Finite Group Databasebeta

← #20277563

Subgroups — #20277563

Order 6784.

792 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133175C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D53C211yesmaximal, characteristic
3C2×C22⋊C4×D53C211yesmaximal, characteristic
4#16810517C211yesmaximal, characteristic, complemented
5#16810516C211yesmaximal, characteristic, complemented
6#16810511C211yesmaximal, characteristic, complemented
7#16810497C211yesmaximal, characteristic, complemented
8#16810494C211yesmaximal, characteristic, complemented
9#16810485C211yesmaximal, characteristic, complemented
10#16810482C211yesmaximal, characteristic, complemented
11#16810478C211yesmaximal, characteristic
12C4×#11719921C211yesmaximal, characteristic, complemented
13C4×#11719921C211yesmaximal, characteristic, complemented
14#16810470C211yesmaximal, characteristic, complemented
15C53×#388C211yesmaximal, characteristic, complemented
16C24×D53C2211yescharacteristic
17C22×C4×D53C2211yescharacteristic
18C22×C4×D53C2211yescharacteristic
19C22×C4×D53C2211yescharacteristic
20C22×C4×D53C2211yescharacteristic
21#13133175—24no—
22C2×C4⋊Dic53C2211yescharacteristic, complemented
23C2×#11719919C2211yescharacteristic, complemented
24C2×#11719919C2211yescharacteristic, complemented
25C2×C22⋊Dic53C2211yescharacteristic
26C2×C22⋊Dic53C2211yescharacteristic
27C2×C22⋊Dic53C2211yescharacteristic
28C2×#11719921C2211yescharacteristic
29C2×#11719921C2211yescharacteristic
30C2×#11719921C2211yescharacteristic
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C22⋊C4×D53—24no—
37C22⋊C4×D53C418yescomplemented
38C2×C4×Dic53C2211yescharacteristic, complemented
39C2×C4×Dic53C2211yescharacteristic, complemented
40C2×C4×Dic53C2211yescharacteristic, complemented
41#13133124C2211yescharacteristic, complemented
42#13133124C2211yescharacteristic, complemented
43#13133124C2211yescharacteristic, complemented
44#13133124C2211yescharacteristic, complemented
45#13133124C2211yescharacteristic, complemented
46#13133124C2211yescharacteristic, complemented
47C2×C4⋊C4×C53C2211yescharacteristic, complemented
48C2×C22⋊C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50C2×C22⋊C4×C53C2211yescharacteristic
51C2×C42×C53C2211yescharacteristic, complemented
52#95×C53C2211yescharacteristic, complemented
53#95×C53C2211yescharacteristic, complemented
54C24×C53C2311yescharacteristic
55C23×D53—22no—
56C23×D53—22no—
57C23×D53—22no—
58C23×D53—22no—
59C23×D53—41no—
60C23×D53—41no—
61C23×D53C2×C414yes—
62C23×D53C2311yescharacteristic
63C23×D53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C22×C4×C53C2311yescharacteristic
68C22×C4×C53C2311yescharacteristic
69C22×C4×C53C2311yescharacteristic
70C22×Dic53C2311yescharacteristic
71C22×Dic53C2311yescharacteristic
72C22×Dic53C2311yescharacteristic
73C22×Dic53C2311yescharacteristic
74C22×Dic53C2311yescharacteristic
75C22×Dic53C2311yescharacteristic
76C2×C4×D53—22no—
77C2×C4×D53—22no—
78C2×C4×D53—22no—
79C2×C4×D53—22no—
80C2×C4×D53—22no—
81C2×C4×D53—22no—
82C2×C4×D53—41no—
83C2×C4×D53—41no—
84C2×C4×D53—41no—
85C2×C4×D53—41no—
86C2×C4×D53C2×C414yescomplemented
87C2×C4×D53C2×C414yescomplemented
88C4⋊C4×C53—22no—
89C22⋊C4×C53—22no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53C2×C414yescomplemented
92C42×C53—41no—
93C22⋊Dic53—22no—
94C22⋊Dic53—41no—
95C22⋊Dic53C2×C414yescomplemented
96#11719921—22no—
97#11719921—22no—
98#11719921—22no—
99#11719921—22no—
100#11719921—22no—
101#11719921C2×C414yes—
102#11719921C2×C414yes—
103#11719921D414yes—
104C4⋊Dic53—22no—
