Finite Group Databasebeta

← #20281145

Subgroups — #20281145

Order 6784.

626 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133189C211yesmaximal, characteristic, complemented
2#16810561C211yesmaximal, characteristic, complemented
3#16810551C211yesmaximal, characteristic, complemented
4#16810549C211yesmaximal, characteristic, complemented
5#16810538C211yesmaximal, characteristic, complemented
6#16810531C211yesmaximal, characteristic, complemented
7#16810531C211yesmaximal, characteristic, complemented
8#16810503C211yesmaximal, characteristic, complemented
9#16810503C211yesmaximal, characteristic, complemented
10#16810503C211yesmaximal, characteristic, complemented
11#16810502C211yesmaximal, characteristic, complemented
12#16810500C211yesmaximal, characteristic, complemented
13#16810496C211yesmaximal, characteristic, complemented
14#16810484C211yesmaximal, characteristic, complemented
15C2×#121×C53C211yesmaximal, characteristic, complemented
16C22×C22⋊D53C2211yescharacteristic
17C22×C22⋊D53C2211yescharacteristic
18C22×C22⋊D53C2211yescharacteristic
19C23×Dic53C2211yescharacteristic
20C23×Dic53C2211yescharacteristic
21#13133189—24no—
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×C22⋊Dic53C2211yescharacteristic
25C2×C22⋊Dic53C2211yescharacteristic
26C2×C22⋊Dic53C2211yescharacteristic
27C2×C22⋊Dic53C2211yescharacteristic
28C2×C22⋊Dic53C2211yescharacteristic
29C2×C22⋊Dic53C2211yescharacteristic
30C2×C22⋊Dic53C2211yescharacteristic
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35#13133124C2211yescharacteristic
36#13133124C2211yescharacteristic
37#13133124C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45C22×D4×C53C2211yescharacteristic
46C22×D4×C53C2211yescharacteristic
47C22×D4×C53C2211yescharacteristic
48C23×C4×C53C2211yescharacteristic
49#121×C53—24no—
50C2×C4⋊C4×C53C2211yescharacteristic
51C2×C22⋊C4×C53C2211yescharacteristic
52C2×C22⋊C4×C53C2211yescharacteristic
53C24×C53C2311yescharacteristic
54C24×C53C2311yescharacteristic
55C24×C53C2311yescharacteristic
56C23×D53C2311yescharacteristic
57C2×D4×C53—22no—
58C2×D4×C53—22no—
59C2×D4×C53—22no—
60C2×D4×C53—22no—
61C2×D4×C53—41no—
62C2×D4×C53—41no—
63C2×D4×C53—41no—
64C2×D4×C53—41no—
65C2×D4×C53D414yescomplemented
66C22×C4×C53—22no—
67C22×C4×C53—41no—
68C22×C4×C53—41no—
69C22×C4×C53C2311yescharacteristic
70C22×C4×C53C2311yescharacteristic
71C22×C4×C53C2311yescharacteristic
72C22×C4×C53C2311yescharacteristic
73C2×C22⋊D53—22no—
74C2×C22⋊D53—22no—
75C2×C22⋊D53—22no—
76C2×C22⋊D53—22no—
77C2×C22⋊D53—41no—
78C2×C22⋊D53—41no—
79C2×C22⋊D53—41no—
80C2×C22⋊D53—41no—
81C2×C22⋊D53—41no—
82C22×Dic53—22no—
83C22×Dic53—22no—
84C22×Dic53—41no—
85C22×Dic53—41no—
86C22×Dic53—41no—
87C22×Dic53—41no—
88C22×Dic53C2311yescharacteristic
89C22×Dic53C2311yescharacteristic
90C22×Dic53C2311yescharacteristic
91C22×Dic53C2311yescharacteristic
92C22×Dic53C2311yescharacteristic
93C22×Dic53C2311yescharacteristic
94C22×Dic53C2311yescharacteristic
95C4⋊C4×C53—22no—
96C22⋊C4×C53—22no—
97C22⋊C4×C53—22no—
98C22⋊Dic53—22no—
99C22⋊Dic53—22no—
100C22⋊Dic53—41no—
101C22⋊Dic53—41no—
102C22⋊Dic53—41no—
103C22⋊Dic53—41no—
104C22⋊Dic53—41no—
