Finite Group Databasebeta

← #20286486

Subgroups — #20286486

Order 6784.

448 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
1#16811438C211yesmaximal, characteristic, complemented
2#16811431C211yesmaximal, characteristic, complemented
3#16811430C211yesmaximal, characteristic, complemented
4#16811016C211yesmaximal, characteristic, complemented
5#16810896C211yesmaximal, characteristic, complemented
6#16810895C211yesmaximal, characteristic, complemented
7#16810888C211yesmaximal, characteristic, complemented
8#16810886C211yesmaximal, characteristic, complemented
9#16810806C211yesmaximal, characteristic, complemented
10#16810804C211yesmaximal, characteristic, complemented
11#16810799C211yesmaximal, characteristic, complemented
12#16810796C211yesmaximal, characteristic, complemented
13#16810745C211yesmaximal, characteristic, complemented
14#16810743C211yesmaximal, characteristic, complemented
15C53×#578C211yesmaximal, characteristic, complemented
16#13133264C2211yescharacteristic
17#13133263C2211yescharacteristic
18#13133249—21no—
19#13133248—21no—
20#13133248—21no—
21#13133248C2211yescharacteristic
22#13133248C2211yescharacteristic
23#13133247—21no—
24#13133247C2211yescharacteristic
25#13133247C2211yescharacteristic, complemented
26#13133242—21no—
27#13133242C2211yescharacteristic
28#13133242C2211yescharacteristic
29#13133241—21no—
30#13133238—21no—
31#13133237—21no—
32#13133237—21no—
33#13133237C2211yescharacteristic
34C2×#11719925—21no—
35C2×#11719925C2211yescharacteristic, complemented
36C2×#11719913C2211yescharacteristic
37C2×#11719924—21no—
38C2×#11719924C2211yescharacteristic
39SD16×D53—21no—
40#13133224C2211yescharacteristic
41#13133224C2211yescharacteristic, complemented
42#13133223C2211yescharacteristic
43#13133215C2211yescharacteristic, complemented
44#13133206C2211yescharacteristic
45#13133153C2211yescharacteristic, complemented
46#13133152C2211yescharacteristic, complemented
47#13133144C2211yescharacteristic
48#13133144C2211yescharacteristic, complemented
49#13133144C2211yescharacteristic, complemented
50#13133143C2211yescharacteristic
51#13133136C2211yescharacteristic, complemented
52#13133136C2211yescharacteristic, complemented
53#13133135C2211yescharacteristic, complemented
54#13133133C2211yescharacteristic
55#13133127C2211yescharacteristic, complemented
562−1+4×C53C2211yescharacteristic
572−1+4×C53C2211yescharacteristic, complemented
58#137×C53—21no—
59#137×C53—21no—
60#137×C53C2211yescharacteristic
61#137×C53C2211yescharacteristic, complemented
62#135×C53—21no—
63#135×C53C2211yescharacteristic
64C2×Q16×C53—21no—
65C2×Q16×C53C2211yescharacteristic, complemented
66#131×C53C2211yescharacteristic, complemented
67#41×C53—21no—
68#41×C53—21no—
69#41×C53—21no—
70#41×C53—21no—
71#41×C53—41no—
72#41×C53—41no—
73#41×C53C2311yescharacteristic
74#41×C53C2311yescharacteristic
75#41×C53C2311yescharacteristic
76C2×Q8×C53—21no—
77C2×Q8×C53—21no—
78C2×Q8×C53—21no—
79C2×Q8×C53—21no—
80C2×Q8×C53C2311yescharacteristic
81C2×Q8×C53C2311yescharacteristic
82#11719949—21no—
83Q8×D53—21no—
84#11719947—21no—
85#11719947—41no—
86D4×D53—21no—
87D4×D53—41no—
88#11719945—21no—
89#11719945—21no—
90#11719945C2311yescharacteristic
91C2×D212—21no—
92C2×D212C2311yescharacteristic
93C2×Dic106—21no—
94C2×Dic106C2311yescharacteristic
95Q16×C53—21no—
96Q16×C53—21no—
97Q16×C53—21no—
98Q16×C53—21no—
99Q16×C53D411yescharacteristic
100SD16×C53—21no—
101SD16×C53—21no—
102SD16×C53D411yescharacteristic
103SD16×C53D411yescharacteristic
104D8×C53D411yescharacteristic
105M16×C53—21no—
106M16×C53C2311yescharacteristic
107C2×C8×C53—21no—
108C2×C8×C53C2311yescharacteristic
109#11719925—21no—
110#11719925—21no—
111#11719925—21no—
