Finite Group Databasebeta

← #20286736

Subgroups — #20286736

Order 6784.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811313C211yesmaximal, characteristic, complemented
2#16810987C211yesmaximal, characteristic, complemented
3#16810944C211yesmaximal, characteristic, complemented
4#16810934C211yesmaximal, characteristic, complemented
5#16810923C211yesmaximal, characteristic, complemented
6#16810867C211yesmaximal, characteristic, complemented
7#16810850C211yesmaximal, characteristic, complemented
8#16810820C211yesmaximal, characteristic, complemented
9#16810774C211yesmaximal, characteristic, complemented
10#16810774C211yesmaximal, characteristic, complemented
11#16810723C211yesmaximal, characteristic, complemented
12#16810695C211yesmaximal, characteristic, complemented
13#16810628C211yesmaximal, characteristic, complemented
14C2×#13133138C211yesmaximal, characteristic, complemented
15D42×C53C211yesmaximal, characteristic
16C2×#11719923C2211yescharacteristic
17C2×#11719923C2211yescharacteristic, complemented
18C2×#11719923C2211yescharacteristic, complemented
19C2×#11719917C2211yescharacteristic, complemented
20C22×#88322C2211yescharacteristic, complemented
21#13133210—21no—
22#13133209C2211yescharacteristic, complemented
23#13133207—21no—
24#13133198—21no—
25#13133196C2211yescharacteristic, complemented
26#13133196C2211yescharacteristic, complemented
27C4×Dic106C2211yescharacteristic, complemented
28#13133176C2211yescharacteristic, complemented
29#13133173—21no—
30C2×C4⋊Dic53C2211yescharacteristic, complemented
31#13133150C2211yescharacteristic, complemented
32#13133150C2211yescharacteristic, complemented
33#13133148C2211yescharacteristic, complemented
34C4⋊#88322C2211yescharacteristic, complemented
35#13133140C2211yescharacteristic, complemented
36#13133138C2211yescharacteristic
37#13133138C2211yescharacteristic
38#13133138C2211yescharacteristic
39#13133138C2211yescharacteristic, complemented
40#13133138C2211yescharacteristic, complemented
41#13133138C2211yescharacteristic, complemented
42#13133138C2211yescharacteristic, complemented
43#13133138C2211yescharacteristic, complemented
44#13133138C2211yescharacteristic, complemented
45C22⋊#88322C2211yescharacteristic, complemented
46C22⋊#88322C2211yescharacteristic, complemented
47#13133128C2211yescharacteristic, complemented
48C22×D4×C53—21no—
49C22×D4×C53C2211yescharacteristic
50C22×D4×C53C2211yescharacteristic
51#127×C53C2211yescharacteristic
52#121×C53—21no—
53#121×C53C2211yescharacteristic
54#121×C53C2211yescharacteristic
55#120×C53—21no—
56#120×C53—21no—
57C4×D4×C53C2211yescharacteristic
58C4×D4×C53C2211yescharacteristic
59C24×C53—21no—
60C24×C53—21no—
61C2×D4×C53—22no—
62C2×D4×C53—22no—
63C2×D4×C53—21no—
64C2×D4×C53—21no—
65C2×D4×C53—21no—
66C2×D4×C53—21no—
67C2×D4×C53—21no—
68C2×D4×C53—21no—
69C2×D4×C53—41no—
70C2×D4×C53—41no—
71C2×D4×C53—44no—
72C2×D4×C53D411yescharacteristic
73C2×D4×C53C2311yescharacteristic
74C2×D4×C53C2311yescharacteristic
75C2×D4×C53C2311yescharacteristic
76C22×C4×C53—21no—
77C22×C4×C53C2311yescharacteristic
78C22×C4×C53C2311yescharacteristic
79C22×Dic53—21no—
80C2×Dic106C2311yescharacteristic
81C4⋊C4×C53D411yescharacteristic
82C4⋊C4×C53C2311yescharacteristic
83C22⋊C4×C53—21no—
