Finite Group Databasebeta

← #20286072

Subgroups — #20286072

Order 6784.

392 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811168C211yesmaximal, characteristic, complemented
2#16811159C211yesmaximal, characteristic, complemented
3#16810926C211yesmaximal, characteristic, complemented
4#16810925C211yesmaximal, characteristic, complemented
5#16810906C211yesmaximal, characteristic, complemented
6#16810902C211yesmaximal, characteristic, complemented
7#16810876C211yesmaximal, characteristic, complemented
8#16810874C211yesmaximal, characteristic, complemented
9#16810871C211yesmaximal, characteristic, complemented
10#16810841C211yesmaximal, characteristic, complemented
11#16810840C211yesmaximal, characteristic, complemented
12#16810821C211yesmaximal, characteristic, complemented
13#16810817C211yesmaximal, characteristic, complemented
14#16810646C211yesmaximal, characteristic, complemented
15C53×#468C211yesmaximal, characteristic, complemented
16C2×#11719947—21no—
17C2×#11719924C2211yescharacteristic
18C2×D424C2211yescharacteristic
19C2×#11719922C2211yescharacteristic
20C2×#11719917C2211yescharacteristic
21#13133216C2211yescharacteristic
22#13133213C2211yescharacteristic
23#13133212—21no—
24#13133208C2211yescharacteristic
25#13133206C2211yescharacteristic
26#13133203—21no—
27#13133201—21no—
28#13133191C2211yescharacteristic
29#13133191C2211yescharacteristic
30#13133189—21no—
31Q8×Dic53C2211yescharacteristic
32D4×Dic53C2211yescharacteristic
33#13133179C2211yescharacteristic
34#13133179C2211yescharacteristic
35#13133177—21no—
36#13133177—21no—
37#13133175—21no—
38C2×C4×Dic53C2211yescharacteristic
39#13133150C2211yescharacteristic
40#13133149C2211yescharacteristic
41#13133148C2211yescharacteristic
42#13133146C2211yescharacteristic
43#13133146C2211yescharacteristic
44#13133146C2211yescharacteristic
45#13133146C2211yescharacteristic
46#13133141C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133139C2211yescharacteristic
49#13133138C2211yescharacteristic
50#13133137C2211yescharacteristic
51#13133137C2211yescharacteristic
52C2×SD16×C53C2211yescharacteristic
53C2×M16×C53C2211yescharacteristic
54C22⋊Q8×C53C2211yescharacteristic
55#121×C53C2211yescharacteristic
56#107×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C2×D4×C53—21no—
61C2×D4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C22×Dic53—21no—
68C22×Dic53D411yescharacteristic
69C22×Dic53D411yescharacteristic
70#11719947—44no—
71C2×D212C2311yescharacteristic
72C2×D212C2311yescharacteristic
73C2×C4×D53—21no—
74C2×C4×D53—21no—
75C2×Dic106—21no—
76SD16×C53—41no—
77M16×C53—41no—
78C2×C8×C53C2311yescharacteristic
79C2×C8×C53C2311yescharacteristic
80C4⋊C4×C53—21no—
81C4⋊C4×C53C2311yescharacteristic
82C4⋊C4×C53C2311yescharacteristic
83C22⋊C4×C53—21no—
84C22⋊C4×C53—21no—
85C22⋊Dic53—21no—
86#11719924—41no—
87#11719922—41no—
88#11719921—21no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94C4⋊Dic53—21no—
95C4⋊Dic53—21no—
96C4⋊Dic53C2311yescharacteristic
97C4⋊Dic53C2311yescharacteristic
98#11719919—21no—
99#11719919—21no—
100C4×Dic53—21no—
101C4×Dic53D411yescharacteristic
102C4×Dic53D411yescharacteristic
103C4×Dic53C2311yescharacteristic
104C4×Dic53C2311yescharacteristic
105#11719917—41no—
106C2×#88322C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108D424—41no—
109C23×C53—21no—
110C23×C53C2×D411yescharacteristic
111C22×D53—21no—
112C22×D53—21no—
113Q8×C53—21no—
114Q8×C53—21no—
115D4×C53—21no—
116D4×C53—21no—
117D4×C53—42no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53C2×D411yescharacteristic
124C2×C4×C53C2411yescharacteristic
125C22⋊D53—42no—
126C22⋊D53—42no—
127C22⋊D53—42no—
128C22⋊D53—42no—
129C2×Dic53—22no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—41no—
133C2×Dic53—42no—
134C2×Dic53C2×D411yescharacteristic
135C2×Dic53C2×D411yescharacteristic
136C2×Dic53C2×D411yescharacteristic
137C2×Dic53C2×D411yescharacteristic
138D212—21no—
139D212—21no—
140D212—21no—
141D212—21no—
142C4×D53—42no—
143C4×D53—42no—
144Dic106—42no—
145C8×C53—21no—
146C8×C53—21no—
147#88322—21no—
148#88322—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53C22×D411yescharacteristic
156C2×D53—41no—
157C2×D53—41no—
158C2×D53—41no—
