Finite Group Databasebeta

← #20285491

Subgroups — #20285491

Order 6784.

420 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811133C211yesmaximal, characteristic
2C22⋊Q8×D53C211yesmaximal, characteristic, complemented
3#16810967C211yesmaximal, characteristic, complemented
4#16810963C211yesmaximal, characteristic, complemented
5#16810955C211yesmaximal, characteristic, complemented
6#16810935C211yesmaximal, characteristic, complemented
7#16810864C211yesmaximal, characteristic, complemented
8#16810862C211yesmaximal, characteristic, complemented
9#16810850C211yesmaximal, characteristic, complemented
10#16810786C211yesmaximal, characteristic, complemented
11#16810782C211yesmaximal, characteristic, complemented
12#16810780C211yesmaximal, characteristic, complemented
13#16810760C211yesmaximal, characteristic, complemented
14C2×#13133129C211yesmaximal, characteristic, complemented
15C53×#465C211yesmaximal, characteristic, complemented
16C2×#11719947C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic
18C22×C4×D53C2211yescharacteristic
19C2×Dic212C2211yescharacteristic, complemented
20C2×#11719925C2211yescharacteristic, complemented
21C2×#11719923C2211yescharacteristic, complemented
22C22×#88322C2211yescharacteristic
23#13133199—21no—
24#13133197C2211yescharacteristic
25#13133197C2211yescharacteristic
26#13133196C2211yescharacteristic
27#13133196C2211yescharacteristic
28#13133195—21no—
29#13133194—21no—
30#13133193C2211yescharacteristic
31#13133188C2211yescharacteristic
32#13133185—21no—
33D4×Dic53—21no—
34#13133177—21no—
35#13133175C2211yescharacteristic
36C4⋊C4×D53—21no—
37C4⋊C4×D53C2211yescharacteristic
38C22⋊C4×D53—21no—
39#13133151C2211yescharacteristic, complemented
40#13133150C2211yescharacteristic, complemented
41#13133148C2211yescharacteristic, complemented
42#13133142C2211yescharacteristic, complemented
43#13133142C2211yescharacteristic, complemented
44#13133141C2211yescharacteristic, complemented
45#13133140C2211yescharacteristic, complemented
46#13133140C2211yescharacteristic, complemented
47#13133138C2211yescharacteristic, complemented
48#13133129C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133129C2211yescharacteristic
52C2×SD16×C53C2211yescharacteristic, complemented
53C22×C8×C53C2211yescharacteristic
54C22⋊Q8×C53C2211yescharacteristic
55#121×C53C2211yescharacteristic
56#106×C53C2211yescharacteristic, complemented
57#103×C53C2211yescharacteristic, complemented
58#102×C53C2211yescharacteristic, complemented
59C23×D53D411yescharacteristic
60C2×Q8×C53C2311yescharacteristic
61C2×D4×C53—21no—
62C2×D4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C22×Dic53—21no—
67C22×Dic53D411yescharacteristic
68Q8×D53—41no—
69Q8×D53—41no—
70#11719947—41no—
71#11719947—41no—
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×C4×D53—41no—
75C2×C4×D53—41no—
76C2×C4×D53D411yescharacteristic
77C2×C4×D53D411yescharacteristic
78C2×C4×D53C2311yescharacteristic
79C2×C4×D53C2311yescharacteristic
80C2×Dic106—21no—
81C2×Dic106C2311yescharacteristic
82C2×Dic106C2311yescharacteristic
83SD16×C53—41no—
84C2×C8×C53—41no—
85C2×C8×C53C2311yescharacteristic
86C2×C8×C53C2311yescharacteristic
87C4⋊C4×C53—21no—
88C4⋊C4×C53C2311yescharacteristic
89C4⋊C4×C53C2311yescharacteristic
90C22⋊C4×C53—21no—
91C22⋊C4×C53—21no—