112#11719925—21no—
113#11719925—21no—
114#11719925—21no—
115#11719925D411yescharacteristic
116#11719924—21no—
117#11719924—21no—
118#11719924—21no—
119#11719924—21no—
120#11719924—21no—
121#11719924—21no—
122#11719924D411yescharacteristic
123#11719923—21no—
124#11719923—21no—
125#11719923D411yescharacteristic
126#11719922—21no—
127#11719922—21no—
128#11719922D411yescharacteristic
129#11719921—41no—
130#11719919—41no—
131C4×Dic53C2311yescharacteristic
132#11719917—21no—
133#11719917—21no—
134#11719917C2311yescharacteristic
135#11719917C2311yescharacteristic
136#11719917C2311yescharacteristic
137C2×#88322—21no—
138C2×#88322—21no—
139C2×#88322C2311yescharacteristic
140#11719913—21no—
141#11719913—21no—
142#11719912—21no—
143C8×D53—21no—
144C22×D53—21no—
145C22×D53—41no—
146Q8×C53—21no—
147Q8×C53—21no—
148Q8×C53—21no—
149Q8×C53—21no—
150Q8×C53—41no—
151Q8×C53—41no—
152Q8×C53C2×D411yescharacteristic
153Q8×C53C2×D411yescharacteristic
154Q8×C53C2×D411yescharacteristic
155D4×C53—21no—
156D4×C53—21no—
157D4×C53—21no—
158D4×C53—41no—
159D4×C53—41no—
160D4×C53C2×D411yescharacteristic
161D4×C53C2×D411yescharacteristic
162D4×C53C2×D411yescharacteristic
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53C2411yescharacteristic
173C22⋊D53—21no—
174C22⋊D53—41no—
175C22⋊D53—41no—
176C2×Dic53—21no—
177C2×Dic53—21no—
178C2×Dic53—41no—
179D212—21no—
180D212—21no—
181D212—41no—
182D212C2×D411yescharacteristic
183C4×D53—21no—
184C4×D53—41no—
185C4×D53—41no—
186Dic106—21no—
187Dic106—21no—
188Dic106—41no—
189Dic106C2×D411yescharacteristic
190C8×C53—21no—
191C8×C53C2×D411yescharacteristic
192C8×C53C2×D411yescharacteristic
193#88322—21no—
194#88322—21no—
195#88322—21no—
196#88322C2×D411yescharacteristic
197#88322C2×D411yescharacteristic
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—21no—
201C22×C53H(5,𝔽2)11yescharacteristic
202C2×D53—21no—
203C2×D53—41no—
204C2×D53—81no—
205C2×D53—161no—
206C4×C53—21no—
207C4×C53—21no—
208C4×C53—21no—
209C4×C53—41no—
210C4×C53—41no—
211C4×C53C22×D411yescharacteristic
212C4×C53C22×D411yescharacteristic
213Dic53—21no—
214Dic53—41no—
215Dic53—41no—
216#3289—531nomaximal, Sylow
217C2×C53—21no—
218C2×C53—41no—
219C2×C53—41no—
220C2×C53—41no—
221C2×C53D4211yescharacteristic
222D53—41no—
223D53—81no—
224#578—531no—
225#578D5311yescharacteristic, complemented
226#577—531no—
227#577—531no—
228#497—531no—
229#473—531no—
230#473—531no—
231#471—531no—
232#471—531no—
233#456—531no—
234#456—531no—
235#455—531no—
236#454—531no—
237#444—531no—
238#444—531no—
239C53#328911yescharacteristic, complemented, Sylow
2402−1+4—531no—
2412−1+4C2×D5311yescharacteristic
2422−1+4C2×D5311yescharacteristic, complemented
243H(5,𝔽2)—531no—
244#137—21no—
245#137—21no—
246#137—531no—
247#137—531no—
248#137—531no—
249#137—531no—
250#137—1061no—
251#137—1061no—
252#137—1061no—
253#137—1061no—
254#137C2×D5311yescharacteristic
255#137C2×D5311yescharacteristic, complemented
256AGL(1,ℤ/8)—531no—
257AGL(1,ℤ/8)—531no—
258AGL(1,ℤ/8)—1061no—
259AGL(1,ℤ/8)—1061no—
260#135—21no—
261#135—531no—
262#135—1061no—
263#135—1061no—
264#135—1061no—
265#135C2×D5311yescharacteristic
266C2×Q16—21no—
267C2×Q16—531no—
268C2×Q16—1061no—
269C2×Q16C2×D5311yescharacteristic, complemented
270C2×SD16—531no—
271C2×SD16—531no—
272C2×SD16—1061no—
273C2×SD16—1061no—
274#131—531no—
275#131—531no—
276#131—531no—
277#131C2×D5311yescharacteristic, complemented
278C4⋊Q8—531no—
279#124—531no—
280#108—531no—
281#108—531no—
282#104—531no—
283#104—531no—
284#104—531no—
285#104—531no—
286#101—531no—
287#101—531no—
288#101—531no—
289#100—531no—
290#97—531no—
291#41—21no—
292#41—21no—
293#41—21no—
294#41—21no—
295#41—41no—
296#41—41no—
297#41—531no—
