Finite Group Databasebeta

← #20287512

Subgroups — #20287512

Order 6784.

384 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811359C211yesmaximal, characteristic, complemented
2#16811288C211yesmaximal, characteristic, complemented
3#16810981C212yesmaximal, complemented
4#16810911C212yesmaximal, complemented
5#16810905C211yesmaximal, characteristic, complemented
6#16810837C212yesmaximal, complemented
7#16810815C211yesmaximal, characteristic
8#16810753C211yesmaximal, characteristic, complemented
9#16810732C212yesmaximal, complemented
10C4×#11719913C211yesmaximal, characteristic, complemented
11C53×#486C211yesmaximal, characteristic, complemented
12C2×Q8×D53—21no—
13C2×#11719913C2211yescharacteristic
14C2×#11719922C2212yes—
15#13133216C2212yes—
16#13133204—21no—
17#13133204C2212yes—
18#13133198—22no—
19#13133195—22no—
20#13133193—21no—
21C4×Dic106C2211yescharacteristic
22C4×Dic106C2211yescharacteristic
23C4×D212C2211yescharacteristic
24D4×Dic53—21no—
25D4×Dic53C2212yes—
26#13133179C2212yes—
27#13133149C2211yescharacteristic
28#13133148C2212yes—
29C4⋊#88322C2211yescharacteristic
30#13133146C2212yes—
31#13133142C2211yescharacteristic
32#13133140C2212yes—
33#13133139C2211yescharacteristic
34#13133138C2212yes—
35#13133127C2212yes—
36C4⋊Q8×C53C2211yescharacteristic
37#127×C53C2211yescharacteristic
38#102×C53C2212yes—
39#102×C53C2212yes—
40C4×C8×C53C2211yescharacteristic
41C2×Q8×C53—21no—
42C2×D4×C53—21no—
43C2×D4×C53—21no—
44C2×D4×C53C2312yes—
45C22×Dic53—22no—
46Q8×D53—44no—
47C2×D212C2311yescharacteristic
48C2×C4×D53—21no—
49C2×C4×D53—22no—
50C2×Dic106—22no—
51C2×Dic106D411yescharacteristic
52C2×Dic106C2311yescharacteristic
53C2×C8×C53C2311yescharacteristic
54C2×C8×C53C2311yescharacteristic
55C4⋊C4×C53—21no—
56C4⋊C4×C53C2312yes—
57C42×C53C2311yescharacteristic
58C22⋊Dic53—22no—
59C22⋊Dic53—22no—
60C22⋊Dic53—22no—
61#11719922—24no—
62#11719921—21no—
63#11719921—22no—
64C4⋊Dic53D411yescharacteristic
65C4⋊Dic53C2311yescharacteristic
66C4⋊Dic53C2311yescharacteristic
67#11719919—21no—
68#11719919—22no—
69#11719919D412yes—
70C4×Dic53—21no—
71C4×Dic53C2312yes—
72C2×#88322C2312yes—
73#11719913—22no—
74C23×C53—22no—
75C22×D53—21no—
76Q8×C53—42no—
77D4×C53—22no—
78D4×C53—22no—
79D4×C53—42no—
80D4×C53—81no—
81C2×C4×C53—22no—
82C2×C4×C53C2×D411yescharacteristic
83C2×C4×C53C2×D411yescharacteristic
84C2×C4×C53C2411yescharacteristic
85C2×Dic53—21no—
86C2×Dic53—21no—
87C2×Dic53—21no—
88C2×Dic53—82no—
89C2×Dic53C2×D412yes—
90C2×Dic53C2×D412yes—
91D212—21no—
92D212#4112yes—
93C4×D53—44no—
94C4×D53—42no—
95Dic106—22no—
96Dic106—21no—
97Dic106—21no—
98Dic106—44no—
99C8×C53—21no—
100C8×C53—21no—
101#88322—22no—
102C22×C53—42no—
103C22×C53—42no—
104C22×C53—42no—
105C22×C53C22×D411yescharacteristic
106C2×D53—22no—
107C2×D53—41no—
108C2×D53—41no—
109C4×C53—21no—
110C4×C53—42no—
111C4×C53AGL(1,ℤ/8)12yes—
112C4×C53C2×#4111yescharacteristic
113C4×C532−1+411yescharacteristic
114Dic53—22no—
115Dic53—22no—
116Dic53—41no—
