Finite Group Databasebeta

← #20287732

Subgroups — #20287732

Order 6784.

406 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811342C211yesmaximal, characteristic, complemented
2#16811293C211yesmaximal, characteristic, complemented
3#16811011C211yesmaximal, characteristic, complemented
4#16810994C211yesmaximal, characteristic, complemented
5#16810836C211yesmaximal, characteristic, complemented
6#16810835C211yesmaximal, characteristic, complemented
7#16810826C211yesmaximal, characteristic, complemented
8#16810821C211yesmaximal, characteristic, complemented
9#16810820C211yesmaximal, characteristic, complemented
10#16810816C211yesmaximal, characteristic, complemented
11#16810751C211yesmaximal, characteristic, complemented
12#16810725C211yesmaximal, characteristic, complemented
13#16810724C211yesmaximal, characteristic, complemented
14D8×Dic53C211yesmaximal, characteristic, complemented
15C53×#496C211yesmaximal, characteristic, complemented
16C2×#11719947—21no—
17C2×#11719913C2211yescharacteristic
18C2×#11719923C2211yescharacteristic
19C2×#11719923C2211yescharacteristic
20C2×#11719924C2211yescharacteristic
21C2×#11719922C2211yescharacteristic
22#13133215C2211yescharacteristic
23#13133213C2211yescharacteristic
24#13133213C2211yescharacteristic
25#13133210—21no—
26#13133210—21no—
27#13133204C2211yescharacteristic
28#13133194—21no—
29#13133189—21no—
30#13133189—21no—
31#13133188—21no—
32Q8×Dic53C2211yescharacteristic
33C4×Dic106C2211yescharacteristic
34C4×Dic106C2211yescharacteristic
35C4×D212C2211yescharacteristic
36D4×Dic53C2211yescharacteristic
37D4×Dic53C2211yescharacteristic
38D4×Dic53C2211yescharacteristic
39#13133173—21no—
40#13133151C2211yescharacteristic
41C4⋊#88322C2211yescharacteristic
42#13133146C2211yescharacteristic
43#13133146C2211yescharacteristic
44#13133142C2211yescharacteristic
45#13133141C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133138C2211yescharacteristic
49#13133138C2211yescharacteristic
50#13133127C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C2×SD16×C53C2211yescharacteristic
53C2×SD16×C53C2211yescharacteristic
54C2×D8×C53C2211yescharacteristic
55C2×D8×C53C2211yescharacteristic
56#127×C53C2211yescharacteristic
57#124×C53C2211yescharacteristic
58#97×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C2×D4×C53—21no—
61C2×D4×C53—21no—
62C2×D4×C53C2311yescharacteristic
63C2×D4×C53C2311yescharacteristic
64C2×D4×C53C2311yescharacteristic
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—21no—
72#11719947—44no—
73C2×D212C2311yescharacteristic
74C2×C4×D53—21no—
75C2×Dic106D411yescharacteristic
76C2×Dic106C2311yescharacteristic
77SD16×C53—22no—
78SD16×C53—41no—
79D8×C53—22no—
80D8×C53—41no—
81C2×C8×C53C2311yescharacteristic
82C2×C8×C53C2311yescharacteristic
83C22⋊C4×C53—21no—
84C22⋊C4×C53—21no—
85C42×C53C2311yescharacteristic
86C22⋊Dic53—21no—
87C22⋊Dic53—21no—
88C22⋊Dic53—21no—
89C22⋊Dic53—21no—
90C22⋊Dic53—21no—
91#11719924—41no—
92#11719923—41no—
93#11719923—41no—
94#11719922—41no—
95#11719921—21no—
96C4⋊Dic53—21no—
97C4⋊Dic53D411yescharacteristic
98C4⋊Dic53C2311yescharacteristic
99C4⋊Dic53C2311yescharacteristic
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103#11719919D411yescharacteristic
104#11719919D411yescharacteristic
105C4×Dic53—21no—
106C4×Dic53—21no—
107C4×Dic53C2311yescharacteristic
108C4×Dic53C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111#11719913—22no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C22×D53—21no—
116Q8×C53—21no—
117Q8×C53—21no—
118D4×C53—21no—
119D4×C53—21no—
120D4×C53—21no—
121D4×C53—21no—
122D4×C53—21no—
123D4×C53—21no—
124D4×C53—42no—
125D4×C53—81no—
126C2×C4×C53—21no—
127C2×C4×C53C2×D411yescharacteristic
128C2×C4×C53C2×D411yescharacteristic
129C2×C4×C53C2411yescharacteristic
130C22⋊D53—42no—
131C22⋊D53—42no—
132C22⋊D53—81no—
133C22⋊D53—81no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—42no—
138C2×Dic53—42no—
139C2×Dic53—42no—
140C2×Dic53C2×D411yescharacteristic
141C2×Dic53C2×D411yescharacteristic
142C2×Dic53C2×D411yescharacteristic
143C2×Dic53C2×D411yescharacteristic
144D212—21no—
145D212—21no—
146C4×D53—42no—
147Dic106—22no—
148Dic106—21no—
149Dic106—21no—
150C8×C53—21no—
151C8×C53#4112yescomplemented
152#88322—21no—
153#88322—21no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53C22×D411yescharacteristic
