Finite Group Databasebeta

← #20285556

Subgroups — #20285556

Order 6784.

418 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811136C211yesmaximal, characteristic, complemented
2#16811130C211yesmaximal, characteristic, complemented
3#16810963C211yesmaximal, characteristic, complemented
4#16810961C211yesmaximal, characteristic, complemented
5#16810948C211yesmaximal, characteristic, complemented
6#16810946C211yesmaximal, characteristic, complemented
7#16810878C211yesmaximal, characteristic, complemented
8#16810875C211yesmaximal, characteristic, complemented
9#16810872C211yesmaximal, characteristic, complemented
10#16810866C211yesmaximal, characteristic, complemented
11#16810784C211yesmaximal, characteristic, complemented
12#16810768C211yesmaximal, characteristic, complemented
13#16810767C211yesmaximal, characteristic, complemented
14#16810585C211yesmaximal, characteristic, complemented
15C53×#452C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C22×C4×D53C2211yescharacteristic
18C2×#11719924C2211yescharacteristic, complemented
19C2×#11719922C2211yescharacteristic, complemented
20C2×#11719912C2211yescharacteristic
21C2×#11719912C2211yescharacteristic
22#13133202—21no—
23#13133198—21no—
24#13133193C2211yescharacteristic
25#13133193C2211yescharacteristic
26#13133193C2211yescharacteristic
27#13133192C2211yescharacteristic
28#13133191C2211yescharacteristic
29#13133188C2211yescharacteristic
30#13133188C2211yescharacteristic
31#13133187—21no—
32#13133185—21no—
33#13133179—21no—
34C4×C22⋊D53—21no—
35C4×C22⋊D53—21no—
36#13133177—21no—
37#13133176C2211yescharacteristic
38#13133175C2211yescharacteristic
39#13133151C2211yescharacteristic, complemented
40#13133150C2211yescharacteristic, complemented
41#13133149C2211yescharacteristic, complemented
42#13133148C2211yescharacteristic, complemented
43#13133141C2211yescharacteristic, complemented
44#13133141C2211yescharacteristic, complemented
45#13133141C2211yescharacteristic, complemented
46#13133139C2211yescharacteristic, complemented
47#13133139C2211yescharacteristic, complemented
48#13133138C2211yescharacteristic, complemented
49C22⋊#88322C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133129C2211yescharacteristic
52C2×#41×C53C2211yescharacteristic
53C2×Q16×C53C2211yescharacteristic, complemented
54C2×SD16×C53C2211yescharacteristic, complemented
55C22⋊Q8×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic, complemented
57#102×C53C2211yescharacteristic, complemented
58C22⋊C8×C53C2211yescharacteristic
59C23×D53D411yescharacteristic
60#41×C53—41no—
61#41×C53—41no—
62C2×Q8×C53C2311yescharacteristic
63C2×Q8×C53C2311yescharacteristic
64C2×D4×C53—21no—
65C2×D4×C53C2311yescharacteristic
66C22×C4×C53—21no—
67C22×C4×C53C2311yescharacteristic
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C22×Dic53D411yescharacteristic
72#11719949—41no—
73#11719949—41no—
74C2×D212—21no—
75C2×D212—21no—
76C2×D212C2311yescharacteristic
77C2×C4×D53—21no—
78C2×C4×D53—41no—
79C2×C4×D53—41no—
80C2×C4×D53D411yescharacteristic
81C2×C4×D53D411yescharacteristic
82C2×C4×D53C2311yescharacteristic
83C2×C4×D53C2311yescharacteristic
84Q16×C53—41no—
85SD16×C53—41no—
86C2×C8×C53C2311yescharacteristic
87C2×C8×C53C2311yescharacteristic
88C4⋊C4×C53—21no—
89C4⋊C4×C53C2311yescharacteristic
90C22⋊C4×C53—21no—
