Finite Group Databasebeta

← #26131573

Subgroups — #26131573

Order 12032.

438 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689246C211yesmaximal, characteristic
2C2×#15688932C211yesmaximal, characteristic
3#20245098C211yesmaximal, characteristic
4C2×#15688901C211yesmaximal, characteristic, complemented
5#20241496C211yesmaximal, characteristic, complemented
6#20241486C211yesmaximal, characteristic, complemented
7C47×#1776C211yesmaximal, characteristic, complemented
8C2×#13094046C2211yescharacteristic
9C2×#13094030—21no—
10C2×#13093964C2211yescharacteristic
11C2×#13093964C2211yescharacteristic
12#15689246—22no—
13#15689246C414yescomplemented
14C2×#13093960C2211yescharacteristic
15C2×#13093952—21no—
16#15688932C414yescomplemented
17#15688926C412yescomplemented
18#15688923C412yescomplemented
19#15688901—22no—
20C2×#107×C47C2211yescharacteristic
21C2×#106×C47C2211yescharacteristic
22C2×#97×C47C2211yescharacteristic
23C4×C4⋊C4×C47—21no—
24C47×#340—21no—
25C2×#10625350—21no—
26#13094046—21no—
27#13094046—21no—
28#13094046C2×C412yes—
29#13094046C2×C412yes—
30C2×Dic188C2×C411yescharacteristic
31C2×#10625325—21no—
32C2×D376C2×C411yescharacteristic
33#13094030—24no—
34#13094030—42no—
35C2×#10625329—21no—
36C2×#10625324—41no—
37C2×C8×D47—41no—
38#13093964—21no—
39#13093964—21no—
40#13093964—21no—
41#13093964C2×C412yes—
42C2×C47⋊C16—21no—
43C2×C47⋊C16C2×C412yes—
44#13093960—21no—
45#13093960C2×C412yes—
46#13093952—24no—
47C22×C8×C47D411yescharacteristic
48C22×C8×C47C2311yescharacteristic
49C2×C4⋊C4×C47—21no—
50C2×C4⋊C4×C47—21no—
51C2×C42×C47—41no—
52C2×C42×C47D411yescharacteristic
53#107×C47—21no—
54#107×C47C2×C412yes—
55#106×C47—21no—
56#106×C47C2×C412yes—
57#97×C47C2×C412yes—
58#97×C47D412yes—
59#95×C47—41no—
60C22×C4×C47—21no—
61C22×C4×C47—41no—
62C22×C4×C47—41no—
63C22×C4×C47C2×D411yescharacteristic
64C2×C22⋊D47—41no—
65#10625350—22no—
66#10625350—22no—
67#10625350—41no—
68#10625350—41no—
69C2×D188—21no—
70C2×C4×D47—41no—
71C2×Dic94—21no—
72C2×C8×C47—21no—
73C2×C8×C47—21no—
74C2×C8×C47C22⋊C412yes—
75C2×C8×C47C22×C411yescharacteristic
76C2×C8×C47C22×C411yescharacteristic
77C2×C8×C47C22×C411yescharacteristic
78C2×C8×C47C2×D411yescharacteristic
79C2×C8×C47#4111yescharacteristic
80C2×C8×C47#4111yescharacteristic
81C4⋊C4×C47—22no—
82C4⋊C4×C47—22no—
83C4⋊C4×C47—41no—
84C4⋊C4×C47—41no—
85C42×C47—21no—
86C42×C47—82no—
87C42×C47C22⋊C412yes—
88#10625329—22no—
89#10625329—41no—
90C2×C47⋊C8—41no—
91Dic188—21no—
92Dic188C4212yescomplemented
93D376—21no—
94D376C4212yescomplemented
95#10625325—22no—
96#10625325—22no—
97#10625324—44no—
98C8×D47—44no—
99C47⋊C16—22no—
100C47⋊C16—22no—
101C23×C47AGL(1,ℤ/8)11yescharacteristic
102C22×D47—41no—
103C2×C4×C47—22no—
104C2×C4×C47—41no—
105C2×C4×C47—41no—
106C2×C4×C47—42no—
107C2×C4×C47—42no—
108C2×C4×C47—81no—
109C2×C4×C47—81no—
110C2×C4×C47—81no—
111C2×C4×C47—81no—
112C2×C4×C47C2×C22⋊C411yescharacteristic
113C2×C4×C47C4×D411yescharacteristic
114C2×C4×C47C4×D411yescharacteristic
115C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
116C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
117C2×C4×C47#13711yescharacteristic
118C22⋊D47—42no—
119C22⋊D47—42no—
120C2×Dic47—41no—
121D188—22no—
122D188—41no—
123C4×D47—42no—
124C4×D47—42no—
125Dic94—22no—
126Dic94—41no—
127C8×C47—22no—
128C8×C47C2×C4211yescharacteristic
129C8×C47#11711yescharacteristic
130C8×C47C4×D411yescharacteristic
131C8×C47C4×D411yescharacteristic
132C47⋊C8—42no—
133#12650—471nomaximal, Sylow
134C22×C47—21no—
135C22×C47—21no—
136C22×C47#41711yescharacteristic
137C22×C47#44211yescharacteristic
138C22×C47#44211yescharacteristic
139C2×D47—42no—
140C2×D47—81no—
141C2×D47—81no—
142C4×C47—42no—
143C4×C47—82no—
144C4×C47—82no—
145C4×C47C4×C22⋊C411yescharacteristic
146C4×C47#41811yescharacteristic
147C4×C47#44011yescharacteristic
148C4×C47#44011yescharacteristic
149Dic47—42no—
150C2×#444—471no—
151C2×#360—471no—
152#2139—471no—
153C2×#347—471no—
154#1776D4711yescharacteristic, complemented
155#1387—471no—
