Finite Group Databasebeta

← #20278964

Subgroups — #20278964

Order 6784.

664 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133212C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D53C211yesmaximal, characteristic
3#16810552C211yesmaximal, characteristic, complemented
4#16810549C211yesmaximal, characteristic, complemented
5#16810548C211yesmaximal, characteristic, complemented
6#16810517C211yesmaximal, characteristic, complemented
7#16810513C211yesmaximal, characteristic, complemented
8#16810511C211yesmaximal, characteristic, complemented
9#16810494C211yesmaximal, characteristic, complemented
10#16810494C211yesmaximal, characteristic, complemented
11#16810480C211yesmaximal, characteristic, complemented
12#16810478C211yesmaximal, characteristic
13#16810478C211yesmaximal, characteristic
14#16810470C211yesmaximal, characteristic, complemented
15C53×#396C211yesmaximal, characteristic, complemented
16C24×D53C2211yescharacteristic
17C22×C4×D53C2211yescharacteristic
18C22×C4×D53C2211yescharacteristic
19#13133212—24no—
20C2×C4⋊Dic53C2211yescharacteristic, complemented
21C2×#11719919C2211yescharacteristic, complemented
22C2×#11719919C2211yescharacteristic, complemented
23C2×C22⋊Dic53C2211yescharacteristic
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
36C22⋊C4×D53—24no—
37C2×C4×Dic53C2211yescharacteristic, complemented
38C2×C4×Dic53C2211yescharacteristic, complemented
39#13133124C2211yescharacteristic, complemented
40#13133124C2211yescharacteristic, complemented
41#13133124C2211yescharacteristic, complemented
42#13133124C2211yescharacteristic, 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—22no—
58C23×D53—22no—
59C23×D53—41no—
60C23×D53—41no—
61C23×D53C2311yescharacteristic
62C23×D53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C22×C4×C53C2311yescharacteristic
68C22×C4×C53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C22×Dic53C2311yescharacteristic
71C22×Dic53C2311yescharacteristic
72C22×Dic53C2311yescharacteristic
73C22×Dic53C2311yescharacteristic
74C22×Dic53C2311yescharacteristic
75C2×C4×D53—22no—
76C2×C4×D53—22no—
77C2×C4×D53—22no—
78C2×C4×D53—22no—
79C2×C4×D53—41no—
80C2×C4×D53—41no—
81C4⋊C4×C53—22no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C22⋊C4×C53—22no—
85C22⋊C4×C53—41no—
86C22⋊C4×C53—41no—
87C22⋊Dic53—22no—
88C22⋊Dic53—22no—
89C22⋊Dic53—22no—
90#11719921—22no—
91#11719921—22no—
92#11719921—22no—
93#11719921—22no—
94#11719921—22no—
95#11719921—22no—
96#11719921—22no—
97#11719921—41no—
98#11719921—41no—
99#11719921D414yes—
100C4⋊Dic53—22no—
101#11719919—22no—
102#11719919—41no—
103C4×Dic53—22no—
104C4×Dic53—41no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53#4112yes—
112C23×C53C2411yescharacteristic
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—21no—
116C22×D53—21no—
117C22×D53—28no—
118C22×D53—21no—
119C22×D53—21no—
120C22×D53—21no—
121C22×D53—21no—
122C22×D53—21no—
123C22×D53—21no—
124C22×D53—44no—
125C22×D53—44no—
126C22×D53—44no—
127C22×D53—44no—
128C22×D53—44no—
129C22×D53—44no—
130C22×D53C2×D412yes—
131C22×D53#4112yes—
132C22×D53#4112yes—
133C22×D53#4112yes—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