92C22⋊Dic53—21no—
93C22⋊Dic53—21no—
94C22⋊Dic53—21no—
95#11719925—41no—
96#11719923—41no—
97#11719921—21no—
98#11719921—21no—
99#11719921—21no—
100#11719921—21no—
101C4⋊Dic53—21no—
102C4⋊Dic53—21no—
103C4⋊Dic53C2311yescharacteristic
104C4⋊Dic53C2311yescharacteristic
105#11719919—21no—
106#11719919—21no—
107#11719919—21no—
108#11719919—21no—
109C4×Dic53—21no—
110C2×#88322—41no—
111C2×#88322C2311yescharacteristic
112C2×#88322C2311yescharacteristic
113Dic212—41no—
114C23×C53—21no—
115C23×C53C2×D411yescharacteristic
116C22×D53—22no—
117C22×D53—22no—
118C22×D53—41no—
119C22×D53C2×D411yescharacteristic
120C22×D53C2×D411yescharacteristic
121Q8×C53—21no—
122Q8×C53—21no—
123D4×C53—21no—
124D4×C53—21no—
125D4×C53—42no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53C2×D411yescharacteristic
132C2×C4×C53C2411yescharacteristic
133C22⋊D53—42no—
134C22⋊D53—42no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—41no—
140C2×Dic53—42no—
141C2×Dic53C2×D411yescharacteristic
142C2×Dic53C2×D411yescharacteristic
143C4×D53—21no—
144C4×D53—21no—
145C4×D53—21no—
146C4×D53—21no—
147C4×D53—41no—
148C4×D53—41no—
149C4×D53—42no—
150C4×D53—42no—
151Dic106—21no—
152Dic106—21no—
153Dic106—21no—
154Dic106—21no—
155Dic106—42no—
156C8×C53—21no—
157C8×C53—21no—
158#88322—21no—
159#88322—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53#13712yes—
166C22×C53C22×D411yescharacteristic
167C2×D53—21no—
168C2×D53—21no—
169C2×D53—21no—
170C2×D53—21no—
171C2×D53—21no—
172C2×D53—44no—
173C2×D53#13512yes—
174C4×C53—21no—
175C4×C53—41no—
176C4×C53—41no—
177C4×C53—41no—
178C4×C53H(5,𝔽2)11yescharacteristic
179C4×C53H(5,𝔽2)11yescharacteristic
180Dic53—21no—
181Dic53—21no—
182Dic53—41no—
183Dic53—41no—
184Dic53—41no—
185Dic53—41no—
186#3190—531nomaximal, Sylow
187C2×C53—22no—
188C2×C53—81no—
189C2×C53#53411yescharacteristic
190C2×C53C2×#13511yescharacteristic
191C2×C53C2×#13711yescharacteristic
192D53—22no—
193D53—41no—
194#525—531no—
195C2×C22⋊Q8—531no—
196#485—531no—
197#485—531no—
198#483—531no—
199#480—531no—
200#467—531no—
201#467—531no—
202#465—531no—
203#465D5311yescharacteristic, complemented
204#452—531no—
205#452—531no—
206#451—531no—
207#448—531no—
208C2×C22⋊C8—531no—
209C53#319011yescharacteristic, complemented, Sylow
210C2×#41—531no—
211C22×Q8—531no—
212C23×C4—531no—
213C2×Q16—531no—
214C2×Q16—531no—
215C2×SD16—531no—
216C2×SD16C2×D5311yescharacteristic, complemented
217C22×C8—531no—
218C22×C8C2×D5311yescharacteristic
219#123—1061no—
220C22⋊Q8—531no—
221C22⋊Q8—531no—
222C22⋊Q8—531no—
223C22⋊Q8—531no—
224C22⋊Q8—531no—
225C22⋊Q8—1061no—
226C22⋊Q8—1061no—
227C22⋊Q8C2×D5311yescharacteristic
228#121—531no—
229#121C2×D5311yescharacteristic
230#120—1061no—
231C4×D4—1061no—
232C4×D4—1061no—
233#117—531no—
234C2×C4⋊C4—531no—
235C2×C4⋊C4—1061no—
236C2×C22⋊C4—1061no—
237#107—531no—
238#107—531no—
239#106—531no—
240#106C2×D5311yescharacteristic, complemented
241#103—531no—
242#103—531no—
243#103—531no—
244#103—531no—
245#103—531no—
246#103C2×D5311yescharacteristic, complemented
247#102—531no—
248#102C2×D5311yescharacteristic, complemented
249C22⋊C8—531no—
250C22⋊C8—531no—
251C22⋊C8—531no—
252C22⋊C8—531no—
253C24—531no—
254#41—2121no—
255#41—2121no—
256C2×Q8—531no—
257C2×Q8—531no—
258C2×Q8—1061no—
259C2×Q8—2121no—
260C2×Q8—2121no—
