Finite Group Databasebeta

← #20284257

Subgroups — #20284257

Order 6784.

504 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133244C211yesmaximal, characteristic, complemented
2C2×#13133240C211yesmaximal, characteristic, complemented
3#16811191C211yesmaximal, characteristic, complemented
4#16811105C211yesmaximal, characteristic, complemented
5#16810847C211yesmaximal, characteristic, complemented
6#16810841C211yesmaximal, characteristic, complemented
7#16810835C211yesmaximal, characteristic, complemented
8#16810823C211yesmaximal, characteristic, complemented
9#16810821C211yesmaximal, characteristic, complemented
10#16810819C211yesmaximal, characteristic, complemented
11#16810817C211yesmaximal, characteristic, complemented
12#16810648C211yesmaximal, characteristic, complemented
13#16810623C211yesmaximal, characteristic, complemented
14#102×D53C211yesmaximal, characteristic, complemented
15C53×#464C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719949C2211yescharacteristic
18C2×D4×D53C2211yescharacteristic
19#13133244—24no—
20#13133240—24no—
21C2×#11719924C2211yescharacteristic
22C2×#11719924C2211yescharacteristic, complemented
23C2×#11719924C2211yescharacteristic, complemented
24C2×D424C2211yescharacteristic
25C2×D424C2211yescharacteristic, complemented
26C2×#11719922C2211yescharacteristic
27C2×#11719912C2211yescharacteristic
28C2×C8×D53C2211yescharacteristic
29#13133214C2211yescharacteristic
30#13133213C2211yescharacteristic
31#13133206C2211yescharacteristic
32#13133190—21no—
33#13133190—21no—
34#13133187—21no—
35#13133187—21no—
36Q8×Dic53C2211yescharacteristic, complemented
37C4×D212—21no—
38C4×D212C2211yescharacteristic, complemented
39#13133179—21no—
40#13133179C2211yescharacteristic, complemented
41#13133174—21no—
42#13133174C2211yescharacteristic
43C4⋊C4×D53—21no—
44C4⋊C4×D53C2211yescharacteristic
45C4⋊#88322C2211yescharacteristic, complemented
46#13133146C2211yescharacteristic, complemented
47#13133146C2211yescharacteristic, complemented
48#13133141C2211yescharacteristic, complemented
49#13133139C2211yescharacteristic
50#13133139C2211yescharacteristic, complemented
51#13133138C2211yescharacteristic
52#13133137C2211yescharacteristic
53#13133137C2211yescharacteristic, complemented
54C2×Q16×C53C2211yescharacteristic, complemented
55C2×SD16×C53C2211yescharacteristic
56#124×C53C2211yescharacteristic
57C4×Q8×C53C2211yescharacteristic, complemented
58C4⋊C8×C53C2211yescharacteristic, complemented
59#103×C53C2211yescharacteristic, complemented
60#102×C53C2211yescharacteristic
61C23×D53—21no—
62C2×Q8×C53C2311yescharacteristic
63C2×Q8×C53C2311yescharacteristic
64C2×D4×C53C2311yescharacteristic
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68#11719949—22no—
69#11719949—22no—
70#11719949—22no—
71#11719949D414yescomplemented
72D4×D53—22no—
73D4×D53—22no—
74C2×D212—21no—
75C2×D212—21no—
76C2×D212—21no—
77C2×D212—21no—
78C2×D212C2311yescharacteristic
79C2×D212C2311yescharacteristic
80C2×D212C2311yescharacteristic
81C2×C4×D53—21no—
82C2×C4×D53—21no—
83C2×C4×D53—21no—
84C2×C4×D53—21no—
85C2×C4×D53—21no—
86C2×C4×D53C2311yescharacteristic
87Q16×C53—22no—
88SD16×C53—22no—
89C2×C8×C53C2311yescharacteristic
90C2×C8×C53C2311yescharacteristic
91C4⋊C4×C53—21no—
92C4⋊C4×C53C2311yescharacteristic
93C22⋊C4×C53—21no—
94C22⋊C4×C53—21no—
95C42×C53—21no—
96C42×C53C2311yescharacteristic
97#11719924—22no—
98#11719924—22no—
99#11719924—22no—
100#11719922—22no—
101#11719921—21no—
