Finite Group Databasebeta

← #20286558

Subgroups — #20286558

Order 6784.

420 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#16811261C211yesmaximal, characteristic, complemented
3#16810979C211yesmaximal, characteristic, complemented
4#16810957C211yesmaximal, characteristic, complemented
5#16810948C211yesmaximal, characteristic, complemented
6#16810925C211yesmaximal, characteristic, complemented
7#16810857C211yesmaximal, characteristic, complemented
8#16810849C211yesmaximal, characteristic, complemented
9#16810840C211yesmaximal, characteristic, complemented
10#16810755C211yesmaximal, characteristic, complemented
11D8×Dic53C211yesmaximal, characteristic, complemented
12#16810686C211yesmaximal, characteristic, complemented
13#16810619C211yesmaximal, characteristic, complemented
14#102×D53C211yesmaximal, characteristic, complemented
15C53×#449C211yesmaximal, characteristic, complemented
16C2×D4×D53C2211yescharacteristic
17C2×#11719922C2211yescharacteristic
18C22×#88322C2211yescharacteristic, complemented
19C2×C8×D53C2211yescharacteristic
20#13133210—21no—
21#13133208C2211yescharacteristic, complemented
22#13133206C2211yescharacteristic
23#13133200—21no—
24#13133198—21no—
25#13133193C2211yescharacteristic
26#13133191C2211yescharacteristic
27#13133189—21no—
28#13133188—21no—
29#13133188C2211yescharacteristic
30#13133185—21no—
31D4×Dic53C2211yescharacteristic
32D4×Dic53C2211yescharacteristic, complemented
33C4×C22⋊D53C2211yescharacteristic
34#13133177—21no—
35#13133176C2211yescharacteristic, complemented
36C4⋊C4×D53C2211yescharacteristic
37C2×#11719921—21no—
38#13133151C2211yescharacteristic, complemented
39#13133150C2211yescharacteristic, complemented
40#13133149C2211yescharacteristic, complemented
41#13133146C2211yescharacteristic, complemented
42#13133141C2211yescharacteristic, complemented
43#13133141C2211yescharacteristic, complemented
44#13133139C2211yescharacteristic
45#13133139C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133138C2211yescharacteristic, complemented
49#13133137C2211yescharacteristic
50#13133129C2211yescharacteristic
51C8×Dic53C2211yescharacteristic, complemented
52C2×#41×C53C2211yescharacteristic
53C2×SD16×C53C2211yescharacteristic, complemented
54C2×D8×C53C2211yescharacteristic
55#121×C53C2211yescharacteristic, complemented
56#103×C53C2211yescharacteristic, complemented
57#102×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60#41×C53—22no—
61#41×C53—41no—
62C2×Q8×C53C2311yescharacteristic
63C2×D4×C53—21no—
64C2×D4×C53—21no—
65C2×D4×C53C2311yescharacteristic
66C2×D4×C53C2311yescharacteristic
67C22×C4×C53—21no—
68C22×C4×C53C2311yescharacteristic
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C2×C22⋊D53—21no—
72C2×C22⋊D53D411yescharacteristic
73C22×Dic53—21no—
74C22×Dic53—21no—
75D4×D53—22no—
76D4×D53—41no—
77C2×D212C2311yescharacteristic
78C2×C4×D53—21no—
79C2×C4×D53C2311yescharacteristic
80SD16×C53—41no—
81D8×C53—22no—
82C2×C8×C53C2311yescharacteristic
83C2×C8×C53C2311yescharacteristic
84C4⋊C4×C53C2311yescharacteristic
85C22⋊C4×C53—21no—
86C22⋊Dic53—21no—
87C22⋊Dic53—21no—
88C22⋊Dic53—21no—
89C22⋊Dic53—21no—
90C22⋊Dic53D411yescharacteristic
91#11719922—41no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921D411yescharacteristic
97C4⋊Dic53—21no—
98C4⋊Dic53C2311yescharacteristic
99C4⋊Dic53C2311yescharacteristic
100C4⋊Dic53C2311yescharacteristic
101#11719919—21no—
102#11719919—21no—
103#11719919—21no—
104#11719919D411yescharacteristic
105C4×Dic53—21no—
106C4×Dic53C2311yescharacteristic
107C2×#88322—41no—
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C8×D53—41no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53C2×D411yescharacteristic
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—21no—
124D4×C53—42no—
125D4×C53—81no—
126D4×C53#4112yes—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—42no—
132C2×C4×C53C2×D411yescharacteristic
133C2×C4×C53C2411yescharacteristic
134C22⋊D53—22no—
135C22⋊D53—42no—
136C22⋊D53—42no—
137C22⋊D53—81no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—81no—
142C2×Dic53—81no—
143C2×Dic53C2×D411yescharacteristic
144C2×Dic53C2×D411yescharacteristic
145C2×Dic53C2×D411yescharacteristic
146D212—21no—
147D212—21no—
148C4×D53—21no—
149C4×D53—21no—
150C4×D53—81no—
151C8×C53—21no—
152C8×C53—21no—
153#88322—21no—
154#88322—21no—
155C22×C53—22no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53C22×D411yescharacteristic
165C2×D53—21no—
166C2×D53—21no—
167C2×D53—41no—
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—84no—
