Finite Group Databasebeta

← #20280489

Subgroups — #20280489

Order 6784.

702 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133187C211yesmaximal, characteristic
2C2×C22⋊C4×D53C211yesmaximal, characteristic, complemented
3#16810552C211yesmaximal, characteristic, complemented
4#16810551C211yesmaximal, characteristic, complemented
5#16810550C211yesmaximal, characteristic, complemented
6#16810541C211yesmaximal, characteristic, complemented
7#16810540C211yesmaximal, characteristic, complemented
8#16810539C211yesmaximal, characteristic, complemented
9#16810538C211yesmaximal, characteristic, complemented
10#16810501C211yesmaximal, characteristic, complemented
11#16810501C211yesmaximal, characteristic, complemented
12#16810497C211yesmaximal, characteristic, complemented
13#16810494C211yesmaximal, characteristic, complemented
14#16810478C211yesmaximal, characteristic, complemented
15C53×#397C211yesmaximal, characteristic, complemented
16C24×D53C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C22×D212C2211yescharacteristic
19C22×C22⋊D53C2211yescharacteristic
20C22×C4×D53C2211yescharacteristic
21C22×C4×D53C2211yescharacteristic
22#13133187—24no—
23C2×C4⋊Dic53C2211yescharacteristic, complemented
24C2×#11719919C2211yescharacteristic, complemented
25C2×#11719919C2211yescharacteristic, complemented
26C2×C22⋊Dic53C2211yescharacteristic
27C2×C22⋊Dic53C2211yescharacteristic
28C2×#11719921C2211yescharacteristic
29C2×#11719921C2211yescharacteristic
30C2×#11719921C2211yescharacteristic
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic, complemented
37C2×#11719921C2211yescharacteristic, complemented
38C2×#11719921C2211yescharacteristic, complemented
39C2×#11719921C2211yescharacteristic, complemented
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×C22⋊C4×C53C2211yescharacteristic
48C2×C22⋊C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50#95×C53C2211yescharacteristic, complemented
51#95×C53C2211yescharacteristic, complemented
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×D53D414yes—
61C23×D53C2311yescharacteristic
62C23×D53C2311yescharacteristic
63C23×D53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C22×C4×C53C2311yescharacteristic
68C22×C4×C53C2311yescharacteristic
69C22×C4×C53C2311yescharacteristic
70C2×C22⋊D53—22no—
71C2×C22⋊D53—41no—
72C2×C22⋊D53—41no—
73C22×Dic53C2311yescharacteristic
74C22×Dic53C2311yescharacteristic
75C22×Dic53C2311yescharacteristic
76C22×Dic53C2311yescharacteristic
77C22×Dic53C2311yescharacteristic
78C2×D212—22no—
79C2×D212—22no—
80C2×D212—22no—
81C2×D212—22no—
82C2×D212—41no—
83C2×D212—41no—
84C2×C4×D53—22no—
85C2×C4×D53—22no—
86C2×C4×D53—41no—
87C2×C4×D53—41no—
88C2×C4×D53—41no—
89C2×C4×D53—41no—
90C4⋊C4×C53—41no—
91C22⋊C4×C53—22no—
92C22⋊C4×C53—22no—
93C22⋊C4×C53—41no—
94C22⋊Dic53—22no—
95C22⋊Dic53—41no—
96#11719921—22no—
97#11719921—22no—
98#11719921—22no—
99#11719921—22no—
100#11719921—22no—
101#11719921—22no—
102#11719921—41no—
103#11719921—41no—
104#11719921—41no—
105#11719921—41no—
106#11719921—41no—
107#11719921—41no—
108C4⋊Dic53—41no—
109#11719919—41no—
110#11719919—41no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53C2×D412yes—
118C23×C53C2411yescharacteristic
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—21no—
128C22×D53—21no—
129C22×D53—28no—
130C22×D53—21no—
131C22×D53—21no—
132C22×D53—21no—
133C22×D53—21no—
134C22×D53—21no—
135C22×D53—21no—
136C22×D53—21no—
137C22×D53—44no—
138C22×D53—44no—
139C22×D53—44no—
140C22×D53—44no—
141C22×D53—44no—
142C22×D53—44no—
143C22×D53C2×D412yes—
