
250D8#1062535012yes—
251M16—22no—
252M16—22no—
253M16—42no—
254M16—41no—
255M16—42no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—42no—
259C2×C8—41no—
260C2×C8—42no—
261C2×C8C2×C22⋊D4711yescharacteristic
262C2×C8C2×C22⋊D4711yescharacteristic
263C2×C8C2×C22⋊D4711yescharacteristic
264C2×C8C23×D4711yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—942no—
267C4⋊C4—1882no—
268C4⋊C4—1882no—
269C4⋊C4—1882no—
270C4⋊C4—1882no—
271C22⋊C4—21no—
272C22⋊C4—1882no—
273C42—1882no—
274C42C2×C22⋊D4711yescharacteristic
275C42C2×C22⋊D4711yescharacteristic
276C16—944no—
277C23—41no—
278C23#1309404911yescharacteristic
279Q8—22no—
280Q8—41no—
281Q8—1882no—
282Q8—1882no—
283D4—22no—
284D4—41no—
285D4—42no—
286D4—42no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—1882no—
295C2×C4—1882no—
296C2×C4#1309404911yescharacteristic
297C2×C4C22×C22⋊D4711yescharacteristic
298C8—21no—
299C8—21no—
300C8—42no—
301C8#1309405011yescharacteristic
302C8#1309405011yescharacteristic
303C8C2×#1062535012yes—
304C22—21no—
305C22—41no—
306C22—42no—
307C22—82no—
308C22C2×#1309404911yescharacteristic
309C4—21no—
310C4—41no—
311C4—41no—
312C4—42no—
313C4—3762no—
314C4—3762no—
315C4#1568963411yescharacteristic
316C4C2×#1309405011yescharacteristic
317C2—21no—
318C2—41no—
319C2—82no—
320C2#2025347911yescharacteristic
subgroupG/H
centerC2#20253479
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC47×#2955C2
socleC2×C47#3109
prime radicalC47×#575C22
hypercenter#2955D47
hypocenterC47#33693
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2955

Complements

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

NH ≅ G/Ntype
1. #20254047C2semidirect
2. #20245607C2semidirect
3. #20245564C2semidirect
4. #20245469C2semidirect
6. #20245279C2semidirect
7. #20245204C2semidirect
11. #15689525C22semidirect
12. #15689503C22semidirect
16. #15689231C22semidirect
18. #15688947C22semidirect
19. #15688944C22semidirect
20. #15688938C22semidirect
23. #15688901C22semidirect
193. C47#33693semidirect

Maximal subgroups

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

subgroupindex
#202540472
#202456072
#202455642
#202454692
#202453392
#202452792
#202452042
C47×#29552
#3369347