Finite Group Databasebeta

← #20286608

Subgroups — #20286608

Order 6784.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811315C211yesmaximal, characteristic, complemented
2#16811279C211yesmaximal, characteristic
3#16810976C211yesmaximal, characteristic, complemented
4#16810967C211yesmaximal, characteristic, complemented
5#16810934C211yesmaximal, characteristic, complemented
6#16810933C211yesmaximal, characteristic, complemented
7#16810863C211yesmaximal, characteristic, complemented
8#16810843C211yesmaximal, characteristic, complemented
9#16810782C211yesmaximal, characteristic, complemented
10#16810778C211yesmaximal, characteristic, complemented
11#16810715C211yesmaximal, characteristic, complemented
12C8×C22⋊D53C211yesmaximal, characteristic, complemented
13#16810616C211yesmaximal, characteristic, complemented
14C2×#13133142C211yesmaximal, characteristic, complemented
15C53×#465C211yesmaximal, characteristic, complemented
16C2×#11719947C2211yescharacteristic
17C22×Dic106C2211yescharacteristic
18C2×#11719925C2211yescharacteristic
19C2×#11719925C2211yescharacteristic
20C2×C8×D53C2211yescharacteristic
21#13133212—21no—
22#13133208—21no—
23#13133208C2211yescharacteristic
24#13133204C2211yescharacteristic
25#13133203—21no—
26#13133201—21no—
27#13133197C2211yescharacteristic
28#13133194—21no—
29#13133193C2211yescharacteristic
30#13133189—21no—
31Q8×Dic53C2211yescharacteristic
32#13133182C2211yescharacteristic
33D4×Dic53—21no—
34C4×C22⋊D53C2211yescharacteristic
35#13133175C2211yescharacteristic
36#13133173—21no—
37C2×C4⋊Dic53C2211yescharacteristic
38#13133150C2211yescharacteristic
39#13133150C2211yescharacteristic
40#13133146C2211yescharacteristic
41#13133142C2211yescharacteristic
42#13133142C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133142C2211yescharacteristic
45#13133140C2211yescharacteristic
46#13133140C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133138C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133129C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C2×SD16×C53C2211yescharacteristic
53C22×C8×C53C2211yescharacteristic
54C22⋊Q8×C53C2211yescharacteristic
55#121×C53C2211yescharacteristic
56#106×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⋊D53D411yescharacteristic
65C22×Dic53—21no—
66C22×Dic53—21no—
67C22×Dic53—21no—
68#11719947—22no—
69#11719947—41no—
70C2×C4×D53C2311yescharacteristic
71C2×Dic106—22no—
72C2×Dic106—21no—
73C2×Dic106—41no—
74C2×Dic106C2311yescharacteristic
75C2×Dic106C2311yescharacteristic
76SD16×C53—41no—
77C2×C8×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—
86C22⋊Dic53—21no—
87C22⋊Dic53D411yescharacteristic
88#11719925—22no—
89#11719925—41no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921D411yescharacteristic
94C4⋊Dic53—21no—
95C4⋊Dic53—21no—
96C4⋊Dic53—21no—
97C4⋊Dic53—21no—
98C4⋊Dic53C2311yescharacteristic
99C4⋊Dic53C2311yescharacteristic
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103#11719919D411yescharacteristic
104C4×Dic53—21no—
105C4×Dic53—21no—
106C4×Dic53—21no—
107C4×Dic53C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C8×D53—41no—
111C23×C53—21no—
112C23×C53C2×D411yescharacteristic
113C22×D53C2×D411yescharacteristic
114Q8×C53—21no—
115Q8×C53—21no—
116D4×C53—21no—
117D4×C53—21no—
118D4×C53—42no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53C2×D411yescharacteristic
125C2×C4×C53C2411yescharacteristic
126C22⋊D53—22no—
127C22⋊D53—42no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—42no—
133C2×Dic53—42no—
134C2×Dic53—42no—
135C2×Dic53C2×D411yescharacteristic
136C2×Dic53C2×D411yescharacteristic
137C2×Dic53C2×D411yescharacteristic
138C4×D53—21no—
139C4×D53—21no—
140Dic106—21no—
141Dic106—21no—
142Dic106—21no—
143Dic106—42no—
144Dic106#4112yes—
145C8×C53—21no—
146C8×C53—21no—
147#88322—21no—
148#88322—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53#13512yes—
155C22×C53C22×D411yescharacteristic
156C2×D53—21no—
157C2×D53—21no—
158C2×D53—21no—
