Finite Group Databasebeta

← #20287788

Subgroups — #20287788

Order 6784.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811356C211yesmaximal, characteristic, complemented
2#16811347C211yesmaximal, characteristic, complemented
3#16811015C211yesmaximal, characteristic, complemented
4#16811009C211yesmaximal, characteristic
5#16811004C211yesmaximal, characteristic, complemented
6#16810848C211yesmaximal, characteristic, complemented
7#16810843C211yesmaximal, characteristic, complemented
8#16810841C211yesmaximal, characteristic, complemented
9#16810839C211yesmaximal, characteristic, complemented
10#16810821C211yesmaximal, characteristic, complemented
11#16810752C211yesmaximal, characteristic, complemented
12#16810733C211yesmaximal, characteristic, complemented
13#16810727C211yesmaximal, characteristic, complemented
14#16810708C211yesmaximal, characteristic, complemented
15C53×#461C211yesmaximal, characteristic, complemented
16C2×#11719925C2211yescharacteristic
17C2×#11719925C2211yescharacteristic
18C2×#11719913C2211yescharacteristic
19C2×#11719924C2211yescharacteristic
20C2×#11719924C2211yescharacteristic
21C2×D424C2211yescharacteristic
22C2×#11719922C2211yescharacteristic
23#13133213C2211yescharacteristic
24#13133212—21no—
25#13133212—21no—
26#13133206C2211yescharacteristic
27#13133206C2211yescharacteristic
28#13133205C2211yescharacteristic
29#13133204C2211yescharacteristic
30#13133203—21no—
31#13133200—21no—
32#13133199—21no—
33#13133197—21no—
34Q8×Dic53C2211yescharacteristic
35Q8×Dic53C2211yescharacteristic
36C4×Dic106C2211yescharacteristic
37#13133182C2211yescharacteristic
38C4×D212C2211yescharacteristic
39#13133179C2211yescharacteristic
40C4×C22⋊D53—21no—
41#13133175—21no—
42#13133148C2211yescharacteristic
43#13133146C2211yescharacteristic
44#13133146C2211yescharacteristic
45#13133142C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133128C2211yescharacteristic
50#13133127C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C2×SD16×C53C2211yescharacteristic
53C2×SD16×C53C2211yescharacteristic
54C4⋊Q8×C53C2211yescharacteristic
55C4×D4×C53C2211yescharacteristic
56C4⋊C8×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58#103×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C2×Q8×C53C2311yescharacteristic
61C2×D4×C53C2311yescharacteristic
62C22×C4×C53—21no—
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C2×D212C2311yescharacteristic
66C2×D212C2311yescharacteristic
67C2×C4×D53—21no—
68C2×C4×D53—21no—
69C2×Dic106D411yescharacteristic
70C2×Dic106C2311yescharacteristic
71SD16×C53—41no—
72SD16×C53—41no—
73C2×C8×C53C2311yescharacteristic
74C2×C8×C53C2311yescharacteristic
75C4⋊C4×C53—21no—
76C4⋊C4×C53—21no—
77C4⋊C4×C53C2311yescharacteristic
78C22⋊C4×C53—21no—
79C42×C53C2311yescharacteristic
80C22⋊Dic53—21no—
81C22⋊Dic53—21no—
82#11719925—22no—
83#11719925—41no—
84#11719924—22no—
85#11719924—41no—
86#11719922—22no—
87#11719921—21no—
88#11719921—21no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95C4⋊Dic53—21no—
96C4⋊Dic53—21no—
97C4⋊Dic53D411yescharacteristic
98C4⋊Dic53C2311yescharacteristic
99#11719919—21no—
100#11719919D411yescharacteristic
101#11719919D411yescharacteristic
102C4×Dic53—21no—
103C4×Dic53—21no—
104C4×Dic53—21no—
105C4×Dic53C2311yescharacteristic
106C4×Dic53C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109D424—41no—
110#11719913—41no—
111C23×C53—21no—
112C22×D53—21no—
113C22×D53—21no—
114Q8×C53—21no—
115Q8×C53—21no—
116Q8×C53—21no—
117Q8×C53—21no—
118D4×C53—21no—
119D4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—81no—
124C2×C4×C53C2×D411yescharacteristic
125C2×C4×C53C2×D411yescharacteristic
126C2×C4×C53C2411yescharacteristic
127C22⋊D53—42no—
128C22⋊D53—81no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53C2×D411yescharacteristic
132C2×Dic53C2×D411yescharacteristic
133C2×Dic53C2×D411yescharacteristic
134C2×Dic53C2×D411yescharacteristic
135D212—21no—
136D212—21no—
137D212—21no—
138D212—21no—
139C4×D53—42no—
140C4×D53—81no—
141Dic106—21no—
142Dic106—21no—
143Dic106—41no—
144C8×C53—21no—
145C8×C53—21no—
146#88322—21no—
147#88322#4112yes—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53C22×D411yescharacteristic
152C2×D53—41no—
153C2×D53—41no—
154C2×D53—41no—
