Finite Group Databasebeta

← #20284338

Subgroups — #20284338

Order 6784.

496 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133246C211yesmaximal, characteristic, complemented
2C22×#11719912C211yesmaximal, characteristic, complemented
3#16811133C211yesmaximal, characteristic
4C22⋊Q8×D53C211yesmaximal, characteristic, complemented
5#16810882C211yesmaximal, characteristic, complemented
6#16810880C211yesmaximal, characteristic, complemented
7#16810868C211yesmaximal, characteristic, complemented
8#16810866C211yesmaximal, characteristic, complemented
9#16810864C211yesmaximal, characteristic, complemented
10#16810862C211yesmaximal, characteristic, complemented
11#16810850C211yesmaximal, characteristic, complemented
12#16810662C211yesmaximal, characteristic, complemented
13#16810630C211yesmaximal, characteristic, complemented
14#16810621C211yesmaximal, characteristic, complemented
15C53×#465C211yesmaximal, characteristic, complemented
16C2×#11719947C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic
18C22×C4×D53C2211yescharacteristic
19#13133246—24no—
20C2×Dic212C2211yescharacteristic, complemented
21C2×#11719925C2211yescharacteristic, complemented
22C2×#11719923C2211yescharacteristic, complemented
23C2×#11719912—24no—
24C2×#11719912C2211yescharacteristic
25C2×#11719912C2211yescharacteristic
26C2×#11719912C2211yescharacteristic
27C2×#11719912C2211yescharacteristic
28C22×#88322C2211yescharacteristic
29#13133199—21no—
30#13133197C2211yescharacteristic
31#13133197C2211yescharacteristic
32#13133196C2211yescharacteristic
33#13133196C2211yescharacteristic
34#13133195—21no—
35#13133194—21no—
36#13133193C2211yescharacteristic
37#13133188C2211yescharacteristic
38#13133185—21no—
39D4×Dic53—21no—
40#13133177—21no—
41#13133175C2211yescharacteristic
42C4⋊C4×D53—21no—
43C4⋊C4×D53C2211yescharacteristic
44C22⋊C4×D53—21no—
45#13133151C2211yescharacteristic, complemented
46#13133150C2211yescharacteristic, complemented
47#13133148C2211yescharacteristic, complemented
48#13133142C2211yescharacteristic, complemented
49#13133142C2211yescharacteristic, complemented
50#13133141C2211yescharacteristic, complemented
51#13133140C2211yescharacteristic, complemented
52#13133140C2211yescharacteristic, complemented
53#13133138C2211yescharacteristic, complemented
54C2×SD16×C53C2211yescharacteristic, complemented
55C22×C8×C53C2211yescharacteristic
56C22⋊Q8×C53C2211yescharacteristic
57#121×C53C2211yescharacteristic
58#106×C53C2211yescharacteristic, complemented
59#103×C53C2211yescharacteristic, complemented
60#102×C53C2211yescharacteristic, complemented
61C23×D53D411yescharacteristic
62C2×Q8×C53C2311yescharacteristic
63C2×D4×C53—21no—
64C2×D4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C22×Dic53—21no—
69C22×Dic53D411yescharacteristic
70Q8×D53—22no—
71Q8×D53—22no—
72#11719947—22no—
73#11719947—22no—
74C2×C4×D53—22no—
75C2×C4×D53—22no—
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53D411yescharacteristic
79C2×C4×D53D411yescharacteristic
80C2×C4×D53C2311yescharacteristic
81C2×C4×D53C2311yescharacteristic
82C2×Dic106—21no—
83C2×Dic106C2311yescharacteristic
84C2×Dic106C2311yescharacteristic
85SD16×C53—22no—
86C2×C8×C53—22no—
87C2×C8×C53C2311yescharacteristic
88C2×C8×C53C2311yescharacteristic
89C4⋊C4×C53—21no—
90C4⋊C4×C53C2311yescharacteristic
91C4⋊C4×C53C2311yescharacteristic
92C22⋊C4×C53—21no—
93C22⋊C4×C53—21no—
94C22⋊Dic53—21no—
95C22⋊Dic53—21no—
