Finite Group Databasebeta

← #26133306

Subgroups — #26133306

Order 12032.

492 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689393C211yesmaximal, characteristic
2C2×#15688935C211yesmaximal, characteristic, complemented
3C2×#15688935C211yesmaximal, characteristic, complemented
4#20245016C211yesmaximal, characteristic, complemented
5#20241501C211yesmaximal, characteristic, complemented
6#20241491C211yesmaximal, characteristic, complemented
7C47×#2916C211yesmaximal, characteristic, complemented
8C2×#13094049—21no—
9C22×D376C2211yescharacteristic
10C2×#13093964C2211yescharacteristic
11C2×#13093964C2211yescharacteristic
12#15689393—22no—
13#15689393C414yes—
14C2×#13093960C2211yescharacteristic
15C2×#13093940—21no—
16#15688935—22no—
17#15688935—22no—
18#15688907—22no—
19C47×#568—21no—
20C2×#131×C47C2211yescharacteristic
21C47×#430—21no—
22C47×#430—22no—
23C2×#108×C47—22no—
24C2×#108×C47C2211yescharacteristic
25C2×#108×C47C2211yescharacteristic
26C22×D188—21no—
27#13094049—24no—
28#13094049—42no—
29C2×#10625336—41no—
30C2×D376—21no—
31C2×D376—21no—
32C2×D376—21no—
33C2×D376—21no—
34C2×D376—21no—
35C2×D376C2×C412yes—
36C2×#10625334—41no—
37C2×#10625329—21no—
38#13093964—21no—
39#13093964—21no—
40#13093964—21no—
41#13093964—21no—
42C2×C47⋊C16—21no—
43C2×C47⋊C16—21no—
44#13093960—21no—
45#13093960C2×C412yes—
46#13093940—24no—
47C2×#41×C47D411yescharacteristic
48#131×C47—21no—
49#131×C47—21no—
50#131×C47—42no—
51#131×C47—42no—
52#131×C47C2×C412yes—
53#131×C47D412yes—
54C2×M16×C47—21no—
55C2×M16×C47—24no—
56C2×M16×C47—21no—
57C2×M16×C47—21no—
58C2×M16×C47D411yescharacteristic
59C22×C8×C47—21no—
60C22×C8×C47C2311yescharacteristic
61#108×C47—24no—
62#108×C47—21no—
63#108×C47—21no—
64#108×C47D412yes—
65#108×C47D412yes—
66C23×D47—41no—
67#41×C47—21no—
68#41×C47—21no—
69#41×C47C22⋊C412yes—
70#41×C47D812yes—
71C2×Q8×C47SD1611yescharacteristic
72C2×D4×C47—21no—
73C2×D4×C47SD1611yescharacteristic
74C22×C4×C47—21no—
75C22×C4×C47C2×D411yescharacteristic
76C2×D188—22no—
77C2×D188—41no—
78C2×D188—41no—
79C2×D188—41no—
80C2×D188—41no—
81C2×D188—41no—
82M16×C47—21no—
83M16×C47—22no—
84M16×C47—22no—
85M16×C47—22no—
86M16×C47—22no—
87M16×C47—44no—
88M16×C47—41no—
89M16×C47—41no—
90M16×C47C22⋊C412yes—
91C2×C8×C47—21no—
92C2×C8×C47—22no—
93C2×C8×C47—22no—
94C2×C8×C47—44no—
95C2×C8×C47—41no—
96C2×C8×C47—41no—
97C2×C8×C47C22×C411yescharacteristic
98C2×C8×C47C2×D411yescharacteristic
99C2×C8×C47C2×D411yescharacteristic
100C2×C8×C47C2×D411yescharacteristic
101C2×C8×C47#4111yescharacteristic
102C2×C8×C47#4111yescharacteristic
103#10625336—44no—
104#10625334—44no—
105#10625329—22no—
106#10625329—41no—
107C2×C47⋊C8—41no—
108D376—22no—
109D376—22no—
110D376—41no—
111D376—41no—
112C47⋊C16—41no—
113C47⋊C16—41no—
114C23×C47—21no—
115C23×C47#13711yescharacteristic
116C22×D47—42no—
117C22×D47—81no—
118C22×D47—81no—
119C22×D47—162no—
120Q8×C47—21no—
121Q8×C47#10212yes—
122D4×C47—22no—
123D4×C47—22no—
124D4×C47—21no—
125D4×C47#10212yes—
126C2×C4×C47—21no—
127C2×C4×C47—22no—
128C2×C4×C47—22no—
129C2×C4×C47C2×C22⋊C411yescharacteristic
130C2×C4×C47#12011yescharacteristic
131C2×C4×C47#12311yescharacteristic
132C2×C4×C47C2×D811yescharacteristic
133C2×C4×C47C2×SD1611yescharacteristic
134C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
135D188—42no—
136D188—42no—
137D188—81no—
138D188—81no—
139C8×C47—22no—
140C8×C47—42no—
141C8×C47—42no—
142C8×C47C4×D411yescharacteristic
143C8×C47C4×D411yescharacteristic
144C8×C47#12111yescharacteristic
145C8×C47#12111yescharacteristic
146C47⋊C8—42no—
147#13007—471nomaximal, Sylow
148C22×C47—22no—
149C22×C47—41no—
150C22×C47—81no—
151C22×C47#41711yescharacteristic
152C22×C47#44811yescharacteristic
153C22×C47#48011yescharacteristic
154C2×D47—82no—
155C2×D47—162no—
156C2×D47—162no—
157C2×D47—162no—
158C4×C47—22no—
159C4×C47#38611yescharacteristic
160C4×C47C2×#10211yescharacteristic
161C4×C47#44711yescharacteristic
162C4×C47#48111yescharacteristic
163C2×#472—471no—
164#2916D4711yescharacteristic, complemented
165C2×#361—471no—
166C2×#361—471no—
167#2117—471no—
168#1388—471no—
169#1385—471no—
170C2×C47—21no—
