Finite Group Databasebeta

← #26133243

Subgroups — #26133243

Order 12032.

512 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×#15689393C211yesmaximal, characteristic
2#20245103C211yesmaximal, characteristic, complemented
3C2×#15688923C211yesmaximal, characteristic, complemented
4C2×#15688907C211yesmaximal, characteristic, complemented
5#20241496C211yesmaximal, characteristic, complemented
6#20241483C211yesmaximal, characteristic, complemented
7C47×#2913C211yesmaximal, characteristic, complemented
8C2×#13094049—21no—
9C22×D376C2211yescharacteristic
10C2×#13093964C2211yescharacteristic
11C22×C47⋊C16C2211yescharacteristic
12#15689393—22no—
13#15689393C414yescomplemented
14C2×#13093960C2211yescharacteristic
15C2×#13093940—21no—
16#15688923—22no—
17#15688923—21no—
18#15688923—21no—
19#15688907—22no—
20C2×#131×C47C2211yescharacteristic
21C47×#520—21no—
22C47×#428—21no—
23C47×#428—22no—
24C47×#427—21no—
25C47×#427C2211yescharacteristic
26C2×#107×C47—21no—
27C2×#107×C47C2211yescharacteristic
28C22×D188—21no—
29#13094049—24no—
30#13094049—42no—
31C2×#10625336—41no—
32C2×D376—21no—
33C2×D376—21no—
34C2×D376—21no—
35C2×D376—21no—
36C2×D376—21no—
37C2×D376C2×C412yescomplemented
38C2×#10625334—41no—
39C2×#10625329—21no—
40#13093964—21no—
41#13093964—21no—
42C2×C47⋊C16—21no—
43C2×C47⋊C16—21no—
44C2×C47⋊C16—21no—
45C2×C47⋊C16—21no—
46#13093960—21no—
47#13093960C2×C412yescomplemented
48#13093940—24no—
49C2×#41×C47D411yescharacteristic
50#131×C47—21no—
51#131×C47—21no—
52#131×C47C2×C412yes—
53#131×C47D412yes—
54C2×M16×C47—21no—
55C2×M16×C47D411yescharacteristic
56C22×C8×C47—21no—
57C22×C8×C47C2311yescharacteristic
58C4×Q8×C47—41no—
59C4×D4×C47—42no—
60C4×D4×C47—41no—
61#117×C47—22no—
62#117×C47—21no—
63C2×C4⋊C4×C47—22no—
64C2×C4⋊C4×C47—21no—
65#107×C47—22no—
66#107×C47—21no—
67#107×C47D411yescharacteristic
68#107×C47D412yes—
69#107×C47D411yescharacteristic
70#106×C47—22no—
71#106×C47—21no—
72C23×D47—41no—
73#41×C47—21no—
74#41×C47—21no—
75#41×C47C22⋊C412yes—
76#41×C47SD1612yes—
77C2×Q8×C47D811yescharacteristic
78C2×D4×C47—21no—
79C2×D4×C47D811yescharacteristic
80C22×C4×C47—21no—
81C22×C4×C47—42no—
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—
90M16×C47—21no—
91M16×C47—22no—
92M16×C47—22no—
93M16×C47C22⋊C412yes—
94C2×C8×C47—21no—
95C2×C8×C47—22no—
96C2×C8×C47—22no—
97C2×C8×C47C22×C411yescharacteristic
98C2×C8×C47C2×D411yescharacteristic
99C2×C8×C47C2×D411yescharacteristic
100C2×C8×C47C2×D411yescharacteristic
101C2×C8×C47#4111yescharacteristic
102C2×C8×C47#4111yescharacteristic
103C4⋊C4×C47—21no—
104C4⋊C4×C47—22no—
105C4⋊C4×C47—21no—
106C4⋊C4×C47—42no—
107C4⋊C4×C47—41no—
108C22⋊C4×C47—42no—
109C22⋊C4×C47—41no—
110C42×C47—42no—
111C42×C47—41no—
112#10625336—44no—
113#10625334—44no—
114#10625329—22no—
115#10625329—41no—
116C2×C47⋊C8—41no—
117D376—22no—
118D376—22no—
119D376—41no—
120D376—41no—
121C47⋊C16—41no—
122C47⋊C16—41no—
123C23×C47—21no—
