Finite Group Databasebeta

← #20280525

Subgroups — #20280525

Order 6784.

668 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133195C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D53C211yesmaximal, characteristic
3#16810559C211yesmaximal, characteristic, complemented
4#16810558C211yesmaximal, characteristic, complemented
5#16810546C211yesmaximal, characteristic, complemented
6#16810545C211yesmaximal, characteristic, complemented
7#16810545C211yesmaximal, characteristic, complemented
8#16810536C211yesmaximal, characteristic, complemented
9#16810535C211yesmaximal, characteristic, complemented
10#16810497C211yesmaximal, characteristic, complemented
11#16810494C211yesmaximal, characteristic, complemented
12#16810480C211yesmaximal, characteristic, complemented
13#16810478C211yesmaximal, characteristic
14#16810478C211yesmaximal, characteristic
15C53×#396C211yesmaximal, characteristic, complemented
16C24×D53C2211yescharacteristic
17C22×Dic106C2211yescharacteristic, complemented
18C22×C4×D53C2211yescharacteristic
19C22×C4×D53C2211yescharacteristic
20#13133195—24no—
21C2×C4⋊Dic53C2211yescharacteristic, complemented
22C2×C4⋊Dic53C2211yescharacteristic, complemented
23C2×#11719919C2211yescharacteristic, complemented
24C2×#11719919C2211yescharacteristic, complemented
25C2×#11719919C2211yescharacteristic, complemented
26C2×#11719919C2211yescharacteristic, complemented
27C2×C22⋊Dic53C2211yescharacteristic
28C2×C22⋊Dic53C2211yescharacteristic
29C2×C22⋊Dic53C2211yescharacteristic
30C2×#11719921C2211yescharacteristic
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C2×#11719921C2211yescharacteristic
40C22⋊C4×D53—24no—
41#13133124C2211yescharacteristic, complemented
42#13133124C2211yescharacteristic, complemented
43#13133124C2211yescharacteristic, complemented
44#13133124C2211yescharacteristic, complemented
45#13133124C2211yescharacteristic, complemented
46C2×C4⋊C4×C53C2211yescharacteristic, complemented
47C2×C4⋊C4×C53C2211yescharacteristic, complemented
48C2×C4⋊C4×C53C2211yescharacteristic, complemented
49C2×C22⋊C4×C53C2211yescharacteristic
50C2×C22⋊C4×C53C2211yescharacteristic
51C2×C22⋊C4×C53C2211yescharacteristic
52#95×C53C2211yescharacteristic, complemented
53C24×C53C2311yescharacteristic
54C23×D53—22no—
55C23×D53—22no—
56C23×D53—41no—
57C23×D53—41no—
58C23×D53—41no—
59C23×D53—41no—
60C23×D53Q814yescomplemented
61C23×D53C2311yescharacteristic
62C23×D53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C22×C4×C53C2311yescharacteristic
68C22×C4×C53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C22×Dic53C2311yescharacteristic
71C22×Dic53C2311yescharacteristic
72C22×Dic53C2311yescharacteristic
73C22×Dic53C2311yescharacteristic
74C22×Dic53C2311yescharacteristic
75C2×C4×D53—22no—
76C2×C4×D53—41no—
77C2×C4×D53—41no—
78C2×C4×D53—41no—
79C2×C4×D53—41no—
80C2×C4×D53D414yes—
81C2×Dic106—22no—
82C2×Dic106—41no—
83C2×Dic106—41no—
84C4⋊C4×C53—22no—
85C4⋊C4×C53—41no—
86C4⋊C4×C53—41no—
87C22⋊C4×C53—22no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C22⋊Dic53—22no—
91C22⋊Dic53—41no—
92C22⋊Dic53—41no—
93#11719921—22no—
94#11719921—22no—
95#11719921—22no—
96#11719921—22no—
97#11719921—22no—
98#11719921—22no—
99#11719921—41no—
100#11719921—41no—
101#11719921—41no—
102#11719921—41no—
103C4⋊Dic53—41no—
104C4⋊Dic53—41no—
105#11719919—22no—
106#11719919—22no—
107#11719919—41no—
108#11719919—41no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53C2×Q812yes—
116C23×C53C2411yescharacteristic
117C22×D53—21no—
118C22×D53—21no—
119C22×D53—21no—
120C22×D53—21no—
