Finite Group Databasebeta

← #20285700

Subgroups — #20285700

Order 6784.

444 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811170C211yesmaximal, characteristic, complemented
2#124×D53C211yesmaximal, characteristic, complemented
3#16811011C211yesmaximal, characteristic, complemented
4#16811009C211yesmaximal, characteristic, complemented
5#16810984C211yesmaximal, characteristic, complemented
6#16810981C211yesmaximal, characteristic, complemented
7#16810839C211yesmaximal, characteristic, complemented
8#16810838C211yesmaximal, characteristic, complemented
9#16810835C211yesmaximal, characteristic, complemented
10#16810763C211yesmaximal, characteristic, complemented
11#16810760C211yesmaximal, characteristic, complemented
12#16810758C211yesmaximal, characteristic, complemented
13#16810755C211yesmaximal, characteristic, complemented
14#16810679C211yesmaximal, characteristic, complemented
15C53×#463C211yesmaximal, characteristic, complemented
16C2×#11719945—21no—
17C2×Q8×D53C2211yescharacteristic
18C2×D4×D53C2211yescharacteristic
19C2×#11719913C2211yescharacteristic, complemented
20C2×#11719923C2211yescharacteristic, complemented
21C2×#11719924C2211yescharacteristic, complemented
22C2×D424C2211yescharacteristic
23C2×#11719922C2211yescharacteristic
24C2×#11719922C2211yescharacteristic
25#13133216C2211yescharacteristic, complemented
26#13133213C2211yescharacteristic, complemented
27#13133206C2211yescharacteristic
28#13133205C2211yescharacteristic
29#13133204C2211yescharacteristic
30#13133203—21no—
31#13133203—21no—
32#13133203—21no—
33#13133192C2211yescharacteristic
34#13133190—21no—
35#13133188C2211yescharacteristic
36C4×D212C2211yescharacteristic, complemented
37C4×C22⋊D53—21no—
38C22⋊C4×D53—21no—
39C22⋊C4×D53—21no—
40C42×D53C2211yescharacteristic
41C4⋊#88322C2211yescharacteristic, complemented
42#13133142C2211yescharacteristic, complemented
43#13133141C2211yescharacteristic, complemented
44#13133140C2211yescharacteristic, complemented
45#13133139C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133129C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133127C2211yescharacteristic, complemented
51#13133127C2211yescharacteristic, complemented
52C2×SD16×C53C2211yescharacteristic, complemented
53C2×D8×C53C2211yescharacteristic
54#124×C53C2211yescharacteristic
55C4×D4×C53C2211yescharacteristic, complemented
56C4⋊C8×C53C2211yescharacteristic, complemented
57#103×C53C2211yescharacteristic, complemented
58#102×C53C2211yescharacteristic
59C23×D53—21no—
60C2×Q8×C53C2311yescharacteristic
61C2×D4×C53C2311yescharacteristic
62C2×D4×C53C2311yescharacteristic
63C22×C4×C53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68Q8×D53—41no—
69Q8×D53—41no—
70D4×D53—41no—
71D4×D53—41no—
72#11719945—44no—
73C2×D212—21no—
74C2×D212C2311yescharacteristic
75C2×D212C2311yescharacteristic
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53D411yescharacteristic
79C2×C4×D53D411yescharacteristic
80C2×C4×D53C2311yescharacteristic
81C2×Dic106—21no—
82C2×Dic106—21no—
83C2×Dic106C2311yescharacteristic
84SD16×C53—41no—
85D8×C53—41no—
86C2×C8×C53C2311yescharacteristic
87C2×C8×C53C2311yescharacteristic
88C4⋊C4×C53C2311yescharacteristic
89C22⋊C4×C53—21no—
90C22⋊C4×C53—21no—
91C22⋊C4×C53—21no—
92C42×C53C2311yescharacteristic
93C22⋊Dic53—21no—
94C22⋊Dic53—21no—
95C22⋊Dic53—21no—
96#11719924—41no—
97#11719923—41no—
