Finite Group Databasebeta

← #20286491

Subgroups — #20286491

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#16811440C211yesmaximal, characteristic, complemented
2#16811437C211yesmaximal, characteristic, complemented
3#16811435C211yesmaximal, characteristic, complemented
4#16811430C211yesmaximal, characteristic, complemented
5#16811017C211yesmaximal, characteristic, complemented
6#16810899C211yesmaximal, characteristic, complemented
7#16810898C211yesmaximal, characteristic, complemented
8#16810887C211yesmaximal, characteristic, complemented
9#16810886C211yesmaximal, characteristic, complemented
10#16810807C211yesmaximal, characteristic, complemented
11#16810807C211yesmaximal, characteristic, complemented
12#16810802C211yesmaximal, characteristic, complemented
13#16810797C211yesmaximal, characteristic, complemented
14#16810746C211yesmaximal, characteristic, complemented
15C53×#444C211yesmaximal, characteristic, complemented
16#13133264C2211yescharacteristic
17#13133264C2211yescharacteristic
18#13133264C2211yescharacteristic, complemented
19#13133263C2211yescharacteristic
20#13133250—21no—
21#13133250—21no—
22#13133250—21no—
23#13133250C2211yescharacteristic
24#13133250C2211yescharacteristic
25#13133250C2211yescharacteristic, complemented
26#13133249—21no—
27#13133247C2211yescharacteristic
28#13133247C2211yescharacteristic
29#13133247C2211yescharacteristic, complemented
30#13133246—21no—
31#13133246—21no—
32#13133245—21no—
33#13133245C2211yescharacteristic
34#13133242C2211yescharacteristic
35#13133241—21no—
36#13133239—21no—
37#13133239—21no—
38#13133239C2211yescharacteristic
39#13133238—21no—
40#13133236—21no—
41C2×Dic212—21no—
42C2×Dic212C2211yescharacteristic, complemented
43C2×Dic212C2211yescharacteristic, complemented
44Q16×D53—21no—
45C2×#11719913—21no—
46C2×#11719913C2211yescharacteristic
47C2×#11719913C2211yescharacteristic
48SD16×D53—21no—
49#13133223C2211yescharacteristic
50#13133223C2211yescharacteristic, complemented
51#13133217C2211yescharacteristic, complemented
52#13133205C2211yescharacteristic
53#13133153C2211yescharacteristic, complemented
54#13133145C2211yescharacteristic
55#13133145C2211yescharacteristic, complemented
56#13133135C2211yescharacteristic, complemented
57#13133135C2211yescharacteristic, complemented
58#13133135C2211yescharacteristic, complemented
59#13133133C2211yescharacteristic
60#135×C53C2211yescharacteristic
61#131×C53C2211yescharacteristic
62#131×C53C2211yescharacteristic, complemented
63#108×C53C2211yescharacteristic, complemented
64#104×C53C2211yescharacteristic
65#104×C53C2211yescharacteristic, complemented
66#97×C53C2211yescharacteristic, complemented
67#41×C53C2311yescharacteristic
68#41×C53C2311yescharacteristic
69#11719949—21no—
70Q8×D53—21no—
71Q8×D53—21no—
72Q8×D53—21no—
73#11719947—21no—
74#11719947—21no—
75#11719947—21no—
76#11719947—41no—
77#11719947—41no—
78#11719947—41no—
79D4×D53—21no—
80D4×D53—41no—
81#11719945—21no—
82#11719945—21no—
83#11719945—21no—
84#11719945—21no—
85#11719945C2311yescharacteristic
86#11719945C2311yescharacteristic
87C2×D212—21no—
88C2×D212C2311yescharacteristic
89C2×Dic106—21no—
90C2×Dic106—21no—
91C2×Dic106—21no—
92C2×Dic106C2311yescharacteristic
93C2×Dic106C2311yescharacteristic
94C2×Dic106C2311yescharacteristic
95Q16×C53D411yescharacteristic
96SD16×C53D411yescharacteristic
97SD16×C53D411yescharacteristic
98D8×C53D411yescharacteristic
99M16×C53—21no—
100M16×C53—21no—
101M16×C53C2311yescharacteristic
102M16×C53C2311yescharacteristic
103C2×C8×C53—21no—
104C2×C8×C53—21no—
105C2×C8×C53C2311yescharacteristic
106C2×C8×C53C2311yescharacteristic
107C42×C53C2311yescharacteristic
108#11719925—21no—
109#11719925—21no—
