Finite Group Databasebeta

← #20285675

Subgroups — #20285675

Order 6784.

410 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811152C211yesmaximal, characteristic, complemented
2#124×D53C211yesmaximal, characteristic, complemented
3#16811005C211yesmaximal, characteristic, complemented
4#16810978C211yesmaximal, characteristic, complemented
5#16810977C211yesmaximal, characteristic, complemented
6#16810847C211yesmaximal, characteristic, complemented
7#16810843C211yesmaximal, characteristic, complemented
8#16810840C211yesmaximal, characteristic, complemented
9#16810836C211yesmaximal, characteristic, complemented
10#16810777C211yesmaximal, characteristic, complemented
11#16810772C211yesmaximal, characteristic, complemented
12#16810760C211yesmaximal, characteristic, complemented
13#16810755C211yesmaximal, characteristic
14#16810674C211yesmaximal, characteristic, complemented
15C53×#495C211yesmaximal, characteristic, complemented
16C2×Q8×D53C2211yescharacteristic
17C2×D4×D53C2211yescharacteristic
18C2×#11719925C2211yescharacteristic, complemented
19C2×#11719923C2211yescharacteristic, complemented
20C2×#11719924C2211yescharacteristic
21C2×#11719922C2211yescharacteristic
22#13133210—21no—
23#13133209C2211yescharacteristic, complemented
24#13133206C2211yescharacteristic
25#13133205C2211yescharacteristic
26#13133204C2211yescharacteristic
27#13133203—21no—
28#13133203—21no—
29#13133199—21no—
30#13133193C2211yescharacteristic
31#13133188C2211yescharacteristic
32Q8×Dic53C2211yescharacteristic, complemented
33D4×Dic53C2211yescharacteristic, complemented
34#13133177—21no—
35#13133173—21no—
36C22⋊C4×D53—21no—
37C22⋊C4×D53—21no—
38C42×D53C2211yescharacteristic
39#13133146C2211yescharacteristic, complemented
40#13133146C2211yescharacteristic, complemented
41#13133142C2211yescharacteristic, complemented
42#13133142C2211yescharacteristic, complemented
43#13133141C2211yescharacteristic, complemented
44#13133141C2211yescharacteristic, complemented
45#13133139C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133138C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133129C2211yescharacteristic
51C4×#88322C2211yescharacteristic, complemented
52C2×Q16×C53C2211yescharacteristic, complemented
53C2×SD16×C53C2211yescharacteristic
54C2×SD16×C53C2211yescharacteristic, complemented
55C2×D8×C53C2211yescharacteristic
56#124×C53C2211yescharacteristic
57#124×C53C2211yescharacteristic, complemented
58C4×C8×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C2×Q8×C53C2311yescharacteristic
61C2×Q8×C53C2311yescharacteristic
62C2×D4×C53C2311yescharacteristic
63C2×D4×C53C2311yescharacteristic
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C22×Dic53—21no—
67Q8×D53—41no—
68Q8×D53—41no—
69D4×D53—41no—
70D4×D53—41no—
71C2×D212C2311yescharacteristic
72C2×C4×D53—21no—
73C2×C4×D53D411yescharacteristic
74C2×C4×D53D411yescharacteristic
75C2×C4×D53C2311yescharacteristic
76C2×Dic106—21no—
77C2×Dic106C2311yescharacteristic
78Q16×C53—41no—
79SD16×C53—41no—
80SD16×C53—41no—
81D8×C53—41no—
82C2×C8×C53C2311yescharacteristic
83C2×C8×C53C2311yescharacteristic
84C22⋊C4×C53—21no—
85C22⋊C4×C53—21no—
86C22⋊C4×C53—21no—
87C22⋊C4×C53—21no—
88C42×C53C2311yescharacteristic