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×C53C2×D412yes—
152C2×Dic53—21no—
153C2×Dic53—21no—
154C2×Dic53—21no—
155C2×Dic53—21no—
156C2×Dic53—21no—
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×Dic53C2×D412yes—
170C4×D53—44no—
171C4×D53—44no—
172C22×C53—22no—
173C22×C53—22no—
174C22×C53—22no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53C22×D411yescharacteristic
180C22×C53C2×#4111yescharacteristic
181C22×C53C2×#4111yescharacteristic
182C22×C53C2×#4111yescharacteristic
183C22×C53C2×#4111yescharacteristic
184C22×C53H(5,𝔽2)11yescharacteristic
185C22×C53H(5,𝔽2)11yescharacteristic
186C2×D53—22no—
187C2×D53—22no—
188C2×D53—22no—
189C2×D53—22no—
190C2×D53—22no—
191C2×D53—22no—
192C2×D53—22no—
193C2×D53—22no—
194C2×D53—22no—
195C2×D53—416no—
196C2×D53—41no—
197C2×D53—41no—
198C2×D53—41no—
199C2×D53—41no—
200C2×D53#12614yescomplemented
201C4×C53—22no—
202C4×C53—41no—
203C4×C53—41no—
204C4×C53—41no—
205C4×C53—41no—
206C4×C53—41no—
207Dic53—22no—
208Dic53—41no—
209Dic53—41no—
210Dic53—41no—
211Dic53—41no—
212Dic53—41no—
213#2480—531nomaximal, Sylow
214C2×C53—42no—
215C2×C53#52511yescharacteristic
216C2×C53C2×#12611yescharacteristic
217C2×C53#53811yescharacteristic
218C2×C53#54611yescharacteristic
219C2×C53#54611yescharacteristic
220C2×C53#55111yescharacteristic
221C2×C53#55111yescharacteristic
222D53—24no—
223D53—42no—
224C2×#126—531no—
225C22×C22⋊C4—531no—
226#397—531no—
227#397—531no—
228#397—531no—
229#396D5311yescharacteristic, complemented
230#388—531no—
231#388—531no—
232#388—531no—
233#385—531no—
234#385—531no—
235#380—531no—
236#379—531no—
237#379—531no—
238C4×C22⋊C4—531no—
239C53#248011yescharacteristic, complemented, Sylow
240C25—531no—
241C23×C4—531no—
242C23×C4—531no—
243#126—1064no—
244C2×C4⋊C4—531no—
245C2×C4⋊C4—531no—
246C2×C4⋊C4—531no—
247C2×C4⋊C4C2×D5311yescharacteristic, complemented
248C2×C4⋊C4C2×D5311yescharacteristic, complemented
249C2×C4⋊C4C2×D5311yescharacteristic, complemented
250C2×C22⋊C4—531no—
251C2×C22⋊C4—531no—
252C2×C22⋊C4—531no—
253C2×C22⋊C4—531no—
254C2×C22⋊C4—531no—
255C2×C22⋊C4—531no—
256C2×C22⋊C4—531no—
257C2×C22⋊C4—531no—
258C2×C22⋊C4—531no—
259C2×C22⋊C4—531no—
260C2×C22⋊C4—531no—
261C2×C22⋊C4—531no—
262C2×C22⋊C4—531no—
263C2×C22⋊C4—1064no—
264C2×C22⋊C4C2×D5311yescharacteristic
265C2×C22⋊C4C2×D5311yescharacteristic
266C2×C22⋊C4C2×D5311yescharacteristic
267C2×C42—531no—
268C2×C42—531no—
269#95—531no—
270#95—531no—
271#95—531no—
272#95—531no—
273#95—531no—
274#95—531no—
275#95—531no—
276#95C2×D5311yescharacteristic, complemented
277C24—531no—
278C24—531no—
279C24—1062no—
280C24—1062no—
281C24—1062no—
282C24—1062no—
283C24—1062no—
284C24—2121no—
285C24—2121no—
286C24C22×D5311yescharacteristic
287C22×C4—531no—
288C22×C4—531no—
289C22×C4—531no—
290C22×C4—531no—
291C22×C4—531no—
292C22×C4—531no—
293C22×C4—1062no—
294C22×C4—1062no—
295C22×C4—1062no—
296C22×C4—1062no—
297C22×C4—2121no—
298C22×C4—2121no—
299C22×C4C22×D5311yescharacteristic
300C22×C4C22×D5311yescharacteristic
301C22×C4C22×D5311yescharacteristic
302C22×C4C22×D5311yescharacteristic
303C22×C4C22×D5311yescharacteristic
304C22×C4C22×D5311yescharacteristic
305C4⋊C4—22no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—1062no—
