Finite Group Databasebeta

← #20287306

Subgroups — #20287306

Order 6784.

386 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#16811336C211yesmaximal, characteristic, complemented
2#16811274C211yesmaximal, characteristic, complemented
3#16811020C211yesmaximal, characteristic, complemented
4#16810950C211yesmaximal, characteristic, complemented
5#16810947C211yesmaximal, characteristic, complemented
6#16810943C211yesmaximal, characteristic, complemented
7#16810941C211yesmaximal, characteristic, complemented
8#16810912C211yesmaximal, characteristic, complemented
9#16810902C211yesmaximal, characteristic, complemented
10#16810877C211yesmaximal, characteristic, complemented
11#16810876C211yesmaximal, characteristic, complemented
12#16810705C211yesmaximal, characteristic, complemented
13#16810699C211yesmaximal, characteristic, complemented
14#16810664C211yesmaximal, characteristic, complemented
15C53×#428C211yesmaximal, characteristic, complemented
16C2×#11719945—21no—
17C2×D424C2211yescharacteristic
18C2×#11719912C2211yescharacteristic
19#13133216C2211yescharacteristic
20#13133216C2211yescharacteristic
21#13133205—21no—
22#13133203—21no—
23#13133202—21no—
24#13133200—21no—
25#13133195—21no—
26#13133192C2211yescharacteristic
27#13133192C2211yescharacteristic
28#13133191C2211yescharacteristic
29#13133191C2211yescharacteristic
30#13133190—21no—
31#13133179C2211yescharacteristic
32#13133179C2211yescharacteristic
33C4×C22⋊D53C2211yescharacteristic
34#13133175C2211yescharacteristic
35#13133174—21no—
36C4⋊C4×D53C2211yescharacteristic
37#13133150C2211yescharacteristic
38#13133149C2211yescharacteristic
39#13133148C2211yescharacteristic
40#13133148C2211yescharacteristic
41#13133148C2211yescharacteristic
42#13133146C2211yescharacteristic
43#13133139C2211yescharacteristic
44#13133139C2211yescharacteristic
45#13133137C2211yescharacteristic
46#13133137C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133129C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C2×M16×C53C2211yescharacteristic
53#117×C53C2211yescharacteristic
54C2×C4⋊C4×C53C2211yescharacteristic
55#107×C53C2211yescharacteristic
56#107×C53C2211yescharacteristic
57#106×C53C2211yescharacteristic
58#106×C53C2211yescharacteristic
59C22×C4×C53—21no—
60C22×C4×C53C2311yescharacteristic
61C2×C22⋊D53—21no—
62C2×C22⋊D53—21no—
63C2×C22⋊D53D411yescharacteristic
64#11719945—44no—
65C2×D212—21no—
66C2×D212—21no—
67C2×D212C2311yescharacteristic
68C2×D212C2311yescharacteristic
69C2×C4×D53—21no—
70C2×C4×D53—21no—
71C2×C4×D53—21no—
72C2×C4×D53C2311yescharacteristic
73C2×Dic106—21no—
74C2×Dic106—21no—
75M16×C53—22no—
76C2×C8×C53C2311yescharacteristic
77C2×C8×C53C2311yescharacteristic
78C4⋊C4×C53—21no—
79C4⋊C4×C53C2311yescharacteristic
80C4⋊C4×C53C2311yescharacteristic
81C4⋊C4×C53C2311yescharacteristic
82C4⋊C4×C53C2311yescharacteristic
83C22⋊C4×C53—21no—
84C42×C53—21no—
85C22⋊Dic53D411yescharacteristic
86#11719921—21no—
87#11719921—21no—
88#11719921—21no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921D411yescharacteristic
94C4⋊Dic53C2311yescharacteristic
95C4⋊Dic53C2311yescharacteristic
96#11719919—21no—
97#11719919—21no—
98#11719919—21no—
99#11719919D411yescharacteristic
100C4×Dic53—21no—
101C4×Dic53C2311yescharacteristic
102C2×#88322C2311yescharacteristic
103C2×#88322C2311yescharacteristic
104D424—22no—
105#11719912—22no—
106C23×C53C2×D411yescharacteristic
107C22×D53—21no—
108C22×D53—21no—
109C22×D53C2×D411yescharacteristic
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—42no—
117C2×C4×C53C2×D411yescharacteristic
118C2×C4×C53C2411yescharacteristic
119C22⋊D53—22no—
120C22⋊D53—42no—
121C22⋊D53—81no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53C2×D411yescharacteristic
125C2×Dic53C2×D411yescharacteristic
126C2×Dic53C2×D411yescharacteristic
127D212—21no—
128D212—21no—
129D212—21no—
130D212—21no—
131D212—42no—
132D212—81no—
133C4×D53—21no—
134C4×D53—21no—
135C4×D53—42no—
136C4×D53—42no—
137C4×D53—42no—
138Dic106—42no—
139Dic106—81no—
140C8×C53—21no—
141C8×C53#4112yescomplemented
142#88322—21no—
143#88322—21no—
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—21no—
147C22×C53C22×D411yescharacteristic
148C2×D53—21no—
149C2×D53—21no—
150C2×D53—21no—
151C2×D53—41no—
