Finite Group Databasebeta

← #20277855

Subgroups — #20277855

Order 6784.

688 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133187C211yesmaximal, characteristic
2#16810525C211yesmaximal, characteristic, complemented
3#16810524C211yesmaximal, characteristic, complemented
4#16810523C211yesmaximal, characteristic, complemented
5#16810516C211yesmaximal, characteristic, complemented
6#16810515C211yesmaximal, characteristic, complemented
7#16810514C211yesmaximal, characteristic, complemented
8#16810513C211yesmaximal, characteristic, complemented
9#16810512C211yesmaximal, characteristic, complemented
10#16810501C211yesmaximal, characteristic, complemented
11#16810501C211yesmaximal, characteristic, complemented
12#16810499C211yesmaximal, characteristic, complemented
13#16810478C211yesmaximal, characteristic, complemented
14#16810478C211yesmaximal, characteristic, complemented
15C53×#388C211yesmaximal, characteristic, complemented
16C24×D53C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C22×D212C2211yescharacteristic
19C22×C22⋊D53C2211yescharacteristic
20C22×C4×D53C2211yescharacteristic, complemented
21#13133187C418yescomplemented
22C2×C4⋊Dic53C2211yescharacteristic, complemented
23C2×#11719919C2211yescharacteristic, complemented
24C2×C22⋊Dic53C2211yescharacteristic
25C2×C22⋊Dic53C2211yescharacteristic
26C2×#11719921C2211yescharacteristic
27C2×#11719921C2211yescharacteristic
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×C4×Dic53C2211yescharacteristic, complemented
40C2×C4×Dic53C2211yescharacteristic, complemented
41#13133124C2211yescharacteristic, complemented
42#13133124C2211yescharacteristic, complemented
43#13133124C2211yescharacteristic, complemented
44#13133124C2211yescharacteristic, complemented
45C2×C4⋊C4×C53C2211yescharacteristic, complemented
46C2×C22⋊C4×C53C2211yescharacteristic
47C2×C22⋊C4×C53C2211yescharacteristic
48C2×C22⋊C4×C53C2211yescharacteristic
49C2×C42×C53C2211yescharacteristic, complemented
50#95×C53C2211yescharacteristic, complemented
51#95×C53C2211yescharacteristic, complemented
52C24×C53C2311yescharacteristic
53C23×D53—41no—
54C23×D53—41no—
55C23×D53—41no—
56C23×D53—41no—
57C23×D53—41no—
58C23×D53—41no—
59C23×D53C2×C414yescomplemented
60C23×D53C2311yescharacteristic
61C23×D53C2311yescharacteristic
62C23×D53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C22×C4×C53C2311yescharacteristic
68C22×C4×C53C2311yescharacteristic
69C2×C22⋊D53—41no—
70C2×C22⋊D53—41no—
71C2×C22⋊D53C2×C414yes—
72C22×Dic53C2311yescharacteristic
73C22×Dic53C2311yescharacteristic
74C22×Dic53C2311yescharacteristic
75C22×Dic53C2311yescharacteristic
76C22×Dic53C2311yescharacteristic
77C2×D212—41no—
78C2×D212—41no—
79C2×D212—41no—
80C2×D212—41no—
81C2×D212C2×C414yes—
82C2×D212C2×C414yes—
83C2×C4×D53—41no—
84C2×C4×D53—41no—
85C2×C4×D53—41no—
86C4⋊C4×C53—41no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53C2×C414yescomplemented
90C42×C53—41no—
91C22⋊Dic53—41no—
92C22⋊Dic53—41no—
93#11719921—41no—
94#11719921—41no—
95#11719921—41no—
96#11719921—41no—
97#11719921—41no—
98#11719921—41no—
99#11719921—41no—
100#11719921—41no—
101#11719921—41no—
102#11719921—41no—
103#11719921—41no—
104#11719921C2×C414yescomplemented
105#11719921C2×C414yescomplemented
106C4⋊Dic53—41no—
107#11719919—41no—
108C4×Dic53—41no—
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—21no—
127C22×D53—21no—
128C22×D53—21no—
129C22×D53—21no—
130C22×D53—21no—
