Finite Group Databasebeta

← #26184724

Subgroups — #26184724

Order 12032.

1308 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20257166C211yesmaximal, characteristic
2#20252499C211yesmaximal, characteristic, complemented
3#20242066C213yesmaximal, complemented
4#20242057C213yesmaximal, complemented
5#20242038C213yesmaximal, complemented
6C2×#15688866C213yesmaximal, complemented
7C47×#2906C211yesmaximal, characteristic, complemented
8C2×#13094070—21no—
9#15689795—21no—
10#15689795C412yes—
11D4×D188—23no—
12D4×D188C416yes—
13#15689692—23no—
14#15689673C2213yes—
15C2×#13094021C2213yes—
16Q8×C47⋊C8—21no—
17D4×C47⋊C8—23no—
18#15689176C2213yescomplemented
19C2×C4⋊(C47⋊C8)C2213yescomplemented
20#15688870C2211yescharacteristic, complemented
21#15688870C2213yescomplemented
22#15688866C2213yescomplemented
23#15688866C2216yescomplemented
24#15688866C2213yescomplemented
25C2×#131×C47—21no—
26C4×#41×C47C2211yescharacteristic
27C47×#424—23no—
28C47×#423C2213yescomplemented
29C2×C4⋊C8×C47C2213yescomplemented
30#13094070—44no—
31#13094070—44no—
32C2×#10625354—22no—
33C2×#10625350—23no—
34C22×D188—23no—
35C22×D188—23no—
36C22×D188C2×C416yes—
37C2×D4×D47—26no—
38C22×C47⋊C8—23no—
39#13094024C2311yescharacteristic
40#13094021—23no—
41#13094021—23no—
42#13094021C2×C416yes—
43#13094021C2313yes—
44#13094012—23no—
45#13093999—26no—
46#13093999—26no—
47#13093998—23no—
48#13093998C2×C416yes—
49#13093994—212no—
50C4×D188—23no—
51C4×D188C2×C416yes—
52C4⋊(C47⋊C8)—23no—
53C4⋊(C47⋊C8)C2313yes—
54C4⋊(C47⋊C8)C2311yescharacteristic
55C22⋊(C47⋊C8)—23no—
56C4×C47⋊C8—23no—
57C2×#41×C47D411yescharacteristic
58#131×C47—44no—
59C2×M16×C47—23no—
60C22×C8×C47—23no—
61C4×Q8×C47C2×C411yescharacteristic
62C4×Q8×C47D411yescharacteristic
63C4×D4×C47C2×C413yes—
64C4×D4×C47D413yes—
65#117×C47D413yes—
66C2×C42×C47C2313yes—
67C4⋊C8×C47—23no—
68C4⋊C8×C47—23no—
69C4⋊C8×C47C2311yescharacteristic
70C4⋊C8×C47C2313yes—
71C23×D47—26no—
72#41×C47D814yes—
73#41×C47SD1614yes—
74C2×Q8×C47C22⋊C411yescharacteristic
75C2×D4×C47C22⋊C413yes—
76C22×C4×C47C22×C413yes—
77C22×C4×C47C2×D413yes—
78C22×C4×C47C2×D413yes—
79C2×C22⋊D47—26no—
80#10625354—48no—
81D4×D47—424no—
82#10625350—46no—
83#10625350—46no—
84C2×D188—26no—
85C2×D188—23no—
86C2×D188—23no—
87C2×D188—26no—
88C2×D188—26no—
89C2×D188—46no—
90C2×D188—46no—
91C2×D188—83no—
92C2×D188—83no—
93C2×D188—86no—
94C2×D188C22×C416yes—
95C2×C4×D47—26no—
96C2×Dic94—21no—
97M16×C47—46no—
98C2×C8×C47—21no—
99C2×C8×C47—23no—
100C2×C8×C47—46no—
101C4⋊C4×C47C22⋊C413yes—
102C4⋊C4×C47C22⋊C413yes—
103C22⋊C4×C47C22⋊C416yes—
104C42×C47C22×C413yes—
105C42×C47C2×D413yes—
106C42×C47C2×D413yes—
107C42×C47C2411yescharacteristic
108(C2×D47)⋊C4—26no—
109(C2×D47)⋊C4—26no—
110C4⋊Dic47—21no—
111C4⋊Dic47C22×C412yes—
112C2×C47⋊C8—21no—
113C2×C47⋊C8—23no—
114C2×C47⋊C8—83no—
115C23×C47C2×C22⋊C413yes—
