Finite Group Databasebeta

← #20286236

Subgroups — #20286236

Order 6784.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133243C211yesmaximal, characteristic, complemented
2C2×#13133237C211yesmaximal, characteristic, complemented
3#16811278C211yesmaximal, characteristic, complemented
4#16811018C211yesmaximal, characteristic, complemented
5#16810879C211yesmaximal, characteristic, complemented
6#16810850C211yesmaximal, characteristic, complemented
7#16810835C211yesmaximal, characteristic, complemented
8#16810828C211yesmaximal, characteristic, complemented
9#16810785C211yesmaximal, characteristic, complemented
10#16810783C211yesmaximal, characteristic, complemented
11#16810774C211yesmaximal, characteristic, complemented
12#16810766C211yesmaximal, characteristic, complemented
13#16810733C211yesmaximal, characteristic, complemented
14#16810695C211yesmaximal, characteristic, complemented
15C53×#546C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×#11719945C2211yescharacteristic, complemented
18#13133243—24no—
19#13133237—24no—
20C2×#11719925C2211yescharacteristic, complemented
21C2×#11719925C2211yescharacteristic, complemented
22C2×#11719923C2211yescharacteristic
23C2×#11719923C2211yescharacteristic
24C2×#11719923C2211yescharacteristic
25C2×#11719923C2211yescharacteristic
26C2×#11719924C2211yescharacteristic, complemented
27C2×#11719922C2211yescharacteristic
28C2×#11719922C2211yescharacteristic
29C2×#11719917C2211yescharacteristic
30C22×#88322C2211yescharacteristic, complemented
31#13133217C2211yescharacteristic, complemented
32#13133199—21no—
33#13133199—21no—
34#13133196C2211yescharacteristic
35#13133196C2211yescharacteristic, complemented
36#13133188—21no—
37C4×D212C2211yescharacteristic, complemented
38C4⋊C4×D53—21no—
39#13133148C2211yescharacteristic, complemented
40C4⋊#88322C2211yescharacteristic, complemented
41#13133141C2211yescharacteristic, complemented
42#13133140C2211yescharacteristic, complemented
43#13133140C2211yescharacteristic, complemented
44#13133138C2211yescharacteristic
45#13133138C2211yescharacteristic
46#13133137C2211yescharacteristic, complemented
47C22⋊#88322C2211yescharacteristic
48C22⋊#88322C2211yescharacteristic, complemented
49#13133128C2211yescharacteristic, complemented
50C2×#41×C53C2211yescharacteristic
51C22×D4×C53C2211yescharacteristic
52#124×C53C2211yescharacteristic
53#123×C53—21no—
54C22⋊Q8×C53C2211yescharacteristic
55#121×C53—21no—
56#121×C53C2211yescharacteristic
57#120×C53—21no—
58C4×D4×C53C2211yescharacteristic
59C4×D4×C53C2211yescharacteristic
60C2×C22⋊C4×C53—21no—
61C24×C53—21no—
62#41×C53—22no—
63#41×C53—41no—
64C2×Q8×C53C2311yescharacteristic
65C2×D4×C53—22no—
66C2×D4×C53—21no—
67C2×D4×C53—21no—
68C2×D4×C53—21no—
69C2×D4×C53—41no—
70C2×D4×C53D411yescharacteristic
71C2×D4×C53C2311yescharacteristic
72C2×D4×C53C2311yescharacteristic
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53C2311yescharacteristic
76C22×C4×C53C2311yescharacteristic
77C2×C22⋊D53—21no—
78C2×C22⋊D53—21no—
79#11719945—22no—
80#11719945—22no—
81#11719945—41no—
82#11719945—41no—
83C2×D212C2311yescharacteristic
84C2×C4×D53—21no—
85C2×C4×D53—21no—
86C2×C4×D53—21no—
87C2×Dic106C2311yescharacteristic
88C2×Dic106C2311yescharacteristic
89C4⋊C4×C53—21no—
90C4⋊C4×C53D411yescharacteristic
91C4⋊C4×C53C2311yescharacteristic
92C22⋊C4×C53—21no—
93C22⋊C4×C53—21no—
94C22⋊C4×C53—21no—
95C22⋊C4×C53—21no—
96C22⋊C4×C53—21no—
97C22⋊C4×C53D411yescharacteristic