96C22⋊Dic53—21no—
97#11719925—22no—
98#11719923—22no—
99#11719921—21no—
100#11719921—21no—
101#11719921—21no—
102#11719921—21no—
103C4⋊Dic53—21no—
104C4⋊Dic53—21no—
105C4⋊Dic53C2311yescharacteristic
106C4⋊Dic53C2311yescharacteristic
107#11719919—21no—
108#11719919—21no—
109#11719919—21no—
110#11719919—21no—
111C4×Dic53—21no—
112C2×#88322—22no—
113C2×#88322C2311yescharacteristic
114C2×#88322C2311yescharacteristic
115Dic212—22no—
116#11719912—22no—
117#11719912—22no—
118#11719912—22no—
119#11719912D414yescomplemented
120C23×C53—21no—
121C23×C53C2×D411yescharacteristic
122C22×D53—22no—
123C22×D53—22no—
124C22×D53—41no—
125C22×D53C2×D411yescharacteristic
126C22×D53C2×D411yescharacteristic
127Q8×C53—21no—
128Q8×C53—21no—
129D4×C53—21no—
130D4×C53—21no—
131D4×C53—42no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53C2×D411yescharacteristic
138C2×C4×C53C2411yescharacteristic
139C22⋊D53—42no—
140C22⋊D53—42no—
141C2×Dic53—22no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—42no—
147C2×Dic53C2×D411yescharacteristic
148C2×Dic53C2×D411yescharacteristic
149C4×D53—22no—
150C4×D53—21no—
151C4×D53—21no—
152C4×D53—22no—
153C4×D53—42no—
154C4×D53—42no—
155C4×D53C2×D412yes—
156C4×D53#4112yes—
157Dic106—21no—
158Dic106—21no—
159Dic106—21no—
160Dic106—21no—
161Dic106—42no—
162C8×C53—21no—
163C8×C53C2×D412yes—
164#88322—21no—
165#88322C2×D412yes—
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53#13712yescomplemented
172C22×C53C22×D411yescharacteristic
173C2×D53—21no—
174C2×D53—21no—
175C2×D53—21no—
176C2×D53—21no—
177C2×D53—21no—
178C2×D53—44no—
179C2×D53#12112yes—
180C4×C53—21no—
181C4×C53—41no—
182C4×C53—41no—
183C4×C53—41no—
184C4×C53C22×D411yescharacteristic
185C4×C53C2×#4111yescharacteristic
186Dic53—21no—
187Dic53—41no—
188Dic53—41no—
189Dic53—41no—
190Dic53—41no—
191Dic53#12112yes—
192#3056—531nomaximal, Sylow
193C2×C53—22no—
194C2×C53—81no—
195C2×C53C2×#12111yescharacteristic
196C2×C53C2×#13711yescharacteristic
197C2×C53#57511yescharacteristic
198D53—22no—
199D53—41no—
200C2×#137—531no—
201C22×M16—531no—
202#525—531no—
203C2×C22⋊Q8—531no—
204#470—531no—
205#470—531no—
206#468—531no—
207#468—531no—
208#467—531no—
209#467—531no—
210#465—531no—
211#465D5311yescharacteristic, complemented
212#428—531no—
213#419—531no—
214#417—531no—
215C53#305611yescharacteristic, complemented, Sylow
216C2×#41—531no—
217C22×Q8—531no—
218C23×C4—531no—
219#137—1064no—
220C2×Q16—531no—
221C2×Q16—531no—
222C2×SD16—531no—
223C2×SD16C2×D5311yescharacteristic, complemented
224C2×M16—531no—
225C2×M16—531no—
226C2×M16—531no—
227C2×M16—531no—
228C2×M16—1064no—
229C22×C8—531no—
230C22×C8C2×D5311yescharacteristic
231#123—1061no—
232C22⋊Q8—531no—
233C22⋊Q8—531no—
234C22⋊Q8—531no—
235C22⋊Q8—531no—
236C22⋊Q8—531no—
237C22⋊Q8—1061no—
238C22⋊Q8—1061no—
239C22⋊Q8C2×D5311yescharacteristic
240#121—531no—
241#121C2×D5311yescharacteristic
242#120—1061no—
243C4×D4—1061no—
244C4×D4—1061no—
245#117—531no—
246C2×C4⋊C4—531no—
247C2×C4⋊C4—1061no—
248C2×C22⋊C4—1061no—
249#107—531no—
250#107—531no—
251#106—531no—
252#106C2×D5311yescharacteristic, complemented
253#103—531no—
254#103—531no—
255#103—531no—
256#103—531no—
257#103—531no—
258#103C2×D5311yescharacteristic, complemented
259#102—531no—
260#102C2×D5311yescharacteristic, complemented
261C24—531no—
262#41—1062no—
263#41—1062no—
264C2×Q8—531no—
265C2×Q8—531no—
266C2×Q8—1061no—
267C2×Q8—1062no—
268C2×Q8—1062no—
269C2×Q8C22×D5311yescharacteristic
