Finite Group Databasebeta

← #20286922

Subgroups — #20286922

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#16811320C211yesmaximal, characteristic, complemented
2#16811264C211yesmaximal, characteristic, complemented
3#16810979C211yesmaximal, characteristic, complemented
4#16810961C211yesmaximal, characteristic, complemented
5#16810938C211yesmaximal, characteristic, complemented
6#16810926C211yesmaximal, characteristic, complemented
7#16810875C211yesmaximal, characteristic, complemented
8#16810840C211yesmaximal, characteristic, complemented
9#16810767C211yesmaximal, characteristic, complemented
10#16810756C211yesmaximal, characteristic, complemented
11#16810703C211yesmaximal, characteristic, complemented
12#16810699C211yesmaximal, characteristic, complemented
13#16810629C211yesmaximal, characteristic, complemented
14#16810623C211yesmaximal, characteristic, complemented
15C53×#468C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C2×#11719922C2211yescharacteristic
19C2×#11719922C2211yescharacteristic
20C2×#11719912C2211yescharacteristic, complemented
21#13133210—21no—
22#13133208C2211yescharacteristic, complemented
23#13133206C2211yescharacteristic
24#13133202—21no—
25#13133198—21no—
26#13133192—21no—
27#13133192C2211yescharacteristic
28#13133188—21no—
29#13133188C2211yescharacteristic, complemented
30#13133187—21no—
31D4×Dic53C2211yescharacteristic, complemented
32#13133179C2211yescharacteristic
33C4×C22⋊D53C2211yescharacteristic
34#13133177—21no—
35#13133176C2211yescharacteristic
36C4⋊C4×D53C2211yescharacteristic, complemented
37C22⋊C4×D53—21no—
38#13133150C2211yescharacteristic, complemented
39#13133148C2211yescharacteristic, complemented
40#13133146C2211yescharacteristic, complemented
41#13133141C2211yescharacteristic, complemented
42#13133141C2211yescharacteristic, complemented
43#13133139C2211yescharacteristic
44#13133139C2211yescharacteristic
45#13133139C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133129C2211yescharacteristic, complemented
51C8×Dic53C2211yescharacteristic, complemented
52C2×SD16×C53C2211yescharacteristic, complemented
53C2×M16×C53C2211yescharacteristic
54C22⋊Q8×C53C2211yescharacteristic
55#121×C53C2211yescharacteristic
56#107×C53C2211yescharacteristic, complemented
57#103×C53C2211yescharacteristic, complemented
58#102×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C2×Q8×C53C2311yescharacteristic
61C2×D4×C53—21no—
62C2×D4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C2×C22⋊D53—21no—
65C2×C22⋊D53D411yescharacteristic
66C22×Dic53—21no—
67#11719949—22no—
68#11719949—41no—
69C2×D212—22no—
70C2×D212—21no—
71C2×D212—21no—
72C2×D212—21no—
73C2×D212—41no—
74C2×D212C2311yescharacteristic
75C2×D212C2311yescharacteristic
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×C4×D53C2311yescharacteristic
80SD16×C53—41no—
81M16×C53—41no—
82C2×C8×C53C2311yescharacteristic
83C2×C8×C53C2311yescharacteristic
84C4⋊C4×C53—21no—
85C4⋊C4×C53C2311yescharacteristic
86C4⋊C4×C53C2311yescharacteristic
87C22⋊C4×C53—21no—
88C22⋊C4×C53—21no—
89C22⋊Dic53—21no—
90C22⋊Dic53—21no—
91C22⋊Dic53—21no—
92C22⋊Dic53D411yescharacteristic
93#11719922—22no—
94#11719922—41no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99#11719921D411yescharacteristic
100C4⋊Dic53—21no—
101C4⋊Dic53C2311yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103#11719919—21no—
104#11719919—21no—
105#11719919D411yescharacteristic
106C4×Dic53—21no—
107C4×Dic53C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110#11719912—41no—
111C23×C53—21no—
112C23×C53C2×D411yescharacteristic
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—42no—
116C22×D53—42no—
117C22×D53—81no—
118C22×D53C2×D411yescharacteristic
119Q8×C53—21no—
120Q8×C53—21no—
121D4×C53—21no—
122D4×C53—21no—
123D4×C53—81no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53C2×D411yescharacteristic
130C2×C4×C53C2411yescharacteristic
131C22⋊D53—41no—
132C22⋊D53—42no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—81no—
136C2×Dic53C2×D411yescharacteristic
137C2×Dic53C2×D411yescharacteristic
138C2×Dic53C2×D411yescharacteristic
139D212—21no—
140D212—21no—
141D212—21no—
142D212—42no—
143D212—42no—
144D212—81no—
145D212#4112yes—
146C4×D53—21no—
147C4×D53—21no—
148C4×D53—42no—
149C4×D53—81no—
150C4×D53—81no—
151C8×C53—21no—
152C8×C53—21no—
153#88322—21no—
154#88322—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53C22×D411yescharacteristic
162C2×D53—21no—
163C2×D53—22no—
164C2×D53—21no—
165C2×D53—21no—
166C2×D53—41no—
167C2×D53—41no—
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—41no—
171C2×D53—84no—
172C4×C53—21no—
173C4×C53—41no—
174C4×C53—41no—
175C4×C53—41no—
176C4×C53C2×#4111yescharacteristic
