Finite Group Databasebeta

← #20286215

Subgroups — #20286215

Order 6784.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133245C211yesmaximal, characteristic, complemented
2C22×#11719913C211yesmaximal, characteristic, complemented
3#16811282C211yesmaximal, characteristic, complemented
4D4×D212C211yesmaximal, characteristic, complemented
5#16811012C211yesmaximal, characteristic, complemented
6#16810869C211yesmaximal, characteristic, complemented
7#16810859C211yesmaximal, characteristic, complemented
8#16810839C211yesmaximal, characteristic, complemented
9#16810824C211yesmaximal, characteristic, complemented
10#16810776C211yesmaximal, characteristic, complemented
11#16810770C211yesmaximal, characteristic, complemented
12#16810764C211yesmaximal, characteristic, complemented
13#16810759C211yesmaximal, characteristic, complemented
14#16810725C211yesmaximal, characteristic, complemented
15C53×#435C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic, complemented
17C22×Dic106C2211yescharacteristic, complemented
18C22×D212C2211yescharacteristic
19C22×D212C2211yescharacteristic
20C2×D4×D53—21no—
21#13133245—24no—
22C2×#11719913—24no—
23C2×#11719913C2211yescharacteristic
24C2×#11719913C2211yescharacteristic
25C2×#11719913C2211yescharacteristic, complemented
26C2×#11719913C2211yescharacteristic, complemented
27C2×#11719913C2211yescharacteristic, complemented
28C2×#11719913C2211yescharacteristic, complemented
29C2×D424C2211yescharacteristic
30C2×D424C2211yescharacteristic, complemented
31C2×D424C2211yescharacteristic, complemented
32#13133214C2211yescharacteristic
33#13133205C2211yescharacteristic, complemented
34#13133203—21no—
35#13133196C2211yescharacteristic, complemented
36#13133195—21no—
37#13133192—21no—
38#13133191C2211yescharacteristic
39#13133191C2211yescharacteristic
40#13133191C2211yescharacteristic, complemented
41#13133189—21no—
42#13133187—21no—
43#13133187—21no—
44C4×Dic106C2211yescharacteristic, complemented
45C4×D212C2211yescharacteristic
46#13133177—21no—
47#13133151C2211yescharacteristic, complemented
48#13133142C2211yescharacteristic, complemented
49#13133142C2211yescharacteristic, complemented
50#13133139C2211yescharacteristic
51#13133139C2211yescharacteristic, complemented
52#13133139C2211yescharacteristic, complemented
53#13133139C2211yescharacteristic, complemented
54C2×M16×C53C2211yescharacteristic, complemented
55C22×C8×C53C2211yescharacteristic, complemented
56C4×D4×C53C2211yescharacteristic
57C4⋊C8×C53C2211yescharacteristic, complemented
58C22⋊C8×C53C2211yescharacteristic, complemented
59C22⋊C8×C53C2211yescharacteristic, complemented
60#97×C53C2211yescharacteristic, complemented
61C23×D53—21no—
62C23×D53—21no—
63C2×D4×C53D411yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C22×Dic53—21no—
71D4×D53—44no—
72#11719945—22no—
73#11719945—41no—
74C2×D212—22no—
75C2×D212—22no—
