Finite Group Databasebeta

← #20286255

Subgroups — #20286255

Order 6784.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133244C211yesmaximal, characteristic, complemented
2#16811292C211yesmaximal, characteristic, complemented
3#16811261C211yesmaximal, characteristic, complemented
4#16811009C211yesmaximal, characteristic, complemented
5#16810866C211yesmaximal, characteristic, complemented
6#16810859C211yesmaximal, characteristic, complemented
7#16810847C211yesmaximal, characteristic, complemented
8#16810816C211yesmaximal, characteristic, complemented
9#16810784C211yesmaximal, characteristic, complemented
10#16810768C211yesmaximal, characteristic, complemented
11#16810768C211yesmaximal, characteristic, complemented
12#16810755C211yesmaximal, characteristic, complemented
13#16810724C211yesmaximal, characteristic, complemented
14#16810694C211yesmaximal, characteristic, complemented
15C2×#135×C53C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×D4×D53C2211yescharacteristic
18#13133244—24no—
19C2×#11719924C2211yescharacteristic
20C2×#11719924C2211yescharacteristic, complemented
21C2×D424C2211yescharacteristic
22C2×#11719922C2211yescharacteristic
23C2×#11719922C2211yescharacteristic
24C2×#11719912C2211yescharacteristic
25#13133213—21no—
26#13133213C2211yescharacteristic, complemented
27#13133206C2211yescharacteristic
28#13133200—21no—
29#13133193C2211yescharacteristic
30#13133191—21no—
31#13133191C2211yescharacteristic
32#13133191C2211yescharacteristic, complemented
33#13133189—21no—
34#13133188—21no—
35#13133188C2211yescharacteristic
36#13133185—21no—
37Q8×Dic53C2211yescharacteristic, complemented
38D4×Dic53C2211yescharacteristic
39C4×C22⋊D53—21no—
40C4×C22⋊D53C2211yescharacteristic
41C2×#11719921—21no—
42#13133151C2211yescharacteristic, complemented
43#13133146C2211yescharacteristic, complemented
44#13133141C2211yescharacteristic, complemented
45#13133141C2211yescharacteristic, complemented
46#13133139C2211yescharacteristic
47#13133139C2211yescharacteristic, complemented
48#13133138C2211yescharacteristic
49#13133138C2211yescharacteristic, complemented
50C22⋊#88322C2211yescharacteristic, complemented
51#13133129C2211yescharacteristic
52#13133127C2211yescharacteristic, complemented
53C2×#41×C53C2211yescharacteristic
54C2×#41×C53C2211yescharacteristic, complemented
55#135×C53—24no—
56C2×Q16×C53C2211yescharacteristic, complemented
57C2×SD16×C53C2211yescharacteristic
58C2×SD16×C53C2211yescharacteristic, complemented
59C2×D8×C53C2211yescharacteristic
60C22×C8×C53C2211yescharacteristic, complemented
61C23×D53—21no—
62#41×C53—22no—
63#41×C53—22no—
64#41×C53—41no—
65#41×C53—41no—
66C2×Q8×C53C2311yescharacteristic
67C2×Q8×C53C2311yescharacteristic
68C2×D4×C53—21no—
69C2×D4×C53—21no—
70C2×D4×C53C2311yescharacteristic
71C2×D4×C53C2311yescharacteristic
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53C2311yescharacteristic
75C2×C22⋊D53—21no—
76C2×C22⋊D53—21no—
77C2×C22⋊D53—21no—
78C2×C22⋊D53—21no—
79C2×C22⋊D53—21no—
80C2×C22⋊D53—21no—
81C2×C22⋊D53D411yescharacteristic
82C22×Dic53—21no—
83#11719949—22no—
84#11719949—41no—
85D4×D53—22no—
86D4×D53—41no—
87C2×D212—21no—
88C2×D212C2311yescharacteristic
89C2×D212C2311yescharacteristic
90C2×C4×D53—21no—
91C2×C4×D53C2311yescharacteristic
92Q16×C53—22no—
93SD16×C53—22no—
94SD16×C53D414yescomplemented
95D8×C53—22no—
96C2×C8×C53—22no—
97C2×C8×C53C2311yescharacteristic
98C2×C8×C53C2311yescharacteristic
99C22⋊Dic53—21no—
100C22⋊Dic53—21no—
101C22⋊Dic53D411yescharacteristic
102#11719924—22no—
103#11719924—41no—
