Finite Group Databasebeta

← #20284643

Subgroups — #20284643

Order 6784.

414 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811133C211yesmaximal, characteristic
2#117×D53C211yesmaximal, characteristic, complemented
3#16810967C211yesmaximal, characteristic, complemented
4#16810966C211yesmaximal, characteristic, complemented
5#16810965C211yesmaximal, characteristic, complemented
6#16810964C211yesmaximal, characteristic, complemented
7#16810946C211yesmaximal, characteristic, complemented
8#16810945C211yesmaximal, characteristic, complemented
9#16810944C211yesmaximal, characteristic, complemented
10#16810664C211yesmaximal, characteristic, complemented
11#16810662C211yesmaximal, characteristic, complemented
12#16810621C211yesmaximal, characteristic, complemented
13#16810621C211yesmaximal, characteristic, complemented
14#16810585C211yesmaximal, characteristic, complemented
15C53×#482C211yesmaximal, characteristic, complemented
16C2×#11719947C2211yescharacteristic
17C22×C4×D53C2211yescharacteristic
18C2×#11719912C2211yescharacteristic
19C2×#11719912C2211yescharacteristic
20#13133199—21no—
21#13133197C2211yescharacteristic
22#13133196C2211yescharacteristic
23#13133188C2211yescharacteristic
24#13133185—21no—
25D4×Dic53—21no—
26#13133177—21no—
27#13133176C2211yescharacteristic
28#13133175C2211yescharacteristic
29#13133175C2211yescharacteristic
30#13133175C2211yescharacteristic
31#13133174—21no—
32#13133173—21no—
33C4⋊C4×D53C2211yescharacteristic
34C4⋊C4×D53C2211yescharacteristic
35C22⋊C4×D53—21no—
36C42×D53—21no—
37#13133151C2211yescharacteristic, complemented
38#13133150C2211yescharacteristic, complemented
39#13133150C2211yescharacteristic, complemented
40#13133149C2211yescharacteristic, complemented
41#13133148C2211yescharacteristic, complemented
42#13133148C2211yescharacteristic, complemented
43#13133142C2211yescharacteristic, complemented
44#13133142C2211yescharacteristic, complemented
45#13133140C2211yescharacteristic, complemented
46#13133140C2211yescharacteristic, complemented
47#13133138C2211yescharacteristic, complemented
48#13133138C2211yescharacteristic, complemented
49C22⋊#88322C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133129C2211yescharacteristic
52#121×C53C2211yescharacteristic
53#117×C53C2211yescharacteristic
54#107×C53C2211yescharacteristic, complemented
55#106×C53C2211yescharacteristic, complemented
56#102×C53C2211yescharacteristic, complemented
57#102×C53C2211yescharacteristic, complemented
58C22⋊C8×C53C2211yescharacteristic
59C23×D53D411yescharacteristic
60C2×D4×C53—21no—
61C2×D4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C22×Dic53—21no—
66C22×Dic53D411yescharacteristic
67#11719947—22no—
68#11719947—22no—
69C2×C4×D53—22no—
70C2×C4×D53—22no—
71C2×C4×D53—21no—
72C2×C4×D53—21no—
73C2×C4×D53D411yescharacteristic
74C2×C4×D53D411yescharacteristic
75C2×C4×D53C2311yescharacteristic
76C2×C4×D53C2311yescharacteristic
77C2×Dic106C2311yescharacteristic
78C2×C8×C53C2311yescharacteristic
79C2×C8×C53C2311yescharacteristic
80C4⋊C4×C53C2311yescharacteristic
81C4⋊C4×C53C2311yescharacteristic
