Finite Group Databasebeta

← #26133405

Subgroups — #26133405

Order 12032.

494 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689558C211yesmaximal, characteristic
2C2×#15688930C211yesmaximal, characteristic, complemented
3C2×#15688923C211yesmaximal, characteristic, complemented
4#20244298C211yesmaximal, characteristic, complemented
5#20241496C211yesmaximal, characteristic, complemented
6#20241477C211yesmaximal, characteristic, complemented
7C47×#2951C211yesmaximal, characteristic, complemented
8C22×D376C2211yescharacteristic
9#15689558—22no—
10#15689558C414yescomplemented
11C2×#13093967—21no—
12C2×#13093965—21no—
13C2×#13093964C2211yescharacteristic
14C22×C47⋊C16C2211yescharacteristic
15C2×#13093960C2211yescharacteristic
16#15689127—21no—
17C2×#13093940—21no—
18#15688930—22no—
19#15688923—22no—
20C2×#135×C47C2211yescharacteristic
21C47×#520—21no—
22C47×#440—22no—
23C4×Q16×C47—21no—
24C4×D8×C47—21no—
25C2×#107×C47C2211yescharacteristic
26C47×#417—21no—
27C47×#405C2211yescharacteristic
28C22×D188—21no—
29C2×D376—21no—
30C2×D376—21no—
31C2×D376—21no—
32C2×D376—21no—
33C2×D376—21no—
34C2×D376C2×C412yescomplemented
35C2×#10625329—21no—
36#13093967—24no—
37#13093965—24no—
38#13093964—21no—
39#13093964C2×C412yescomplemented
40C2×C47⋊C16—21no—
41C2×C47⋊C16—21no—
42C2×C47⋊C16—21no—
43C2×C47⋊C16—21no—
44#13093960—21no—
45#13093960—21no—
46#13093944—41no—
47#13093944—41no—
48#13093940—42no—
49C2×#41×C47—21no—
50#135×C47—21no—
51#135×C47—21no—
52#135×C47C2×C412yes—
53#135×C47D412yes—
54C2×Q16×C47D411yescharacteristic
55C2×SD16×C47—21no—
56C2×D8×C47D411yescharacteristic
57C2×M16×C47D411yescharacteristic
58C22×C8×C47C2311yescharacteristic
59C4×Q8×C47—22no—
60C4×D4×C47—22no—
61C4×D4×C47—42no—
62#117×C47—41no—
63#117×C47D411yescharacteristic
64C2×C4⋊C4×C47—21no—
65C2×C4⋊C4×C47—41no—
66#107×C47—21no—
67#107×C47D412yes—
68#103×C47—22no—
69#102×C47—22no—
70#97×C47—21no—
71C4×C8×C47—21no—
72C23×D47—41no—
73#41×C47—22no—
74#41×C47—22no—
75#41×C47—41no—
76#41×C47—41no—
77C2×Q8×C47—21no—
78C2×D4×C47—21no—
79C2×D4×C47—41no—
80C22×C4×C47—42no—
81C22×C4×C47—41no—
82C22×C4×C47—41no—
83C22×C4×C47C2×D411yescharacteristic
84C2×D188—22no—
85C2×D188—41no—
86C2×D188—41no—
87C2×D188—41no—
88C2×D188—41no—
89C2×D188—41no—
90Q16×C47—21no—
91Q16×C47C22⋊C412yes—
92SD16×C47—22no—
93SD16×C47—22no—
94D8×C47—21no—
95D8×C47C22⋊C412yes—
96M16×C47—21no—
97M16×C47—21no—
98C2×C8×C47—21no—
99C2×C8×C47C22×C411yescharacteristic
100C2×C8×C47C2×D411yescharacteristic
101C2×C8×C47C2×D411yescharacteristic
102C2×C8×C47C2×D411yescharacteristic
103C2×C8×C47#4111yescharacteristic
104C2×C8×C47#4111yescharacteristic
105C4⋊C4×C47—22no—
106C4⋊C4×C47—42no—
107C4⋊C4×C47—41no—
108C4⋊C4×C47D811yescharacteristic
109C4⋊C4×C47D811yescharacteristic
110C22⋊C4×C47—21no—
111C22⋊C4×C47—42no—
112C42×C47—21no—
113C42×C47—42no—
114#10625329—41no—
115#10625329—41no—
116C2×C47⋊C8—41no—
117D376—22no—
118D376—22no—
119D376—41no—
120D376—41no—
121C47⋊C16—22no—
122C47⋊C16—41no—
123C23×C47—41no—
