Finite Group Databasebeta

← #20286878

Subgroups — #20286878

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#16811338C211yesmaximal, characteristic, complemented
2#16811266C211yesmaximal, characteristic, complemented
3#16810982C211yesmaximal, characteristic, complemented
4#16810963C211yesmaximal, characteristic, complemented
5#16810945C211yesmaximal, characteristic, complemented
6#16810904C211yesmaximal, characteristic, complemented
7#16810866C211yesmaximal, characteristic, complemented
8#16810856C211yesmaximal, characteristic, complemented
9#16810767C211yesmaximal, characteristic, complemented
10#16810755C211yesmaximal, characteristic, complemented
11#16810733C211yesmaximal, characteristic, complemented
12#16810693C211yesmaximal, characteristic, complemented
13#16810626C211yesmaximal, characteristic, complemented
14#16810618C211yesmaximal, characteristic, complemented
15C53×#463C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×D4×D53C2211yescharacteristic
18C2×#11719922C2211yescharacteristic
19C2×#11719912C2211yescharacteristic
20C2×C8×D53C2211yescharacteristic, complemented
21#13133217C2211yescharacteristic, complemented
22#13133203—21no—
23#13133203—21no—
24#13133199—21no—
25#13133193C2211yescharacteristic
26#13133191—21no—
27#13133190—21no—
28#13133188C2211yescharacteristic
29#13133188C2211yescharacteristic
30#13133188C2211yescharacteristic, complemented
31#13133187—21no—
32C4×D212C2211yescharacteristic
33C4×D212C2211yescharacteristic
34#13133176—21no—
35#13133175C2211yescharacteristic
36#13133175C2211yescharacteristic, complemented
37C22⋊C4×D53—21no—
38#13133151C2211yescharacteristic, complemented
39#13133151C2211yescharacteristic, complemented
40#13133148C2211yescharacteristic, complemented
41#13133141C2211yescharacteristic, complemented
42#13133141C2211yescharacteristic, complemented
43#13133139C2211yescharacteristic
44#13133139C2211yescharacteristic
45#13133138C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133138C2211yescharacteristic, complemented
48#13133137C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133129C2211yescharacteristic, complemented
51#13133128C2211yescharacteristic, 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—
68#11719949—22no—
69#11719949—41no—
70D4×D53—22no—
71D4×D53—41no—
72C2×D212—21no—
73C2×D212D411yescharacteristic
74C2×D212C2311yescharacteristic
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×C4×D53C2311yescharacteristic
78C2×C4×D53C2311yescharacteristic
79C2×Dic106—21no—
80SD16×C53—41no—
81D8×C53—41no—
82C2×C8×C53C2311yescharacteristic
83C2×C8×C53C2311yescharacteristic
84C4⋊C4×C53C2311yescharacteristic
85C22⋊C4×C53—21no—
86C22⋊C4×C53—21no—
87C22⋊C4×C53—21no—
88C42×C53C2311yescharacteristic
89C22⋊Dic53—21no—
90C22⋊Dic53—21no—
91C22⋊Dic53—21no—
92#11719922—22no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97#11719921D411yescharacteristic
98#11719921D411yescharacteristic
99C4⋊Dic53—21no—
100C4⋊Dic53D411yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103C4⋊Dic53C2311yescharacteristic
104#11719919—21no—
105C4×Dic53—21no—
106C4×Dic53—21no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109#11719912—41no—
110C8×D53—41no—
111C23×C53—21no—
112C23×C53—21no—
113C22×D53—21no—
114C22×D53—42no—
115C22×D53—42no—
116C22×D53—81no—
117C22×D53C2×D411yescharacteristic
118C22×D53C2×D411yescharacteristic
119Q8×C53—21no—
120Q8×C53—21no—
121D4×C53—21no—
122D4×C53—21no—
123D4×C53—21no—
124D4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—42no—
128C2×C4×C53C2×D411yescharacteristic
129C2×C4×C53C2×D411yescharacteristic
130C2×C4×C53C2411yescharacteristic
131C22⋊D53—42no—
132C22⋊D53—81no—
133C22⋊D53—81no—
134C22⋊D53—81no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53C2×D411yescharacteristic
139C2×Dic53C2×D411yescharacteristic
140D212—21no—
141D212—41no—
142D212—42no—
143D212#4112yes—
144C4×D53—21no—
145C4×D53—21no—
146C4×D53—21no—
147C4×D53—21no—
148C4×D53—42no—
149C4×D53—81no—
150Dic106—81no—
151C8×C53—21no—
152C8×C53—21no—
153#88322—21no—
154#88322—21no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53C22×D411yescharacteristic
162C2×D53—21no—
163C2×D53—21no—
164C2×D53—22no—
165C2×D53—21no—
166C2×D53—21no—
167C2×D53—21no—
168C2×D53—21no—
169C2×D53—41no—
170C2×D53—41no—
171C2×D53—84no—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—41no—