82C4⋊C4×C53C2311yescharacteristic
83C22⋊C4×C53—21no—
84C22⋊C4×C53—21no—
85C42×C53—21no—
86C22⋊Dic53—21no—
87C22⋊Dic53—21no—
88C22⋊Dic53—21no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93C4⋊Dic53—21no—
94C4⋊Dic53C2311yescharacteristic
95C4⋊Dic53C2311yescharacteristic
96C4⋊Dic53C2311yescharacteristic
97#11719919—21no—
98#11719919—21no—
99#11719919—21no—
100C4×Dic53—21no—
101C4×Dic53—21no—
102C4×Dic53—21no—
103C4×Dic53—21no—
104C2×#88322C2311yescharacteristic
105C2×#88322C2311yescharacteristic
106#11719912—22no—
107#11719912—22no—
108C23×C53—21no—
109C23×C53C2×D411yescharacteristic
110C22×D53—22no—
111C22×D53—41no—
112C22×D53—41no—
113C22×D53C2×D411yescharacteristic
114C22×D53C2×D411yescharacteristic
115D4×C53—21no—
116D4×C53—21no—
117D4×C53—42no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53C2×D411yescharacteristic
124C2×C4×C53C2411yescharacteristic
125C22⋊D53—42no—
126C22⋊D53—42no—
127C2×Dic53—22no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—42no—
133C2×Dic53C2×D411yescharacteristic
134C2×Dic53C2×D411yescharacteristic
135C4×D53—22no—
136C4×D53—21no—
137C4×D53—21no—
138C4×D53—22no—
139C4×D53—42no—
140C4×D53—42no—
141C4×D53#4112yes—
142C4×D53#4112yes—
143Dic106—21no—
144Dic106—21no—
145C8×C53—21no—
146C8×C53—21no—
147#88322—21no—
148#88322—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53C22×D411yescharacteristic
156C2×D53—21no—
157C2×D53—21no—
158C2×D53—21no—
159C2×D53—21no—
160C2×D53—21no—
161C2×D53—44no—
162C2×D53#12312yes—
163C4×C53—21no—
164C4×C53—41no—
165C4×C53—41no—
166C4×C53—41no—
167C4×C53C2×#4111yescharacteristic
168C4×C53C2×#4111yescharacteristic
169Dic53—21no—
170Dic53—41no—
171Dic53—41no—
172Dic53—41no—
173Dic53—41no—
174Dic53#12312yes—
175#3093—531nomaximal, Sylow
176C2×C53—41no—
177C2×C53—81no—
178C2×C53C2×#12311yescharacteristic
179C2×C53#57511yescharacteristic
180C2×C53#57511yescharacteristic
181D53—22no—
182D53—41no—
183#525—531no—
184C2×#117—531no—
185#485—531no—
186#485—531no—
187#485—531no—
188#485—531no—
189#482—531no—
190#482—531no—
191#482—531no—
192#482D5311yescharacteristic, complemented
193#428—531no—
194#428—531no—
195#417—531no—
196#417—531no—
197#407—531no—
198C53#309311yescharacteristic, complemented, Sylow
199C2×#41—531no—
200C23×C4—531no—
201C2×M16—531no—
202C2×M16—531no—
203#123—1061no—
204C22⋊Q8—531no—
205C22⋊Q8—531no—
206#121—531no—
207#121C2×D5311yescharacteristic
208#120—1061no—
209C4×D4—1061no—
210C4×D4—1061no—
211#117—531no—
212#117—531no—
213#117—531no—
214#117—531no—
215#117—1061no—
216#117—1061no—
217#117C2×D5311yescharacteristic
218C2×C4⋊C4—531no—
219C2×C4⋊C4—531no—
220C2×C22⋊C4—1061no—
221C2×C42—1061no—
222#107—531no—
223#107—531no—
224#107—531no—
225#107C2×D5311yescharacteristic, complemented
226#106—531no—
227#106—531no—
228#106—531no—
229#106C2×D5311yescharacteristic, complemented
230#103—531no—
231#103—531no—
232#103—531no—
233#103—531no—
234#102—531no—
235#102—531no—
236#102C2×D5311yescharacteristic, complemented
237#102C2×D5311yescharacteristic, complemented
238C22⋊C8—531no—
239C22⋊C8—531no—
240C22⋊C8—531no—