124C23×C47AGL(1,ℤ/8)11yescharacteristic
125C22×D47—42no—
126C22×D47—81no—
127C22×D47—81no—
128C22×D47—161no—
129C22×D47—161no—
130Q8×C47—22no—
131Q8×C47—41no—
132D4×C47—22no—
133D4×C47—42no—
134D4×C47—42no—
135D4×C47—41no—
136C2×C4×C47—21no—
137C2×C4×C47—21no—
138C2×C4×C47—41no—
139C2×C4×C47—42no—
140C2×C4×C47—42no—
141C2×C4×C47—42no—
142C2×C4×C47—162no—
143C2×C4×C47—161no—
144C2×C4×C47C4×D411yescharacteristic
145C2×C4×C47#12011yescharacteristic
146C2×C4×C47#12111yescharacteristic
147C2×C4×C47C2×D811yescharacteristic
148C2×C4×C47#13511yescharacteristic
149C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
150D188—42no—
151D188—42no—
152D188—81no—
153D188—81no—
154C8×C47—41no—
155C8×C47C2×C22⋊C411yescharacteristic
156C8×C47C4×D411yescharacteristic
157C8×C47#12111yescharacteristic
158C8×C47#12311yescharacteristic
159C47⋊C8—81no—
160#13022—471nomaximal, Sylow
161C22×C47—21no—
162C22×C47—21no—
163C22×C47—42no—
164C22×C47—81no—
165C22×C47—81no—
166C22×C47#44211yescharacteristic
167C22×C47#44711yescharacteristic
168C22×C47#46311yescharacteristic
169C2×D47—82no—
170C2×D47—162no—
171C2×D47—161no—
172C2×D47—161no—
173C2×D47—162no—
174C4×C47—42no—
175C4×C47—41no—
176C4×C47—41no—
177C4×C47—82no—
178C4×C47#38611yescharacteristic
179C4×C47C4×D811yescharacteristic
180C4×C47#44911yescharacteristic
181C4×C47#45911yescharacteristic
182C2×#509—471no—
183#2951D4711yescharacteristic, complemented
184C2×#359—471no—
185C2×#357—471no—
186#1978—471no—
187#1387—471no—
188#1379—471no—
189C2×C47—21no—
190C2×C47—21no—
191C2×C47—82no—
192C2×C47#187911yescharacteristic
193C2×C47#219211yescharacteristic
194C2×C47#219311yescharacteristic
195D47—162no—
196C2×#135C2×D4711yescharacteristic
197C22×D8—471no—
198#520—21no—
199#509—474no—
200#509—942no—
201C2×SD32—941no—
202C2×D16—941no—
203C2×M32—471no—
204C22×C16—471no—
205#440—22no—
206C4×Q16—21no—
207C4×D8—21no—
208C2×#108—471no—
209C2×#107C2×D4711yescharacteristic
210#418—941no—
211#417—21no—
212C2×#100—941no—
213#405C2×D4711yescharacteristic
214#359—942no—
215#357—942no—
216C47#1302211yescharacteristic, complemented, Sylow
217C2×#41—21no—
218C22×D4—941no—
219#135—21no—
220#135—21no—
221#135C4×D4712yes—
222#135D18812yes—
223C2×Q16C22⋊D4711yescharacteristic
224C2×SD16—21no—
225C2×D8—472no—
226C2×D8—941no—
227C2×D8—941no—
228C2×D8—941no—
229C2×D8—941no—
230C2×D8—941no—
231C2×D8C22⋊D4711yescharacteristic
232C2×M16—941no—
233C2×M16C22⋊D4711yescharacteristic
234C22×C8C22×D4711yescharacteristic
235C4×Q8—22no—
236C4×D4—22no—
237C4×D4—42no—
238#117—41no—
239#117C22⋊D4711yescharacteristic
240C2×C4⋊C4—21no—
241C2×C4⋊C4—41no—
242SD32—944no—
243D16—944no—
244M32—472no—
245M32—941no—
246C2×C16—941no—
247C2×C16—941no—
248C2×C16—941no—
249C2×C16—941no—
250#108—941no—
251#108—941no—
252#107—21no—
253#107C22⋊D4712yes—
254#103—22no—
255#102—22no—
256#102—1881no—
257#102—1881no—
258#100—1882no—
259#97—21no—
260C4×C8—21no—
261C24—1881no—
262#41—22no—
263#41—22no—
264#41—41no—
265#41—41no—
266C2×Q8—21no—
267C2×D4—21no—
