Finite Group Databasebeta

← #20278376

Subgroups — #20278376

Order 6784.

780 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133196C211yesmaximal, characteristic, complemented
2C2×#13133194C211yesmaximal, characteristic, complemented
3C22×C22⋊Dic53C211yesmaximal, characteristic
4#16810559C211yesmaximal, characteristic, complemented
5#16810546C211yesmaximal, characteristic, complemented
6#16810546C211yesmaximal, characteristic, complemented
7#16810535C211yesmaximal, characteristic, complemented
8#16810498C211yesmaximal, characteristic, complemented
9#16810496C211yesmaximal, characteristic, complemented
10#16810496C211yesmaximal, characteristic, complemented
11#16810495C211yesmaximal, characteristic, complemented
12#16810483C211yesmaximal, characteristic, complemented
13#16810481C211yesmaximal, characteristic, complemented
14#16810479C211yesmaximal, characteristic
15C22×C22⋊C4×C53C211yesmaximal, characteristic
16C22×Dic106C2211yescharacteristic, complemented
17C23×Dic53C2211yescharacteristic
18C23×Dic53C2211yescharacteristic
19#13133196—24no—
20#13133194—24no—
21C2×C4⋊Dic53C2211yescharacteristic, complemented
22C2×C4⋊Dic53C2211yescharacteristic, complemented
23C2×C4⋊Dic53C2211yescharacteristic, complemented
24C2×#11719919C2211yescharacteristic, complemented
25C2×#11719919C2211yescharacteristic, complemented
26C2×#11719919C2211yescharacteristic, complemented
27C2×#11719919C2211yescharacteristic, complemented
28C2×#11719919C2211yescharacteristic, complemented
29C2×#11719919C2211yescharacteristic, complemented
30C2×C22⋊Dic53—24no—
31C2×C22⋊Dic53C2211yescharacteristic
32C2×C22⋊Dic53C2211yescharacteristic
33C2×C22⋊Dic53C2211yescharacteristic
34C2×C22⋊Dic53C2211yescharacteristic
35C2×C22⋊Dic53C2211yescharacteristic
36C2×C22⋊Dic53C2211yescharacteristic
37C2×C22⋊Dic53C2211yescharacteristic
38C2×C22⋊Dic53C2211yescharacteristic
39C2×C22⋊Dic53C2211yescharacteristic
40C2×C22⋊Dic53C2211yescharacteristic
41#13133124C2211yescharacteristic, complemented
42#13133124C2211yescharacteristic, complemented
43#13133124C2211yescharacteristic, complemented
44#13133124C2211yescharacteristic, complemented
45#13133124C2211yescharacteristic, complemented
46#13133124C2211yescharacteristic, complemented
47C25×C53C2211yescharacteristic
48C23×C4×C53C2211yescharacteristic
49C23×C4×C53C2211yescharacteristic
50C2×C22⋊C4×C53—24no—
51C2×C22⋊C4×C53C2211yescharacteristic
52C2×C22⋊C4×C53C2211yescharacteristic
53C2×C22⋊C4×C53C2211yescharacteristic
54C2×C22⋊C4×C53C2211yescharacteristic
55C24×C53—22no—
56C24×C53—22no—
57C24×C53—22no—
58C24×C53—22no—
59C24×C53—41no—
60C24×C53—41no—
61C24×C53Q814yescomplemented
62C24×C53C2311yescharacteristic
63C24×C53C2311yescharacteristic
64C24×C53C2311yescharacteristic
65C22×C4×C53—22no—
66C22×C4×C53—22no—
67C22×C4×C53—22no—
68C22×C4×C53—22no—
69C22×C4×C53—41no—
70C22×C4×C53—41no—
71C22×C4×C53C2311yescharacteristic
72C22×C4×C53C2311yescharacteristic
73C22×C4×C53C2311yescharacteristic
74C22×C4×C53C2311yescharacteristic
75C22×Dic53—22no—
76C22×Dic53—22no—
77C22×Dic53—22no—
78C22×Dic53—22no—
79C22×Dic53—41no—
80C22×Dic53—41no—
81C22×Dic53C2311yescharacteristic
82C22×Dic53C2311yescharacteristic
83C22×Dic53C2311yescharacteristic
84C22×Dic53C2311yescharacteristic
85C22×Dic53C2311yescharacteristic
86C22×Dic53C2311yescharacteristic
87C22×Dic53C2311yescharacteristic
88C22×Dic53C2311yescharacteristic
89C2×Dic106—22no—
90C2×Dic106—22no—
91C2×Dic106—41no—
92C22⋊C4×C53—22no—
93C22⋊C4×C53—22no—
94C22⋊C4×C53—22no—
95C22⋊C4×C53D414yes—