131C22×D53—21no—
132C22×D53—21no—
133C22×D53—21no—
134C22×D53—21no—
135C22×D53—21no—
136C22×D53—44no—
137C22×D53—44no—
138C22×D53—44no—
139C22×D53—44no—
140C22×D53—44no—
141C22×D53—44no—
142C22×D53—44no—
143C22×D53C22×C412yes—
144C22×D53C22×C412yes—
145C22×D53C22×C412yes—
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×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×C53C22×C412yes—
163C2×C4×C53C22×C412yes—
164C22⋊D53—44no—
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×Dic53—21no—
174C2×Dic53—21no—
175C2×Dic53—21no—
176C2×Dic53—21no—
177C2×Dic53—21no—
178C2×Dic53—21no—
179C2×Dic53C22×C412yes—
180D212—44no—
181D212—44no—
182C4×D53—44no—
183C22×C53—22no—
184C22×C53—22no—
185C22×C53—22no—
186C22×C53—41no—
187C22×C53—41no—
188C22×C53—41no—
189C22×C53—41no—
190C22×C53C23×C411yescharacteristic
191C22×C53H(5,𝔽2)11yescharacteristic
192C22×C53H(5,𝔽2)11yescharacteristic
193C22×C53H(5,𝔽2)11yescharacteristic
194C22×C53H(5,𝔽2)11yescharacteristic
195C22×C53H(5,𝔽2)11yescharacteristic
196C22×C53H(5,𝔽2)11yescharacteristic
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—22no—
206C2×D53—416no—
207C2×D53—41no—
208C2×D53—41no—
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—
219C4×C53—22no—
220C4×C53—22no—
221C4×C53—41no—
222C4×C53—41no—
223C4×C53—41no—
224C4×C53—41no—
225Dic53—22no—
226Dic53—41no—
227Dic53—41no—
228Dic53—41no—
229Dic53—41no—
230#2375—531nomaximal, Sylow
231C2×C53—42no—
232C2×C53#51811yescharacteristic
233C2×C53#51811yescharacteristic
234C2×C53#51811yescharacteristic
235C2×C53#56011yescharacteristic
236C2×C53#56011yescharacteristic
237C2×C53#56011yescharacteristic
238C2×C53H(3,𝔽4)11yescharacteristic
239D53—42no—
240D53—42no—
241D53—42no—
242C2×#120—531no—
243#390—531no—
244#390—531no—
245#390—531no—
246#388—531no—
247#388—531no—
248#388—531no—
249#388—531no—
250#388—531no—
251#388D5311yescharacteristic, complemented
252#386—531no—
253#386—531no—
254#386—531no—
255#379—531no—
256#379—531no—
257C53#237511yescharacteristic, complemented, Sylow
258C25—531no—
259C22×D4—531no—
260C22×D4—531no—
261C22×D4—531no—
262C23×C4—531no—
263#120—538no—
264C2×C4⋊C4—531no—
265C2×C4⋊C4—531no—
266C2×C4⋊C4C2×D5311yescharacteristic, complemented
267C2×C22⋊C4—531no—
268C2×C22⋊C4—531no—
269C2×C22⋊C4—531no—
270C2×C22⋊C4—531no—
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⋊C4C2×D5311yescharacteristic
283C2×C22⋊C4C2×D5311yescharacteristic
284C2×C22⋊C4C2×D5311yescharacteristic
285C2×C42—531no—
286C2×C42—531no—
287C2×C42C2×D5311yescharacteristic, complemented
288#95—531no—
289#95—531no—
290#95—531no—
291#95—531no—
292#95C2×D5311yescharacteristic, complemented
293#95C2×D5311yescharacteristic, complemented
294C24—531no—
295C24—531no—
296C24—531no—
297C24—534no—
298C24—2121no—
299C24—2121no—
300C24—2121no—
301C24—2121no—
302C24—2121no—
303C24—2121no—
304C24C22×D5311yescharacteristic
305C2×D4—534no—
306C2×D4—534no—
307C2×D4—534no—
308C2×D4—2121no—
309C2×D4—2121no—
310C2×D4—2121no—
311C2×D4—2121no—
312C2×D4—2121no—
313C2×D4—2121no—
314C22×C4—531no—
315C22×C4—531no—
316C22×C4—531no—
317C22×C4—531no—
318C22×C4—531no—
319C22×C4—2121no—
320C22×C4—2121no—
321C22×C4—2121no—
322C22×C4C22×D5311yescharacteristic
323C22×C4C22×D5311yescharacteristic
324C22×C4C22×D5311yescharacteristic
325C22×C4C22×D5311yescharacteristic
326C22×C4C22×D5311yescharacteristic
327C22×C4C22×D5311yescharacteristic
328C4⋊C4—41no—
329C4⋊C4—2121no—
330C4⋊C4—2121no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—534no—
