Finite Group Databasebeta

← #20284773

Subgroups — #20284773

Order 6784.

434 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811170C211yesmaximal, characteristic, complemented
2C4×#11719949C211yesmaximal, characteristic, complemented
3#16810949C211yesmaximal, characteristic, complemented
4#16810945C211yesmaximal, characteristic, complemented
5#16810916C211yesmaximal, characteristic, complemented
6#16810912C211yesmaximal, characteristic, complemented
7#16810821C211yesmaximal, characteristic, complemented
8#16810817C211yesmaximal, characteristic, complemented
9#16810771C211yesmaximal, characteristic, complemented
10#16810767C211yesmaximal, characteristic, complemented
11C4×#11719924C211yesmaximal, characteristic, complemented
12C4×D424C211yesmaximal, characteristic, complemented
13C4×#11719922C211yesmaximal, characteristic, complemented
14#16810674C211yesmaximal, characteristic, complemented
15C4×SD16×C53C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719945—21no—
18C2×#11719924C2211yescharacteristic
19C2×D424C2211yescharacteristic
20C2×#11719922C2211yescharacteristic
21#13133216C2211yescharacteristic
22#13133213C2211yescharacteristic
23#13133203—21no—
24#13133192C2211yescharacteristic
25#13133190—21no—
26#13133188C2211yescharacteristic
27Q8×Dic53C2211yescharacteristic
28C4×D212—21no—
29C4×D212C2211yescharacteristic
30C4×D212C2211yescharacteristic
31#13133179—21no—
32#13133179C2211yescharacteristic
33C4×C22⋊D53—21no—
34#13133175—21no—
35#13133175C2211yescharacteristic
36C42×D53—21no—
37C42×D53C2211yescharacteristic
38#13133151C2211yescharacteristic
39#13133150C2211yescharacteristic
40#13133148C2211yescharacteristic
41#13133146C2211yescharacteristic
42#13133146C2211yescharacteristic
43#13133141C2211yescharacteristic
44#13133139C2211yescharacteristic
45#13133139C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133129C2211yescharacteristic
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#11719949—22no—
66#11719949—22no—
67#11719945—44no—
68C2×D212—21no—
69C2×D212—21no—
70C2×D212C2311yescharacteristic
71C2×D212C2311yescharacteristic
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×C4×D53—21no—
75C2×C4×D53—21no—
76C2×C4×D53D411yescharacteristic
77C2×C4×D53D411yescharacteristic
78C2×C4×D53C2311yescharacteristic
79C2×Dic106—21no—
80SD16×C53—22no—
81C2×C8×C53C2311yescharacteristic
82C2×C8×C53C2311yescharacteristic
83C4⋊C4×C53—21no—
84C4⋊C4×C53C2311yescharacteristic
85C4⋊C4×C53C2311yescharacteristic
86C22⋊C4×C53—21no—
87C42×C53—21no—
88C42×C53C2311yescharacteristic
89C22⋊Dic53—21no—
90#11719924—22no—
91#11719922—22no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97C4⋊Dic53—21no—
98C4⋊Dic53C2311yescharacteristic
99C4⋊Dic53C2311yescharacteristic
100#11719919—21no—
101C4×Dic53—21no—
102C4×Dic53—21no—
103C4×Dic53D411yescharacteristic
104C4×Dic53D411yescharacteristic
105C4×Dic53C2311yescharacteristic
106C2×#88322C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108D424—22no—
109C23×C53—21no—
110C22×D53—21no—
111C22×D53—21no—
112C22×D53C2×D411yescharacteristic
113Q8×C53—21no—
114Q8×C53#4112yescomplemented
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—24no—
133D212—21no—
134D212—21no—
135D212—21no—
136D212—42no—
137D212#4112yescomplemented
138C4×D53—24no—
139C4×D53—22no—
140C4×D53—21no—
141C4×D53—21no—
142C4×D53—22no—
143C4×D53—42no—
144C4×D53—42no—
145C4×D53—42no—
146Dic106—42no—
147C8×C53—21no—
148C8×C53—21no—
149#88322—21no—
150#88322—21no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53C22×D411yescharacteristic
155C2×D53—21no—
156C2×D53—22no—
157C2×D53—21no—
158C2×D53—21no—
159C2×D53—41no—
160C2×D53—41no—
161C2×D53—41no—
162C2×D53—41no—
163C2×D53—41no—
164C4×C53—22no—
165C4×C53—21no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53AGL(1,ℤ/8)12yes—
169C4×C53C2×#4111yescharacteristic
170C4×C53C2×#4111yescharacteristic
171Dic53—21no—
172Dic53—21no—
173Dic53—41no—
174Dic53—41no—
175Dic53#13512yes—
176#3106—531nomaximal, Sylow
177C2×C53—81no—
178C2×C53#52511yescharacteristic
179C2×C53C2×#13511yescharacteristic
