Finite Group Databasebeta

← #20287182

Subgroups — #20287182

Order 6784.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811326C211yesmaximal, characteristic, complemented
2#16811278C211yesmaximal, characteristic, complemented
3#16811033C211yesmaximal, characteristic, complemented
4#16810946C211yesmaximal, characteristic, complemented
5#16810943C211yesmaximal, characteristic, complemented
6#16810940C211yesmaximal, characteristic, complemented
7#16810938C211yesmaximal, characteristic, complemented
8#16810914C211yesmaximal, characteristic, complemented
9#16810875C211yesmaximal, characteristic, complemented
10#16810858C211yesmaximal, characteristic, complemented
11#16810733C211yesmaximal, characteristic, complemented
12#16810693C211yesmaximal, characteristic, complemented
13#16810664C211yesmaximal, characteristic, complemented
14#106×D53C211yesmaximal, characteristic, complemented
15C53×#478C211yesmaximal, characteristic, complemented
16C2×#11719945—21no—
17C2×#11719922C2211yescharacteristic
18C2×#11719912C2211yescharacteristic
19C2×C8×D53C2211yescharacteristic
20#13133217C2211yescharacteristic
21#13133212—21no—
22#13133202—21no—
23#13133199—21no—
24#13133199—21no—
25#13133196—21no—
26#13133192—21no—
27#13133192C2211yescharacteristic
28#13133192C2211yescharacteristic
29#13133188C2211yescharacteristic
30#13133188C2211yescharacteristic
31C4×D212C2211yescharacteristic
32C4×D212C2211yescharacteristic
33#13133179—21no—
34#13133175C2211yescharacteristic
35C4⋊C4×D53C2211yescharacteristic
36C4⋊C4×D53C2211yescharacteristic
37C4⋊C4×D53C2211yescharacteristic
38#13133150C2211yescharacteristic
39#13133150C2211yescharacteristic
40#13133149C2211yescharacteristic
41#13133149C2211yescharacteristic
42#13133148C2211yescharacteristic
43#13133148C2211yescharacteristic
44#13133148C2211yescharacteristic
45#13133139C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133128C2211yescharacteristic
52#125×C53C2211yescharacteristic
53C4×D4×C53C2211yescharacteristic
54#107×C53C2211yescharacteristic
55#106×C53C2211yescharacteristic
56C4⋊C8×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59C2×D4×C53C2311yescharacteristic
60C22×C4×C53—21no—
61C2×C22⋊D53—21no—
62C2×C22⋊D53—21no—
63#11719945—44no—
64C2×D212—21no—
65C2×D212—21no—
66C2×D212D411yescharacteristic
67C2×D212C2311yescharacteristic
68C2×C4×D53—21no—
69C2×C4×D53—21no—
70C2×C4×D53—21no—
71C2×C4×D53—21no—
72C2×C4×D53C2311yescharacteristic
73C2×C4×D53C2311yescharacteristic
74C2×Dic106—21no—
75C2×C8×C53C2311yescharacteristic
76C2×C8×C53C2311yescharacteristic
77C4⋊C4×C53—21no—
78C4⋊C4×C53—21no—
79C4⋊C4×C53C2311yescharacteristic
80C4⋊C4×C53C2311yescharacteristic
81C4⋊C4×C53C2311yescharacteristic
82C22⋊C4×C53—21no—
83C42×C53C2311yescharacteristic
84C22⋊Dic53—21no—
85C22⋊Dic53—21no—
86#11719922—22no—
87#11719921—21no—
88#11719921—21no—
89#11719921—21no—
90#11719921—21no—
91#11719921D411yescharacteristic
92#11719921D411yescharacteristic
93C4⋊Dic53—21no—
94C4⋊Dic53D411yescharacteristic
95C4⋊Dic53C2311yescharacteristic
96C4⋊Dic53C2311yescharacteristic
97C4⋊Dic53C2311yescharacteristic
98#11719919—21no—
99#11719919—21no—
100#11719919—21no—
101C4×Dic53—21no—
102C2×#88322C2311yescharacteristic
103C2×#88322C2311yescharacteristic
104#11719912—22no—
105C8×D53—22no—
106C23×C53—21no—
107C22×D53—21no—
108C22×D53C2×D411yescharacteristic
109C22×D53C2×D411yescharacteristic
110D4×C53—21no—
111D4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—42no—
116C2×C4×C53C2×D411yescharacteristic
117C2×C4×C53C2×D411yescharacteristic
118C2×C4×C53C2411yescharacteristic
119C22⋊D53—42no—
120C22⋊D53—42no—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53C2×D411yescharacteristic
125C2×Dic53C2×D411yescharacteristic
126D212—22no—
127D212—21no—
128D212—21no—
129D212—42no—
130D212—81no—
131C4×D53—21no—
132C4×D53—21no—
133C4×D53—21no—
134C4×D53—21no—
135C4×D53—42no—
136C4×D53—42no—
137C4×D53—81no—
138C4×D53—81no—
139Dic106—42no—
140C8×C53—21no—
141C8×C53—21no—
142#88322—21no—
143#88322#4112yescomplemented
144C22×C53—41no—
145C22×C53—41no—
146C22×C53—41no—
147C22×C53C22×D411yescharacteristic
148C2×D53—21no—
149C2×D53—21no—
150C2×D53—21no—
151C2×D53—21no—
152C2×D53—21no—
153C2×D53—41no—
154C2×D53—41no—
155C2×D53—41no—
156C2×D53AGL(1,ℤ/8)12yes—
157C4×C53—21no—
