Finite Group Databasebeta

← #20286678

Subgroups — #20286678

Order 6784.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811277C211yesmaximal, characteristic, complemented
2#16811269C211yesmaximal, characteristic, complemented
3#16810983C211yesmaximal, characteristic, complemented
4#16810954C211yesmaximal, characteristic, complemented
5#16810942C211yesmaximal, characteristic, complemented
6#16810914C211yesmaximal, characteristic, complemented
7#16810876C211yesmaximal, characteristic, complemented
8#16810852C211yesmaximal, characteristic, complemented
9#16810839C211yesmaximal, characteristic, complemented
10#16810770C211yesmaximal, characteristic, complemented
11#16810759C211yesmaximal, characteristic, complemented
12#16810736C211yesmaximal, characteristic, complemented
13#16810625C211yesmaximal, characteristic, complemented
14C2×#13133139C211yesmaximal, characteristic, complemented
15C53×#435C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×#11719947—21no—
18C22×D212C2211yescharacteristic
19C2×#11719913C2211yescharacteristic, complemented
20C2×D424C2211yescharacteristic
21C2×D424C2211yescharacteristic
22#13133217C2211yescharacteristic, complemented
23#13133205C2211yescharacteristic
24#13133202—21no—
25#13133201—21no—
26#13133201—21no—
27#13133197—21no—
28#13133196C2211yescharacteristic
29#13133191C2211yescharacteristic
30#13133191C2211yescharacteristic
31#13133191C2211yescharacteristic, complemented
32#13133190—21no—
33#13133185—21no—
34C4×D212C2211yescharacteristic
35C4×D212C2211yescharacteristic, complemented
36C2×C4⋊Dic53C2211yescharacteristic
37C2×C4⋊Dic53C2211yescharacteristic, complemented
38C22⋊C4×D53—21no—
39#13133149C2211yescharacteristic, complemented
40#13133149C2211yescharacteristic, complemented
41#13133149C2211yescharacteristic, complemented
42#13133142C2211yescharacteristic, complemented
43#13133142C2211yescharacteristic, complemented
44#13133142C2211yescharacteristic, complemented
45#13133139C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133139C2211yescharacteristic
49#13133139C2211yescharacteristic
50#13133139C2211yescharacteristic
51#13133139C2211yescharacteristic, complemented
52C2×M16×C53C2211yescharacteristic, complemented
53C22×C8×C53C2211yescharacteristic
54C4×D4×C53C2211yescharacteristic
55C4⋊C8×C53C2211yescharacteristic, complemented
56C22⋊C8×C53C2211yescharacteristic
57C22⋊C8×C53C2211yescharacteristic, complemented
58#97×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C2×D4×C53D411yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C22×Dic53—21no—
68C22×Dic53—21no—
69#11719947—44no—
70#11719945—22no—
71#11719945—41no—
72C2×D212—22no—
73C2×D212—21no—
74C2×D212—41no—
75C2×D212C2311yescharacteristic
76C2×D212C2311yescharacteristic
77C2×D212C2311yescharacteristic
78C2×C4×D53—21no—
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×Dic106—21no—
82C2×Dic106C2311yescharacteristic
83M16×C53—41no—
84C2×C8×C53—41no—
85C2×C8×C53C2311yescharacteristic
86C2×C8×C53C2311yescharacteristic
87C2×C8×C53C2311yescharacteristic
88C2×C8×C53C2311yescharacteristic
89C4⋊C4×C53D411yescharacteristic
90C22⋊C4×C53D411yescharacteristic
91C22⋊C4×C53D411yescharacteristic
92C42×C53C2311yescharacteristic
93C22⋊Dic53—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99#11719921—21no—
