Finite Group Databasebeta

← #20257467

Subgroups — #20257467

Order 6016.

1104 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15689856C213yesmaximal, complemented
2Q8×Dic94C211yesmaximal, characteristic, complemented
3#15689828C213yesmaximal, complemented
4#15689792C211yesmaximal, characteristic, complemented
5#15689785C211yesmaximal, characteristic, complemented
6#15689780C211yesmaximal, characteristic, complemented
7#15689778C213yesmaximal, complemented
8#15689774C213yesmaximal, complemented
9#15689744C213yesmaximal, complemented
10#15689738C213yesmaximal, complemented
11#15689670C213yesmaximal, complemented
12#15689651C213yesmaximal, complemented
13#15689605C211yesmaximal, characteristic, complemented
14C4×Q8×D47C211yesmaximal, characteristic, complemented
15C47×#520C211yesmaximal, characteristic, complemented
16C2×#10625354C2211yescharacteristic
17C2×#10625350C2216yes—
18C2×Q8×D47C2211yescharacteristic
19#13094024C2213yes—
20#13094023C2213yes—
21#13094023C2213yes—
22#13094020C2213yes—
23#13094019C2213yes—
24#13094017C2213yes—
25#13094014C2213yes—
26#13094012C2213yes—
27#13094011C2213yes—
28#13094010C2216yes—
29#13094010C2213yes—
30#13094009C2213yes—
31#13094007C2216yes—
32#13094006C2216yes—
33#13094006C2213yes—
34#13094004C2213yes—
35#13094003C2216yes—
36#13094003C2213yes—
37#13094002C2216yes—
38#13094002C2211yescharacteristic
39#13093999C2213yes—
40#13093998C2213yes—
41#13093997C2213yes—
42#13093995C2213yes—
43#13093995C2213yes—
44Q8×Dic47C2211yescharacteristic
45Q8×Dic47C2211yescharacteristic
46C4×Dic94C2213yes—
47C4×Dic94C2213yes—
48#13093989C2213yes—
49C4×D188C2213yes—
50C4×D188C2213yes—
51#13093986C2213yes—
52C4×C22⋊D47C2213yes—
53C4×C22⋊D47C2213yes—
54C4×C22⋊D47C2216yes—
55#13093981C2213yes—
56C4⋊C4×D47C2213yes—
57C4⋊C4×D47C2213yes—
58C42×D47C2213yes—
59C2×#41×C47C2211yescharacteristic
60C4×Q8×C47C2211yescharacteristic
61C4×Q8×C47C2211yescharacteristic
62C4×D4×C47C2213yes—
63C4×D4×C47C2213yes—
64#117×C47C2213yes—
65C2×C4⋊C4×C47C2213yes—
66#41×C47—24no—
67C2×Q8×C47C2311yescharacteristic
68C2×D4×C47C2313yes—
69C22×C4×C47C2313yes—
70C22×C4×C47C2316yes—
71C2×C22⋊D47C2313yes—
72C2×C22⋊D47C2313yes—
73C2×C22⋊D47C2316yes—
74#10625354—24no—
75Q8×D47—24no—
76#10625350—412no—
77C2×D188C2313yes—
78C2×D188C2313yes—
79C2×C4×D47C2311yescharacteristic
80C2×C4×D47C2313yes—
81C2×C4×D47C2313yes—
82C2×C4×D47C2313yes—
83C2×C4×D47C2316yes—
84C2×Dic94C2313yes—
85C2×Dic94C2311yescharacteristic
86C2×Dic94C2316yes—
87C4⋊C4×C47C2313yes—
88C4⋊C4×C47C2313yes—
89C4⋊C4×C47C2313yes—
90C4⋊C4×C47C2311yescharacteristic
91C22⋊C4×C47C2313yes—
92C22⋊C4×C47C2313yes—
93C42×C47C2313yes—
94C42×C47C2313yes—
95C22⋊Dic47C2313yes—
96C22⋊Dic47C2316yes—
97C22⋊Dic47C2313yes—
98(C2×D47)⋊C4C2313yes—
99(C2×D47)⋊C4C2313yes—
100(C2×D47)⋊C4C2311yescharacteristic
