Finite Group Databasebeta

← #20284814

Subgroups — #20284814

Order 6784.

444 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 133 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#16811180C211yesmaximal, characteristic, complemented
2#16810958C212yesmaximal, complemented
3#16810903C212yesmaximal, complemented
4#16810828C212yesmaximal, complemented
5#16810774C212yesmaximal
6#16810723C214yesmaximal, complemented
7#16810680C211yesmaximal, characteristic, complemented
8C2×C4×D4×C53C211yesmaximal, characteristic
9C2×#11719923C2214yes—
10#13133217C2211yescharacteristic, complemented
11#13133217C2211yescharacteristic, complemented
12#13133209—21no—
13#13133196—21no—
14#13133196C2212yes—
15C4×Dic106—21no—
16C4×Dic106C2212yescomplemented
17#13133176—21no—
18#13133150C2214yescomplemented
19C4⋊#88322C2212yescomplemented
20#13133140C2214yescomplemented
21#13133138C2214yes—
22C22⋊#88322C2212yes—
23#13133128C2212yescomplemented
24C22×D4×C53C2211yescharacteristic
25C23×C4×C53—21no—
26C4×D4×C53—21no—
27C4×D4×C53—21no—
28C4×D4×C53C2212yes—
29C4×D4×C53C2212yes—
30C2×C4⋊C4×C53C2211yescharacteristic
31C2×C22⋊C4×C53—21no—
32C2×C42×C53C2211yescharacteristic
33C24×C53—21no—
34C2×D4×C53—24no—
35C2×D4×C53—21no—
36C2×D4×C53C2312yes—
37C22×C4×C53—21no—
38C22×C4×C53—22no—
39C22×C4×C53—22no—
40C22×C4×C53—44no—
41C22×C4×C53D411yescharacteristic
42C22×C4×C53D411yescharacteristic
43C22×C4×C53C2311yescharacteristic
44C2×Dic106—21no—
45C2×Dic106C2312yes—
46C4⋊C4×C53—21no—
47C4⋊C4×C53C2312yes—
48C22⋊C4×C53—22no—
49C22⋊C4×C53—22no—
50C42×C53D411yescharacteristic
51C42×C53D411yescharacteristic
52C42×C53C2311yescharacteristic
53C42×C53C2311yescharacteristic
54C22⋊Dic53—22no—
55#11719923—28no—
56C4⋊Dic53—21no—
57C4⋊Dic53—21no—
58C4⋊Dic53—21no—
59C4⋊Dic53C2312yes—
60#11719919—22no—
61#11719919—22no—
62C4×Dic53—22no—
63C2×#88322C2314yes—
64C23×C53—24no—
65C23×C53—22no—
66C23×C53—42no—
67C23×C53—42no—
68C23×C53C2×D411yescharacteristic
69D4×C53—24no—
70D4×C53—22no—
71D4×C53#4114yes—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—44no—
77C2×C4×C53—42no—
78C2×C4×C53—44no—
79C2×C4×C53C2×D411yescharacteristic
80C2×C4×C53C2×D411yescharacteristic
81C2×C4×C53C2×D411yescharacteristic
82C2×C4×C53C2×D411yescharacteristic
83C2×C4×C53C2×D411yescharacteristic
84C2×C4×C53C2411yescharacteristic
85C2×Dic53—22no—
86C2×Dic53—22no—
87Dic106—24no—
88Dic106—42no—
89#88322—24no—
90C22×C53—21no—
91C22×C53—21no—
92C22×C53—24no—
93C22×C53—48no—
94C22×C53—42no—
95C22×C53—42no—
96C22×C53#13712yes—
97C22×C53C22×D411yescharacteristic
98C4×C53—21no—
99C4×C53—21no—
100C4×C53—21no—
101C4×C53—21no—
102C4×C53—42no—
103C4×C53AGL(1,ℤ/8)12yes—
104C4×C53C2×#4112yes—
105Dic53—42no—
106Dic53—42no—
107#3110—531nomaximal, Sylow
108C2×C53—22no—
109C2×C53—44no—
110C2×C53#52511yescharacteristic