164C2×D53—41no—
165C2×D53—41no—
166C2×D53—41no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—41no—
170C4×C53C2×#4111yescharacteristic
171C4×C53H(5,𝔽2)11yescharacteristic
172Dic53—21no—
173Dic53—21no—
174Dic53—21no—
175Dic53—41no—
176Dic53—41no—
177Dic53—41no—
178Dic53AGL(1,ℤ/8)12yes—
179#3397—531nomaximal, Sylow
180C2×C53—81no—
181C2×C53—81no—
182C2×C53—81no—
183C2×C53#55011yescharacteristic
184C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C53#57711yescharacteristic
186D53—81no—
187#556—531no—
188#550—531no—
189#496—531no—
190#496D5311yescharacteristic, complemented
191#492—531no—
192#463—531no—
193#463—531no—
194#461—531no—
195#460—531no—
196#460—531no—
197#459—531no—
198#446—531no—
199#440—531no—
200#440—531no—
201C4×D8—531no—
202C53#339711yescharacteristic, complemented, Sylow
203C2×#41—1061no—
204C2×SD16—531no—
205C2×SD16—531no—
206C2×SD16—531no—
207C2×SD16—531no—
208C2×SD16C2×D5311yescharacteristic
209C2×SD16C2×D5311yescharacteristic
210C2×D8—531no—
211C2×D8C2×D5311yescharacteristic
212C2×D8C2×D5311yescharacteristic
213C4⋊Q8—531no—
214#127—531no—
215#127—531no—
216#127C2×D5311yescharacteristic
217#126—1061no—
218#126—1061no—
219#124—531no—
220#124C2×D5311yescharacteristic
221C22⋊Q8—1061no—
222#121—1061no—
223#121—1061no—
224#121—1061no—
225C4×Q8—531no—
226C4×Q8—531no—
227C4×Q8—531no—
228C4×D4—531no—
229C4×D4—531no—
230C4×D4—531no—
231C4×D4—531no—
232#117—1061no—
233#107—531no—
234C4⋊C8—531no—
235C4⋊C8—531no—
236C4⋊C8—531no—
237#103—531no—
238#103—531no—
239#102—531no—
240#102—531no—
241#102—531no—
242#102—531no—
243#97—531no—
244#97C2×D5311yescharacteristic
245C4×C8—531no—
246#41—2124no—
247C2×Q8—531no—
248C2×Q8—531no—
249C2×Q8C22×D5311yescharacteristic
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—531no—
253C2×D4—1061no—
254C2×D4—1061no—
255C2×D4—1061no—
256C2×D4—1061no—
257C2×D4C22×D5311yescharacteristic
258C2×D4C22×D5311yescharacteristic
259C2×D4C22×D5311yescharacteristic
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—1061no—
264SD16—22no—
265SD16—41no—
266SD16—1062no—
267SD16—2121no—
268SD16—2121no—
269SD16—2121no—
270D8—22no—
271D8—41no—
272D8—2121no—
273C2×C8—531no—
274C2×C8—531no—
275C2×C8C22×D5311yescharacteristic
276C2×C8C22×D5311yescharacteristic
277C4⋊C4—531no—
278C4⋊C4—531no—
279C4⋊C4—531no—
280C4⋊C4—531no—
281C4⋊C4—531no—
282C4⋊C4—1061no—
283C4⋊C4—1061no—
284C4⋊C4—1061no—
285C4⋊C4—1061no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C22⋊C4—1061no—
294C42—531no—
295C42—531no—
296C42—1061no—
297C42—1061no—
298C42C22×D5311yescharacteristic
299C23—21no—
300C23—21no—
301C23—21no—
302C23—1061no—
303Q8—21no—
304Q8—21no—
305Q8—1062no—
306Q8—1061no—
307Q8—1061no—
308D4—21no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—42no—
315D4—81no—
316D4—1061no—
317D4—1061no—
318D4—4241no—
319D4—4241no—
320D4—2122no—
321D4—2122no—
322C2×C4—21no—
323C2×C4—531no—
324C2×C4—531no—
325C2×C4—531no—
326C2×C4—531no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—2122no—
331C2×C4—2122no—
332C2×C4—2122no—
333C2×C4—2122no—
334C2×C4D4×D5311yescharacteristic
335C2×C4D4×D5311yescharacteristic
336C2×C4C23×D5311yescharacteristic
337C8—21no—
338C8—1061no—
339C8—1061no—
340C8#1171994712yescomplemented
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—2121no—
351C22—2121no—
352C22—2121no—
353C22C2×D4×D5311yescharacteristic
354C4—21no—
355C4—21no—
356C4—41no—
357C4—532no—
358C4—1061no—
359C4—1061no—
360C4—1061no—
361C4—2121no—
362C4—2121no—
363C4—2121no—
364C4C2×#1171994711yescharacteristic
365C4#1313326211yescharacteristic
366C2—81no—
367C2—81no—
368C2—81no—
369C2—4241no—
370C2#1681129311yescharacteristic
371C2AGL(1,ℤ/8)×D5311yescharacteristic
372C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#496C2
socleC22×C53C22×D4
hypercenter#496D53
hypocenterC53#3397
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #16811342C2semidirect
2. #16811293C2semidirect
3. #16811011C2semidirect
4. #16810994C2semidirect
5. #16810836C2semidirect
6. #16810835C2semidirect
7. #16810826C2semidirect
8. #16810821C2semidirect
9. #16810820C2semidirect
10. #16810816C2semidirect
11. #16810751C2semidirect
12. #16810725C2semidirect
13. #16810724C2semidirect
14. D8×Dic53C2semidirect
15. C53×#496C2semidirect
151. C8×C53#41semidirect
190. #496D53semidirect
202. C53#3397semidirect
340. C8#11719947semidirect