Finite Group Databasebeta

← #20284913

Subgroups — #20284913

Order 6784.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811227C211yesmaximal, characteristic, complemented
2#16811105C211yesmaximal, characteristic, complemented
3#16810940C211yesmaximal, characteristic, complemented
4#16810935C211yesmaximal, characteristic, complemented
5#16810930C211yesmaximal, characteristic, complemented
6#16810924C211yesmaximal, characteristic, complemented
7#16810815C211yesmaximal, characteristic, complemented
8#16810767C211yesmaximal, characteristic, complemented
9#16810767C211yesmaximal, characteristic, complemented
10#16810732C211yesmaximal, characteristic, complemented
11#16810724C211yesmaximal, characteristic, complemented
12SD16×Dic53C211yesmaximal, characteristic, complemented
13#16810703C211yesmaximal, characteristic, complemented
14#16810683C211yesmaximal, characteristic, complemented
15C53×#460C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719922C2211yescharacteristic
18C2×#11719922C2211yescharacteristic
19#13133209C2211yescharacteristic
20#13133198—21no—
21#13133198—21no—
22#13133189—21no—
23#13133189—21no—
24#13133188C2211yescharacteristic
25#13133188C2211yescharacteristic
26Q8×Dic53C2211yescharacteristic
27C4×D212—21no—
28C4×D212C2211yescharacteristic
29D4×Dic53C2211yescharacteristic
30D4×Dic53C2211yescharacteristic
31#13133179—21no—
32#13133179C2211yescharacteristic
33#13133174—21no—
34#13133174C2211yescharacteristic
35C4⋊C4×D53—21no—
36C4⋊C4×D53C2211yescharacteristic
37#13133151C2211yescharacteristic
38#13133150C2211yescharacteristic
39#13133149C2211yescharacteristic
40#13133148C2211yescharacteristic
41C4⋊#88322C2211yescharacteristic
42#13133141C2211yescharacteristic
43#13133141C2211yescharacteristic
44#13133139C2211yescharacteristic
45#13133139C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133129C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133127C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C2×SD16×C53C2211yescharacteristic
53C2×SD16×C53C2211yescharacteristic
54#127×C53C2211yescharacteristic
55C4×Q8×C53C2211yescharacteristic
56C4⋊C8×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C2×D4×C53—21no—
61C2×D4×C53—21no—
62C2×D4×C53C2311yescharacteristic
63C2×D4×C53C2311yescharacteristic
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C22×Dic53—21no—
67C22×Dic53—21no—
68#11719949—22no—
69#11719949—22no—
70C2×D212—21no—
71C2×D212C2311yescharacteristic
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×C4×D53—21no—
75C2×C4×D53—21no—
76C2×C4×D53C2311yescharacteristic
77SD16×C53—22no—
78SD16×C53—22no—
79C2×C8×C53C2311yescharacteristic
80C2×C8×C53C2311yescharacteristic
81C4⋊C4×C53—21no—
82C4⋊C4×C53C2311yescharacteristic
83C42×C53—21no—
84C42×C53C2311yescharacteristic
85C22⋊Dic53—21no—
86C22⋊Dic53—21no—
87C22⋊Dic53—21no—
88C22⋊Dic53—21no—
89#11719922—22no—
90#11719922—22no—
91#11719921—21no—
92#11719921—21no—
93#11719921D411yescharacteristic
94#11719921D411yescharacteristic
95C4⋊Dic53—21no—
96C4⋊Dic53C2311yescharacteristic
97C4⋊Dic53C2311yescharacteristic
98C4⋊Dic53C2311yescharacteristic
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102#11719919D411yescharacteristic
103#11719919D411yescharacteristic
104C4×Dic53—21no—
105C4×Dic53C2311yescharacteristic
106C2×#88322C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108C23×C53—21no—
109C23×C53—21no—
110C22×D53—21no—
111C22×D53C2×D411yescharacteristic
112Q8×C53—21no—
113Q8×C53#4112yes—
114D4×C53—21no—
115D4×C53—21no—
116D4×C53—21no—
117D4×C53—21no—
118D4×C53—81no—
119D4×C53—81no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53C2×D411yescharacteristic
123C2×C4×C53C2×D411yescharacteristic
124C2×C4×C53C2411yescharacteristic
125C22⋊D53—81no—
126C22⋊D53—81no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—81no—
131C2×Dic53—81no—
132C2×Dic53C2×D411yescharacteristic
133C2×Dic53C2×D411yescharacteristic
134C2×Dic53C2×D411yescharacteristic
135D212—24no—
136D212—21no—
137D212#4112yes—
138C4×D53—24no—
139C4×D53—21no—
140C4×D53—21no—
141C4×D53—81no—
142C4×D53—81no—
143C4×D53—81no—
144C8×C53—21no—
145C8×C53—21no—
146#88322—21no—
147#88322—21no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53C22×D411yescharacteristic
155C2×D53—21no—
