Finite Group Databasebeta

← #20281752

Subgroups — #20281752

Order 6784.

552 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16810559C211yesmaximal, characteristic, complemented
2#16810551C211yesmaximal, characteristic, complemented
3#16810551C211yesmaximal, characteristic, complemented
4#16810548C211yesmaximal, characteristic, complemented
5#16810546C211yesmaximal, characteristic, complemented
6#16810546C211yesmaximal, characteristic, complemented
7#16810517C211yesmaximal, characteristic, complemented
8#16810517C211yesmaximal, characteristic, complemented
9#16810516C211yesmaximal, characteristic, complemented
10#16810516C211yesmaximal, characteristic, complemented
11#16810511C211yesmaximal, characteristic, complemented
12#16810511C211yesmaximal, characteristic, complemented
13#16810479C211yesmaximal, characteristic
14#16810479C211yesmaximal, characteristic
15C53×#379C211yesmaximal, characteristic
16C2×C4⋊Dic53C2211yescharacteristic, complemented
17C2×C4⋊Dic53C2211yescharacteristic, complemented
18C2×#11719919C2211yescharacteristic, complemented
19C2×#11719919C2211yescharacteristic, complemented
20C2×#11719919C2211yescharacteristic, complemented
21C2×#11719919C2211yescharacteristic, complemented
22C2×C22⋊Dic53C2211yescharacteristic
23C2×C22⋊Dic53C2211yescharacteristic
24C2×C22⋊Dic53C2211yescharacteristic
25C2×C22⋊Dic53C2211yescharacteristic
26C2×C22⋊Dic53C2211yescharacteristic
27C2×C22⋊Dic53C2211yescharacteristic
28C2×C22⋊Dic53C2211yescharacteristic
29C2×C22⋊Dic53C2211yescharacteristic
30C2×C22⋊Dic53C2211yescharacteristic
31C2×C22⋊Dic53C2211yescharacteristic
32C2×C22⋊Dic53C2211yescharacteristic
33C2×C22⋊Dic53C2211yescharacteristic
34C2×C4×Dic53C2211yescharacteristic, complemented
35C2×C4×Dic53C2211yescharacteristic, complemented
36#13133124C2211yescharacteristic, complemented
37#13133124C2211yescharacteristic, complemented
38#13133124C2211yescharacteristic, complemented
39#13133124C2211yescharacteristic, complemented
40#13133124C2211yescharacteristic, complemented
41#13133124C2211yescharacteristic, complemented
42#13133124C2211yescharacteristic, complemented
43#13133124C2211yescharacteristic, complemented
44C25×C53C2211yescharacteristic
45C2×C22⋊C4×C53C2211yescharacteristic
46C2×C22⋊C4×C53C2211yescharacteristic
47C2×C22⋊C4×C53C2211yescharacteristic
48C2×C22⋊C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50C2×C22⋊C4×C53C2211yescharacteristic
51C24×C53—22no—
52C24×C53—22no—
53C24×C53—22no—
54C24×C53—41no—
55C24×C53—41no—
56C24×C53—41no—
57C24×C53—41no—
58C24×C53C2311yescharacteristic
59C24×C53C2311yescharacteristic
60C24×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C22×Dic53C2311yescharacteristic
66C22×Dic53C2311yescharacteristic
67C22×Dic53C2311yescharacteristic
68C22×Dic53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C22×Dic53C2311yescharacteristic
71C22×Dic53C2311yescharacteristic
72C22×Dic53C2311yescharacteristic
73C22⋊C4×C53—22no—
74C22⋊C4×C53—22no—
75C22⋊C4×C53—41no—
76C22⋊C4×C53—41no—
77C22⋊C4×C53—41no—
78C22⋊C4×C53—41no—
79C22⋊Dic53—22no—
80C22⋊Dic53—22no—
81C22⋊Dic53—22no—
82C22⋊Dic53—22no—
