Finite Group Databasebeta

← #20283987

Subgroups — #20283987

Order 6784.

752 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133262C211yesmaximal, characteristic
2C2×#13133246C211yesmaximal, characteristic, complemented
3C2×SD16×D53C211yesmaximal, characteristic, complemented
4#16811196C211yesmaximal, characteristic, complemented
5#16810787C211yesmaximal, characteristic, complemented
6#16810786C211yesmaximal, characteristic, complemented
7#16810783C211yesmaximal, characteristic, complemented
8#16810782C211yesmaximal, characteristic, complemented
9#16810775C211yesmaximal, characteristic, complemented
10#16810774C211yesmaximal, characteristic, complemented
11#16810772C211yesmaximal, characteristic, complemented
12#16810626C211yesmaximal, characteristic, complemented
13#16810616C211yesmaximal, characteristic, complemented
14#16810588C211yesmaximal, characteristic, complemented
15C53×#450C211yesmaximal, characteristic, complemented
16#13133262—24no—
17#13133262—24no—
18C2×#11719945C2211yescharacteristic
19C2×#11719945C2211yescharacteristic
20C2×#11719947—21no—
21C2×#11719947C2211yescharacteristic
22C2×#11719947C2211yescharacteristic
23C2×Q8×D53C2211yescharacteristic, complemented
24C22×C22⋊D53—21no—
25C22×C22⋊D53—21no—
26C2×D4×D53—21no—
27C2×D4×D53C2211yescharacteristic
28C2×D4×D53C2211yescharacteristic
29#13133246—24no—
30C2×Dic212C2211yescharacteristic, complemented
31C2×#11719925C2211yescharacteristic, complemented
32C2×#11719913C2211yescharacteristic, complemented
33C2×#11719923C2211yescharacteristic, complemented
34C2×#11719923C2211yescharacteristic, complemented
35C2×#11719924C2211yescharacteristic, complemented
36SD16×D53—24no—
37C2×#11719912C2211yescharacteristic, complemented
38C2×C8×D53C2211yescharacteristic, complemented
39#13133216—21no—
40#13133203—21no—
41#13133202—21no—
42#13133199—21no—
43#13133197C2211yescharacteristic, complemented
44#13133196C2211yescharacteristic, complemented
45#13133193C2211yescharacteristic, complemented
46#13133175C2211yescharacteristic, complemented
47#13133142C2211yescharacteristic, complemented
48#13133140C2211yescharacteristic, complemented
49#13133139C2211yescharacteristic, complemented
50#13133138C2211yescharacteristic, complemented
51#13133138C2211yescharacteristic, complemented
52#13133137C2211yescharacteristic, complemented
53C22⋊#88322C2211yescharacteristic, complemented
54#13133129C2211yescharacteristic, complemented
55#13133129C2211yescharacteristic, complemented
56C22×D4×C53C2211yescharacteristic
57C2×SD16×C53C2211yescharacteristic, complemented
58C2×SD16×C53C2211yescharacteristic, complemented
59C22⋊Q8×C53C2211yescharacteristic, complemented
60#102×C53C2211yescharacteristic, complemented
61#102×C53C2211yescharacteristic, complemented
62C22⋊C8×C53C2211yescharacteristic, complemented
63C24×C53—21no—
64C23×D53—21no—
65C23×D53—21no—
66C2×Q8×C53C2311yescharacteristic
67C2×D4×C53—22no—
68C2×D4×C53—22no—
69C2×D4×C53—21no—
70C2×D4×C53C2311yescharacteristic
71C2×D4×C53C2311yescharacteristic
72C22×C4×C53C2311yescharacteristic