144C22×D53C2×D412yes—
145C22×D53#4112yes—
146C22×D53#4112yes—
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×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C22⋊D53—44no—
166C2×Dic53—21no—
167C2×Dic53—21no—
168C2×Dic53—21no—
169C2×Dic53—21no—
170C2×Dic53—21no—
171C2×Dic53—21no—
172C2×Dic53—21no—
173C2×Dic53—21no—
174C2×Dic53—21no—
175C2×Dic53—21no—
176C2×Dic53—21no—
177C2×Dic53—21no—
178C2×Dic53—21no—
179C2×Dic53—21no—
180C2×Dic53—21no—
181D212—44no—
182D212—44no—
183C4×D53—44no—
184C4×D53—44no—
185C22×C53—22no—
186C22×C53—22no—
187C22×C53—22no—
188C22×C53—41no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53C22×D411yescharacteristic
193C22×C53C2×#4111yescharacteristic
194C22×C53C2×#4111yescharacteristic
195C22×C53H(5,𝔽2)11yescharacteristic
196C22×C53H(5,𝔽2)11yescharacteristic
197C22×C53H(5,𝔽2)11yescharacteristic
198C22×C53H(5,𝔽2)11yescharacteristic
199C2×D53—22no—
200C2×D53—22no—
201C2×D53—22no—
202C2×D53—22no—
203C2×D53—22no—
204C2×D53—22no—
205C2×D53—22no—
206C2×D53—22no—
207C2×D53—22no—
208C2×D53—416no—
209C2×D53—41no—
210C2×D53—41no—
211C2×D53—41no—
212C2×D53—41no—
213C2×D53—41no—
214C2×D53—41no—
215C2×D53—41no—
216C2×D53—41no—
217C2×D53—41no—
218C2×D53—41no—
219C2×D53—41no—
220C2×D53#12314yes—
221C4×C53—41no—
222C4×C53—41no—
223C4×C53—41no—
224C4×C53—41no—
225C4×C53—41no—
226C4×C53—41no—
227Dic53—41no—
228Dic53—41no—
229Dic53—41no—
230Dic53—41no—
231Dic53—41no—
232#2622—531nomaximal, Sylow
233C2×C53—42no—
234C2×C53C2×#12311yescharacteristic
235C2×C53#53411yescharacteristic
236C2×C53#53411yescharacteristic
237C2×C53#53811yescharacteristic
238C2×C53#53811yescharacteristic
239C2×C53#53811yescharacteristic
240C2×C53#53811yescharacteristic
241D53—24no—
242D53—42no—
243D53—81no—
244C2×#120—531no—
245C22×C22⋊C4—531no—
246#397—531no—
247#397—531no—
248#397—531no—
249#397D5311yescharacteristic, complemented
250#394—531no—
251#394—531no—
252#394—531no—
253#394—531no—
254#386—531no—
255#386—531no—
256#385—531no—
257#385—531no—
258#379—531no—
259C53#262211yescharacteristic, complemented, Sylow
260C25—531no—
261C22×D4—531no—
262C22×D4—531no—
263C22×D4—531no—
264C23×C4—531no—
265C23×C4—531no—
266#120—1064no—
267C2×C4⋊C4—531no—
268C2×C4⋊C4—531no—
269C2×C4⋊C4—531no—
270C2×C4⋊C4C2×D5311yescharacteristic, complemented
271C2×C22⋊C4—531no—
272C2×C22⋊C4—531no—
273C2×C22⋊C4—531no—
274C2×C22⋊C4—531no—
275C2×C22⋊C4—531no—
276C2×C22⋊C4—531no—
277C2×C22⋊C4—531no—
278C2×C22⋊C4—531no—
279C2×C22⋊C4—531no—
280C2×C22⋊C4—531no—
281C2×C22⋊C4—531no—
282C2×C22⋊C4—531no—
283C2×C22⋊C4—531no—
284C2×C22⋊C4—531no—
285C2×C22⋊C4—1064no—
286C2×C22⋊C4C2×D5311yescharacteristic
287C2×C22⋊C4C2×D5311yescharacteristic
288C2×C22⋊C4C2×D5311yescharacteristic
289#95—531no—
290#95—531no—
291#95—531no—
292#95—531no—
293#95—531no—
294#95C2×D5311yescharacteristic, complemented
295#95C2×D5311yescharacteristic, complemented
296#95C2×D5311yescharacteristic, complemented
297C24—531no—
298C24—531no—
299C24—534no—
300C24—531no—
301C24—1062no—
302C24—1062no—
303C24—2121no—
304C24—2121no—
305C24—2121no—
306C24—2121no—
307C24C22×D5311yescharacteristic
308C2×D4—1062no—
309C2×D4—1062no—
310C2×D4—1062no—
311C2×D4—1062no—
312C2×D4—1062no—
313C2×D4—2121no—
314C2×D4—2121no—
315C2×D4—2121no—
316C2×D4—2121no—
317C22×C4—531no—
318C22×C4—531no—
319C22×C4—531no—
320C22×C4—531no—
321C22×C4—531no—
322C22×C4—1062no—
323C22×C4—1062no—
324C22×C4—2121no—
325C22×C4—2121no—
326C22×C4—2121no—
327C22×C4—2121no—
328C22×C4C22×D5311yescharacteristic
329C22×C4C22×D5311yescharacteristic
330C22×C4C22×D5311yescharacteristic