102#11719921—21no—
103#11719921—21no—
104#11719921—21no—
105#11719921D411yescharacteristic
106#11719921D411yescharacteristic
107C4⋊Dic53—21no—
108C4⋊Dic53C2311yescharacteristic
109#11719919—21no—
110#11719919D411yescharacteristic
111#11719919D411yescharacteristic
112C4×Dic53—21no—
113C4×Dic53C2311yescharacteristic
114C2×#88322C2311yescharacteristic
115C2×#88322C2311yescharacteristic
116D424—22no—
117D424—22no—
118#11719912—22no—
119C8×D53—22no—
120C23×C53—21no—
121C22×D53—21no—
122C22×D53—21no—
123C22×D53—21no—
124C22×D53—42no—
125C22×D53—81no—
126C22×D53—81no—
127C22×D53C2×D411yescharacteristic
128Q8×C53—21no—
129Q8×C53—21no—
130Q8×C53—21no—
131Q8×C53C2×D412yes—
132D4×C53—21no—
133D4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53C2×D411yescharacteristic
138C2×C4×C53C2×D411yescharacteristic
139C2×C4×C53C2411yescharacteristic
140C22⋊D53—42no—
141C22⋊D53—81no—
142C22⋊D53—81no—
143C2×Dic53—21no—
144C2×Dic53C2×D411yescharacteristic
145C2×Dic53C2×D411yescharacteristic
146C2×Dic53C2×D411yescharacteristic
147D212—24no—
148D212—21no—
149D212—21no—
150D212—21no—
151D212—21no—
152D212—21no—
153D212—42no—
154D212—81no—
155D212—81no—
156D212C2×D412yes—
157C4×D53—24no—
158C4×D53—42no—
159C4×D53—81no—
160C4×D53—81no—
161C4×D53—81no—
162C4×D53C2×D412yes—
163C4×D53#4112yes—
164C8×C53—21no—
165C8×C53—21no—
166#88322—21no—
167#88322—21no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53C22×D411yescharacteristic
172C2×D53—21no—
173C2×D53—21no—
174C2×D53—22no—
175C2×D53—41no—
176C2×D53—41no—
177C2×D53—41no—
178C2×D53—41no—
179C2×D53—41no—
180C2×D53—41no—
181C2×D53—41no—
182C2×D53—41no—
183C2×D53—84no—
184C2×D53#12112yes—
185C4×C53—22no—
186C4×C53—21no—
187C4×C53—21no—
188C4×C53—41no—
189C4×C53—41no—
190C4×C53C22×D411yescharacteristic
191C4×C53C2×#4111yescharacteristic
192Dic53—21no—
193Dic53—21no—
194Dic53—41no—
195Dic53#12112yes—
196#3045—531nomaximal, Sylow
197C2×C53—81no—
198C2×C53C2×#12111yescharacteristic
199C2×C53#57611yescharacteristic
200C2×C53#57711yescharacteristic
201D53—22no—
202D53—42no—
203D53—81no—
204D53—81no—
205C2×AGL(1,ℤ/8)—531no—
206C2×#135—531no—
207#535—531no—
208#520—531no—
209#464—531no—
210#464D5311yescharacteristic, complemented
211#463—531no—
212#463—531no—
213#460—531no—
214#460—531no—
215#459—531no—
216#459—531no—
217#424—531no—
218#417—531no—
219C2×#102—531no—
220C53#304511yescharacteristic, complemented, Sylow
221C2×#41—531no—
222C2×#41—531no—
223C22×D4—531no—
224AGL(1,ℤ/8)—1064no—
225#135—1064no—
226C2×Q16C2×D5311yescharacteristic, complemented
227C2×SD16—531no—
228C2×SD16—531no—
229C2×SD16—531no—
230C2×SD16C2×D5311yescharacteristic
231C2×D8—531no—
232C2×D8—531no—
233C2×D8—531no—
234C2×M16—531no—
235C22×C8—531no—
236#127—531no—
237#127—531no—
238#124—531no—
239#124C2×D5311yescharacteristic
240#121—1061no—
241#121—1061no—
242#120—1061no—
243#120—1061no—
244C4×Q8—531no—
245C4×Q8C2×D5311yescharacteristic, complemented
246C4×D4—531no—
247C4×D4—531no—
248C4×D4—1061no—
249C4×D4—1061no—
250#117—531no—
251#117—1061no—
252C2×C4⋊C4—531no—
253C2×C4⋊C4—1061no—
254C4⋊C8—531no—
255C4⋊C8—531no—
256C4⋊C8—531no—
257C4⋊C8C2×D5311yescharacteristic, complemented
258#103—531no—
259#103C2×D5311yescharacteristic, complemented
260#102—531no—
261#102—531no—
262#102—531no—
263#102—531no—
264#102—531no—
265#102C2×D5311yescharacteristic
266C24—1061no—
267#41—534no—
268#41—1062no—
269#41—1062no—
270#41—1062no—
271C2×Q8C22×D5311yescharacteristic
272C2×Q8C22×D5311yescharacteristic