91C22⋊Dic53—21no—
92C22⋊Dic53—21no—
93C22⋊Dic53—21no—
94#11719924—41no—
95#11719922—41no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99#11719921—21no—
100#11719921—21no—
101#11719921—21no—
102C4⋊Dic53C2311yescharacteristic
103C4⋊Dic53C2311yescharacteristic
104C4⋊Dic53C2311yescharacteristic
105#11719919—21no—
106#11719919—21no—
107#11719919—21no—
108C4×Dic53—21no—
109C4×Dic53—21no—
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112#11719912—41no—
113#11719912—41no—
114C23×C53—21no—
115C23×C53C2×D411yescharacteristic
116C22×D53—21no—
117C22×D53—41no—
118C22×D53—41no—
119C22×D53—41no—
120C22×D53C2×D411yescharacteristic
121C22×D53C2×D411yescharacteristic
122Q8×C53—21no—
123Q8×C53—21no—
124Q8×C53—21no—
125Q8×C53—21no—
126D4×C53—21no—
127D4×C53—21no—
128D4×C53—42no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—42no—
135C2×C4×C53C2×D411yescharacteristic
136C2×C4×C53C2411yescharacteristic
137C22⋊D53—42no—
138C22⋊D53—42no—
139C22⋊D53—42no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—41no—
144C2×Dic53C2×D411yescharacteristic
145C2×Dic53C2×D411yescharacteristic
146D212—21no—
147D212—21no—
148D212—42no—
149D212—42no—
150C4×D53—21no—
151C4×D53—21no—
152C4×D53—21no—
153C4×D53—21no—
154C4×D53—41no—
155C4×D53—41no—
156C4×D53—42no—
157C8×C53—21no—
158C8×C53—21no—
159#88322—21no—
160#88322—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53C22×D411yescharacteristic
168C2×D53—21no—
169C2×D53—21no—
170C2×D53—21no—
171C2×D53—21no—
172C2×D53—21no—
173C2×D53—21no—
174C2×D53—44no—
175C2×D53—41no—
176C2×D53—41no—
177C2×D53—41no—
178C4×C53—21no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53—41no—
182C4×C53H(5,𝔽2)11yescharacteristic
183C4×C53H(5,𝔽2)11yescharacteristic
184Dic53—21no—
185Dic53—21no—
186Dic53—41no—
187Dic53—41no—
188Dic53—41no—
189#3197—531nomaximal, Sylow
190C2×C53—41no—
191C2×C53—81no—
192C2×C53#53411yescharacteristic
193C2×C53#57511yescharacteristic
194C2×C53#57511yescharacteristic
195D53—41no—
196D53—41no—
197D53—81no—
198#525—531no—
199#525—531no—
200#485—531no—
201#485—531no—
202#482—531no—
203#482—531no—
204#470—531no—
205#469—531no—
206#468—531no—
207#468—531no—
208#452—531no—
209#452D5311yescharacteristic, complemented
210#449—531no—
211#449—531no—
212#407—531no—
213C53#319711yescharacteristic, complemented, Sylow
214C2×#41—531no—
215C2×#41C2×D5311yescharacteristic
216C23×C4—531no—
217C2×Q16C2×D5311yescharacteristic, complemented
218C2×SD16—531no—
219C2×SD16C2×D5311yescharacteristic, complemented
220C2×D8—531no—
221C2×M16—531no—
222C2×M16—531no—
223#123—1061no—
224#123—1061no—
225C22⋊Q8—531no—
226C22⋊Q8—531no—
227C22⋊Q8—531no—
228C22⋊Q8C2×D5311yescharacteristic
229#121—531no—
230#121—531no—
231#121—531no—
232#121—531no—
233#120—1061no—
234#120—1061no—
235C4×D4—1061no—
236C4×D4—1061no—
237C4×D4—1061no—
238C4×D4—1061no—
239#117—531no—
240#117—531no—
241#107—531no—
242#107—531no—
243#106—531no—
244#106—531no—
245#103—531no—
246#103—531no—
247#103—531no—
248#103C2×D5311yescharacteristic, complemented
249#102—531no—
250#102—531no—
251#102—531no—
252#102C2×D5311yescharacteristic, complemented
253C22⋊C8—531no—
254C22⋊C8—531no—
255C22⋊C8—531no—
256C22⋊C8C2×D5311yescharacteristic
257C24—531no—
258#41—41no—
259#41—41no—
260#41—2121no—
261#41—2121no—
262C2×Q8C22×D5311yescharacteristic