270C2×D4—21no—
271C2×D4—1061no—
272C2×D4—1061no—
273C2×D4C22×D5311yescharacteristic
274C22×C4—531no—
275C22×C4—531no—
276C22×C4—531no—
277C22×C4—531no—
278C22×C4—531no—
279C22×C4—1062no—
280C22×C4—1062no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—1061no—
284C22×C4C22×D5311yescharacteristic
285Q16—1062no—
286Q16—1062no—
287SD16—22no—
288SD16—1062no—
289M16—534no—
290M16—1062no—
291M16—1062no—
292M16—1062no—
293C2×C8—22no—
294C2×C8—531no—
295C2×C8—531no—
296C2×C8—1062no—
297C2×C8C22×D5311yescharacteristic
298C2×C8C22×D5311yescharacteristic
299C4⋊C4—21no—
300C4⋊C4—531no—
301C4⋊C4—531no—
302C4⋊C4—1061no—
303C4⋊C4—1061no—
304C4⋊C4—1061no—
305C4⋊C4—1061no—
306C4⋊C4—1061no—
307C4⋊C4—1061no—
308C4⋊C4C22×D5311yescharacteristic
309C4⋊C4C22×D5311yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—1061no—
313C22⋊C4—1061no—
314C22⋊C4—1061no—
315C22⋊C4—1061no—
316C22⋊C4—1061no—
317C22⋊C4—1061no—
318C22⋊C4—1061no—
319C42—1061no—
320C23—21no—
321C23—531no—
322C23—531no—
323C23—1062no—
324C23—1062no—
325C23—2121no—
326C23D4×D5311yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—1061no—
330Q8—1061no—
331Q8—1061no—
332Q8—1061no—
333Q8—2122no—
334D4—21no—
335D4—21no—
336D4—42no—
337D4—2122no—
338D4—2122no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—531no—
345C2×C4—532no—
346C2×C4—532no—
347C2×C4—531no—
348C2×C4—1062no—
349C2×C4—1062no—
350C2×C4—1062no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—1061no—
356C2×C4—1061no—
357C2×C4—2122no—
358C2×C4—2122no—
359C2×C4—2122no—
360C2×C4D4×D5311yescharacteristic
361C2×C4C23×D5311yescharacteristic
362C8—21no—
363C8—532no—
364C8—1061no—
365C8D4×D5312yes—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—532no—
372C22—1061no—
373C22—1061no—
374C22—1061no—
375C22—1061no—
376C22—1061no—
377C22—2124no—
378C22#1313324612yescomplemented
379C22C2×D4×D5311yescharacteristic
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—532no—
385C4—1061no—
386C4—2121no—
387C4—2121no—
388C4—2121no—
389C4—2121no—
390C4C2×D4×D5311yescharacteristic
391C4#41×D5311yescharacteristic
392C2—22no—
393C2—81no—
394C2—1062no—
395C2—2121no—
396C2#121×D5311yescharacteristic
397C2C2×#1313324611yescharacteristic
398C2#1681142011yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#465C2
socleC22×C53C22×D4
hypercenter#465D53
hypocenterC53#3056
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#13133246C2semidirect
2. C22×#11719912C2semidirect
4. C22⋊Q8×D53C2semidirect
5. #16810882C2semidirect
6. #16810880C2semidirect
7. #16810868C2semidirect
8. #16810866C2semidirect
9. #16810864C2semidirect
10. #16810862C2semidirect
11. #16810850C2semidirect
12. #16810662C2semidirect
13. #16810630C2semidirect
14. #16810621C2semidirect
15. C53×#465C2semidirect
20. C2×Dic212C22semidirect
21. C2×#11719925C22semidirect
22. C2×#11719923C22semidirect
45. #13133151C22semidirect
46. #13133150C22semidirect
47. #13133148C22semidirect
48. #13133142C22semidirect
49. #13133142C22semidirect
50. #13133141C22semidirect
51. #13133140C22semidirect
52. #13133140C22semidirect
53. #13133138C22semidirect
54. C2×SD16×C53C22semidirect
58. #106×C53C22semidirect
59. #103×C53C22semidirect
60. #102×C53C22semidirect
119. #11719912D4semidirect
171. C22×C53#137semidirect
211. #465D53semidirect
215. C53#3056semidirect
223. C2×SD16C2×D53semidirect
252. #106C2×D53semidirect
258. #103C2×D53semidirect
260. #102C2×D53semidirect
378. C22#13133246semidirect