Finite Group Databasebeta

← #20284450

Subgroups — #20284450

Order 6784.

446 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811422C211yesmaximal, characteristic
2C2×#13133249C211yesmaximal, characteristic, complemented
3#137×D53C211yesmaximal, characteristic, complemented
4#16811388C211yesmaximal, characteristic, complemented
5#16810897C211yesmaximal, characteristic, complemented
6#16810896C211yesmaximal, characteristic, complemented
7#16810895C211yesmaximal, characteristic, complemented
8#16810889C211yesmaximal, characteristic, complemented
9#16810888C211yesmaximal, characteristic, complemented
10#16810887C211yesmaximal, characteristic, complemented
11#16810886C211yesmaximal, characteristic, complemented
12#16810670C211yesmaximal, characteristic, complemented
13#16810606C211yesmaximal, characteristic, complemented
14#101×D53C211yesmaximal, characteristic, complemented
15C53×#473C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17#41×D53C2211yescharacteristic
18C2×Q8×D53C2211yescharacteristic
19#13133250C2211yescharacteristic
20#13133249—21no—
21#13133249—21no—
22#13133249—21no—
23#13133249—21no—
24#13133249—21no—
25#13133249—21no—
26#13133248C2211yescharacteristic
27#13133248C2211yescharacteristic
28#13133247C2211yescharacteristic
29#13133246—21no—
30#13133245C2211yescharacteristic
31#13133244—21no—
32#13133242C2211yescharacteristic
33#13133240—21no—
34#13133238—21no—
35C2×#11719925C2211yescharacteristic, complemented
36Q16×D53—21no—
37C2×#11719913C2211yescharacteristic, complemented
38C2×#11719924C2211yescharacteristic, complemented
39SD16×D53—21no—
40#13133225—21no—
41#13133225C2211yescharacteristic
42#13133224C2211yescharacteristic
43#13133223—21no—
44#13133223C2211yescharacteristic
45C2×#11719912—21no—
46C2×#11719912C2211yescharacteristic
47M16×D53—21no—
48M16×D53C2211yescharacteristic
49M16×D53C2211yescharacteristic
50M16×D53C2211yescharacteristic
51#13133153C2211yescharacteristic, complemented
52#13133152C2211yescharacteristic, complemented
53#13133152C2211yescharacteristic, complemented
54#13133136C2211yescharacteristic, complemented
55#13133136C2211yescharacteristic, complemented
56#13133135C2211yescharacteristic, complemented
57#13133135C2211yescharacteristic, complemented
58#13133133C2211yescharacteristic, complemented
59#13133133C2211yescharacteristic, complemented
60#137×C53C2211yescharacteristic
61#137×C53C2211yescharacteristic
62C2×Q16×C53C2211yescharacteristic, complemented
63#131×C53C2211yescharacteristic
64#108×C53C2211yescharacteristic, complemented
65#101×C53C2211yescharacteristic, complemented
66#101×C53C2211yescharacteristic, complemented
67#41×C53C2311yescharacteristic
68C2×Q8×C53C2311yescharacteristic
69C2×Q8×C53C2311yescharacteristic
70#11719949—21no—
71#11719949—21no—
72#11719949—21no—
73#11719949—21no—
74#11719949D411yescharacteristic
75Q8×D53—21no—
76Q8×D53—21no—
77Q8×D53—21no—
78Q8×D53—21no—
79Q8×D53D411yescharacteristic
80#11719947—21no—
81#11719947D411yescharacteristic
82D4×D53—21no—
83D4×D53D411yescharacteristic
84#11719945—21no—
85#11719945C2311yescharacteristic
86C2×D212—21no—
87C2×D212C2311yescharacteristic
88C2×C4×D53—21no—
89C2×C4×D53—21no—
90C2×C4×D53—21no—
91C2×C4×D53C2311yescharacteristic
92C2×Dic106—21no—
93C2×Dic106C2311yescharacteristic
94Q16×C53—21no—
95Q16×C53—21no—
96Q16×C53—21no—
97Q16×C53—21no—
98SD16×C53—21no—
99SD16×C53—21no—
100M16×C53—21no—
101M16×C53C2311yescharacteristic
102M16×C53C2311yescharacteristic
103M16×C53C2311yescharacteristic
104C2×C8×C53—21no—
105C2×C8×C53C2311yescharacteristic
106#11719925—21no—
107#11719925—21no—
108#11719925—21no—
109#11719925—21no—
110#11719925—21no—
111#11719924—21no—
112#11719924—21no—
113#11719924—21no—
114#11719924—21no—
115#11719924—21no—
116#11719923—21no—
117#11719922—21no—
118#11719917—21no—
119#11719917C2311yescharacteristic
120#11719917C2311yescharacteristic
121#11719917C2311yescharacteristic
122C2×#88322—21no—
123C2×#88322C2311yescharacteristic
124Dic212—21no—
125D424—21no—
126#11719913—21no—
127#11719913—21no—
128#11719913—21no—
129#11719913—21no—
130#11719912—21no—
131#11719912—21no—
132#11719912—21no—
133#11719912—21no—
134#11719912—21no—
135#11719912D411yescharacteristic
136#11719912D411yescharacteristic
137#11719912D411yescharacteristic
138#11719912D411yescharacteristic
139C8×D53—21no—
140C8×D53—21no—
141C8×D53—21no—
142C8×D53—21no—
143C8×D53—21no—
144C22×D53—21no—
145C22×D53—21no—
146C22×D53#4111yescharacteristic
147Q8×C53—21no—
148Q8×C53—21no—
149Q8×C53—21no—
150Q8×C53—21no—
151Q8×C53C2×D411yescharacteristic
152D4×C53—21no—
153D4×C53C2×D411yescharacteristic
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53C2411yescharacteristic
158C22⋊D53—21no—
159C22⋊D53—21no—
160C22⋊D53—21no—
161C2×Dic53—21no—
162C2×Dic53—21no—
163C2×Dic53#4111yescharacteristic
164D212—21no—
165D212—21no—
166D212—21no—
167D212—41no—
168D212—41no—
169D212C2×D411yescharacteristic
170C4×D53—21no—
171C4×D53—21no—
172C4×D53—21no—
173C4×D53—41no—
174C4×D53—41no—
175C4×D53—41no—
176C4×D53—41no—
177C4×D53C2×D411yescharacteristic
178C4×D53C2×D411yescharacteristic
179C4×D53C2×D411yescharacteristic