96C22⋊Dic53—22no—
97C22⋊Dic53—22no—
98C22⋊Dic53—22no—
99C22⋊Dic53—22no—
100C22⋊Dic53—22no—
101C22⋊Dic53—22no—
102C22⋊Dic53—22no—
103C22⋊Dic53—41no—
104C22⋊Dic53—41no—
105C22⋊Dic53D414yes—
106C4⋊Dic53—22no—
107C4⋊Dic53—22no—
108C4⋊Dic53—41no—
109#11719919—22no—
110#11719919—22no—
111#11719919—22no—
112#11719919—22no—
113#11719919—41no—
114#11719919—41no—
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—28no—
124C23×C53—28no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53—21no—
128C23×C53—21no—
129C23×C53—21no—
130C23×C53—21no—
131C23×C53—44no—
132C23×C53—44no—
133C23×C53—44no—
134C23×C53—44no—
135C23×C53—44no—
136C23×C53C2×D412yes—
137C23×C53C2×D412yes—
138C23×C53C2×Q812yes—
139C23×C53C2×Q812yes—
140C23×C53C2×Q812yes—
141C23×C53#4112yes—
142C23×C53#4112yes—
143C23×C53C2411yescharacteristic
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×C53—44no—
156C2×C4×C53C2×D412yes—
157C2×C4×C53C2×D412yes—
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×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—44no—
181C2×Dic53—44no—
182C2×Dic53C2×D412yes—
183C2×Dic53C2×D412yes—
184Dic106—44no—
185C22×C53—22no—
186C22×C53—22no—
187C22×C53—22no—
188C22×C53—22no—
189C22×C53—22no—
190C22×C53—22no—
191C22×C53—22no—
192C22×C53—22no—
193C22×C53—22no—
194C22×C53—22no—
195C22×C53—22no—
196C22×C53—22no—
197C22×C53—22no—
198C22×C53—22no—
199C22×C53—22no—
200C22×C53—416no—
201C22×C53—41no—
202C22×C53—41no—
203C22×C53—41no—
204C22×C53—41no—
205C22×C53C22⋊Q814yescomplemented
206C22×C53C22⋊Q814yescomplemented
207C22×C53C22×D411yescharacteristic
208C22×C53C22×D411yescharacteristic
209C22×C53C22×Q811yescharacteristic
210C22×C53C2×#4111yescharacteristic
211C22×C53C2×#4111yescharacteristic
212C22×C53H(5,𝔽2)11yescharacteristic
213C22×C53H(5,𝔽2)11yescharacteristic
214C4×C53—22no—
215C4×C53—22no—
216C4×C53—41no—
217C4×C53—41no—
218Dic53—22no—
219Dic53—22no—
220Dic53—41no—
221Dic53—41no—
222Dic53—41no—
223Dic53—41no—
224Dic53—41no—
225Dic53—41no—
226#2437—531nomaximal, Sylow
227C2×C53—24no—
228C2×C53—24no—
229C2×C53—42no—
230C2×C53C2×C22⋊Q811yescharacteristic
231C2×C53C2×C22⋊Q811yescharacteristic
232C2×C53#54311yescharacteristic
233C2×C53D4211yescharacteristic
234C2×C53#54611yescharacteristic
235C2×C53#54611yescharacteristic
236C2×C53#55111yescharacteristic
237C2×C22⋊Q8—531no—
238C2×C22⋊Q8—531no—
239C22×C22⋊C4—531no—
240C22×C22⋊C4D5311yescharacteristic
241#399—531no—
242#396—531no—
243#396—531no—
244#393—531no—
245#385—531no—
246#385—531no—
247#385—531no—
248#385—531no—
249#380—531no—
250#380—531no—
251#379—531no—
252C53#243711yescharacteristic, complemented, Sylow
253C25C2×D5311yescharacteristic
254C22×Q8—531no—
255C23×C4—531no—
256C23×C4—531no—
257C23×C4C2×D5311yescharacteristic
258C23×C4C2×D5311yescharacteristic
259C22⋊Q8—1064no—
260C22⋊Q8—1064no—
261C2×C4⋊C4—531no—
262C2×C4⋊C4—531no—
263C2×C4⋊C4—531no—
264C2×C4⋊C4—531no—
265C2×C4⋊C4—531no—
266C2×C4⋊C4—531no—
267C2×C4⋊C4—531no—
268C2×C4⋊C4—531no—
269C2×C4⋊C4—531no—
270C2×C22⋊C4—24no—
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—1064no—
282C2×C22⋊C4C2×D5311yescharacteristic
283C2×C22⋊C4C2×D5311yescharacteristic
284C2×C22⋊C4C2×D5311yescharacteristic
285C2×C22⋊C4C2×D5311yescharacteristic
286#95—531no—
287#95—531no—
288#95—531no—
289#95—531no—
290#95—531no—
291#95—531no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—41no—
297C24—41no—
298C24Dic10614yescomplemented
299C24C22×D5311yescharacteristic
300C24C22×D5311yescharacteristic
301C24C22×D5311yescharacteristic
