Finite Group Databasebeta

← #20287745

Subgroups — #20287745

Order 6784.

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

normal subgroups only

This group has 350 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#16811351C211yesmaximal, characteristic, complemented
2#16811347C211yesmaximal, characteristic
3#16811016C212yesmaximal
4#16811002C211yesmaximal, characteristic, complemented
5#16810848C212yesmaximal
6#16810831C212yesmaximal, complemented
7#16810829C211yesmaximal, characteristic
8#16810750C211yesmaximal, characteristic, complemented
9#16810727C212yesmaximal, complemented
10C4×#11719924C211yesmaximal, characteristic
11C53×#462C211yesmaximal, characteristic, complemented
12C2×#11719925C2212yes—
13C2×#11719925C2212yes—
14C2×#11719913C2212yes—
15C2×#11719924C2211yescharacteristic
16#13133217C2211yescharacteristic
17#13133215C2212yes—
18#13133212—22no—
19#13133209—21no—
20#13133207—22no—
21#13133206C2212yes—
22#13133197—21no—
23Q8×Dic53C2212yes—
24C4×Dic106—21no—
25C4×Dic106C2211yescharacteristic
26#13133182C2212yes—
27C4×D212C2211yescharacteristic
28#13133175—21no—
29#13133148C2211yescharacteristic
30#13133146C2212yes—
31#13133142C2212yes—
32#13133141C2211yescharacteristic
33#13133137C2211yescharacteristic
34#13133127C2212yes—
35C4×#88322C2211yescharacteristic
36C2×Q16×C53C2212yes—
37C4⋊Q8×C53C2211yescharacteristic
38C4×Q8×C53C2211yescharacteristic
39C4⋊C8×C53C2211yescharacteristic
40#103×C53C2212yes—
41C2×Q8×C53C2312yes—
42C2×Q8×C53C2311yescharacteristic
43C2×D212C2311yescharacteristic
44C2×C4×D53—21no—
45C2×Dic106D411yescharacteristic
46C2×Dic106C2312yes—
47Q16×C53—42no—
48C2×C8×C53C2312yes—
49C4⋊C4×C53—21no—
50C4⋊C4×C53—21no—
51C4⋊C4×C53—21no—
52C4⋊C4×C53C2311yescharacteristic
53C42×C53—21no—
54C42×C53C2311yescharacteristic
55#11719925—24no—
56#11719925—42no—
57#11719924—22no—
58#11719921—21no—
59#11719921—22no—
60#11719921—22no—
61C4⋊Dic53—21no—
62C4⋊Dic53—21no—
63C4⋊Dic53—22no—
64C4⋊Dic53D411yescharacteristic
65C4⋊Dic53C2311yescharacteristic
66#11719919—22no—
67#11719919—22no—
68#11719919—22no—
69#11719919D412yes—
70C4×Dic53—22no—
71C4×Dic53—22no—
72C4×Dic53C2312yes—
73C2×#88322C2311yescharacteristic
74C2×#88322C2311yescharacteristic
75#11719913—42no—
76C22×D53—21no—
77Q8×C53—21no—
78Q8×C53—22no—
79Q8×C53—22no—
80Q8×C53—21no—
81C2×C4×C53—22no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53C2×D411yescharacteristic
85C2×C4×C53C2×D411yescharacteristic
86C2×C4×C53C2411yescharacteristic
87C2×Dic53—22no—
88C2×Dic53—21no—
89C2×Dic53C2×D412yes—
90C2×Dic53C2×D412yes—
91D212—21no—
92D212—21no—
93C4×D53—42no—
94Dic106—22no—
95Dic106—22no—
96Dic106—22no—
97C8×C53—22no—
98#88322—21no—
99#88322#4112yes—
100C22×C53C22×D411yescharacteristic
101C2×D53—41no—
102C2×D53—41no—
103C2×D53—41no—