263C2×Q8C22×D5311yescharacteristic
264C2×D4—21no—
265C2×D4—531no—
266C2×D4—1061no—
267C2×D4—1061no—
268C2×D4—1061no—
269C2×D4—1061no—
270C2×D4—1061no—
271C2×D4C22×D5311yescharacteristic
272C22×C4—21no—
273C22×C4—531no—
274C22×C4—531no—
275C22×C4—531no—
276C22×C4—531no—
277C22×C4—531no—
278C22×C4—1061no—
279C22×C4—2121no—
280C22×C4—2121no—
281C22×C4C22×D5311yescharacteristic
282Q16—41no—
283SD16—41no—
284SD16—2121no—
285D8—2121no—
286M16—2121no—
287M16—2121no—
288C2×C8—531no—
289C2×C8—531no—
290C2×C8C22×D5311yescharacteristic
291C2×C8C22×D5311yescharacteristic
292C4⋊C4—21no—
293C4⋊C4—531no—
294C4⋊C4—531no—
295C4⋊C4—531no—
296C4⋊C4—1061no—
297C4⋊C4—1061no—
298C4⋊C4—1061no—
299C4⋊C4C22×D5311yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—1061no—
302C22⋊C4—1061no—
303C22⋊C4—1061no—
304C22⋊C4—1061no—
305C22⋊C4—1061no—
306C22⋊C4—1061no—
307C22⋊C4—1061no—
308C22⋊C4—1061no—
309C22⋊C4—1061no—
310C42—1061no—
311C42—1061no—
312C23—21no—
313C23—531no—
314C23—531no—
315C23—1061no—
316C23—2121no—
317C23—2121no—
318C23—2121no—
319C23D4×D5311yescharacteristic
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—21no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—1061no—
328D4—1061no—
329D4—2122no—
330D4—2122no—
331D4—2122no—
332D4—2122no—
333D4—2122no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—531no—
341C2×C4—531no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—2121no—
350C2×C4—2121no—
351C2×C4—2122no—
352C2×C4—2121no—
353C2×C4D4×D5311yescharacteristic
354C2×C4C23×D5311yescharacteristic
355C8—21no—
356C8—21no—
357C8—1061no—
358C8—1061no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—1061no—
366C22—1061no—
367C22—1061no—
368C22—1061no—
369C22—1061no—
370C22—1061no—
371C22—2124no—
372C22—2121no—
373C22—2121no—
374C22—2121no—
375C22C2×D4×D5311yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—1061no—
381C4—1061no—
382C4—2121no—
383C4—2121no—
384C4—2121no—
385C4#1313326211yescharacteristic
386C4#1313326211yescharacteristic
387C2—41no—
388C2—81no—
389C2—4241no—
390C2—2121no—
391C2—2121no—
392C2#1681118311yescharacteristic
393C2#1681142011yescharacteristic
394C2#1681142211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#452C2
socleC22×C53C22×D4
hypercenter#452D53
hypocenterC53#3197
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #16811136C2semidirect
2. #16811130C2semidirect
3. #16810963C2semidirect
4. #16810961C2semidirect
5. #16810948C2semidirect
6. #16810946C2semidirect
7. #16810878C2semidirect
8. #16810875C2semidirect
9. #16810872C2semidirect
10. #16810866C2semidirect
11. #16810784C2semidirect
12. #16810768C2semidirect
13. #16810767C2semidirect
14. #16810585C2semidirect
15. C53×#452C2semidirect
18. C2×#11719924C22semidirect
19. C2×#11719922C22semidirect
39. #13133151C22semidirect
40. #13133150C22semidirect
41. #13133149C22semidirect
42. #13133148C22semidirect
43. #13133141C22semidirect
44. #13133141C22semidirect
45. #13133141C22semidirect
46. #13133139C22semidirect
47. #13133139C22semidirect
48. #13133138C22semidirect
53. C2×Q16×C53C22semidirect
54. C2×SD16×C53C22semidirect
56. #103×C53C22semidirect
57. #102×C53C22semidirect
209. #452D53semidirect
213. C53#3197semidirect
217. C2×Q16C2×D53semidirect
219. C2×SD16C2×D53semidirect
248. #103C2×D53semidirect
252. #102C2×D53semidirect