84C22⋊C4×C53—21no—
85C22⋊C4×C53—21no—
86C22⋊C4×C53D411yescharacteristic
87C22⋊C4×C53D411yescharacteristic
88C42×C53C2311yescharacteristic
89C22⋊Dic53—21no—
90C22⋊Dic53—21no—
91C22⋊Dic53—21no—
92#11719923—22no—
93#11719923—41no—
94#11719923—41no—
95C4⋊Dic53—21no—
96C4⋊Dic53C2311yescharacteristic
97C4⋊Dic53C2311yescharacteristic
98C4⋊Dic53C2311yescharacteristic
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103#11719919—21no—
104C4×Dic53—21no—
105C4×Dic53—21no—
106#11719917—41no—
107C2×#88322—41no—
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—42no—
116C23×C53—42no—
117C23×C53—42no—
118C23×C53—42no—
119C23×C53—81no—
120C23×C53—81no—
121C23×C53C2×D411yescharacteristic
122C23×C53C2×D411yescharacteristic
123D4×C53—22no—
124D4×C53—21no—
125D4×C53—21no—
126D4×C53—21no—
127D4×C53—21no—
128D4×C53—21no—
129D4×C53—42no—
130D4×C53—42no—
131D4×C53—42no—
132D4×C53—42no—
133D4×C53—42no—
134D4×C53—81no—
135D4×C53#4112yes—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—42no—
142C2×C4×C53C2×D411yescharacteristic
143C2×C4×C53C2×D411yescharacteristic
144C2×C4×C53C2×D411yescharacteristic
145C2×C4×C53C2×D411yescharacteristic
146C2×C4×C53C2411yescharacteristic
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—42no—
152Dic106—21no—
153Dic106—21no—
154#88322—21no—
155#88322—21no—
156#88322—21no—
157#88322—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—22no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—84no—
173C22×C53—84no—
174C22×C53AGL(1,ℤ/8)12yes—
175C22×C53C22×D411yescharacteristic
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—41no—
181C4×C53C2×#4111yescharacteristic
182C4×C53H(5,𝔽2)11yescharacteristic
183Dic53—41no—
184Dic53—41no—
185Dic53—41no—
186Dic53—41no—
187#3310—531nomaximal, Sylow
188C2×C53—22no—
189C2×C53—41no—
190C2×C53—42no—
191C2×C53—81no—
192C2×C53—81no—
193C2×C53#54611yescharacteristic
194C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
195C2×C53#57711yescharacteristic
196#552—531no—
197D42D5311yescharacteristic
198#490—531no—
199#482—531no—
200#480—531no—
201#478—531no—
202#468—531no—
203#465—531no—
204#460—531no—
205#450—531no—
206#450—531no—
207#440—531no—
208#435—531no—
209#418—531no—
210C2×#102—531no—
211C53#331011yescharacteristic, complemented, Sylow
212C22×D4—21no—
213C22×D4C2×D5311yescharacteristic
214C22×D4C2×D5311yescharacteristic
215C2×SD16—531no—
216C2×SD16—531no—
217C2×SD16—531no—
218C2×M16—531no—
219C22×C8—531no—
220#127C2×D5311yescharacteristic
221#126—1061no—
222#125—531no—
223#125—1061no—
224#123—1061no—
225C22⋊Q8—531no—
226C22⋊Q8—531no—
227#121—21no—
228#121C2×D5311yescharacteristic
229#121C2×D5311yescharacteristic
230#120—21no—
231#120—21no—
232C4×Q8—531no—
233C4×D4C2×D5311yescharacteristic
234C4×D4C2×D5311yescharacteristic
235#117—531no—
236#117—1061no—
237C2×C4⋊C4—531no—
238#107—531no—
239#107—531no—
240#106—531no—
241C4⋊C8—531no—
242#103—531no—
243#102—531no—
244#102—531no—
245#102—531no—
246#102—531no—
247#102—531no—
248#102—531no—
249#102—531no—
250#102—531no—
251#102—531no—
252C22⋊C8—531no—
253C22⋊C8—531no—
254#97—531no—
255C24—21no—
256C24—21no—
257C2×Q8—531no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—44no—
269C2×D4C22⋊D5311yescharacteristic
270C2×D4C22×D5311yescharacteristic