116C22×D47—26no—
117C22×D47—412no—
118C22×D47—86no—
119C22×D47—86no—
120Q8×C47#10214yes—
121D4×C47#102112yes—
122C2×C4×C47—23no—
123C2×C4×C47—23no—
124C2×C4×C47—43no—
125C2×C4×C47C2×C22⋊C413yes—
126C2×C4×C47C2×C22⋊C413yes—
127C2×C4×C47C2×C22⋊C413yes—
128C2×C4×C47C2×D816yes—
129C2×C4×C47C2×SD1616yes—
130C2×C4×C47C23×C411yescharacteristic
131C2×C4×C47C22×D411yescharacteristic
132C2×C4×C47C22×D411yescharacteristic
133C22⋊D47—412no—
134C2×Dic47—22no—
135D188—46no—
136D188—43no—
137D188—43no—
138D188—43no—
139D188—43no—
140D188—412no—
141D188—86no—
142D188—86no—
143C4×D47—412no—
144Dic94—41no—
145Dic94—41no—
146C8×C47—41no—
147C8×C47—43no—
148C47⋊C8—43no—
149C47⋊C8—41no—
150#17720—471nomaximal, Sylow
151C22×C47—23no—
152C22×C47—23no—
153C22×C47C2×#10216yes—
154C22×C47C22×C22⋊C411yescharacteristic
155C2×D47—46no—
156C2×D47—46no—
157C2×D47—46no—
158C2×D47—1612no—
159C4×C47—21no—
160C4×C47—23no—
161C4×C47—23no—
162C4×C47C2×#10216yes—
163C4×C47C22×D811yescharacteristic
164C4×C47C22×SD1611yescharacteristic
165C4×C47#57611yescharacteristic
166C4×C47#57711yescharacteristic
167Dic47—42no—
168#3498—471no—
169#2988—471no—
170#2906D4711yescharacteristic, complemented
171#1547—473no—
172#1545—473no—
173#1540—473no—
174C2×#331—473no—
175C2×C47—26no—
176C2×C47#288811yescharacteristic
177C2×C47C22×#10211yescharacteristic
178C2×C47#289711yescharacteristic
179D47—86no—
180C2×H(5,𝔽2)—941no—
181C2×#131—21no—
182#550—472no—
183#550—941no—
184D42—476no—
185D42—943no—
186#535—943no—
187#532—473no—
188C2×#127—473no—
189C4×#41C2×D4711yescharacteristic
190C8×Q8—941no—
191C8×D4—943no—
192#431—473no—
193#424—23no—
194#423C2×D4713yescomplemented
195C2×C4⋊C8—473no—
196C2×C4⋊C8C2×D4713yescomplemented
197#332—471no—
198#332—473no—
199#331—473no—
200#331—476no—
201#331—473no—
202C47#1772011yescharacteristic, complemented, Sylow
203H(5,𝔽2)—1884no—
204H(5,𝔽2)—1884no—
205C2×#41—942no—
206C2×#41—943no—
207C2×#41C22⋊D4711yescharacteristic
208C22×D4—476no—
209C22×D4—946no—
210C22×D4—943no—
211C22×D4—943no—
212#131—44no—
213C2×M16—23no—
214C22×C8—23no—
215C22×C8—943no—
216C4⋊Q8—471no—
217#127—476no—
218#127—473no—
219#127—943no—
220#127—943no—
221#124—943no—
222#121—476no—
223#121—943no—
224#121—946no—
225#121—946no—
226#120—9412no—
227C4×Q8C4×D4711yescharacteristic
228C4×Q8D18811yescharacteristic
229C4×D4—476no—
230C4×D4—943no—
231C4×D4C4×D4713yes—
232C4×D4D18813yes—
233#117C22⋊D4713yes—
234C2×C42C22×D4713yes—
235C4⋊C8—23no—
236C4⋊C8—23no—
237C4⋊C8—473no—
238C4⋊C8—471no—
239C4⋊C8—943no—
240C4⋊C8C22×D4711yescharacteristic
241C4⋊C8C22×D4713yes—
242C22⋊C8—943no—
243C4×C8—943no—
244C24—946no—
245#41—1888no—
246#41—1886no—
247#41—1886no—
248#41#1062533414yes—
249#41#1062533614yes—
250C2×Q8—941no—
251C2×Q8(C2×D47)⋊C411yescharacteristic
252C2×D4—476no—
253C2×D4—946no—
254C2×D4—946no—
255C2×D4—946no—
256C2×D4—946no—
