Finite Group Databasebeta

← #20284899

Subgroups — #20284899

Order 6784.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811244C211yesmaximal, characteristic, complemented
2#16810964C211yesmaximal, characteristic, complemented
3#16810947C211yesmaximal, characteristic, complemented
4#16810913C211yesmaximal, characteristic, complemented
5#16810903C211yesmaximal, characteristic, complemented
6#16810838C211yesmaximal, characteristic, complemented
7#16810838C211yesmaximal, characteristic, complemented
8#16810774C211yesmaximal, characteristic, complemented
9#16810757C211yesmaximal, characteristic
10#16810733C211yesmaximal, characteristic, complemented
11#16810723C211yesmaximal, characteristic, complemented
12C4×#11719923C211yesmaximal, characteristic, complemented
13C4×#11719922C211yesmaximal, characteristic, complemented
14#16810684C211yesmaximal, characteristic, complemented
15C53×#518C211yesmaximal, characteristic, complemented
16C2×#11719923C2211yescharacteristic
17C2×#11719923C2211yescharacteristic
18C2×#11719922C2211yescharacteristic
19C2×#11719922C2211yescharacteristic
20#13133217C2211yescharacteristic, complemented
21#13133212—21no—
22#13133205C2211yescharacteristic, complemented
23#13133203—21no—
24#13133199—21no—
25#13133196C2211yescharacteristic
26#13133195—21no—
27#13133191C2211yescharacteristic
28C4×Dic106C2211yescharacteristic, complemented
29C4×D212C2211yescharacteristic, complemented
30#13133150C2211yescharacteristic, complemented
31#13133150C2211yescharacteristic, complemented
32#13133148C2211yescharacteristic, complemented
33#13133148C2211yescharacteristic, complemented
34C4⋊#88322C2211yescharacteristic, complemented
35C4⋊#88322C2211yescharacteristic, complemented
36#13133140C2211yescharacteristic, complemented
37#13133140C2211yescharacteristic, complemented
38#13133138C2211yescharacteristic
39#13133138C2211yescharacteristic
40#13133138C2211yescharacteristic
41#13133138C2211yescharacteristic
42#13133137C2211yescharacteristic, complemented
43#13133137C2211yescharacteristic, complemented
44C22⋊#88322C2211yescharacteristic
45C22⋊#88322C2211yescharacteristic
46#13133128C2211yescharacteristic, complemented
47C4×#88322C2211yescharacteristic, complemented
48C22×D4×C53C2211yescharacteristic
49C4×D4×C53—21no—
50C4×D4×C53—21no—
51C4×D4×C53C2211yescharacteristic
52C4×D4×C53C2211yescharacteristic
53C4×D4×C53C2211yescharacteristic
54C4×D4×C53C2211yescharacteristic
55#117×C53C2211yescharacteristic
56#117×C53C2211yescharacteristic
57C2×C22⋊C4×C53—21no—
58C2×C22⋊C4×C53—21no—
59C24×C53—21no—
60C2×D4×C53—22no—
61C2×D4×C53—22no—
62C2×D4×C53—21no—
63C2×D4×C53C2311yescharacteristic
64C2×D4×C53C2311yescharacteristic
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53C2311yescharacteristic
70C2×C22⋊D53—21no—
71C2×D212C2311yescharacteristic
72C2×C4×D53—21no—
73C2×Dic106—21no—
74C2×Dic106C2311yescharacteristic
75C4⋊C4×C53D411yescharacteristic
76C4⋊C4×C53D411yescharacteristic
77C4⋊C4×C53C2311yescharacteristic
78C4⋊C4×C53C2311yescharacteristic
79C22⋊C4×C53—21no—
80C22⋊C4×C53—21no—
81C22⋊C4×C53—21no—
82C22⋊C4×C53—21no—
83C22⋊C4×C53—21no—
84C22⋊C4×C53D411yescharacteristic
85C22⋊C4×C53D411yescharacteristic
86C42×C53—21no—
87C42×C53C2311yescharacteristic
88C42×C53C2311yescharacteristic
89C22⋊Dic53—21no—
90#11719923—22no—
91#11719923—22no—
92#11719922—22no—
93#11719922—22no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98C4⋊Dic53—21no—
99C4⋊Dic53C2311yescharacteristic
100C4⋊Dic53C2311yescharacteristic
101#11719919—21no—
102#11719919—21no—
103C4×Dic53—21no—
104C2×#88322C2311yescharacteristic
105C2×#88322C2311yescharacteristic
106C2×#88322C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108C23×C53—24no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—81no—
112C23×C53—81no—
113C23×C53C2×D411yescharacteristic
114C22×D53—21no—
115D4×C53—24no—
116D4×C53—21no—
117D4×C53—21no—
118D4×C53#4112yes—
119D4×C53#4112yes—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—81no—
125C2×C4×C53—81no—
126C2×C4×C53—81no—
127C2×C4×C53—81no—
128C2×C4×C53C2×D411yescharacteristic
129C2×C4×C53C2×D411yescharacteristic
130C2×C4×C53C2×D411yescharacteristic
131C2×C4×C53C2×D411yescharacteristic
132C2×C4×C53C2×D411yescharacteristic
133C2×C4×C53C2411yescharacteristic
134C22⋊D53—81no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138D212—21no—
139D212—21no—
140C4×D53—81no—
141Dic106—21no—
142Dic106—21no—
143Dic106—81no—
144#88322—21no—
145#88322—21no—
146#88322—21no—
147#88322—21no—
148C22×C53—21no—
149C22×C53—22no—
