Finite Group Databasebeta

← #20286872

Subgroups — #20286872

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#16811336C211yesmaximal, characteristic, complemented
2#16811267C211yesmaximal, characteristic, complemented
3#16810981C211yesmaximal, characteristic, complemented
4#16810964C211yesmaximal, characteristic, complemented
5#16810945C211yesmaximal, characteristic, complemented
6#16810901C211yesmaximal, characteristic, complemented
7#16810869C211yesmaximal, characteristic, complemented
8#16810858C211yesmaximal, characteristic, complemented
9#16810837C211yesmaximal, characteristic, complemented
10#16810769C211yesmaximal, characteristic, complemented
11#16810757C211yesmaximal, characteristic, complemented
12#16810732C211yesmaximal, characteristic, complemented
13#16810692C211yesmaximal, characteristic, complemented
14#16810616C211yesmaximal, characteristic, complemented
15C53×#418C211yesmaximal, characteristic, complemented
16C2×#11719947C2211yescharacteristic
17C2×#11719913C2211yescharacteristic, complemented
18C2×#11719922C2211yescharacteristic
19C2×#11719922C2211yescharacteristic
20C2×C8×D53C2211yescharacteristic, complemented
21#13133216C2211yescharacteristic, complemented
22#13133205—21no—
23#13133204C2211yescharacteristic
24#13133203—21no—
25#13133198—21no—
26#13133196C2211yescharacteristic
27#13133192C2211yescharacteristic, complemented
28#13133191C2211yescharacteristic
29#13133190—21no—
30#13133189—21no—
31#13133188C2211yescharacteristic
32#13133186—21no—
33D4×Dic53C2211yescharacteristic
34#13133179C2211yescharacteristic, complemented
35C4×C22⋊D53C2211yescharacteristic
36#13133175C2211yescharacteristic, complemented
37#13133174—21no—
38C2×C22⋊Dic53—21no—
39#13133151C2211yescharacteristic, complemented
40#13133150C2211yescharacteristic, complemented
41#13133148C2211yescharacteristic, complemented
42#13133146C2211yescharacteristic, complemented
43#13133142C2211yescharacteristic, complemented
44#13133140C2211yescharacteristic, complemented
45#13133140C2211yescharacteristic, complemented
46#13133139C2211yescharacteristic, complemented
47#13133138C2211yescharacteristic
48#13133138C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133129C2211yescharacteristic, complemented
51#13133127C2211yescharacteristic, complemented
52C22×D4×C53C2211yescharacteristic
53C2×M16×C53C2211yescharacteristic
54#117×C53C2211yescharacteristic
55#102×C53C2211yescharacteristic
56#102×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59C24×C53—21no—
60C2×D4×C53—22no—
61C2×D4×C53—21no—
62C2×D4×C53—41no—
63C2×D4×C53C2311yescharacteristic
64C2×D4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53D411yescharacteristic
70C22×Dic53—21no—
71C22×Dic53—21no—
72#11719947—22no—
73#11719947—41no—
74C2×D212—21no—
75C2×D212C2311yescharacteristic
76C2×C4×D53—21no—
77C2×C4×D53C2311yescharacteristic
78C2×Dic106—21no—
79C2×Dic106C2311yescharacteristic
80M16×C53—41no—
81C2×C8×C53C2311yescharacteristic
82C2×C8×C53C2311yescharacteristic
83C4⋊C4×C53C2311yescharacteristic
84C4⋊C4×C53C2311yescharacteristic
85C22⋊C4×C53—21no—
86C42×C53—21no—
87C22⋊Dic53—21no—
88C22⋊Dic53—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⋊Dic53C2311yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102#11719919—21no—
103#11719919—21no—
104#11719919D411yescharacteristic
105C4×Dic53—21no—
106C4×Dic53C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109#11719913—41no—
110C8×D53—41no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—42no—
114C23×C53—42no—
115C23×C53—81no—
116C23×C53C2×D411yescharacteristic
117C22×D53—21no—
118C22×D53C2×D411yescharacteristic
119D4×C53—21no—
120D4×C53—21no—
121D4×C53—21no—
122D4×C53—42no—
123D4×C53#4112yes—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53C2×D411yescharacteristic
129C2×C4×C53C2411yescharacteristic
130C22⋊D53—41no—
131C22⋊D53—42no—
132C22⋊D53—81no—
133C22⋊D53—81no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—42no—
138C2×Dic53—81no—
139C2×Dic53C2×D411yescharacteristic
140C2×Dic53C2×D411yescharacteristic
141C2×Dic53C2×D411yescharacteristic
142D212—21no—
143D212—21no—
144D212—81no—
145C4×D53—21no—
146C4×D53—21no—
147C4×D53—42no—