156#1383—471no—
157C2×C47—21no—
158C2×C47—21no—
159C2×C47#176211yescharacteristic
160C2×C47#218011yescharacteristic
161C2×C47AGL(1,ℤ/16)11yescharacteristic
162D47—82no—
163C2×#135—471no—
164C2×#131—941no—
165C2×M32—471no—
166C2×M32—471no—
167#444—474no—
168#444—942no—
169C2×#108—471no—
170C2×#107C2×D4711yescharacteristic
171C2×#106C2×D4711yescharacteristic
172C2×#104—941no—
173C2×#97C2×D4711yescharacteristic
174C4×C4⋊C4—21no—
175#360—474no—
176#358—472no—
177#357—472no—
178#347—942no—
179#340—21no—
180C47#1265011yescharacteristic, complemented, Sylow
181C2×#41—941no—
182#135—472no—
183#135—472no—
184#135—941no—
185#135—941no—
186C2×Q16—471no—
187C2×SD16—941no—
188C2×D8—471no—
189#131—944no—
190#131—1882no—
191C2×M16—941no—
192C2×M16—1881no—
193C22×C8—1881no—
194C22×C8C22⋊D4711yescharacteristic
195C22×C8C22×D4711yescharacteristic
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C42—41no—
199C2×C42C22⋊D4711yescharacteristic
200M32—472no—
201M32—941no—
202M32—941no—
203M32—941no—
204C2×C16—472no—
205C2×C16—941no—
206#108—472no—
207#108—941no—
208#107—21no—
209#107C4×D4712yes—
210#106—21no—
211#106C4×D4712yes—
212#104—944no—
213#97C4×D4712yes—
214#97D18812yes—
215#95—41no—
216#41—942no—
217#41—942no—
218#41—1881no—
219#41—1881no—
220C2×Q8—941no—
221C2×D4—941no—
222C2×D4—1881no—
223C22×C4—21no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—1881no—
227C22×C4C2×C22⋊D4711yescharacteristic
228Q16—472no—
229Q16—941no—
230SD16—942no—
231SD16—942no—
232D8—472no—
233D8—941no—
234M16—942no—
235M16—1884no—
236M16—1881no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—1884no—
240C2×C8—1881no—
241C2×C8(C2×D47)⋊C412yes—
242C2×C8C2×C4×D4711yescharacteristic
243C2×C8C2×C4×D4711yescharacteristic
244C2×C8C2×C4×D4711yescharacteristic
245C2×C8C2×D18811yescharacteristic
246C2×C8#1062535011yescharacteristic
247C2×C8#1062535011yescharacteristic
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C42—21no—
253C42—82no—
254C42(C2×D47)⋊C412yes—
255C16—942no—
256C16—942no—
257C23—1881no—
258C23#1309405011yescharacteristic
259Q8—942no—
260Q8—1881no—
261D4—942no—
262D4—1882no—
263D4—1882no—
264D4—1881no—
265C2×C4—22no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—1881no—
275C2×C4—1882no—
276C2×C4—1882no—
277C2×C4C2×(C2×D47)⋊C411yescharacteristic
278C2×C4C4×C22⋊D4711yescharacteristic
279C2×C4C4×C22⋊D4711yescharacteristic
280C2×C4#1309404911yescharacteristic
281C2×C4#1309404911yescharacteristic
282C2×C4#1309405511yescharacteristic
283C8—22no—
284C8—1882no—
285C8C42×D4711yescharacteristic
286C8#1309398111yescharacteristic
287C8C4×D18811yescharacteristic
288C8C4×D18811yescharacteristic
289C22—21no—
290C22—21no—
291C22—3761no—
292C22—3761no—
293C22—1882no—
294C22#1568912211yescharacteristic
295C22#1568923311yescharacteristic
296C22#1568923311yescharacteristic
297C4—42no—
298C4—82no—
299C4—82no—
300C4—1882no—
301C4C4×(C2×D47)⋊C411yescharacteristic
302C4#1568912711yescharacteristic
303C4#1568922111yescharacteristic
304C4#1568922111yescharacteristic
305C2—21no—
306C2—21no—
307C2—3762no—
308C2#2024321611yescharacteristic
309C2#2024527711yescharacteristic
310C2#2024528411yescharacteristic
subgroupG/H
centerC22#15689233
commutatorC8×C47C2×C42
FrattiniC22×C8C22×D47
FittingC47×#1776C2
socleC22×C47#442
prime radicalC2×#13094046C22
cosocleC22×C8×C47C23
hypercenter#1776D47
hypocenterC47#12650
LCSG ▷ C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ #1776

Complements

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

NH ≅ G/Ntype
4. C2×#15688901C2semidirect
5. #20241496C2semidirect
6. #20241486C2semidirect
7. C47×#1776C2semidirect
13. #15689246C4semidirect
16. #15688932C4semidirect
17. #15688926C4semidirect
18. #15688923C4semidirect
92. Dic188C42semidirect
94. D376C42semidirect
154. #1776D47semidirect
180. C47#12650semidirect

Maximal subgroups

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

subgroupindex
C2×#156892462
C2×#156889322
#202450982
C2×#156889012
#202414962
#202414862
C47×#17762
#1265047