334C22⋊C4—534no—
335C22⋊C4—2121no—
336C22⋊C4—2121no—
337C22⋊C4—2121no—
338C22⋊C4—2121no—
339C22⋊C4—2121no—
340C22⋊C4—2121no—
341C22⋊C4—2121no—
342C22⋊C4—2121no—
343C22⋊C4—2121no—
344C22⋊C4—2121no—
345C22⋊C4—2121no—
346C22⋊C4—2121no—
347C22⋊C4—2121no—
348C22⋊C4C4×D5314yescomplemented
349C42—41no—
350C42—2121no—
351C42—2121no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—532no—
359C23—532no—
360C23—532no—
361C23—1061no—
362C23—1061no—
363C23—1061no—
364C23—1061no—
365C23—1061no—
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—2124no—
380C23—2124no—
381C23—2124no—
382C23—2124no—
383C23—2124no—
384C23—2124no—
385C23—2124no—
386C23C2×C4×D5312yes—
387C23C23×D5311yescharacteristic
388D4—2124no—
389D4—2124no—
390D4—2124no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
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—532no—
408C2×C4—1061no—
409C2×C4—1061no—
410C2×C4—1061no—
411C2×C4—1061no—
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—2124no—
423C2×C4C2×C4×D5312yes—
424C2×C4C2×C4×D5312yes—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—1062no—
433C22—1062no—
434C22—1062no—
435C22—1062no—
436C22—1062no—
437C22—1062no—
438C22—1062no—
439C22—1062no—
440C22—1062no—
441C22—21216no—
442C22—2121no—
443C22—2121no—
444C22—2121no—
445C22—2121no—
446C22—2121no—
447C22—2121no—
448C22—2121no—
449C22—2121no—
450C22—2121no—
451C22—2121no—
452C22—2121no—
453C22—2121no—
454C22C22×C4×D5311yescharacteristic
455C22#1313326211yescharacteristic
456C22#1313326211yescharacteristic
457C22#1313326311yescharacteristic
458C22#1313326311yescharacteristic
459C22#1313326311yescharacteristic
460C22#1313326311yescharacteristic
461C4—22no—
462C4—22no—
463C4—41no—
464C4—41no—
465C4—41no—
466C4—41no—
467C4—1062no—
468C4—2121no—
469C4—2121no—
470C4—2121no—
471C4—2121no—
472C2—42no—
473C2—2122no—
474C2—2122no—
475C2—2122no—
476C2#1681109511yescharacteristic
477C2#1681109511yescharacteristic
478C2#1681109711yescharacteristic
479C2#1681136111yescharacteristic
480C2#1681136311yescharacteristic
481C2#1681136411yescharacteristic
482C2#1681136511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC53×#388C2
socleC23×C53C24
hypercenter#388D53
hypocenterC53#2375
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #16810525C2semidirect
3. #16810524C2semidirect
4. #16810523C2semidirect
5. #16810516C2semidirect
6. #16810515C2semidirect
7. #16810514C2semidirect
8. #16810513C2semidirect
9. #16810512C2semidirect
10. #16810501C2semidirect
11. #16810501C2semidirect
12. #16810499C2semidirect
13. #16810478C2semidirect
14. #16810478C2semidirect
15. C53×#388C2semidirect
20. C22×C4×D53C22semidirect
21. #13133187C4semidirect
22. C2×C4⋊Dic53C22semidirect
23. C2×#11719919C22semidirect
36. C2×#11719921C22semidirect
37. C2×#11719921C22semidirect
38. C2×#11719921C22semidirect
39. C2×C4×Dic53C22semidirect
40. C2×C4×Dic53C22semidirect
41. #13133124C22semidirect
42. #13133124C22semidirect
43. #13133124C22semidirect
44. #13133124C22semidirect
45. C2×C4⋊C4×C53C22semidirect
49. C2×C42×C53C22semidirect
50. #95×C53C22semidirect
51. #95×C53C22semidirect
59. C23×D53C2×C4semidirect
89. C22⋊C4×C53C2×C4semidirect
104. #11719921C2×C4semidirect
105. #11719921C2×C4semidirect
251. #388D53semidirect
257. C53#2375semidirect
266. C2×C4⋊C4C2×D53semidirect
287. C2×C42C2×D53semidirect
292. #95C2×D53semidirect
293. #95C2×D53semidirect
348. C22⋊C4C4×D53semidirect