180C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
181D53—41no—
182D53—42no—
183D53—81no—
184#532—531no—
185C4×#41—531no—
186#482—531no—
187#482—531no—
188#477—531no—
189#476—531no—
190#460—531no—
191#459—531no—
192#449—531no—
193#449—531no—
194C4×SD16—531no—
195C4×SD16D5311yescharacteristic, complemented
196C4×D8—531no—
197C4×D8—531no—
198#431—531no—
199C53#310611yescharacteristic, complemented, Sylow
200C2×#41—531no—
201C2×#41—1061no—
202C2×SD16—531no—
203C2×SD16C2×D5311yescharacteristic
204C2×D8—531no—
205C2×D8—531no—
206C4⋊Q8—531no—
207#127—531no—
208#124—1061no—
209#121—531no—
210#121—531no—
211#121—1061no—
212C4×Q8—531no—
213C4×Q8C2×D5311yescharacteristic
214C4×D4—531no—
215C4×D4—531no—
216C4×D4—531no—
217C4×D4—1061no—
218C4×D4—1061no—
219C4×D4—1061no—
220C4×D4C2×D5311yescharacteristic
221#117—531no—
222#117—1061no—
223C2×C42—531no—
224C2×C42—1061no—
225#107—531no—
226#107—531no—
227#106—531no—
228#106C2×D5311yescharacteristic
229C4⋊C8—531no—
230C4⋊C8—531no—
231#103—531no—
232#103C2×D5311yescharacteristic
233#102—531no—
234#102—531no—
235#102—531no—
236#102—531no—
237#102—531no—
238#102C2×D5311yescharacteristic
239C22⋊C8—531no—
240C22⋊C8—531no—
241C4×C8—531no—
242C4×C8C2×D5311yescharacteristic
243#41—1062no—
244#41—1062no—
245#41—2124no—
246C2×Q8—1061no—
247C2×Q8C22×D5311yescharacteristic
248C2×D4—531no—
249C2×D4—531no—
250C2×D4—1061no—
251C2×D4—1061no—
252C2×D4—1061no—
253C2×D4—1061no—
254C2×D4—1061no—
255C2×D4C22×D5311yescharacteristic
256C22×C4—21no—
257C22×C4—531no—
258C22×C4—531no—
259C22×C4—531no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—1061no—
264SD16—22no—
265SD16—1062no—
266D8—1062no—
267D8—1062no—
268C2×C8—531no—
269C2×C8—531no—
270C2×C8C22×D5311yescharacteristic
271C2×C8C22×D5311yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—531no—
274C4⋊C4—531no—
275C4⋊C4—1061no—
276C4⋊C4—1061no—
277C4⋊C4C22×D5311yescharacteristic
278C4⋊C4C22×D5311yescharacteristic
279C22⋊C4—21no—
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—
290C42—1061no—
291C42—1061no—
292C42C22×D5311yescharacteristic
293C23—21no—
294C23—531no—
295C23—1061no—
296C23—1061no—
297Q8—21no—
298Q8—2122no—
299Q8#1171994512yescomplemented
300D4—21no—
301D4—21no—
302D4—532no—
303D4—1064no—
304D4—1061no—
305D4—1061no—
306D4—1061no—
307D4—2122no—
308D4—2122no—
309D4—2122no—
310D4—2122no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—531no—
316C2×C4—531no—
317C2×C4—531no—
318C2×C4—1062no—
319C2×C4—1064no—
320C2×C4—1062no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—2122no—
326C2×C4—2122no—
327C2×C4—2122no—
328C2×C4D4×D5311yescharacteristic
329C2×C4D4×D5311yescharacteristic
330C2×C4C23×D5311yescharacteristic
331C8—21no—
332C8—21no—
333C8—1061no—
334C8—1061no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—1062no—
339C22—1061no—
340C22—1061no—
341C22—1061no—
342C22—2121no—
343C22—2121no—
344C22—2121no—
345C22—2121no—
346C22—2121no—
347C22C2×D4×D5311yescharacteristic
348C4—22no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—532no—
353C4—1061no—
354C4—1061no—
355C4—2121no—
356C4—2121no—
357C4#1313324412yes—
358C4#41×D5311yescharacteristic
359C4C2×#1171994511yescharacteristic
360C2—81no—
361C2—4241no—
362C2—2121no—
363C2—2122no—
364C2#1681113211yescharacteristic
365C2#135×D5311yescharacteristic
366C2C2×#1313324411yescharacteristic
subgroupG/H
centerC2×C4D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC4×SD16×C53C2
socleC22×C53C22×D4
hypercenterC4×SD16D53
hypocenterC53#3106
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. C4×#11719949C2semidirect
3. #16810949C2semidirect
4. #16810945C2semidirect
5. #16810916C2semidirect
6. #16810912C2semidirect
7. #16810821C2semidirect
8. #16810817C2semidirect
9. #16810771C2semidirect
10. #16810767C2semidirect
11. C4×#11719924C2semidirect
12. C4×D424C2semidirect
13. C4×#11719922C2semidirect
14. #16810674C2semidirect
15. C4×SD16×C53C2semidirect
114. Q8×C53#41semidirect
137. D212#41semidirect
195. C4×SD16D53semidirect
199. C53#3106semidirect
299. Q8#11719945semidirect