105#11719919—41no—
106#11719919—41no—
107C4×Dic53—22no—
108C4×Dic53—22no—
109C4×Dic53—41no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53C22×C412yes—
117C23×C53C2411yescharacteristic
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—28no—
127C22×D53—28no—
128C22×D53—44no—
129C22×D53—44no—
130C22×D53—44no—
131C22×D53—44no—
132C22×D53—44no—
133C22×D53C22×C412yes—
134C22×D53C22×C412yes—
135C22×D53C2×D412yes—
136C22×D53#4112yes—
137C22×D53#4112yes—
138C22×D53#4112yes—
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×C53C22×C412yes—
155C2×C4×C53C22×C412yes—
156C2×C4×C53C2×D412yes—
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×Dic53C22×C412yes—
173C2×Dic53C22×C412yes—
174C2×Dic53C2×D412yes—
175C4×D53—28no—
176C4×D53—44no—
177C4×D53—44no—
178C4×D53—44no—
179C22×C53—22no—
180C22×C53—22no—
181C22×C53—22no—
182C22×C53—41no—
183C22×C53—41no—
184C22×C53—41no—
185C22×C53—41no—
186C22×C53C23×C411yescharacteristic
187C22×C53C22×D411yescharacteristic
188C22×C53C2×#4111yescharacteristic
189C22×C53C2×#4111yescharacteristic
190C22×C53C2×#4111yescharacteristic
191C22×C53H(5,𝔽2)11yescharacteristic
192C22×C53H(5,𝔽2)11yescharacteristic
193C2×D53—22no—
194C2×D53—22no—
195C2×D53—22no—
196C2×D53—22no—
197C2×D53—22no—
198C2×D53—22no—
199C2×D53—22no—
200C2×D53—22no—
201C2×D53—22no—
202C2×D53—22no—
203C2×D53—22no—
204C2×D53—22no—
205C2×D53—416no—
206C2×D53#11714yescomplemented
207C2×D53C4×D414yes—
208C4×C53—22no—
209C4×C53—22no—
210C4×C53—22no—
211C4×C53—41no—
212C4×C53—41no—
213C4×C53—41no—
214Dic53—22no—
215Dic53—22no—
216Dic53—22no—
217Dic53—41no—
218Dic53—41no—
219Dic53—41no—
220#2342—531nomaximal, Sylow
221C2×C53—42no—
222C2×C53C2×#11711yescharacteristic
223C2×C53C2×C4×D411yescharacteristic
224C2×C53#51811yescharacteristic
225C2×C53#54611yescharacteristic
226C2×C53#54611yescharacteristic
227C2×C53#55111yescharacteristic
228C2×C53#55111yescharacteristic
229D53—24no—
230D53—24no—
231C2×#117—531no—
232C22×C22⋊C4—531no—
233C22×C22⋊C4—531no—
234#388—531no—
235#388—531no—
236#388—531no—
237#388D5311yescharacteristic, complemented
238#385—531no—
239#385—531no—
240#381—531no—
241#380—531no—
242#379—531no—
243C4×C22⋊C4—531no—
244C4×C22⋊C4—531no—
245C4×C22⋊C4—531no—
246C53#234211yescharacteristic, complemented, Sylow
247C25—531no—
248C23×C4—531no—
249C23×C4—531no—
250C23×C4—531no—
251C23×C4—531no—
252#117—1064no—
253C2×C4⋊C4—531no—
254C2×C4⋊C4—531no—
255C2×C4⋊C4—531no—
256C2×C4⋊C4C2×D5311yescharacteristic, complemented
257C2×C22⋊C4—531no—
258C2×C22⋊C4—531no—
259C2×C22⋊C4—531no—
260C2×C22⋊C4—531no—
261C2×C22⋊C4—538no—
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—1064no—
270C2×C22⋊C4C2×D5311yescharacteristic
271C2×C22⋊C4C2×D5311yescharacteristic
272C2×C22⋊C4C2×D5311yescharacteristic
273C2×C42—531no—
274C2×C42—531no—
275C2×C42—531no—
276C2×C42C2×D5311yescharacteristic, complemented
277#95—531no—
278#95—531no—
279#95—531no—
280#95—531no—
281#95—531no—
282#95—531no—
283#95C2×D5311yescharacteristic, complemented
284#95C2×D5311yescharacteristic, complemented
285C24—531no—
286C24—531no—
287C24—534no—
288C24—1062no—
289C24—1062no—
290C24—1062no—
291C24—1062no—
292C24—2121no—
293C24—2121no—
294C24C22×D5311yescharacteristic
295C22×C4—531no—
296C22×C4—531no—
297C22×C4—531no—
298C22×C4—534no—
299C22×C4—534no—
300C22×C4—531no—
301C22×C4—531no—
302C22×C4—531no—
303C22×C4—1062no—
304C22×C4—1062no—
305C22×C4—1062no—
306C22×C4—1062no—
307C22×C4—1062no—
308C22×C4—1062no—
309C22×C4—2121no—
310C22×C4—2121no—
311C22×C4—2121no—
312C22×C4—2121no—