180C4×D53C2×D411yescharacteristic
181Dic106—21no—
182Dic106—21no—
183Dic106—21no—
184Dic106—41no—
185Dic106—41no—
186Dic106C2×D411yescharacteristic
187C8×C53—21no—
188C8×C53—21no—
189C8×C53—21no—
190C8×C53C2×D411yescharacteristic
191C8×C53C2×D411yescharacteristic
192#88322—21no—
193#88322—21no—
194#88322—21no—
195#88322C2×D411yescharacteristic
196#88322C2×D411yescharacteristic
197C22×C53—21no—
198C22×C53C2×#4111yescharacteristic
199C2×D53—21no—
200C2×D53—41no—
201C2×D53—41no—
202C2×D53—81no—
203C2×D53—81no—
204C2×D53#12111yescharacteristic
205C2×D53#12111yescharacteristic
206C4×C53—21no—
207C4×C53—41no—
208C4×C53—41no—
209C4×C53C22×D411yescharacteristic
210C4×C53C22×D411yescharacteristic
211Dic53—21no—
212Dic53—41no—
213Dic53#12111yescharacteristic
214Dic53#12111yescharacteristic
215#3069—531nomaximal, Sylow
216C2×C53—21no—
217C2×C53—41no—
218C2×C53C2×#12111yescharacteristic
219D53—21no—
220D53—21no—
221D53—41no—
222D53—81no—
223#575—531no—
224C2×#137—531no—
225C2×#137—531no—
226#568—531no—
227#473—531no—
228#473—531no—
229#473—531no—
230#473D5311yescharacteristic, complemented
231#471—531no—
232#471—531no—
233#471—531no—
234#471—531no—
235#430—531no—
236#413—531no—
237C2×#101—531no—
238C53#306911yescharacteristic, complemented, Sylow
239C2×#41—531no—
240C2×#41—531no—
241C22×Q8—531no—
242#137—531no—
243#137—531no—
244#137—531no—
245#137—531no—
246#137—1061no—
247#137—1061no—
248#137—1061no—
249#137—1061no—
250#137—1061no—
251#137—1061no—
252#137—1061no—
253#137C2×D5311yescharacteristic
254#137C2×D5311yescharacteristic
255AGL(1,ℤ/8)—531no—
256AGL(1,ℤ/8)—531no—
257AGL(1,ℤ/8)—1061no—
258#135—1061no—
259#135—1061no—
260C2×Q16—531no—
261C2×Q16—1061no—
262C2×Q16C2×D5311yescharacteristic, complemented
263C2×SD16—531no—
264C2×SD16—531no—
265C2×SD16—1061no—
266#131—531no—
267#131—531no—
268#131—531no—
269#131—1061no—
270#131—1061no—
271#131C2×D5311yescharacteristic
272C2×M16—531no—
273C2×M16—531no—
274C2×M16—531no—
275C2×M16—531no—
276C2×M16—1061no—
277C2×M16—1061no—
278#108—531no—
279#108—531no—
280#108—531no—
281#108C2×D5311yescharacteristic, complemented
282#101—531no—
283#101—531no—
284#101—531no—
285#101—531no—
286#101C2×D5311yescharacteristic, complemented
287#101C2×D5311yescharacteristic, complemented
288#100—531no—
289#100—531no—
290#41—531no—
291#41—531no—
292#41—531no—
293#41—1061no—
294#41—1061no—
295#41—1061no—
296#41—1061no—
297#41—1061no—
298#41—1061no—
299#41C22×D5311yescharacteristic
300C2×Q8—531no—
301C2×Q8—531no—
302C2×Q8—1061no—
303C2×Q8—1061no—
304C2×Q8—1061no—
305C2×Q8—1061no—
306C2×Q8—1061no—
307C2×Q8C22×D5311yescharacteristic
308C2×Q8C22×D5311yescharacteristic
309C2×D4—531no—
310C2×D4—531no—
311C2×D4—1061no—
312C2×D4—1061no—
313C22×C4—531no—
314C22×C4—1061no—
315C22×C4—1061no—
316C22×C4—1061no—
317Q16—21no—
318Q16—21no—
319Q16—21no—
320Q16—21no—
321Q16—1061no—
322Q16—1061no—
323Q16—1061no—
324Q16—1061no—
325Q16—1061no—
326Q16—1061no—
327SD16—21no—
328SD16—21no—
329SD16—1061no—
330SD16—1061no—
331SD16—1061no—
332SD16—1061no—
333SD16—1061no—
334SD16—1061no—
335SD16—1061no—
336SD16—1061no—
337SD16—1061no—
338SD16—1061no—
339D8—1061no—
340D8—1061no—
341M16—21no—
342M16—531no—
343M16—531no—
344M16—531no—
345M16—531no—
346M16—531no—
347M16—531no—
348M16—531no—
349M16—1061no—
350M16—1061no—
351M16—1061no—
352M16—1061no—
353M16—1061no—
354M16—1061no—
355M16C22×D5311yescharacteristic
356M16C22×D5311yescharacteristic
357M16C22×D5311yescharacteristic
358C2×C8—21no—
359C2×C8—531no—
360C2×C8—1061no—
361C2×C8—1061no—
362C2×C8—1061no—
363C2×C8—1061no—
364C2×C8—1061no—
365C2×C8—1061no—
366C2×C8C22×D5311yescharacteristic
367C23—531no—
368C23—1061no—
369C23—1061no—
370Q8—21no—
371Q8—21no—
372Q8—21no—
373Q8—21no—
374Q8—531no—
375Q8—1061no—
376Q8—1061no—
377Q8—1061no—
378Q8—2121no—
379Q8—2121no—
380Q8D4×D5311yescharacteristic
381D4—21no—
382D4—531no—
383D4—1061no—
384D4—1061no—
385D4—1061no—
386D4—1061no—
387D4—1061no—
388D4—1061no—
389D4—2121no—
390D4—2121no—
391D4D4×D5311yescharacteristic
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—531no—
396C2×C4—531no—
397C2×C4—531no—
398C2×C4—531no—
399C2×C4—531no—
400C2×C4—1061no—
401C2×C4—1061no—
402C2×C4—1061no—
403C2×C4—1061no—
404C2×C4—1061no—
405C2×C4—2121no—
406C2×C4—2121no—
407C2×C4—2121no—
408C2×C4—2121no—
409C2×C4C23×D5311yescharacteristic
410C8—21no—
411C8—21no—
412C8—21no—
413C8—531no—
414C8—531no—
415C8—1061no—
416C8—1061no—
417C8—1061no—
418C8D4×D5311yescharacteristic
419C8D4×D5311yescharacteristic
420C22—21no—
421C22—531no—
422C22—531no—
423C22—1061no—
424C22—4241no—
425C22—4241no—
426C22—2121no—
427C22—2121no—
428C22#41×D5311yescharacteristic
429C4—21no—
430C4—41no—
431C4—41no—
432C4—531no—
433C4—531no—
434C4—1061no—
435C4—2121no—
436C4C2×D4×D5311yescharacteristic
437C4C2×D4×D5311yescharacteristic
438C2—21no—
439C2—41no—
440C2—1061no—
441C2—1061no—
442C2—4241no—
443C2—2121no—
444C2#121×D5311yescharacteristic
subgroupG/H
centerC2#121×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#473C2
socleC2×C53C2×#121
hypercenter#473D53
hypocenterC53#3069
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #473

