Finite Group Databasebeta

← #20285190

Subgroups — #20285190

Order 6784.

564 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133249C211yesmaximal, characteristic, complemented
2C2×#13133244C211yesmaximal, characteristic, complemented
3C2×#13133240C211yesmaximal, characteristic, complemented
4C2×SD16×D53C211yesmaximal, characteristic, complemented
5#16811198C211yesmaximal, characteristic, complemented
6#16811191C211yesmaximal, characteristic, complemented
7#16811016C211yesmaximal, characteristic, complemented
8#16811012C211yesmaximal, characteristic, complemented
9#16811009C211yesmaximal, characteristic, complemented
10#16811008C211yesmaximal, characteristic, complemented
11#16811005C211yesmaximal, characteristic, complemented
12#16810997C211yesmaximal, characteristic, complemented
13#16810994C211yesmaximal, characteristic, complemented
14#16810580C211yesmaximal, characteristic, complemented
15C53×#497C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719949C2211yescharacteristic
18C2×Q8×D53C2211yescharacteristic
19C2×D4×D53C2211yescharacteristic
20#13133249—24no—
21#13133244—24no—
22#13133240—24no—
23C2×#11719925C2211yescharacteristic, complemented
24C2×#11719913C2211yescharacteristic
25C2×#11719913C2211yescharacteristic, complemented
26C2×#11719923C2211yescharacteristic
27C2×#11719924C2211yescharacteristic
28C2×#11719924C2211yescharacteristic
29C2×#11719924C2211yescharacteristic, complemented
30C2×#11719924C2211yescharacteristic, complemented
31C2×#11719924C2211yescharacteristic, complemented
32SD16×D53—24no—
33C2×D424C2211yescharacteristic
34C2×D424C2211yescharacteristic, complemented
35C2×#11719922C2211yescharacteristic
36C2×#11719912C2211yescharacteristic
37C2×C8×D53C2211yescharacteristic
38#13133215C2211yescharacteristic, complemented
39#13133214C2211yescharacteristic
40#13133213C2211yescharacteristic
41#13133206C2211yescharacteristic
42#13133206C2211yescharacteristic, complemented
43#13133205C2211yescharacteristic, complemented
44#13133202—22no—
45#13133195—22no—
46#13133190—22no—
47#13133187—22no—
48#13133174C2211yescharacteristic
49#13133127C2211yescharacteristic, complemented
50C4×#88322C2211yescharacteristic, complemented
51C8×Dic53C2211yescharacteristic, complemented
52C2×Q16×C53C2211yescharacteristic, complemented
53C2×Q16×C53C2211yescharacteristic, complemented
54C2×SD16×C53C2211yescharacteristic
55C2×SD16×C53C2211yescharacteristic
56C4⋊Q8×C53C2211yescharacteristic, complemented
57#124×C53C2211yescharacteristic
58#97×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C2×Q8×C53C2311yescharacteristic
61C2×Q8×C53C2311yescharacteristic
62C2×Q8×C53C2311yescharacteristic
63C2×D4×C53C2311yescharacteristic
64C2×C22⋊D53—21no—
65C2×C22⋊D53—22no—
66#11719949—22no—
67#11719949—22no—
68#11719949—22no—
69#11719949—22no—
70Q8×D53—22no—
71Q8×D53—22no—
72D4×D53—22no—
73D4×D53—22no—
74C2×D212—21no—
75C2×D212—21no—
76C2×D212—22no—
77C2×D212C2311yescharacteristic
78C2×D212C2311yescharacteristic
79C2×D212C2311yescharacteristic
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×C4×D53—21no—
83C2×C4×D53C2311yescharacteristic
84C2×Dic106—21no—
85C2×Dic106C2311yescharacteristic
86Q16×C53—22no—
87Q16×C53—22no—
88SD16×C53—22no—
89SD16×C53—22no—
90C2×C8×C53C2311yescharacteristic
91C2×C8×C53C2311yescharacteristic
92C4⋊C4×C53—22no—