76C2×D212—21no—
77C2×D212—21no—
78C2×D212—21no—
79C2×D212—21no—
80C2×D212—41no—
81C2×D212—41no—
82C2×D212C2311yescharacteristic
83C2×D212C2311yescharacteristic
84C2×D212C2311yescharacteristic
85C2×D212C2311yescharacteristic
86C2×C4×D53—21no—
87C2×C4×D53—21no—
88C2×Dic106—22no—
89C2×Dic106—21no—
90C2×Dic106—41no—
91C2×Dic106C2311yescharacteristic
92C2×Dic106C2311yescharacteristic
93M16×C53—22no—
94C2×C8×C53—22no—
95C2×C8×C53C2311yescharacteristic
96C2×C8×C53C2311yescharacteristic
97C2×C8×C53C2311yescharacteristic
98C2×C8×C53C2311yescharacteristic
99C4⋊C4×C53D411yescharacteristic
100C22⋊C4×C53D411yescharacteristic
101C22⋊C4×C53D411yescharacteristic
102C42×C53C2311yescharacteristic
103C22⋊Dic53—21no—
104#11719921—21no—
105#11719921—21no—
106#11719921—21no—
107#11719921—21no—
108#11719921—21no—
109#11719921—21no—
110C4⋊Dic53C2311yescharacteristic
111C4⋊Dic53C2311yescharacteristic
112#11719919—21no—
113#11719919—21no—
114C4×Dic53—21no—
115D424—22no—
116D424—22no—
117D424—41no—
118#11719913—22no—
119#11719913—22no—
120#11719913—22no—
121#11719913—22no—
122#11719913—41no—
123#11719913D414yescomplemented
124C23×C53C2×D411yescharacteristic
125C23×C53C2×D411yescharacteristic
126C22×D53—21no—
127C22×D53—21no—
128C22×D53—21no—
129C22×D53—21no—
130C22×D53—42no—
131C22×D53—42no—
132C22×D53—42no—
133C22×D53—42no—
134C22×D53—81no—
135C22×D53—81no—
136D4×C53—22no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53C2×D411yescharacteristic
142C2×C4×C53C2×D411yescharacteristic
143C2×C4×C53C2×D411yescharacteristic
144C2×C4×C53C2×D411yescharacteristic
145C2×C4×C53C2411yescharacteristic
146C22⋊D53—42no—
147C22⋊D53—42no—
148C22⋊D53—42no—
149C22⋊D53—42no—
150C2×Dic53—21no—
151C2×Dic53—21no—
152C2×Dic53—21no—
153C2×Dic53—21no—
154C2×Dic53—42no—
155D212—21no—
156D212—21no—
157D212—21no—
158D212—21no—
159D212—21no—
160D212—21no—
161D212—21no—
162D212—42no—
163D212—42no—
164D212—42no—
165D212—81no—
166D212C2×D412yes—
167C4×D53—42no—
168C4×D53—42no—
169Dic106—21no—
170Dic106—21no—
171Dic106—21no—
172Dic106—42no—
173Dic106C2×D412yescomplemented
174C8×C53—21no—
175C8×C53—21no—
176C8×C53—21no—
177C8×C53C2×D412yescomplemented
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53AGL(1,ℤ/8)12yescomplemented
184C22×C53C22×D411yescharacteristic
185C2×D53—22no—
186C2×D53—41no—
187C2×D53—41no—
188C2×D53—41no—
189C2×D53—41no—
190C2×D53—41no—
191C2×D53—41no—
192C2×D53—41no—
193C2×D53—41no—
194C2×D53—41no—
195C2×D53—41no—
196C2×D53—41no—
197C2×D53—84no—
198C2×D53—84no—
199C4×C53—21no—
200C4×C53—21no—
201C4×C53—21no—
202C4×C53—21no—
203C4×C53C22×D411yescharacteristic
204C4×C53H(5,𝔽2)11yescharacteristic
205Dic53—22no—
206Dic53—41no—
207Dic53—41no—
208Dic53—41no—
209#3265—531nomaximal, Sylow
210C2×C53—22no—
211C2×C53—41no—
212C2×C53D4211yescharacteristic
213C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C53#57711yescharacteristic
