Finite Group Databasebeta

← #20287780

Subgroups — #20287780

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#16811355C211yesmaximal, characteristic, complemented
2#16811347C211yesmaximal, characteristic, complemented
3#16811009C211yesmaximal, characteristic
4#16811008C211yesmaximal, characteristic, complemented
5#16810847C211yesmaximal, characteristic, complemented
6#16810847C211yesmaximal, characteristic, complemented
7#16810840C211yesmaximal, characteristic, complemented
8#16810835C211yesmaximal, characteristic, complemented
9#16810825C211yesmaximal, characteristic, complemented
10#16810821C211yesmaximal, characteristic, complemented
11#16810751C211yesmaximal, characteristic, complemented
12#16810736C211yesmaximal, characteristic, complemented
13#16810730C211yesmaximal, characteristic, complemented
14SD16×Dic53C211yesmaximal, characteristic, complemented
15C53×#497C211yesmaximal, characteristic, complemented
16C2×#11719924C2211yescharacteristic
17C2×#11719924C2211yescharacteristic
18C2×#11719924C2211yescharacteristic
19C2×D424C2211yescharacteristic
20C2×#11719922C2211yescharacteristic
21#13133213C2211yescharacteristic
22#13133212—21no—
23#13133212—21no—
24#13133206C2211yescharacteristic
25#13133206C2211yescharacteristic
26#13133206C2211yescharacteristic
27#13133203—21no—
28#13133201—21no—
29#13133199—21no—
30#13133197—21no—
31Q8×Dic53C2211yescharacteristic
32Q8×Dic53C2211yescharacteristic
33Q8×Dic53C2211yescharacteristic
34C4×Dic106C2211yescharacteristic
35C4×D212C2211yescharacteristic
36C4×D212C2211yescharacteristic
37D4×Dic53C2211yescharacteristic
38#13133177—21no—
39#13133175—21no—
40#13133149C2211yescharacteristic
41C4⋊#88322C2211yescharacteristic
42#13133146C2211yescharacteristic
43#13133146C2211yescharacteristic
44#13133141C2211yescharacteristic
45#13133141C2211yescharacteristic
46#13133141C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133139C2211yescharacteristic
49#13133138C2211yescharacteristic
50#13133127C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C2×Q16×C53C2211yescharacteristic
53C2×Q16×C53C2211yescharacteristic
54C2×SD16×C53C2211yescharacteristic
55C2×SD16×C53C2211yescharacteristic
56C4⋊Q8×C53C2211yescharacteristic
57#124×C53C2211yescharacteristic
58#97×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C2×Q8×C53C2311yescharacteristic
61C2×Q8×C53C2311yescharacteristic
62C2×D4×C53C2311yescharacteristic
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C22×Dic53—21no—
66C2×D212C2311yescharacteristic
67C2×D212C2311yescharacteristic
68C2×C4×D53—21no—
69C2×C4×D53—21no—
70C2×Dic106D411yescharacteristic
71Q16×C53—22no—
72Q16×C53—41no—
73SD16×C53—22no—
74SD16×C53—41no—
75C2×C8×C53C2311yescharacteristic
76C2×C8×C53C2311yescharacteristic
77C4⋊C4×C53—21no—
78C4⋊C4×C53—21no—
79C22⋊C4×C53—21no—
80C22⋊C4×C53—21no—
81C42×C53C2311yescharacteristic
82C22⋊Dic53—21no—
83#11719924—41no—
84#11719924—41no—
85#11719924—41no—
86#11719922—41no—
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⋊Dic53—21no—
98C4⋊Dic53D411yescharacteristic
99C4⋊Dic53C2311yescharacteristic
100C4⋊Dic53C2311yescharacteristic
101#11719919D411yescharacteristic
102#11719919D411yescharacteristic
103C4×Dic53—21no—
104C4×Dic53—21no—
105C4×Dic53—21no—
106C4×Dic53C2311yescharacteristic
107C4×Dic53C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110D424—22no—