98#11719922—41no—
99#11719922—41no—
100#11719921—21no—
101#11719921—21no—
102#11719921—21no—
103#11719921—21no—
104#11719921—21no—
105#11719921—21no—
106C4⋊Dic53C2311yescharacteristic
107#11719919—21no—
108C4×Dic53D411yescharacteristic
109C4×Dic53D411yescharacteristic
110C4×Dic53C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112C2×#88322C2311yescharacteristic
113D424—41no—
114#11719913—41no—
115C23×C53—21no—
116C23×C53—21no—
117C22×D53—21no—
118C22×D53—21no—
119C22×D53—42no—
120C22×D53—42no—
121C22×D53—42no—
122C22×D53C2×D411yescharacteristic
123Q8×C53—21no—
124Q8×C53—21no—
125D4×C53—21no—
126D4×C53—21no—
127D4×C53—21no—
128D4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—42no—
132C2×C4×C53C2×D411yescharacteristic
133C2×C4×C53C2×D411yescharacteristic
134C2×C4×C53C2411yescharacteristic
135C22⋊D53—42no—
136C22⋊D53—42no—
137C22⋊D53—42no—
138C22⋊D53—42no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53C2×D411yescharacteristic
142C2×Dic53C2×D411yescharacteristic
143C2×Dic53C2×D411yescharacteristic
144D212—21no—
145D212—21no—
146D212—21no—
147D212—21no—
148D212—42no—
149C4×D53—22no—
150C4×D53—21no—
151C4×D53—21no—
152C4×D53—41no—
153C4×D53—42no—
154C4×D53—42no—
155Dic106—21no—
156Dic106—21no—
157Dic106—42no—
158Dic106—42no—
159C8×C53—21no—
160C8×C53—21no—
161#88322—21no—
162#88322—21no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53C22×D411yescharacteristic
170C2×D53—21no—
171C2×D53—21no—
172C2×D53—41no—
173C2×D53—41no—
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—41no—
177C2×D53—41no—
178C2×D53—84no—
179C2×D53AGL(1,ℤ/8)12yes—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—41no—
183C4×C53—41no—
184C4×C53H(5,𝔽2)11yescharacteristic
185C4×C53H(5,𝔽2)11yescharacteristic
186Dic53—21no—
187Dic53—21no—
188Dic53—21no—
189Dic53—41no—
190Dic53—41no—
191#3213—531nomaximal, Sylow
192C2×C53—81no—
193C2×C53—81no—
194C2×C53#53511yescharacteristic
195C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
196C2×C53#57511yescharacteristic
197D53—22no—
198D53—81no—
199D53—81no—
200#532—531no—
201C2×#124—531no—
202#496—531no—
203#496—531no—
204#489—531no—
205#489—531no—
206#463—531no—
207#463—531no—
208#463—531no—
209#463D5311yescharacteristic, complemented
210#448—531no—
211#448—531no—
212#447—531no—
213#447—531no—
214#432—531no—
215C53#321311yescharacteristic, complemented, Sylow
216C2×#41—1061no—
217C22×Q8—531no—
218C22×D4—531no—
219C2×SD16—531no—
220C2×SD16—531no—
221C2×SD16—531no—
222C2×SD16C2×D5311yescharacteristic, complemented
223C2×D8—531no—
224C2×D8—531no—
225C2×D8—531no—
226C2×D8C2×D5311yescharacteristic
227C4⋊Q8—531no—
228#127—531no—
229#124—531no—
230#124—531no—
231#124—531no—
232#124—1061no—
233#124—1061no—
234#124—1061no—
235#124C2×D5311yescharacteristic
236#121—531no—
237#121—531no—
238#121—1061no—
239C4×D4—531no—
240C4×D4—1061no—
241C4×D4C2×D5311yescharacteristic, complemented
242C2×C22⋊C4—1061no—
243C2×C22⋊C4—1061no—
244C2×C42—531no—
245C4⋊C8—531no—
246C4⋊C8C2×D5311yescharacteristic, complemented
247#103—531no—
248#103—531no—
249#103—531no—
250#103C2×D5311yescharacteristic, complemented
251#102—531no—
252#102—531no—
253#102—531no—
254#102C2×D5311yescharacteristic
255C22⋊C8—531no—
256C22⋊C8—531no—
257#97—531no—
258#97—531no—
259C24—1061no—
260#41—2124no—
261C2×Q8—531no—
262C2×Q8—1061no—
263C2×Q8—1061no—
264C2×Q8—2121no—
265C2×Q8—2121no—
266C2×Q8C22×D5311yescharacteristic