73C2×C22⋊D53—28no—
74C2×C22⋊D53—28no—
75C2×C22⋊D53—21no—
76C2×C22⋊D53—21no—
77C2×C22⋊D53—21no—
78C2×C22⋊D53—21no—
79C2×C22⋊D53—21no—
80C2×C22⋊D53D411yescharacteristic
81C2×C22⋊D53D411yescharacteristic
82C22×Dic53—21no—
83C22×Dic53—21no—
84Q8×D53—22no—
85Q8×D53—22no—
86#11719947—28no—
87#11719947—22no—
88#11719947—22no—
89#11719947—22no—
90#11719947D414yes—
91D4×D53—28no—
92D4×D53—22no—
93D4×D53—22no—
94D4×D53—22no—
95D4×D53D414yes—
96#11719945—22no—
97#11719945—22no—
98#11719945—22no—
99#11719945—22no—
100C2×D212D411yescharacteristic
101C2×D212C2311yescharacteristic
102C2×C4×D53—21no—
103C2×C4×D53—21no—
104C2×C4×D53C2311yescharacteristic
105C2×C4×D53C2311yescharacteristic
106C2×Dic106—21no—
107C2×Dic106D411yescharacteristic
108C2×Dic106C2311yescharacteristic
109C2×Dic106C2311yescharacteristic
110SD16×C53—22no—
111SD16×C53—22no—
112C2×C8×C53C2311yescharacteristic
113C2×C8×C53C2311yescharacteristic
114C4⋊C4×C53—21no—
115C4⋊C4×C53C2311yescharacteristic
116C22⋊C4×C53—21no—
117C22⋊Dic53—21no—
118#11719925—22no—
119#11719924—22no—
120#11719923—22no—
121#11719923—22no—
122#11719921—21no—
123#11719921—21no—
124#11719921—21no—
125C4⋊Dic53—21no—
126C4⋊Dic53C2311yescharacteristic
127#11719919—21no—
128#11719919—21no—
129C4×Dic53—21no—
130C2×#88322C2311yescharacteristic
131C2×#88322C2311yescharacteristic
132Dic212—22no—
133#11719913—22no—
134#11719912—22no—
135C8×D53—22no—
136C23×C53—24no—
137C23×C53—21no—
138C23×C53—21no—
139C23×C53—81no—
140C23×C53—81no—
141C23×C53C2×D411yescharacteristic
142C22×D53—21no—
143C22×D53—24no—
144C22×D53—24no—
145C22×D53—81no—
146C22×D53—81no—
147C22×D53—81no—
148C22×D53—81no—
149C22×D53C2×D411yescharacteristic
150C22×D53C2×D411yescharacteristic
151Q8×C53—21no—
152Q8×C53—21no—
153D4×C53—24no—
154D4×C53—21no—
155D4×C53—21no—
156D4×C53C2×D412yes—
157D4×C53C2×D412yes—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53C2×D411yescharacteristic
163C2×C4×C53C2411yescharacteristic
164C22⋊D53—24no—
165C22⋊D53—24no—
166C22⋊D53—24no—
167C22⋊D53—22no—
168C22⋊D53—24no—
169C22⋊D53—22no—
170C22⋊D53—24no—
171C2×Dic53—24no—
172C2×Dic53—21no—
173C2×Dic53—24no—
174C2×Dic53—21no—
175C2×Dic53—21no—
176C2×Dic53C2×D411yescharacteristic
177C2×Dic53C2×D411yescharacteristic
178D212—22no—
179D212—21no—
180D212C2×D412yes—
181C4×D53—24no—
182C4×D53—21no—
183C4×D53—21no—
184C4×D53—42no—
185C4×D53C2×D412yes—
186C4×D53C2×D412yes—
187Dic106—21no—
188Dic106—22no—
189Dic106—21no—
190Dic106—21no—
191Dic106—42no—
192Dic106C2×D412yes—
193C8×C53—21no—
194C8×C53—21no—
195#88322—21no—
196#88322—21no—
197C22×C53—22no—
198C22×C53—22no—
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—21no—
202C22×C53—41no—
203C22×C53—41no—
204C22×C53—41no—
205C22×C53—41no—
206C22×C53—84no—
207C22×C53C22×D411yescharacteristic
208C2×D53—21no—
209C2×D53—21no—
210C2×D53—21no—
211C2×D53—22no—
212C2×D53—21no—
213C2×D53—21no—
214C2×D53—41no—
215C2×D53—41no—
216C2×D53—84no—
217C2×D53—84no—
218C2×D53#12012yes—
219C4×C53—21no—
220C4×C53—41no—
221C4×C53—41no—
222C4×C53C22×D411yescharacteristic
223C4×C53C22×D411yescharacteristic