89C22⋊Dic53—21no—
90C22⋊Dic53—21no—
91C22⋊Dic53—21no—
92C22⋊Dic53—21no—
93#11719925—41no—
94#11719924—41no—
95#11719923—41no—
96#11719922—41no—
97#11719921—21no—
98#11719921—21no—
99#11719921—21no—
100#11719921—21no—
101#11719921—21no—
102C4⋊Dic53—21no—
103C4⋊Dic53C2311yescharacteristic
104C4⋊Dic53C2311yescharacteristic
105#11719919—21no—
106#11719919—21no—
107#11719919—21no—
108C4×Dic53—21no—
109C4×Dic53D411yescharacteristic
110C4×Dic53D411yescharacteristic
111C4×Dic53C2311yescharacteristic
112C2×#88322C2311yescharacteristic
113C2×#88322C2311yescharacteristic
114C23×C53—21no—
115C23×C53—21no—
116C22×D53—21no—
117C22×D53—42no—
118C22×D53—42no—
119C22×D53—42no—
120C22×D53C2×D411yescharacteristic
121Q8×C53—21no—
122Q8×C53—21no—
123Q8×C53—21no—
124Q8×C53—21no—
125D4×C53—21no—
126D4×C53—21no—
127D4×C53—21no—
128D4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53C2×D411yescharacteristic
132C2×C4×C53C2×D411yescharacteristic
133C2×C4×C53C2411yescharacteristic
134C22⋊D53—42no—
135C22⋊D53—42no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—42no—
140C2×Dic53C2×D411yescharacteristic
141C2×Dic53C2×D411yescharacteristic
142C2×Dic53C2×D411yescharacteristic
143D212—21no—
144D212—21no—
145C4×D53—22no—
146C4×D53—21no—
147C4×D53—21no—
148C4×D53—41no—
149C4×D53—42no—
150Dic106—21no—
151Dic106—21no—
152Dic106—42no—
153C8×C53—21no—
154C8×C53—21no—
155#88322—21no—
156#88322—21no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53C22×D411yescharacteristic
164C2×D53—21no—
165C2×D53—21no—
166C2×D53—41no—
167C2×D53—41no—
168C2×D53—41no—
169C2×D53—84no—
170C2×D53#13512yes—
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—41no—
174C4×C53—41no—
175C4×C53H(5,𝔽2)11yescharacteristic
176C4×C53H(5,𝔽2)11yescharacteristic
177Dic53—21no—
178Dic53—21no—
179Dic53—21no—
180Dic53—41no—
181Dic53—41no—
182Dic53—41no—
183#3211—531nomaximal, Sylow
184C2×C53—81no—
185C2×C53—81no—
186C2×C53#53511yescharacteristic
187C2×C53C2×#13511yescharacteristic
188C2×C53#57511yescharacteristic
189D53—22no—
190D53—81no—
191#529—531no—
192C2×#124—531no—
193#495—531no—
194#495D5311yescharacteristic, complemented
195#488—531no—
196#488—531no—
197#464—531no—
198#464—531no—
199#463—531no—
200#463—531no—
201#451—531no—
202#450—531no—
203#448—531no—
204#447—531no—
205#431—531no—
206C53#321111yescharacteristic, complemented, Sylow
207C22×Q8—531no—
208C22×D4—531no—
209C2×Q16—531no—
210C2×Q16C2×D5311yescharacteristic, complemented
211C2×SD16—531no—
212C2×SD16—531no—
213C2×SD16C2×D5311yescharacteristic
214C2×SD16C2×D5311yescharacteristic, complemented
215C2×D8—531no—
216C2×D8C2×D5311yescharacteristic
217#126—1061no—
218#125—531no—
219#124—531no—
220#124—531no—
221#124—531no—
222#124—1061no—
223#124—1061no—
224#124C2×D5311yescharacteristic
225#124C2×D5311yescharacteristic, complemented
226#123—1061no—
227C22⋊Q8—531no—
228#121—531no—
229C4×Q8—531no—
230C4×D4—531no—
231C4×D4—1061no—
232#117—1061no—
233C2×C22⋊C4—1061no—
234C2×C22⋊C4—1061no—
235C2×C42—531no—
236C4⋊C8—531no—
237C4⋊C8—531no—
238#103—531no—
239#103—531no—
240#103—531no—
241#103—531no—
242#102—531no—
243#102—531no—
244#102—531no—
245#102—531no—
246C22⋊C8—531no—
247C22⋊C8—531no—
248C4×C8—531no—
249C4×C8C2×D5311yescharacteristic, complemented
