Finite Group Databasebeta

← #20286552

Subgroups — #20286552

Order 6784.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811313C211yesmaximal, characteristic, complemented
2#16811269C211yesmaximal, characteristic
3#16810978C211yesmaximal, characteristic, complemented
4#16810954C211yesmaximal, characteristic, complemented
5#16810948C211yesmaximal, characteristic, complemented
6#16810924C211yesmaximal, characteristic, complemented
7#16810864C211yesmaximal, characteristic, complemented
8#16810852C211yesmaximal, characteristic, complemented
9#16810846C211yesmaximal, characteristic, complemented
10#16810787C211yesmaximal, characteristic, complemented
11#16810776C211yesmaximal, characteristic, complemented
12C4×#11719913C211yesmaximal, characteristic, complemented
13#16810620C211yesmaximal, characteristic, complemented
14C2×#13133139C211yesmaximal, characteristic, complemented
15C8×D4×C53C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C2×Dic212C2211yescharacteristic, complemented
19C2×#11719913C2211yescharacteristic
20C2×#11719913C2211yescharacteristic
21#13133210—21no—
22#13133209C2211yescharacteristic, complemented
23#13133207—21no—
24#13133205C2211yescharacteristic
25#13133202—21no—
26#13133198—21no—
27#13133196C2211yescharacteristic
28#13133196C2211yescharacteristic
29#13133196C2211yescharacteristic, complemented
30#13133191C2211yescharacteristic
31#13133190—21no—
32#13133185—21no—
33C4×Dic106C2211yescharacteristic, complemented
34C4×D212C2211yescharacteristic
35#13133176C2211yescharacteristic, complemented
36#13133173—21no—
37C2×C4⋊Dic53C2211yescharacteristic
38C22⋊C4×D53—21no—
39#13133151C2211yescharacteristic, complemented
40#13133149C2211yescharacteristic, complemented
41#13133149C2211yescharacteristic, complemented
42#13133142C2211yescharacteristic, complemented
43#13133142C2211yescharacteristic, complemented
44#13133142C2211yescharacteristic, complemented
45#13133142C2211yescharacteristic, complemented
46#13133139C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133139C2211yescharacteristic
49#13133139C2211yescharacteristic
50#13133139C2211yescharacteristic
51#13133139C2211yescharacteristic
52C22×C8×C53C2211yescharacteristic
53C22×C8×C53C2211yescharacteristic, complemented
54C4×D4×C53C2211yescharacteristic
55C4⋊C8×C53C2211yescharacteristic, complemented
56C22⋊C8×C53C2211yescharacteristic
57C22⋊C8×C53C2211yescharacteristic, complemented
58C4×C8×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C2×D4×C53D411yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C22×Dic53—21no—
66#11719945—22no—
67#11719945—41no—
68C2×D212—22no—
69C2×D212—21no—
70C2×D212—41no—
71C2×D212C2311yescharacteristic
72C2×D212C2311yescharacteristic
73C2×C4×D53—21no—
74C2×C4×D53—21no—
75C2×Dic106C2311yescharacteristic
76C2×Dic106C2311yescharacteristic
77C2×C8×C53—41no—
78C2×C8×C53—41no—
79C2×C8×C53C2311yescharacteristic
80C2×C8×C53C2311yescharacteristic
81C2×C8×C53C2311yescharacteristic
82C2×C8×C53C2311yescharacteristic
83C4⋊C4×C53D411yescharacteristic
84C22⋊C4×C53D411yescharacteristic
85C22⋊C4×C53D411yescharacteristic
86C42×C53C2311yescharacteristic
87C22⋊Dic53—21no—