101(C2×D47)⋊C4C2313yes—
102(C2×D47)⋊C4C2316yes—
103(C2×D47)⋊C4C2311yescharacteristic
104(C2×D47)⋊C4C2313yes—
105C4⋊Dic47C2313yes—
106C4⋊Dic47C2313yes—
107C4⋊Dic47C2313yes—
108C4⋊Dic47C2313yes—
109Dic47⋊C4C2313yes—
110Dic47⋊C4C2316yes—
111Dic47⋊C4C2311yescharacteristic
112Dic47⋊C4C2311yescharacteristic
113Dic47⋊C4C2313yes—
114Dic47⋊C4C2313yes—
115Dic47⋊C4C2313yes—
116C4×Dic47C2313yes—
117C4×Dic47C2313yes—
118C4×Dic47C2313yes—
119C4×Dic47C2313yes—
120C23×C47C2413yes—
121C22×D47C2413yes—
122C22×D47C2411yescharacteristic
123Q8×C47#4114yescomplemented
124D4×C47—26no—
125C2×C4×C47—26no—
126C2×C4×C47—46no—
127C2×C4×C47C2411yescharacteristic
128C2×C4×C47C2413yes—
129C2×C4×C47C2411yescharacteristic
130C2×C4×C47C2413yes—
131C2×C4×C47C2411yescharacteristic
132C2×C4×C47C2413yes—
133C22⋊D47—26no—
134C22⋊D47—26no—
135C22⋊D47—46no—
136C2×Dic47C2411yescharacteristic
137C2×Dic47C2413yes—
138C2×Dic47C2413yes—
139C2×Dic47C2413yes—
140C2×Dic47C2411yescharacteristic
141C2×Dic47C2411yescharacteristic
142D188—26no—
143D188—26no—
144C4×D47—26no—
145C4×D47—26no—
146C4×D47—26no—
147C4×D47—22no—
148C4×D47—46no—
149Dic94—26no—
150Dic94—22no—
151Dic94—46no—
152C22×C47—23no—
153C22×C47—23no—
154C22×C47—23no—
155C22×C47C2511yescharacteristic
156C2×D47—21no—
157C2×D47—21no—
158C2×D47—23no—
159C2×D47—23no—
160C2×D47—23no—
161C2×D47H(5,𝔽2)12yes—
162C4×C47—21no—
163C4×C47—23no—
164C4×C47—23no—
165C4×C47—21no—
166C4×C47—21no—
167C4×C47C2×#4116yes—
168Dic47—23no—
169Dic47—21no—
170Dic47—23no—
171Dic47—23no—
172Dic47—21no—
173Dic472−1+412yes—
174#3522—471nomaximal, Sylow
175C2×C47—43no—
176C2×C47C22×#4111yescharacteristic
177C2×C47C2×H(5,𝔽2)11yescharacteristic
178C2×C47C2×2−1+411yescharacteristic
179D47—22no—
180D47—43no—
181#559—473no—
182Q82—471no—
183#554—473no—
184#550—471no—
185#548—471no—
186#548—471no—
187#547—473no—
188#547—473no—
189#542—473no—
190#542—473no—
191#532—473no—
192#529—473no—
193#520—471no—
194#520D4711yescharacteristic, complemented
195C2×C4×Q8—471no—
196C47#352211yescharacteristic, complemented, Sylow
197C2×#41—471no—
198C2×#41—476no—
199C2×#41C2×D4711yescharacteristic
200C22×Q8—471no—
201C4⋊Q8—473no—
202C4⋊Q8—473no—
203C4⋊Q8—473no—
204#127—473no—
205#126—473no—
206#126—473no—
207#125—473no—
208#124—473no—
209#124—473no—
210#124—476no—
211#124—473no—
212#123—476no—
213#123—473no—
214#123—473no—
215#123—476no—
216C22⋊Q8—476no—
217C22⋊Q8—473no—
218C22⋊Q8—471no—
219C22⋊Q8—476no—
220C22⋊Q8—473no—
221#121—473no—
222#121—473no—
223#121—473no—
224#121—473no—
225#121—473no—
226C4×Q8—471no—
227C4×Q8—471no—
228C4×Q8—473no—
229C4×Q8—473no—
230C4×Q8—473no—
231C4×Q8C2×D4711yescharacteristic
232C4×Q8C2×D4711yescharacteristic
233C4×D4—473no—
234C4×D4—473no—
235C4×D4—476no—
236C4×D4—473no—
237C4×D4—473no—
238C4×D4—473no—
239C4×D4C2×D4713yes—