298#41—1061no—
299#41—1061no—
300#41—1061no—
301#41—1061no—
302#41—2121no—
303#41C22×D5311yescharacteristic
304#41C22×D5311yescharacteristic
305#41C22×D5311yescharacteristic
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—531no—
311C2×Q8—1061no—
312C2×Q8—1061no—
313C2×Q8C22×D5311yescharacteristic
314C2×Q8C22×D5311yescharacteristic
315C2×D4—531no—
316C2×D4—1061no—
317C2×D4—1061no—
318C2×D4—2121no—
319Q16—21no—
320Q16—21no—
321Q16—21no—
322Q16—21no—
323Q16—531no—
324Q16—1061no—
325Q16—1061no—
326Q16—1061no—
327Q16—1061no—
328Q16—1061no—
329Q16—1061no—
330Q16C22⋊D5311yescharacteristic
331SD16—21no—
332SD16—21no—
333SD16—531no—
334SD16—531no—
335SD16—1061no—
336SD16—1061no—
337SD16—1061no—
338SD16—1061no—
339SD16—1061no—
340SD16—1061no—
341SD16—1061no—
342SD16—1061no—
343SD16—1061no—
344SD16—1061no—
345SD16C22⋊D5311yescharacteristic
346SD16C22⋊D5311yescharacteristic
347D8—531no—
348D8—1061no—
349D8—1061no—
350D8C22⋊D5311yescharacteristic
351M16—21no—
352M16—531no—
353M16—531no—
354M16—531no—
355M16—1061no—
356M16—1061no—
357M16—1061no—
358M16C22×D5311yescharacteristic
359C2×C8—21no—
360C2×C8—531no—
361C2×C8—1061no—
362C2×C8—1061no—
363C2×C8—1061no—
364C2×C8C22×D5311yescharacteristic
365C4⋊C4—2121no—
366C22⋊C4—2121no—
367C42—531no—
368C23—1061no—
369C23—2121no—
370Q8—21no—
371Q8—21no—
372Q8—21no—
373Q8—21no—
374Q8—41no—
375Q8—41no—
376Q8—531no—
377Q8—1061no—
378Q8—1061no—
379Q8—2121no—
380Q8D4×D5311yescharacteristic
381Q8C2×C22⋊D5311yescharacteristic
382Q8C2×C22⋊D5311yescharacteristic
383D4—21no—
384D4—21no—
385D4—21no—
386D4—41no—
387D4—41no—
388D4—531no—
389D4—1061no—
390D4—1061no—
391D4—1061no—
392D4—2121no—
393D4—2121no—
394D4—2121no—
395D4D4×D5311yescharacteristic
396D4C2×C22⋊D5311yescharacteristic
397D4C2×C22⋊D5311yescharacteristic
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—1061no—
408C2×C4—1061no—
409C2×C4—1061no—
410C2×C4—2121no—
411C2×C4—2121no—
412C2×C4—2121no—
413C2×C4C23×D5311yescharacteristic
414C8—21no—
415C8—531no—
416C8—531no—
417C8—1061no—
418C8—1061no—
419C8—1061no—
420C8C2×C22⋊D5311yescharacteristic
421C8C2×C22⋊D5311yescharacteristic
422C22—21no—
423C22—21no—
424C22—21no—
425C22—8481no—
426C22—1061no—
427C22—4241no—
428C22—2121no—
429C22#1313326211yescharacteristic
430C4—21no—
431C4—21no—
432C4—21no—
433C4—41no—
434C4—41no—
435C4—1061no—
436C4—2121no—
437C4—2121no—
438C4C2×D4×D5311yescharacteristic
439C4C22×C22⋊D5311yescharacteristic
440C2—21no—
441C2—41no—
442C2—41no—
443C2—41no—
444C2—4241no—
445C2—2121no—
446C2D4×C22⋊D5311yescharacteristic
subgroupG/H
centerC2D4×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#578C2
socleC2×C53D42
hypercenter#578D53
hypocenterC53#3289
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #16811438C2semidirect
2. #16811431C2semidirect
3. #16811430C2semidirect
4. #16811016C2semidirect
5. #16810896C2semidirect
6. #16810895C2semidirect
7. #16810888C2semidirect
8. #16810886C2semidirect
9. #16810806C2semidirect
10. #16810804C2semidirect
11. #16810799C2semidirect
12. #16810796C2semidirect
13. #16810745C2semidirect
14. #16810743C2semidirect
15. C53×#578C2semidirect
25. #13133247C22semidirect
35. C2×#11719925C22semidirect
41. #13133224C22semidirect
43. #13133215C22semidirect
45. #13133153C22semidirect
46. #13133152C22semidirect
48. #13133144C22semidirect
49. #13133144C22semidirect
51. #13133136C22semidirect
52. #13133136C22semidirect
53. #13133135C22semidirect
55. #13133127C22semidirect
57. 2−1+4×C53C22semidirect
61. #137×C53C22semidirect
65. C2×Q16×C53C22semidirect
66. #131×C53C22semidirect
225. #578D53semidirect
239. C53#3289semidirect
242. 2−1+4C2×D53semidirect
255. #137C2×D53semidirect
269. C2×Q16C2×D53semidirect
277. #131C2×D53semidirect