Finite Group Databasebeta

← #20286260

Subgroups — #20286260

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×#13133245C211yesmaximal, characteristic, complemented
2C2×#13133239C211yesmaximal, characteristic, complemented
3#16811294C211yesmaximal, characteristic, complemented
4#16811269C211yesmaximal, characteristic, complemented
5#16811012C211yesmaximal, characteristic, complemented
6#16810869C211yesmaximal, characteristic, complemented
7#16810859C211yesmaximal, characteristic, complemented
8#16810846C211yesmaximal, characteristic, complemented
9#16810819C211yesmaximal, characteristic, complemented
10#16810787C211yesmaximal, characteristic, complemented
11#16810770C211yesmaximal, characteristic, complemented
12#16810770C211yesmaximal, characteristic, complemented
13#16810759C211yesmaximal, characteristic, complemented
14#16810725C211yesmaximal, characteristic, complemented
15C53×#435C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×#11719945C2211yescharacteristic
18C2×#11719945C2211yescharacteristic, complemented
19C22×D212C2211yescharacteristic
20#13133245—24no—
21#13133239—24no—
22C2×Dic212C2211yescharacteristic, complemented
23C2×#11719913C2211yescharacteristic
24C2×#11719913C2211yescharacteristic
25C2×#11719913C2211yescharacteristic, complemented
26C2×#11719913C2211yescharacteristic, complemented
27C2×D424C2211yescharacteristic
28C2×D424C2211yescharacteristic
29C2×D424C2211yescharacteristic
30C2×D424C2211yescharacteristic
31#13133214C2211yescharacteristic, complemented
32#13133213—21no—
33#13133205C2211yescharacteristic
34#13133202—21no—
35#13133196C2211yescharacteristic
36#13133191C2211yescharacteristic
37#13133191C2211yescharacteristic
38#13133191C2211yescharacteristic, complemented
39#13133190—21no—
40#13133190—21no—
41#13133190—21no—
42#13133185—21no—
43C4×Dic106C2211yescharacteristic, complemented
44C4×D212C2211yescharacteristic
45#13133179—21no—
46C22⋊C4×D53—21no—
47#13133151C2211yescharacteristic, complemented
48#13133142C2211yescharacteristic, complemented
49#13133142C2211yescharacteristic, complemented
50#13133139C2211yescharacteristic
51#13133139C2211yescharacteristic
52#13133139C2211yescharacteristic, complemented
53#13133139C2211yescharacteristic, complemented
54C2×M16×C53C2211yescharacteristic
55C22×C8×C53C2211yescharacteristic, complemented
56C4×D4×C53C2211yescharacteristic
57C4⋊C8×C53C2211yescharacteristic, complemented
58C22⋊C8×C53C2211yescharacteristic
59C22⋊C8×C53C2211yescharacteristic, complemented
60#97×C53C2211yescharacteristic, complemented
61C23×D53—21no—
62C2×D4×C53D411yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71#11719945—22no—
72#11719945—22no—
73#11719945—22no—
74#11719945—41no—
75#11719945—41no—
76#11719945—41no—
77C2×D212—22no—
78C2×D212—21no—
79C2×D212—21no—
80C2×D212—21no—
81C2×D212—41no—
82C2×D212C2311yescharacteristic
83C2×D212C2311yescharacteristic
84C2×D212C2311yescharacteristic
85C2×D212C2311yescharacteristic
86C2×C4×D53—21no—
87C2×C4×D53—21no—
88C2×C4×D53—21no—
89C2×C4×D53—21no—
90C2×Dic106C2311yescharacteristic
91C2×Dic106C2311yescharacteristic
92M16×C53—22no—
93C2×C8×C53—22no—
94C2×C8×C53C2311yescharacteristic
95C2×C8×C53C2311yescharacteristic
96C2×C8×C53C2311yescharacteristic
97C2×C8×C53C2311yescharacteristic
98C4⋊C4×C53D411yescharacteristic
99C22⋊C4×C53D411yescharacteristic
100C22⋊C4×C53D411yescharacteristic
101C42×C53C2311yescharacteristic