159C2×D53—41no—
160C2×D53—41no—
161C2×D53—41no—
162C4×C53—21no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53—41no—
166C4×C53H(5,𝔽2)11yescharacteristic
167C4×C532−1+411yescharacteristic
168Dic53—21no—
169Dic53—21no—
170Dic53—21no—
171Dic53—41no—
172Dic53—41no—
173Dic53AGL(1,ℤ/8)12yes—
174#3251—531nomaximal, Sylow
175C2×C53—41no—
176C2×C53—81no—
177C2×C53#53711yescharacteristic
178C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
179C2×C53#57511yescharacteristic
180D53—81no—
181D53—81no—
182#532—531no—
183#529—531no—
184#478—531no—
185#478—531no—
186#475—531no—
187#474—531no—
188#469—531no—
189#469—531no—
190#468—531no—
191#468D5311yescharacteristic, complemented
192#463—531no—
193#463—531no—
194#460—531no—
195#459—531no—
196#423—531no—
197C53#325111yescharacteristic, complemented, Sylow
198C2×#41—1061no—
199C2×SD16—531no—
200C2×SD16C2×D5311yescharacteristic
201C2×D8—531no—
202C2×D8—531no—
203C2×M16—531no—
204C2×M16C2×D5311yescharacteristic
205C4⋊Q8—531no—
206#127—531no—
207#126—1061no—
208#125—531no—
209#124—531no—
210#124—1061no—
211#123—1061no—
212C22⋊Q8C2×D5311yescharacteristic
213#121—531no—
214#121—531no—
215#121—1061no—
216#121C2×D5311yescharacteristic
217C4×Q8—531no—
218C4×D4—531no—
219C4×D4—531no—
220C4×D4—531no—
221C4×D4—1061no—
222C4×D4—1061no—
223#117—1061no—
224C2×C42—531no—
225#107—531no—
226#107C2×D5311yescharacteristic
227#106—531no—
228#106—531no—
229C4⋊C8—531no—
230C4⋊C8—531no—
231C4⋊C8—531no—
232C4⋊C8—531no—
233#103—531no—
234#103C2×D5311yescharacteristic
235#102—531no—
236#102—531no—
237#102—531no—
238#102—531no—
239#102—531no—
240#102C2×D5311yescharacteristic
241#41—2124no—
242C2×Q8—1061no—
243C2×Q8C22×D5311yescharacteristic
244C2×D4—21no—
245C2×D4—531no—
246C2×D4—531no—
247C2×D4—1061no—
248C2×D4—1061no—
249C2×D4—1061no—
250C2×D4—1061no—
251C2×D4C22×D5311yescharacteristic
252C22×C4—531no—
253C22×C4—531no—
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4C22×D5311yescharacteristic
258SD16—41no—
259SD16—2121no—
260D8—2121no—
261D8—2121no—
262M16—41no—
263M16—2121no—
264C2×C8—531no—
265C2×C8—531no—
266C2×C8C22×D5311yescharacteristic
267C2×C8C22×D5311yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—531no—
270C4⋊C4—531no—
271C4⋊C4—1061no—
272C4⋊C4—1061no—
273C4⋊C4—1061no—
274C4⋊C4—1061no—
275C4⋊C4C22×D5311yescharacteristic
276C4⋊C4C22×D5311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C42—531no—
287C42—531no—
288C42—531no—
289C42—531no—
290C42—1061no—
291C23—21no—
292C23—1061no—
293C23—1061no—
294C23D4×D5311yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—2122no—
298D4—21no—
299D4—21no—
300D4—42no—
301D4—1061no—
302D4—1061no—
303D4—1061no—
304D4—1061no—
305D4—2122no—
306D4—2122no—
307D4—2122no—
308D4—2122no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—531no—
315C2×C4—531no—
316C2×C4—531no—
317C2×C4—531no—
318C2×C4—1062no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—2122no—
322C2×C4—2122no—
323C2×C4—2121no—
324C2×C4—2122no—
325C2×C4D4×D5311yescharacteristic
326C2×C4C23×D5311yescharacteristic
327C8—21no—
328C8—21no—
329C8—1061no—
330C8—1061no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—2121no—
338C22—2121no—
339C22—2121no—
340C22—2121no—
341C22—2121no—
342C22—2121no—
343C22C2×D4×D5311yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—532no—
349C4—1061no—
350C4—1061no—
351C4—1061no—
352C4—2121no—
353C4—2121no—
354C4#1313326211yescharacteristic
355C4#1313326411yescharacteristic
356C2—41no—
357C2—81no—
358C2—4241no—
359C2—4241no—
360C2#1681120011yescharacteristic
361C2AGL(1,ℤ/8)×D5311yescharacteristic
362C2#1681142211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#468C2
socleC22×C53C22×D4
hypercenter#468D53
hypocenterC53#3251
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #16811168C2semidirect
2. #16811159C2semidirect
3. #16810926C2semidirect
4. #16810925C2semidirect
5. #16810906C2semidirect
6. #16810902C2semidirect
7. #16810876C2semidirect
8. #16810874C2semidirect
9. #16810871C2semidirect
10. #16810841C2semidirect
11. #16810840C2semidirect
12. #16810821C2semidirect
13. #16810817C2semidirect
14. #16810646C2semidirect
15. C53×#468C2semidirect
191. #468D53semidirect
197. C53#3251semidirect