Finite Group Databasebeta

← #20286913

Subgroups — #20286913

Order 6784.

414 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811326C211yesmaximal, characteristic, complemented
2#16811268C211yesmaximal, characteristic, complemented
3#16810980C211yesmaximal, characteristic, complemented
4#16810967C211yesmaximal, characteristic, complemented
5#16810938C211yesmaximal, characteristic, complemented
6#16810877C211yesmaximal, characteristic, complemented
7#16810870C211yesmaximal, characteristic, complemented
8#16810839C211yesmaximal, characteristic, complemented
9#16810769C211yesmaximal, characteristic, complemented
10#16810758C211yesmaximal, characteristic, complemented
11C4×#11719922C211yesmaximal, characteristic, complemented
12#16810700C211yesmaximal, characteristic, complemented
13#16810626C211yesmaximal, characteristic, complemented
14#16810621C211yesmaximal, characteristic, complemented
15C53×#478C211yesmaximal, characteristic, complemented
16C2×#11719947C2211yescharacteristic
17C2×D4×D53C2211yescharacteristic
18C2×#11719913C2211yescharacteristic, complemented
19C2×D424C2211yescharacteristic
20C2×#11719922C2211yescharacteristic
21C2×#11719912C2211yescharacteristic
22C2×#11719912C2211yescharacteristic, complemented
23#13133212—21no—
24#13133205C2211yescharacteristic
25#13133202—21no—
26#13133201—21no—
27#13133197C2211yescharacteristic
28#13133192—21no—
29#13133192C2211yescharacteristic
30#13133192C2211yescharacteristic
31#13133192C2211yescharacteristic, complemented
32#13133191—21no—
33#13133187—21no—
34C4×D212C2211yescharacteristic
35C4×D212C2211yescharacteristic, complemented
36#13133179—21no—
37#13133175C2211yescharacteristic
38C4⋊C4×D53C2211yescharacteristic, complemented
39C2×#11719921—21no—
40#13133150C2211yescharacteristic, complemented
41#13133142C2211yescharacteristic, complemented
42#13133140C2211yescharacteristic, complemented
43#13133140C2211yescharacteristic, complemented
44#13133139C2211yescharacteristic
45#13133138C2211yescharacteristic
46#13133137C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic, complemented
49#13133129C2211yescharacteristic
50#13133129C2211yescharacteristic, complemented
51C4×#88322C2211yescharacteristic, complemented
52#125×C53C2211yescharacteristic, complemented
53C4×D4×C53C2211yescharacteristic
54#107×C53C2211yescharacteristic, complemented
55#106×C53C2211yescharacteristic, complemented
56C4⋊C8×C53C2211yescharacteristic, complemented
57#102×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59C23×D53—21no—
60C2×D4×C53C2311yescharacteristic
61C22×C4×C53—21no—
62C2×C22⋊D53—21no—
63C2×C22⋊D53—21no—
64C22×Dic53—21no—
65#11719947—22no—
66#11719947—41no—
67D4×D53—22no—
68D4×D53—41no—
69C2×D212—21no—
70C2×D212—21no—
71C2×D212—21no—
72C2×D212D411yescharacteristic
73C2×D212C2311yescharacteristic
74C2×D212C2311yescharacteristic
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×C4×D53C2311yescharacteristic
78C2×C4×D53C2311yescharacteristic
79C2×Dic106C2311yescharacteristic
80C2×C8×C53C2311yescharacteristic
81C2×C8×C53C2311yescharacteristic
82C4⋊C4×C53—21no—
83C4⋊C4×C53—21no—
84C4⋊C4×C53C2311yescharacteristic
85C4⋊C4×C53C2311yescharacteristic
86C4⋊C4×C53C2311yescharacteristic
87C22⋊C4×C53—21no—
88C42×C53C2311yescharacteristic
