Finite Group Databasebeta

← #20283520

Subgroups — #20283520

Order 6784.

548 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133244C211yesmaximal, characteristic
2C4×#11719949C211yesmaximal, characteristic, complemented
3C4×D4×D53C211yesmaximal, characteristic, complemented
4#16810736C211yesmaximal, characteristic, complemented
5#16810733C211yesmaximal, characteristic, complemented
6#16810721C211yesmaximal, characteristic, complemented
7SD16×Dic53C211yesmaximal, characteristic, complemented
8#16810708C211yesmaximal, characteristic, complemented
9#16810705C211yesmaximal, characteristic, complemented
10#16810703C211yesmaximal, characteristic, complemented
11#16810664C211yesmaximal, characteristic, complemented
12#16810626C211yesmaximal, characteristic, complemented
13#16810623C211yesmaximal, characteristic, complemented
14#16810575C211yesmaximal, characteristic, complemented
15C53×#440C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×D4×D53C2211yescharacteristic
18C22×C4×D53—21no—
19#13133244C418yescomplemented
20C2×#11719924C2211yescharacteristic
21C2×D424C2211yescharacteristic
22C2×#11719922C2211yescharacteristic
23C2×#11719912C2211yescharacteristic
24C2×#11719912C2211yescharacteristic
25Q8×Dic53C2211yescharacteristic, complemented
26C4×D212—21no—
27C4×D212C2211yescharacteristic
28C4×D212C2211yescharacteristic, complemented
29D4×Dic53C2211yescharacteristic
30#13133179—21no—
31#13133179C2211yescharacteristic
32#13133179C2211yescharacteristic, complemented
33C4×C22⋊D53—21no—
34#13133177—21no—
35#13133175—21no—
36#13133175C2211yescharacteristic
37C4⋊C4×D53C2211yescharacteristic
38C22⋊C4×D53—21no—
39C42×D53—21no—
40C42×D53C2211yescharacteristic
41#13133150C2211yescharacteristic, complemented
42#13133149C2211yescharacteristic, complemented
43#13133148C2211yescharacteristic, complemented
44#13133141C2211yescharacteristic, complemented
45#13133139C2211yescharacteristic
46#13133139C2211yescharacteristic, complemented
47#13133138C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133137C2211yescharacteristic, complemented
50#13133128C2211yescharacteristic, complemented
51C8×Dic53C2211yescharacteristic, complemented
52C8×Dic53C2211yescharacteristic, complemented
53C2×SD16×C53C2211yescharacteristic
54C4×Q8×C53C2211yescharacteristic, complemented
55C4×D4×C53C2211yescharacteristic
56#107×C53C2211yescharacteristic, complemented
57#103×C53C2211yescharacteristic, complemented
58#102×C53C2211yescharacteristic
59#97×C53C2211yescharacteristic, complemented
60C23×D53—21no—
61C2×Q8×C53C2311yescharacteristic
62C2×D4×C53C2311yescharacteristic
63C22×C4×C53—21no—
64C2×C22⋊D53—21no—
65C22×Dic53—21no—
66#11719949—22no—
67#11719949C2×C414yescomplemented
68D4×D53—22no—
69D4×D53C2×C414yescomplemented
70C2×D212—21no—
71C2×D212C2311yescharacteristic
72C2×D212C2311yescharacteristic
73C2×C4×D53—21no—
74C2×C4×D53—21no—
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×C4×D53—44no—
80C2×C4×D53D411yescharacteristic
81C2×C4×D53D411yescharacteristic
82C2×C4×D53C2311yescharacteristic
83SD16×C53C2×C414yescomplemented
84C2×C8×C53C2311yescharacteristic
85C2×C8×C53C2311yescharacteristic
86C4⋊C4×C53—21no—
87C4⋊C4×C53C2311yescharacteristic
88C4⋊C4×C53C2311yescharacteristic
89C22⋊C4×C53—21no—
90C42×C53—21no—
91C42×C53C2311yescharacteristic
92C22⋊Dic53—21no—
93#11719924C2×C414yescomplemented
94#11719922C2×C414yescomplemented
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99#11719921—21no—
100C4⋊Dic53—21no—
101C4⋊Dic53C2311yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103#11719919—21no—
104C4×Dic53—21no—