83C22⋊Dic53—41no—
84C22⋊Dic53—41no—
85C22⋊Dic53—41no—
86C22⋊Dic53—41no—
87C22⋊Dic53—41no—
88C22⋊Dic53—41no—
89C22⋊Dic53—41no—
90C22⋊Dic53—41no—
91C4⋊Dic53—41no—
92C4⋊Dic53—41no—
93#11719919—41no—
94#11719919—41no—
95#11719919—41no—
96#11719919—41no—
97C4×Dic53—41no—
98C4×Dic53—41no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—44no—
118C23×C53—44no—
119C23×C53—44no—
120C23×C53—44no—
121C23×C53—44no—
122C23×C53—44no—
123C23×C53—44no—
124C23×C53#4112yes—
125C23×C53#4112yes—
126C23×C53#4112yes—
127C23×C53C2411yescharacteristic
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—21no—
152C2×Dic53—21no—
153C2×Dic53—21no—
154C2×Dic53—21no—
155C2×Dic53—21no—
156C2×Dic53—21no—
157C2×Dic53—21no—
158C2×Dic53—21no—
159C2×Dic53—21no—
160C2×Dic53—21no—
161C2×Dic53—21no—
162C2×Dic53—21no—
163C2×Dic53—21no—
164C22×C53—22no—
165C22×C53—22no—
166C22×C53—22no—
167C22×C53—22no—
168C22×C53—22no—
169C22×C53—22no—
170C22×C53—22no—
171C22×C53—22no—
172C22×C53—22no—
173C22×C53—416no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53—41no—
184C22×C53—41no—
185C22×C53—41no—
186C22×C53C2×#4111yescharacteristic
187C22×C53C2×#4111yescharacteristic
188C22×C53C2×#4111yescharacteristic
189C22×C53H(5,𝔽2)11yescharacteristic
190C22×C53H(5,𝔽2)11yescharacteristic
191C22×C53H(5,𝔽2)11yescharacteristic
192C22×C53H(5,𝔽2)11yescharacteristic
193C4×C53—41no—
194C4×C53—41no—
195C4×C53—41no—
196C4×C53—41no—
197Dic53—41no—
198Dic53—41no—
199Dic53—41no—
200Dic53—41no—
201Dic53—41no—
202Dic53—41no—
203Dic53—41no—
204Dic53—41no—
205#2735—531nomaximal, Sylow
206C2×C53—42no—
207C2×C53—42no—
208C2×C53—42no—
209C2×C53#53811yescharacteristic
210C2×C53#53811yescharacteristic
211C2×C53#53811yescharacteristic
212C2×C53#55111yescharacteristic
213C2×C53#55111yescharacteristic
214C2×C53#55111yescharacteristic
215C2×C53#56011yescharacteristic
216#399—531no—
217#397—531no—
218#397—531no—
219#397—531no—
220#396—531no—
221#396—531no—
222#388—531no—
223#388—531no—
224#388—531no—
225#388—531no—
226#388—531no—
227#388—531no—
228#379—531no—
229#379—531no—
230#379D5311yescharacteristic
231C53#273511yescharacteristic, complemented, Sylow
232C25C2×D5311yescharacteristic
233C2×C4⋊C4—531no—
234C2×C4⋊C4—531no—
235C2×C4⋊C4—531no—
236C2×C4⋊C4—531no—
237C2×C4⋊C4—531no—
238C2×C4⋊C4—531no—
239C2×C22⋊C4—531no—
240C2×C22⋊C4—531no—
241C2×C22⋊C4—531no—
242C2×C22⋊C4—531no—
243C2×C22⋊C4—531no—
244C2×C22⋊C4—531no—
245C2×C22⋊C4—531no—
246C2×C22⋊C4—531no—
247C2×C22⋊C4—531no—
248C2×C22⋊C4—531no—
249C2×C22⋊C4—531no—
250C2×C22⋊C4—531no—
251C2×C22⋊C4C2×D5311yescharacteristic
252C2×C22⋊C4C2×D5311yescharacteristic
253C2×C22⋊C4C2×D5311yescharacteristic
254C2×C22⋊C4C2×D5311yescharacteristic
255C2×C22⋊C4C2×D5311yescharacteristic
256C2×C22⋊C4C2×D5311yescharacteristic
257C2×C42—531no—
258C2×C42—531no—
259#95—531no—
260#95—531no—
261#95—531no—
262#95—531no—
263#95—531no—
264#95—531no—
265#95—531no—
266#95—531no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—41no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24C22×D5311yescharacteristic
275C24C22×D5311yescharacteristic
276C24C22×D5311yescharacteristic