224Dic53—21no—
225Dic53—22no—
226Dic53—41no—
227Dic53—41no—
228Dic53#12012yes—
229#3009—531nomaximal, Sylow
230C2×C53—42no—
231C2×C53—42no—
232C2×C53—41no—
233C2×C53C2×#12011yescharacteristic
234C2×C53#57711yescharacteristic
235C2×C53#57711yescharacteristic
236D53—22no—
237D53—41no—
238D53—42no—
239C2×H(5,𝔽2)—531no—
240C2×#137—531no—
241C22×SD16—531no—
242#536—531no—
243#452—531no—
244#452—531no—
245#452—531no—
246#452—531no—
247#450—531no—
248#450—531no—
249#450—531no—
250#450D5311yescharacteristic, complemented
251#418—531no—
252#416—531no—
253#408—531no—
254C53#300911yescharacteristic, complemented, Sylow
255H(5,𝔽2)—1064no—
256H(5,𝔽2)—1064no—
257C2×#41—531no—
258C2×#41—531no—
259C2×#41—531no—
260C2×#41—531no—
261C2×#41—1061no—
262C22×Q8—531no—
263C22×D4—531no—
264C22×D4—531no—
265C22×D4—1061no—
266C22×D4—1061no—
267C22×D4—1061no—
268C22×D4C2×D5311yescharacteristic
269#137—1064no—
270C2×Q16—531no—
271C2×Q16—531no—
272C2×SD16—531no—
273C2×SD16—531no—
274C2×SD16—531no—
275C2×SD16—531no—
276C2×SD16—1064no—
277C2×SD16C2×D5311yescharacteristic, complemented
278C2×SD16C2×D5311yescharacteristic, complemented
279C2×M16—531no—
280C22×C8—531no—
281C4⋊Q8—1061no—
282#124—1061no—
283#123—1061no—
284#123—1061no—
285C22⋊Q8—531no—
286C22⋊Q8—531no—
287C22⋊Q8—531no—
288C22⋊Q8C2×D5311yescharacteristic, complemented
289#117—531no—
290#103—531no—
291#103—531no—
292#102—531no—
293#102—531no—
294#102—531no—
295#102—531no—
296#102C2×D5311yescharacteristic, complemented
297#102C2×D5311yescharacteristic, complemented
298C22⋊C8—531no—
299C22⋊C8—531no—
300C22⋊C8—531no—
301C22⋊C8C2×D5311yescharacteristic, complemented
302C24—21no—
303C24—1061no—
304C24—1061no—
305#41—534no—
306#41—1062no—
307#41—1062no—
308#41—1062no—
309#41—1062no—
310#41—1062no—
311#41—1062no—
312#41—1068no—
313#41—1062no—
314C2×Q8—531no—
315C2×Q8—531no—
316C2×Q8—531no—
317C2×Q8—1061no—
318C2×Q8—1062no—
319C2×Q8—1062no—
320C2×Q8C22×D5311yescharacteristic
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—21no—
324C2×D4—534no—
325C2×D4—531no—
326C2×D4—531no—
327C2×D4—531no—
328C2×D4—531no—
329C2×D4—1062no—
330C2×D4—1061no—
331C2×D4—1061no—
332C2×D4—1062no—
333C2×D4—1068no—
334C2×D4—1061no—
335C2×D4—1061no—
336C2×D4—1068no—
337C2×D4—1068no—
338C2×D4—1061no—
339C2×D4—1062no—
340C2×D4C22×D5311yescharacteristic
341C2×D4C22×D5311yescharacteristic
342C22×C4—531no—
343C22×C4—531no—
344C22×C4—1061no—
345C22×C4—1061no—
346C22×C4—1061no—
347C22×C4—1061no—
348C22×C4C22×D5311yescharacteristic
349Q16—1062no—
350Q16—1062no—
351SD16—22no—
352SD16—22no—
353SD16—1062no—
354SD16—1062no—
355SD16—1062no—
356SD16—1062no—
357M16—1062no—
358C2×C8—531no—
359C2×C8—531no—
360C2×C8—1062no—
361C2×C8C22×D5311yescharacteristic
362C2×C8C22×D5311yescharacteristic
363C4⋊C4—21no—
364C4⋊C4—531no—
365C4⋊C4—1061no—
366C4⋊C4—1061no—
367C4⋊C4—1061no—
368C4⋊C4C22×D5311yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—1061no—
371C22⋊C4—1061no—
372C22⋊C4—1061no—
373C22⋊C4—1061no—
374C42—1061no—
375C23—24no—
376C23—21no—
377C23—21no—
378C23—81no—
379C23—81no—