105C4×Dic53—21no—
106C4×Dic53D411yescharacteristic
107C4×Dic53D411yescharacteristic
108C4×Dic53C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111D424C2×C414yescomplemented
112#11719912—22no—
113#11719912C2×C414yescomplemented
114C23×C53—21no—
115C22×D53—21no—
116C22×D53—21no—
117C22×D53—24no—
118C22×D53—42no—
119C22×D53—42no—
120C22×D53C2×D411yescharacteristic
121Q8×C53—21no—
122Q8×C53C22×C412yescomplemented
123D4×C53—21no—
124D4×C53C22×C412yes—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—42no—
129C2×C4×C53C2×D411yescharacteristic
130C2×C4×C53C2×D411yescharacteristic
131C2×C4×C53C2411yescharacteristic
132C22⋊D53—24no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—42no—
136C2×Dic53C2×D411yescharacteristic
137C2×Dic53C2×D411yescharacteristic
138C2×Dic53C2×D411yescharacteristic
139D212—24no—
140D212—21no—
141D212—21no—
142D212C22×C412yes—
143D212C22×C412yescomplemented
144C4×D53—24no—
145C4×D53—22no—
146C4×D53—41no—
147C4×D53—42no—
148C4×D53—42no—
149C4×D53—42no—
150C4×D53—42no—
151C4×D53—42no—
152C4×D53C22×C412yes—
153C4×D53#4112yes—
154C8×C53—21no—
155C8×C53C22×C412yescomplemented
156#88322—21no—
157#88322C22×C412yescomplemented
158C22×C53—22no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53C22×D411yescharacteristic
162C2×D53—21no—
163C2×D53—21no—
164C2×D53—22no—
165C2×D53—22no—
166C2×D53—48no—
167C2×D53—41no—
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—41no—
171C2×D53C4×D412yes—
172C4×C53—22no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—41no—
176C4×C53—41no—
177C4×C53C23×C411yescharacteristic
178C4×C53C2×#4111yescharacteristic
179Dic53—21no—
180Dic53—41no—
181Dic53—41no—
182Dic53C4×D412yes—
183Dic53AGL(1,ℤ/8)12yescomplemented
184C4×AGL(1,ℤ/8)—531nomaximal, Sylow
185C2×C53—42no—
186C2×C53C2×C4×D411yescharacteristic
187C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
188C2×C53#57511yescharacteristic
189D53—22no—
190D53—42no—
191D53—42no—
192C2×AGL(1,ℤ/8)—531no—
193C4×#41—531no—
194C2×C4×D4—531no—
195#442—531no—
196#442—531no—
197#440—531no—
198#440D5311yescharacteristic, complemented
199C4×SD16—531no—
200C4×SD16—531no—
201C4×D8—531no—
202C4×D8—531no—
203#428—531no—
204#418—531no—
205#417—531no—
206C4×M16—531no—
207C53C4×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
208C2×#41—531no—
209C22×D4—531no—
210C23×C4—1061no—
211AGL(1,ℤ/8)—538no—
212C2×SD16—531no—
213C2×SD16C2×D5311yescharacteristic
214C2×D8—531no—
215C2×D8—531no—
216C2×M16—531no—
217C2×M16—531no—
218C4×Q8—531no—
219C4×Q8C2×D5311yescharacteristic, complemented
220C4×D4—531no—
221C4×D4—531no—
222C4×D4—531no—
223C4×D4—531no—
224C4×D4—531no—
225C4×D4—1061no—
226C4×D4—1061no—
227C4×D4—1061no—
228C4×D4—1061no—
229C4×D4C2×D5311yescharacteristic
230#117—531no—
231#117—1061no—
232C2×C4⋊C4—531no—
233C2×C22⋊C4—1061no—
234C2×C42—531no—
235C2×C42—1061no—
236#107—531no—
237#107C2×D5311yescharacteristic, complemented
238#106—531no—
239#106—531no—
240#103—531no—
241#103C2×D5311yescharacteristic, complemented
242#102—531no—
243#102—531no—
244#102—531no—
245#102—531no—
246#102—531no—
247#102C2×D5311yescharacteristic
248#97—531no—
249#97C2×D5311yescharacteristic, complemented
250C4×C8—531no—
251C4×C8—531no—
252C24—1061no—
253#41—534no—
254#41—1062no—
255C2×Q8C22×D5311yescharacteristic
256C2×D4—534no—
257C2×D4—531no—
258C2×D4—531no—
259C2×D4—1061no—
260C2×D4—1062no—
261C2×D4—1061no—
262C2×D4C22×D5311yescharacteristic
263C22×C4—21no—
264C22×C4—531no—
265C22×C4—531no—
266C22×C4—531no—
267C22×C4—1061no—
268C22×C4—1061no—
269C22×C4—1061no—
270C22×C4—1061no—
271C22×C4—1061no—
272C22×C4—1061no—
273C22×C4—1061no—