277C22×C4—531no—
278C22×C4—531no—
279C22×C4—531no—
280C22×C4—531no—
281C22×C4—531no—
282C22×C4—531no—
283C22×C4—531no—
284C22×C4—531no—
285C22×C4C22×D5311yescharacteristic
286C22×C4C22×D5311yescharacteristic
287C22×C4C22×D5311yescharacteristic
288C22×C4C22×D5311yescharacteristic
289C4⋊C4—2121no—
290C4⋊C4—2121no—
291C4⋊C4—2121no—
292C4⋊C4—2121no—
293C4⋊C4—2121no—
294C4⋊C4—2121no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—1062no—
302C22⋊C4—1062no—
303C22⋊C4—1062no—
304C22⋊C4—1062no—
305C22⋊C4—2121no—
306C22⋊C4—2121no—
307C22⋊C4—2121no—
308C22⋊C4—2121no—
309C22⋊C4—2121no—
310C22⋊C4—2121no—
311C22⋊C4—2121no—
312C22⋊C4—2121no—
313C42—2121no—
314C42—2121no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—44no—
334C23—44no—
335C23—44no—
336C23—44no—
337C23—44no—
338C23—44no—
339C23—44no—
340C23#1171994512yes—
341C23#1171994712yes—
342C23#1171994712yes—
343C23C23×D5311yescharacteristic
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—1061no—
357C2×C4—1061no—
358C2×C4—1061no—
359C2×C4—1061no—
360C2×C4—1061no—
361C2×C4—1061no—
362C2×C4—1061no—
363C2×C4—1061no—
364C2×C4—1061no—
365C2×C4—1061no—
366C2×C4—1061no—
367C2×C4—1061no—
368C2×C4—1061no—
369C2×C4—1061no—
370C2×C4—1061no—
371C2×C4—1061no—
372C2×C4—1061no—
373C2×C4—1061no—
374C2×C4—1061no—
375C2×C4—1061no—
376C2×C4—1061no—
377C2×C4—1061no—
378C2×C4—1061no—
379C2×C4—1061no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—416no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22C2×#1171994711yescharacteristic
403C22C2×#1171994711yescharacteristic
404C22C2×#1171994511yescharacteristic
405C22#1313326211yescharacteristic
406C22#1313326211yescharacteristic
407C22#1313326211yescharacteristic
408C22#1313326211yescharacteristic
409C4—41no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—2121no—
414C4—2121no—
415C4—2121no—
416C4—2121no—
417C4—2121no—
418C4—2121no—
419C4—2121no—
420C4—2121no—
421C2—42no—
422C2—42no—
423C2—42no—
424C2#1681120811yescharacteristic
425C2#1681120811yescharacteristic
426C2#1681121111yescharacteristic
427C2#1681129911yescharacteristic
428C2#1681129911yescharacteristic
429C2#1681130411yescharacteristic
430C2#1681136211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#379C2
hypercenter#379D53
hypocenterC53#2735
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
1. #16810559C2semidirect
2. #16810551C2semidirect
3. #16810551C2semidirect
4. #16810548C2semidirect
5. #16810546C2semidirect
6. #16810546C2semidirect
7. #16810517C2semidirect
8. #16810517C2semidirect
9. #16810516C2semidirect
10. #16810516C2semidirect
11. #16810511C2semidirect
12. #16810511C2semidirect
16. C2×C4⋊Dic53C22semidirect
17. C2×C4⋊Dic53C22semidirect
18. C2×#11719919C22semidirect
19. C2×#11719919C22semidirect
20. C2×#11719919C22semidirect
21. C2×#11719919C22semidirect
34. C2×C4×Dic53C22semidirect
35. C2×C4×Dic53C22semidirect
36. #13133124C22semidirect
37. #13133124C22semidirect
38. #13133124C22semidirect
39. #13133124C22semidirect
40. #13133124C22semidirect
41. #13133124C22semidirect
42. #13133124C22semidirect
43. #13133124C22semidirect
231. C53#2735semidirect