88C22⋊Dic53—21no—
89C22⋊Dic53—21no—
90C22⋊Dic53—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95C4⋊Dic53—21no—
96C4⋊Dic53C2311yescharacteristic
97C4⋊Dic53C2311yescharacteristic
98C4⋊Dic53C2311yescharacteristic
99C4⋊Dic53C2311yescharacteristic
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106C4×Dic53—21no—
107C4×Dic53—21no—
108Dic212—41no—
109#11719913—22no—
110#11719913—41no—
111C23×C53C2×D411yescharacteristic
112C23×C53C2×D411yescharacteristic
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—42no—
116C22×D53—42no—
117C22×D53—81no—
118D4×C53—22no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53C2×D411yescharacteristic
124C2×C4×C53C2×D411yescharacteristic
125C2×C4×C53C2×D411yescharacteristic
126C2×C4×C53C2×D411yescharacteristic
127C2×C4×C53C2411yescharacteristic
128C22⋊D53—42no—
129C22⋊D53—81no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—42no—
137D212—21no—
138D212—21no—
139D212—21no—
140D212—42no—
141D212#4112yes—
142C4×D53—42no—
143C4×D53—81no—
144Dic106—21no—
145Dic106—21no—
146Dic106—21no—
147Dic106—21no—
148C8×C53—21no—
149C8×C53—21no—
150C8×C53—21no—
151C8×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53#13512yes—
158C22×C53C22×D411yescharacteristic
159C2×D53—22no—
160C2×D53—41no—
161C2×D53—41no—
162C2×D53—41no—
163C2×D53—41no—
164C2×D53—41no—
165C2×D53—84no—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53C2×#4111yescharacteristic
171C4×C53H(5,𝔽2)11yescharacteristic
172Dic53—41no—
173Dic53—41no—
174Dic53—41no—
175Dic53—41no—
176Dic53—41no—
177Dic53—41no—
178#3296—531nomaximal, Sylow
179C2×C53—22no—
180C2×C53—41no—
181C2×C53#54611yescharacteristic
182C2×C53C2×#13511yescharacteristic
183C2×C53#57711yescharacteristic
184D53—42no—
185D53—81no—
186#552—531no—
187#546—531no—
188#488—531no—
189#483—531no—
190#482—531no—
191#478—531no—
192#467—531no—
193#465—531no—
194#464—531no—
195#452—531no—
196#450—531no—
197C4×SD16—531no—
198C8×D4D5311yescharacteristic, complemented
199#416—531no—
200C2×#102—531no—
201C53#329611yescharacteristic, complemented, Sylow
202C2×#41—531no—
203C22×D4—531no—
204C2×Q16—531no—
205C2×SD16—531no—
206C2×SD16—531no—
207C22×C8C2×D5311yescharacteristic
208C22×C8C2×D5311yescharacteristic, complemented
209#126—1061no—
210#125—531no—
211#125—1061no—
212#124—531no—
213#123—1061no—
214#123—1061no—
215C22⋊Q8—531no—
216C22⋊Q8—531no—
217C22⋊Q8—531no—
218#121—531no—
219#121—1061no—
220#120—1061no—
221C4×Q8—531no—
222C4×D4—531no—
223C4×D4C2×D5311yescharacteristic
224#117—531no—
225#117—1061no—
226C2×C4⋊C4—531no—
227C2×C22⋊C4—1061no—
228#107—531no—
229#106—531no—
230#106—531no—
231C4⋊C8C2×D5311yescharacteristic, complemented
232#103—531no—
233#103—531no—
234#103—531no—
235#103—531no—
236#102—531no—
237#102—531no—
238#102—531no—
239#102—531no—
240#102—531no—
241#102—531no—
242C22⋊C8C2×D5311yescharacteristic
243C22⋊C8C2×D5311yescharacteristic, complemented
244C4×C8C2×D5311yescharacteristic, complemented
245C24—1061no—
246#41—1062no—
247#41—2121no—
248C2×Q8—531no—
249C2×Q8—531no—
250C2×D4—531no—
251C2×D4—531no—
252C2×D4—1062no—
253C2×D4—1061no—
254C2×D4—1061no—
255C2×D4—1061no—
256C2×D4—2121no—
257C2×D4D21211yescharacteristic