121C22×D53—21no—
122C22×D53—21no—
123C22×D53—21no—
124C22×D53—21no—
125C22×D53—21no—
126C22×D53—21no—
127C22×D53—28no—
128C22×D53—44no—
129C22×D53—44no—
130C22×D53—44no—
131C22×D53—44no—
132C22×D53—44no—
133C22×D53—44no—
134C22×D53C2×D412yes—
135C22×D53C2×Q812yes—
136C22×D53C2×Q812yes—
137C22×D53#4112yes—
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×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53C2×D412yes—
156C2×Dic53—21no—
157C2×Dic53—21no—
158C2×Dic53—21no—
159C2×Dic53—21no—
160C2×Dic53—21no—
161C2×Dic53—21no—
162C2×Dic53—21no—
163C2×Dic53—21no—
164C2×Dic53—21no—
165C2×Dic53—21no—
166C2×Dic53—21no—
167C2×Dic53—21no—
168C2×Dic53—21no—
169C2×Dic53—21no—
170C2×Dic53—21no—
171C2×Dic53—21no—
172C2×Dic53—21no—
173C2×Dic53C2×D412yes—
174C4×D53—44no—
175C4×D53—44no—
176Dic106—44no—
177C22×C53—22no—
178C22×C53—22no—
179C22×C53—22no—
180C22×C53—41no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53—41no—
184C22×C53C22×D411yescharacteristic
185C22×C53C22×Q811yescharacteristic
186C22×C53C2×#4111yescharacteristic
187C22×C53H(5,𝔽2)11yescharacteristic
188C22×C53H(5,𝔽2)11yescharacteristic
189C22×C53H(5,𝔽2)11yescharacteristic
190C22×C53H(5,𝔽2)11yescharacteristic
191C2×D53—22no—
192C2×D53—22no—
193C2×D53—22no—
194C2×D53—22no—
195C2×D53—22no—
196C2×D53—22no—
197C2×D53—22no—
198C2×D53—22no—
199C2×D53—22no—
200C2×D53—416no—
201C2×D53—41no—
202C2×D53—41no—
203C2×D53—41no—
204C2×D53—41no—
205C2×D53C22⋊Q814yes—
206C4×C53—22no—
207C4×C53—41no—
208C4×C53—41no—
209C4×C53—41no—
210C4×C53—41no—
211C4×C53—41no—
212Dic53—22no—
213Dic53—41no—
214Dic53—41no—
215Dic53—41no—
216Dic53—41no—
217Dic53—41no—
218#2626—531nomaximal, Sylow
219C2×C53—42no—
220C2×C53C2×C22⋊Q811yescharacteristic
221C2×C53#53411yescharacteristic
222C2×C53#53511yescharacteristic
223C2×C53#53811yescharacteristic
224C2×C53#53811yescharacteristic
225C2×C53#54311yescharacteristic
226C2×C53#54311yescharacteristic
227D53—24no—
228D53—42no—
229C2×C22⋊Q8—531no—
230C22×C22⋊C4—531no—
231#399—531no—
232#399—531no—
233#396—531no—
234#396—531no—
235#396—531no—
236#396D5311yescharacteristic, complemented
237#393—531no—
238#393—531no—
239#385—531no—
240#385—531no—
241#380—531no—
242#379—531no—
243#379—531no—
244C53#262611yescharacteristic, complemented, Sylow
245C25—531no—
246C22×Q8—531no—
247C23×C4—531no—
248C23×C4—531no—
249C22⋊Q8—1064no—
250C2×C4⋊C4—531no—
251C2×C4⋊C4—531no—
252C2×C4⋊C4—531no—
253C2×C4⋊C4—531no—
254C2×C4⋊C4—531no—
255C2×C4⋊C4—531no—
256C2×C4⋊C4C2×D5311yescharacteristic, complemented
257C2×C4⋊C4C2×D5311yescharacteristic, complemented
258C2×C4⋊C4C2×D5311yescharacteristic, complemented
259C2×C22⋊C4—531no—
260C2×C22⋊C4—531no—
261C2×C22⋊C4—531no—
262C2×C22⋊C4—531no—
263C2×C22⋊C4—531no—
264C2×C22⋊C4—531no—
265C2×C22⋊C4—531no—
266C2×C22⋊C4—531no—
267C2×C22⋊C4—531no—
268C2×C22⋊C4—531no—
269C2×C22⋊C4—531no—
270C2×C22⋊C4—531no—
271C2×C22⋊C4—531no—
272C2×C22⋊C4—1064no—
273C2×C22⋊C4C2×D5311yescharacteristic
274C2×C22⋊C4C2×D5311yescharacteristic
275C2×C22⋊C4C2×D5311yescharacteristic
276#95—531no—
277#95—531no—
278#95—531no—
279#95—531no—
280#95—531no—
281#95C2×D5311yescharacteristic, complemented
282C24—531no—
283C24—531no—
284C24—534no—
285C24—1062no—
286C24—1062no—
287C24—2121no—
288C24—2121no—
289C24—2121no—
290C24—2121no—
291C24C22×D5311yescharacteristic
292C2×Q8—1062no—
293C2×Q8—2121no—
294C2×Q8—2121no—
295C22×C4—531no—
296C22×C4—534no—
297C22×C4—531no—
298C22×C4—531no—
299C22×C4—531no—
300C22×C4—531no—
301C22×C4—531no—
302C22×C4—1062no—
303C22×C4—2121no—
304C22×C4—2121no—
305C22×C4—2121no—
306C22×C4—2121no—
