Finite Group Databasebeta

← #20284430

Subgroups — #20284430

Order 6784.

452 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811422C211yesmaximal, characteristic, complemented
2#16811419C211yesmaximal, characteristic, complemented
3C2×#13133238C211yesmaximal, characteristic, complemented
4#131×D53C211yesmaximal, characteristic, complemented
5#16810897C211yesmaximal, characteristic, complemented
6#16810896C211yesmaximal, characteristic, complemented
7#16810892C211yesmaximal, characteristic, complemented
8#16810891C211yesmaximal, characteristic, complemented
9#16810887C211yesmaximal, characteristic, complemented
10#16810886C211yesmaximal, characteristic, complemented
11#16810883C211yesmaximal, characteristic, complemented
12#108×D53C211yesmaximal, characteristic, complemented
13#16810606C211yesmaximal, characteristic, complemented
14#16810604C211yesmaximal, characteristic, complemented
15C53×#471C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719947C2211yescharacteristic
18#41×D53C2211yescharacteristic
19#13133250C2211yescharacteristic
20#13133249—21no—
21#13133248C2211yescharacteristic
22#13133247C2211yescharacteristic
23#13133246—21no—
24#13133245C2211yescharacteristic
25#13133244—21no—
26#13133243C2211yescharacteristic
27#13133242C2211yescharacteristic
28#13133241—21no—
29#13133240—21no—
30#13133238—21no—
31#13133238—21no—
32#13133238—21no—
33#13133238—21no—
34#13133238—21no—
35#13133238—21no—
36#13133236—21no—
37C2×#11719925C2211yescharacteristic, complemented
38C2×#11719913C2211yescharacteristic, complemented
39C2×#11719922C2211yescharacteristic, complemented
40#13133225—21no—
41#13133225C2211yescharacteristic
42#13133224C2211yescharacteristic
43#13133223—21no—
44#13133223C2211yescharacteristic
45M16×D53—21no—
46M16×D53C2211yescharacteristic
47M16×D53C2211yescharacteristic
48M16×D53C2211yescharacteristic
49C2×C8×D53—21no—
50C2×C8×D53C2211yescharacteristic
51#13133153C2211yescharacteristic, complemented
52#13133152C2211yescharacteristic, complemented
53#13133152C2211yescharacteristic, complemented
54#13133136C2211yescharacteristic, complemented
55#13133135C2211yescharacteristic, complemented
56#13133135C2211yescharacteristic, complemented
57#13133134C2211yescharacteristic, complemented
58#13133133C2211yescharacteristic, complemented
59#13133133C2211yescharacteristic, complemented
60#137×C53C2211yescharacteristic
61AGL(1,ℤ/8)×C53C2211yescharacteristic
62C2×SD16×C53C2211yescharacteristic, complemented
63#131×C53C2211yescharacteristic
64#108×C53C2211yescharacteristic, complemented
65#101×C53C2211yescharacteristic, complemented
66#100×C53C2211yescharacteristic, complemented
67#41×C53C2311yescharacteristic
68C2×Q8×C53C2311yescharacteristic
69C2×D4×C53C2311yescharacteristic
70C2×C22⋊D53—21no—
71C22×Dic53—21no—
72#11719949—21no—
73#11719949—21no—
74#11719949—21no—
75#11719949—21no—
76#11719949D411yescharacteristic
77Q8×D53D411yescharacteristic
78#11719947—21no—
79#11719947—21no—
80#11719947—21no—
81#11719947—21no—
82#11719947—21no—
83#11719947D411yescharacteristic
84D4×D53—21no—
85D4×D53D411yescharacteristic
86#11719945—21no—
87#11719945C2311yescharacteristic
88C2×D212—21no—
89C2×D212C2311yescharacteristic
90C2×C4×D53—21no—
91C2×C4×D53—21no—
92C2×C4×D53C2311yescharacteristic
93C2×Dic106C2311yescharacteristic
94Q16×C53—21no—
95SD16×C53—21no—
96SD16×C53—21no—
97SD16×C53—21no—
98SD16×C53—21no—
99D8×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#11719924—21no—
111#11719924—21no—
112#11719923—21no—
113#11719923—21no—
114#11719922—21no—
115#11719922—21no—
116#11719922—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—
135C8×D53—21no—
136C8×D53—21no—
137C8×D53—21no—
138C8×D53—21no—
139C8×D53—21no—
140C8×D53D411yescharacteristic
141C8×D53D411yescharacteristic
142C8×D53D411yescharacteristic
143C8×D53D411yescharacteristic
144C23×C53—21no—
145C22×D53—21no—
146C22×D53—21no—
147C22×D53#4111yescharacteristic
148Q8×C53—21no—
149Q8×C53—21no—
150Q8×C53C2×D411yescharacteristic
151D4×C53—21no—
152D4×C53—21no—
153D4×C53—21no—
154D4×C53C2×D411yescharacteristic
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53C2411yescharacteristic
158C22⋊D53—21no—
159C22⋊D53—21no—
160C22⋊D53—21no—
161C22⋊D53—41no—
162C22⋊D53—41no—
163C2×Dic53—21no—
164C2×Dic53—21no—
165C2×Dic53—41no—
166C2×Dic53—41no—
167C2×Dic53#4111yescharacteristic
168D212—21no—
169D212—21no—
170D212—21no—
171D212—41no—
172D212—41no—
173D212C2×D411yescharacteristic
174C4×D53—21no—
175C4×D53—21no—
176C4×D53—21no—
177C4×D53—41no—
178C4×D53—41no—
179C4×D53C2×D411yescharacteristic
180C4×D53C2×D411yescharacteristic
181C4×D53C2×D411yescharacteristic
182C4×D53C2×D411yescharacteristic
183Dic106—21no—
184Dic106—21no—
185Dic106—21no—
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×C53—41no—