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4—1061no—
261C22×C4C22×D5311yescharacteristic
262C22×C4C22×D5311yescharacteristic
263Q16—2121no—
264SD16—1062no—
265SD16—2121no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8C22×D5311yescharacteristic
269C2×C8C22×D5311yescharacteristic
270C2×C8C22×D5311yescharacteristic
271C2×C8C22×D5311yescharacteristic
272C4⋊C4—531no—
273C4⋊C4—531no—
274C4⋊C4—531no—
275C4⋊C4—531no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C4⋊C4—1061no—
279C4⋊C4—1061no—
280C4⋊C4—1061no—
281C4⋊C4—1061no—
282C4⋊C4—1061no—
283C4⋊C4D21211yescharacteristic
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C22⋊C4—1061no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C22⋊C4D21211yescharacteristic
293C22⋊C4D21211yescharacteristic
294C42—1061no—
295C42—1061no—
296C42C22×D5311yescharacteristic
297C23—1061no—
298C23—1061no—
299C23—4241no—
300C23—2122no—
301C23—2122no—
302C23C2×D21211yescharacteristic
303C23C2×D21211yescharacteristic
304Q8—1061no—
305Q8—1061no—
306Q8—1061no—
307Q8—1061no—
308D4—22no—
309D4—532no—
310D4—1061no—
311D4—1061no—
312D4—1061no—
313D4—4241no—
314D4—2122no—
315D4—2122no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—4241no—
327C2×C4—2122no—
328C2×C4—2122no—
329C2×C4C2×D21211yescharacteristic
330C2×C4C2×D21211yescharacteristic
331C2×C4C2×D21211yescharacteristic
332C2×C4C2×D21211yescharacteristic
333C2×C4C23×D5311yescharacteristic
334C8—21no—
335C8—21no—
336C8—21no—
337C8—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—1062no—
344C22—4244no—
345C22—2121no—
346C22—2121no—
347C22—2121no—
348C22—2121no—
349C22—2121no—
350C22#1313323912yes—
351C22C22×D21211yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—2121no—
357C4—2121no—
358C4—2121no—
359C4—2121no—
360C4—2121no—
361C4—2121no—
362C4#41×D5311yescharacteristic
363C4#1313326211yescharacteristic
364C2—22no—
365C2—41no—
366C2—4241no—
367C2—2122no—
368C2#1681126811yescharacteristic
369C2C2×#1313323911yescharacteristic
370C2#1681143511yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC8×D4×C53C2
socleC22×C53C22×D4
hypercenterC8×D4D53
hypocenterC53#3296
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #16811313C2semidirect
3. #16810978C2semidirect
4. #16810954C2semidirect
5. #16810948C2semidirect
6. #16810924C2semidirect
7. #16810864C2semidirect
8. #16810852C2semidirect
9. #16810846C2semidirect
10. #16810787C2semidirect
11. #16810776C2semidirect
12. C4×#11719913C2semidirect
13. #16810620C2semidirect
14. C2×#13133139C2semidirect
15. C8×D4×C53C2semidirect
18. C2×Dic212C22semidirect
22. #13133209C22semidirect
29. #13133196C22semidirect
33. C4×Dic106C22semidirect
35. #13133176C22semidirect
39. #13133151C22semidirect
40. #13133149C22semidirect
41. #13133149C22semidirect
42. #13133142C22semidirect
43. #13133142C22semidirect
44. #13133142C22semidirect
45. #13133142C22semidirect
53. C22×C8×C53C22semidirect
55. C4⋊C8×C53C22semidirect
57. C22⋊C8×C53C22semidirect
58. C4×C8×C53C22semidirect
198. C8×D4D53semidirect
201. C53#3296semidirect
208. C22×C8C2×D53semidirect
231. C4⋊C8C2×D53semidirect
243. C22⋊C8C2×D53semidirect
244. C4×C8C2×D53semidirect