267C2×D4—531no—
268C2×D4—531no—
269C2×D4—1061no—
270C2×D4—1061no—
271C2×D4—1061no—
272C2×D4—1061no—
273C2×D4—1061no—
274C2×D4—2121no—
275C2×D4—2121no—
276C2×D4C22×D5311yescharacteristic
277C2×D4C22×D5311yescharacteristic
278C22×C4—21no—
279C22×C4—531no—
280C22×C4—531no—
281C22×C4—531no—
282C22×C4—1061no—
283C22×C4—1061no—
284SD16—41no—
285SD16—2121no—
286SD16—2121no—
287SD16—2121no—
288D8—41no—
289D8—2121no—
290D8—2121no—
291D8—2121no—
292C2×C8—531no—
293C2×C8—531no—
294C2×C8C22×D5311yescharacteristic
295C2×C8C22×D5311yescharacteristic
296C4⋊C4—531no—
297C4⋊C4—1061no—
298C4⋊C4C22×D5311yescharacteristic
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—1061no—
303C22⋊C4—1061no—
304C22⋊C4—1061no—
305C22⋊C4—1061no—
306C22⋊C4—1061no—
307C22⋊C4—1061no—
308C22⋊C4—1061no—
309C22⋊C4—1061no—
310C22⋊C4—1061no—
311C42—531no—
312C42—531no—
313C42—531no—
314C42C22×D5311yescharacteristic
315C23—21no—
316C23—21no—
317C23—531no—
318C23—1061no—
319C23—1061no—
320C23—2122no—
321C23—2122no—
322C23—2122no—
323Q8—21no—
324Q8—21no—
325Q8—1061no—
326Q8—1061no—
327Q8—2122no—
328Q8—2122no—
329D4—21no—
330D4—21no—
331D4—21no—
332D4—21no—
333D4—1061no—
334D4—1061no—
335D4—1061no—
336D4—1061no—
337D4—2122no—
338D4—2122no—
339D4—2122no—
340D4—2122no—
341D4—2122no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—42no—
345C2×C4—531no—
346C2×C4—531no—
347C2×C4—531no—
348C2×C4—1062no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—2121no—
354C2×C4—2122no—
355C2×C4—2122no—
356C2×C4D4×D5311yescharacteristic
357C2×C4D4×D5311yescharacteristic
358C2×C4C23×D5311yescharacteristic
359C8—21no—
360C8—21no—
361C8—1061no—
362C8—1061no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—532no—
370C22—1061no—
371C22—1061no—
372C22—4244no—
373C22—2121no—
374C22—2121no—
375C22—2121no—
376C22—2121no—
377C22—2121no—
378C22—2121no—
379C22C2×D4×D5311yescharacteristic
380C4—21no—
381C4—21no—
382C4—41no—
383C4—41no—
384C4—1061no—
385C4—1061no—
386C4—1061no—
387C4—2121no—
388C4—2121no—
389C4#1313326211yescharacteristic
390C4#1313326311yescharacteristic
391C2—81no—
392C2—81no—
393C2—1062no—
394C2—4241no—
395C2—4241no—
396C2#1681118911yescharacteristic
397C2AGL(1,ℤ/8)×D5311yescharacteristic
398C2#1681142011yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#463C2
socleC22×C53C22×D4
hypercenter#463D53
hypocenterC53#3213
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #16811170C2semidirect
2. #124×D53C2semidirect
3. #16811011C2semidirect
4. #16811009C2semidirect
5. #16810984C2semidirect
6. #16810981C2semidirect
7. #16810839C2semidirect
8. #16810838C2semidirect
9. #16810835C2semidirect
10. #16810763C2semidirect
11. #16810760C2semidirect
12. #16810758C2semidirect
13. #16810755C2semidirect
14. #16810679C2semidirect
15. C53×#463C2semidirect
19. C2×#11719913C22semidirect
20. C2×#11719923C22semidirect
21. C2×#11719924C22semidirect
25. #13133216C22semidirect
26. #13133213C22semidirect
36. C4×D212C22semidirect
41. C4⋊#88322C22semidirect
42. #13133142C22semidirect
43. #13133141C22semidirect
44. #13133140C22semidirect
50. #13133127C22semidirect
51. #13133127C22semidirect
52. C2×SD16×C53C22semidirect
55. C4×D4×C53C22semidirect
56. C4⋊C8×C53C22semidirect
57. #103×C53C22semidirect
209. #463D53semidirect
215. C53#3213semidirect
222. C2×SD16C2×D53semidirect
241. C4×D4C2×D53semidirect
246. C4⋊C8C2×D53semidirect
250. #103C2×D53semidirect