380C23—531no—
381C23—531no—
382C23—1064no—
383C23—1064no—
384C23—1061no—
385C23—4241no—
386C23—4241no—
387C23—4241no—
388C23—4241no—
389C23D4×D5311yescharacteristic
390Q8—21no—
391Q8—21no—
392Q8—532no—
393Q8—1062no—
394Q8—1061no—
395Q8—1061no—
396Q8—1061no—
397Q8—2122no—
398D4—24no—
399D4—21no—
400D4—21no—
401D4—532no—
402D4—1062no—
403D4—1062no—
404D4—1064no—
405D4—1062no—
406D4—1064no—
407D4—1064no—
408D4—1064no—
409D4—1064no—
410D4—1061no—
411D4D4×D5312yes—
412D4D4×D5312yes—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—531no—
418C2×C4—532no—
419C2×C4—532no—
420C2×C4—531no—
421C2×C4—1064no—
422C2×C4—1064no—
423C2×C4—1064no—
424C2×C4—1061no—
425C2×C4—1061no—
426C2×C4—1061no—
427C2×C4—1061no—
428C2×C4—1061no—
429C2×C4—2122no—
430C2×C4D4×D5311yescharacteristic
431C2×C4C23×D5311yescharacteristic
432C8—21no—
433C8—21no—
434C8—1061no—
435C8—1061no—
436C22—21no—
437C22—22no—
438C22—22no—
439C22—21no—
440C22—21no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—84no—
446C22—532no—
447C22—1061no—
448C22—1061no—
449C22—1062no—
450C22—1061no—
451C22—1061no—
452C22—1061no—
453C22—4244no—
454C22—4244no—
455C22—2121no—
456C22—2121no—
457C22C2×D4×D5311yescharacteristic
458C4—21no—
459C4—41no—
460C4—41no—
461C4—532no—
462C4—1061no—
463C4—1062no—
464C4—2121no—
465C4—2121no—
466C4C2×D4×D5311yescharacteristic
467C4C2×D4×D5311yescharacteristic
468C2—41no—
469C2—42no—
470C2—42no—
471C2—1062no—
472C2—2121no—
473C2—2122no—
474C2#120×D5311yescharacteristic
475C2#1681142911yescharacteristic
476C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#450C2
socleC22×C53C22×D4
hypercenter#450D53
hypocenterC53#3009
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×#13133246C2semidirect
3. C2×SD16×D53C2semidirect
4. #16811196C2semidirect
5. #16810787C2semidirect
6. #16810786C2semidirect
7. #16810783C2semidirect
8. #16810782C2semidirect
9. #16810775C2semidirect
10. #16810774C2semidirect
11. #16810772C2semidirect
12. #16810626C2semidirect
13. #16810616C2semidirect
14. #16810588C2semidirect
15. C53×#450C2semidirect
23. C2×Q8×D53C22semidirect
30. C2×Dic212C22semidirect
31. C2×#11719925C22semidirect
32. C2×#11719913C22semidirect
33. C2×#11719923C22semidirect
34. C2×#11719923C22semidirect
35. C2×#11719924C22semidirect
37. C2×#11719912C22semidirect
38. C2×C8×D53C22semidirect
43. #13133197C22semidirect
44. #13133196C22semidirect
45. #13133193C22semidirect
46. #13133175C22semidirect
47. #13133142C22semidirect
48. #13133140C22semidirect
49. #13133139C22semidirect
50. #13133138C22semidirect
51. #13133138C22semidirect
52. #13133137C22semidirect
53. C22⋊#88322C22semidirect
54. #13133129C22semidirect
55. #13133129C22semidirect
57. C2×SD16×C53C22semidirect
58. C2×SD16×C53C22semidirect
59. C22⋊Q8×C53C22semidirect
60. #102×C53C22semidirect
61. #102×C53C22semidirect
62. C22⋊C8×C53C22semidirect
250. #450D53semidirect
254. C53#3009semidirect
277. C2×SD16C2×D53semidirect
278. C2×SD16C2×D53semidirect
288. C22⋊Q8C2×D53semidirect
296. #102C2×D53semidirect
297. #102C2×D53semidirect
301. C22⋊C8C2×D53semidirect