Finite Group Databasebeta

← #20281111

Subgroups — #20281111

Order 6784.

666 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×#13133212C211yesmaximal, characteristic, complemented
2C2×#13133185C211yesmaximal, characteristic
3#16810559C211yesmaximal, characteristic, complemented
4#16810545C211yesmaximal, characteristic, complemented
5#16810539C211yesmaximal, characteristic, complemented
6#16810539C211yesmaximal, characteristic, complemented
7#16810538C211yesmaximal, characteristic, complemented
8#16810526C211yesmaximal, characteristic, complemented
9#16810523C211yesmaximal, characteristic, complemented
10#16810516C211yesmaximal, characteristic, complemented
11#16810515C211yesmaximal, characteristic, complemented
12#16810515C211yesmaximal, characteristic, complemented
13#16810478C211yesmaximal, characteristic, complemented
14#16810478C211yesmaximal, characteristic, complemented
15C53×#394C211yesmaximal, characteristic, complemented
16C24×D53C2211yescharacteristic
17C22×C22⋊D53C2211yescharacteristic
18C22×C22⋊D53C2211yescharacteristic
19#13133212—24no—
20#13133185—24no—
21C2×C4⋊Dic53C2211yescharacteristic, complemented
22C2×C4⋊Dic53C2211yescharacteristic, complemented
23C2×C4⋊Dic53C2211yescharacteristic, complemented
24C2×#11719919C2211yescharacteristic, complemented
25C2×#11719919C2211yescharacteristic, complemented
26C2×C22⋊Dic53C2211yescharacteristic
27C2×C22⋊Dic53C2211yescharacteristic
28C2×C22⋊Dic53C2211yescharacteristic
29C2×C22⋊Dic53C2211yescharacteristic, complemented
30C2×C22⋊Dic53C2211yescharacteristic, complemented
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C2×#11719921C2211yescharacteristic
40C2×#11719921C2211yescharacteristic
41C2×C4×Dic53C2211yescharacteristic, complemented
42C2×C4×Dic53C2211yescharacteristic, complemented
43#13133124C2211yescharacteristic, complemented
44#13133124C2211yescharacteristic, complemented
45#13133124C2211yescharacteristic, complemented
46C22×D4×C53C2211yescharacteristic
47C2×C4⋊C4×C53C2211yescharacteristic, complemented
48C2×C22⋊C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50C2×C22⋊C4×C53C2211yescharacteristic, complemented
51C2×C22⋊C4×C53C2211yescharacteristic, complemented
52#95×C53C2211yescharacteristic, complemented
53C24×C53C2311yescharacteristic
54C24×C53C2311yescharacteristic
55C23×D53—22no—
56C23×D53—22no—
57C23×D53—22no—
58C23×D53—41no—
59C23×D53—41no—
60C23×D53—41no—
61C23×D53—41no—
62C23×D53C2311yescharacteristic
63C23×D53C2311yescharacteristic
64C2×D4×C53—22no—
65C2×D4×C53—22no—
66C2×D4×C53—41no—
67C22×C4×C53C2311yescharacteristic
68C22×C4×C53C2311yescharacteristic
69C22×C4×C53C2311yescharacteristic
70C22×C4×C53C2311yescharacteristic
71C22×C4×C53C2311yescharacteristic
72C2×C22⋊D53—22no—
73C2×C22⋊D53—22no—
74C2×C22⋊D53—22no—
75C2×C22⋊D53—41no—
76C2×C22⋊D53—41no—
77C2×C22⋊D53—41no—
78C22×Dic53C2311yescharacteristic
79C22×Dic53C2311yescharacteristic
80C22×Dic53C2311yescharacteristic
81C22×Dic53C2311yescharacteristic
82C22×Dic53C2311yescharacteristic
83C22×Dic53C2311yescharacteristic
84C4⋊C4×C53—22no—
85C22⋊C4×C53—22no—
86C22⋊C4×C53—41no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C22⋊Dic53—22no—
90C22⋊Dic53—41no—
91C22⋊Dic53—41no—
92C22⋊Dic53—41no—
93C22⋊Dic53—41no—
94#11719921—22no—
95#11719921—22no—
96#11719921—22no—
97#11719921—22no—
