Finite Group Databasebeta

← #20281113

Subgroups — #20281113

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×#13133187C211yesmaximal, characteristic
3#16810558C211yesmaximal, characteristic, complemented
4#16810541C211yesmaximal, characteristic, complemented
5#16810540C211yesmaximal, characteristic, complemented
6#16810540C211yesmaximal, characteristic, complemented
7#16810539C211yesmaximal, characteristic, complemented
8#16810524C211yesmaximal, characteristic, complemented
9#16810523C211yesmaximal, characteristic, complemented
10#16810515C211yesmaximal, characteristic, complemented
11#16810513C211yesmaximal, characteristic, complemented
12#16810511C211yesmaximal, characteristic, complemented
13#16810478C211yesmaximal, characteristic, complemented
14#16810478C211yesmaximal, characteristic, complemented
15C53×#396C211yesmaximal, characteristic, complemented
16C24×D53C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C22×D212C2211yescharacteristic
19C22×C22⋊D53C2211yescharacteristic
20#13133212—24no—
21#13133187—24no—
22C2×C4⋊Dic53C2211yescharacteristic, complemented
23C2×#11719919C2211yescharacteristic, complemented
24C2×#11719919C2211yescharacteristic, complemented
25C2×C22⋊Dic53C2211yescharacteristic
26C2×C22⋊Dic53C2211yescharacteristic
27C2×#11719921C2211yescharacteristic
28C2×#11719921C2211yescharacteristic
29C2×#11719921C2211yescharacteristic
30C2×#11719921C2211yescharacteristic
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic, complemented
38C2×#11719921C2211yescharacteristic, complemented
39C2×#11719921C2211yescharacteristic, complemented
40C2×#11719921C2211yescharacteristic, complemented
41C2×C4×Dic53C2211yescharacteristic, complemented
42C2×C4×Dic53C2211yescharacteristic, 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—22no—
57C23×D53—41no—
58C23×D53—41no—
59C23×D53—41no—
60C23×D53—41no—
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—22no—
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—41no—
82C2×D212—41no—
83C2×D212—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—
92#11719921—22no—
93#11719921—22no—
94#11719921—22no—
95#11719921—22no—
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#11719921—41no—
105#11719921D414yes—
106C4⋊Dic53—22no—
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#4112yes—
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—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—
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×C53C2×D412yes—
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×Dic53C2×D412yes—
181D212—44no—
182D212—44no—
183C22×C53—22no—
184C22×C53—22no—
185C22×C53—22no—
186C22×C53—41no—
187C22×C53—41no—
188C22×C53—41no—
189C22×C53—41no—
190C22×C53C22×D411yescharacteristic
191C22×C53C2×#4111yescharacteristic
192C22×C53C2×#4111yescharacteristic
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—416no—
204C2×D53—41no—
205C2×D53—41no—
206C2×D53—41no—
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—41no—
221C4×C53—41no—
222C4×C53—41no—
223C4×C53—41no—
224C4×C53—41no—
225Dic53—22no—
226Dic53—41no—
227Dic53—41no—
228Dic53—41no—
229Dic53—41no—
230#2685—531nomaximal, Sylow
231C2×C53—42no—
232C2×C53#53511yescharacteristic
233C2×C53#53811yescharacteristic
234C2×C53#53811yescharacteristic
235C2×C53#54611yescharacteristic
236C2×C53#54611yescharacteristic
237C2×C53#55111yescharacteristic
238C2×C53#56011yescharacteristic
239D53—42no—
240D53—42no—
241D53—81no—
242C2×#126—531no—
243C2×#120—531no—
244#399—531no—
245#396D5311yescharacteristic, complemented
246#394—531no—
247#394—531no—
248#394—531no—
249#394—531no—
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×D4—531no—
262#126—1064no—
263#120—1064no—
264C2×C4⋊C4—531no—
265C2×C4⋊C4—531no—
266C2×C4⋊C4—531no—
267C2×C4⋊C4C2×D5311yescharacteristic, complemented
