Finite Group Databasebeta

← #20287751

Subgroups — #20287751

Order 6784.

350 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 117 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#16811348C211yesmaximal, characteristic, complemented
2#16811347C211yesmaximal, characteristic
3#16811016C212yesmaximal
4#16810847C212yesmaximal
5#16810831C212yesmaximal, complemented
6#16810830C211yesmaximal, characteristic, complemented
7#16810825C211yesmaximal, characteristic
8#16810753C211yesmaximal, characteristic, complemented
9#16810730C212yesmaximal, complemented
10C4×#11719913C211yesmaximal, characteristic
11C53×#494C211yesmaximal, characteristic, complemented
12C2×#11719925C2212yes—
13C2×#11719913C2211yescharacteristic
14C2×#11719924C2212yes—
15#13133215C2212yes—
16#13133212—22no—
17#13133208—21no—
18#13133207—22no—
19#13133206C2212yes—
20#13133197—21no—
21Q8×Dic53C2212yes—
22Q8×Dic53C2212yes—
23C4×Dic106C2211yescharacteristic
24C4×Dic106C2211yescharacteristic
25#13133182—21no—
26C4×D212C2211yescharacteristic
27#13133175—21no—
28#13133149C2211yescharacteristic
29C4⋊#88322C2211yescharacteristic
30#13133146C2212yes—
31#13133142C2211yescharacteristic
32#13133141C2212yes—
33#13133141C2212yes—
34#13133139C2211yescharacteristic
35#13133127C2212yes—
36C2×Q16×C53C2212yes—
37C2×Q16×C53C2212yes—
38C4⋊Q8×C53C2211yescharacteristic
39C4⋊Q8×C53C2211yescharacteristic
40C4×C8×C53C2211yescharacteristic
41C2×Q8×C53C2312yes—
42C2×Q8×C53C2312yes—
43C2×D212C2311yescharacteristic
44C2×C4×D53—21no—
45C2×Dic106D411yescharacteristic
46C2×Dic106C2311yescharacteristic
47Q16×C53—24no—
48Q16×C53—42no—
49C2×C8×C53C2311yescharacteristic
50C2×C8×C53C2311yescharacteristic
51C4⋊C4×C53—21no—
52C4⋊C4×C53—21no—
53C4⋊C4×C53—21no—
54C4⋊C4×C53—21no—
55C42×C53C2311yescharacteristic
56#11719925—42no—
57#11719924—42no—
58#11719921—21no—
59#11719921—22no—
60#11719921—22no—
61C4⋊Dic53—22no—
62C4⋊Dic53—22no—
63C4⋊Dic53D411yescharacteristic
64C4⋊Dic53C2311yescharacteristic
65C4⋊Dic53C2311yescharacteristic
66#11719919—21no—
67#11719919—22no—
68#11719919—22no—
69#11719919D412yes—
70C4×Dic53—21no—
71C4×Dic53—22no—
72C4×Dic53—22no—
73C4×Dic53C2312yes—
74C2×#88322C2312yes—
75#11719913—22no—
76C22×D53—21no—
77Q8×C53—22no—
78Q8×C53—22no—
79Q8×C53—22no—
80Q8×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53C2×D411yescharacteristic
84C2×C4×C53C2×D411yescharacteristic
85C2×C4×C53C2411yescharacteristic
86C2×Dic53—21no—
87C2×Dic53—21no—
88C2×Dic53—21no—
89C2×Dic53C2×D412yes—
90C2×Dic53C2×D412yes—
91D212—21no—
92D212—21no—
93C4×D53—42no—
94Dic106—22no—
95Dic106—21no—
96Dic106—21no—
97C8×C53—21no—
98C8×C53#4112yes—
99#88322—22no—
100C22×C53C22×D411yescharacteristic
101C2×D53—41no—
102C2×D53—41no—
103C2×D53—41no—
104C4×C53—21no—
105C4×C53—42no—
106C4×C53—42no—
107C4×C53#13712yes—
108C4×C53C2×#4111yescharacteristic
