Finite Group Databasebeta

← #20286738

Subgroups — #20286738

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
2D4×D212C211yesmaximal, characteristic
3#16810988C211yesmaximal, characteristic, complemented
4#16810948C211yesmaximal, characteristic, complemented
5#16810937C211yesmaximal, characteristic, complemented
6#16810924C211yesmaximal, characteristic, complemented
7#16810869C211yesmaximal, characteristic, complemented
8#16810852C211yesmaximal, characteristic, complemented
9#16810824C211yesmaximal, characteristic, complemented
10#16810776C211yesmaximal, characteristic, complemented
11#16810776C211yesmaximal, characteristic, complemented
12#16810725C211yesmaximal, characteristic, complemented
13#16810629C211yesmaximal, characteristic, complemented
14C2×#13133139C211yesmaximal, characteristic, complemented
15C53×#435C211yesmaximal, characteristic, complemented
16C22×D212C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C2×D4×D53—21no—
19C2×#11719913C2211yescharacteristic
20C2×#11719913C2211yescharacteristic, complemented
21C2×#11719913C2211yescharacteristic, complemented
22#13133214C2211yescharacteristic
23#13133210—21no—
24#13133209C2211yescharacteristic, complemented
25#13133207—21no—
26#13133198—21no—
27#13133196C2211yescharacteristic, complemented
28#13133196C2211yescharacteristic, complemented
29#13133192—21no—
30#13133191C2211yescharacteristic
31#13133191C2211yescharacteristic
32#13133187—21no—
33#13133187—21no—
34C4×Dic106C2211yescharacteristic, complemented
35C4×D212C2211yescharacteristic
36#13133176C2211yescharacteristic, complemented
37#13133173—21no—
38C2×C4⋊Dic53C2211yescharacteristic, complemented
39#13133151C2211yescharacteristic, complemented
40#13133151C2211yescharacteristic, complemented
41#13133149C2211yescharacteristic, complemented
42#13133142C2211yescharacteristic, complemented
43#13133139C2211yescharacteristic
44#13133139C2211yescharacteristic
45#13133139C2211yescharacteristic
46#13133139C2211yescharacteristic, complemented
47#13133139C2211yescharacteristic, complemented
48#13133139C2211yescharacteristic, complemented
49#13133139C2211yescharacteristic, complemented
50#13133139C2211yescharacteristic, complemented
51#13133139C2211yescharacteristic, complemented
52C2×M16×C53C2211yescharacteristic, complemented
53C22×C8×C53C2211yescharacteristic, complemented
54C4×D4×C53C2211yescharacteristic
55C4⋊C8×C53C2211yescharacteristic, complemented
56C22⋊C8×C53C2211yescharacteristic, complemented
57C22⋊C8×C53C2211yescharacteristic, complemented
58#97×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C23×D53—21no—
61C2×D4×C53D411yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C22×Dic53—21no—
67D4×D53—44no—
68C2×D212—22no—
69C2×D212—22no—
70C2×D212—21no—
71C2×D212—21no—
72C2×D212—21no—
73C2×D212—21no—
74C2×D212—41no—
75C2×D212—41no—
76C2×D212C2311yescharacteristic
77C2×D212C2311yescharacteristic
78C2×D212C2311yescharacteristic
79C2×C4×D53—21no—
80C2×Dic106C2311yescharacteristic
81M16×C53—41no—
82C2×C8×C53—41no—
83C2×C8×C53C2311yescharacteristic
84C2×C8×C53C2311yescharacteristic
85C2×C8×C53C2311yescharacteristic
86C2×C8×C53C2311yescharacteristic
87C4⋊C4×C53D411yescharacteristic
88C22⋊C4×C53D411yescharacteristic