250C24—1061no—
251C2×Q8—531no—
252C2×Q8—1061no—
253C2×Q8—2121no—
254C2×Q8—2121no—
255C2×Q8C22×D5311yescharacteristic
256C2×Q8C22×D5311yescharacteristic
257C2×D4—531no—
258C2×D4—1061no—
259C2×D4—1061no—
260C2×D4—2121no—
261C2×D4—2121no—
262C2×D4C22×D5311yescharacteristic
263C2×D4C22×D5311yescharacteristic
264C22×C4—531no—
265C22×C4—531no—
266C22×C4—531no—
267C22×C4—1061no—
268C22×C4—1061no—
269Q16—41no—
270Q16—2121no—
271SD16—41no—
272SD16—41no—
273SD16—2121no—
274SD16—2121no—
275D8—41no—
276D8—2121no—
277C2×C8—531no—
278C2×C8—531no—
279C2×C8C22×D5311yescharacteristic
280C2×C8C22×D5311yescharacteristic
281C4⋊C4—531no—
282C4⋊C4—531no—
283C4⋊C4—1061no—
284C4⋊C4—1061no—
285C4⋊C4—1061no—
286C4⋊C4—1061no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—21no—
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—531no—
301C42—531no—
302C42—531no—
303C42—1061no—
304C42C22×D5311yescharacteristic
305C23—21no—
306C23—21no—
307C23—531no—
308C23—1061no—
309C23—2122no—
310C23—2122no—
311C23—2122no—
312Q8—21no—
313Q8—21no—
314Q8—21no—
315Q8—21no—
316Q8—1061no—
317Q8—1061no—
318Q8—2122no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—1061no—
324D4—1061no—
325D4—2122no—
326D4—2122no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—531no—
330C2×C4—531no—
331C2×C4—531no—
332C2×C4—1062no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—2121no—
339C2×C4—2122no—
340C2×C4—2122no—
341C2×C4D4×D5311yescharacteristic
342C2×C4D4×D5311yescharacteristic
343C2×C4C23×D5311yescharacteristic
344C8—21no—
345C8—21no—
346C8—1061no—
347C8—1061no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—532no—
355C22—1061no—
356C22—1061no—
357C22—4244no—
358C22—2121no—
359C22—2121no—
360C22—2121no—
361C22C2×D4×D5311yescharacteristic
362C4—21no—
363C4—21no—
364C4—41no—
365C4—41no—
366C4—1061no—
367C4—1061no—
368C4—1061no—
369C4—2121no—
370C4—2121no—
371C4—2121no—
372C4#1313326211yescharacteristic
373C4#1313326211yescharacteristic
374C2—81no—
375C2—81no—
376C2—1062no—
377C2—4241no—
378C2#1681118611yescharacteristic
379C2#135×D5311yescharacteristic
380C2#1681142011yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#495C2
socleC22×C53C22×D4
hypercenter#495D53
hypocenterC53#3211
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #16811152C2semidirect
2. #124×D53C2semidirect
3. #16811005C2semidirect
4. #16810978C2semidirect
5. #16810977C2semidirect
6. #16810847C2semidirect
7. #16810843C2semidirect
8. #16810840C2semidirect
9. #16810836C2semidirect
10. #16810777C2semidirect
11. #16810772C2semidirect
12. #16810760C2semidirect
14. #16810674C2semidirect
15. C53×#495C2semidirect
18. C2×#11719925C22semidirect
19. C2×#11719923C22semidirect
23. #13133209C22semidirect
32. Q8×Dic53C22semidirect
33. D4×Dic53C22semidirect
39. #13133146C22semidirect
40. #13133146C22semidirect
41. #13133142C22semidirect
42. #13133142C22semidirect
43. #13133141C22semidirect
44. #13133141C22semidirect
51. C4×#88322C22semidirect
52. C2×Q16×C53C22semidirect
54. C2×SD16×C53C22semidirect
57. #124×C53C22semidirect
58. C4×C8×C53C22semidirect
194. #495D53semidirect
206. C53#3211semidirect
210. C2×Q16C2×D53semidirect
214. C2×SD16C2×D53semidirect
225. #124C2×D53semidirect
249. C4×C8C2×D53semidirect