240C4×D4C2×D4713yes—
241#117—473no—
242#117C2×D4713yes—
243C2×C4⋊C4—473no—
244C2×C4⋊C4—473no—
245C2×C4⋊C4C2×D4713yes—
246C2×C42—473no—
247#41—24no—
248#41—944no—
249#41—18812no—
250C2×Q8—471no—
251C2×Q8—473no—
252C2×Q8—476no—
253C2×Q8—944no—
254C2×Q8C22×D4711yescharacteristic
255C2×D4—473no—
256C2×D4—473no—
257C2×D4—476no—
258C2×D4—473no—
259C2×D4—473no—
260C2×D4C22×D4713yes—
261C22×C4—471no—
262C22×C4—473no—
263C22×C4—473no—
264C22×C4—473no—
265C22×C4—476no—
266C22×C4C22×D4713yes—
267C22×C4C22×D4716yes—
268C4⋊C4—473no—
269C4⋊C4—476no—
270C4⋊C4—471no—
271C4⋊C4—471no—
272C4⋊C4—473no—
273C4⋊C4—473no—
274C4⋊C4—473no—
275C4⋊C4—473no—
276C4⋊C4—473no—
277C4⋊C4—473no—
278C4⋊C4—473no—
279C4⋊C4C22×D4713yes—
280C4⋊C4C22×D4713yes—
281C4⋊C4C22×D4713yes—
282C4⋊C4C22×D4711yescharacteristic
283C22⋊C4—473no—
284C22⋊C4—476no—
285C22⋊C4—473no—
286C22⋊C4—473no—
287C22⋊C4—473no—
288C22⋊C4—471no—
289C22⋊C4—471no—
290C22⋊C4—473no—
291C22⋊C4—476no—
292C22⋊C4—473no—
293C22⋊C4C22×D4713yes—
294C22⋊C4C22×D4713yes—
295C42—473no—
296C42—473no—
297C42—473no—
298C42—473no—
299C42C22×D4713yes—
300C42C22×D4713yes—
301C23—471no—
302C23—473no—
303C23C23×D4713yes—
304Q8—942no—
305Q8—946no—
306Q8—1886no—
307Q8#1062535014yescomplemented
308D4—26no—
309D4—946no—
310D4—946no—
311D4—946no—
312D4—946no—
313D4—1886no—
314C2×C4—26no—
315C2×C4—46no—
316C2×C4—471no—
317C2×C4—471no—
318C2×C4—473no—
319C2×C4—473no—
320C2×C4—473no—
321C2×C4—471no—
322C2×C4—942no—
323C2×C4—946no—
324C2×C4—946no—
325C2×C4—946no—
326C2×C4—1886no—
327C2×C4C23×D4711yescharacteristic
328C2×C4C23×D4713yes—
329C2×C4C23×D4711yescharacteristic
330C2×C4C23×D4713yes—
331C2×C4C23×D4711yescharacteristic
332C2×C4C23×D4713yes—
333C22—23no—
334C22—23no—
335C22—23no—
336C22—472no—
337C22—941no—
338C22—941no—
339C22—943no—
340C22—943no—
341C22—943no—
342C22C24×D4711yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—21no—
348C4—472no—
349C4—941no—
350C4—943no—
351C4—943no—
352C4—943no—
353C4—941no—
354C4C2×#1062535016yes—
355C2—43no—
356C2—942no—
357C2—1883no—
358C2C22×#1062535011yescharacteristic
359C2H(5,𝔽2)×D4711yescharacteristic
360C22−1+4×D4711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D47
commutator, socleC22×C47C25
FittingC47×#520C2
hypercenter#520D47
hypocenterC47#3522
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #15689856C2semidirect
2. Q8×Dic94C2semidirect
3. #15689828C2semidirect
4. #15689792C2semidirect
5. #15689785C2semidirect
6. #15689780C2semidirect
7. #15689778C2semidirect
8. #15689774C2semidirect
9. #15689744C2semidirect
10. #15689738C2semidirect
11. #15689670C2semidirect
12. #15689651C2semidirect
13. #15689605C2semidirect
14. C4×Q8×D47C2semidirect
15. C47×#520C2semidirect
123. Q8×C47#41semidirect
194. #520D47semidirect
196. C47#3522semidirect
307. Q8#10625350semidirect