104#11719922—22no—
105#11719922—41no—
106#11719921—21no—
107#11719921—21no—
108#11719921—21no—
109#11719921—21no—
110#11719921—21no—
111#11719921D411yescharacteristic
112C4⋊Dic53—21no—
113C4⋊Dic53C2311yescharacteristic
114C4⋊Dic53C2311yescharacteristic
115#11719919—21no—
116#11719919D411yescharacteristic
117C4×Dic53—21no—
118C4×Dic53C2311yescharacteristic
119C2×#88322C2311yescharacteristic
120C2×#88322C2311yescharacteristic
121D424—22no—
122#11719912—22no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53C2×D411yescharacteristic
126C22×D53—21no—
127C22×D53—21no—
128C22×D53—42no—
129C22×D53—42no—
130C22×D53—81no—
131C22×D53C2×D411yescharacteristic
132Q8×C53—21no—
133Q8×C53—21no—
134Q8×C53—21no—
135Q8×C53C2×D412yes—
136D4×C53—21no—
137D4×C53—21no—
138D4×C53—21no—
139D4×C53—42no—
140D4×C53—42no—
141D4×C53C2×D412yes—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53C2×D411yescharacteristic
149C2×C4×C53C2411yescharacteristic
150C22⋊D53—41no—
151C22⋊D53—42no—
152C22⋊D53—42no—
153C22⋊D53—81no—
154C22⋊D53—81no—
155C22⋊D53—81no—
156C22⋊D53—81no—
157C2×Dic53—21no—
158C2×Dic53—21no—
159C2×Dic53—81no—
160C2×Dic53C2×D411yescharacteristic
161C2×Dic53C2×D411yescharacteristic
162C2×Dic53C2×D411yescharacteristic
163D212—21no—
164D212—21no—
165D212—21no—
166D212—21no—
167D212—42no—
168C4×D53—21no—
169C4×D53—21no—
170C4×D53—42no—
171C8×C53—21no—
172C8×C53C2×D412yes—
173#88322—21no—
174#88322—21no—
175C22×C53—22no—
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53—41no—
184C22×C53C22×D411yescharacteristic
185C2×D53—21no—
186C2×D53—21no—
187C2×D53—21no—
188C2×D53—41no—
189C2×D53—41no—
190C2×D53—41no—
191C2×D53—41no—
192C2×D53—41no—
193C2×D53—41no—
194C2×D53—84no—
195C4×C53—22no—
196C4×C53—21no—
197C4×C53—41no—
198C4×C53C22×D411yescharacteristic
199C4×C53H(5,𝔽2)11yescharacteristic
200Dic53—21no—
201Dic53—21no—
202Dic53—21no—
203Dic53—41no—
204Dic53—41no—
205#3268—531nomaximal, Sylow
206C2×C53—42no—
207C2×C53—41no—
208C2×C53—81no—
209C2×C53D4211yescharacteristic
210C2×C53#57511yescharacteristic
211C2×C53#57611yescharacteristic
212D53—41no—
213D53—81no—
214D53—81no—
215C2×AGL(1,ℤ/8)—531no—
216C2×#135D5311yescharacteristic, complemented
217#550—531no—
218#546—531no—
219#496—531no—
220#468—531no—
221#466—531no—
222#464—531no—
223#459—531no—
224#452—531no—
225#449—531no—
226#449—531no—
227#447—531no—
228#440—531no—
229#435—531no—
230C53#326811yescharacteristic, complemented, Sylow
231C2×#41—531no—
232C2×#41C2×D5311yescharacteristic
233C2×#41C2×D5311yescharacteristic, complemented
234C22×D4—531no—
235AGL(1,ℤ/8)—1064no—
236#135—24no—
237C2×Q16C2×D5311yescharacteristic, complemented
238C2×SD16—531no—
239C2×SD16—531no—
240C2×SD16C2×D5311yescharacteristic
241C2×SD16C2×D5311yescharacteristic, complemented
242C2×D8—531no—
243C2×D8—531no—
244C2×D8—531no—
245C2×D8C2×D5311yescharacteristic
246C2×M16—531no—
247C22×C8C2×D5311yescharacteristic, complemented
248#127—531no—
249#127—1061no—
250#124—531no—
251#123—1061no—
252C22⋊Q8—531no—
253#121—531no—
254#121—531no—
255#121—531no—
256#121—1061no—
257#121—1061no—
258#121—1061no—
259#120—1061no—
260C4×Q8—531no—
261C4×D4—531no—
262C4×D4—531no—
263C4×D4—1061no—
264C2×C22⋊C4—1061no—
265#107—531no—
266C4⋊C8—531no—
267#103—531no—
268#103—531no—
269#102—531no—
270#102—531no—
271#102—531no—
272#102—531no—
273C22⋊C8—531no—
274C22⋊C8—531no—
275#97—531no—
276C24—1061no—
277#41—22no—
278#41—22no—
279#41—41no—
280#41—41no—
281#41—1062no—
282#41—2121no—
283C2×Q8C22×D5311yescharacteristic
284C2×Q8C22×D5311yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—531no—