215D53—42no—
216D53—81no—
217D53—81no—
218D53—81no—
219C2×AGL(1,ℤ/8)—531no—
220C22×SD16—531no—
221#548—531no—
222D42—531no—
223#496—531no—
224#468—531no—
225#466—531no—
226#463—531no—
227#460—531no—
228#450—531no—
229#449—531no—
230#448—531no—
231#447—531no—
232#440—531no—
233#435D5311yescharacteristic, complemented
234C53#326511yescharacteristic, complemented, Sylow
235C2×#41—531no—
236C22×Q8—531no—
237C22×D4—531no—
238C22×D4—531no—
239C22×D4—1061no—
240AGL(1,ℤ/8)—1064no—
241C2×SD16—531no—
242C2×SD16—531no—
243C2×SD16—531no—
244C2×SD16—531no—
245C2×SD16—531no—
246C2×SD16—531no—
247C2×SD16—1064no—
248C2×D8—531no—
249C2×D8—531no—
250C2×D8—531no—
251C2×M16C2×D5311yescharacteristic, complemented
252C22×C8C2×D5311yescharacteristic, complemented
253#127—531no—
254#124—531no—
255#124—1061no—
256C22⋊Q8—531no—
257C22⋊Q8—1061no—
258#121—531no—
259#121—531no—
260#121—531no—
261#121—1061no—
262#121—1061no—
263#120—1061no—
264#120—1061no—
265C4×Q8—531no—
266C4×D4—531no—
267C4×D4—1061no—
268C4×D4C2×D5311yescharacteristic
269#107—531no—
270C4⋊C8C2×D5311yescharacteristic, complemented
271#103—531no—
272#103—531no—
273#102—531no—
274#102—531no—
275#102—531no—
276#102—531no—
277C22⋊C8C2×D5311yescharacteristic, complemented
278C22⋊C8C2×D5311yescharacteristic, complemented
279#97C2×D5311yescharacteristic, complemented
280C24—1061no—
281C24—1061no—
282#41—1062no—
283#41—2121no—
284C2×Q8—531no—
285C2×Q8—531no—
286C2×Q8—1062no—
287C2×Q8—1061no—
288C2×Q8—2121no—
289C2×D4—531no—
290C2×D4—531no—
291C2×D4—531no—
292C2×D4—531no—
293C2×D4—1062no—
294C2×D4—1062no—
295C2×D4—1061no—
296C2×D4—1061no—
297C2×D4—1061no—
298C2×D4—1061no—
299C2×D4—1061no—
300C2×D4—1061no—
301C2×D4—1061no—
302C2×D4—1061no—
303C2×D4—2121no—
304C2×D4—2121no—
305C2×D4—2124no—
306C2×D4D21211yescharacteristic
307C22×C4—1061no—
308C22×C4—1061no—
309C22×C4—1061no—
310C22×C4C22×D5311yescharacteristic
311C22×C4C22×D5311yescharacteristic
312SD16—534no—
313SD16—1062no—
314SD16—1062no—
315SD16—1062no—
316SD16—1062no—
317SD16—2121no—
318D8—1062no—
319D8—1062no—
320D8—2121no—
321M16—22no—
322C2×C8—22no—
323C2×C8C22×D5311yescharacteristic
324C2×C8C22×D5311yescharacteristic
325C2×C8C22×D5311yescharacteristic
326C2×C8C22×D5311yescharacteristic
327C4⋊C4—531no—
328C4⋊C4—531no—
329C4⋊C4—1061no—
330C4⋊C4—1061no—
331C4⋊C4D21211yescharacteristic
332C22⋊C4—1061no—
333C22⋊C4—1061no—
334C22⋊C4—1061no—
335C22⋊C4—1061no—
336C22⋊C4—1061no—
337C22⋊C4—1061no—
338C22⋊C4—1061no—
339C22⋊C4D21211yescharacteristic
340C22⋊C4D21211yescharacteristic
341C42—1061no—
342C42C22×D5311yescharacteristic
343C23—1061no—
344C23—1061no—
345C23—1061no—
346C23—1061no—
347C23—4241no—
348C23—4241no—
349C23—2122no—
350C23—2122no—
351C23—2122no—
352C23—2122no—
353C23C2×D21211yescharacteristic
354C23C2×D21211yescharacteristic
355Q8—532no—
356Q8—1061no—
357Q8—1061no—
358Q8—1061no—
359Q8—2122no—
360D4—22no—