105#11719921—22no—
106#11719921—41no—
107#11719921—41no—
108#11719921—41no—
109#11719919—22no—
110#11719919—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—21no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53—21no—
128C23×C53—21no—
129C23×C53C2×D412yes—
130C23×C53C2×D412yes—
131C23×C53#4112yes—
132C23×C53C2411yescharacteristic
133C22×D53—21no—
134C22×D53—21no—
135C22×D53—21no—
136C22×D53—21no—
137C22×D53—21no—
138C22×D53—21no—
139C22×D53—21no—
140D4×C53—44no—
141D4×C53—44no—
142D4×C53—44no—
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×C53—44no—
155C2×C4×C53C2×D412yes—
156C22⋊D53—44no—
157C22⋊D53—44no—
158C22⋊D53—44no—
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×Dic53—21no—
173C2×Dic53—21no—
174C2×Dic53—21no—
175C2×Dic53—21no—
176C2×Dic53—21no—
177C2×Dic53—21no—
178C2×Dic53—21no—
179C2×Dic53—44no—
180C2×Dic53—44no—
181C2×Dic53#4112yes—
182C22×C53—22no—
183C22×C53—22no—
184C22×C53—22no—
185C22×C53—22no—
186C22×C53—22no—
187C22×C53—22no—
188C22×C53—22no—
189C22×C53—22no—
190C22×C53—22no—
191C22×C53—41no—
192C22×C53—41no—
193C22×C53—41no—
194C22×C53—41no—
195C22×C53—41no—
196C22×C53—41no—
197C22×C53—41no—
198C22×C53—41no—
199C22×C53—41no—
200C22×C53—41no—
201C22×C53—41no—
202C22×C53—41no—
203C22×C53C22×D411yescharacteristic
204C22×C53C2×#4111yescharacteristic
205C22×C53C2×#4111yescharacteristic
206C22×C53H(5,𝔽2)11yescharacteristic
207C22×C53H(5,𝔽2)11yescharacteristic
208C22×C53H(5,𝔽2)11yescharacteristic
209C22×C53H(5,𝔽2)11yescharacteristic
210C2×D53—41no—
211C2×D53—41no—
212C2×D53—41no—
213C2×D53—41no—
214C2×D53—41no—
215C2×D53—41no—
216C2×D53—41no—
217C4×C53—22no—
218C4×C53—41no—
219C4×C53—41no—
220C4×C53—41no—
221Dic53—22no—
222Dic53—41no—
223Dic53—41no—
224Dic53—41no—
225Dic53—41no—
226Dic53—41no—
227Dic53—41no—
228#2688—531nomaximal, Sylow
229C2×C53—42no—
230C2×C53—42no—
231C2×C53—42no—
232C2×C53#53411yescharacteristic
233C2×C53#53811yescharacteristic
234C2×C53#54011yescharacteristic
235C2×C53#54611yescharacteristic
236C2×C53#54611yescharacteristic
237C2×C53#55311yescharacteristic
238C2×C53#56011yescharacteristic
239D53—81no—
240C2×#121—531no—
241C2×#121D5311yescharacteristic, complemented
242#400—531no—
243#397—531no—
244#397—531no—
245#394—531no—
246#392—531no—
247#392—531no—
248#386—531no—
249#386—531no—
250#386—531no—
251#386—531no—
252#386—531no—
253#385—531no—
254#381—531no—
255C53#268811yescharacteristic, complemented, Sylow
256C22×D4—531no—
257C22×D4—531no—
258C22×D4—531no—
259C22×D4C2×D5311yescharacteristic
260C22×D4C2×D5311yescharacteristic
261C22×D4C2×D5311yescharacteristic
262C23×C4—531no—
263C23×C4—531no—
264C23×C4C2×D5311yescharacteristic
265#121—24no—
266#121—1064no—
267C2×C4⋊C4—531no—
268C2×C4⋊C4—531no—
269C2×C4⋊C4C2×D5311yescharacteristic
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⋊C4C2×D5311yescharacteristic
282C2×C22⋊C4C2×D5311yescharacteristic
283#95—531no—
284#95—531no—
285#95—531no—
286#95—531no—
287#95—531no—
288#95—531no—
289#95—531no—
290#95—531no—
291#95—531no—
292#95—531no—
293C24—531no—
294C24C22×D5311yescharacteristic
295C24C22×D5311yescharacteristic
296C24C22×D5311yescharacteristic
