Finite Group Databasebeta

← #20281755

Subgroups — #20281755

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#16810560C211yesmaximal, characteristic, complemented
2#16810552C211yesmaximal, characteristic, complemented
3#16810552C211yesmaximal, characteristic, complemented
4#16810546C211yesmaximal, characteristic, complemented
5#16810545C211yesmaximal, characteristic, complemented
6#16810516C211yesmaximal, characteristic, complemented
7#16810516C211yesmaximal, characteristic, complemented
8#16810515C211yesmaximal, characteristic, complemented
9#16810515C211yesmaximal, characteristic, complemented
10#16810513C211yesmaximal, characteristic, complemented
11#16810512C211yesmaximal, characteristic, complemented
12#16810478C211yesmaximal, characteristic
13#16810478C211yesmaximal, characteristic
14#16810478C211yesmaximal, characteristic
15C53×#397C211yesmaximal, characteristic, complemented
16C24×D53C2211yescharacteristic
17C2×C4⋊Dic53C2211yescharacteristic, complemented
18C2×C4⋊Dic53C2211yescharacteristic, complemented
19C2×C4⋊Dic53C2211yescharacteristic, complemented
20C2×#11719919C2211yescharacteristic, complemented
21C2×#11719919C2211yescharacteristic, complemented
22C2×C22⋊Dic53C2211yescharacteristic
23C2×C22⋊Dic53C2211yescharacteristic
24C2×C22⋊Dic53C2211yescharacteristic
25C2×#11719921C2211yescharacteristic
26C2×#11719921C2211yescharacteristic
27C2×#11719921C2211yescharacteristic
28C2×#11719921C2211yescharacteristic
29C2×#11719921C2211yescharacteristic
30C2×#11719921C2211yescharacteristic
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×C4×Dic53C2211yescharacteristic, complemented
38C2×C4×Dic53C2211yescharacteristic, complemented
39#13133124C2211yescharacteristic, complemented
40#13133124C2211yescharacteristic, complemented
41#13133124C2211yescharacteristic, complemented
42#13133124C2211yescharacteristic, complemented
43#13133124C2211yescharacteristic, complemented
44C2×C4⋊C4×C53C2211yescharacteristic, complemented
45C2×C22⋊C4×C53C2211yescharacteristic
46C2×C22⋊C4×C53C2211yescharacteristic
47C2×C22⋊C4×C53C2211yescharacteristic
48#95×C53C2211yescharacteristic, complemented
49#95×C53C2211yescharacteristic, complemented
50#95×C53C2211yescharacteristic, complemented
51C24×C53C2311yescharacteristic
52C23×D53—22no—
53C23×D53—22no—
54C23×D53—22no—
55C23×D53—41no—
56C23×D53—41no—
57C23×D53—41no—
58C23×D53—41no—
59C23×D53C2311yescharacteristic
60C23×D53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C22×Dic53C2311yescharacteristic
68C22×Dic53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C22×Dic53C2311yescharacteristic
71C22×Dic53C2311yescharacteristic
72C22×Dic53C2311yescharacteristic
73C4⋊C4×C53—41no—
74C22⋊C4×C53—22no—
75C22⋊C4×C53—22no—
76C22⋊C4×C53—41no—
77C22⋊Dic53—41no—
78C22⋊Dic53—41no—
79C22⋊Dic53—41no—
80#11719921—22no—
81#11719921—22no—
82#11719921—22no—
83#11719921—22no—
84#11719921—41no—
85#11719921—41no—
86#11719921—41no—
87#11719921—41no—
88#11719921—41no—
89#11719921—41no—
90#11719921—41no—
91#11719921—41no—
92C4⋊Dic53—41no—
93C4⋊Dic53—41no—
94C4⋊Dic53—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#4112yes—
106C23×C53C2411yescharacteristic
107C22×D53—21no—
108C22×D53—21no—
109C22×D53—21no—
110C22×D53—21no—
111C22×D53—21no—
112C22×D53—21no—
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—21no—
116C22×D53—21no—
117C22×D53—21no—
118C22×D53—21no—
119C22×D53—44no—
120C22×D53—44no—
121C22×D53—44no—
122C22×D53—44no—
123C22×D53—44no—
124C22×D53—44no—
125C22×D53—44no—
126C22×D53#4112yes—
127C22×D53#4112yes—
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×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—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—41no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53C2×#4111yescharacteristic
172C22×C53C2×#4111yescharacteristic
173C22×C53C2×#4111yescharacteristic
174C22×C53H(5,𝔽2)11yescharacteristic
175C22×C53H(5,𝔽2)11yescharacteristic
176C22×C53H(5,𝔽2)11yescharacteristic
177C22×C53H(5,𝔽2)11yescharacteristic
178C2×D53—22no—
179C2×D53—22no—
180C2×D53—22no—
181C2×D53—22no—
182C2×D53—22no—
183C2×D53—22no—
184C2×D53—416no—
185C2×D53—41no—
186C2×D53—41no—
187C2×D53—41no—
188C2×D53—41no—
189C2×D53—41no—
190C2×D53—41no—
191C2×D53—41no—
192C2×D53—41no—
193C4×C53—41no—
194C4×C53—41no—
195C4×C53—41no—