124C23×C47AGL(1,ℤ/8)11yescharacteristic
125C22×D47—42no—
126C22×D47—81no—
127C22×D47—81no—
128C22×D47—161no—
129C22×D47—161no—
130Q8×C47—21no—
131Q8×C47#10212yes—
132D4×C47—22no—
133D4×C47—22no—
134D4×C47—21no—
135D4×C47#10212yes—
136C2×C4×C47—21no—
137C2×C4×C47—22no—
138C2×C4×C47—22no—
139C2×C4×C47—41no—
140C2×C4×C47—42no—
141C2×C4×C47—41no—
142C2×C4×C47—161no—
143C2×C4×C47—162no—
144C2×C4×C47C2×C22⋊C411yescharacteristic
145C2×C4×C47#12011yescharacteristic
146C2×C4×C47#12311yescharacteristic
147C2×C4×C47C2×D811yescharacteristic
148C2×C4×C47C2×SD1611yescharacteristic
149C2×C4×C47#13711yescharacteristic
150D188—42no—
151D188—42no—
152D188—81no—
153D188—81no—
154C8×C47—22no—
155C8×C47C4×D411yescharacteristic
156C8×C47C4×D411yescharacteristic
157C8×C47#12111yescharacteristic
158C8×C47#12111yescharacteristic
159C47⋊C8—42no—
160#12996—471nomaximal, Sylow
161C22×C47—21no—
162C22×C47—21no—
163C22×C47—22no—
164C22×C47—41no—
165C22×C47—41no—
166C22×C47#41711yescharacteristic
167C22×C47#44711yescharacteristic
168C22×C47#48111yescharacteristic
169C2×D47—82no—
170C2×D47—162no—
171C2×D47—161no—
172C2×D47—161no—
173C2×D47—162no—
174C4×C47—22no—
175C4×C47—82no—
176C4×C47—81no—
177C4×C47—81no—
178C4×C47#38611yescharacteristic
179C4×C47C2×#10211yescharacteristic
180C4×C47#44811yescharacteristic
181C4×C47#48011yescharacteristic
182C2×#472—471no—
183#2913D4711yescharacteristic, complemented
184#2140—471no—
185C2×#357—471no—
186C2×#350—471no—
187#1387—471no—
188#1382—471no—
189C2×C47—21no—
190C2×C47—21no—
191C2×C47—42no—
192C2×C47#187411yescharacteristic
193C2×C47#219211yescharacteristic
194C2×C47#219311yescharacteristic
195D47—162no—
196C2×AGL(1,ℤ/8)—941no—
197C22×D8—471no—
198C2×#131C2×D4711yescharacteristic
199#520—21no—
200C2×M32—471no—
201C22×C16—471no—
202#472—474no—
203#472—942no—
204C2×#108—471no—
205#428—21no—
206#428—22no—
207#427—21no—
208#427C2×D4711yescharacteristic
209C2×#107—21no—
210C2×#107C2×D4711yescharacteristic
211C2×#100—941no—
212#357—942no—
213#357—941no—
214#357—941no—
215#350—942no—
216C47#1299611yescharacteristic, complemented, Sylow
217C2×#41C22⋊D4711yescharacteristic
218C22×D4—941no—
219AGL(1,ℤ/8)—944no—
220AGL(1,ℤ/8)—1882no—
221C2×SD16—1881no—
222C2×D8—472no—
223C2×D8—941no—
224C2×D8—941no—
225C2×D8—941no—
226C2×D8—941no—
227C2×D8—941no—
228C2×D8—1881no—
229#131—21no—
230#131—21no—
231#131C4×D4712yes—
232#131D18812yes—
233C2×M16—21no—
234C2×M16—941no—
235C2×M16C22⋊D4711yescharacteristic
236C22×C8—21no—
237C22×C8C22×D4711yescharacteristic
238C4×Q8—41no—
239C4×D4—42no—
240C4×D4—41no—
241#117—22no—
242#117—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245M32—941no—
246M32—941no—
247C2×C16—941no—
248C2×C16—941no—
249C2×C16—941no—
250C2×C16—941no—
251#108—472no—
252#108—941no—
253#107—22no—
254#107—21no—
255#107C22⋊D4711yescharacteristic
256#107C22⋊D4712yes—
257#107C22⋊D4711yescharacteristic
258#106—22no—
259#106—21no—
260#100—944no—
261C24—1881no—
262#41—21no—