313C22×C4C22×D5311yescharacteristic
314C22×C4C22×D5311yescharacteristic
315C22×C4C22×D5311yescharacteristic
316C22×C4C22×D5311yescharacteristic
317C22×C4C22×D5311yescharacteristic
318C22×C4C22×D5311yescharacteristic
319C4⋊C4—22no—
320C4⋊C4—1062no—
321C4⋊C4—2121no—
322C4⋊C4—2121no—
323C22⋊C4—22no—
324C22⋊C4—41no—
325C22⋊C4—534no—
326C22⋊C4—534no—
327C22⋊C4—534no—
328C22⋊C4—534no—
329C22⋊C4—1062no—
330C22⋊C4—1062no—
331C22⋊C4—1062no—
332C22⋊C4—1062no—
333C22⋊C4—1062no—
334C22⋊C4—1062no—
335C22⋊C4—2121no—
336C22⋊C4C4×D5314yescomplemented
337C42—41no—
338C42—1062no—
339C42—1062no—
340C42—2121no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—532no—
348C23—532no—
349C23—532no—
350C23—532no—
351C23—532no—
352C23—532no—
353C23—1068no—
354C23—1068no—
355C23—1061no—
356C23—1061no—
357C23—1061no—
358C23—1061no—
359C23—1061no—
360C23—1061no—
361C23—1061no—
362C23—1061no—
363C23—2124no—
364C23—2124no—
365C23—2124no—
366C23—2124no—
367C23—2124no—
368C23C2×C4×D5312yes—
369C23C23×D5311yescharacteristic
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—532no—
387C2×C4—532no—
388C2×C4—1068no—
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—1061no—
404C2×C4—2124no—
405C2×C4—2124no—
406C2×C4—2124no—
407C2×C4C2×C4×D5312yes—
408C2×C4C2×C4×D5312yes—
409C2×C4D4×D5312yes—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—534no—
418C22—534no—
419C22—1062no—
420C22—1062no—
421C22—1062no—
422C22—1062no—
423C22—1062no—
424C22—1062no—
425C22—1062no—
426C22—1062no—
427C22—1062no—
428C22—1062no—
429C22—1062no—
430C22—1062no—
431C22—21216no—
432C22C22×C4×D5311yescharacteristic
433C22C2×D4×D5311yescharacteristic
434C22#41×D5311yescharacteristic
435C22#41×D5311yescharacteristic
436C22#41×D5311yescharacteristic
437C22#1313326211yescharacteristic
438C22#1313326311yescharacteristic
439C4—22no—
440C4—22no—
441C4—22no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—1062no—
446C4—1062no—
447C4—1062no—
448C4—2121no—
449C4—2121no—
450C4—2121no—
451C2—42no—
452C2—1064no—
453C2—1064no—
454C2#117×D5311yescharacteristic
455C2C4×D4×D5311yescharacteristic
456C2#1681109511yescharacteristic
457C2#1681126211yescharacteristic
458C2#1681126611yescharacteristic
459C2#1681129811yescharacteristic
460C2#1681130111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC53×#388C2
socleC23×C53C24
hypercenter#388D53
hypocenterC53#2342
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#13133175C2semidirect
4. #16810517C2semidirect
5. #16810516C2semidirect
6. #16810511C2semidirect
7. #16810497C2semidirect
8. #16810494C2semidirect
9. #16810485C2semidirect
10. #16810482C2semidirect
12. C4×#11719921C2semidirect
13. C4×#11719921C2semidirect
14. #16810470C2semidirect
15. C53×#388C2semidirect
22. C2×C4⋊Dic53C22semidirect
23. C2×#11719919C22semidirect
24. C2×#11719919C22semidirect
37. C22⋊C4×D53C4semidirect
38. C2×C4×Dic53C22semidirect
39. C2×C4×Dic53C22semidirect
40. C2×C4×Dic53C22semidirect
41. #13133124C22semidirect
42. #13133124C22semidirect
43. #13133124C22semidirect
44. #13133124C22semidirect
45. #13133124C22semidirect
46. #13133124C22semidirect
47. C2×C4⋊C4×C53C22semidirect
51. C2×C42×C53C22semidirect
52. #95×C53C22semidirect
53. #95×C53C22semidirect
86. C2×C4×D53C2×C4semidirect
87. C2×C4×D53C2×C4semidirect
91. C22⋊C4×C53C2×C4semidirect
95. C22⋊Dic53C2×C4semidirect
206. C2×D53#117semidirect
237. #388D53semidirect
246. C53#2342semidirect
256. C2×C4⋊C4C2×D53semidirect
276. C2×C42C2×D53semidirect
283. #95C2×D53semidirect
284. #95C2×D53semidirect
336. C22⋊C4C4×D53semidirect