309C4⋊C4—1062no—
310C4⋊C4—2121no—
311C22⋊C4—22no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—534no—
315C22⋊C4—1062no—
316C22⋊C4—1062no—
317C22⋊C4—1062no—
318C22⋊C4—1062no—
319C22⋊C4—1062no—
320C22⋊C4—1062no—
321C22⋊C4—1062no—
322C22⋊C4—1062no—
323C22⋊C4—1062no—
324C22⋊C4—1062no—
325C22⋊C4—2121no—
326C22⋊C4—2121no—
327C42—1062no—
328C42—2121no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—532no—
336C23—532no—
337C23—532no—
338C23—532no—
339C23—1068no—
340C23—1061no—
341C23—1061no—
342C23—1061no—
343C23—1061no—
344C23—1061no—
345C23—1061no—
346C23—1061no—
347C23—1061no—
348C23—1061no—
349C23—1061no—
350C23—2124no—
351C23—2124no—
352C23—2124no—
353C23—2124no—
354C23—2124no—
355C23—2124no—
356C23#1171994912yes—
357C23C23×D5311yescharacteristic
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—532no—
376C2×C4—1061no—
377C2×C4—1061no—
378C2×C4—1061no—
379C2×C4—1061no—
380C2×C4—1061no—
381C2×C4—1061no—
382C2×C4—1061no—
383C2×C4—1061no—
384C2×C4—1061no—
385C2×C4—1061no—
386C2×C4—1061no—
387C2×C4—1061no—
388C2×C4—1061no—
389C2×C4—1061no—
390C2×C4—1061no—
391C2×C4—1061no—
392C2×C4—1061no—
393C2×C4—2124no—
394C2×C4—2124no—
395C2×C4D4×D5312yes—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—534no—
404C22—1062no—
405C22—1062no—
406C22—1062no—
407C22—1062no—
408C22—1062no—
409C22—1062no—
410C22—1062no—
411C22—1062no—
412C22—1062no—
413C22—21216no—
414C22—2121no—
415C22—2121no—
416C22—2121no—
417C22—2121no—
418C22C2×D4×D5311yescharacteristic
419C22#41×D5311yescharacteristic
420C22#41×D5311yescharacteristic
421C22#41×D5311yescharacteristic
422C22C2×#1171994911yescharacteristic
423C22#1313326211yescharacteristic
424C22#1313326211yescharacteristic
425C4—22no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—41no—
430C4—41no—
431C4—1062no—
432C4—2121no—
433C4—2121no—
434C4—2121no—
435C4—2121no—
436C4—2121no—
437C2—42no—
438C2—1064no—
439C2—2122no—
440C2#1681113611yescharacteristic
441C2#126×D5311yescharacteristic
442C2#1681120511yescharacteristic
443C2#1681126211yescharacteristic
444C2#1681127011yescharacteristic
445C2#1681130011yescharacteristic
446C2#1681130011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#396C2
hypercenter#396D53
hypocenterC53#2480
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. #16810552C2semidirect
4. #16810549C2semidirect
5. #16810548C2semidirect
6. #16810517C2semidirect
7. #16810513C2semidirect
8. #16810511C2semidirect
9. #16810494C2semidirect
10. #16810494C2semidirect
11. #16810480C2semidirect
14. #16810470C2semidirect
15. C53×#396C2semidirect
20. C2×C4⋊Dic53C22semidirect
21. C2×#11719919C22semidirect
22. C2×#11719919C22semidirect
37. C2×C4×Dic53C22semidirect
38. C2×C4×Dic53C22semidirect
39. #13133124C22semidirect
40. #13133124C22semidirect
41. #13133124C22semidirect
42. #13133124C22semidirect
43. #13133124C22semidirect
44. #13133124C22semidirect
45. #13133124C22semidirect
46. C2×C4⋊C4×C53C22semidirect
47. C2×C4⋊C4×C53C22semidirect
48. C2×C4⋊C4×C53C22semidirect
52. #95×C53C22semidirect
200. C2×D53#126semidirect
229. #396D53semidirect
239. C53#2480semidirect
247. C2×C4⋊C4C2×D53semidirect
248. C2×C4⋊C4C2×D53semidirect
249. C2×C4⋊C4C2×D53semidirect
276. #95C2×D53semidirect