177C4×C53H(5,𝔽2)11yescharacteristic
178Dic53—21no—
179Dic53—21no—
180Dic53—21no—
181Dic53—41no—
182Dic53—41no—
183#3322—531nomaximal, Sylow
184C2×C53—41no—
185C2×C53—81no—
186C2×C53#54611yescharacteristic
187C2×C53#57511yescharacteristic
188C2×C53#57611yescharacteristic
189D53—41no—
190D53—42no—
191D53—81no—
192#553—531no—
193#546—531no—
194#488—531no—
195#485—531no—
196#480—531no—
197#478—531no—
198#469—531no—
199#468D5311yescharacteristic, complemented
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×SD16C2×D5311yescharacteristic, complemented
211C2×D8—531no—
212C2×D8—531no—
213C2×M16—531no—
214C2×M16C2×D5311yescharacteristic
215#126—1061no—
216#125—531no—
217#124—531no—
218#123—1061no—
219#123—1061no—
220C22⋊Q8C2×D5311yescharacteristic
221#121—531no—
222#121—531no—
223#121—1061no—
224#121—1061no—
225#121C2×D5311yescharacteristic
226#120—1061no—
227C4×D4—531no—
228C4×D4—531no—
229C4×D4—531no—
230C4×D4—1061no—
231#117—531no—
232C2×C4⋊C4—531no—
233C2×C22⋊C4—1061no—
234#107—531no—
235#107C2×D5311yescharacteristic, complemented
236#106—531no—
237C4⋊C8—531no—
238#103—531no—
239#103—531no—
240#103C2×D5311yescharacteristic, complemented
241#102—531no—
242#102—531no—
243#102—531no—
244#102—531no—
245#102—531no—
246#102—531no—
247#102C2×D5311yescharacteristic, complemented
248C22⋊C8—531no—
249C22⋊C8—531no—
250C4×C8—531no—
251C24—1061no—
252#41—1062no—
253#41—2121no—
254C2×Q8C22×D5311yescharacteristic
255C2×D4—21no—
256C2×D4—531no—
257C2×D4—531no—
258C2×D4—531no—
259C2×D4—1062no—
260C2×D4—1061no—
261C2×D4—1061no—
262C2×D4—1061no—
263C2×D4—1061no—
264C2×D4—2121no—
265C2×D4C22×D5311yescharacteristic
266C22×C4—531no—
267C22×C4—1061no—
268C22×C4—1061no—
269C22×C4—1061no—
270C22×C4—1061no—
271C22×C4C22×D5311yescharacteristic
272SD16—41no—
273D8—1062no—
274D8—2121no—
275M16—41no—
276M16—2121no—
277C2×C8—531no—
278C2×C8—531no—
279C2×C8C22×D5311yescharacteristic
280C2×C8C22×D5311yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—531no—
283C4⋊C4—531no—
284C4⋊C4—531no—
285C4⋊C4—1061no—
286C4⋊C4—1061no—
287C4⋊C4—1061no—
288C4⋊C4C22×D5311yescharacteristic
289C4⋊C4C22×D5311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—531no—
293C22⋊C4—531no—
294C22⋊C4—1061no—
295C22⋊C4—1061no—
296C22⋊C4—1061no—
297C22⋊C4—1061no—
298C22⋊C4—1061no—
299C22⋊C4—1061no—
300C22⋊C4—1061no—
301C42—531no—
302C42—1061no—
303C23—21no—
304C23—531no—
305C23—1061no—
306C23—1061no—
307C23—4241no—
308C23—2122no—
309C23—2122no—
310C23D4×D5311yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—81no—
316D4—532no—
317D4—1061no—
318D4—1061no—
319D4—1061no—
320D4—4241no—
321D4—2122no—
322D4—2122no—
323D4—2121no—
324D4—2122no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—531no—
331C2×C4—531no—
332C2×C4—531no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—4241no—
338C2×C4—4241no—
339C2×C4—4241no—
340C2×C4—2122no—
341C2×C4D4×D5311yescharacteristic
342C2×C4C23×D5311yescharacteristic
343C8—21no—
344C8—21no—
345C8—1061no—
346C8—1061no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—1061no—
354C22—1061no—
355C22—1061no—
356C22—1062no—
357C22—4244no—
358C22—2121no—
359C22—2121no—
360C22—2121no—
361C22—2121no—
362C22—2121no—
363C22C2×D4×D5311yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—1061no—
369C4—1061no—
370C4—1061no—
371C4—2121no—
372C4—2121no—
373C4#41×D5311yescharacteristic
374C4#1313326211yescharacteristic
375C2—41no—
376C2—81no—
377C2—4241no—
378C2—2121no—
379C2—2122no—
380C2#1681126211yescharacteristic
381C2#1681142211yescharacteristic
382C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#468C2
socleC22×C53C22×D4
hypercenter#468D53
hypocenterC53#3322
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #16811320C2semidirect
2. #16811264C2semidirect
3. #16810979C2semidirect
4. #16810961C2semidirect
5. #16810938C2semidirect
6. #16810926C2semidirect
7. #16810875C2semidirect
8. #16810840C2semidirect
9. #16810767C2semidirect
10. #16810756C2semidirect
11. #16810703C2semidirect
12. #16810699C2semidirect
13. #16810629C2semidirect
14. #16810623C2semidirect
15. C53×#468C2semidirect
20. C2×#11719912C22semidirect
22. #13133208C22semidirect
29. #13133188C22semidirect
31. D4×Dic53C22semidirect
36. C4⋊C4×D53C22semidirect
38. #13133150C22semidirect
39. #13133148C22semidirect
40. #13133146C22semidirect
41. #13133141C22semidirect
42. #13133141C22semidirect
50. #13133129C22semidirect
51. C8×Dic53C22semidirect
52. C2×SD16×C53C22semidirect
56. #107×C53C22semidirect
57. #103×C53C22semidirect
58. #102×C53C22semidirect
199. #468D53semidirect
207. C53#3322semidirect
210. C2×SD16C2×D53semidirect
235. #107C2×D53semidirect
240. #103C2×D53semidirect
247. #102C2×D53semidirect