100#11719921—21no—
101C4⋊Dic53—21no—
102C4⋊Dic53—21no—
103C4⋊Dic53—21no—
104C4⋊Dic53C2311yescharacteristic
105C4⋊Dic53C2311yescharacteristic
106C4⋊Dic53C2311yescharacteristic
107C4⋊Dic53C2311yescharacteristic
108#11719919—21no—
109D424—22no—
110D424—41no—
111#11719913—41no—
112C23×C53C2×D411yescharacteristic
113C23×C53C2×D411yescharacteristic
114C22×D53—21no—
115C22×D53—21no—
116C22×D53—21no—
117C22×D53—42no—
118C22×D53—42no—
119C22×D53—81no—
120D4×C53—22no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53C2×D411yescharacteristic
126C2×C4×C53C2×D411yescharacteristic
127C2×C4×C53C2×D411yescharacteristic
128C2×C4×C53C2×D411yescharacteristic
129C2×C4×C53C2411yescharacteristic
130C22⋊D53—42no—
131C22⋊D53—42no—
132C22⋊D53—42no—
133C22⋊D53—81no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—42no—
140C2×Dic53—42no—
141D212—21no—
142D212—21no—
143D212—21no—
144D212—21no—
145D212—21no—
146D212—42no—
147D212#4112yescomplemented
148C4×D53—42no—
149C4×D53—42no—
150C4×D53—81no—
151Dic106—21no—
152Dic106—21no—
153Dic106—42no—
154C8×C53—21no—
155C8×C53—21no—
156C8×C53—21no—
157C8×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53AGL(1,ℤ/8)12yes—
164C22×C53C22×D411yescharacteristic
165C2×D53—22no—
166C2×D53—41no—
167C2×D53—41no—
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—41no—
171C2×D53—41no—
172C2×D53—41no—
173C2×D53—41no—
174C2×D53—84no—
175C4×C53—21no—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53C2×#4111yescharacteristic
180C4×C53H(5,𝔽2)11yescharacteristic
181Dic53—41no—
182Dic53—41no—
183Dic53—41no—
184Dic53—41no—
185Dic53—41no—
186#3306—531nomaximal, Sylow
187C2×C53—22no—
188C2×C53—41no—
189C2×C53#54611yescharacteristic
190C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C53#57711yescharacteristic
192D53—42no—
193D53—81no—
194D53—81no—
195#547—531no—
196#546—531no—
197#489—531no—
198#483—531no—
199#481—531no—
200#476—531no—
201#469—531no—
202#465—531no—
203#463—531no—
204#449—531no—
205#447—531no—
206#442—531no—
207#435D5311yescharacteristic, complemented
208#417—531no—
209C2×#102—531no—
210C53#330611yescharacteristic, complemented, Sylow
211C2×#41—531no—
212C2×#41—1061no—
213C22×D4—531no—
214C2×SD16—531no—
215C2×D8—531no—
216C2×D8—531no—
217C2×M16C2×D5311yescharacteristic, complemented
218C22×C8C2×D5311yescharacteristic
219C4⋊Q8—531no—
220#124—531no—
221#123—1061no—
222#123—1061no—
223#123—1061no—
224C22⋊Q8—531no—
225C22⋊Q8—1061no—
226#121—531no—
227#121—531no—
228#121—531no—
229#121—1061no—
230#120—1061no—
231C4×D4—531no—
232C4×D4—531no—
233C4×D4C2×D5311yescharacteristic
234C2×C4⋊C4—531no—
235C2×C4⋊C4—531no—
236C2×C22⋊C4—1061no—
237#106—531no—
238#106—531no—
239#106—531no—
240C4⋊C8C2×D5311yescharacteristic, complemented
241#103—531no—
242#103—531no—
243#103—531no—
244#102—531no—
245#102—531no—
246#102—531no—
247#102—531no—
248#102—531no—
249#102—531no—
250#102—531no—
251C22⋊C8C2×D5311yescharacteristic
252C22⋊C8C2×D5311yescharacteristic, complemented
253#97C2×D5311yescharacteristic, complemented
254C24—1061no—
255#41—1062no—
256#41—2121no—
257#41—2124no—
258C2×Q8—531no—
259C2×Q8—1061no—
260C2×D4—531no—
261C2×D4—531no—
262C2×D4—531no—
263C2×D4—1062no—
264C2×D4—1061no—
265C2×D4—1061no—
266C2×D4—1061no—
267C2×D4—1061no—
268C2×D4—1061no—
269C2×D4—2121no—