98C22⋊C4×C53D411yescharacteristic
99C42×C53C2311yescharacteristic
100C22⋊Dic53—21no—
101C22⋊Dic53—21no—
102#11719925—22no—
103#11719925—41no—
104#11719924—22no—
105#11719923—22no—
106#11719923—22no—
107#11719923—22no—
108#11719923—41no—
109#11719922—22no—
110#11719922D414yescomplemented
111#11719921—21no—
112C4⋊Dic53—21no—
113C4⋊Dic53—21no—
114C4⋊Dic53C2311yescharacteristic
115#11719919—21no—
116#11719919—21no—
117#11719917—22no—
118C2×#88322—22no—
119C2×#88322C2311yescharacteristic
120C2×#88322C2311yescharacteristic
121C2×#88322C2311yescharacteristic
122C2×#88322C2311yescharacteristic
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—42no—
126C23×C53—42no—
127C23×C53—81no—
128C23×C53C2×D411yescharacteristic
129C23×C53C2×D411yescharacteristic
130C22×D53—21no—
131Q8×C53—21no—
132Q8×C53—21no—
133D4×C53—21no—
134D4×C53—21no—
135D4×C53—21no—
136D4×C53—41no—
137D4×C53—42no—
138D4×C53—42no—
139D4×C53—81no—
140D4×C53C2×D412yes—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—42no—
148C2×C4×C53—81no—
149C2×C4×C53C2×D411yescharacteristic
150C2×C4×C53C2×D411yescharacteristic
151C2×C4×C53C2×D411yescharacteristic
152C2×C4×C53C2×D411yescharacteristic
153C2×C4×C53C2411yescharacteristic
154C22⋊D53—42no—
155C22⋊D53—42no—
156C2×Dic53—21no—
157C2×Dic53—21no—
158C2×Dic53—21no—
159D212—21no—
160D212C2×D412yes—
161C4×D53—42no—
162C4×D53—42no—
163C4×D53—42no—
164Dic106—21no—
165Dic106—21no—
166Dic106—21no—
167Dic106—21no—
168#88322—21no—
169#88322—21no—
170#88322—21no—
171#88322C2×D412yes—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—22no—
175C22×C53—21no—
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×C53—84no—
185C22×C53C22×D411yescharacteristic
186C2×D53—22no—
187C2×D53—41no—
188C2×D53—41no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—41no—
194C4×C53—41no—
195C4×C53C22×D411yescharacteristic
196C4×C53H(5,𝔽2)11yescharacteristic
197Dic53—41no—
198Dic53—41no—
199Dic53—41no—
200#3267—531nomaximal, Sylow
201C2×C53—41no—
202C2×C53—41no—
203C2×C53—42no—
204C2×C53—81no—
205C2×C53D4211yescharacteristic
206C2×C53#57511yescharacteristic
207C2×C53#57711yescharacteristic
208D53—42no—
209C2×AGL(1,ℤ/8)—531no—
210C2×#135—531no—
211#547—531no—
212#546D5311yescharacteristic, complemented
213#497—531no—
214#470—531no—
215#465—531no—
216#463—531no—
217#461—531no—
218#452—531no—
219#452—531no—
220#450—531no—
221#449—531no—
222#442—531no—
223#435—531no—
224C53#326711yescharacteristic, complemented, Sylow
225C2×#41—531no—
226C2×#41—531no—
227C2×#41C2×D5311yescharacteristic
228C22×D4C2×D5311yescharacteristic
229AGL(1,ℤ/8)—1064no—
230#135—1064no—
231C2×Q16—531no—
232C2×Q16—531no—
233C2×SD16—531no—
234C2×SD16—531no—
235C2×SD16—531no—
236C2×SD16—531no—
237C2×SD16—531no—
238C2×D8—531no—
239C2×D8—531no—
240C2×M16—531no—
241C22×C8—531no—
242C4⋊Q8—531no—
243#124C2×D5311yescharacteristic
244#123—21no—
245#123—1061no—
246#123—1061no—
247C22⋊Q8—531no—
248C22⋊Q8—531no—
249C22⋊Q8C2×D5311yescharacteristic
250#121—21no—
251#121—1061no—
252#121C2×D5311yescharacteristic
253#120—21no—
254C4×D4—531no—
255C4×D4C2×D5311yescharacteristic
256C4×D4C2×D5311yescharacteristic
257C2×C4⋊C4—1061no—
258C2×C22⋊C4—21no—
259#106—531no—
260C4⋊C8—531no—
261#103—531no—
262#103—531no—
263#103—531no—
264#102—531no—
265#102—531no—
266#102—531no—
267C22⋊C8—531no—
268C22⋊C8—531no—
269#97—531no—
270C24—21no—
271#41—22no—
272#41—41no—
273#41—1062no—
274#41—1062no—
275#41—2121no—
276#41—2121no—