104C4×C53—21no—
105C4×C53—42no—
106C4×C53—41no—
107C4×C53—41no—
108C4×C53#13712yes—
109C4×C53C2×#4111yescharacteristic
110C4×C53H(5,𝔽2)11yescharacteristic
111Dic53—22no—
112Dic53—22no—
113Dic53—42no—
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#494—531no—
124#464—532no—
125#462—532no—
126#462D5311yescharacteristic, complemented
127#461—531no—
128#446—531no—
129#441—532no—
130C4×SD16—531no—
131C53#339811yescharacteristic, complemented, Sylow
132C2×Q16—532no—
133C2×Q16—532no—
134C2×Q16C2×D5312yes—
135C2×SD16—531no—
136C2×SD16—532no—
137C4⋊Q8—532no—
138C4⋊Q8—531no—
139C4⋊Q8C2×D5311yescharacteristic
140#126—1062no—
141#125—1061no—
142#125—1062no—
143#124—532no—
144C22⋊Q8—1061no—
145C4×Q8—531no—
146C4×Q8—532no—
147C4×Q8—532no—
148C4×Q8—1061no—
149C4×Q8C2×D5311yescharacteristic
150C4×D4—531no—
151#117—1061no—
152#106—531no—
153C4⋊C8—532no—
154C4⋊C8C2×D5311yescharacteristic
155#103—531no—
156#103—532no—
157#103C2×D5312yes—
158#102—531no—
159#97—532no—
160C4×C8—531no—
161C2×Q8—531no—
162C2×Q8—532no—
163C2×Q8C22×D5311yescharacteristic
164C2×Q8C22×D5312yes—
165C2×D4—531no—
166C22×C4—1061no—
167Q16—42no—
168Q16—1064no—
169Q16—2122no—
170SD16—1062no—
171SD16—2122no—
172C2×C8—531no—
173C2×C8—531no—
174C2×C8C22×D5312yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—531no—
179C4⋊C4—532no—
180C4⋊C4—531no—
181C4⋊C4—1062no—
182C4⋊C4—1062no—
183C4⋊C4—1062no—
184C4⋊C4—1062no—
185C4⋊C4—1061no—
186C4⋊C4—1061no—
187C4⋊C4C22×D5311yescharacteristic
188C22⋊C4—1062no—
189C22⋊C4—1062no—
190C22⋊C4—1061no—
191C42—21no—
192C42—532no—
193C42—1062no—
194C42—1062no—
195C42C22×D5311yescharacteristic
196C23—1061no—
197Q8—21no—
198Q8—22no—
199Q8—21no—
200Q8—22no—
201Q8—1062no—
202Q8—1062no—
203Q8—1062no—
204D4—1061no—
205D4—1061no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—532no—
210C2×C4—532no—
211C2×C4—1061no—
212C2×C4—1062no—
213C2×C4—2122no—
214C2×C4D4×D5311yescharacteristic
215C2×C4D4×D5311yescharacteristic
216C2×C4C23×D5311yescharacteristic
217C8—22no—
218C8—532no—
219C8—1061no—
220C22—2121no—
221C22—2121no—
222C22—2121no—
223C22C2×D4×D5311yescharacteristic
224C4—21no—
225C4—41no—
226C4—42no—
227C4—41no—
228C4—1062no—
229C4—1062no—
230C4—2122no—
231C4—2121no—
232C4#1313324912yes—
233C4#41×D5311yescharacteristic
234C4#1313326211yescharacteristic
235C2—4241no—
236C2#1681129111yescharacteristic
237C2C2×#1313324911yescharacteristic
238C2#1681143911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#462C2
socleC22×C53C22×D4
hypercenter#462D53
hypocenterC53#3398
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
1. #16811351C2semidirect
4. #16811002C2semidirect
6. #16810831C2semidirect
8. #16810750C2semidirect
9. #16810727C2semidirect
11. C53×#462C2semidirect
126. #462D53semidirect
131. C53#3398semidirect