Finite Group Databasebeta

← #20284231

Subgroups — #20284231

Order 6784.

490 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133246C211yesmaximal, characteristic, complemented
2C2×SD16×D53C211yesmaximal, characteristic, complemented
3#16811186C211yesmaximal, characteristic
4#16811099C211yesmaximal, characteristic, complemented
5#16810846C211yesmaximal, characteristic, complemented
6#16810845C211yesmaximal, characteristic, complemented
7#16810844C211yesmaximal, characteristic, complemented
8#16810843C211yesmaximal, characteristic, complemented
9#16810822C211yesmaximal, characteristic, complemented
10#16810821C211yesmaximal, characteristic, complemented
11#16810820C211yesmaximal, characteristic, complemented
12#16810648C211yesmaximal, characteristic, complemented
13#16810626C211yesmaximal, characteristic, complemented
14#16810616C211yesmaximal, characteristic, complemented
15C53×#460C211yesmaximal, characteristic, complemented
16C2×#11719947C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic
18C2×D4×D53C2211yescharacteristic
19#13133246—24no—
20C2×Dic212C2211yescharacteristic, complemented
21C2×#11719925C2211yescharacteristic, complemented
22C2×#11719913C2211yescharacteristic
23C2×#11719923C2211yescharacteristic
24C2×#11719923C2211yescharacteristic, complemented
25C2×#11719924C2211yescharacteristic
26SD16×D53—24no—
27C2×#11719912C2211yescharacteristic
28C2×C8×D53C2211yescharacteristic
29#13133213C2211yescharacteristic
30#13133205C2211yescharacteristic
31#13133204C2211yescharacteristic
32#13133189—21no—
33#13133189—21no—
34#13133185—21no—
35#13133185—21no—
36Q8×Dic53C2211yescharacteristic, complemented
37C4×Dic106—21no—
38C4×Dic106C2211yescharacteristic, complemented
39#13133182—21no—
40#13133182C2211yescharacteristic, complemented
41#13133175—21no—
42#13133175C2211yescharacteristic
43#13133174—21no—
44#13133174C2211yescharacteristic
45C4⋊#88322C2211yescharacteristic, complemented
46#13133146C2211yescharacteristic, complemented
47#13133146C2211yescharacteristic, complemented
48#13133142C2211yescharacteristic, complemented
49#13133140C2211yescharacteristic, complemented
50#13133139C2211yescharacteristic
51#13133138C2211yescharacteristic
52#13133138C2211yescharacteristic, complemented
53#13133137C2211yescharacteristic
54C2×SD16×C53C2211yescharacteristic
55C2×SD16×C53C2211yescharacteristic, complemented
56#127×C53C2211yescharacteristic
57C4×Q8×C53C2211yescharacteristic, complemented
58C4⋊C8×C53C2211yescharacteristic, complemented
59#102×C53C2211yescharacteristic
60#102×C53C2211yescharacteristic, complemented
61C23×D53—21no—
62C2×Q8×C53C2311yescharacteristic
63C2×D4×C53—21no—
64C2×D4×C53—21no—
65C2×D4×C53C2311yescharacteristic
66C2×D4×C53C2311yescharacteristic
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C22×Dic53—21no—
72Q8×D53—22no—
73Q8×D53D414yescomplemented
74#11719947—22no—
75#11719947—22no—
76D4×D53—22no—
77D4×D53—22no—
78C2×D212C2311yescharacteristic
79C2×C4×D53—21no—
80C2×C4×D53C2311yescharacteristic
81C2×Dic106—21no—
82C2×Dic106C2311yescharacteristic
83C2×Dic106C2311yescharacteristic
84SD16×C53—22no—
85SD16×C53—22no—
86C2×C8×C53C2311yescharacteristic
87C2×C8×C53C2311yescharacteristic
88C4⋊C4×C53—21no—
89C4⋊C4×C53C2311yescharacteristic
90C42×C53—21no—
91C42×C53C2311yescharacteristic
92C22⋊Dic53—21no—
93C22⋊Dic53—21no—
94#11719925—22no—
95#11719924—22no—
96#11719923—22no—
97#11719923—22no—