263#41—21no—
264#41(C2×D47)⋊C412yes—
265#41#1062533612yes—
266C2×Q8#1062533411yescharacteristic
267C2×D4—21no—
268C2×D4—942no—
269C2×D4—1881no—
270C2×D4—1881no—
271C2×D4—1881no—
272C2×D4—1881no—
273C2×D4—1881no—
274C2×D4#1062533411yescharacteristic
275C22×C4—21no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4C2×C22⋊D4711yescharacteristic
279SD16—1884no—
280D8—942no—
281D8—942no—
282D8—1884no—
283D8—1881no—
284D8—1881no—
285M16—21no—
286M16—22no—
287M16—22no—
288M16—942no—
289M16—1881no—
290M16(C2×D47)⋊C412yes—
291C2×C8—21no—
292C2×C8—22no—
293C2×C8—22no—
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—21no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C22⋊C4—42no—
307C22⋊C4—41no—
308C42—42no—
309C42—41no—
310C16—1881no—
311C16—1881no—
312C23—21no—
313C23—3761no—
314C23—3761no—
315C23—1882no—
316C23—7521no—
317C23—7521no—
318C23#1309405011yescharacteristic
319Q8—21no—
320Q8#1309394412yes—
321D4—22no—
322D4—22no—
323D4—21no—
324D4—3761no—
325D4—3761no—
326D4—1882no—
327D4—1882no—
328D4#1309394412yes—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—161no—
336C2×C4—162no—
337C2×C4C2×(C2×D47)⋊C411yescharacteristic
338C2×C4#1309399311yescharacteristic
339C2×C4#1309400711yescharacteristic
340C2×C4C2×#1062533411yescharacteristic
341C2×C4C2×#1062533611yescharacteristic
342C2×C4#1309405511yescharacteristic
343C8—22no—
344C8—1882no—
345C8C4×C22⋊D4711yescharacteristic
346C8C4×C22⋊D4711yescharacteristic
347C8#1309399811yescharacteristic
348C8#1309399811yescharacteristic
349C22—21no—
350C22—21no—
351C22—22no—
352C22—41no—
353C22—41no—
354C22—3762no—
355C22—7522no—
356C22—7522no—
357C22—7521no—
358C22—7521no—
359C22#1568912211yescharacteristic
360C22#1568925711yescharacteristic
361C22#1568944111yescharacteristic
362C4—22no—
363C4—82no—
364C4—81no—
365C4—81no—
366C4#1568900211yescharacteristic
367C4C2×#1309394411yescharacteristic
368C4#1568926211yescharacteristic
369C4#1568943611yescharacteristic
370C2—21no—
371C2—21no—
372C2—42no—
373C2—7522no—
374C2#2024370511yescharacteristic
375C2#2024534511yescharacteristic
376C2#2024535211yescharacteristic
subgroupG/H
centerC22#15689257
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC47×#2913C2
socleC22×C47#447
prime radicalC2×#15689393C2
cosocleC22×C8×C47C23
hypercenter#2913D47
hypocenterC47#12996
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ #2913

Complements

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

NH ≅ G/Ntype
2. #20245103C2semidirect
3. C2×#15688923C2semidirect
4. C2×#15688907C2semidirect
5. #20241496C2semidirect
6. #20241483C2semidirect
7. C47×#2913C2semidirect
13. #15689393C4semidirect
37. C2×D376C2×C4semidirect
47. #13093960C2×C4semidirect
183. #2913D47semidirect
216. C47#12996semidirect

Maximal subgroups

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

subgroupindex
C2×#156893932
#202451032
C2×#156889232
C2×#156889072
#202414962
#202414832
C47×#29132
#1299647