171C2×C47—21no—
172C2×C47—21no—
173C2×C47—42no—
174C2×C47#187411yescharacteristic
175D47—162no—
176C2×AGL(1,ℤ/8)—941no—
177C22×D8—471no—
178#568—21no—
179C2×#131C2×D4711yescharacteristic
180C2×M32—471no—
181C2×M32—471no—
182#472—474no—
183#472—942no—
184#430—21no—
185#430—22no—
186C2×#108—22no—
187C2×#108—471no—
188C2×#108C2×D4711yescharacteristic
189C2×#108C2×D4711yescharacteristic
190C2×#100—941no—
191#361—942no—
192#361—942no—
193#350—942no—
194C47#1300711yescharacteristic, complemented, Sylow
195C2×#41C22⋊D4711yescharacteristic
196C22×D4—941no—
197AGL(1,ℤ/8)—944no—
198AGL(1,ℤ/8)—1882no—
199C2×SD16—1881no—
200C2×D8—472no—
201C2×D8—941no—
202C2×D8—941no—
203C2×D8—941no—
204C2×D8—941no—
205C2×D8—941no—
206C2×D8—1881no—
207#131—21no—
208#131—21no—
209#131—42no—
210#131—42no—
211#131C4×D4712yes—
212#131D18812yes—
213C2×M16—21no—
214C2×M16—24no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—941no—
218C2×M16C22⋊D4711yescharacteristic
219C22×C8—21no—
220C22×C8C22×D4711yescharacteristic
221M32—941no—
222M32—941no—
223M32—941no—
224M32—941no—
225C2×C16—941no—
226C2×C16—941no—
227#108—24no—
228#108—21no—
229#108—21no—
230#108—472no—
231#108—941no—
232#108C22⋊D4712yes—
233#108C22⋊D4712yes—
234#100—944no—
235C24—1881no—
236#41—21no—
237#41—21no—
238#41(C2×D47)⋊C412yes—
239#41#1062533412yes—
240C2×Q8#1062533611yescharacteristic
241C2×D4—21no—
242C2×D4—942no—
243C2×D4—1881no—
244C2×D4—1881no—
245C2×D4—1881no—
246C2×D4—1881no—
247C2×D4—1881no—
248C2×D4#1062533611yescharacteristic
249C22×C4—21no—
250C22×C4C2×C22⋊D4711yescharacteristic
251SD16—1884no—
252D8—942no—
253D8—942no—
254D8—1884no—
255D8—1881no—
256D8—1881no—
257M16—21no—
258M16—22no—
259M16—22no—
260M16—22no—
261M16—22no—
262M16—44no—
263M16—41no—
264M16—41no—
265M16—942no—
266M16—1881no—
267M16(C2×D47)⋊C412yes—
268C2×C8—21no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—44no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—1881no—
275C2×C8C2×C4×D4711yescharacteristic
276C2×C8C2×D18811yescharacteristic
277C2×C8#1062535011yescharacteristic
278C2×C8#1062535011yescharacteristic
279C2×C8C2×C22⋊D4711yescharacteristic
280C2×C8C2×C22⋊D4711yescharacteristic
281C16—1881no—
282C16—1881no—
283C23—21no—
284C23—3761no—
285C23—3761no—
286C23—1882no—
287C23—7522no—
288C23#1309405511yescharacteristic
289Q8—21no—
290Q8#1309394412yes—
291D4—22no—
292D4—22no—
293D4—21no—
294D4—3761no—
295D4—3761no—
296D4—1882no—
297D4—1882no—
298D4#1309394412yes—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4C2×(C2×D47)⋊C411yescharacteristic
303C2×C4#1309399311yescharacteristic
304C2×C4#1309400711yescharacteristic
305C2×C4C2×#1062533411yescharacteristic
306C2×C4C2×#1062533611yescharacteristic
307C2×C4#1309405011yescharacteristic
308C8—22no—
309C8—42no—
310C8—42no—
311C8—1882no—
312C8C4×C22⋊D4711yescharacteristic
313C8C4×C22⋊D4711yescharacteristic
314C8#1309399811yescharacteristic
315C8#1309399811yescharacteristic
316C22—22no—
317C22—41no—
318C22—81no—
319C22—3762no—
320C22—7522no—
321C22—7522no—
322C22—7522no—
323C22#1568912211yescharacteristic
324C22#1568926211yescharacteristic
325C22#1568943611yescharacteristic
326C4—22no—
327C4#1568900211yescharacteristic
328C4C2×#1309394411yescharacteristic
329C4#1568925711yescharacteristic
330C4#1568944111yescharacteristic
331C2—21no—
332C2—21no—
333C2—21no—
334C2—42no—
335C2—7522no—
336C2#2024370511yescharacteristic
subgroupG/H
centerC2#20243705
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC47×#2916C2
socleC2×C47#1874
prime radicalC2×#15689393C2
cosocleC22×C8×C47C23
hypercenter#2916D47
hypocenterC47#13007
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #2916

Complements

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

NH ≅ G/Ntype
2. C2×#15688935C2semidirect
3. C2×#15688935C2semidirect
4. #20245016C2semidirect
5. #20241501C2semidirect
6. #20241491C2semidirect
7. C47×#2916C2semidirect
164. #2916D47semidirect
194. C47#13007semidirect

Maximal subgroups

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

subgroupindex
C2×#156893932
C2×#156889352
#202450162
#202415012
#202414912
C47×#29162
#1300747