89C22⋊C4×C53D411yescharacteristic
90C42×C53C2311yescharacteristic
91C22⋊Dic53—21no—
92C22⋊Dic53—21no—
93C22⋊Dic53—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97C4⋊Dic53—21no—
98C4⋊Dic53C2311yescharacteristic
99C4⋊Dic53C2311yescharacteristic
100C4⋊Dic53C2311yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102#11719919—21no—
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106#11719919—21no—
107C4×Dic53—21no—
108C4×Dic53—21no—
109#11719913—22no—
110#11719913—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—42no—
120C22×D53—42no—
121C22×D53—81no—
122C22×D53—81no—
123D4×C53—22no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53C2×D411yescharacteristic
129C2×C4×C53C2×D411yescharacteristic
130C2×C4×C53C2×D411yescharacteristic
131C2×C4×C53C2×D411yescharacteristic
132C2×C4×C53C2411yescharacteristic
133C22⋊D53—42no—
134C22⋊D53—42no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—42no—
141D212—21no—
142D212—21no—
143D212—21no—
144D212—21no—
145D212—21no—
146D212—42no—
147D212—42no—
148D212—42no—
149D212—81no—
150D212#4112yes—
151C4×D53—42no—
152Dic106—21no—
153Dic106—21no—
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—
175C2×D53—84no—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53C2×#4111yescharacteristic
181C4×C53H(5,𝔽2)11yescharacteristic
182Dic53—41no—
183Dic53—41no—
184Dic53—41no—
185Dic53—41no—
186Dic53—41no—
187#3310—531nomaximal, Sylow
188C2×C53—22no—
189C2×C53—41no—
190C2×C53#54611yescharacteristic
191C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C53#57711yescharacteristic
193D53—42no—
194D53—81no—
195D53—81no—
196#552—531no—
197D42—531no—
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#435D5311yescharacteristic, complemented
209#418—531no—
210C2×#102—531no—
211C53#331011yescharacteristic, complemented, Sylow
212C22×D4—531no—
213C22×D4—531no—
214C22×D4—1061no—
215C2×SD16—531no—
216C2×SD16—531no—
217C2×SD16—531no—
218C2×M16C2×D5311yescharacteristic, complemented
219C22×C8C2×D5311yescharacteristic, complemented
220#127—531no—
221#126—1061no—
222#125—531no—
223#125—1061no—
224#123—1061no—
225C22⋊Q8—531no—
226C22⋊Q8—531no—
227#121—531no—
228#121—531no—
229#121—1061no—
230#120—1061no—
231#120—1061no—
232C4×Q8—531no—
233C4×D4—531no—
234C4×D4C2×D5311yescharacteristic
235#117—531no—
236#117—1061no—
237C2×C4⋊C4—531no—
238#107—531no—
239#107—531no—
240#106—531no—
241C4⋊C8C2×D5311yescharacteristic, complemented
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⋊C8C2×D5311yescharacteristic, complemented
253C22⋊C8C2×D5311yescharacteristic, complemented
254#97C2×D5311yescharacteristic, complemented
255C24—1061no—
256C24—1061no—
257C2×Q8—531no—
258C2×D4—531no—
259C2×D4—531no—
260C2×D4—531no—
261C2×D4—1062no—
262C2×D4—1062no—
263C2×D4—1061no—
264C2×D4—1061no—
265C2×D4—1061no—
266C2×D4—1061no—
267C2×D4—1061no—
268C2×D4—1061no—
269C2×D4—2121no—
270C2×D4—2121no—
271C2×D4—2124no—
272C2×D4D21211yescharacteristic
273C22×C4—1061no—
274C22×C4—1061no—
275C22×C4C22×D5311yescharacteristic
276C22×C4C22×D5311yescharacteristic
277SD16—1062no—
278SD16—2121no—
279SD16—2121no—
280M16—41no—
281C2×C8—41no—
282C2×C8C22×D5311yescharacteristic
283C2×C8C22×D5311yescharacteristic