155C2×D53—41no—
156C2×D53—41no—
157C2×D53—41no—
158C4×C53—21no—
159C4×C53—21no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53C2×#4111yescharacteristic
164C4×C53H(5,𝔽2)11yescharacteristic
165Dic53—21no—
166Dic53—21no—
167Dic53—21no—
168Dic53—21no—
169Dic53—41no—
170Dic53—41no—
171#3401—531nomaximal, Sylow
172C2×C53—81no—
173C2×C53#55011yescharacteristic
174C2×C53#57511yescharacteristic
175C2×C53#57711yescharacteristic
176D53—81no—
177D53—81no—
178#559—531no—
179#556—531no—
180#497—531no—
181#496—531no—
182#495—531no—
183#464—531no—
184#464—531no—
185#463—531no—
186#463—531no—
187#461D5311yescharacteristic, complemented
188#460—531no—
189#446—531no—
190#442—531no—
191#441—531no—
192C4×SD16—531no—
193C53#340111yescharacteristic, complemented, Sylow
194C2×Q16—531no—
195C2×Q16—531no—
196C2×SD16—531no—
197C2×SD16—531no—
198C2×SD16—531no—
199C2×SD16C2×D5311yescharacteristic
200C2×SD16C2×D5311yescharacteristic
201C2×D8—531no—
202C2×D8—531no—
203C4⋊Q8C2×D5311yescharacteristic
204#127—531no—
205#126—1061no—
206#126—1061no—
207#124—531no—
208#124—531no—
209#124—531no—
210#124—531no—
211#124—1061no—
212#123—1061no—
213#123—1061no—
214C22⋊Q8—1061no—
215C4×Q8—531no—
216C4×Q8—531no—
217C4×Q8—531no—
218C4×Q8—531no—
219C4×D4—531no—
220C4×D4—531no—
221C4×D4—1061no—
222C4×D4C2×D5311yescharacteristic
223#117—1061no—
224#106—531no—
225C4⋊C8—531no—
226C4⋊C8—531no—
227C4⋊C8C2×D5311yescharacteristic
228#103—531no—
229#103C2×D5311yescharacteristic
230#103C2×D5311yescharacteristic
231#102—531no—
232#102—531no—
233#102—531no—
234#97—531no—
235#97—531no—
236C4×C8—531no—
237C2×Q8—531no—
238C2×Q8—531no—
239C2×Q8C22×D5311yescharacteristic
240C2×Q8C22×D5311yescharacteristic
241C2×D4—531no—
242C2×D4—531no—
243C2×D4—1061no—
244C2×D4—1061no—
245C2×D4C22×D5311yescharacteristic
246C22×C4—21no—
247C22×C4—1061no—
248C22×C4—1061no—
249Q16—1062no—
250Q16—2121no—
251SD16—41no—
252SD16—41no—
253SD16—1062no—
254SD16—2121no—
255SD16—2121no—
256D8—1062no—
257D8—2121no—
258C2×C8—531no—
259C2×C8—531no—
260C2×C8C22×D5311yescharacteristic
261C2×C8C22×D5311yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—531no—
265C4⋊C4—531no—
266C4⋊C4—531no—
267C4⋊C4—531no—
268C4⋊C4—1061no—
269C4⋊C4—1061no—
270C4⋊C4—1061no—
271C4⋊C4C22×D5311yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—1061no—
274C22⋊C4—1061no—
275C22⋊C4—1061no—
276C22⋊C4—1061no—
277C22⋊C4—1061no—
278C22⋊C4—1061no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C42—531no—
284C42—531no—
285C42—1061no—
286C42—1061no—
287C42—1061no—
288C42C22×D5311yescharacteristic
289C23—21no—
290C23—1061no—
291C23—1061no—
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—1061no—
297Q8—1061no—
298Q8—2121no—
299D4—21no—
300D4—21no—
301D4—1061no—
302D4—1061no—
303D4—1061no—
304D4—1061no—
305D4—4241no—
306D4—2122no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—81no—
311C2×C4—531no—
312C2×C4—531no—
313C2×C4—531no—
314C2×C4—531no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—4241no—
318C2×C4—2122no—
319C2×C4D4×D5311yescharacteristic
320C2×C4D4×D5311yescharacteristic
321C2×C4C23×D5311yescharacteristic
322C8—21no—
323C8—21no—
324C8—532no—
325C8—1061no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—2121no—
330C22—2121no—
331C22—2121no—
332C22—2121no—
333C22—2121no—
334C22—2121no—
335C22C2×D4×D5311yescharacteristic
336C4—21no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—1061no—
342C4—1061no—
343C4—1061no—
344C4—1061no—
345C4—2121no—
346C4—2121no—
347C4#41×D5311yescharacteristic
348C4#1313326211yescharacteristic
349C2—81no—
350C2—4241no—
351C2—4241no—
352C2#1681129111yescharacteristic
353C2#1681142211yescharacteristic
354C2#1681142911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#461C2
socleC22×C53C22×D4
hypercenter#461D53
hypocenterC53#3401
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #16811356C2semidirect
2. #16811347C2semidirect
3. #16811015C2semidirect
5. #16811004C2semidirect
6. #16810848C2semidirect
7. #16810843C2semidirect
8. #16810841C2semidirect
9. #16810839C2semidirect
10. #16810821C2semidirect
11. #16810752C2semidirect
12. #16810733C2semidirect
13. #16810727C2semidirect
14. #16810708C2semidirect
15. C53×#461C2semidirect
187. #461D53semidirect
193. C53#3401semidirect