274C22×C4—2124no—
275SD16—534no—
276SD16C4×D5314yescomplemented
277D8—534no—
278D8—534no—
279M16—534no—
280M16—1062no—
281C2×C8—531no—
282C2×C8—531no—
283C2×C8C22×D5311yescharacteristic
284C2×C8C22×D5311yescharacteristic
285C4⋊C4—21no—
286C4⋊C4—531no—
287C4⋊C4—531no—
288C4⋊C4—1061no—
289C4⋊C4—1061no—
290C4⋊C4C22×D5311yescharacteristic
291C4⋊C4C22×D5311yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—1061no—
294C22⋊C4—1061no—
295C22⋊C4—1061no—
296C22⋊C4—1061no—
297C22⋊C4—1061no—
298C22⋊C4—1061no—
299C42—21no—
300C42—531no—
301C42—531no—
302C42—531no—
303C42—1061no—
304C42—1061no—
305C42C22×D5311yescharacteristic
306C23—21no—
307C23—531no—
308C23—1064no—
309C23—1061no—
310C23—1061no—
311C23—2122no—
312C23—2122no—
313Q8—21no—
314Q8C2×C4×D5312yescomplemented
315D4—21no—
316D4—532no—
317D4—532no—
318D4—1064no—
319D4—1064no—
320D4—1061no—
321D4—1061no—
322D4C2×C4×D5312yes—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—531no—
328C2×C4—531no—
329C2×C4—532no—
330C2×C4—532no—
331C2×C4—531no—
332C2×C4—1062no—
333C2×C4—1064no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—2121no—
337C2×C4—2122no—
338C2×C4—2122no—
339C2×C4—2122no—
340C2×C4—2122no—
341C2×C4—2122no—
342C2×C4—2122no—
343C2×C4D4×D5311yescharacteristic
344C2×C4D4×D5311yescharacteristic
345C2×C4C23×D5311yescharacteristic
346C8—21no—
347C8—532no—
348C8—1061no—
349C8C2×C4×D5312yescomplemented
350C22—22no—
351C22—41no—
352C22—41no—
353C22—532no—
354C22—1061no—
355C22—1061no—
356C22—1062no—
357C22—1062no—
358C22—2128no—
359C22—2121no—
360C22—2121no—
361C22—2121no—
362C22—2121no—
363C22C2×D4×D5311yescharacteristic
364C4—22no—
365C4—21no—
366C4—21no—
367C4—41no—
368C4—41no—
369C4—532no—
370C4—532no—
371C4—1061no—
372C4—2121no—
373C4—2121no—
374C4C22×C4×D5311yescharacteristic
375C4#41×D5311yescharacteristic
376C2—42no—
377C2—1062no—
378C2—2122no—
379C2—2122no—
380C2C4×D4×D5311yescharacteristic
381C2AGL(1,ℤ/8)×D5311yescharacteristic
382C2#1681142211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC4×C53C23×C4
FrattiniC2×C4C23×D53
FittingC53×#440C2
socleC22×C53C22×D4
hypercenter#440D53
hypocenterC53C4×AGL(1,ℤ/8)
LCSG ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. C4×#11719949C2semidirect
3. C4×D4×D53C2semidirect
4. #16810736C2semidirect
5. #16810733C2semidirect
6. #16810721C2semidirect
7. SD16×Dic53C2semidirect
8. #16810708C2semidirect
9. #16810705C2semidirect
10. #16810703C2semidirect
11. #16810664C2semidirect
12. #16810626C2semidirect
13. #16810623C2semidirect
14. #16810575C2semidirect
15. C53×#440C2semidirect
19. #13133244C4semidirect
25. Q8×Dic53C22semidirect
28. C4×D212C22semidirect
32. #13133179C22semidirect
41. #13133150C22semidirect
42. #13133149C22semidirect
43. #13133148C22semidirect
44. #13133141C22semidirect
46. #13133139C22semidirect
49. #13133137C22semidirect
50. #13133128C22semidirect
51. C8×Dic53C22semidirect
52. C8×Dic53C22semidirect
54. C4×Q8×C53C22semidirect
56. #107×C53C22semidirect
57. #103×C53C22semidirect
59. #97×C53C22semidirect
67. #11719949C2×C4semidirect
69. D4×D53C2×C4semidirect
83. SD16×C53C2×C4semidirect
93. #11719924C2×C4semidirect
94. #11719922C2×C4semidirect
111. D424C2×C4semidirect
113. #11719912C2×C4semidirect
122. Q8×C53C22×C4semidirect
143. D212C22×C4semidirect
155. C8×C53C22×C4semidirect
157. #88322C22×C4semidirect
183. Dic53AGL(1,ℤ/8)semidirect
198. #440D53semidirect
207. C53C4×AGL(1,ℤ/8)semidirect
219. C4×Q8C2×D53semidirect
237. #107C2×D53semidirect
241. #103C2×D53semidirect
249. #97C2×D53semidirect
276. SD16C4×D53semidirect
314. Q8C2×C4×D53semidirect
349. C8C2×C4×D53semidirect