271C2×D4C22×D5311yescharacteristic
272C2×D4C22×D5311yescharacteristic
273C22×C4—21no—
274C22×C4—1061no—
275C22×C4C22×D5311yescharacteristic
276C22×C4C22×D5311yescharacteristic
277SD16—1062no—
278SD16—2121no—
279SD16—2121no—
280M16—2121no—
281C2×C8—531no—
282C2×C8—531no—
283C2×C8—531no—
284C2×C8—531no—
285C2×C8—2121no—
286C4⋊C4—531no—
287C4⋊C4—531no—
288C4⋊C4—531no—
289C4⋊C4—1061no—
290C4⋊C4—1061no—
291C4⋊C4—1061no—
292C4⋊C4—1061no—
293C4⋊C4—1061no—
294C4⋊C4—1061no—
295C4⋊C4C22⋊D5311yescharacteristic
296C4⋊C4C22×D5311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—1061no—
301C22⋊C4—1061no—
302C22⋊C4—1061no—
303C22⋊C4C22⋊D5311yescharacteristic
304C22⋊C4C22⋊D5311yescharacteristic
305C42—1061no—
306C42—1061no—
307C42C22×D5311yescharacteristic
308C23—21no—
309C23—21no—
310C23—21no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—81no—
316C23—81no—
317C23C2×C22⋊D5311yescharacteristic
318C23C2×C22⋊D5311yescharacteristic
319Q8—1061no—
320Q8—1061no—
321D4—22no—
322D4—21no—
323D4—21no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—42no—
328D4—42no—
329D4—42no—
330D4—42no—
331D4—42no—
332D4—81no—
333D4#1171994712yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—1061no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—2122no—
345C2×C4C2×C22⋊D5311yescharacteristic
346C2×C4C2×C22⋊D5311yescharacteristic
347C2×C4C2×C22⋊D5311yescharacteristic
348C2×C4C2×C22⋊D5311yescharacteristic
349C2×C4C23×D5311yescharacteristic
350C8—1061no—
351C8—1061no—
352C8—1061no—
353C8—1061no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—22no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—84no—
369C22—84no—
370C22#1313324312yes—
371C22C22×C22⋊D5311yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—2121no—
378C4—2121no—
379C4—2121no—
380C4—2121no—
381C4C2×#1171994711yescharacteristic
382C4#1313326211yescharacteristic
383C2—22no—
384C2—41no—
385C2—42no—
386C2—81no—
387C2—81no—
388C2#1681126711yescharacteristic
389C2C2×#1313324311yescharacteristic
390C2#1681143411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingD42×C53C2
socleC22×C53C22×D4
hypercenterD42D53
hypocenterC53#3310
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #16811313C2semidirect
2. #16810987C2semidirect
3. #16810944C2semidirect
4. #16810934C2semidirect
5. #16810923C2semidirect
6. #16810867C2semidirect
7. #16810850C2semidirect
8. #16810820C2semidirect
9. #16810774C2semidirect
10. #16810774C2semidirect
11. #16810723C2semidirect
12. #16810695C2semidirect
13. #16810628C2semidirect
14. C2×#13133138C2semidirect
17. C2×#11719923C22semidirect
18. C2×#11719923C22semidirect
19. C2×#11719917C22semidirect
20. C22×#88322C22semidirect
22. #13133209C22semidirect
25. #13133196C22semidirect
26. #13133196C22semidirect
27. C4×Dic106C22semidirect
28. #13133176C22semidirect
30. C2×C4⋊Dic53C22semidirect
31. #13133150C22semidirect
32. #13133150C22semidirect
33. #13133148C22semidirect
34. C4⋊#88322C22semidirect
35. #13133140C22semidirect
39. #13133138C22semidirect
40. #13133138C22semidirect
41. #13133138C22semidirect
42. #13133138C22semidirect
43. #13133138C22semidirect
44. #13133138C22semidirect
45. C22⋊#88322C22semidirect
46. C22⋊#88322C22semidirect
47. #13133128C22semidirect
211. C53#3310semidirect