98#11719921—21no—
99#11719921—21no—
100#11719921—21no—
101#11719921D411yescharacteristic
102#11719921D411yescharacteristic
103C4⋊Dic53—21no—
104C4⋊Dic53C2311yescharacteristic
105#11719919—21no—
106#11719919—21no—
107#11719919D411yescharacteristic
108#11719919D411yescharacteristic
109C4×Dic53—21no—
110C4×Dic53—21no—
111C4×Dic53—21no—
112C4×Dic53—21no—
113C4×Dic53C2311yescharacteristic
114C2×#88322C2311yescharacteristic
115C2×#88322C2311yescharacteristic
116Dic212—22no—
117#11719913—22no—
118#11719912—22no—
119C8×D53—22no—
120C23×C53—21no—
121C23×C53—21no—
122C22×D53—21no—
123C22×D53—42no—
124C22×D53—81no—
125C22×D53—81no—
126C22×D53C2×D411yescharacteristic
127Q8×C53—21no—
128Q8×C53C2×D412yes—
129D4×C53—21no—
130D4×C53—21no—
131D4×C53—21no—
132D4×C53—21no—
133D4×C53—81no—
134D4×C53—81no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53C2×D411yescharacteristic
138C2×C4×C53C2×D411yescharacteristic
139C2×C4×C53C2411yescharacteristic
140C22⋊D53—42no—
141C22⋊D53—42no—
142C22⋊D53—81no—
143C22⋊D53—81no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—42no—
148C2×Dic53C2×D411yescharacteristic
149C2×Dic53C2×D411yescharacteristic
150C2×Dic53C2×D411yescharacteristic
151D212—21no—
152D212—21no—
153C4×D53—24no—
154C4×D53C2×D412yes—
155C4×D53#4112yes—
156Dic106—21no—
157Dic106—24no—
158Dic106—21no—
159Dic106—21no—
160Dic106C2×D412yes—
161C8×C53—21no—
162C8×C53—21no—
163#88322—21no—
164#88322—21no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53C22×D411yescharacteristic
172C2×D53—21no—
173C2×D53—21no—
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—41no—
177C2×D53—84no—
178C2×D53#12112yes—
179C4×C53—22no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—41no—
183C4×C53C22×D411yescharacteristic
184C4×C53C2×#4111yescharacteristic
185Dic53—22no—
186Dic53—21no—
187Dic53—21no—
188Dic53—41no—
189Dic53—41no—
190Dic53#12112yes—
191#3043—531nomaximal, Sylow
192C2×C53—81no—
193C2×C53—81no—
194C2×C53C2×#12111yescharacteristic
195C2×C53#57711yescharacteristic
196C2×C53#57711yescharacteristic
197D53—22no—
198D53—81no—
199C2×#137—531no—
200C22×SD16—531no—
201#535—531no—
202#519—531no—
203#464—531no—
204#464—531no—
205#464—531no—
206#464—531no—
207#460—531no—
208#460—531no—
209#460—531no—
210#460D5311yescharacteristic, complemented
211#424—531no—
212#418—531no—
213#416—531no—
214C53#304311yescharacteristic, complemented, Sylow
215C2×#41—531no—
216C22×Q8—531no—
217C22×D4—531no—
218#137—1064no—
219C2×Q16—531no—
220C2×Q16—531no—
221C2×SD16—531no—
222C2×SD16—531no—
223C2×SD16—531no—
224C2×SD16—531no—
225C2×SD16—1064no—
226C2×SD16C2×D5311yescharacteristic
227C2×SD16C2×D5311yescharacteristic, complemented
228C2×M16—531no—
229C22×C8—531no—
230#127—531no—
231#127C2×D5311yescharacteristic
232#124—531no—
233#124—531no—
234#121—1061no—
235#121—1061no—
236#120—1061no—
237#120—1061no—
238C4×Q8—531no—
239C4×Q8—531no—
240C4×Q8—531no—
241C4×Q8—1061no—
242C4×Q8—1061no—
243C4×Q8C2×D5311yescharacteristic, complemented
244#117—531no—
245#117—531no—
246#117—1061no—
247#117—1061no—
248C4⋊C8—531no—
249C4⋊C8—531no—
250C4⋊C8—531no—
251C4⋊C8C2×D5311yescharacteristic, complemented
252#103—531no—
253#103—531no—
254#102—531no—
255#102—531no—
256#102—531no—
257#102—531no—
258#102C2×D5311yescharacteristic
259#102C2×D5311yescharacteristic, complemented
260C24—1061no—
261#41—1062no—
262#41—1062no—
263C2×Q8—534no—
264C2×Q8—531no—
265C2×Q8—531no—
266C2×Q8—1061no—