196C4×C53—41no—
197C4×C53—41no—
198C4×C53—41no—
199Dic53—41no—
200Dic53—41no—
201Dic53—41no—
202Dic53—41no—
203Dic53—41no—
204Dic53—41no—
205#2735—531nomaximal, Sylow
206C2×C53—42no—
207C2×C53#53811yescharacteristic
208C2×C53#53811yescharacteristic
209C2×C53#53811yescharacteristic
210C2×C53#55111yescharacteristic
211C2×C53#55111yescharacteristic
212C2×C53#55111yescharacteristic
213C2×C53#56011yescharacteristic
214D53—42no—
215D53—42no—
216#399—531no—
217#397—531no—
218#397—531no—
219#397D5311yescharacteristic, complemented
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#379—531no—
231C53#273511yescharacteristic, complemented, Sylow
232C25—531no—
233C2×C4⋊C4—531no—
234C2×C4⋊C4—531no—
235C2×C4⋊C4—531no—
236C2×C4⋊C4—531no—
237C2×C4⋊C4—531no—
238C2×C4⋊C4C2×D5311yescharacteristic, complemented
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⋊C4—531no—
252C2×C22⋊C4—531no—
253C2×C22⋊C4—531no—
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#95C2×D5311yescharacteristic, complemented
265#95C2×D5311yescharacteristic, complemented
266#95C2×D5311yescharacteristic, complemented
267C24—531no—
268C24—531no—
269C24—1062no—
270C24—1062no—
271C24—1062no—
272C24—2121no—
273C24—2121no—
274C24—2121no—
275C24—2121no—
276C24C22×D5311yescharacteristic
277C22×C4—531no—
278C22×C4—531no—
279C22×C4—531no—
280C22×C4—531no—
281C22×C4—531no—
282C22×C4—531no—
283C22×C4C22×D5311yescharacteristic
284C22×C4C22×D5311yescharacteristic
285C22×C4C22×D5311yescharacteristic
286C22×C4C22×D5311yescharacteristic
287C22×C4C22×D5311yescharacteristic
288C22×C4C22×D5311yescharacteristic
289C4⋊C4—41no—
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—1062no—
299C22⋊C4—1062no—
300C22⋊C4—1062no—
301C22⋊C4—1062no—
302C22⋊C4—2121no—
303C22⋊C4—2121no—
304C22⋊C4—2121no—
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—532no—
322C23—532no—
323C23—1061no—
324C23—1061no—
325C23—1061no—
326C23—1061no—
327C23—1061no—
328C23—1061no—
329C23—1061no—
330C23—1061no—
331C23—1061no—
332C23—1061no—
333C23—1061no—
334C23—1061no—
335C23—2124no—
336C23—2124no—
337C23—2124no—
338C23—2124no—
339C23—2124no—
340C23—2124no—
341C23—2124no—
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—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
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—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—1062no—
388C22—1062no—
389C22—1062no—
390C22—1062no—
391C22—1062no—
392C22—1062no—
393C22—21216no—
394C22—2121no—
395C22—2121no—
396C22—2121no—
397C22—2121no—
398C22—2121no—
399C22—2121no—
400C22—2121no—
401C22—2121no—
402C22#41×D5311yescharacteristic
403C22#41×D5311yescharacteristic
404C22C2×#1171994711yescharacteristic
405C22#1313326211yescharacteristic
406C22#1313326311yescharacteristic
407C22#1313326311yescharacteristic
408C22#1313326311yescharacteristic
409C4—41no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—2121no—
416C4—2121no—
417C4—2121no—
418C4—2121no—
419C4—2121no—
420C4—2121no—
421C2—42no—
422C2—2122no—
423C2—2122no—
424C2#1681120711yescharacteristic
425C2#1681120911yescharacteristic
426C2#1681121011yescharacteristic
427C2#1681129611yescharacteristic
428C2#1681129611yescharacteristic
429C2#1681129811yescharacteristic
430C2#1681136411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#397C2
hypercenter#397D53
hypocenterC53#2735
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. #16810560C2semidirect
2. #16810552C2semidirect
3. #16810552C2semidirect
4. #16810546C2semidirect
5. #16810545C2semidirect
6. #16810516C2semidirect
7. #16810516C2semidirect
8. #16810515C2semidirect
9. #16810515C2semidirect
10. #16810513C2semidirect
11. #16810512C2semidirect
15. C53×#397C2semidirect
17. C2×C4⋊Dic53C22semidirect
18. C2×C4⋊Dic53C22semidirect
19. C2×C4⋊Dic53C22semidirect
20. C2×#11719919C22semidirect
21. C2×#11719919C22semidirect
37. C2×C4×Dic53C22semidirect
38. C2×C4×Dic53C22semidirect
39. #13133124C22semidirect
40. #13133124C22semidirect
41. #13133124C22semidirect
42. #13133124C22semidirect
43. #13133124C22semidirect
44. C2×C4⋊C4×C53C22semidirect
48. #95×C53C22semidirect
49. #95×C53C22semidirect
50. #95×C53C22semidirect
219. #397D53semidirect
231. C53#2735semidirect
238. C2×C4⋊C4C2×D53semidirect
264. #95C2×D53semidirect
265. #95C2×D53semidirect
266. #95C2×D53semidirect