159C4×C53—21no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53C2×#4111yescharacteristic
164C4×C53H(5,𝔽2)11yescharacteristic
165Dic53—22no—
166Dic53—21no—
167Dic53—21no—
168Dic53—21no—
169Dic53—41no—
170Dic53—41no—
171Dic53—41no—
172#3300—531nomaximal, Sylow
173C2×C53—22no—
174C2×C53—81no—
175C2×C53#54611yescharacteristic
176C2×C53C2×#13511yescharacteristic
177C2×C53#57811yescharacteristic
178D53—41no—
179#552—531no—
180#548—531no—
181#488—531no—
182#485—531no—
183#480—531no—
184#479—531no—
185#467—531no—
186#465D5311yescharacteristic, complemented
187#464—531no—
188#452—531no—
189#451—531no—
190C4×Q16—531no—
191C8×D4—531no—
192#416—531no—
193C2×#103—531no—
194C53#330011yescharacteristic, complemented, Sylow
195C2×#41—531no—
196C22×Q8—531no—
197C2×Q16—531no—
198C2×Q16—531no—
199C2×SD16C2×D5311yescharacteristic
200C22×C8—531no—
201C22×C8C2×D5311yescharacteristic
202#126—1061no—
203#125—531no—
204#125—1061no—
205#124—531no—
206#124—1061no—
207#123—1061no—
208C22⋊Q8—531no—
209C22⋊Q8—531no—
210C22⋊Q8—1061no—
211C22⋊Q8C2×D5311yescharacteristic
212#121—1061no—
213#121C2×D5311yescharacteristic
214C4×Q8—531no—
215C4×Q8—531no—
216C4×D4—531no—
217C4×D4—1061no—
218#117—531no—
219#117—1061no—
220C2×C4⋊C4—531no—
221#107—531no—
222#107—531no—
223#106C2×D5311yescharacteristic
224C4⋊C8—531no—
225#103—531no—
226#103—531no—
227#103—531no—
228#103—531no—
229#103—531no—
230#103—531no—
231#103C2×D5311yescharacteristic
232#102—531no—
233#102—531no—
234#102C2×D5311yescharacteristic
235C22⋊C8—531no—
236C22⋊C8—531no—
237C4×C8—531no—
238#41—1062no—
239#41—2121no—
240C2×Q8—531no—
241C2×Q8—531no—
242C2×Q8—1062no—
243C2×Q8—1061no—
244C2×Q8—2121no—
245C2×Q8C22×D5311yescharacteristic
246C2×D4—21no—
247C2×D4—531no—
248C2×D4—1061no—
249C2×D4C22×D5311yescharacteristic
250C22×C4—531no—
251C22×C4—1061no—
252C22×C4—1061no—
253C22×C4—1061no—
254C22×C4C22×D5311yescharacteristic
255Q16—1062no—
256Q16—2121no—
257SD16—41no—
258C2×C8—41no—
259C2×C8—531no—
260C2×C8—531no—
261C2×C8—2121no—
262C2×C8C22×D5311yescharacteristic
263C2×C8C22×D5311yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—531no—
266C4⋊C4—531no—
267C4⋊C4—531no—
268C4⋊C4—1061no—
269C4⋊C4—1061no—
270C4⋊C4—1061no—
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—531no—
280C22⋊C4—531no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C42—531no—
287C42—1061no—
288C42—1061no—
289C42—1061no—
290C23—21no—
291C23—531no—
292C23D4×D5311yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—532no—
296Q8—1061no—
297Q8—1061no—
298Q8—1061no—
299Q8—2122no—
300D4—21no—
301D4—21no—
302D4—42no—
303D4—1062no—
304D4—2122no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—531no—
311C2×C4—531no—
312C2×C4—531no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—2122no—
320C2×C4—2122no—
321C2×C4—2122no—
322C2×C4D4×D5311yescharacteristic
323C2×C4C23×D5311yescharacteristic
324C8—21no—
325C8—21no—
326C8—1061no—
327C8—1061no—
328C22—21no—
329C22—21no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—1061no—
334C22—1061no—
335C22—1061no—
336C22#1313323812yes—
337C22C2×D4×D5311yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—1061no—
343C4—1061no—
344C4—1061no—
345C4—1062no—
346C4—2121no—
347C4—2121no—
348C4—2121no—
349C4#41×D5311yescharacteristic
350C4#1313326211yescharacteristic
351C2—22no—
352C2—81no—
353C2—2121no—
354C2#1681126211yescharacteristic
355C2C2×#1313323811yescharacteristic
356C2#1681143711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#465C2
socleC22×C53C22×D4
hypercenter#465D53
hypocenterC53#3300
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #16811315C2semidirect
3. #16810976C2semidirect
4. #16810967C2semidirect
5. #16810934C2semidirect
6. #16810933C2semidirect
7. #16810863C2semidirect
8. #16810843C2semidirect
9. #16810782C2semidirect
10. #16810778C2semidirect
11. #16810715C2semidirect
12. C8×C22⋊D53C2semidirect
13. #16810616C2semidirect
14. C2×#13133142C2semidirect
15. C53×#465C2semidirect
186. #465D53semidirect
194. C53#3300semidirect