111C23×C53—21no—
112C22×D53—21no—
113C22×D53—21no—
114Q8×C53—21no—
115Q8×C53—21no—
116Q8×C53—21no—
117Q8×C53—21no—
118Q8×C53—21no—
119Q8×C53—21no—
120D4×C53—21no—
121D4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53C2×D411yescharacteristic
126C2×C4×C53C2×D411yescharacteristic
127C2×C4×C53C2411yescharacteristic
128C22⋊D53—42no—
129C22⋊D53—81no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—81no—
133C2×Dic53C2×D411yescharacteristic
134C2×Dic53C2×D411yescharacteristic
135C2×Dic53C2×D411yescharacteristic
136C2×Dic53C2×D411yescharacteristic
137D212—21no—
138D212—21no—
139D212—21no—
140D212—21no—
141C4×D53—42no—
142C4×D53—81no—
143Dic106—41no—
144C8×C53—21no—
145C8×C53#4112yes—
146#88322—21no—
147#88322—21no—
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#497D5311yescharacteristic, complemented
181#496—531no—
182#495—531no—
183#464—531no—
184#464—531no—
185#463—531no—
186#463—531no—
187#461—531no—
188#460—531no—
189#446—531no—
190#442—531no—
191#441—531no—
192C4×SD16—531no—
193C53#340111yescharacteristic, complemented, Sylow
194C2×Q16C2×D5311yescharacteristic
195C2×Q16C2×D5311yescharacteristic
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—1061no—
211#124C2×D5311yescharacteristic
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—531no—
222C4×D4—1061no—
223#117—1061no—
224#106—531no—
225C4⋊C8—531no—
226C4⋊C8—531no—
227C4⋊C8—531no—
228#103—531no—
229#103—531no—
230#103—531no—
231#102—531no—
232#102—531no—
233#102—531no—
234#97—531no—
235#97C2×D5311yescharacteristic
236C4×C8—531no—
237C2×Q8—531no—
238C2×Q8C22×D5311yescharacteristic
239C2×Q8C22×D5311yescharacteristic
240C2×Q8C22×D5311yescharacteristic
241C2×D4—531no—
242C2×D4—531no—
243C2×D4—1061no—
244C2×D4—1061no—
245C2×D4C22×D5311yescharacteristic
246C22×C4—1061no—
247C22×C4—1061no—
248C22×C4—1061no—
249Q16—22no—
250Q16—41no—
251SD16—22no—
252SD16—41no—
253SD16—2121no—
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—531no—
269C4⋊C4—1061no—
270C4⋊C4—1061no—
271C4⋊C4—1061no—
272C22⋊C4—21no—
273C22⋊C4—21no—
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—21no—
297Q8—21no—
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—531no—
311C2×C4—531no—
312C2×C4—531no—
313C2×C4—531no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—4241no—
317C2×C4—4241no—
318C2×C4—2122no—
319C2×C4D4×D5311yescharacteristic
320C2×C4D4×D5311yescharacteristic
321C2×C4C23×D5311yescharacteristic
322C8—21no—
323C8—1061no—
324C8—1061no—
325C8#1171994712yes—
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—
347C4C2×#1171994711yescharacteristic
348C4#1313326211yescharacteristic
349C2—81no—
350C2—4241no—
351C2—4241no—
352C2#1681129311yescharacteristic
353C2#1681142211yescharacteristic
354C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#497C2
socleC22×C53C22×D4
hypercenter#497D53
hypocenterC53#3401
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #16811355C2semidirect
2. #16811347C2semidirect
4. #16811008C2semidirect
5. #16810847C2semidirect
6. #16810847C2semidirect
7. #16810840C2semidirect
8. #16810835C2semidirect
9. #16810825C2semidirect
10. #16810821C2semidirect
11. #16810751C2semidirect
12. #16810736C2semidirect
13. #16810730C2semidirect
14. SD16×Dic53C2semidirect
15. C53×#497C2semidirect
180. #497D53semidirect
193. C53#3401semidirect