117Dic53—41no—
118Dic53—41no—
119#3373—531nomaximal, Sylow
120C2×C53—82no—
121C2×C53#54811yescharacteristic
122C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
123C2×C53#57711yescharacteristic
124D53—42no—
125#559—531no—
126D4×Q8—531no—
127#489—532no—
128#486D5311yescharacteristic, complemented
129#476—532no—
130#475—531no—
131#463—532no—
132#459—531no—
133#446—531no—
134#442—532no—
135C4×SD16—531no—
136C53#337311yescharacteristic, complemented, Sylow
137C22×Q8—1061no—
138C2×SD16—531no—
139C2×D8—532no—
140C4⋊Q8—532no—
141C4⋊Q8C2×D5311yescharacteristic
142#127C2×D5311yescharacteristic
143#124—532no—
144#124—1061no—
145#123—1062no—
146C22⋊Q8—1062no—
147C22⋊Q8—1061no—
148C4×Q8—531no—
149C4×Q8—531no—
150C4×D4—532no—
151C4×D4—532no—
152C4×D4—531no—
153C4×D4—1061no—
154#106—531no—
155#106—532no—
156C4⋊C8—532no—
157C4⋊C8—531no—
158#103—531no—
159#103—532no—
160#102—532no—
161#102—531no—
162#102C2×D5312yes—
163#102C2×D5312yes—
164#97—532no—
165C4×C8C2×D5311yescharacteristic
166C2×Q8—21no—
167C2×Q8—531no—
168C2×Q8—531no—
169C2×Q8—1062no—
170C2×Q8—2124no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—531no—
174C2×D4C22×D5312yes—
175C22×C4—1062no—
176C22×C4—1062no—
177C22×C4—1061no—
178SD16—1062no—
179D8—1064no—
180C2×C8—532no—
181C2×C8C22×D5311yescharacteristic
182C2×C8C22×D5311yescharacteristic
183C4⋊C4—21no—
184C4⋊C4—531no—
185C4⋊C4—532no—
186C4⋊C4—531no—
187C4⋊C4—531no—
188C4⋊C4—1062no—
189C4⋊C4—1061no—
190C4⋊C4C22×D5312yes—
191C22⋊C4—1062no—
192C22⋊C4—1062no—
193C22⋊C4—1062no—
194C22⋊C4—1062no—
195C22⋊C4—1061no—
196C42—532no—
197C42—1061no—
198C42C22×D5311yescharacteristic
199C23—22no—
200C23—1061no—
201Q8—42no—
202Q8—1062no—
203Q8—1061no—
204Q8—1061no—
205Q8—2124no—
206D4—22no—
207D4—22no—
208D4—42no—
209D4—81no—
210D4—532no—
211D4—1061no—
212C2×C4—22no—
213C2×C4—532no—
214C2×C4—532no—
215C2×C4—1061no—
216C2×C4—1061no—
217C2×C4—1061no—
218C2×C4—4242no—
219C2×C4—2124no—
220C2×C4—2122no—
221C2×C4D4×D5311yescharacteristic
222C2×C4D4×D5311yescharacteristic
223C2×C4C23×D5311yescharacteristic
224C8—21no—
225C8—21no—
226C8—1062no—
227C22—42no—
228C22—42no—
229C22—42no—
230C22—1062no—
231C22—2121no—
232C22—2121no—
233C22C2×D4×D5311yescharacteristic
234C4—21no—
235C4—42no—
236C4—1062no—
237C4—1062no—
238C4—2121no—
239C4—2121no—
240C4—2121no—
241C4#1313324112yes—
242C4#41×D5311yescharacteristic
243C4#1313326411yescharacteristic
244C2—82no—
245C2—2122no—
246C2#1681128111yescharacteristic
247C2C2×#1313324111yescharacteristic
248C2#1681142911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#486C2
socleC22×C53C22×D4
hypercenter#486D53
hypocenterC53#3373
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #16811359C2semidirect
2. #16811288C2semidirect
3. #16810981C2semidirect
4. #16810911C2semidirect
5. #16810905C2semidirect
6. #16810837C2semidirect
8. #16810753C2semidirect
9. #16810732C2semidirect
10. C4×#11719913C2semidirect
11. C53×#486C2semidirect
128. #486D53semidirect
136. C53#3373semidirect