110#11719925—21no—
111#11719924—21no—
112#11719923—21no—
113#11719923—21no—
114#11719923—21no—
115#11719922—21no—
116#11719921—41no—
117C4⋊Dic53—41no—
118#11719917C2311yescharacteristic
119#11719917C2311yescharacteristic
120Dic212—21no—
121Dic212—21no—
122Dic212—21no—
123Dic212—21no—
124Dic212—21no—
125Dic212—21no—
126Dic212—21no—
127Dic212D411yescharacteristic
128D424—21no—
129D424D411yescharacteristic
130#11719913—21no—
131#11719913—21no—
132#11719913—21no—
133#11719913—21no—
134#11719913—21no—
135#11719913—21no—
136#11719913—21no—
137#11719913—21no—
138#11719913D411yescharacteristic
139#11719913D411yescharacteristic
140#11719912—21no—
141#11719912—21no—
142C8×D53—21no—
143C8×D53—21no—
144C22×D53—21no—
145C22×D53—41no—
146Q8×C53C2×D411yescharacteristic
147Q8×C53C2×D411yescharacteristic
148D4×C53—21no—
149D4×C53—21no—
150D4×C53C2×D411yescharacteristic
151D4×C53C2×D411yescharacteristic
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53C2411yescharacteristic
156C22⋊D53—21no—
157C22⋊D53—21no—
158C22⋊D53—41no—
159C22⋊D53—41no—
160C22⋊D53—41no—
161C22⋊D53—41no—
162C2×Dic53—21no—
163C2×Dic53—21no—
164C2×Dic53—21no—
165C2×Dic53—41no—
166C2×Dic53—41no—
167C2×Dic53—41no—
168D212—21no—
169D212—21no—
170D212—41no—
171D212C2×D411yescharacteristic
172D212C2×D411yescharacteristic
173C4×D53—21no—
174C4×D53—21no—
175C4×D53—41no—
176C4×D53—41no—
177C4×D53—41no—
178C4×D53—41no—
179Dic106—21no—
180Dic106—21no—
181Dic106—21no—
182Dic106—21no—
183Dic106—21no—
184Dic106—21no—
185Dic106—41no—
186Dic106—41no—
187Dic106—41no—
188Dic106C2×D411yescharacteristic
189Dic106C2×D411yescharacteristic
190C8×C53—21no—
191C8×C53—21no—
192C8×C53C2×D411yescharacteristic
193C8×C53C2×D411yescharacteristic
194C8×C53C2×D411yescharacteristic
195C8×C53C2×D411yescharacteristic
196#88322—21no—
197#88322—21no—
198C22×C53—21no—
199C22×C53—21no—
200C22×C53H(5,𝔽2)11yescharacteristic
201C2×D53—21no—
202C2×D53—21no—
203C2×D53—41no—
204C2×D53—81no—
205C2×D53—161no—
206C4×C53—21no—
207C4×C53—21no—
208C4×C53—41no—
209C4×C53C22×D411yescharacteristic
210C4×C53C22×D411yescharacteristic
211Dic53—21no—
212Dic53—21no—
213Dic53—41no—
214Dic53—41no—
215Dic53—41no—
216#3289—531nomaximal, Sylow
217C2×C53—21no—
218C2×C53—41no—
219C2×C53—41no—
220C2×C53D4211yescharacteristic
221D53—41no—
222D53—41no—
223D53—81no—
224#578—531no—
225#578—531no—
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#444D5311yescharacteristic, complemented
239C53#328911yescharacteristic, complemented, Sylow
2402−1+4—531no—
2412−1+4—531no—
2422−1+4—531no—
243H(5,𝔽2)—531no—
244#137—531no—
245#137—531no—
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#137—1061no—
255#137—1061no—
256AGL(1,ℤ/8)—531no—
257AGL(1,ℤ/8)—531no—
258AGL(1,ℤ/8)—1061no—
259AGL(1,ℤ/8)—1061no—
260#135—531no—
261#135—1061no—
262#135—1061no—
263#135—1061no—
264#135—1061no—
265#135C2×D5311yescharacteristic
266C2×Q16—531no—
267C2×Q16—531no—
268C2×Q16—1061no—
269C2×Q16—1061no—
270C2×SD16—531no—
271C2×SD16—531no—
272C2×SD16—1061no—
273C2×SD16—1061no—
274#131—531no—
275#131—531no—
276#131C2×D5311yescharacteristic
277#131C2×D5311yescharacteristic, complemented
278C4⋊Q8—531no—
279#124—531no—
280#108—531no—
281#108C2×D5311yescharacteristic, complemented
282#104—531no—
283#104—531no—
284#104C2×D5311yescharacteristic
285#104C2×D5311yescharacteristic, complemented
286#101—531no—
287#101—531no—
288#101—531no—
289#100—531no—
290#97C2×D5311yescharacteristic, complemented
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#41—1061no—
300#41—1061no—
301#41—2121no—
302#41—2121no—
303#41—2121no—
304#41C22×D5311yescharacteristic
305#41C22×D5311yescharacteristic
306C2×Q8—531no—
307C2×Q8—531no—
308C2×Q8—531no—
309C2×Q8—1061no—