307C22×C4C22×D5311yescharacteristic
308C22×C4C22×D5311yescharacteristic
309C22×C4C22×D5311yescharacteristic
310C22×C4C22×D5311yescharacteristic
311C22×C4C22×D5311yescharacteristic
312C22×C4C22×D5311yescharacteristic
313C4⋊C4—22no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—1062no—
317C4⋊C4—1062no—
318C4⋊C4—2121no—
319C4⋊C4—2121no—
320C4⋊C4—2121no—
321C4⋊C4—2121no—
322C22⋊C4—22no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—1062no—
326C22⋊C4—1062no—
327C22⋊C4—1062no—
328C22⋊C4—1062no—
329C22⋊C4—1062no—
330C22⋊C4—1062no—
331C22⋊C4—1062no—
332C22⋊C4—2121no—
333C22⋊C4—2121no—
334C22⋊C4—2121no—
335C22⋊C4—2121no—
336C22⋊C4—2121no—
337C22⋊C4—2121no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—532no—
345C23—532no—
346C23—532no—
347C23—532no—
348C23—1068no—
349C23—1061no—
350C23—1061no—
351C23—1061no—
352C23—1061no—
353C23—1061no—
354C23—1061no—
355C23—1061no—
356C23—1061no—
357C23—1061no—
358C23—1061no—
359C23—2124no—
360C23—2124no—
361C23—2124no—
362C23—2124no—
363C23—2124no—
364C23—2124no—
365C23Q8×D5312yes—
366C23C23×D5311yescharacteristic
367Q8—2124no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—532no—
386C2×C4—1061no—
387C2×C4—1061no—
388C2×C4—1061no—
389C2×C4—1061no—
390C2×C4—1061no—
391C2×C4—1061no—
392C2×C4—1061no—
393C2×C4—1061no—
394C2×C4—1061no—
395C2×C4—1061no—
396C2×C4—1061no—
397C2×C4—1061no—
398C2×C4—1061no—
399C2×C4—1061no—
400C2×C4—1061no—
401C2×C4—1061no—
402C2×C4—1061no—
403C2×C4—2124no—
404C2×C4—2124no—
405C2×C4D4×D5312yes—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—534no—
414C22—1062no—
415C22—1062no—
416C22—1062no—
417C22—1062no—
418C22—1062no—
419C22—1062no—
420C22—1062no—
421C22—1062no—
422C22—1062no—
423C22—21216no—
424C22—2121no—
425C22—2121no—
426C22—2121no—
427C22—2121no—
428C22C2×D4×D5311yescharacteristic
429C22C2×Q8×D5311yescharacteristic
430C22#41×D5311yescharacteristic
431C22#1313326211yescharacteristic
432C22#1313326211yescharacteristic
433C22#1313326311yescharacteristic
434C22#1313326311yescharacteristic
435C4—22no—
436C4—41no—
437C4—41no—
438C4—41no—
439C4—41no—
440C4—41no—
441C4—1062no—
442C4—2121no—
443C4—2121no—
444C4—2121no—
445C4—2121no—
446C4—2121no—
447C2—42no—
448C2—1064no—
449C2—2122no—
450C2C22⋊Q8×D5311yescharacteristic
451C2#1681118111yescharacteristic
452C2#1681118911yescharacteristic
453C2#1681120511yescharacteristic
454C2#1681121211yescharacteristic
455C2#1681124811yescharacteristic
456C2#1681124811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#396C2
hypercenter#396D53
hypocenterC53#2626
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#13133195C2semidirect
3. #16810559C2semidirect
4. #16810558C2semidirect
5. #16810546C2semidirect
6. #16810545C2semidirect
7. #16810545C2semidirect
8. #16810536C2semidirect
9. #16810535C2semidirect
10. #16810497C2semidirect
11. #16810494C2semidirect
12. #16810480C2semidirect
15. C53×#396C2semidirect
17. C22×Dic106C22semidirect
21. C2×C4⋊Dic53C22semidirect
22. C2×C4⋊Dic53C22semidirect
23. C2×#11719919C22semidirect
24. C2×#11719919C22semidirect
25. C2×#11719919C22semidirect
26. C2×#11719919C22semidirect
41. #13133124C22semidirect
42. #13133124C22semidirect
43. #13133124C22semidirect
44. #13133124C22semidirect
45. #13133124C22semidirect
46. C2×C4⋊C4×C53C22semidirect
47. C2×C4⋊C4×C53C22semidirect
48. C2×C4⋊C4×C53C22semidirect
52. #95×C53C22semidirect
60. C23×D53Q8semidirect
236. #396D53semidirect
244. C53#2626semidirect
256. C2×C4⋊C4C2×D53semidirect
257. C2×C4⋊C4C2×D53semidirect
258. C2×C4⋊C4C2×D53semidirect
281. #95C2×D53semidirect