Finite Group Databasebeta

← #20287425

Subgroups — #20287425

Order 6784.

396 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811329C211yesmaximal, characteristic, complemented
2#16811294C211yesmaximal, characteristic, complemented
3#16810989C211yesmaximal, characteristic, complemented
4#16810986C211yesmaximal, characteristic, complemented
5#16810968C211yesmaximal, characteristic, complemented
6#16810926C211yesmaximal, characteristic, complemented
7#16810912C211yesmaximal, characteristic, complemented
8#16810819C211yesmaximal, characteristic, complemented
9#16810817C211yesmaximal, characteristic, complemented
10#16810817C211yesmaximal, characteristic, complemented
11#16810752C211yesmaximal, characteristic, complemented
12#16810734C211yesmaximal, characteristic, complemented
13#16810733C211yesmaximal, characteristic, complemented
14#16810705C211yesmaximal, characteristic, complemented
15C53×#478C211yesmaximal, characteristic, complemented
16C2×#11719945—21no—
17C2×D424C2211yescharacteristic
18C2×D424C2211yescharacteristic
19C2×#11719922C2211yescharacteristic
20#13133216C2211yescharacteristic
21#13133214C2211yescharacteristic
22#13133213C2211yescharacteristic
23#13133213C2211yescharacteristic
24#13133208C2211yescharacteristic
25#13133197—21no—
26#13133195—21no—
27#13133195—21no—
28#13133191—21no—
29#13133190—21no—
30#13133190—21no—
31C4×Dic106C2211yescharacteristic
32C4×D212C2211yescharacteristic
33#13133179C2211yescharacteristic
34#13133179C2211yescharacteristic
35#13133179C2211yescharacteristic
36#13133179C2211yescharacteristic
37C4⋊C4×D53—21no—
38#13133148C2211yescharacteristic
39#13133146C2211yescharacteristic
40#13133146C2211yescharacteristic
41#13133139C2211yescharacteristic
42#13133139C2211yescharacteristic
43#13133138C2211yescharacteristic
44#13133137C2211yescharacteristic
45#13133137C2211yescharacteristic
46#13133137C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133128C2211yescharacteristic
50#13133127C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52#125×C53C2211yescharacteristic
53C4×D4×C53C2211yescharacteristic
54#107×C53C2211yescharacteristic
55#106×C53C2211yescharacteristic
56C4⋊C8×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59C2×D4×C53C2311yescharacteristic
60C22×C4×C53—21no—
61C2×C22⋊D53—21no—
62C2×C22⋊D53—21no—
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65#11719945—44no—
66C2×D212—21no—
67C2×D212—21no—
68C2×D212C2311yescharacteristic
69C2×D212C2311yescharacteristic
70C2×D212C2311yescharacteristic
71C2×C4×D53—21no—
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×C4×D53—21no—
75C2×C4×D53—21no—
76C2×Dic106—21no—
77C2×Dic106D411yescharacteristic
78C2×C8×C53C2311yescharacteristic
79C2×C8×C53C2311yescharacteristic
80C4⋊C4×C53—21no—
81C4⋊C4×C53—21no—
82C4⋊C4×C53C2311yescharacteristic
83C4⋊C4×C53C2311yescharacteristic
84C4⋊C4×C53C2311yescharacteristic
85C22⋊C4×C53—21no—
86C42×C53C2311yescharacteristic
87#11719922—22no—
88#11719921—21no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93C4⋊Dic53—21no—
94C4⋊Dic53D411yescharacteristic
95C4⋊Dic53C2311yescharacteristic
96#11719919—21no—
97#11719919—21no—
98#11719919D411yescharacteristic
99#11719919D411yescharacteristic
100C4×Dic53C2311yescharacteristic
101C4×Dic53C2311yescharacteristic
102C2×#88322C2311yescharacteristic
103C2×#88322C2311yescharacteristic
104D424—22no—
105D424—22no—
106C23×C53—21no—
107C22×D53—21no—
108C22×D53—21no—
109C22×D53—21no—
110D4×C53—21no—
111D4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—42no—
116C2×C4×C53C2×D411yescharacteristic
117C2×C4×C53C2×D411yescharacteristic
118C2×C4×C53C2411yescharacteristic
119C22⋊D53—42no—
120C22⋊D53—42no—
121C22⋊D53—81no—
122C22⋊D53—81no—
123C2×Dic53—21no—
124C2×Dic53C2×D411yescharacteristic
125C2×Dic53C2×D411yescharacteristic
126C2×Dic53C2×D411yescharacteristic
127C2×Dic53C2×D411yescharacteristic
128D212—21no—
129D212—21no—
130D212—21no—
131D212—21no—
132D212—21no—
133D212—42no—
134D212—81no—
135D212#4112yescomplemented
136C4×D53—42no—
137C4×D53—42no—
138C4×D53—42no—
139C4×D53—81no—
140C4×D53—81no—
141Dic106—22no—
142Dic106—81no—
143C8×C53—21no—
144C8×C53—21no—
145#88322—21no—
146#88322—21no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53C22×D411yescharacteristic
151C2×D53—22no—
152C2×D53—41no—
153C2×D53—41no—
154C2×D53—41no—
155C2×D53—41no—