288C2×D4—531no—
289C2×D4—531no—
290C2×D4—1061no—
291C2×D4—1061no—
292C2×D4—1061no—
293C2×D4—1061no—
294C2×D4—1062no—
295C2×D4—1061no—
296C2×D4—1061no—
297C2×D4—1061no—
298C2×D4—2121no—
299C2×D4C22×D5311yescharacteristic
300C2×D4C22×D5311yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—531no—
304C22×C4—1061no—
305C22×C4—1061no—
306C22×C4C22×D5311yescharacteristic
307Q16—22no—
308SD16—22no—
309SD16—1062no—
310SD16—2121no—
311SD16C22⋊D5314yescomplemented
312D8—22no—
313D8—1062no—
314D8—1062no—
315D8—2121no—
316M16—1062no—
317C2×C8—22no—
318C2×C8—531no—
319C2×C8—531no—
320C2×C8C22×D5311yescharacteristic
321C2×C8C22×D5311yescharacteristic
322C4⋊C4—531no—
323C4⋊C4—531no—
324C4⋊C4—531no—
325C4⋊C4—1061no—
326C4⋊C4—1061no—
327C22⋊C4—531no—
328C22⋊C4—531no—
329C22⋊C4—1061no—
330C22⋊C4—1061no—
331C22⋊C4—1061no—
332C22⋊C4—1061no—
333C22⋊C4—1061no—
334C22⋊C4—1061no—
335C22⋊C4—1061no—
336C42—531no—
337C42—1061no—
338C23—21no—
339C23—21no—
340C23—531no—
341C23—1061no—
342C23—1061no—
343C23—4241no—
344C23—2122no—
345C23—2122no—
346C23D4×D5311yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—21no—
350Q8C2×C22⋊D5312yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—42no—
355D4—42no—
356D4—1061no—
357D4—1061no—
358D4—1061no—
359D4—1061no—
360D4—4241no—
361D4—4241no—
362D4—4241no—
363D4—4241no—
364D4—2122no—
365D4—2121no—
366D4—2122no—
367D4—2122no—
368D4C2×C22⋊D5312yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—531no—
376C2×C4—531no—
377C2×C4—531no—
378C2×C4—1061no—
379C2×C4—1061no—
380C2×C4—1061no—
381C2×C4—1061no—
382C2×C4—4241no—
383C2×C4—2122no—
384C2×C4D4×D5311yescharacteristic
385C2×C4C23×D5311yescharacteristic
386C8—21no—
387C8—1061no—
388C8—1061no—
389C8C2×C22⋊D5312yes—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—1061no—
400C22—1061no—
401C22—1061no—
402C22—4244no—
403C22—2121no—
404C22—2121no—
405C22—2121no—
406C22—2121no—
407C22—2121no—
408C22—2121no—
409C22C2×D4×D5311yescharacteristic
410C4—21no—
411C4—22no—
412C4—41no—
413C4—1061no—
414C4—1061no—
415C4—1061no—
416C4—2121no—
417C4—2121no—
418C4C22×C22⋊D5311yescharacteristic
419C4#1313326211yescharacteristic
420C2—41no—
421C2—42no—
422C2—81no—
423C2—4241no—
424C2—4241no—
425C2—2121no—
426C2D4×C22⋊D5311yescharacteristic
427C2#1681142011yescharacteristic
428C2#1681142511yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×#135×C53C2
socleC22×C53C22×D4
hypercenterC2×#135D53
hypocenterC53#3268
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#13133244C2semidirect
2. #16811292C2semidirect
3. #16811261C2semidirect
4. #16811009C2semidirect
5. #16810866C2semidirect
6. #16810859C2semidirect
7. #16810847C2semidirect
8. #16810816C2semidirect
9. #16810784C2semidirect
10. #16810768C2semidirect
11. #16810768C2semidirect
12. #16810755C2semidirect
13. #16810724C2semidirect
14. #16810694C2semidirect
15. C2×#135×C53C2semidirect
20. C2×#11719924C22semidirect
26. #13133213C22semidirect
32. #13133191C22semidirect
37. Q8×Dic53C22semidirect
42. #13133151C22semidirect
43. #13133146C22semidirect
44. #13133141C22semidirect
45. #13133141C22semidirect
47. #13133139C22semidirect
49. #13133138C22semidirect
50. C22⋊#88322C22semidirect
52. #13133127C22semidirect
54. C2×#41×C53C22semidirect
56. C2×Q16×C53C22semidirect
58. C2×SD16×C53C22semidirect
60. C22×C8×C53C22semidirect
94. SD16×C53D4semidirect
216. C2×#135D53semidirect
230. C53#3268semidirect
233. C2×#41C2×D53semidirect
237. C2×Q16C2×D53semidirect
241. C2×SD16C2×D53semidirect
247. C22×C8C2×D53semidirect
311. SD16C22⋊D53semidirect