270C2×D4D21211yescharacteristic
271C22×C4—1061no—
272C22×C4—1061no—
273C22×C4—1061no—
274C22×C4—1061no—
275C22×C4—1061no—
276C22×C4C22×D5311yescharacteristic
277C22×C4C22×D5311yescharacteristic
278SD16—2121no—
279D8—1062no—
280D8—2121no—
281M16—41no—
282C2×C8—41no—
283C2×C8C22×D5311yescharacteristic
284C2×C8C22×D5311yescharacteristic
285C2×C8C22×D5311yescharacteristic
286C2×C8C22×D5311yescharacteristic
287C4⋊C4—531no—
288C4⋊C4—531no—
289C4⋊C4—531no—
290C4⋊C4—531no—
291C4⋊C4—1061no—
292C4⋊C4—1061no—
293C4⋊C4—1061no—
294C4⋊C4—1061no—
295C4⋊C4D21211yescharacteristic
296C22⋊C4—1061no—
297C22⋊C4—1061no—
298C22⋊C4—1061no—
299C22⋊C4—1061no—
300C22⋊C4—1061no—
301C22⋊C4—1061no—
302C22⋊C4—1061no—
303C22⋊C4—1061no—
304C22⋊C4D21211yescharacteristic
305C22⋊C4D21211yescharacteristic
306C42C22×D5311yescharacteristic
307C23—1061no—
308C23—1061no—
309C23—1061no—
310C23—4241no—
311C23—2122no—
312C23—2122no—
313C23C2×D21211yescharacteristic
314C23C2×D21211yescharacteristic
315Q8—1061no—
316Q8—1061no—
317Q8—2122no—
318D4—22no—
319D4—532no—
320D4—1061no—
321D4—1061no—
322D4—1061no—
323D4—1061no—
324D4—1061no—
325D4—4241no—
326D4—2122no—
327D4—2122no—
328D4—2122no—
329D4—2122no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—1061no—
339C2×C4—4241no—
340C2×C4—2122no—
341C2×C4—2122no—
342C2×C4—2122no—
343C2×C4—2122no—
344C2×C4C2×D21211yescharacteristic
345C2×C4C2×D21211yescharacteristic
346C2×C4C2×D21211yescharacteristic
347C2×C4C2×D21211yescharacteristic
348C2×C4C23×D5311yescharacteristic
349C8—21no—
350C8—21no—
351C8—21no—
352C8—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—1062no—
359C22—4244no—
360C22—2121no—
361C22—2121no—
362C22—2121no—
363C22—2121no—
364C22—2121no—
365C22—2121no—
366C22—2121no—
367C22—2121no—
368C22#1313324512yes—
369C22C22×D21211yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—2121no—
375C4—2121no—
376C4—2121no—
377C4—2121no—
378C4—2121no—
379C4#41×D5311yescharacteristic
380C4#1313326211yescharacteristic
381C2—22no—
382C2—41no—
383C2—4241no—
384C2—4241no—
385C2—2122no—
386C2#1681126811yescharacteristic
387C2C2×#1313324511yescharacteristic
388C2#1681143511yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#435C2
socleC22×C53C22×D4
hypercenter#435D53
hypocenterC53#3306
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #16811277C2semidirect
2. #16811269C2semidirect
3. #16810983C2semidirect
4. #16810954C2semidirect
5. #16810942C2semidirect
6. #16810914C2semidirect
7. #16810876C2semidirect
8. #16810852C2semidirect
9. #16810839C2semidirect
10. #16810770C2semidirect
11. #16810759C2semidirect
12. #16810736C2semidirect
13. #16810625C2semidirect
14. C2×#13133139C2semidirect
15. C53×#435C2semidirect
19. C2×#11719913C22semidirect
22. #13133217C22semidirect
31. #13133191C22semidirect
35. C4×D212C22semidirect
37. C2×C4⋊Dic53C22semidirect
39. #13133149C22semidirect
40. #13133149C22semidirect
41. #13133149C22semidirect
42. #13133142C22semidirect
43. #13133142C22semidirect
44. #13133142C22semidirect
51. #13133139C22semidirect
52. C2×M16×C53C22semidirect
55. C4⋊C8×C53C22semidirect
57. C22⋊C8×C53C22semidirect
58. #97×C53C22semidirect
147. D212#41semidirect
207. #435D53semidirect
210. C53#3306semidirect
217. C2×M16C2×D53semidirect
240. C4⋊C8C2×D53semidirect
252. C22⋊C8C2×D53semidirect
253. #97C2×D53semidirect