257C2×D4—943no—
258C2×D4—943no—
259C2×D4—3763no—
260C2×D4—3763no—
261C2×D4—3766no—
262C2×D4—18824no—
263C2×D4—1886no—
264C2×D4—1886no—
265C2×D4(C2×D47)⋊C413yes—
266C22×C4—946no—
267C22×C4C2×C4×D4713yes—
268C22×C4C2×D18813yes—
269C22×C4C2×C22⋊D4713yes—
270M16—46no—
271C2×C8—21no—
272C2×C8—23no—
273C2×C8—46no—
274C2×C8—941no—
275C2×C8—943no—
276C2×C8—3763no—
277C4⋊C4—472no—
278C4⋊C4—941no—
279C4⋊C4(C2×D47)⋊C413yes—
280C4⋊C4(C2×D47)⋊C413yes—
281C22⋊C4—946no—
282C22⋊C4—946no—
283C22⋊C4(C2×D47)⋊C416yes—
284C42C2×C4×D4713yes—
285C42C2×D18813yes—
286C42C2×C22⋊D4713yes—
287C42C23×D4711yescharacteristic
288C23—946no—
289C23—3766no—
290C23—3766no—
291C23—18812no—
292C23C2×(C2×D47)⋊C413yes—
293Q8—1881no—
294Q8—1881no—
295Q8#1309394414yes—
296D4—3766no—
297D4—3766no—
298D4—1886no—
299D4—18812no—
300D4—18812no—
301D4—1883no—
302D4—1883no—
303D4—1883no—
304D4—1883no—
305D4#13093944112yes—
306C2×C4—23no—
307C2×C4—23no—
308C2×C4—43no—
309C2×C4—942no—
310C2×C4—18812no—
311C2×C4C2×(C2×D47)⋊C413yes—
312C2×C4C2×(C2×D47)⋊C413yes—
313C2×C4C2×(C2×D47)⋊C413yes—
314C2×C4C2×#1062533416yes—
315C2×C4C2×#1062533616yes—
316C2×C4C22×C4×D4711yescharacteristic
317C2×C4C22×C22⋊D4711yescharacteristic
318C2×C4C22×D18811yescharacteristic
319C8—41no—
320C8—43no—
321C8—1883no—
322C8—1881no—
323C22—23no—
324C22—23no—
325C22—1886no—
326C22—1886no—
327C22—1886no—
328C22—75212no—
329C22C2×#1309394416yes—
330C22C22×(C2×D47)⋊C411yescharacteristic
331C4—23no—
332C4—21no—
333C4—23no—
334C4—1882no—
335C4C2×#1309394416yes—
336C4C22×#1062533411yescharacteristic
337C4C22×#1062533611yescharacteristic
338C4#1568992811yescharacteristic
339C4#1568993511yescharacteristic
340C2—26no—
341C2—3766no—
342C2#2025186611yescharacteristic
343C2C22×#1309394411yescharacteristic
344C2#2025191511yescharacteristic
subgroupG/H
centerC22C22×(C2×D47)⋊C4
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#2906C2
socleC22×C47C22×C22⋊C4
prime radical#20257166C2
cosocleC42×C47C24
hypercenter#2906D47
hypocenterC47#17720
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C4×#41 ◁ #2906

Complements

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

NH ≅ G/Ntype
2. #20252499C2semidirect
3. #20242066C2semidirect
4. #20242057C2semidirect
5. #20242038C2semidirect
6. C2×#15688866C2semidirect
7. C47×#2906C2semidirect
18. #15689176C22semidirect
19. C2×C4⋊(C47⋊C8)C22semidirect
20. #15688870C22semidirect
21. #15688870C22semidirect
22. #15688866C22semidirect
23. #15688866C22semidirect
24. #15688866C22semidirect
28. C47×#423C22semidirect
29. C2×C4⋊C8×C47C22semidirect
170. #2906D47semidirect
194. #423C2×D47semidirect
196. C2×C4⋊C8C2×D47semidirect
202. C47#17720semidirect

Maximal subgroups

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

subgroupindex
#202571662
#202524992
#202420662
#202420572
#202420382
C2×#156888662
C47×#29062
#1772047