175C4×C53—41no—
176C4×C53C2×#4111yescharacteristic
177C4×C53H(5,𝔽2)11yescharacteristic
178Dic53—21no—
179Dic53—21no—
180Dic53—41no—
181Dic53—41no—
182Dic53—41no—
183#3319—531nomaximal, Sylow
184C2×C53—81no—
185C2×C53—81no—
186C2×C53#54611yescharacteristic
187C2×C53#57511yescharacteristic
188C2×C53#57611yescharacteristic
189D53—41no—
190D53—41no—
191D53—42no—
192#555—531no—
193#546—531no—
194#489—531no—
195#485—531no—
196#482—531no—
197#474—531no—
198#468—531no—
199#466—531no—
200#463D5311yescharacteristic, complemented
201#449—531no—
202#447—531no—
203#442—531no—
204#435—531no—
205#418—531no—
206#416—531no—
207C53#331911yescharacteristic, complemented, Sylow
208C2×#41—531no—
209C22×D4—531no—
210C2×SD16C2×D5311yescharacteristic, complemented
211C2×D8—531no—
212C2×D8C2×D5311yescharacteristic
213C2×M16—531no—
214C22×C8—531no—
215C4⋊Q8—531no—
216#124—1061no—
217#124—1061no—
218#124C2×D5311yescharacteristic
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×D4C2×D5311yescharacteristic, complemented
230#117—531no—
231#117—531no—
232#117—1061no—
233C2×C22⋊C4—1061no—
234#107—531no—
235#107—531no—
236#106—531no—
237C4⋊C8C2×D5311yescharacteristic, complemented
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⋊C8—531no—
250#97—531no—
251C24—1061no—
252#41—1062no—
253#41—2121no—
254C2×Q8—1061no—
255C2×Q8C22×D5311yescharacteristic
256C2×D4—531no—
257C2×D4—531no—
258C2×D4—1061no—
259C2×D4—1061no—
260C2×D4—1062no—
261C2×D4—1061no—
262C2×D4—1061no—
263C2×D4—1061no—
264C2×D4—2121no—
265C2×D4C22×D5311yescharacteristic
266C2×D4C22×D5311yescharacteristic
267C22×C4—21no—
268C22×C4—531no—
269C22×C4—531no—
270C22×C4—1061no—
271C22×C4—1061no—
272SD16—41no—
273D8—41no—
274D8—1062no—
275M16—2121no—
276C2×C8—531no—
277C2×C8—531no—
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⋊C4C22×D5311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
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—
300C42—1061no—
301C42—1061no—
302C42C22×D5311yescharacteristic
303C23—21no—
304C23—21no—
305C23—531no—
306C23—531no—
307C23—1061no—
308C23—4241no—
309C23—2122no—
310C23—2122no—
311Q8—21no—
312Q8—21no—
313Q8—4241no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—21no—
318D4—532no—
319D4—1061no—
320D4—4241no—
321D4—4241no—
322D4—4241no—
323D4—2121no—
324D4—2122no—
325D4—2122no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—42no—
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—1061no—
338C2×C4—4241no—
339C2×C4—2122no—
340C2×C4D4×D5311yescharacteristic
341C2×C4D4×D5311yescharacteristic
342C2×C4C23×D5311yescharacteristic
343C8—21no—
344C8—21no—
345C8—1061no—
346C8—1061no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—1061no—
354C22—1061no—
355C22—1062no—
356C22—1061no—
357C22—1061no—
358C22—1061no—
359C22—1061no—
360C22—4244no—
361C22—2121no—
362C22—2121no—
363C22C2×D4×D5311yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—1061no—
369C4—1061no—
370C4—2121no—
371C4—2121no—
372C4—2121no—
373C4#41×D5311yescharacteristic
374C4#1313326211yescharacteristic
375C2—81no—
376C2—81no—
377C2—2121no—
378C2—2121no—
379C2—2122no—
380C2#1681127011yescharacteristic
381C2#1681141911yescharacteristic
382C2#1681142511yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#463C2
socleC22×C53C22×D4
hypercenter#463D53
hypocenterC53#3319
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. #16811338C2semidirect
2. #16811266C2semidirect
3. #16810982C2semidirect
4. #16810963C2semidirect
5. #16810945C2semidirect
6. #16810904C2semidirect
7. #16810866C2semidirect
8. #16810856C2semidirect
9. #16810767C2semidirect
10. #16810755C2semidirect
11. #16810733C2semidirect
12. #16810693C2semidirect
13. #16810626C2semidirect
14. #16810618C2semidirect
15. C53×#463C2semidirect
20. C2×C8×D53C22semidirect
21. #13133217C22semidirect
30. #13133188C22semidirect
36. #13133175C22semidirect
38. #13133151C22semidirect
39. #13133151C22semidirect
40. #13133148C22semidirect
41. #13133141C22semidirect
42. #13133141C22semidirect
47. #13133138C22semidirect
50. #13133129C22semidirect
51. #13133128C22semidirect
52. C2×SD16×C53C22semidirect
55. C4×D4×C53C22semidirect
56. C4⋊C8×C53C22semidirect
57. #103×C53C22semidirect
200. #463D53semidirect
207. C53#3319semidirect
210. C2×SD16C2×D53semidirect
229. C4×D4C2×D53semidirect
237. C4⋊C8C2×D53semidirect
240. #103C2×D53semidirect