302C2×Q8—1062no—
303C2×Q8—1062no—
304C2×Q8—2121no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—531no—
312C22×C4—531no—
313C22×C4—531no—
314C22×C4—531no—
315C22×C4—531no—
316C22×C4—531no—
317C22×C4—531no—
318C22×C4—531no—
319C22×C4—1062no—
320C22×C4—1062no—
321C22×C4—1062no—
322C22×C4—1062no—
323C22×C4—2121no—
324C22×C4—2121no—
325C22×C4C22×D5311yescharacteristic
326C22×C4C22×D5311yescharacteristic
327C22×C4C22×D5311yescharacteristic
328C22×C4C22×D5311yescharacteristic
329C4⋊C4—1062no—
330C4⋊C4—1062no—
331C4⋊C4—1062no—
332C4⋊C4—1062no—
333C4⋊C4—1062no—
334C4⋊C4—1062no—
335C4⋊C4—2121no—
336C4⋊C4—2121no—
337C4⋊C4—2121no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—534no—
342C22⋊C4—1062no—
343C22⋊C4—1062no—
344C22⋊C4—1062no—
345C22⋊C4—1062no—
346C22⋊C4—1062no—
347C22⋊C4—1062no—
348C22⋊C4—1062no—
349C22⋊C4—2121no—
350C22⋊C4—2121no—
351C22⋊C4C22⋊D5314yes—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—28no—
361C23—28no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—44no—
369C23—44no—
370C23—44no—
371C23—44no—
372C23—44no—
373C23C2×Dic10612yes—
374C23C2×Dic10612yes—
375C23C2×Dic10612yes—
376C23#1171994512yes—
377C23D4×D5312yes—
378C23#1171994712yes—
379C23C2×C22⋊D5312yes—
380C23C23×D5311yescharacteristic
381Q8—2124no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—532no—
395C2×C4—532no—
396C2×C4—1061no—
397C2×C4—1061no—
398C2×C4—1061no—
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—2124no—
419C2×C4—2124no—
420C2×C4C2×C22⋊D5312yes—
421C2×C4C2×C22⋊D5312yes—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—416no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22#1313319414yescomplemented
443C22#1313319614yescomplemented
444C22C2×D4×D5311yescharacteristic
445C22C22×C22⋊D5311yescharacteristic
446C22C22×Dic10611yescharacteristic
447C22C2×#1171994711yescharacteristic
448C22C2×#1171994511yescharacteristic
449C22#1313326211yescharacteristic
450C22#1313326211yescharacteristic
451C4—22no—
452C4—22no—
453C4—41no—
454C4—41no—
455C4—1062no—
456C4—1062no—
457C4—2121no—
458C4—2121no—
459C4—2121no—
460C4—2121no—
461C4—2121no—
462C4—2121no—
463C2—24no—
464C2—24no—
465C2—42no—
466C2C2×#1313319411yescharacteristic
467C2C2×#1313319611yescharacteristic
468C2#1681124911yescharacteristic
469C2D4×C22⋊D5311yescharacteristic
470C2#1681126311yescharacteristic
471C2#1681126711yescharacteristic
472C2#1681129911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC22×C22⋊C4×C53C2
hypercenterC22×C22⋊C4D53
hypocenterC53#2437
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#13133196C2semidirect
2. C2×#13133194C2semidirect
4. #16810559C2semidirect
5. #16810546C2semidirect
6. #16810546C2semidirect
7. #16810535C2semidirect
8. #16810498C2semidirect
9. #16810496C2semidirect
10. #16810496C2semidirect
11. #16810495C2semidirect
12. #16810483C2semidirect
13. #16810481C2semidirect
16. C22×Dic106C22semidirect
21. C2×C4⋊Dic53C22semidirect
22. C2×C4⋊Dic53C22semidirect
23. C2×C4⋊Dic53C22semidirect
24. C2×#11719919C22semidirect
25. C2×#11719919C22semidirect
26. C2×#11719919C22semidirect
27. C2×#11719919C22semidirect
28. C2×#11719919C22semidirect
29. C2×#11719919C22semidirect
41. #13133124C22semidirect
42. #13133124C22semidirect
43. #13133124C22semidirect
44. #13133124C22semidirect
45. #13133124C22semidirect
46. #13133124C22semidirect
61. C24×C53Q8semidirect
205. C22×C53C22⋊Q8semidirect
206. C22×C53C22⋊Q8semidirect
252. C53#2437semidirect
298. C24Dic106semidirect
442. C22#13133194semidirect
443. C22#13133196semidirect