310C2×Q8—1061no—
311C2×Q8—1061no—
312C2×Q8—1061no—
313C2×Q8—1061no—
314C2×Q8—1061no—
315C2×D4—531no—
316C2×D4—1061no—
317C2×D4—1061no—
318C2×D4—2121no—
319Q16—531no—
320Q16—1061no—
321Q16—1061no—
322Q16—1061no—
323Q16—1061no—
324Q16—1061no—
325Q16—1061no—
326Q16—1061no—
327Q16—1061no—
328Q16—1061no—
329Q16—1061no—
330Q16D21211yescharacteristic
331SD16—531no—
332SD16—531no—
333SD16—1061no—
334SD16—1061no—
335SD16—1061no—
336SD16—1061no—
337SD16—1061no—
338SD16—1061no—
339SD16—1061no—
340SD16—1061no—
341SD16—1061no—
342SD16—1061no—
343SD16—1061no—
344SD16—1061no—
345SD16D21211yescharacteristic
346SD16D21211yescharacteristic
347D8—531no—
348D8—1061no—
349D8—1061no—
350D8D21211yescharacteristic
351M16—21no—
352M16—21no—
353M16—531no—
354M16—531no—
355M16—1061no—
356M16—1061no—
357M16C22×D5311yescharacteristic
358M16C22×D5311yescharacteristic
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—1061no—
362C2×C8—1061no—
363C2×C8C22×D5311yescharacteristic
364C2×C8C22×D5311yescharacteristic
365C4⋊C4—2121no—
366C22⋊C4—2121no—
367C42C22×D5311yescharacteristic
368C23—1061no—
369C23—2121no—
370Q8—531no—
371Q8—531no—
372Q8—1061no—
373Q8—1061no—
374Q8—1061no—
375Q8—1061no—
376Q8—1061no—
377Q8—1061no—
378Q8—2121no—
379Q8—2121no—
380Q8—2121no—
381Q8C2×D21211yescharacteristic
382Q8C2×D21211yescharacteristic
383D4—21no—
384D4—21no—
385D4—531no—
386D4—531no—
387D4—1061no—
388D4—1061no—
389D4—1061no—
390D4—1061no—
391D4—2121no—
392D4—2121no—
393D4—2121no—
394D4—2121no—
395D4—2121no—
396D4C2×D21211yescharacteristic
397D4C2×D21211yescharacteristic
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—1061no—
402C2×C4—1061no—
403C2×C4—1061no—
404C2×C4—1061no—
405C2×C4—1061no—
406C2×C4—2121no—
407C2×C4—2121no—
408C2×C4—2121no—
409C2×C4—2121no—
410C2×C4—2121no—
411C2×C4—2121no—
412C2×C4—2121no—
413C2×C4C23×D5311yescharacteristic
414C8—21no—
415C8—21no—
416C8—1061no—
417C8—1061no—
418C8C2×D21211yescharacteristic
419C8C2×D21211yescharacteristic
420C8D4×D5311yescharacteristic
421C8D4×D5311yescharacteristic
422C22—21no—
423C22—21no—
424C22—8481no—
425C22—1061no—
426C22—1061no—
427C22—4241no—
428C22—2121no—
429C22#1313326311yescharacteristic
430C4—21no—
431C4—21no—
432C4—41no—
433C4—1061no—
434C4—1061no—
435C4—2121no—
436C4—2121no—
437C4—2121no—
438C4C2×D4×D5311yescharacteristic
439C4C22×D21211yescharacteristic
440C2—21no—
441C2—41no—
442C2—41no—
443C2—4241no—
444C2—2121no—
445C2—2121no—
446C2D4×D21211yescharacteristic
subgroupG/H
centerC2D4×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#444C2
socleC2×C53D42
hypercenter#444D53
hypocenterC53#3289
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #444

Complements

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

NH ≅ G/Ntype
1. #16811440C2semidirect
2. #16811437C2semidirect
3. #16811435C2semidirect
4. #16811430C2semidirect
5. #16811017C2semidirect
6. #16810899C2semidirect
7. #16810898C2semidirect
8. #16810887C2semidirect
9. #16810886C2semidirect
10. #16810807C2semidirect
11. #16810807C2semidirect
12. #16810802C2semidirect
13. #16810797C2semidirect
14. #16810746C2semidirect
15. C53×#444C2semidirect
18. #13133264C22semidirect
25. #13133250C22semidirect
29. #13133247C22semidirect
42. C2×Dic212C22semidirect
43. C2×Dic212C22semidirect
50. #13133223C22semidirect
51. #13133217C22semidirect
53. #13133153C22semidirect
55. #13133145C22semidirect
56. #13133135C22semidirect
57. #13133135C22semidirect
58. #13133135C22semidirect
62. #131×C53C22semidirect
63. #108×C53C22semidirect
65. #104×C53C22semidirect
66. #97×C53C22semidirect
238. #444D53semidirect
239. C53#3289semidirect
277. #131C2×D53semidirect
281. #108C2×D53semidirect
285. #104C2×D53semidirect
290. #97C2×D53semidirect