277C2×Q8—531no—
278C2×Q8—531no—
279C2×Q8C22×D5311yescharacteristic
280C2×D4—22no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—41no—
285C2×D4—531no—
286C2×D4—1061no—
287C2×D4—1061no—
288C2×D4C22⋊D5311yescharacteristic
289C2×D4C22×D5311yescharacteristic
290C2×D4C22×D5311yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—1061no—
294C22×C4—1061no—
295C22×C4—1061no—
296C22×C4C22×D5311yescharacteristic
297C22×C4C22×D5311yescharacteristic
298Q16—1062no—
299Q16—2121no—
300SD16—1062no—
301SD16—1062no—
302SD16—1062no—
303SD16—1062no—
304SD16—2121no—
305D8—534no—
306D8—1062no—
307M16—1062no—
308C2×C8—531no—
309C2×C8—531no—
310C2×C8—531no—
311C2×C8—531no—
312C2×C8—1062no—
313C4⋊C4—21no—
314C4⋊C4—531no—
315C4⋊C4—1061no—
316C4⋊C4—1061no—
317C4⋊C4—1061no—
318C4⋊C4—1061no—
319C4⋊C4C22⋊D5311yescharacteristic
320C4⋊C4C22×D5311yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—21no—
326C22⋊C4—1061no—
327C22⋊C4—1061no—
328C22⋊C4—1061no—
329C22⋊C4C22⋊D5311yescharacteristic
330C22⋊C4C22⋊D5311yescharacteristic
331C42C22×D5311yescharacteristic
332C23—21no—
333C23—21no—
334C23—42no—
335C23—42no—
336C23—81no—
337C23—1061no—
338C23C2×C22⋊D5311yescharacteristic
339C23C2×C22⋊D5311yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—1061no—
343Q8—1061no—
344Q8—1061no—
345Q8—1061no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—41no—
350D4—42no—
351D4—42no—
352D4—81no—
353D4—532no—
354D4—1061no—
355D4—2122no—
356D4—2122no—
357D4D4×D5312yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—81no—
366C2×C4—1061no—
367C2×C4—1061no—
368C2×C4—1061no—
369C2×C4—2122no—
370C2×C4—2122no—
371C2×C4—2122no—
372C2×C4C2×C22⋊D5311yescharacteristic
373C2×C4C2×C22⋊D5311yescharacteristic
374C2×C4C2×C22⋊D5311yescharacteristic
375C2×C4C2×C22⋊D5311yescharacteristic
376C2×C4C23×D5311yescharacteristic
377C8—532no—
378C8—1061no—
379C8—1061no—
380C8—1061no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—22no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—84no—
394C22—1062no—
395C22—2121no—
396C22—2121no—
397C22C22×C22⋊D5311yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—2121no—
405C4—2121no—
406C4—2121no—
407C4C2×D4×D5311yescharacteristic
408C4#1313326211yescharacteristic
409C2—41no—
410C2—41no—
411C2—42no—
412C2—81no—
413C2—2122no—
414C2D4×C22⋊D5311yescharacteristic
415C2#1681142111yescharacteristic
416C2#1681143411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#546C2
socleC22×C53C22×D4
hypercenter#546D53
hypocenterC53#3267
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#13133243C2semidirect
2. C2×#13133237C2semidirect
3. #16811278C2semidirect
4. #16811018C2semidirect
5. #16810879C2semidirect
6. #16810850C2semidirect
7. #16810835C2semidirect
8. #16810828C2semidirect
9. #16810785C2semidirect
10. #16810783C2semidirect
11. #16810774C2semidirect
12. #16810766C2semidirect
13. #16810733C2semidirect
14. #16810695C2semidirect
15. C53×#546C2semidirect
17. C2×#11719945C22semidirect
20. C2×#11719925C22semidirect
21. C2×#11719925C22semidirect
26. C2×#11719924C22semidirect
30. C22×#88322C22semidirect
31. #13133217C22semidirect
35. #13133196C22semidirect
37. C4×D212C22semidirect
39. #13133148C22semidirect
40. C4⋊#88322C22semidirect
41. #13133141C22semidirect
42. #13133140C22semidirect
43. #13133140C22semidirect
46. #13133137C22semidirect
48. C22⋊#88322C22semidirect
49. #13133128C22semidirect
110. #11719922D4semidirect
212. #546D53semidirect
224. C53#3267semidirect