150C22×C53—22no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—84no—
158C22×C53C22×D411yescharacteristic
159C2×D53—41no—
160C2×D53—41no—
161C2×D53—41no—
162C4×C53—21no—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53C2×#4111yescharacteristic
170C4×C53C2×#4111yescharacteristic
171Dic53—41no—
172Dic53—41no—
173Dic53—41no—
174#3118—531nomaximal, Sylow
175C2×C53—41no—
176C2×C53—42no—
177C2×C53—42no—
178C2×C53#52511yescharacteristic
179C2×C53#57611yescharacteristic
180C2×C53#57711yescharacteristic
181D53—81no—
182#542—531no—
183#518D5311yescharacteristic, complemented
184#485—531no—
185#482—531no—
186#476—531no—
187#474—531no—
188#463—531no—
189#463—531no—
190#450—531no—
191#447—531no—
192#442—531no—
193#440—531no—
194C4×SD16—531no—
195C4×D8—531no—
196#433—531no—
197C53#311811yescharacteristic, complemented, Sylow
198C22×D4C2×D5311yescharacteristic
199C2×SD16—531no—
200C2×SD16—531no—
201C2×D8—531no—
202C2×D8—531no—
203C4⋊Q8—531no—
204#126—1061no—
205#124—531no—
206#124—1061no—
207#123—1061no—
208C22⋊Q8—531no—
209C22⋊Q8—1061no—
210#121—531no—
211C4×Q8—531no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—531no—
215C4×D4C2×D5311yescharacteristic
216C4×D4C2×D5311yescharacteristic
217C4×D4C2×D5311yescharacteristic
218C4×D4C2×D5311yescharacteristic
219#117C2×D5311yescharacteristic
220#117C2×D5311yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223#107—531no—
224#107—531no—
225#106—531no—
226#106—531no—
227C4⋊C8—531no—
228C4⋊C8—531no—
229#103—531no—
230#103—531no—
231#102—531no—
232#102—531no—
233#102—531no—
234#102—531no—
235#102—531no—
236#102—531no—
237C22⋊C8—531no—
238C22⋊C8—531no—
239#97—531no—
240C4×C8—531no—
241C24—21no—
242C2×Q8—531no—
243C2×Q8—1061no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—531no—
248C2×D4—1061no—
249C2×D4C22×D5311yescharacteristic
250C2×D4C22×D5311yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—1061no—
256C22×C4C22×D5311yescharacteristic
257SD16—1062no—
258SD16—1062no—
259D8—1062no—
260D8—1062no—
261C2×C8—531no—
262C2×C8—531no—
263C2×C8—531no—
264C2×C8—531no—
265C4⋊C4—531no—
266C4⋊C4—531no—
267C4⋊C4—1061no—
268C4⋊C4—1061no—
269C4⋊C4—1061no—
270C4⋊C4C22⋊D5311yescharacteristic
271C4⋊C4C22⋊D5311yescharacteristic
272C4⋊C4C22×D5311yescharacteristic
273C4⋊C4C22×D5311yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C22⋊C4C22⋊D5311yescharacteristic
285C22⋊C4C22⋊D5311yescharacteristic
286C42—21no—
287C42—1061no—
288C42C22×D5311yescharacteristic
289C42C22×D5311yescharacteristic
290C23—24no—
291C23—21no—
292C23—21no—
293C23—81no—
294C23—81no—
295C23—1061no—
296C23C2×C22⋊D5311yescharacteristic
297Q8—1061no—
298Q8—1061no—
299Q8—4241no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—1061no—
304D4—1061no—
305D4—4241no—
306D4#1171994512yes—
307D4#1171994512yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—4241no—
320C2×C4C2×C22⋊D5311yescharacteristic
321C2×C4C2×C22⋊D5311yescharacteristic
322C2×C4C2×C22⋊D5311yescharacteristic
323C2×C4C2×C22⋊D5311yescharacteristic
324C2×C4C2×C22⋊D5311yescharacteristic
325C2×C4C23×D5311yescharacteristic
326C8—1061no—
327C8—1061no—
328C8—1061no—
329C8—1061no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—84no—
340C22—2121no—
341C22—2121no—
342C22—2121no—
343C22C22×C22⋊D5311yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—2121no—
352C4—2121no—
353C4—2121no—
354C4C2×#1171994511yescharacteristic
355C4C2×#1171994511yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—4241no—
360C2#1681113411yescharacteristic
361C2#1681142611yescharacteristic
362C2#1681143411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#518C2
socleC22×C53C22×D4
hypercenter#518D53
hypocenterC53#3118
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #16811244C2semidirect
2. #16810964C2semidirect
3. #16810947C2semidirect
4. #16810913C2semidirect
5. #16810903C2semidirect
6. #16810838C2semidirect
7. #16810838C2semidirect
8. #16810774C2semidirect
10. #16810733C2semidirect
11. #16810723C2semidirect
12. C4×#11719923C2semidirect
13. C4×#11719922C2semidirect
14. #16810684C2semidirect
15. C53×#518C2semidirect
20. #13133217C22semidirect
22. #13133205C22semidirect
28. C4×Dic106C22semidirect
29. C4×D212C22semidirect
30. #13133150C22semidirect
31. #13133150C22semidirect
32. #13133148C22semidirect
33. #13133148C22semidirect
34. C4⋊#88322C22semidirect
35. C4⋊#88322C22semidirect
36. #13133140C22semidirect
37. #13133140C22semidirect
42. #13133137C22semidirect
43. #13133137C22semidirect
46. #13133128C22semidirect
47. C4×#88322C22semidirect
183. #518D53semidirect
197. C53#3118semidirect