89#11719922—22no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98#11719921D411yescharacteristic
99#11719921D411yescharacteristic
100C4⋊Dic53—21no—
101C4⋊Dic53D411yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103#11719919—21no—
104C4×Dic53—21no—
105C2×#88322C2311yescharacteristic
106C2×#88322C2311yescharacteristic
107D424—41no—
108#11719913—41no—
109#11719912—41no—
110#11719912—41no—
111C23×C53—21no—
112C22×D53—21no—
113C22×D53—21no—
114C22×D53—42no—
115C22×D53—42no—
116C22×D53—81no—
117C22×D53C2×D411yescharacteristic
118C22×D53C2×D411yescharacteristic
119D4×C53—21no—
120D4×C53#4112yes—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—81no—
125C2×C4×C53C2×D411yescharacteristic
126C2×C4×C53C2×D411yescharacteristic
127C2×C4×C53C2411yescharacteristic
128C22⋊D53—42no—
129C22⋊D53—42no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—42no—
133C2×Dic53C2×D411yescharacteristic
134C2×Dic53C2×D411yescharacteristic
135D212—21no—
136D212—21no—
137D212—21no—
138D212—21no—
139D212—41no—
140D212—42no—
141D212—81no—
142D212—81no—
143C4×D53—21no—
144C4×D53—21no—
145C4×D53—21no—
146C4×D53—21no—
147C4×D53—81no—
148C4×D53—81no—
149Dic106—21no—
150Dic106—21no—
151C8×C53—21no—
152C8×C53—21no—
153#88322—21no—
154#88322—21no—
155C22×C53—22no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53C22×D411yescharacteristic
159C2×D53—21no—
160C2×D53—21no—
161C2×D53—21no—
162C2×D53—21no—
163C2×D53—21no—
164C2×D53—21no—
165C2×D53—41no—
166C2×D53—41no—
167C2×D53—41no—
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—41no—
171C2×D53—84no—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—41no—
175C4×C53—41no—
176C4×C53—41no—
177C4×C53C2×#4111yescharacteristic
178C4×C53H(5,𝔽2)11yescharacteristic
179Dic53—21no—
180Dic53—21no—
181Dic53—41no—
182Dic53—41no—
183#3322—531nomaximal, Sylow
184C2×C53—42no—
185C2×C53#54611yescharacteristic
186C2×C53#57511yescharacteristic
187C2×C53#57611yescharacteristic
188D53—41no—
189D53—41no—
190D53—81no—
191D53—81no—
192#553—531no—
193#546—531no—
194#488—531no—
195#485—531no—
196#480—531no—
197#478D5311yescharacteristic, complemented
198#469—531no—
199#468—531no—
200#463—531no—
201#449—531no—
202#447—531no—
203C4×D8—531no—
204#436—531no—
205#418—531no—
206#417—531no—
207C53#332211yescharacteristic, complemented, Sylow
208C2×#41—531no—
209C22×D4—531no—
210C2×SD16—531no—
211C2×D8—531no—
212C2×D8—531no—
213C2×M16—531no—
214C2×M16—531no—
215#126—1061no—
216#125C2×D5311yescharacteristic, complemented
217#124—531no—
218#123—1061no—
219#123—1061no—
220C22⋊Q8—531no—
221#121—531no—
222#121—531no—
223#121—531no—
224#121—1061no—
225#121—1061no—
226#120—1061no—
227C4×D4—531no—
228C4×D4—531no—
229C4×D4—1061no—
230C4×D4C2×D5311yescharacteristic
231#117—531no—
232C2×C4⋊C4—531no—
233C2×C22⋊C4—1061no—
234#107—531no—
235#107C2×D5311yescharacteristic, complemented
236#106C2×D5311yescharacteristic, complemented
237C4⋊C8C2×D5311yescharacteristic, complemented
238#103—531no—
239#103—531no—
240#103—531no—
241#102—531no—
242#102—531no—
243#102—531no—
244#102—531no—
245#102—531no—
246#102C2×D5311yescharacteristic
247#102C2×D5311yescharacteristic
248C22⋊C8—531no—
249C22⋊C8—531no—
250C4×C8—531no—
251C24—1061no—
252#41—1062no—
253#41—2121no—
254C2×Q8—531no—
255C2×D4—531no—
256C2×D4—531no—
257C2×D4—531no—
258C2×D4—1061no—