102C22⋊Dic53—21no—
103#11719921—21no—
104#11719921—21no—
105#11719921—21no—
106#11719921—21no—
107#11719921—21no—
108#11719921—21no—
109C4⋊Dic53C2311yescharacteristic
110C4⋊Dic53C2311yescharacteristic
111#11719919—21no—
112#11719919—21no—
113C4×Dic53—21no—
114Dic212—22no—
115D424—22no—
116D424—22no—
117D424—22no—
118D424—41no—
119#11719913—22no—
120#11719913—22no—
121#11719913—41no—
122#11719913D414yescomplemented
123C23×C53C2×D411yescharacteristic
124C23×C53C2×D411yescharacteristic
125C22×D53—21no—
126C22×D53—21no—
127C22×D53—21no—
128C22×D53—21no—
129C22×D53—42no—
130C22×D53—42no—
131C22×D53—81no—
132D4×C53—41no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53C2×D411yescharacteristic
138C2×C4×C53C2×D411yescharacteristic
139C2×C4×C53C2×D411yescharacteristic
140C2×C4×C53C2×D411yescharacteristic
141C2×C4×C53C2411yescharacteristic
142C22⋊D53—42no—
143C22⋊D53—42no—
144C22⋊D53—42no—
145C22⋊D53—81no—
146C22⋊D53—81no—
147C22⋊D53—81no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—21no—
152D212—21no—
153D212—21no—
154D212—21no—
155D212—21no—
156D212—21no—
157D212—21no—
158D212—21no—
159D212—42no—
160D212—42no—
161D212—81no—
162D212C2×D412yes—
163C4×D53—42no—
164C4×D53—42no—
165C4×D53—42no—
166C4×D53—81no—
167Dic106—21no—
168Dic106—21no—
169Dic106—21no—
170Dic106C2×D412yes—
171C8×C53—21no—
172C8×C53—21no—
173C8×C53—21no—
174C8×C53C2×D412yes—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—21no—
181C22×C53C22×D411yescharacteristic
182C2×D53—22no—
183C2×D53—41no—
184C2×D53—41no—
185C2×D53—41no—
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—84no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—21no—
198C4×C53—21no—
199C4×C53C22×D411yescharacteristic
200C4×C53H(5,𝔽2)11yescharacteristic
201Dic53—22no—
202Dic53—41no—
203Dic53—41no—
204Dic53—41no—
205#3268—531nomaximal, Sylow
206C2×C53—41no—
207C2×C53—41no—
208C2×C53D4211yescharacteristic
209C2×C53#57511yescharacteristic
210C2×C53#57611yescharacteristic
211D53—42no—
212D53—81no—
213D53—81no—
214D53—81no—
215C2×AGL(1,ℤ/8)—531no—
216C2×#135—531no—
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#435D5311yescharacteristic, complemented
230C53#326811yescharacteristic, complemented, Sylow
231C2×#41—531no—
232C2×#41—531no—
233C2×#41—531no—
234C22×D4—531no—
235AGL(1,ℤ/8)—1064no—
236#135—1064no—
237C2×Q16—531no—
238C2×SD16—531no—
239C2×SD16—531no—
240C2×SD16—531no—
241C2×SD16—531no—
242C2×D8—531no—
243C2×D8—531no—
244C2×D8—531no—
245C2×D8—531no—
246C2×M16C2×D5311yescharacteristic
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—1061no—
263C4×D4C2×D5311yescharacteristic
264C2×C22⋊C4—1061no—
265#107—531no—
266C4⋊C8C2×D5311yescharacteristic, complemented
267#103—531no—
268#103—531no—
269#102—531no—
270#102—531no—
271#102—531no—
272#102—531no—
273C22⋊C8C2×D5311yescharacteristic
274C22⋊C8C2×D5311yescharacteristic, complemented
275#97C2×D5311yescharacteristic, complemented
276C24—1061no—
277#41—1062no—
278#41—1062no—
279#41—1062no—
280#41—2121no—
281#41—2121no—
282#41—2121no—
283C2×Q8—531no—
284C2×Q8—531no—
285C2×D4—531no—
286C2×D4—531no—
287C2×D4—531no—
288C2×D4—531no—
289C2×D4—1062no—
290C2×D4—1061no—
291C2×D4—1061no—
292C2×D4—1061no—
293C2×D4—1061no—
294C2×D4—1061no—
295C2×D4—1061no—
296C2×D4—1061no—
297C2×D4—1061no—
298C2×D4—1061no—
299C2×D4—2121no—
300C2×D4D21211yescharacteristic
301C22×C4—1061no—
302C22×C4—1061no—
303C22×C4—1061no—