171C2×D53#13512yes—
172C4×C53—21no—
173C4×C53—41no—
174C4×C53—41no—
175C4×C53C2×#4111yescharacteristic
176C4×C53H(5,𝔽2)11yescharacteristic
177Dic53—21no—
178Dic53—21no—
179Dic53—21no—
180Dic53—41no—
181Dic53—41no—
182Dic53—41no—
183#3297—531nomaximal, Sylow
184C2×C53—42no—
185C2×C53—41no—
186C2×C53—81no—
187C2×C53#54611yescharacteristic
188C2×C53C2×#13511yescharacteristic
189C2×C53#57611yescharacteristic
190D53—22no—
191D53—81no—
192#553—531no—
193#546—531no—
194#488—531no—
195#484—531no—
196#482—531no—
197#478—531no—
198#466—531no—
199#465—531no—
200#463—531no—
201#449D5311yescharacteristic, complemented
202#447—531no—
203C4×D8—531no—
204C8×D4—531no—
205#416—531no—
206C2×#102—531no—
207C53#329711yescharacteristic, complemented, Sylow
208C2×#41C2×D5311yescharacteristic
209C22×D4—531no—
210C2×SD16C2×D5311yescharacteristic, complemented
211C2×D8—531no—
212C2×D8C2×D5311yescharacteristic
213C22×C8—531no—
214C22×C8—531no—
215#126—1061no—
216#125—531no—
217#124—531no—
218#123—1061no—
219#123—1061no—
220C22⋊Q8—531no—
221#121—531no—
222#121—531no—
223#121—1061no—
224#121—1061no—
225#121C2×D5311yescharacteristic, complemented
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#107—531no—
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
248C22⋊C8—531no—
249C22⋊C8C2×D5311yescharacteristic, complemented
250C4×C8—531no—
251C24—1061no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D5311yescharacteristic
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—531no—
258C2×D4—531no—
259C2×D4—1061no—
260C2×D4—1062no—
261C2×D4—1061no—
262C2×D4—1061no—
263C2×D4—2121no—
264C2×D4C22×D5311yescharacteristic
265C2×D4C22×D5311yescharacteristic
266C22×C4—21no—
267C22×C4—531no—
268C22×C4—1061no—
269C22×C4—1061no—
270C22×C4—1061no—
271C22×C4C22×D5311yescharacteristic
272SD16—41no—
273D8—22no—
274D8—2121no—
275C2×C8—531no—
276C2×C8—531no—
277C2×C8—2121no—
278C2×C8—2121no—
279C2×C8C22×D5311yescharacteristic
280C2×C8C22×D5311yescharacteristic
281C4⋊C4—531no—
282C4⋊C4—531no—
283C4⋊C4—531no—
284C4⋊C4—531no—
285C4⋊C4—1061no—
286C4⋊C4—1061no—
287C4⋊C4—1061no—
288C4⋊C4—1061no—
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—531no—
302C42—1061no—
303C23—21no—
304C23—21no—
305C23—531no—
306C23—1061no—
307C23—4241no—
308C23—2122no—
309C23—2122no—
310C23D4×D5311yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—81no—
318D4—1062no—
319D4—1061no—
320D4—1061no—
321D4—4241no—
322D4—2122no—
323D4—2122no—
324D4#1171994712yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—42no—
330C2×C4—531no—
331C2×C4—531no—
332C2×C4—531no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—4241no—
339C2×C4—4241no—
340C2×C4—4241no—
341C2×C4D4×D5311yescharacteristic
342C2×C4C23×D5311yescharacteristic
343C8—21no—
344C8—21no—
345C8—1061no—
346C8—1061no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—22no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—532no—
357C22—1061no—
358C22—1061no—
359C22—4244no—
360C22—2121no—
361C22—2121no—
362C22—2121no—
363C22C2×D4×D5311yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—1061no—
368C4—1061no—
369C4—1061no—
370C4—2121no—
371C4—2121no—
372C4—2121no—
373C4C2×#1171994711yescharacteristic
374C4#1313326211yescharacteristic
375C2—41no—
376C2—42no—
377C2—81no—
378C2—1062no—
379C2—4241no—
380C2#1681126011yescharacteristic
381C2#135×D5311yescharacteristic
382C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#449C2
socleC22×C53C22×D4
hypercenter#449D53
hypocenterC53#3297
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #16811320C2semidirect
2. #16811261C2semidirect
3. #16810979C2semidirect
4. #16810957C2semidirect
5. #16810948C2semidirect
6. #16810925C2semidirect
7. #16810857C2semidirect
8. #16810849C2semidirect
9. #16810840C2semidirect
10. #16810755C2semidirect
11. D8×Dic53C2semidirect
12. #16810686C2semidirect
13. #16810619C2semidirect
14. #102×D53C2semidirect
15. C53×#449C2semidirect
18. C22×#88322C22semidirect
21. #13133208C22semidirect
32. D4×Dic53C22semidirect
35. #13133176C22semidirect
38. #13133151C22semidirect
39. #13133150C22semidirect
40. #13133149C22semidirect
41. #13133146C22semidirect
42. #13133141C22semidirect
43. #13133141C22semidirect
48. #13133138C22semidirect
51. C8×Dic53C22semidirect
53. C2×SD16×C53C22semidirect
55. #121×C53C22semidirect
56. #103×C53C22semidirect
58. C22⋊C8×C53C22semidirect
201. #449D53semidirect
207. C53#3297semidirect
210. C2×SD16C2×D53semidirect
225. #121C2×D53semidirect
240. #103C2×D53semidirect
249. C22⋊C8C2×D53semidirect