156C2×D53—22no—
157C2×D53—21no—
158C2×D53—21no—
159C2×D53—41no—
160C2×D53—41no—
161C4×C53—22no—
162C4×C53—21no—
163C4×C53—21no—
164C4×C53—41no—
165C4×C53C2×#4111yescharacteristic
166C4×C53C2×#4111yescharacteristic
167Dic53—21no—
168Dic53—21no—
169Dic53—21no—
170Dic53—41no—
171Dic53—41no—
172Dic53—41no—
173#3119—531nomaximal, Sylow
174C2×C53—81no—
175C2×C53—81no—
176C2×C53#52511yescharacteristic
177C2×C53#57611yescharacteristic
178C2×C53#57711yescharacteristic
179D53—41no—
180D53—42no—
181#540—531no—
182#520—531no—
183#481—531no—
184#480—531no—
185#479—531no—
186#478—531no—
187#460D5311yescharacteristic, complemented
188#459—531no—
189#449—531no—
190#449—531no—
191#442—531no—
192#440—531no—
193C4×SD16—531no—
194C4×D8—531no—
195#433—531no—
196C53#311911yescharacteristic, complemented, Sylow
197C2×#41—531no—
198C2×SD16C2×D5311yescharacteristic
199C2×SD16C2×D5311yescharacteristic
200C2×D8—531no—
201C2×D8—531no—
202#127C2×D5311yescharacteristic
203#125—531no—
204#123—1061no—
205#123—1061no—
206#121—531no—
207#121—531no—
208#121—1061no—
209#121—1061no—
210C4×Q8—531no—
211C4×Q8C2×D5311yescharacteristic
212C4×D4—531no—
213C4×D4—531no—
214C4×D4—531no—
215C4×D4—531no—
216C4×D4—1061no—
217C4×D4—1061no—
218#117—531no—
219#117—1061no—
220C2×C4⋊C4—531no—
221C2×C4⋊C4—1061no—
222#107—531no—
223#107—531no—
224#106—531no—
225#106—531no—
226C4⋊C8—531no—
227C4⋊C8C2×D5311yescharacteristic
228#103—531no—
229#103—531no—
230#102—531no—
231#102—531no—
232#102—531no—
233#102—531no—
234#102C2×D5311yescharacteristic
235#102C2×D5311yescharacteristic
236C22⋊C8—531no—
237C22⋊C8—531no—
238#97—531no—
239C4×C8—531no—
240#41—1062no—
241#41—1062no—
242C2×Q8C22×D5311yescharacteristic
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—531no—
246C2×D4—1061no—
247C2×D4—1061no—
248C2×D4—1061no—
249C2×D4C22×D5311yescharacteristic
250C2×D4C22×D5311yescharacteristic
251C22×C4—531no—
252C22×C4—1061no—
253C22×C4—1061no—
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4—1061no—
258SD16—22no—
259SD16—22no—
260D8—1062no—
261D8—1062no—
262C2×C8—531no—
263C2×C8—531no—
264C2×C8C22×D5311yescharacteristic
265C2×C8C22×D5311yescharacteristic
266C4⋊C4—21no—
267C4⋊C4—531no—
268C4⋊C4—531no—
269C4⋊C4—531no—
270C4⋊C4—531no—
271C4⋊C4—531no—
272C4⋊C4—1061no—
273C4⋊C4—1061no—
274C4⋊C4—1061no—
275C4⋊C4—1061no—
276C4⋊C4C22×D5311yescharacteristic
277C22⋊C4—531no—
278C22⋊C4—531no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C42—21no—
286C42—531no—
287C42—1061no—
288C42C22×D5311yescharacteristic
289C23—21no—
290C23—21no—
291C23—531no—
292C23—1061no—
293Q8—21no—
294Q8#1171994712yes—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—81no—
300D4—81no—
301D4—532no—
302D4—1064no—
303D4—1061no—
304D4—4241no—
305D4—4241no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—531no—
309C2×C4—531no—
310C2×C4—531no—
311C2×C4—1064no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—4241no—
318C2×C4—4241no—
319C2×C4—4241no—
320C2×C4—4241no—
321C2×C4—4241no—
322C2×C4D4×D5311yescharacteristic
323C2×C4D4×D5311yescharacteristic
324C2×C4C23×D5311yescharacteristic
325C8—21no—
326C8—21no—
327C8—1061no—
328C8—1061no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—1062no—
336C22—1061no—
337C22—1061no—
338C22—1061no—
339C22—2121no—
340C22—2121no—
341C22C2×D4×D5311yescharacteristic
342C4—22no—
343C4—21no—
344C4—21no—
345C4—41no—
346C4—1061no—
347C4—1061no—
348C4—1061no—
349C4—2121no—
350C4—2121no—
351C4—2121no—
352C4#41×D5311yescharacteristic
353C4C2×#1171994711yescharacteristic
354C2—81no—
355C2—81no—
356C2—2121no—
357C2—2122no—
358C2#1681113311yescharacteristic
359C2#1681142711yescharacteristic
360C2#1681142911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#460C2
socleC22×C53C22×D4
hypercenter#460D53
hypocenterC53#3119
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. #16811227C2semidirect
2. #16811105C2semidirect
3. #16810940C2semidirect
4. #16810935C2semidirect
5. #16810930C2semidirect
6. #16810924C2semidirect
7. #16810815C2semidirect
8. #16810767C2semidirect
9. #16810767C2semidirect
10. #16810732C2semidirect
11. #16810724C2semidirect
12. SD16×Dic53C2semidirect
13. #16810703C2semidirect
14. #16810683C2semidirect
15. C53×#460C2semidirect
187. #460D53semidirect
196. C53#3119semidirect