158C4×C53—21no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53C2×#4111yescharacteristic
163C4×C532−1+411yescharacteristic
164Dic53—21no—
165Dic53—21no—
166Dic53—41no—
167Dic53—41no—
168Dic53—41no—
169#3345—531nomaximal, Sylow
170C2×C53—81no—
171C2×C53#54711yescharacteristic
172C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
173C2×C53#57711yescharacteristic
174D53—22no—
175D53—41no—
176D53—81no—
177#553—531no—
178#547—531no—
179#501—531no—
180#482—531no—
181#481—531no—
182#481—531no—
183#480—531no—
184#478D5311yescharacteristic, complemented
185#476—531no—
186#469—531no—
187#466—531no—
188#442—531no—
189#435—531no—
190#428—531no—
191C2×#106—531no—
192C53#334511yescharacteristic, complemented, Sylow
193C2×#41—1061no—
194C2×D8—531no—
195C2×M16—531no—
196C22×C8—531no—
197C4⋊Q8—531no—
198#126—1061no—
199#125C2×D5311yescharacteristic
200#123—1061no—
201#123—1061no—
202#123—1061no—
203C22⋊Q8—1061no—
204#121—531no—
205#121—531no—
206#121—531no—
207#121—531no—
208#121—1061no—
209C4×D4—531no—
210C4×D4—531no—
211C4×D4—1061no—
212C4×D4C2×D5311yescharacteristic
213#117—531no—
214C2×C4⋊C4—531no—
215C2×C4⋊C4—531no—
216C2×C4⋊C4—531no—
217#107—531no—
218#107—531no—
219#107C2×D5311yescharacteristic
220#106—531no—
221#106—531no—
222#106—531no—
223#106—531no—
224#106—531no—
225#106C2×D5311yescharacteristic
226C4⋊C8C2×D5311yescharacteristic
227#102—531no—
228#102—531no—
229#102—531no—
230#102—531no—
231#102C2×D5311yescharacteristic
232#102C2×D5311yescharacteristic
233C22⋊C8—531no—
234C22⋊C8—531no—
235#97—531no—
236#41—2124no—
237C2×Q8—1061no—
238C2×D4—531no—
239C2×D4—531no—
240C2×D4—1061no—
241C2×D4—1061no—
242C2×D4—1061no—
243C2×D4—1061no—
244C2×D4C22×D5311yescharacteristic
245C22×C4—21no—
246C22×C4—531no—
247C22×C4—531no—
248C22×C4—1061no—
249C22×C4—1061no—
250C22×C4—1061no—
251C22×C4—1061no—
252D8—1062no—
253M16—1062no—
254C2×C8—531no—
255C2×C8—531no—
256C2×C8—1062no—
257C2×C8C22×D5311yescharacteristic
258C2×C8C22×D5311yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—531no—
262C4⋊C4—531no—
263C4⋊C4—531no—
264C4⋊C4—531no—
265C4⋊C4—1061no—
266C4⋊C4—1061no—
267C4⋊C4—1061no—
268C4⋊C4—1061no—
269C4⋊C4C22×D5311yescharacteristic
270C4⋊C4C22×D5311yescharacteristic
271C4⋊C4C22×D5311yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—531no—
274C22⋊C4—531no—
275C22⋊C4—1061no—
276C22⋊C4—1061no—
277C22⋊C4—1061no—
278C22⋊C4—1061no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C42—1061no—
282C42C22×D5311yescharacteristic
283C23—21no—
284C23—531no—
285C23—531no—
286C23—1061no—
287Q8—2122no—
288D4—21no—
289D4—21no—
290D4—1062no—
291D4—1061no—
292D4—1061no—
293D4—4241no—
294D4—2122no—
295D4—2122no—
296D4—2122no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—42no—
301C2×C4—531no—
302C2×C4—531no—
303C2×C4—1061no—
304C2×C4—1061no—
305C2×C4—1061no—
306C2×C4—1061no—
307C2×C4—1061no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—4241no—
311C2×C4—4241no—
312C2×C4—2122no—
313C2×C4—2122no—
314C2×C4D4×D5311yescharacteristic
315C2×C4D4×D5311yescharacteristic
316C2×C4C23×D5311yescharacteristic
317C8—21no—
318C8—21no—
319C8—532no—
320C8—1061no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—532no—
325C22—1061no—
326C22—1061no—
327C22—1061no—
328C22—1061no—
329C22—1061no—
330C22—2121no—
331C22—2121no—
332C22—2121no—
333C22C2×D4×D5311yescharacteristic
334C4—21no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—1061no—
340C4—1061no—
341C4—2121no—
342C4—2121no—
343C4—2121no—
344C4#41×D5311yescharacteristic
345C4#1313326511yescharacteristic
346C2—81no—
347C2—1062no—
348C2—4241no—
349C2—2121no—
350C2#1681127511yescharacteristic
351C2AGL(1,ℤ/8)×D5311yescharacteristic
352C2#1681143011yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#478C2
socleC22×C53C22×D4
hypercenter#478D53
hypocenterC53#3345
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #16811326C2semidirect
2. #16811278C2semidirect
3. #16811033C2semidirect
4. #16810946C2semidirect
5. #16810943C2semidirect
6. #16810940C2semidirect
7. #16810938C2semidirect
8. #16810914C2semidirect
9. #16810875C2semidirect
10. #16810858C2semidirect
11. #16810733C2semidirect
12. #16810693C2semidirect
13. #16810664C2semidirect
14. #106×D53C2semidirect
15. C53×#478C2semidirect
143. #88322#41semidirect
184. #478D53semidirect
192. C53#3345semidirect