148Dic106—21no—
149Dic106—21no—
150Dic106—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×C53—84no—
165C22×C53C22×D411yescharacteristic
166C2×D53—21no—
167C2×D53—21no—
168C2×D53—21no—
169C2×D53—41no—
170C2×D53—41no—
171C2×D53—41no—
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#3319—531nomaximal, Sylow
184C2×C53—41no—
185C2×C53—42no—
186C2×C53—81no—
187C2×C53#54611yescharacteristic
188C2×C53#57511yescharacteristic
189C2×C53#57611yescharacteristic
190D53—41no—
191D53—81no—
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#463—531no—
201#449—531no—
202#447—531no—
203#442—531no—
204#435—531no—
205#418D5311yescharacteristic, complemented
206#416—531no—
207C53#331911yescharacteristic, complemented, Sylow
208C2×#41—531no—
209C22×D4C2×D5311yescharacteristic
210C2×SD16—531no—
211C2×D8—531no—
212C2×D8—531no—
213C2×M16C2×D5311yescharacteristic
214C22×C8—531no—
215C4⋊Q8—531no—
216#124—531no—
217#124—1061no—
218#124—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—531no—
230#117—531no—
231#117—1061no—
232#117C2×D5311yescharacteristic
233C2×C22⋊C4—1061no—
234#107—531no—
235#107—531no—
236#106—531no—
237C4⋊C8—531no—
238#103—531no—
239#103—531no—
240#103—531no—
241#102—531no—
242#102—531no—
243#102—531no—
244#102C2×D5311yescharacteristic
245#102C2×D5311yescharacteristic
246#102C2×D5311yescharacteristic
247#102C2×D5311yescharacteristic
248C22⋊C8—531no—
249C22⋊C8—531no—
250#97—531no—
251C24—21no—
252#41—1062no—
253#41—2121no—
254C2×Q8—531no—
255C2×Q8—1061no—
256C2×D4—22no—
257C2×D4—21no—
258C2×D4—41no—
259C2×D4—531no—
260C2×D4—531no—
261C2×D4—1061no—
262C2×D4—1061no—
263C2×D4—1061no—
264C2×D4—1061no—
265C2×D4C22×D5311yescharacteristic
266C2×D4C22×D5311yescharacteristic
267C22×C4—531no—
268C22×C4—1061no—
269C22×C4—1061no—
270C22×C4—1061no—
271C22×C4C22×D5311yescharacteristic
272SD16—2121no—
273D8—1062no—
274D8—2121no—
275M16—41no—
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—1061no—
285C4⋊C4—1061no—
286C4⋊C4C22×D5311yescharacteristic
287C4⋊C4C22×D5311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—531no—
290C22⋊C4—531no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C22⋊C4—1061no—
294C22⋊C4—1061no—
295C22⋊C4—1061no—
296C22⋊C4—1061no—
297C22⋊C4—1061no—
298C22⋊C4—1061no—
299C22⋊C4—1061no—
300C42—21no—
301C42—531no—
302C42—1061no—
303C23—21no—
304C23—21no—
305C23—42no—
306C23—42no—
307C23—81no—
308C23—531no—
309C23—1061no—
310C23D4×D5311yescharacteristic
311Q8—1061no—
312Q8—1061no—
313Q8—4241no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—42no—
318D4—1061no—
319D4—1061no—
320D4—4241no—
321D4—4241no—
322D4—4241no—
323D4—2121no—
324D4—2122no—
325D4#1171994512yes—
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—1061no—
338C2×C4—4241no—
339C2×C4—2122no—
340C2×C4—2122no—
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—84no—
357C22—1061no—
358C22—1061no—
359C22—1061no—
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×#1171994511yescharacteristic
374C4#1313326211yescharacteristic
375C2—41no—
376C2—42no—
377C2—81no—
378C2—4241no—
379C2—2121no—
380C2#1681126311yescharacteristic
381C2#1681141911yescharacteristic
382C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#418C2
socleC22×C53C22×D4
hypercenter#418D53
hypocenterC53#3319
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #16811336C2semidirect
2. #16811267C2semidirect
3. #16810981C2semidirect
4. #16810964C2semidirect
5. #16810945C2semidirect
6. #16810901C2semidirect
7. #16810869C2semidirect
8. #16810858C2semidirect
9. #16810837C2semidirect
10. #16810769C2semidirect
11. #16810757C2semidirect
12. #16810732C2semidirect
13. #16810692C2semidirect
14. #16810616C2semidirect
15. C53×#418C2semidirect
17. C2×#11719913C22semidirect
20. C2×C8×D53C22semidirect
21. #13133216C22semidirect
27. #13133192C22semidirect
34. #13133179C22semidirect
36. #13133175C22semidirect
39. #13133151C22semidirect
40. #13133150C22semidirect
41. #13133148C22semidirect
42. #13133146C22semidirect
43. #13133142C22semidirect
44. #13133140C22semidirect
45. #13133140C22semidirect
46. #13133139C22semidirect
50. #13133129C22semidirect
51. #13133127C22semidirect
205. #418D53semidirect
207. C53#3319semidirect