268C2×D4—41no—
269C2×D4—942no—
270C2×D4—1881no—
271C2×D4—1881no—
272C2×D4—1881no—
273C2×D4—1881no—
274C2×D4—1881no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4C2×C22⋊D4711yescharacteristic
279Q16—21no—
280Q16(C2×D47)⋊C412yes—
281SD16—22no—
282SD16—22no—
283D8—21no—
284D8—942no—
285D8—942no—
286D8—1881no—
287D8—1881no—
288D8(C2×D47)⋊C412yes—
289M16—21no—
290M16—21no—
291M16—1881no—
292M16—1881no—
293C2×C8—21no—
294C2×C8—1881no—
295C2×C8C2×C4×D4711yescharacteristic
296C2×C8C2×D18811yescharacteristic
297C2×C8#1062535011yescharacteristic
298C2×C8#1062535011yescharacteristic
299C2×C8C2×C22⋊D4711yescharacteristic
300C2×C8C2×C22⋊D4711yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4#1062533411yescharacteristic
305C4⋊C4#1062533411yescharacteristic
306C22⋊C4—21no—
307C22⋊C4—42no—
308C42—21no—
309C42—42no—
310C16—942no—
311C16—1881no—
312C23—41no—
313C23—3761no—
314C23—3761no—
315C23—1882no—
316C23—7521no—
317C23—7521no—
318C23#1309405011yescharacteristic
319Q8—22no—
320Q8—41no—
321D4—22no—
322D4—42no—
323D4—42no—
324D4—41no—
325D4—3761no—
326D4—3761no—
327D4—1882no—
328D4—1882no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—162no—
336C2×C4—161no—
337C2×C4C4×C22⋊D4711yescharacteristic
338C2×C4#1309399311yescharacteristic
339C2×C4#1309399811yescharacteristic
340C2×C4C2×#1062533411yescharacteristic
341C2×C4#1309404411yescharacteristic
342C2×C4#1309404911yescharacteristic
343C8—41no—
344C8—3761no—
345C8C2×(C2×D47)⋊C411yescharacteristic
346C8C4×C22⋊D4711yescharacteristic
347C8#1309399811yescharacteristic
348C8#1309400711yescharacteristic
349C22—21no—
350C22—21no—
351C22—42no—
352C22—81no—
353C22—81no—
354C22—3762no—
355C22—7522no—
356C22—7522no—
357C22—7521no—
358C22—7521no—
359C22#1568923311yescharacteristic
360C22#1568925711yescharacteristic
361C22#1568933811yescharacteristic
362C4—42no—
363C4—41no—
364C4—41no—
365C4—82no—
366C4#1568900211yescharacteristic
367C4C4×#1062533411yescharacteristic
368C4#1568926811yescharacteristic
369C4#1568931811yescharacteristic
370C2—21no—
371C2—21no—
372C2—82no—
373C2—7522no—
374C2#2024373611yescharacteristic
375C2#2024534511yescharacteristic
376C2#2024535211yescharacteristic
subgroupG/H
centerC22#15689257
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC47×#2951C2
socleC22×C47#447
prime radicalC2×#15689558C2
cosocleC22×C8×C47C23
hypercenter#2951D47
hypocenterC47#13022
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ #2951

Complements

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

NH ≅ G/Ntype
2. C2×#15688930C2semidirect
3. C2×#15688923C2semidirect
4. #20244298C2semidirect
5. #20241496C2semidirect
6. #20241477C2semidirect
7. C47×#2951C2semidirect
10. #15689558C4semidirect
34. C2×D376C2×C4semidirect
39. #13093964C2×C4semidirect
183. #2951D47semidirect
216. C47#13022semidirect

Maximal subgroups

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

subgroupindex
C2×#156895582
C2×#156889302
C2×#156889232
#202442982
#202414962
#202414772
C47×#29512
#1302247