284C2×C8C22×D5311yescharacteristic
285C2×C8C22×D5311yescharacteristic
286C4⋊C4—531no—
287C4⋊C4—531no—
288C4⋊C4—531no—
289C4⋊C4—531no—
290C4⋊C4—1061no—
291C4⋊C4—1061no—
292C4⋊C4—1061no—
293C4⋊C4—1061no—
294C4⋊C4—1061no—
295C4⋊C4—1061no—
296C4⋊C4D21211yescharacteristic
297C22⋊C4—1061no—
298C22⋊C4—1061no—
299C22⋊C4—1061no—
300C22⋊C4—1061no—
301C22⋊C4—1061no—
302C22⋊C4—1061no—
303C22⋊C4D21211yescharacteristic
304C22⋊C4D21211yescharacteristic
305C42—1061no—
306C42—1061no—
307C42C22×D5311yescharacteristic
308C23—1061no—
309C23—1061no—
310C23—1061no—
311C23—4241no—
312C23—4241no—
313C23—2122no—
314C23—2122no—
315C23—2122no—
316C23—2122no—
317C23C2×D21211yescharacteristic
318C23C2×D21211yescharacteristic
319Q8—1061no—
320Q8—1061no—
321D4—22no—
322D4—532no—
323D4—1061no—
324D4—1061no—
325D4—1061no—
326D4—1061no—
327D4—1061no—
328D4—4241no—
329D4—2122no—
330D4—2122no—
331D4—2122no—
332D4—2122no—
333D4—2122no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—2122no—
344C2×C4—2122no—
345C2×C4C2×D21211yescharacteristic
346C2×C4C2×D21211yescharacteristic
347C2×C4C2×D21211yescharacteristic
348C2×C4C2×D21211yescharacteristic
349C2×C4C23×D5311yescharacteristic
350C8—21no—
351C8—21no—
352C8—21no—
353C8—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—1062no—
360C22—4244no—
361C22—4244no—
362C22—2121no—
363C22—2121no—
364C22—2121no—
365C22—2121no—
366C22—2121no—
367C22—2121no—
368C22—2121no—
369C22—2121no—
370C22#1313324512yes—
371C22C22×D21211yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—2121no—
377C4—2121no—
378C4—2121no—
379C4—2121no—
380C4—2121no—
381C4#41×D5311yescharacteristic
382C4#1313326211yescharacteristic
383C2—22no—
384C2—41no—
385C2—4241no—
386C2—4241no—
387C2—2122no—
388C2#1681126811yescharacteristic
389C2C2×#1313324511yescharacteristic
390C2#1681143511yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#435C2
socleC22×C53C22×D4
hypercenter#435D53
hypocenterC53#3310
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. #16811313C2semidirect
3. #16810988C2semidirect
4. #16810948C2semidirect
5. #16810937C2semidirect
6. #16810924C2semidirect
7. #16810869C2semidirect
8. #16810852C2semidirect
9. #16810824C2semidirect
10. #16810776C2semidirect
11. #16810776C2semidirect
12. #16810725C2semidirect
13. #16810629C2semidirect
14. C2×#13133139C2semidirect
15. C53×#435C2semidirect
20. C2×#11719913C22semidirect
21. C2×#11719913C22semidirect
24. #13133209C22semidirect
27. #13133196C22semidirect
28. #13133196C22semidirect
34. C4×Dic106C22semidirect
36. #13133176C22semidirect
38. C2×C4⋊Dic53C22semidirect
39. #13133151C22semidirect
40. #13133151C22semidirect
41. #13133149C22semidirect
42. #13133142C22semidirect
46. #13133139C22semidirect
47. #13133139C22semidirect
48. #13133139C22semidirect
49. #13133139C22semidirect
50. #13133139C22semidirect
51. #13133139C22semidirect
52. C2×M16×C53C22semidirect
53. C22×C8×C53C22semidirect
55. C4⋊C8×C53C22semidirect
56. C22⋊C8×C53C22semidirect
57. C22⋊C8×C53C22semidirect
58. #97×C53C22semidirect
208. #435D53semidirect
211. C53#3310semidirect
218. C2×M16C2×D53semidirect
219. C22×C8C2×D53semidirect
241. C4⋊C8C2×D53semidirect
252. C22⋊C8C2×D53semidirect
253. C22⋊C8C2×D53semidirect
254. #97C2×D53semidirect