199C22×C53—81no—
200C22×C53C2×#4111yescharacteristic
201C2×D53—21no—
202C2×D53—41no—
203C2×D53—42no—
204C2×D53—41no—
205C2×D53—81no—
206C2×D53#12111yescharacteristic
207C2×D53#12111yescharacteristic
208C4×C53—21no—
209C4×C53—41no—
210C4×C53C22×D411yescharacteristic
211C4×C53C22×D411yescharacteristic
212Dic53—21no—
213Dic53—41no—
214Dic53#12111yescharacteristic
215Dic53#12111yescharacteristic
216#3067—531nomaximal, Sylow
217C2×C53—21no—
218C2×C53—41no—
219C2×C53—81no—
220C2×C53C2×#12111yescharacteristic
221D53—21no—
222D53—21no—
223D53—41no—
224D53—81no—
225#575—531no—
226#575—531no—
227C2×#135—531no—
228C2×#131—531no—
229#473—531no—
230#473—531no—
231#472—531no—
232#472—531no—
233#471—531no—
234#471—531no—
235#471—531no—
236#471D5311yescharacteristic, complemented
237C2×#108—531no—
238#413—531no—
239#413—531no—
240C53#306711yescharacteristic, complemented, Sylow
241C2×#41—531no—
242C2×#41—531no—
243C2×#41—531no—
244#137—531no—
245#137—531no—
246#137—531no—
247#137—1061no—
248#137—1061no—
249#137C2×D5311yescharacteristic
250AGL(1,ℤ/8)—531no—
251AGL(1,ℤ/8)—531no—
252AGL(1,ℤ/8)—531no—
253AGL(1,ℤ/8)—1061no—
254AGL(1,ℤ/8)—1061no—
255AGL(1,ℤ/8)C2×D5311yescharacteristic
256#135—1061no—
257#135—1061no—
258#135—1061no—
259#135—1061no—
260#135—1061no—
261#135—1061no—
262#135—1061no—
263#135—1061no—
264C2×Q16—531no—
265C2×SD16—531no—
266C2×SD16C2×D5311yescharacteristic, complemented
267C2×D8—531no—
268#131—531no—
269#131—531no—
270#131—531no—
271#131—1061no—
272#131—1061no—
273#131C2×D5311yescharacteristic
274C2×M16—531no—
275C2×M16—531no—
276C2×M16—531no—
277C2×M16—1061no—
278C22×C8—531no—
279C22×C8—1061no—
280#108—531no—
281#108—531no—
282#108—531no—
283#108C2×D5311yescharacteristic, complemented
284#101—531no—
285#101—531no—
286#101—531no—
287#101C2×D5311yescharacteristic, complemented
288#100—531no—
289#100—531no—
290#100—531no—
291#100C2×D5311yescharacteristic, complemented
292#41—531no—
293#41—531no—
294#41—531no—
295#41—1061no—
296#41—1061no—
297#41—1061no—
298#41—1061no—
299#41—1061no—
300#41—1061no—
301#41—1061no—
302#41—1061no—
303#41—1061no—
304#41—1061no—
305#41C22×D5311yescharacteristic
306C2×Q8—531no—
307C2×Q8—531no—
308C2×Q8C22×D5311yescharacteristic
309C2×D4—531no—
310C2×D4—531no—
311C2×D4—1061no—
312C2×D4—1061no—
313C2×D4—1061no—
314C2×D4C22×D5311yescharacteristic
315C22×C4—531no—
316C22×C4—1061no—
317C22×C4—1061no—
318C22×C4—1061no—
319Q16—21no—
320Q16—1061no—
321Q16—1061no—
322Q16—1061no—
323Q16—1061no—
324Q16—1061no—
325SD16—21no—
326SD16—21no—
327SD16—21no—
328SD16—21no—
329SD16—1061no—
330SD16—1061no—
331SD16—1061no—
332SD16—1061no—
333SD16—1061no—
334SD16—1061no—
335SD16—1061no—
336SD16—1061no—
337D8—21no—
338D8—1061no—
339D8—1061no—
340D8—1061no—
341D8—1061no—
342D8—1061no—
343M16—21no—
344M16—531no—
345M16—531no—
346M16—531no—
347M16—1061no—
348M16—1061no—
349M16—1061no—
350M16—1061no—
351M16—1061no—
352M16—1061no—
353M16C22×D5311yescharacteristic
354M16C22×D5311yescharacteristic
355M16C22×D5311yescharacteristic
356C2×C8—21no—
357C2×C8—531no—
358C2×C8—531no—
359C2×C8—531no—
360C2×C8—531no—
361C2×C8—531no—
362C2×C8—1061no—
363C2×C8—1061no—
364C2×C8—1061no—
365C2×C8—1061no—
366C2×C8—1061no—
367C2×C8—1061no—
368C2×C8C22×D5311yescharacteristic
369C23—21no—
370C23—531no—
371C23—1061no—
372C23—1061no—
373Q8—21no—
374Q8—21no—
375Q8—531no—
376Q8—1061no—
377Q8—1061no—
378Q8—1061no—
379Q8D4×D5311yescharacteristic
380D4—21no—
381D4—21no—
382D4—21no—
383D4—531no—
384D4—1061no—
385D4—1061no—
386D4—1061no—
387D4—1061no—
388D4—1061no—
389D4—1061no—
390D4—2121no—
391D4—2121no—
392D4—2121no—
393D4—2121no—
394D4D4×D5311yescharacteristic
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—531no—
398C2×C4—531no—
399C2×C4—531no—
400C2×C4—531no—
401C2×C4—531no—
402C2×C4—1061no—
403C2×C4—1061no—
404C2×C4—1061no—
405C2×C4—1061no—
406C2×C4—1061no—
407C2×C4—2121no—
408C2×C4—2121no—
409C2×C4—2121no—
410C2×C4—2121no—
411C2×C4C23×D5311yescharacteristic
412C8—21no—
413C8—21no—
414C8—21no—
415C8—531no—
416C8—531no—
417C8—1061no—
418C8—1061no—
419C8—1061no—
420C8D4×D5311yescharacteristic
421C8D4×D5311yescharacteristic
422C22—21no—
423C22—41no—
424C22—81no—
425C22—531no—
426C22—531no—
427C22—1061no—
428C22—4241no—
429C22—2121no—
430C22—2122no—
431C22—2121no—
432C22#41×D5311yescharacteristic
433C4—21no—
434C4—41no—
435C4—531no—
436C4—531no—
437C4—1061no—
438C4—2121no—
439C4C2×D4×D5311yescharacteristic
440C4C2×D4×D5311yescharacteristic
441C2—21no—
442C2—41no—
443C2—81no—
444C2—1061no—
445C2—1061no—
446C2—4241no—
447C2—2121no—
448C2#121×D5311yescharacteristic
subgroupG/H
centerC2#121×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#471C2
socleC2×C53C2×#121
hypercenter#471D53
hypocenterC53#3067
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #471