267C2×Q8—1062no—
268C2×Q8C22×D5311yescharacteristic
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—531no—
272C2×D4—1062no—
273C2×D4—1062no—
274C2×D4—1061no—
275C2×D4—1061no—
276C2×D4—1061no—
277C2×D4—1061no—
278C2×D4C22×D5311yescharacteristic
279C2×D4C22×D5311yescharacteristic
280C22×C4—531no—
281C22×C4—1061no—
282C22×C4—1061no—
283Q16—1062no—
284Q16—1062no—
285SD16—22no—
286SD16—22no—
287SD16—1062no—
288SD16—1062no—
289SD16—1062no—
290SD16—1062no—
291M16—1062no—
292C2×C8—531no—
293C2×C8—531no—
294C2×C8—1062no—
295C2×C8C22×D5311yescharacteristic
296C2×C8C22×D5311yescharacteristic
297C4⋊C4—21no—
298C4⋊C4—531no—
299C4⋊C4—531no—
300C4⋊C4—531no—
301C4⋊C4—1061no—
302C4⋊C4—1061no—
303C4⋊C4—1061no—
304C4⋊C4C22×D5311yescharacteristic
305C22⋊C4—531no—
306C22⋊C4—531no—
307C22⋊C4—1061no—
308C22⋊C4—1061no—
309C22⋊C4—1061no—
310C22⋊C4—1061no—
311C22⋊C4—1061no—
312C42—21no—
313C42—531no—
314C42—1061no—
315C42—1061no—
316C42—1061no—
317C42—1061no—
318C42C22×D5311yescharacteristic
319C23—21no—
320C23—21no—
321C23—531no—
322C23—1061no—
323C23—4241no—
324C23—4241no—
325C23—2122no—
326Q8—21no—
327Q8—532no—
328Q8—1064no—
329Q8—1061no—
330Q8—1061no—
331Q8—1061no—
332Q8D4×D5312yes—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—21no—
337D4—81no—
338D4—81no—
339D4—1061no—
340D4—1061no—
341D4—4241no—
342D4—4241no—
343D4—2122no—
344D4—2122no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—531no—
348C2×C4—532no—
349C2×C4—532no—
350C2×C4—531no—
351C2×C4—531no—
352C2×C4—1064no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—1061no—
356C2×C4—2122no—
357C2×C4D4×D5311yescharacteristic
358C2×C4D4×D5311yescharacteristic
359C2×C4C23×D5311yescharacteristic
360C8—21no—
361C8—21no—
362C8—1061no—
363C8—1061no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—532no—
371C22—1061no—
372C22—1061no—
373C22—4244no—
374C22—2121no—
375C22—2121no—
376C22—2121no—
377C22C2×D4×D5311yescharacteristic
378C4—22no—
379C4—21no—
380C4—21no—
381C4—41no—
382C4—532no—
383C4—1062no—
384C4—1061no—
385C4—1061no—
386C4—2121no—
387C4—2121no—
388C4C2×D4×D5311yescharacteristic
389C4#41×D5311yescharacteristic
390C2—81no—
391C2—81no—
392C2—1062no—
393C2—4241no—
394C2#121×D5311yescharacteristic
395C2#1681142911yescharacteristic
396C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#460C2
socleC22×C53C22×D4
hypercenter#460D53
hypocenterC53#3043
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#13133246C2semidirect
2. C2×SD16×D53C2semidirect
4. #16811099C2semidirect
5. #16810846C2semidirect
6. #16810845C2semidirect
7. #16810844C2semidirect
8. #16810843C2semidirect
9. #16810822C2semidirect
10. #16810821C2semidirect
11. #16810820C2semidirect
12. #16810648C2semidirect
13. #16810626C2semidirect
14. #16810616C2semidirect
15. C53×#460C2semidirect
20. C2×Dic212C22semidirect
21. C2×#11719925C22semidirect
24. C2×#11719923C22semidirect
36. Q8×Dic53C22semidirect
38. C4×Dic106C22semidirect
40. #13133182C22semidirect
45. C4⋊#88322C22semidirect
46. #13133146C22semidirect
47. #13133146C22semidirect
48. #13133142C22semidirect
49. #13133140C22semidirect
52. #13133138C22semidirect
55. C2×SD16×C53C22semidirect
57. C4×Q8×C53C22semidirect
58. C4⋊C8×C53C22semidirect
60. #102×C53C22semidirect
73. Q8×D53D4semidirect
210. #460D53semidirect
214. C53#3043semidirect
227. C2×SD16C2×D53semidirect
243. C4×Q8C2×D53semidirect
251. C4⋊C8C2×D53semidirect
259. #102C2×D53semidirect