273C2×D4—531no—
274C2×D4—531no—
275C2×D4—531no—
276C2×D4—1061no—
277C2×D4—1061no—
278C2×D4—1062no—
279C2×D4—1062no—
280C2×D4—1061no—
281C2×D4—1061no—
282C2×D4—1061no—
283C2×D4—1061no—
284C2×D4—1061no—
285C2×D4C22×D5311yescharacteristic
286C22×C4—531no—
287C22×C4—1061no—
288C22×C4—1061no—
289C22×C4—1061no—
290C22×C4—1061no—
291C22×C4—1061no—
292Q16—22no—
293SD16—22no—
294SD16—1062no—
295SD16—1062no—
296SD16—1062no—
297D8—1062no—
298D8—1062no—
299D8—1062no—
300M16—1062no—
301C2×C8—531no—
302C2×C8—531no—
303C2×C8—1062no—
304C2×C8C22×D5311yescharacteristic
305C2×C8C22×D5311yescharacteristic
306C4⋊C4—21no—
307C4⋊C4—531no—
308C4⋊C4—531no—
309C4⋊C4—531no—
310C4⋊C4—1061no—
311C4⋊C4—1061no—
312C4⋊C4C22×D5311yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—531no—
316C22⋊C4—531no—
317C22⋊C4—1061no—
318C22⋊C4—1061no—
319C22⋊C4—1061no—
320C22⋊C4—1061no—
321C42—21no—
322C42—531no—
323C42—1061no—
324C42C22×D5311yescharacteristic
325C23—21no—
326C23—531no—
327C23—1061no—
328C23—1061no—
329C23—1061no—
330C23—4241no—
331C23—4241no—
332C23—2122no—
333Q8—21no—
334Q8—21no—
335Q8—21no—
336Q8D4×D5312yes—
337D4—21no—
338D4—21no—
339D4—532no—
340D4—1064no—
341D4—1061no—
342D4—1061no—
343D4—1061no—
344D4—1061no—
345D4—1061no—
346D4—4241no—
347D4—4241no—
348D4—4241no—
349D4—4241no—
350D4—2122no—
351D4—2122no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—531no—
356C2×C4—532no—
357C2×C4—532no—
358C2×C4—531no—
359C2×C4—531no—
360C2×C4—1064no—
361C2×C4—1061no—
362C2×C4—4241no—
363C2×C4—4241no—
364C2×C4—4241no—
365C2×C4—2122no—
366C2×C4D4×D5311yescharacteristic
367C2×C4D4×D5311yescharacteristic
368C2×C4C23×D5311yescharacteristic
369C8—21no—
370C8—21no—
371C8—1061no—
372C8—1061no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—532no—
377C22—1061no—
378C22—1061no—
379C22—1062no—
380C22—4244no—
381C22—2121no—
382C22—2121no—
383C22—2121no—
384C22—2121no—
385C22—2121no—
386C22—2121no—
387C22—2121no—
388C22—2121no—
389C22C2×D4×D5311yescharacteristic
390C4—22no—
391C4—21no—
392C4—21no—
393C4—41no—
394C4—41no—
395C4—532no—
396C4—1061no—
397C4—1061no—
398C4—2121no—
399C4C2×D4×D5311yescharacteristic
400C4#41×D5311yescharacteristic
401C2—81no—
402C2—1062no—
403C2—4241no—
404C2—4241no—
405C2—2122no—
406C2#121×D5311yescharacteristic
407C2#1681142511yescharacteristic
408C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#464C2
socleC22×C53C22×D4
hypercenter#464D53
hypocenterC53#3045
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#13133244C2semidirect
2. C2×#13133240C2semidirect
3. #16811191C2semidirect
4. #16811105C2semidirect
5. #16810847C2semidirect
6. #16810841C2semidirect
7. #16810835C2semidirect
8. #16810823C2semidirect
9. #16810821C2semidirect
10. #16810819C2semidirect
11. #16810817C2semidirect
12. #16810648C2semidirect
13. #16810623C2semidirect
14. #102×D53C2semidirect
15. C53×#464C2semidirect
22. C2×#11719924C22semidirect
23. C2×#11719924C22semidirect
25. C2×D424C22semidirect
36. Q8×Dic53C22semidirect
38. C4×D212C22semidirect
40. #13133179C22semidirect
45. C4⋊#88322C22semidirect
46. #13133146C22semidirect
47. #13133146C22semidirect
48. #13133141C22semidirect
50. #13133139C22semidirect
53. #13133137C22semidirect
54. C2×Q16×C53C22semidirect
57. C4×Q8×C53C22semidirect
58. C4⋊C8×C53C22semidirect
59. #103×C53C22semidirect
71. #11719949D4semidirect
210. #464D53semidirect
220. C53#3045semidirect
226. C2×Q16C2×D53semidirect
245. C4×Q8C2×D53semidirect
257. C4⋊C8C2×D53semidirect
259. #103C2×D53semidirect