Finite Group Databasebeta

← #20285306

Subgroups — #20285306

Order 6784.

408 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#16811170C211yesmaximal, characteristic, complemented
2#16811160C211yesmaximal, characteristic, complemented
3#16811030C211yesmaximal, characteristic, complemented
4#16811009C211yesmaximal, characteristic, complemented
5#16810986C211yesmaximal, characteristic, complemented
6#16810984C211yesmaximal, characteristic, complemented
7#16810877C211yesmaximal, characteristic, complemented
8#16810875C211yesmaximal, characteristic, complemented
9#16810872C211yesmaximal, characteristic, complemented
10#16810866C211yesmaximal, characteristic, complemented
11C4×#11719924C211yesmaximal, characteristic, complemented
12C4×D424C211yesmaximal, characteristic, complemented
13C4×#11719922C211yesmaximal, characteristic, complemented
14C4×#11719912C211yesmaximal, characteristic, complemented
15C4×SD16×C53C211yesmaximal, characteristic, complemented
16C2×#11719945—21no—
17C2×#11719924C2211yescharacteristic
18C2×D424C2211yescharacteristic
19C2×#11719922C2211yescharacteristic
20C2×#11719912C2211yescharacteristic
21C2×#11719912C2211yescharacteristic
22#13133216C2211yescharacteristic
23#13133213C2211yescharacteristic
24#13133212—21no—
25#13133208C2211yescharacteristic
26#13133206C2211yescharacteristic
27#13133203—21no—
28#13133202—21no—
29#13133193C2211yescharacteristic
30#13133192C2211yescharacteristic
31#13133192C2211yescharacteristic
32#13133190—21no—
33#13133188C2211yescharacteristic
34C4×D212—21no—
35C4×D212C2211yescharacteristic
36C4×D212C2211yescharacteristic
37C4×C22⋊D53—21no—
38#13133174—21no—
39C42×D53C2211yescharacteristic
40#13133151C2211yescharacteristic
41#13133150C2211yescharacteristic
42#13133148C2211yescharacteristic
43#13133141C2211yescharacteristic
44#13133139C2211yescharacteristic
45#13133139C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133127C2211yescharacteristic
50#13133127C2211yescharacteristic
51C4×#88322C2211yescharacteristic
52C2×SD16×C53C2211yescharacteristic
53C4×Q8×C53C2211yescharacteristic
54C4×D4×C53C2211yescharacteristic
55#106×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58C4×C8×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C2×D4×C53C2311yescharacteristic
61C22×C4×C53—21no—
62C2×C22⋊D53—21no—
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65#11719945—44no—
66C2×D212—21no—
67C2×D212—21no—
68C2×D212C2311yescharacteristic
69C2×D212C2311yescharacteristic
70C2×C4×D53—21no—
71C2×C4×D53—21no—
72C2×C4×D53D411yescharacteristic
73C2×C4×D53D411yescharacteristic
74C2×C4×D53C2311yescharacteristic
75C2×Dic106—21no—
76SD16×C53—22no—
77C2×C8×C53C2311yescharacteristic
78C2×C8×C53C2311yescharacteristic
79C4⋊C4×C53—21no—
80C4⋊C4×C53C2311yescharacteristic
81C4⋊C4×C53C2311yescharacteristic
82C22⋊C4×C53—21no—
83C42×C53—21no—
84C42×C53C2311yescharacteristic
85C22⋊Dic53—21no—
86#11719924—22no—
87#11719922—22no—
88#11719921—21no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95C4⋊Dic53—21no—
96C4⋊Dic53C2311yescharacteristic
97C4⋊Dic53C2311yescharacteristic
98#11719919—21no—
99#11719919—21no—
100#11719919—21no—
101C4×Dic53D411yescharacteristic
102C4×Dic53D411yescharacteristic
103C4×Dic53C2311yescharacteristic
104C2×#88322C2311yescharacteristic
105C2×#88322C2311yescharacteristic
106D424—22no—
107#11719912—22no—
108#11719912—22no—
109C23×C53—21no—
110C22×D53—21no—
111C22×D53—21no—
112C22×D53C2×D411yescharacteristic
113Q8×C53—21no—
114Q8×C53—21no—
115D4×C53—21no—
116D4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—42no—
121C2×C4×C53C2×D411yescharacteristic
122C2×C4×C53C2×D411yescharacteristic
123C2×C4×C53C2411yescharacteristic
124C22⋊D53—42no—
125C22⋊D53—42no—
126C22⋊D53—42no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53C2×D411yescharacteristic
130C2×Dic53C2×D411yescharacteristic
131C2×Dic53C2×D411yescharacteristic
132D212—21no—
133D212—21no—
134D212—21no—
135D212—21no—
136D212—42no—
137D212—42no—
138C4×D53—22no—
139C4×D53—21no—
140C4×D53—21no—
141C4×D53—41no—
142C4×D53—42no—
143C4×D53—42no—
144Dic106—42no—
145C8×C53—21no—
146C8×C53#4112yescomplemented
147#88322—21no—
148#88322#4112yescomplemented