Complements

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

NH ≅ G/Ntype
1. #16811422C2semidirect
2. #16811419C2semidirect
3. C2×#13133238C2semidirect
4. #131×D53C2semidirect
5. #16810897C2semidirect
6. #16810896C2semidirect
7. #16810892C2semidirect
8. #16810891C2semidirect
9. #16810887C2semidirect
10. #16810886C2semidirect
11. #16810883C2semidirect
12. #108×D53C2semidirect
13. #16810606C2semidirect
14. #16810604C2semidirect
15. C53×#471C2semidirect
37. C2×#11719925C22semidirect
38. C2×#11719913C22semidirect
39. C2×#11719922C22semidirect
51. #13133153C22semidirect
52. #13133152C22semidirect
53. #13133152C22semidirect
54. #13133136C22semidirect
55. #13133135C22semidirect
56. #13133135C22semidirect
57. #13133134C22semidirect
58. #13133133C22semidirect
59. #13133133C22semidirect
62. C2×SD16×C53C22semidirect
64. #108×C53C22semidirect
65. #101×C53C22semidirect
66. #100×C53C22semidirect
236. #471D53semidirect
240. C53#3067semidirect
266. C2×SD16C2×D53semidirect
283. #108C2×D53semidirect
287. #101C2×D53semidirect
291. #100C2×D53semidirect