268C2×C4⋊C4C2×D5311yescharacteristic, complemented
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⋊C4—531no—
286C2×C22⋊C4C2×D5311yescharacteristic
287C2×C22⋊C4C2×D5311yescharacteristic
288C2×C22⋊C4C2×D5311yescharacteristic
289C2×C42—531no—
290C2×C42—531no—
291#95—531no—
292#95—531no—
293#95—531no—
294#95C2×D5311yescharacteristic, complemented
295C24—531no—
296C24—531no—
297C24—531no—
298C24—1062no—
299C24—1062no—
300C24—1062no—
301C24—2121no—
302C24—2121no—
303C24—2121no—
304C24—2121no—
305C24C22×D5311yescharacteristic
306C2×D4—1062no—
307C2×D4—1062no—
308C2×D4—1062no—
309C2×D4—1062no—
310C2×D4—1062no—
311C2×D4—2121no—
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×C4C22×D5311yescharacteristic
321C22×C4C22×D5311yescharacteristic
322C22×C4C22×D5311yescharacteristic
323C22×C4C22×D5311yescharacteristic
324C22×C4C22×D5311yescharacteristic
325C22×C4C22×D5311yescharacteristic
326C4⋊C4—22no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—1062no—
330C4⋊C4—1062no—
331C4⋊C4—2121no—
332C22⋊C4—22no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—534no—
336C22⋊C4—1062no—
337C22⋊C4—1062no—
338C22⋊C4—1062no—
339C22⋊C4—1062no—
340C22⋊C4—1062no—
341C22⋊C4—2121no—
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—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—1061no—
380C23—2124no—
381C23—2124no—
382C23—2124no—
383C23—2124no—
384C23—2124no—
385C23—2124no—
386C23—2124no—
387C23#1171994912yes—
388C23C23×D5311yescharacteristic
389D4—2124no—
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—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—532no—
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—1062no—
433C22—1062no—
434C22—1062no—
435C22—1062no—
436C22—1062no—
437C22—1062no—
438C22—21216no—
439C22—2121no—
440C22—2121no—
441C22—2121no—
442C22—2121no—
443C22—2121no—
444C22—2121no—
445C22—2121no—
446C22—2121no—
447C22—2121no—
448C22—2121no—
449C22—2121no—
450C22—2121no—
451C22—2121no—
452C22—2121no—
453C22—2121no—
454C22C2×D4×D5311yescharacteristic
455C22#41×D5311yescharacteristic
456C22C2×#1171994911yescharacteristic
457C22#1313326211yescharacteristic
458C22#1313326211yescharacteristic
459C22#1313326311yescharacteristic
460C22#1313326311yescharacteristic
461C4—22no—
462C4—41no—
463C4—41no—
464C4—41no—
465C4—41no—
466C4—41no—
467C4—1062no—
468C4—2121no—
469C4—2121no—
470C4—2121no—
471C4—2121no—
472C2—42no—
473C2—4241no—
474C2—2122no—
475C2—2122no—
476C2#1681118911yescharacteristic
477C2#1681121211yescharacteristic
478C2#1681121311yescharacteristic
479C2#1681126211yescharacteristic
480C2#1681126411yescharacteristic
481C2#1681130011yescharacteristic
482C2#1681136111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#396C2
hypercenter#396D53
hypocenterC53#2685
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#13133212C2semidirect
3. #16810558C2semidirect
4. #16810541C2semidirect
5. #16810540C2semidirect
6. #16810540C2semidirect
7. #16810539C2semidirect
8. #16810524C2semidirect
9. #16810523C2semidirect
10. #16810515C2semidirect
11. #16810513C2semidirect
12. #16810511C2semidirect
13. #16810478C2semidirect
14. #16810478C2semidirect
15. C53×#396C2semidirect
22. C2×C4⋊Dic53C22semidirect
23. C2×#11719919C22semidirect
24. C2×#11719919C22semidirect
37. C2×#11719921C22semidirect
38. C2×#11719921C22semidirect
39. C2×#11719921C22semidirect
40. C2×#11719921C22semidirect
41. C2×C4×Dic53C22semidirect
42. C2×C4×Dic53C22semidirect
43. #13133124C22semidirect
44. #13133124C22semidirect
45. #13133124C22semidirect
46. C2×C4⋊C4×C53C22semidirect
47. C2×C4⋊C4×C53C22semidirect
48. C2×C4⋊C4×C53C22semidirect
52. #95×C53C22semidirect
245. #396D53semidirect
257. C53#2685semidirect
267. C2×C4⋊C4C2×D53semidirect
268. C2×C4⋊C4C2×D53semidirect
269. C2×C4⋊C4C2×D53semidirect
294. #95C2×D53semidirect