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53C22×D411yescharacteristic
153C2×D53—21no—
154C2×D53—21no—
155C2×D53—21no—
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×C53AGL(1,ℤ/8)12yes—
167C4×C53C2×#4111yescharacteristic
168C4×C53C2×#4111yescharacteristic
169Dic53—21no—
170Dic53—21no—
171Dic53—21no—
172Dic53—41no—
173Dic53—41no—
174#3171—531nomaximal, Sylow
175C2×C53—81no—
176C2×C53#53211yescharacteristic
177C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
178C2×C53#57511yescharacteristic
179D53—41no—
180D53—81no—
181D53—81no—
182#532—531no—
183#529—531no—
184#501—531no—
185#496—531no—
186#490—531no—
187#489—531no—
188#469—531no—
189#469—531no—
190#468—531no—
191#468—531no—
192C4×SD16—531no—
193C4×SD16D5311yescharacteristic, complemented
194C4×D8—531no—
195C4×D8—531no—
196C4×M16—531no—
197C53#317111yescharacteristic, complemented, Sylow
198C2×#41—1061no—
199C2×SD16—531no—
200C2×SD16C2×D5311yescharacteristic
201C2×D8—531no—
202C2×D8—531no—
203C2×M16—531no—
204C2×M16—531no—
205C4⋊Q8—531no—
206#127—531no—
207#126—1061no—
208#125—531no—
209#124—531no—
210#124—1061no—
211#123—1061no—
212C22⋊Q8—531no—
213#121—531no—
214#121—531no—
215#121—531no—
216#121—1061no—
217C4×Q8C2×D5311yescharacteristic
218C4×D4—531no—
219C4×D4—531no—
220C4×D4—1061no—
221C4×D4—1061no—
222C4×D4C2×D5311yescharacteristic
223#117—1061no—
224C2×C42—531no—
225#107—531no—
226#107—531no—
227#106—531no—
228#106C2×D5311yescharacteristic
229#103—531no—
230#103C2×D5311yescharacteristic
231#102—531no—
232#102—531no—
233#102—531no—
234#102—531no—
235#102—531no—
236#102C2×D5311yescharacteristic
237#97—531no—
238#97—531no—
239C4×C8—531no—
240C4×C8C2×D5311yescharacteristic
241#41—2124no—
242C2×Q8—1061no—
243C2×Q8C22×D5311yescharacteristic
244C2×D4—531no—
245C2×D4—531no—
246C2×D4—1061no—
247C2×D4—1061no—
248C2×D4—1061no—
249C2×D4—1061no—
250C2×D4—1061no—
251C2×D4C22×D5311yescharacteristic
252C22×C4—21no—
253C22×C4—531no—
254C22×C4—531no—
255C22×C4—531no—
256C22×C4—1061no—
257C22×C4—1061no—
258SD16—22no—
259SD16—1062no—
260D8—1062no—
261D8—1062no—
262M16—1062no—
263M16—1062no—
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—1061no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C42—21no—
287C42—531no—
288C42—531no—
289C42—531no—
290C42C22×D5311yescharacteristic
291C23—21no—
292C23—531no—
293C23—1061no—
294C23—1061no—
295Q8—21no—
296Q8—21no—
297Q8—2122no—
298D4—21no—
299D4—21no—
300D4—1061no—
301D4—1061no—
302D4—1061no—
303D4—1061no—
304D4—2122no—
305D4—2122no—
306D4—2122no—
307D4—2122no—
308D4—2122no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—531no—
314C2×C4—531no—
315C2×C4—531no—
316C2×C4—1062no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—2121no—
322C2×C4—2122no—
323C2×C4—2122no—
324C2×C4D4×D5311yescharacteristic
325C2×C4D4×D5311yescharacteristic
326C2×C4C23×D5311yescharacteristic
327C8—21no—
328C8—532no—
329C8—1061no—
330C8#1171994512yescomplemented
331C22—41no—
332C22—41no—
333C22—41no—
334C22—1061no—
335C22—1061no—
336C22—1061no—
337C22—2121no—
338C22—2121no—
339C22—2121no—
340C22—2121no—
341C22—2121no—
342C22—2121no—
343C22C2×D4×D5311yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—1061no—
349C4—1061no—
350C4—1061no—
351C4—2121no—
352C4—2121no—
353C4#1313324412yes—
354C4#41×D5311yescharacteristic
355C4C2×#1171994511yescharacteristic
356C2—81no—
357C2—4241no—
358C2—4241no—
359C2—2121no—
360C2#1681117011yescharacteristic
361C2C2×#1313324411yescharacteristic
362C2#1681142011yescharacteristic
subgroupG/H
centerC2×C4D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC4×SD16×C53C2
socleC22×C53C22×D4
hypercenterC4×SD16D53
hypocenterC53#3171
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #16811170C2semidirect
2. #16811160C2semidirect
3. #16811030C2semidirect
4. #16811009C2semidirect
5. #16810986C2semidirect
6. #16810984C2semidirect
7. #16810877C2semidirect
8. #16810875C2semidirect
9. #16810872C2semidirect
10. #16810866C2semidirect
11. C4×#11719924C2semidirect
12. C4×D424C2semidirect
13. C4×#11719922C2semidirect
14. C4×#11719912C2semidirect
15. C4×SD16×C53C2semidirect
146. C8×C53#41semidirect
148. #88322#41semidirect
193. C4×SD16D53semidirect
197. C53#3171semidirect
330. C8#11719945semidirect