156C2×D53—41no—
157C2×D53—41no—
158C2×D53—41no—
159C2×D53—41no—
160C4×C53—21no—
161C4×C53—21no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53C2×#4111yescharacteristic
166C4×C532−1+411yescharacteristic
167Dic53—21no—
168Dic53—21no—
169Dic53—21no—
170Dic53—41no—
171Dic53AGL(1,ℤ/8)12yes—
172#3366—531nomaximal, Sylow
173C2×C53—81no—
174C2×C53#54811yescharacteristic
175C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
176C2×C53#57611yescharacteristic
177D53—42no—
178D53—81no—
179D53—81no—
180#554—531no—
181#550—531no—
182#490—531no—
183#490—531no—
184#486—531no—
185#478—531no—
186#478D5311yescharacteristic, complemented
187#476—531no—
188#459—531no—
189#459—531no—
190#459—531no—
191#446—531no—
192#442—531no—
193#442—531no—
194C4×D8—531no—
195C53#336611yescharacteristic, complemented, Sylow
196C2×#41—1061no—
197C2×D8—531no—
198C2×D8—531no—
199C2×D8—531no—
200C4⋊Q8—531no—
201#127—531no—
202#127—531no—
203#127—531no—
204#125—531no—
205#125C2×D5311yescharacteristic
206C22⋊Q8—1061no—
207C22⋊Q8—1061no—
208C22⋊Q8—1061no—
209#121—1061no—
210#121—1061no—
211#121—1061no—
212C4×Q8—531no—
213C4×D4—531no—
214C4×D4—531no—
215C4×D4—531no—
216C4×D4—531no—
217C4×D4—531no—
218C4×D4C2×D5311yescharacteristic
219C2×C4⋊C4—1061no—
220#107C2×D5311yescharacteristic
221#106—531no—
222#106C2×D5311yescharacteristic
223C4⋊C8—531no—
224C4⋊C8—531no—
225C4⋊C8C2×D5311yescharacteristic
226#102—531no—
227#102—531no—
228#102—531no—
229#102—531no—
230#102—531no—
231#102—531no—
232#102—531no—
233#102—531no—
234#102C2×D5311yescharacteristic
235#102C2×D5311yescharacteristic
236#97—531no—
237#97—531no—
238C4×C8—531no—
239#41—2124no—
240C2×Q8—531no—
241C2×Q8—1061no—
242C2×D4—531no—
243C2×D4—531no—
244C2×D4—531no—
245C2×D4—1061no—
246C2×D4—1061no—
247C2×D4—1061no—
248C2×D4—1061no—
249C2×D4—1061no—
250C2×D4—1061no—
251C2×D4C22×D5311yescharacteristic
252C22×C4—21no—
253C22×C4—1061no—
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4—1061no—
258D8—1062no—
259D8—1062no—
260D8—1062no—
261C2×C8—531no—
262C2×C8—531no—
263C2×C8C22×D5311yescharacteristic
264C2×C8C22×D5311yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—531no—
268C4⋊C4—531no—
269C4⋊C4—531no—
270C4⋊C4—531no—
271C4⋊C4—1061no—
272C4⋊C4—1061no—
273C4⋊C4—1061no—
274C4⋊C4C22×D5311yescharacteristic
275C4⋊C4C22×D5311yescharacteristic
276C4⋊C4C22×D5311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—1061no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C42—531no—
284C42—531no—
285C42C22×D5311yescharacteristic
286C23—21no—
287C23—1061no—
288C23—1061no—
289C23—1061no—
290Q8—1062no—
291Q8—4241no—
292D4—21no—
293D4—21no—
294D4—532no—
295D4—1061no—
296D4—1061no—
297D4—1061no—
298D4—1061no—
299D4—1061no—
300D4—4241no—
301D4—4241no—
302D4—4241no—
303D4—2122no—
304D4—2122no—
305D4—2122no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—531no—
311C2×C4—531no—
312C2×C4—531no—
313C2×C4—531no—
314C2×C4—1061no—
315C2×C4—4241no—
316C2×C4—4241no—
317C2×C4—2122no—
318C2×C4—2122no—
319C2×C4—2122no—
320C2×C4D4×D5311yescharacteristic
321C2×C4D4×D5311yescharacteristic
322C2×C4C23×D5311yescharacteristic
323C8—21no—
324C8—21no—
325C8—1061no—
326C8—1061no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—1062no—
331C22—2121no—
332C22—2121no—
333C22—2121no—
334C22—2121no—
335C22—2121no—
336C22—2121no—
337C22—2121no—
338C22—2121no—
339C22C2×D4×D5311yescharacteristic
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—532no—
346C4—1061no—
347C4—1061no—
348C4—1061no—
349C4—2121no—
350C4#41×D5311yescharacteristic
351C4#1313326511yescharacteristic
352C2—81no—
353C2—4241no—
354C2—4241no—
355C2—2122no—
356C2#1681128311yescharacteristic
357C2AGL(1,ℤ/8)×D5311yescharacteristic
358C2#1681142511yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#478C2
socleC22×C53C22×D4
hypercenter#478D53
hypocenterC53#3366
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #16811329C2semidirect
2. #16811294C2semidirect
3. #16810989C2semidirect
4. #16810986C2semidirect
5. #16810968C2semidirect
6. #16810926C2semidirect
7. #16810912C2semidirect
8. #16810819C2semidirect
9. #16810817C2semidirect
10. #16810817C2semidirect
11. #16810752C2semidirect
12. #16810734C2semidirect
13. #16810733C2semidirect
14. #16810705C2semidirect
15. C53×#478C2semidirect
135. D212#41semidirect
186. #478D53semidirect
195. C53#3366semidirect