93C22⋊C4×C53—22no—
94C42×C53C2311yescharacteristic
95#11719925—22no—
96#11719924—22no—
97#11719924—22no—
98#11719924—22no—
99#11719924—22no—
100#11719924—22no—
101#11719923—22no—
102#11719922—22no—
103#11719921—22no—
104#11719921—22no—
105#11719921—22no—
106#11719921D412yes—
107#11719919—22no—
108#11719919D412yes—
109C4×Dic53C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112D424—22no—
113D424—22no—
114#11719913—22no—
115#11719913—22no—
116#11719912D414yescomplemented
117C8×D53D414yescomplemented
118C23×C53—21no—
119C22×D53—21no—
120C22×D53—21no—
121C22×D53—21no—
122C22×D53—42no—
123C22×D53—82no—
124C22×D53C2×D411yescharacteristic
125Q8×C53—21no—
126Q8×C53—21no—
127Q8×C53—21no—
128Q8×C53—21no—
129Q8×C53—21no—
130Q8×C53—21no—
131D4×C53—21no—
132D4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53C2×D412yes—
137C2×C4×C53C2411yescharacteristic
138C22⋊D53—42no—
139C22⋊D53—82no—
140C2×Dic53—21no—
141C2×Dic53C2×D412yes—
142C2×Dic53C2×D411yescharacteristic
143D212—21no—
144D212—21no—
145D212—21no—
146D212—21no—
147D212—21no—
148D212—21no—
149D212—42no—
150D212—42no—
151D212—82no—
152C4×D53—42no—
153C4×D53—42no—
154C4×D53—42no—
155C4×D53C2×D412yes—
156C4×D53C2×D412yes—
157Dic106—21no—
158Dic106—21no—
159Dic106—42no—
160C8×C53C2×D412yes—
161C8×C53C2×D412yes—
162#88322C2×D412yes—
163#88322C2×D412yes—
164C22×C53—41no—
165C22×C53—42no—
166C22×C53C22×D411yescharacteristic
167C2×D53—22no—
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—41no—
171C2×D53—42no—
172C2×D53—42no—
173C2×D53—42no—
174C2×D53—84no—
175C2×D53#12712yes—
176C4×C53—22no—
177C4×C53—41no—
178C4×C53—41no—
179C4×C53—41no—
180C4×C53C22×D411yescharacteristic
181C4×C53C22×D411yescharacteristic
182Dic53—22no—
183Dic53—41no—
184Dic53#12712yes—
185#3154—531nomaximal, Sylow
186C2×C53—81no—
187C2×C53C2×#12711yescharacteristic
188C2×C53#57712yes—
189D53—22no—
190D53—81no—
191D53—81no—
192D53—81no—
193C2×#137—531no—
194C2×AGL(1,ℤ/8)—531no—
195C2×#135—531no—
196C22×SD16—531no—
197#536—531no—
198#535—531no—
199#497—531no—
200#497D5311yescharacteristic, complemented
201#496—531no—
202#496—531no—
203#495—531no—
204#495—531no—
205#492—531no—
206#492—531no—
207#405—531no—
208C53#315411yescharacteristic, complemented, Sylow
209C2×#41—531no—
210C2×#41—531no—
211C22×Q8—531no—
212C22×D4—531no—
213#137—1064no—
214AGL(1,ℤ/8)—1064no—
215#135—1064no—
216C2×Q16—531no—
217C2×Q16C2×D5311yescharacteristic, complemented
218C2×Q16C2×D5311yescharacteristic, complemented
219C2×SD16—531no—
220C2×SD16—531no—
221C2×SD16—531no—
222C2×SD16—531no—
223C2×SD16—531no—
224C2×SD16—531no—
225C2×SD16—531no—
226C2×SD16—531no—
227C2×SD16—1064no—
228C2×SD16C2×D5311yescharacteristic
229C2×SD16C2×D5311yescharacteristic
230C2×D8—531no—
231C2×D8—531no—
232C2×D8—531no—
233C2×M16—531no—
234C22×C8—531no—
235C4⋊Q8—531no—
236C4⋊Q8C2×D5311yescharacteristic, complemented
237#127—531no—
238#127—531no—
239#124—531no—
240#124—531no—
241#124—531no—
242#124C2×D5311yescharacteristic
243#123—1062no—
244C22⋊Q8—1062no—
245#121—1062no—
246#120—1062no—
247#117—531no—
248#97—531no—
249#97C2×D5311yescharacteristic, complemented
250C4×C8—531no—
251C4×C8—531no—
252C24—1061no—
253#41—1062no—
254#41—1062no—
255#41—1062no—
256#41—1062no—
257C2×Q8—531no—
258C2×Q8—1061no—
259C2×Q8—1062no—
260C2×Q8—1062no—
261C2×Q8C22×D5311yescharacteristic