241C22⋊C8C2×D5311yescharacteristic
242C24—531no—
243#41—1062no—
244#41—1062no—
245C2×Q8—531no—
246C2×D4—21no—
247C2×D4—1061no—
248C2×D4—1061no—
249C2×D4C22×D5311yescharacteristic
250C22×C4—531no—
251C22×C4—531no—
252C22×C4—531no—
253C22×C4—531no—
254C22×C4—531no—
255C22×C4—1062no—
256C22×C4—1062no—
257C22×C4—1061no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4C22×D5311yescharacteristic
261M16—1062no—
262M16—1062no—
263C2×C8—531no—
264C2×C8—531no—
265C2×C8C22×D5311yescharacteristic
266C2×C8C22×D5311yescharacteristic
267C4⋊C4—531no—
268C4⋊C4—531no—
269C4⋊C4—531no—
270C4⋊C4—1061no—
271C4⋊C4—1061no—
272C4⋊C4—1061no—
273C4⋊C4—1061no—
274C4⋊C4C22×D5311yescharacteristic
275C4⋊C4C22×D5311yescharacteristic
276C4⋊C4C22×D5311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C42—21no—
287C42—1061no—
288C42—1061no—
289C42—1061no—
290C42—1061no—
291C23—21no—
292C23—531no—
293C23—531no—
294C23—1062no—
295C23—2121no—
296C23—2121no—
297C23D4×D5311yescharacteristic
298Q8—1061no—
299Q8—1061no—
300D4—21no—
301D4—21no—
302D4—42no—
303D4—2122no—
304D4—2122no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—531no—
311C2×C4—532no—
312C2×C4—532no—
313C2×C4—531no—
314C2×C4—1062no—
315C2×C4—1062no—
316C2×C4—1062no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—2122no—
324C2×C4—2122no—
325C2×C4—2122no—
326C2×C4D4×D5311yescharacteristic
327C2×C4C23×D5311yescharacteristic
328C8—21no—
329C8—21no—
330C8—1061no—
331C8—1061no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—532no—
339C22—1061no—
340C22—1061no—
341C22—1061no—
342C22—1061no—
343C22—1061no—
344C22—2124no—
345C22C2×D4×D5311yescharacteristic
346C4—21no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—532no—
351C4—1061no—
352C4—2121no—
353C4—2121no—
354C4—2121no—
355C4—2121no—
356C4#41×D5311yescharacteristic
357C4#41×D5311yescharacteristic
358C2—41no—
359C2—81no—
360C2—1062no—
361C2—2121no—
362C2#123×D5311yescharacteristic
363C2#1681141911yescharacteristic
364C2#1681142011yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#482C2
socleC22×C53C22×D4
hypercenter#482D53
hypocenterC53#3093
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
2. #117×D53C2semidirect
3. #16810967C2semidirect
4. #16810966C2semidirect
5. #16810965C2semidirect
6. #16810964C2semidirect
7. #16810946C2semidirect
8. #16810945C2semidirect
9. #16810944C2semidirect
10. #16810664C2semidirect
11. #16810662C2semidirect
12. #16810621C2semidirect
13. #16810621C2semidirect
14. #16810585C2semidirect
15. C53×#482C2semidirect
37. #13133151C22semidirect
38. #13133150C22semidirect
39. #13133150C22semidirect
40. #13133149C22semidirect
41. #13133148C22semidirect
42. #13133148C22semidirect
43. #13133142C22semidirect
44. #13133142C22semidirect
45. #13133140C22semidirect
46. #13133140C22semidirect
47. #13133138C22semidirect
48. #13133138C22semidirect
54. #107×C53C22semidirect
55. #106×C53C22semidirect
56. #102×C53C22semidirect
57. #102×C53C22semidirect
192. #482D53semidirect
198. C53#3093semidirect
225. #107C2×D53semidirect
229. #106C2×D53semidirect
236. #102C2×D53semidirect
237. #102C2×D53semidirect