109C4×C53H(5,𝔽2)11yescharacteristic
110Dic53—22no—
111Dic53—22no—
112Dic53—41no—
113Dic53—41no—
114Dic53—41no—
115#3398—531nomaximal, Sylow
116C2×C53#55011yescharacteristic
117C2×C53C2×#13711yescharacteristic
118C2×C53#57811yescharacteristic
119D53—81no—
120#557—531no—
121#556—531no—
122#497—532no—
123#494D5311yescharacteristic, complemented
124#464—532no—
125#462—532no—
126#462—531no—
127#461—531no—
128#446—531no—
129#441—532no—
130C4×SD16—531no—
131C53#339811yescharacteristic, complemented, Sylow
132C2×Q16—532no—
133C2×Q16C2×D5312yes—
134C2×Q16C2×D5312yes—
135C2×SD16—532no—
136C2×SD16—531no—
137C4⋊Q8—532no—
138C4⋊Q8C2×D5311yescharacteristic
139C4⋊Q8C2×D5311yescharacteristic
140#126—1062no—
141#125—1062no—
142#125—1061no—
143#124—532no—
144C22⋊Q8—1061no—
145C4×Q8—531no—
146C4×Q8—532no—
147C4×Q8—532no—
148C4×Q8—531no—
149C4×Q8—1061no—
150C4×D4—531no—
151#117—1061no—
152#106—531no—
153C4⋊C8—532no—
154C4⋊C8—531no—
155#103—532no—
156#103—532no—
157#103—531no—
158#102—531no—
159#97—532no—
160C4×C8C2×D5311yescharacteristic
161C2×Q8—531no—
162C2×Q8—531no—
163C2×Q8C22×D5312yes—
164C2×Q8C22×D5312yes—
165C2×D4—531no—
166C22×C4—1061no—
167Q16—24no—
168Q16—42no—
169Q16—2122no—
170SD16—1062no—
171SD16—2122no—
172C2×C8—532no—
173C2×C8C22×D5311yescharacteristic
174C2×C8C22×D5311yescharacteristic
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—531no—
180C4⋊C4—532no—
181C4⋊C4—531no—
182C4⋊C4—531no—
183C4⋊C4—1062no—
184C4⋊C4—1062no—
185C4⋊C4—1062no—
186C4⋊C4—1062no—
187C4⋊C4—1061no—
188C22⋊C4—1062no—
189C22⋊C4—1062no—
190C22⋊C4—1061no—
191C42—532no—
192C42—1062no—
193C42—1062no—
194C42—1061no—
195C42C22×D5311yescharacteristic
196C23—1061no—
197Q8—22no—
198Q8—22no—
199Q8—22no—
200Q8—22no—
201Q8—1062no—
202Q8—1061no—
203Q8—1061no—
204D4—1061no—
205D4—1061no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—532no—
209C2×C4—532no—
210C2×C4—1061no—
211C2×C4—1061no—
212C2×C4—1061no—
213C2×C4—2122no—
214C2×C4D4×D5311yescharacteristic
215C2×C4D4×D5311yescharacteristic
216C2×C4C23×D5311yescharacteristic
217C8—21no—
218C8—1062no—
219C8#1171994712yes—
220C22—2121no—
221C22—2121no—
222C22—2121no—
223C22C2×D4×D5311yescharacteristic
224C4—21no—
225C4—42no—
226C4—42no—
227C4—1062no—
228C4—1062no—
229C4—2121no—
230C4—2121no—
231C4—2121no—
232C4#1313324912yes—
233C4C2×#1171994711yescharacteristic
234C4#1313326211yescharacteristic
235C2—4241no—
236C2#1681129311yescharacteristic
237C2C2×#1313324911yescharacteristic
238C2#1681143611yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#494C2
socleC22×C53C22×D4
hypercenter#494D53
hypocenterC53#3398
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
1. #16811348C2semidirect
5. #16810831C2semidirect
6. #16810830C2semidirect
8. #16810753C2semidirect
9. #16810730C2semidirect
11. C53×#494C2semidirect
123. #494D53semidirect
131. C53#3398semidirect