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—1062no—
306C2×D4—1062no—
307C2×D4—1062no—
308C2×D4—1062no—
309C2×D4—2121no—
310C2×D4—2121no—
311C2×D4—2121no—
312C2×D4—2121no—
313C2×D4—2121no—
314C2×D4C22⋊D5314yescomplemented
315C22×C4—22no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—531no—
319C22×C4—531no—
320C22×C4—531no—
321C22×C4—531no—
322C22×C4—531no—
323C22×C4—531no—
324C22×C4—531no—
325C22×C4—1062no—
326C22×C4—1062no—
327C22×C4—2121no—
328C22×C4—2121no—
329C22×C4—2121no—
330C22×C4—2121no—
331C22×C4C22×D5311yescharacteristic
332C22×C4C22×D5311yescharacteristic
333C22×C4C22×D5311yescharacteristic
334C22×C4C22×D5311yescharacteristic
335C4⋊C4—22no—
336C4⋊C4—1062no—
337C4⋊C4—2121no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—1062no—
341C22⋊C4—1062no—
342C22⋊C4—1062no—
343C22⋊C4—2121no—
344C22⋊C4—2121no—
345C22⋊C4—2121no—
346C22⋊C4—2121no—
347C22⋊C4—2121no—
348C22⋊C4—2121no—
349C22⋊C4—2121no—
350C22⋊C4—2121no—
351C23—21no—
352C23—21no—
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—21no—
367C23—21no—
368C23—21no—
369C23—1061no—
370C23—1061no—
371C23—1061no—
372C23—1061no—
373C23—1061no—
374C23—1061no—
375C23—1061no—
376C23#1171994712yes—
377C23C2×C22⋊D5312yes—
378C23C2×C22⋊D5312yes—
379C23C23×D5311yescharacteristic
380D4—44no—
381D4—44no—
382D4—44no—
383D4—2124no—
384D4—2124no—
385D4—2124no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—44no—
398C2×C4—532no—
399C2×C4—1061no—
400C2×C4—1061no—
401C2×C4—1061no—
402C2×C4—1061no—
403C2×C4—1061no—
404C2×C4—1061no—
405C2×C4—1061no—
406C2×C4—1061no—
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—2124no—
420C2×C4—2124no—
421C2×C4C2×C22⋊D5312yes—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—2121no—
444C22—2121no—
445C22—2121no—
446C22—2121no—
447C22—2121no—
448C22—2121no—
449C22—2121no—
450C22C22×C22⋊D5311yescharacteristic
451C22#41×D5311yescharacteristic
452C22C2×#1171994711yescharacteristic
453C22#1313326211yescharacteristic
454C22#1313326211yescharacteristic
455C22#1313326211yescharacteristic
456C22#1313326311yescharacteristic
457C4—22no—
458C4—41no—
459C4—41no—
460C4—41no—
461C4—1062no—
462C4—2121no—
463C4—2121no—
464C4—2121no—
465C4—2121no—
466C4—2121no—
467C4—2121no—
468C2—42no—
469C2—42no—
470C2—42no—
471C2—4241no—
472C2#1681118211yescharacteristic
473C2#1681121011yescharacteristic
474C2#1681122311yescharacteristic
475C2#1681126111yescharacteristic
476C2#1681126711yescharacteristic
477C2#1681131611yescharacteristic
478C2#1681136111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#121×C53C2
hypercenterC2×#121D53
hypocenterC53#2688
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#13133189C2semidirect
2. #16810561C2semidirect
3. #16810551C2semidirect
4. #16810549C2semidirect
5. #16810538C2semidirect
6. #16810531C2semidirect
7. #16810531C2semidirect
8. #16810503C2semidirect
9. #16810503C2semidirect
10. #16810503C2semidirect
11. #16810502C2semidirect
12. #16810500C2semidirect
13. #16810496C2semidirect
14. #16810484C2semidirect
15. C2×#121×C53C2semidirect
65. C2×D4×C53D4semidirect
241. C2×#121D53semidirect
255. C53#2688semidirect
314. C2×D4C22⋊D53semidirect