331C22×C4C22×D5311yescharacteristic
332C22×C4C22×D5311yescharacteristic
333C22×C4C22×D5311yescharacteristic
334C4⋊C4—41no—
335C4⋊C4—2121no—
336C4⋊C4—2121no—
337C4⋊C4—2121no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—41no—
341C22⋊C4—1062no—
342C22⋊C4—1062no—
343C22⋊C4—1062no—
344C22⋊C4—1062no—
345C22⋊C4—1062no—
346C22⋊C4—1062no—
347C22⋊C4—1062no—
348C22⋊C4—2121no—
349C22⋊C4—2121no—
350C22⋊C4—2121no—
351C22⋊C4—2121no—
352C22⋊C4—2121no—
353C22⋊C4—2121no—
354C22⋊C4—2121no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—532no—
362C23—532no—
363C23—532no—
364C23—532no—
365C23—1068no—
366C23—1061no—
367C23—1061no—
368C23—1061no—
369C23—1061no—
370C23—1061no—
371C23—1061no—
372C23—1061no—
373C23—1061no—
374C23—1061no—
375C23—1061no—
376C23—1061no—
377C23—1061no—
378C23—1061no—
379C23—1061no—
380C23—1061no—
381C23—1061no—
382C23—1061no—
383C23—2124no—
384C23—2124no—
385C23—2124no—
386C23—2124no—
387C23—2124no—
388C23—2124no—
389C23D4×D5312yes—
390C23C23×D5311yescharacteristic
391D4—2124no—
392D4—2124no—
393D4—2124no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—1061no—
413C2×C4—1061no—
414C2×C4—1061no—
415C2×C4—1061no—
416C2×C4—1061no—
417C2×C4—1061no—
418C2×C4—1061no—
419C2×C4—1061no—
420C2×C4—1061no—
421C2×C4—1061no—
422C2×C4—1061no—
423C2×C4—1061no—
424C2×C4—1061no—
425C2×C4—1061no—
426C2×C4—1061no—
427C2×C4—2124no—
428C2×C4—2124no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—534no—
437C22—1062no—
438C22—1062no—
439C22—1062no—
440C22—1062no—
441C22—1062no—
442C22—1062no—
443C22—1062no—
444C22—1062no—
445C22—1062no—
446C22—21216no—
447C22—2121no—
448C22—2121no—
449C22—2121no—
450C22—2121no—
451C22—2121no—
452C22—2121no—
453C22—2121no—
454C22—2121no—
455C22—2121no—
456C22—2121no—
457C22—2121no—
458C22C2×D4×D5311yescharacteristic
459C22#41×D5311yescharacteristic
460C22#41×D5311yescharacteristic
461C22#1313326211yescharacteristic
462C22#1313326311yescharacteristic
463C22#1313326311yescharacteristic
464C22#1313326311yescharacteristic
465C4—41no—
466C4—41no—
467C4—41no—
468C4—41no—
469C4—41no—
470C4—41no—
471C4—2121no—
472C4—2121no—
473C4—2121no—
474C4—2121no—
475C4—2121no—
476C2—42no—
477C2—1064no—
478C2—4241no—
479C2—2122no—
480C2#123×D5311yescharacteristic
481C2#1681118111yescharacteristic
482C2#1681118411yescharacteristic
483C2#1681120711yescharacteristic
484C2#1681120911yescharacteristic
485C2#1681121211yescharacteristic
486C2#1681121211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#397C2
hypercenter#397D53
hypocenterC53#2622
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊C4×D53C2semidirect
3. #16810552C2semidirect
4. #16810551C2semidirect
5. #16810550C2semidirect
6. #16810541C2semidirect
7. #16810540C2semidirect
8. #16810539C2semidirect
9. #16810538C2semidirect
10. #16810501C2semidirect
11. #16810501C2semidirect
12. #16810497C2semidirect
13. #16810494C2semidirect
14. #16810478C2semidirect
15. C53×#397C2semidirect
23. C2×C4⋊Dic53C22semidirect
24. C2×#11719919C22semidirect
25. C2×#11719919C22semidirect
36. C2×#11719921C22semidirect
37. C2×#11719921C22semidirect
38. C2×#11719921C22semidirect
39. C2×#11719921C22semidirect
41. #13133124C22semidirect
42. #13133124C22semidirect
43. #13133124C22semidirect
44. #13133124C22semidirect
45. #13133124C22semidirect
46. C2×C4⋊C4×C53C22semidirect
50. #95×C53C22semidirect
51. #95×C53C22semidirect
52. #95×C53C22semidirect
249. #397D53semidirect
259. C53#2622semidirect
270. C2×C4⋊C4C2×D53semidirect
294. #95C2×D53semidirect
295. #95C2×D53semidirect
296. #95C2×D53semidirect