262C2×Q8C22×D5311yescharacteristic
263C2×Q8C22×D5311yescharacteristic
264C2×D4—531no—
265C2×D4—531no—
266C2×D4—531no—
267C2×D4—1061no—
268C2×D4—1061no—
269C2×D4—1062no—
270C2×D4—1062no—
271C2×D4—1062no—
272C2×D4—1061no—
273C2×D4—1062no—
274C2×D4C22×D5311yescharacteristic
275C22×C4—531no—
276C22×C4—1061no—
277C22×C4—1061no—
278C22×C4—1061no—
279Q16—22no—
280Q16—22no—
281Q16—1062no—
282SD16—22no—
283SD16—22no—
284SD16—1062no—
285SD16—1062no—
286SD16—1062no—
287SD16—1062no—
288SD16—1062no—
289SD16—1062no—
290SD16—1062no—
291SD16—1062no—
292D8—1062no—
293D8—1062no—
294D8—1062no—
295M16—534no—
296C2×C8—531no—
297C2×C8—531no—
298C2×C8—534no—
299C2×C8C22×D5311yescharacteristic
300C2×C8C22×D5311yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—532no—
303C4⋊C4—1062no—
304C22⋊C4—22no—
305C22⋊C4—532no—
306C22⋊C4—1062no—
307C22⋊C4—1062no—
308C22⋊C4—1062no—
309C42—531no—
310C42C22×D5311yescharacteristic
311C23—21no—
312C23—531no—
313C23—1061no—
314C23—1061no—
315C23—1061no—
316C23—4242no—
317C23—2122no—
318Q8—21no—
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—1061no—
325Q8—1061no—
326Q8—2122no—
327D4—21no—
328D4—21no—
329D4—1061no—
330D4—1061no—
331D4—1061no—
332D4—1061no—
333D4—1061no—
334D4—1061no—
335D4—4242no—
336D4—4242no—
337D4—2122no—
338D4—2122no—
339D4—2122no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—531no—
344C2×C4—532no—
345C2×C4—532no—
346C2×C4—532no—
347C2×C4—1061no—
348C2×C4—2122no—
349C2×C4—2122no—
350C2×C4—2122no—
351C2×C4D4×D5312yes—
352C2×C4C23×D5311yescharacteristic
353C8—532no—
354C8—532no—
355C8D4×D5312yes—
356C8D4×D5312yes—
357C22—41no—
358C22—42no—
359C22—532no—
360C22—1062no—
361C22—4244no—
362C22—2121no—
363C22—2121no—
364C22—2121no—
365C22—2122no—
366C22—2122no—
367C22—2122no—
368C22C2×D4×D5311yescharacteristic
369C4—22no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—532no—
374C4—1062no—
375C4—2121no—
376C4C2×D4×D5311yescharacteristic
377C4C2×D4×D5311yescharacteristic
378C2—81no—
379C2—1062no—
380C2—4241no—
381C2—4241no—
382C2—4241no—
383C2#127×D5311yescharacteristic
384C2#1681143212yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#497C2
socleC22×C53C22×D4
hypercenter#497D53
hypocenterC53#3154
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#13133249C2semidirect
2. C2×#13133244C2semidirect
3. C2×#13133240C2semidirect
4. C2×SD16×D53C2semidirect
5. #16811198C2semidirect
6. #16811191C2semidirect
7. #16811016C2semidirect
8. #16811012C2semidirect
9. #16811009C2semidirect
10. #16811008C2semidirect
11. #16811005C2semidirect
12. #16810997C2semidirect
13. #16810994C2semidirect
14. #16810580C2semidirect
15. C53×#497C2semidirect
23. C2×#11719925C22semidirect
25. C2×#11719913C22semidirect
29. C2×#11719924C22semidirect
30. C2×#11719924C22semidirect
31. C2×#11719924C22semidirect
34. C2×D424C22semidirect
38. #13133215C22semidirect
42. #13133206C22semidirect
43. #13133205C22semidirect
49. #13133127C22semidirect
50. C4×#88322C22semidirect
51. C8×Dic53C22semidirect
52. C2×Q16×C53C22semidirect
53. C2×Q16×C53C22semidirect
56. C4⋊Q8×C53C22semidirect
58. #97×C53C22semidirect
116. #11719912D4semidirect
117. C8×D53D4semidirect
200. #497D53semidirect
208. C53#3154semidirect
217. C2×Q16C2×D53semidirect
218. C2×Q16C2×D53semidirect
236. C4⋊Q8C2×D53semidirect
249. #97C2×D53semidirect