Complements

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

NH ≅ G/Ntype
2. C2×#13133249C2semidirect
3. #137×D53C2semidirect
4. #16811388C2semidirect
5. #16810897C2semidirect
6. #16810896C2semidirect
7. #16810895C2semidirect
8. #16810889C2semidirect
9. #16810888C2semidirect
10. #16810887C2semidirect
11. #16810886C2semidirect
12. #16810670C2semidirect
13. #16810606C2semidirect
14. #101×D53C2semidirect
15. C53×#473C2semidirect
35. C2×#11719925C22semidirect
37. C2×#11719913C22semidirect
38. C2×#11719924C22semidirect
51. #13133153C22semidirect
52. #13133152C22semidirect
53. #13133152C22semidirect
54. #13133136C22semidirect
55. #13133136C22semidirect
56. #13133135C22semidirect
57. #13133135C22semidirect
58. #13133133C22semidirect
59. #13133133C22semidirect
62. C2×Q16×C53C22semidirect
64. #108×C53C22semidirect
65. #101×C53C22semidirect
66. #101×C53C22semidirect
230. #473D53semidirect
238. C53#3069semidirect
262. C2×Q16C2×D53semidirect
281. #108C2×D53semidirect
286. #101C2×D53semidirect
287. #101C2×D53semidirect