Finite Group Databasebeta

← #20287303

Subgroups — #20287303

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#16811337C211yesmaximal, characteristic, complemented
2#16811274C211yesmaximal, characteristic, complemented
3#16811020C211yesmaximal, characteristic, complemented
4#16810949C211yesmaximal, characteristic, complemented
5#16810948C211yesmaximal, characteristic, complemented
6#16810940C211yesmaximal, characteristic, complemented
7#16810912C211yesmaximal, characteristic, complemented
8#16810901C211yesmaximal, characteristic, complemented
9#16810875C211yesmaximal, characteristic, complemented
10#16810874C211yesmaximal, characteristic, complemented
11#16810703C211yesmaximal, characteristic, complemented
12#16810698C211yesmaximal, characteristic, complemented
13#16810663C211yesmaximal, characteristic, complemented
14#16810662C211yesmaximal, characteristic, complemented
15C53×#481C211yesmaximal, characteristic, complemented
16C2×#11719945—21no—
17C2×#11719922C2211yescharacteristic
18C2×#11719917C2211yescharacteristic
19C2×#11719912C2211yescharacteristic
20#13133216C2211yescharacteristic
21#13133216C2211yescharacteristic
22#13133204—21no—
23#13133203—21no—
24#13133202—21no—
25#13133200—21no—
26#13133195—21no—
27#13133192C2211yescharacteristic
28#13133191C2211yescharacteristic
29#13133189—21no—
30#13133188C2211yescharacteristic
31D4×Dic53C2211yescharacteristic
32#13133179C2211yescharacteristic
33C4×C22⋊D53C2211yescharacteristic
34#13133176C2211yescharacteristic
35#13133175C2211yescharacteristic
36#13133173—21no—
37C4⋊C4×D53C2211yescharacteristic
38#13133151C2211yescharacteristic
39#13133150C2211yescharacteristic
40#13133150C2211yescharacteristic
41#13133149C2211yescharacteristic
42#13133148C2211yescharacteristic
43#13133148C2211yescharacteristic
44#13133148C2211yescharacteristic
45#13133146C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133138C2211yescharacteristic
49#13133137C2211yescharacteristic
50#13133129C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52#121×C53C2211yescharacteristic
53C2×C4⋊C4×C53C2211yescharacteristic
54#106×C53C2211yescharacteristic
55#106×C53C2211yescharacteristic
56#102×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59C2×D4×C53—21no—
60C2×D4×C53C2311yescharacteristic
61C22×C4×C53—21no—
62C22×C4×C53C2311yescharacteristic
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53D411yescharacteristic
66C22×Dic53—21no—
67#11719945—44no—
68C2×D212—21no—
69C2×D212C2311yescharacteristic
70C2×C4×D53—21no—
71C2×C4×D53—21no—
72C2×C4×D53C2311yescharacteristic
73C2×Dic106—21no—
74C2×Dic106—21no—
75C2×C8×C53C2311yescharacteristic
76C2×C8×C53C2311yescharacteristic
77C4⋊C4×C53—21no—
78C4⋊C4×C53C2311yescharacteristic
79C4⋊C4×C53C2311yescharacteristic
80C4⋊C4×C53C2311yescharacteristic
81C22⋊C4×C53—21no—
82C22⋊Dic53—21no—
83C22⋊Dic53—21no—
84C22⋊Dic53—21no—
85C22⋊Dic53D411yescharacteristic
86#11719922—22no—
87#11719921—21no—
88#11719921—21no—
89#11719921—21no—
90#11719921—21no—
91#11719921D411yescharacteristic
92C4⋊Dic53C2311yescharacteristic
93C4⋊Dic53C2311yescharacteristic
94C4⋊Dic53C2311yescharacteristic
95#11719919—21no—
96#11719919—21no—
97#11719919—21no—
98#11719919D411yescharacteristic
99C4×Dic53—21no—
100C4×Dic53—21no—
101C4×Dic53C2311yescharacteristic
102#11719917—22no—
103C2×#88322C2311yescharacteristic
104C2×#88322C2311yescharacteristic
105#11719912—22no—
106C23×C53—21no—
107C23×C53C2×D411yescharacteristic
108C22×D53—21no—
109C22×D53C2×D411yescharacteristic
110D4×C53—21no—
111D4×C53—21no—
112D4×C53—81no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—42no—
119C2×C4×C53C2×D411yescharacteristic
120C2×C4×C53C2411yescharacteristic
121C22⋊D53—22no—
122C22⋊D53—42no—
123C22⋊D53—81no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—42no—
128C2×Dic53C2×D411yescharacteristic
129C2×Dic53C2×D411yescharacteristic