259C2×D4—1061no—
260C2×D4—1061no—
261C2×D4—1061no—
262C2×D4—1062no—
263C2×D4—1061no—
264C2×D4—2121no—
265C2×D4C22×D5311yescharacteristic
266C22×C4—21no—
267C22×C4—531no—
268C22×C4—531no—
269C22×C4—1061no—
270C22×C4—1061no—
271C22×C4—1061no—
272SD16—2121no—
273D8—1062no—
274D8—2121no—
275M16—2121no—
276M16—2121no—
277C2×C8—531no—
278C2×C8—531no—
279C2×C8C22×D5311yescharacteristic
280C2×C8C22×D5311yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—531no—
284C4⋊C4—531no—
285C4⋊C4—1061no—
286C4⋊C4—1061no—
287C4⋊C4C22×D5311yescharacteristic
288C4⋊C4C22×D5311yescharacteristic
289C4⋊C4C22×D5311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—531no—
292C22⋊C4—531no—
293C22⋊C4—1061no—
294C22⋊C4—1061no—
295C22⋊C4—1061no—
296C22⋊C4—1061no—
297C22⋊C4—1061no—
298C22⋊C4—1061no—
299C22⋊C4—1061no—
300C22⋊C4—1061no—
301C42—1061no—
302C42C22×D5311yescharacteristic
303C23—21no—
304C23—531no—
305C23—531no—
306C23—1061no—
307C23—1061no—
308C23—4241no—
309C23—2122no—
310C23—2122no—
311Q8—1061no—
312Q8—1061no—
313D4—21no—
314D4—1061no—
315D4—1061no—
316D4—1061no—
317D4—1061no—
318D4—4241no—
319D4—4241no—
320D4—2121no—
321D4—2122no—
322D4—2122no—
323D4—2122no—
324D4#1171994912yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—81no—
329C2×C4—531no—
330C2×C4—531no—
331C2×C4—1061no—
332C2×C4—1061no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—4241no—
338C2×C4—4241no—
339C2×C4—2122no—
340C2×C4D4×D5311yescharacteristic
341C2×C4D4×D5311yescharacteristic
342C2×C4C23×D5311yescharacteristic
343C8—21no—
344C8—21no—
345C8—1061no—
346C8—1061no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—1061no—
351C22—1061no—
352C22—1061no—
353C22—1061no—
354C22—1061no—
355C22—1061no—
356C22—4244no—
357C22—2121no—
358C22—2121no—
359C22—2121no—
360C22—2121no—
361C22—2121no—
362C22—2121no—
363C22C2×D4×D5311yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—1061no—
370C4—1061no—
371C4—2121no—
372C4—2121no—
373C4C2×#1171994911yescharacteristic
374C4#1313326311yescharacteristic
375C2—42no—
376C2—4241no—
377C2—4241no—
378C2—2121no—
379C2—2121no—
380C2#1681126411yescharacteristic
381C2#1681142011yescharacteristic
382C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#478C2
socleC22×C53C22×D4
hypercenter#478D53
hypocenterC53#3322
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #16811326C2semidirect
2. #16811268C2semidirect
3. #16810980C2semidirect
4. #16810967C2semidirect
5. #16810938C2semidirect
6. #16810877C2semidirect
7. #16810870C2semidirect
8. #16810839C2semidirect
9. #16810769C2semidirect
10. #16810758C2semidirect
11. C4×#11719922C2semidirect
12. #16810700C2semidirect
13. #16810626C2semidirect
14. #16810621C2semidirect
15. C53×#478C2semidirect
18. C2×#11719913C22semidirect
22. C2×#11719912C22semidirect
31. #13133192C22semidirect
35. C4×D212C22semidirect
38. C4⋊C4×D53C22semidirect
40. #13133150C22semidirect
41. #13133142C22semidirect
42. #13133140C22semidirect
43. #13133140C22semidirect
48. #13133137C22semidirect
50. #13133129C22semidirect
51. C4×#88322C22semidirect
52. #125×C53C22semidirect
54. #107×C53C22semidirect
55. #106×C53C22semidirect
56. C4⋊C8×C53C22semidirect
197. #478D53semidirect
207. C53#3322semidirect
216. #125C2×D53semidirect
235. #107C2×D53semidirect
236. #106C2×D53semidirect
237. C4⋊C8C2×D53semidirect