98#11719921—41no—
99#11719921—41no—
100#11719921—41no—
101#11719921—41no—
102#11719921—41no—
103#11719921D414yes—
104C4⋊Dic53—22no—
105C4⋊Dic53—41no—
106C4⋊Dic53—41no—
107#11719919—22no—
108#11719919—41no—
109C4×Dic53—22no—
110C4×Dic53—41no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—21no—
124C23×C53#4112yes—
125C23×C53C2411yescharacteristic
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—21no—
137C22×D53—21no—
138C22×D53—44no—
139C22×D53—44no—
140C22×D53—44no—
141C22×D53—44no—
142C22×D53—44no—
143C22×D53—44no—
144C22×D53—44no—
145C22×D53C2×D412yes—
146C22×D53#4112yes—
147D4×C53—44no—
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×C53C2×D412yes—
163C22⋊D53—44no—
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×Dic53—21no—
180C2×Dic53—21no—
181C2×Dic53—21no—
182C2×Dic53C2×D412yes—
183C22×C53—22no—
184C22×C53—22no—
185C22×C53—22no—
186C22×C53—41no—
187C22×C53—41no—
188C22×C53—41no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53—41no—
193C22×C53—41no—
194C22×C53—41no—
195C22×C53—41no—
196C22×C53—41no—
197C22×C53C22×D411yescharacteristic
198C22×C53C2×#4111yescharacteristic
199C22×C53C2×#4111yescharacteristic
200C22×C53H(5,𝔽2)11yescharacteristic
201C22×C53H(5,𝔽2)11yescharacteristic
202C22×C53H(5,𝔽2)11yescharacteristic
203C22×C53H(5,𝔽2)11yescharacteristic
204C2×D53—22no—
205C2×D53—22no—
206C2×D53—22no—
207C2×D53—22no—
208C2×D53—22no—
209C2×D53—22no—
210C2×D53—416no—
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—41no—
221C4×C53—41no—
222C4×C53—41no—
223C4×C53—41no—
224Dic53—22no—
225Dic53—41no—
226Dic53—41no—
227Dic53—41no—
228Dic53—41no—
229Dic53—41no—
230#2685—531nomaximal, Sylow
231C2×C53—42no—
232C2×C53—81no—
233C2×C53#53511yescharacteristic
234C2×C53#53811yescharacteristic
235C2×C53#53811yescharacteristic
236C2×C53#54611yescharacteristic
237C2×C53#54611yescharacteristic
238C2×C53#55111yescharacteristic
239C2×C53#56011yescharacteristic
240D53—42no—
241D53—42no—
242C2×#126—531no—
243C2×#120—531no—
244#399—531no—
245#396—531no—
246#394—531no—
247#394—531no—
248#394—531no—
249#394D5311yescharacteristic, complemented
250#390—531no—
251#390—531no—
252#388—531no—
253#388—531no—
254#388—531no—
255#379—531no—
256#379—531no—
257C53#268511yescharacteristic, complemented, Sylow
258C25—531no—
259C22×D4—531no—
260C22×D4—531no—
261C22×D4C2×D5311yescharacteristic
262#126—1064no—
263#120—1064no—
264C2×C4⋊C4—531no—
265C2×C4⋊C4—531no—
266C2×C4⋊C4—531no—
267C2×C4⋊C4—531no—
268C2×C4⋊C4—531no—
269C2×C4⋊C4C2×D5311yescharacteristic, complemented
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⋊C4—531no—
283C2×C22⋊C4—531no—
284C2×C22⋊C4—531no—
285C2×C22⋊C4C2×D5311yescharacteristic
286C2×C22⋊C4C2×D5311yescharacteristic
287C2×C22⋊C4C2×D5311yescharacteristic, complemented
288C2×C22⋊C4C2×D5311yescharacteristic, complemented
289C2×C42—531no—
290C2×C42—531no—
291#95—531no—
292#95—531no—
293#95—531no—
294#95C2×D5311yescharacteristic, complemented
295C24—531no—
296C24—531no—
297C24—1062no—
298C24—1062no—
299C24—1062no—
300C24—2121no—
301C24—2121no—
302C24—2121no—
303C24—2121no—
304C24C22×D5311yescharacteristic
305C24C22×D5311yescharacteristic
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—41no—
309C2×D4—1062no—
310C2×D4—1062no—
311C2×D4—1062no—
312C2×D4—2121no—
313C2×D4—2121no—
314C2×D4—2121no—
315C22×C4—531no—