130C2×Dic53C2×D411yescharacteristic
131D212—21no—
132D212—21no—
133D212—42no—
134C4×D53—21no—
135C4×D53—21no—
136C4×D53—42no—
137C4×D53—42no—
138Dic106—42no—
139Dic106—81no—
140C8×C53—21no—
141C8×C53—21no—
142#88322—21no—
143#88322#4112yescomplemented
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53C22×D411yescharacteristic
151C2×D53—21no—
152C2×D53—21no—
153C2×D53—21no—
154C2×D53—41no—
155C2×D53—41no—
156C2×D53—41no—
157C4×C53—21no—
158C4×C53—41no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53C2×#4111yescharacteristic
162C4×C532−1+411yescharacteristic
163Dic53—21no—
164Dic53—21no—
165Dic53—41no—
166Dic53—41no—
167Dic53—41no—
168Dic53AGL(1,ℤ/8)12yes—
169#3354—531nomaximal, Sylow
170C2×C53—41no—
171C2×C53—81no—
172C2×C53#54711yescharacteristic
173C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
174C2×C53#57711yescharacteristic
175D53—41no—
176D53—81no—
177#555—531no—
178#547—531no—
179#498—531no—
180#482—531no—
181#482—531no—
182#481—531no—
183#481D5311yescharacteristic, complemented
184#476—531no—
185#474—531no—
186#469—531no—
187#469—531no—
188C4×D8—531no—
189#436—531no—
190#428—531no—
191#428—531no—
192C53#335411yescharacteristic, complemented, Sylow
193C2×#41—1061no—
194C2×D8—531no—
195C2×M16—531no—
196C2×M16—531no—
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—1061no—
208#121C2×D5311yescharacteristic
209C4×D4—531no—
210C4×D4—531no—
211C4×D4—531no—
212#117—531no—
213#117—531no—
214#117—1061no—
215C2×C4⋊C4—531no—
216C2×C4⋊C4C2×D5311yescharacteristic
217#107—531no—
218#107—531no—
219#107—531no—
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#102C2×D5311yescharacteristic
232#102C2×D5311yescharacteristic
233C22⋊C8—531no—
234C22⋊C8C2×D5311yescharacteristic
235C4×C8—531no—
236#41—2124no—
237C2×Q8—1061no—
238C2×Q8—1061no—
239C2×D4—21no—
240C2×D4—531no—
241C2×D4—531no—
242C2×D4—1061no—
243C2×D4—1061no—
244C2×D4—1061no—
245C2×D4C22×D5311yescharacteristic
246C22×C4—21no—
247C22×C4—531no—
248C22×C4—1061no—
249C22×C4—1061no—
250C22×C4—1061no—
251C22×C4C22×D5311yescharacteristic
252D8—1062no—
253M16—1062no—
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—531no—
264C4⋊C4—1061no—
265C4⋊C4—1061no—
266C4⋊C4—1061no—
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—531no—
281C42—1061no—
282C42—1061no—
283C23—21no—
284C23—531no—
285C23—1061no—
286C23D4×D5311yescharacteristic
287Q8—4241no—
288Q8—2122no—
289D4—21no—
290D4—21no—
291D4—81no—
292D4—1062no—
293D4—1061no—
294D4—1061no—
295D4—4241no—
296D4—2122no—
297D4—2122no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—42no—
304C2×C4—531no—
305C2×C4—531no—
306C2×C4—531no—
307C2×C4—1061no—
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—21no—
319C8—532no—
320C8—1061no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—1061no—
328C22—1061no—
329C22—1061no—
330C22—2121no—
331C22—2121no—
332C22—2121no—
333C22C2×D4×D5311yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—532no—
339C4—1061no—
340C4—1061no—
341C4—2121no—
342C4—2121no—
343C4—2121no—
344C4#41×D5311yescharacteristic
345C4#1313326411yescharacteristic
346C2—41no—
347C2—81no—
348C2—4241no—
349C2—2121no—
350C2#1681127111yescharacteristic
351C2AGL(1,ℤ/8)×D5311yescharacteristic
352C2#1681142911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#481C2
socleC22×C53C22×D4
hypercenter#481D53
hypocenterC53#3354
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #16811337C2semidirect
2. #16811274C2semidirect
3. #16811020C2semidirect
4. #16810949C2semidirect
5. #16810948C2semidirect
6. #16810940C2semidirect
7. #16810912C2semidirect
8. #16810901C2semidirect
9. #16810875C2semidirect
10. #16810874C2semidirect
11. #16810703C2semidirect
12. #16810698C2semidirect
13. #16810663C2semidirect
14. #16810662C2semidirect
15. C53×#481C2semidirect
143. #88322#41semidirect
183. #481D53semidirect
192. C53#3354semidirect