261C2×Q8C22×D5311yescharacteristic
262C2×D4—21no—
263C2×D4—1061no—
264C2×D4—1061no—
265C2×D4C22×D5311yescharacteristic
266C22×C4—531no—
267C22×C4—531no—
268C22×C4—531no—
269C22×C4—531no—
270C22×C4—531no—
271C22×C4—1061no—
272C22×C4—1061no—
273C22×C4—1061no—
274C22×C4—2121no—
275C22×C4—2121no—
276C22×C4C22×D5311yescharacteristic
277Q16—2121no—
278Q16—2121no—
279SD16—41no—
280SD16—2121no—
281C2×C8—41no—
282C2×C8—531no—
283C2×C8—531no—
284C2×C8—2121no—
285C2×C8C22×D5311yescharacteristic
286C2×C8C22×D5311yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—531no—
289C4⋊C4—531no—
290C4⋊C4—1061no—
291C4⋊C4—1061no—
292C4⋊C4—1061no—
293C4⋊C4—1061no—
294C4⋊C4—1061no—
295C4⋊C4—1061no—
296C4⋊C4C22×D5311yescharacteristic
297C4⋊C4C22×D5311yescharacteristic
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—1061no—
301C22⋊C4—1061no—
302C22⋊C4—1061no—
303C22⋊C4—1061no—
304C22⋊C4—1061no—
305C22⋊C4—1061no—
306C22⋊C4—1061no—
307C42—1061no—
308C23—21no—
309C23—531no—
310C23—531no—
311C23—1062no—
312C23—1062no—
313C23—2121no—
314C23D4×D5311yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—1061no—
318Q8—1061no—
319Q8—1061no—
320Q8—1061no—
321Q8—2122no—
322D4—21no—
323D4—21no—
324D4—42no—
325D4—2122no—
326D4—2122no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—531no—
333C2×C4—531no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—1061no—
342C2×C4—2121no—
343C2×C4—2121no—
344C2×C4—2122no—
345C2×C4—2122no—
346C2×C4—2121no—
347C2×C4—2122no—
348C2×C4D4×D5311yescharacteristic
349C2×C4C23×D5311yescharacteristic
350C8—21no—
351C8—21no—
352C8—1061no—
353C8—1061no—
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—532no—
360C22—1061no—
361C22—1061no—
362C22—1061no—
363C22—1061no—
364C22—1061no—
365C22—2124no—
366C22#1313324612yes—
367C22C2×D4×D5311yescharacteristic
368C4—21no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—1061no—
373C4—1061no—
374C4—2121no—
375C4—2121no—
376C4—2121no—
377C4—2121no—
378C4#1313326211yescharacteristic
379C4#1313326311yescharacteristic
380C2—22no—
381C2—81no—
382C2—1062no—
383C2—2121no—
384C2#1681118111yescharacteristic
385C2#135×D5311yescharacteristic
386C2C2×#1313324611yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#465C2
socleC22×C53C22×D4
hypercenter#465D53
hypocenterC53#3190
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
2. C22⋊Q8×D53C2semidirect
3. #16810967C2semidirect
4. #16810963C2semidirect
5. #16810955C2semidirect
6. #16810935C2semidirect
7. #16810864C2semidirect
8. #16810862C2semidirect
9. #16810850C2semidirect
10. #16810786C2semidirect
11. #16810782C2semidirect
12. #16810780C2semidirect
13. #16810760C2semidirect
14. C2×#13133129C2semidirect
15. C53×#465C2semidirect
19. C2×Dic212C22semidirect
20. C2×#11719925C22semidirect
21. C2×#11719923C22semidirect
39. #13133151C22semidirect
40. #13133150C22semidirect
41. #13133148C22semidirect
42. #13133142C22semidirect
43. #13133142C22semidirect
44. #13133141C22semidirect
45. #13133140C22semidirect
46. #13133140C22semidirect
47. #13133138C22semidirect
52. C2×SD16×C53C22semidirect
56. #106×C53C22semidirect
57. #103×C53C22semidirect
58. #102×C53C22semidirect
203. #465D53semidirect
209. C53#3190semidirect
216. C2×SD16C2×D53semidirect
240. #106C2×D53semidirect
246. #103C2×D53semidirect
248. #102C2×D53semidirect