Finite Group Databasebeta

← #26371548

Subgroups — #26371548

Order 12032.

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

normal subgroups only

This group has 536 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#20254038C211yesmaximal, characteristic, complemented
2#20245602C212yesmaximal, complemented
3#20245564C212yesmaximal, complemented
4#20245464C212yesmaximal, complemented
5#20245334C212yesmaximal, complemented
6#20245286C214yesmaximal, complemented
7#20245204C211yesmaximal, characteristic, complemented
8C47×#2955C211yesmaximal, characteristic, complemented
9#15689728—21no—
10#15689558C2214yes—
11#15689525C2211yescharacteristic, complemented
12#15689500C2211yescharacteristic, complemented
13#15689488—21no—
14#15689376C2212yes—
15#15689359—21no—
16#15689236C2212yescomplemented
17C4×D376—21no—
18#15688947C2214yescomplemented
19#15688944C2212yescomplemented
20#15688935C2214yescomplemented
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×D376—21no—
34C2×D376C2312yes—
35#13094021—21no—
36#13094016—21no—
37#13094009—42no—
38#13093998—22no—
39#13093997—42no—
40C4×D188—22no—
41#13093967—24no—
42#13093965—24no—
43#13093964C2314yes—
44#13093958—21no—
45#13093956C2312yes—
46#13093946—22no—
47#13093946—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×C22⋊D47—42no—
81C2×D188—22no—
82C2×D188—42no—
83C2×C4×D47—42no—
84Q16×C47—21no—
85Q16×C47#4112yes—
86SD16×C47—22no—
87SD16×C47—22no—
88D8×C47—21no—
89D8×C47#4112yes—
90M16×C47—22no—
91M16×C47—22no—
92M16×C47—42no—
93M16×C47—41no—
94M16×C47—42no—
95C2×C8×C47—21no—
96C2×C8×C47—21no—
97C2×C8×C47—42no—
98C2×C8×C47—41no—
99C2×C8×C47—42no—
100C2×C8×C47C2×D411yescharacteristic
101C2×C8×C47C2×D411yescharacteristic
102C2×C8×C47C2×D411yescharacteristic
103C2×C8×C47C2411yescharacteristic
104C4⋊C4×C47—21no—
105C22⋊C4×C47—21no—
106C42×C47C2×D411yescharacteristic
107C42×C47C2×D411yescharacteristic
108(C2×D47)⋊C4—42no—
109(C2×D47)⋊C4—42no—
110C4⋊Dic47—22no—
111Dic47⋊C4—42no—
112D376—24no—
113D376—41no—
114D376—41no—
115C47⋊C16—24no—
116C23×C47—41no—
117C23×C47AGL(1,ℤ/8)11yescharacteristic
118C22×D47—42no—
119Q8×C47—22no—
120Q8×C47—41no—
121D4×C47—22no—
122D4×C47—41no—
123D4×C47—42no—
124D4×C47—42no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—42no—
130C2×C4×C47—41no—
131C2×C4×C47—42no—
132C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C47C22×D411yescharacteristic
134C22⋊D47—162no—
135C2×Dic47—42no—
136D188—42no—
137D188—42no—
138D188—162no—
139C4×D47—162no—
140C8×C47—21no—
141C8×C47—21no—
142C8×C47—42no—
143C8×C47AGL(1,ℤ/8)11yescharacteristic
144C8×C47AGL(1,ℤ/8)11yescharacteristic
145C8×C47C2×#4112yes—
146#33692—471nomaximal, Sylow
147C22×C47—21no—
148C22×C47—41no—
149C22×C47—42no—
150C22×C47—82no—
151C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
152C2×D47—82no—
153C2×D47—82no—
154C2×D47—82no—
155C4×C47—21no—
156C4×C47—41no—
157C4×C47—41no—
158C4×C47—42no—
159C4×C47#52511yescharacteristic
160C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
161Dic47—82no—
162#3176—471no—
163#2955D4711yescharacteristic, complemented
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—82no—
173C2×C47#310911yescharacteristic
174D47—162no—
175#575C2×D4711yescharacteristic
176C2×#131—21no—
177#540—941no—
178#509—474no—
179#499—471no—
180#493—471no—
181#490—941no—
182#469—472no—
183#466—941no—
184#444—21no—
185#444C2×D4712yes—
186#443—21no—