361D4—532no—
362D4—1061no—
363D4—1061no—
364D4—1061no—
365D4—1061no—
366D4—1061no—
367D4—1061no—
368D4—1061no—
369D4—4241no—
370D4—2122no—
371D4—2122no—
372D4—2122no—
373D4—2122no—
374D4—2122no—
375D4—2122no—
376D4—2122no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—1061no—
382C2×C4—1061no—
383C2×C4—1061no—
384C2×C4—1061no—
385C2×C4—2122no—
386C2×C4—2122no—
387C2×C4—2122no—
388C2×C4C2×D21211yescharacteristic
389C2×C4C2×D21211yescharacteristic
390C2×C4C2×D21211yescharacteristic
391C2×C4C2×D21211yescharacteristic
392C2×C4C23×D5311yescharacteristic
393C8—21no—
394C8—21no—
395C8—21no—
396C8D4×D5312yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—1062no—
403C22—4244no—
404C22—4244no—
405C22—2121no—
406C22—2121no—
407C22—2121no—
408C22—2121no—
409C22—2121no—
410C22—2121no—
411C22—2121no—
412C22—2121no—
413C22—2121no—
414C22—2121no—
415C22—2121no—
416C22#1313324512yescomplemented
417C22C22×D21211yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—1062no—
423C4—2121no—
424C4—2121no—
425C4—2121no—
426C4C2×D4×D5311yescharacteristic
427C4#1313326311yescharacteristic
428C2—22no—
429C2—41no—
430C2—4241no—
431C2—4241no—
432C2—4241no—
433C2—2122no—
434C2D4×D21211yescharacteristic
435C2C2×#1313324511yescharacteristic
436C2#1681143511yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#435C2
socleC22×C53C22×D4
hypercenter#435D53
hypocenterC53#3265
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#13133245C2semidirect
2. C22×#11719913C2semidirect
3. #16811282C2semidirect
4. D4×D212C2semidirect
5. #16811012C2semidirect
6. #16810869C2semidirect
7. #16810859C2semidirect
8. #16810839C2semidirect
9. #16810824C2semidirect
10. #16810776C2semidirect
11. #16810770C2semidirect
12. #16810764C2semidirect
13. #16810759C2semidirect
14. #16810725C2semidirect
15. C53×#435C2semidirect
16. C2×#11719945C22semidirect
17. C22×Dic106C22semidirect
25. C2×#11719913C22semidirect
26. C2×#11719913C22semidirect
27. C2×#11719913C22semidirect
28. C2×#11719913C22semidirect
30. C2×D424C22semidirect
31. C2×D424C22semidirect
33. #13133205C22semidirect
35. #13133196C22semidirect
40. #13133191C22semidirect
44. C4×Dic106C22semidirect
47. #13133151C22semidirect
48. #13133142C22semidirect
49. #13133142C22semidirect
51. #13133139C22semidirect
52. #13133139C22semidirect
53. #13133139C22semidirect
54. C2×M16×C53C22semidirect
55. C22×C8×C53C22semidirect
57. C4⋊C8×C53C22semidirect
58. C22⋊C8×C53C22semidirect
59. C22⋊C8×C53C22semidirect
60. #97×C53C22semidirect
123. #11719913D4semidirect
173. Dic106C2×D4semidirect
177. C8×C53C2×D4semidirect
183. C22×C53AGL(1,ℤ/8)semidirect
233. #435D53semidirect
234. C53#3265semidirect
251. C2×M16C2×D53semidirect
252. C22×C8C2×D53semidirect
270. C4⋊C8C2×D53semidirect
277. C22⋊C8C2×D53semidirect
278. C22⋊C8C2×D53semidirect
279. #97C2×D53semidirect
396. C8D4×D53semidirect
416. C22#13133245semidirect