304C22×C4—1061no—
305C22×C4C22×D5311yescharacteristic
306C22×C4C22×D5311yescharacteristic
307Q16—1062no—
308SD16—534no—
309SD16—1062no—
310SD16—1062no—
311SD16—2121no—
312D8—1062no—
313D8—1062no—
314D8—1062no—
315D8—2121no—
316M16—22no—
317C2×C8—22no—
318C2×C8C22×D5311yescharacteristic
319C2×C8C22×D5311yescharacteristic
320C2×C8C22×D5311yescharacteristic
321C2×C8C22×D5311yescharacteristic
322C4⋊C4—531no—
323C4⋊C4—531no—
324C4⋊C4—1061no—
325C4⋊C4—1061no—
326C4⋊C4D21211yescharacteristic
327C22⋊C4—1061no—
328C22⋊C4—1061no—
329C22⋊C4—1061no—
330C22⋊C4—1061no—
331C22⋊C4—1061no—
332C22⋊C4—1061no—
333C22⋊C4—1061no—
334C22⋊C4D21211yescharacteristic
335C22⋊C4D21211yescharacteristic
336C42—1061no—
337C42C22×D5311yescharacteristic
338C23—1061no—
339C23—1061no—
340C23—1061no—
341C23—1061no—
342C23—4241no—
343C23—2122no—
344C23—2122no—
345C23C2×D21211yescharacteristic
346C23C2×D21211yescharacteristic
347Q8—532no—
348Q8—1061no—
349Q8—1061no—
350Q8—1061no—
351D4—41no—
352D4—532no—
353D4—1061no—
354D4—1061no—
355D4—1061no—
356D4—1061no—
357D4—1061no—
358D4—1061no—
359D4—1061no—
360D4—4241no—
361D4—4241no—
362D4—4241no—
363D4—4241no—
364D4—2122no—
365D4—2122no—
366D4—2122no—
367D4—2122no—
368D4—2122no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—1061no—
374C2×C4—1061no—
375C2×C4—1061no—
376C2×C4—1061no—
377C2×C4—4241no—
378C2×C4—2122no—
379C2×C4—2122no—
380C2×C4—2122no—
381C2×C4C2×D21211yescharacteristic
382C2×C4C2×D21211yescharacteristic
383C2×C4C2×D21211yescharacteristic
384C2×C4C2×D21211yescharacteristic
385C2×C4C23×D5311yescharacteristic
386C8—21no—
387C8—21no—
388C8—21no—
389C8D4×D5312yes—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—1062no—
397C22—4244no—
398C22—2121no—
399C22—2121no—
400C22—2121no—
401C22—2121no—
402C22—2121no—
403C22—2121no—
404C22—2121no—
405C22—2121no—
406C22—2121no—
407C22—2121no—
408C22—2121no—
409C22C22×D21211yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—1062no—
415C4—2121no—
416C4—2121no—
417C4—2121no—
418C4C2×D4×D5311yescharacteristic
419C4#1313326311yescharacteristic
420C2—41no—
421C2—41no—
422C2—4241no—
423C2—4241no—
424C2—4241no—
425C2—2122no—
426C2D4×D21211yescharacteristic
427C2#1681142311yescharacteristic
428C2#1681142811yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#435C2
socleC22×C53C22×D4
hypercenter#435D53
hypocenterC53#3268
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. C2×#13133239C2semidirect
3. #16811294C2semidirect
4. #16811269C2semidirect
5. #16811012C2semidirect
6. #16810869C2semidirect
7. #16810859C2semidirect
8. #16810846C2semidirect
9. #16810819C2semidirect
10. #16810787C2semidirect
11. #16810770C2semidirect
12. #16810770C2semidirect
13. #16810759C2semidirect
14. #16810725C2semidirect
15. C53×#435C2semidirect
18. C2×#11719945C22semidirect
22. C2×Dic212C22semidirect
25. C2×#11719913C22semidirect
26. C2×#11719913C22semidirect
31. #13133214C22semidirect
38. #13133191C22semidirect
43. C4×Dic106C22semidirect
47. #13133151C22semidirect
48. #13133142C22semidirect
49. #13133142C22semidirect
52. #13133139C22semidirect
53. #13133139C22semidirect
55. C22×C8×C53C22semidirect
57. C4⋊C8×C53C22semidirect
59. C22⋊C8×C53C22semidirect
60. #97×C53C22semidirect
122. #11719913D4semidirect
229. #435D53semidirect
230. C53#3268semidirect
247. C22×C8C2×D53semidirect
266. C4⋊C8C2×D53semidirect
274. C22⋊C8C2×D53semidirect
275. #97C2×D53semidirect