152C2×D53—41no—
153C2×D53—41no—
154C2×D53—41no—
155C2×D53—41no—
156C2×D53—41no—
157C4×C53—21no—
158C4×C53—41no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53C2×#4111yescharacteristic
163C4×C532−1+411yescharacteristic
164Dic53—21no—
165Dic53—21no—
166Dic53—41no—
167Dic53—41no—
168Dic53AGL(1,ℤ/8)12yes—
169#3354—531nomaximal, Sylow
170C2×C53—41no—
171C2×C53#54711yescharacteristic
172C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
173C2×C53#57711yescharacteristic
174D53—41no—
175D53—81no—
176D53—81no—
177#555—531no—
178#547—531no—
179#498—531no—
180#482—531no—
181#482—531no—
182#481—531no—
183#481—531no—
184#476—531no—
185#474—531no—
186#469—531no—
187#469—531no—
188C4×D8—531no—
189#436—531no—
190#428—531no—
191#428D5311yescharacteristic, complemented
192C53#335411yescharacteristic, complemented, Sylow
193C2×#41—1061no—
194C2×D8—531no—
195C2×M16—531no—
196C2×M16C2×D5311yescharacteristic
197C4⋊Q8—531no—
198C4⋊Q8—531no—
199#124—1061no—
200#124—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—531no—
212#117—531no—
213#117—1061no—
214#117C2×D5311yescharacteristic
215C2×C4⋊C4—531no—
216C2×C4⋊C4C2×D5311yescharacteristic
217#107—531no—
218#107C2×D5311yescharacteristic
219#107C2×D5311yescharacteristic
220#106—531no—
221#106—531no—
222#106—531no—
223#106—531no—
224#106C2×D5311yescharacteristic
225#106C2×D5311yescharacteristic
226C4⋊C8—531no—
227#102—531no—
228#102—531no—
229#102—531no—
230#102—531no—
231#102—531no—
232#102—531no—
233C22⋊C8—531no—
234C22⋊C8—531no—
235C4×C8—531no—
236#41—2124no—
237C2×Q8—1061no—
238C2×Q8—1061no—
239C2×D4—531no—
240C2×D4—531no—
241C2×D4—531no—
242C2×D4—1061no—
243C2×D4—1061no—
244C2×D4—1061no—
245C2×D4—1061no—
246C22×C4—21no—
247C22×C4—531no—
248C22×C4—1061no—
249C22×C4—1061no—
250C22×C4—1061no—
251C22×C4C22×D5311yescharacteristic
252D8—1062no—
253M16—22no—
254M16—1062no—
255C2×C8—531no—
256C2×C8—531no—
257C2×C8C22×D5311yescharacteristic
258C2×C8C22×D5311yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—531no—
261C4⋊C4—531no—
262C4⋊C4—531no—
263C4⋊C4—1061no—
264C4⋊C4—1061no—
265C4⋊C4—1061no—
266C4⋊C4C22×D5311yescharacteristic
267C4⋊C4C22×D5311yescharacteristic
268C4⋊C4C22×D5311yescharacteristic
269C4⋊C4C22×D5311yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—531no—
272C22⋊C4—531no—
273C22⋊C4—1061no—
274C22⋊C4—1061no—
275C22⋊C4—1061no—
276C22⋊C4—1061no—
277C22⋊C4—1061no—
278C22⋊C4—1061no—
279C22⋊C4—1061no—
280C42—21no—
281C42—531no—
282C42—1061no—
283C23—531no—
284C23—1061no—
285C23—1061no—
286C23D4×D5311yescharacteristic
287Q8—4241no—
288Q8—2122no—
289D4—1062no—
290D4—1061no—
291D4—1061no—
292D4—1061no—
293D4—1061no—
294D4—4241no—
295D4—4241no—
296D4—2122no—
297D4—2122no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—42no—
305C2×C4—531no—
306C2×C4—531no—
307C2×C4—531no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—1061no—
312C2×C4—2122no—
313C2×C4—2122no—
314C2×C4—2122no—
315C2×C4D4×D5311yescharacteristic
316C2×C4C23×D5311yescharacteristic
317C8—21no—
318C8—1061no—
319C8—1061no—
320C8#1171994512yescomplemented
321C22—21no—
322C22—21no—
323C22—21no—
324C22—1061no—
325C22—1061no—
326C22—1061no—
327C22—2121no—
328C22—2121no—
329C22—2121no—
330C22—2121no—
331C22—2121no—
332C22—2121no—
333C22C2×D4×D5311yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—532no—
340C4—1061no—
341C4—1061no—
342C4—2121no—
343C4—2121no—
344C4C2×#1171994511yescharacteristic
345C4#1313326511yescharacteristic
346C2—41no—
347C2—4241no—
348C2—4241no—
349C2—2121no—
350C2#1681127411yescharacteristic
351C2AGL(1,ℤ/8)×D5311yescharacteristic
352C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#428C2
socleC22×C53C22×D4
hypercenter#428D53
hypocenterC53#3354
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #16811336C2semidirect
2. #16811274C2semidirect
3. #16811020C2semidirect
4. #16810950C2semidirect
5. #16810947C2semidirect
6. #16810943C2semidirect
7. #16810941C2semidirect
8. #16810912C2semidirect
9. #16810902C2semidirect
10. #16810877C2semidirect
11. #16810876C2semidirect
12. #16810705C2semidirect
13. #16810699C2semidirect
14. #16810664C2semidirect
15. C53×#428C2semidirect
141. C8×C53#41semidirect
191. #428D53semidirect
192. C53#3354semidirect
320. C8#11719945semidirect