111C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
112C2×C53C2×#13711yescharacteristic
113#533—531no—
114C2×C4×D4D5311yescharacteristic
115#484—532no—
116#474—532no—
117#461—532no—
118#450—532no—
119#440—534no—
120#432—531no—
121C53#311011yescharacteristic, complemented, Sylow
122C22×D4C2×D5311yescharacteristic
123C23×C4—21no—
124C2×SD16—534no—
125C4⋊Q8—531no—
126C4⋊Q8—531no—
127#125—1061no—
128C22⋊Q8—532no—
129C22⋊Q8—1061no—
130C4×Q8—532no—
131C4×Q8—1061no—
132C4×D4—21no—
133C4×D4—21no—
134C4×D4C2×D5312yes—
135C4×D4C2×D5312yes—
136#117—1061no—
137C2×C4⋊C4C2×D5311yescharacteristic
138C2×C22⋊C4—21no—
139C2×C42C2×D5311yescharacteristic
140#107—534no—
141C4⋊C8—532no—
142#103—534no—
143#102—534no—
144C22⋊C8—532no—
145#97—532no—
146C24—21no—
147C2×Q8—532no—
148C2×Q8—1061no—
149C2×D4—24no—
150C2×D4—21no—
151C2×D4C22×D5312yes—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—44no—
156C22×C4C22⋊D5311yescharacteristic
157C22×C4C22⋊D5311yescharacteristic
158C22×C4C22×D5311yescharacteristic
159SD16—1068no—
160C2×C8—534no—
161C4⋊C4—21no—
162C4⋊C4—532no—
163C4⋊C4—1061no—
164C4⋊C4—1061no—
165C4⋊C4—1062no—
166C4⋊C4—1062no—
167C4⋊C4—1061no—
168C4⋊C4C22×D5312yes—
169C22⋊C4—22no—
170C22⋊C4—22no—
171C22⋊C4—1062no—
172C42—1062no—
173C42C22⋊D5311yescharacteristic
174C42C22⋊D5311yescharacteristic
175C42C22×D5311yescharacteristic
176C42C22×D5311yescharacteristic
177C23—24no—
178C23—22no—
179C23—42no—
180C23—42no—
181C23C2×C22⋊D5311yescharacteristic
182Q8—1064no—
183Q8—2122no—
184D4—24no—
185D4—22no—
186D4#1171994514yes—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—44no—
192C2×C4—42no—
193C2×C4—44no—
194C2×C4—1062no—
195C2×C4—1062no—
196C2×C4C2×C22⋊D5311yescharacteristic
197C2×C4C2×C22⋊D5311yescharacteristic
198C2×C4C2×C22⋊D5311yescharacteristic
199C2×C4C2×C22⋊D5311yescharacteristic
200C2×C4C2×C22⋊D5311yescharacteristic
201C2×C4C23×D5311yescharacteristic
202C8—1064no—
203C22—21no—
204C22—21no—
205C22—24no—
206C22—48no—
207C22—42no—
208C22—42no—
209C22#1313324712yes—
210C22C22×C22⋊D5311yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—21no—
215C4—42no—
216C4—2122no—
217C4—2122no—
218C4#1313324312yes—
219C4C2×#1171994512yes—
220C2—22no—
221C2—44no—
222C2#1681113411yescharacteristic
223C2C2×#1313324311yescharacteristic
224C2C2×#1313324711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×C4×D4×C53C2
socleC22×C53C22×D4
hypercenterC2×C4×D4D53
hypocenterC53#3110
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #16811180C2semidirect
2. #16810958C2semidirect
3. #16810903C2semidirect
4. #16810828C2semidirect
6. #16810723C2semidirect
7. #16810680C2semidirect
10. #13133217C22semidirect
11. #13133217C22semidirect
16. C4×Dic106C22semidirect
18. #13133150C22semidirect
19. C4⋊#88322C22semidirect
20. #13133140C22semidirect
23. #13133128C22semidirect
121. C53#3110semidirect