187#443C2×D4712yes—
188#442—472no—
189C4×D8—941no—
190#430C2×D4711yescharacteristic
191#421—21no—
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—21no—
201#137—22no—
202AGL(1,ℤ/8)—22no—
203#135—21no—
204#135C22×D4712yes—
205C2×D8—472no—
206C2×D8—941no—
207#131—22no—
208#131—22no—
209#131—41no—
210#131—41no—
211C2×M16—21no—
212C2×M16—41no—
213C2×M16C22×D4711yescharacteristic
214C22×C8—41no—
215C22×C8C22⋊D4711yescharacteristic
216#127—941no—
217#125—941no—
218#123—1882no—
219#121—942no—
220#121—1882no—
221C4×D4—942no—
222#117C22⋊D4711yescharacteristic
223SD32—944no—
224D16—944no—
225M32—474no—
226#108—21no—
227#108C22×D4712yes—
228#107—941no—
229#106—472no—
230#104—22no—
231#104—22no—
232#102—942no—
233#102—942no—
234#97C22⋊D4711yescharacteristic
235#97C22×D4711yescharacteristic
236C4×C8C22⋊D4711yescharacteristic
237C4×C8C22×D4711yescharacteristic
238#41—22no—
239#41—22no—
240#41—41no—
241#41—41no—
242C2×Q8—21no—
243C2×D4—21no—
244C2×D4—41no—
245C2×D4—942no—
246C2×D4—1882no—
247C2×D4—1882no—
248C22×C4—41no—
249C22×C4—1882no—
250C22×C4C2×C22⋊D4711yescharacteristic
251Q16—21no—
252Q16#1062535012yes—
253SD16—22no—
254SD16—22no—
255D8—21no—
256D8—944no—
257D8—1881no—
258D8—1881no—
259D8#1062535012yes—
260M16—22no—
261M16—22no—
262M16—42no—
263M16—41no—
264M16—42no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—42no—
268C2×C8—41no—
269C2×C8—42no—
270C2×C8C2×C22⋊D4711yescharacteristic
271C2×C8C2×C22⋊D4711yescharacteristic
272C2×C8C2×C22⋊D4711yescharacteristic
273C2×C8C23×D4711yescharacteristic
274C4⋊C4—21no—
275C4⋊C4—942no—
276C4⋊C4—1882no—
277C22⋊C4—21no—
278C22⋊C4—1882no—
279C22⋊C4—1882no—
280C42C2×C22⋊D4711yescharacteristic
281C42C2×C22⋊D4711yescharacteristic
282C16—944no—
283C23—41no—
284C23—1882no—
285C23#1309404911yescharacteristic
286Q8—22no—
287Q8—41no—
288D4—22no—
289D4—41no—
290D4—42no—
291D4—42no—
292D4—1882no—
293D4—1882no—
294D4—7522no—
295D4—7522no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—1882no—
304C2×C4—7522no—
305C2×C4#1309404911yescharacteristic
306C2×C4C22×C22⋊D4711yescharacteristic
307C8—21no—
308C8—21no—
309C8—42no—
310C8#1309405011yescharacteristic
311C8#1309405011yescharacteristic
312C8C2×#1062535012yes—
313C22—21no—
314C22—41no—
315C22—42no—
316C22—82no—
317C22—3762no—
318C22—3762no—
319C22—3762no—
320C22C2×#1309404911yescharacteristic
321C4—21no—
322C4—41no—
323C4—41no—
324C4—42no—
325C4—3762no—
326C4#1568963411yescharacteristic
327C4C2×#1309405011yescharacteristic
328C2—21no—
329C2—41no—
330C2—82no—
331C2—7522no—
332C2#2025347911yescharacteristic
subgroupG/H
centerC2#20253479
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC47×#2955C2
socleC2×C47#3109
hypercenter#2955D47
hypocenterC47#33692
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. #20254038C2semidirect
2. #20245602C2semidirect
3. #20245564C2semidirect
4. #20245464C2semidirect
5. #20245334C2semidirect
6. #20245286C2semidirect
7. #20245204C2semidirect
8. C47×#2955C2semidirect
11. #15689525C22semidirect
12. #15689500C22semidirect
16. #15689236C22semidirect
18. #15688947C22semidirect
19. #15688944C22semidirect
20. #15688935C22semidirect
23. #15688901C22semidirect
163. #2955D47semidirect
199. C47#33692semidirect

Maximal subgroups

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

subgroupindex
#202540382
#202456022
#202455642
#202454642
#202453342
#202452862
#202452042
C47×#29552
#3369247