316C22×C4—531no—
317C22×C4—531no—
318C22×C4—531no—
319C22×C4—531no—
320C22×C4—531no—
321C22×C4C22×D5311yescharacteristic
322C22×C4C22×D5311yescharacteristic
323C22×C4C22×D5311yescharacteristic
324C22×C4C22×D5311yescharacteristic
325C22×C4C22×D5311yescharacteristic
326C4⋊C4—22no—
327C4⋊C4—1062no—
328C4⋊C4—1062no—
329C4⋊C4—2121no—
330C4⋊C4—2121no—
331C4⋊C4—2121no—
332C22⋊C4—22no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—534no—
337C22⋊C4—1062no—
338C22⋊C4—1062no—
339C22⋊C4—1062no—
340C22⋊C4—1062no—
341C22⋊C4—1062no—
342C22⋊C4—2121no—
343C22⋊C4—2121no—
344C22⋊C4—2121no—
345C22⋊C4—2121no—
346C22⋊C4—2121no—
347C22⋊C4—2121no—
348C22⋊C4—2121no—
349C22⋊C4—2121no—
350C22⋊C4—2121no—
351C42—1062no—
352C42—2121no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—532no—
367C23—532no—
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—2124no—
381C23—2124no—
382C23—2124no—
383C23—2124no—
384C23—2124no—
385C23—2124no—
386C23—2124no—
387C23#1171994712yes—
388C23C23×D5311yescharacteristic
389D4—44no—
390D4—2124no—
391D4—2124no—
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—532no—
407C2×C4—1061no—
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—1061no—
423C2×C4—1061no—
424C2×C4D4×D5312yes—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—1062no—
440C22—1062no—
441C22—1062no—
442C22—1062no—
443C22—1062no—
444C22—1062no—
445C22—21216no—
446C22—2121no—
447C22—2121no—
448C22—2121no—
449C22—2121no—
450C22—2121no—
451C22—2121no—
452C22—2121no—
453C22—2121no—
454C22C2×D4×D5311yescharacteristic
455C22#41×D5311yescharacteristic
456C22C2×#1171994711yescharacteristic
457C22#1313326211yescharacteristic
458C22#1313326211yescharacteristic
459C22#1313326311yescharacteristic
460C22#1313326311yescharacteristic
461C4—22no—
462C4—41no—
463C4—41no—
464C4—41no—
465C4—41no—
466C4—1062no—
467C4—2121no—
468C4—2121no—
469C4—2121no—
470C4—2121no—
471C4—2121no—
472C2—42no—
473C2—81no—
474C2—2122no—
475C2—2122no—
476C2#1681118911yescharacteristic
477C2#1681120511yescharacteristic
478C2#1681121011yescharacteristic
479C2#1681126011yescharacteristic
480C2#1681126611yescharacteristic
481C2#1681129611yescharacteristic
482C2#1681136311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#394C2
hypercenter#394D53
hypocenterC53#2685
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#13133212C2semidirect
3. #16810559C2semidirect
4. #16810545C2semidirect
5. #16810539C2semidirect
6. #16810539C2semidirect
7. #16810538C2semidirect
8. #16810526C2semidirect
9. #16810523C2semidirect
10. #16810516C2semidirect
11. #16810515C2semidirect
12. #16810515C2semidirect
13. #16810478C2semidirect
14. #16810478C2semidirect
15. C53×#394C2semidirect
21. C2×C4⋊Dic53C22semidirect
22. C2×C4⋊Dic53C22semidirect
23. C2×C4⋊Dic53C22semidirect
24. C2×#11719919C22semidirect
25. C2×#11719919C22semidirect
29. C2×C22⋊Dic53C22semidirect
30. C2×C22⋊Dic53C22semidirect
41. C2×C4×Dic53C22semidirect
42. C2×C4×Dic53C22semidirect
43. #13133124C22semidirect
44. #13133124C22semidirect
45. #13133124C22semidirect
47. C2×C4⋊C4×C53C22semidirect
50. C2×C22⋊C4×C53C22semidirect
51. C2×C22⋊C4×C53C22semidirect
52. #95×C53C22semidirect
249. #394D53semidirect
257. C53#2685semidirect
269. C2×C4⋊C4C2×D53semidirect
287. C2×C22⋊C4C2×D53semidirect
288. C2×C22⋊C4C2×D53semidirect
294. #95C2×D53semidirect