Finite Group Databasebeta

← #20284648

Subgroups — #20284648

Order 6784.

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

normal subgroups only

This group has 414 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#16811134C211yesmaximal, characteristic
2C2×#13133176C211yesmaximal, characteristic, complemented
3#16810965C214yesmaximal, complemented
4#16810948C214yesmaximal, complemented
5#16810665C212yesmaximal, complemented
6#16810625C212yesmaximal, complemented
7C53×#407C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic
9#13133200—21no—
10#13133196C2212yes—
11#13133191C2212yes—
12#13133186—21no—
13C4×C22⋊D53—22no—
14#13133176—22no—
15#13133176C2214yes—
16#13133176C2211yescharacteristic
17C2×C4⋊Dic53C2212yes—
18C2×C22⋊Dic53—21no—
19C2×C4×Dic53—21no—
20#13133151C2214yescomplemented
21#13133149C2214yescomplemented
22#13133142C2214yescomplemented
23#13133139C2214yescomplemented
24C23×C4×C53C2211yescharacteristic
25C2×M16×C53C2212yes—
26C22⋊C8×C53C2214yes—
27C24×C53D411yescharacteristic
28C22×C4×C53—24no—
29C22×C4×C53D411yescharacteristic
30C22×C4×C53D412yes—
31C22×C4×C53C2312yes—
32C22×C4×C53C2311yescharacteristic
33C2×C22⋊D53—22no—
34C2×C22⋊D53—21no—
35C22×Dic53—22no—
36#11719945—24no—
37C2×D212C2311yescharacteristic
38C2×C4×D53—21no—
39C2×Dic106C2311yescharacteristic
40M16×C53—24no—
41C2×C8×C53C2314yes—
42C22⋊Dic53—24no—
43C22⋊Dic53—21no—
44C22⋊Dic53—22no—
45#11719921—22no—
46C4⋊Dic53—22no—
47C4⋊Dic53C2314yes—
48C4⋊Dic53C2312yes—
49#11719919—22no—
50C4×Dic53—24no—
51C4×Dic53—21no—
52C23×C53—22no—
53C23×C53—42no—
54C23×C53C2×D412yes—
55C23×C53C2×D411yescharacteristic
56C22×D53—21no—
57C2×C4×C53—22no—
58C2×C4×C53—24no—
59C2×C4×C53—24no—
60C2×C4×C53C2×D412yes—
61C2×C4×C53C2×D411yescharacteristic
62C2×C4×C53#4114yes—
63C2×C4×C53C2411yescharacteristic
64C22⋊D53—44no—
65C22⋊D53—42no—
66C2×Dic53—21no—
67C2×Dic53—24no—
68C2×Dic53—22no—
69C2×Dic53—44no—
70D212—22no—
71C4×D53—42no—
72Dic106—22no—
73C8×C53—24no—
74C22×C53—22no—
75C22×C53—22no—
76C22×C53—22no—
77C22×C53—22no—
78C22×C53—44no—
79C22×C53#12312yes—
80C22×C53C22×D411yescharacteristic
81C2×D53—41no—
82C2×D53—42no—
83C4×C53—22no—
84C4×C53#12312yes—
85C4×C53C2×#4112yes—
86Dic53—41no—
87Dic53—44no—
88Dic53—42no—
89#3093—531nomaximal, Sylow
90C2×C53—22no—
91C2×C53—42no—
92C2×C53C2×#12311yescharacteristic
93C2×C53#57512yes—
94D53—81no—
95#525—531no—
96C2×#117—531no—
97#485—534no—
98#482—534no—
99#428—532no—
100#417—532no—
101#407D5311yescharacteristic, complemented
102C53#309311yescharacteristic, complemented, Sylow
103C2×#41—531no—
104C23×C4C2×D5311yescharacteristic
105C2×M16C2×D5312yes—
106#123—1061no—
107C22⋊Q8—532no—
108#121—532no—
109#120—1061no—
110C4×D4—1062no—
111#117—534no—
112#117—531no—
113#117—1062no—
114C2×C4⋊C4—532no—
115C2×C22⋊C4—1061no—
116C2×C42—1061no—
117#107—534no—
118#106—534no—
119#103—534no—
120#102—534no—
121C22⋊C8C2×D5314yes—
122C24D21211yescharacteristic
123#41—1064no—
124C2×Q8—531no—
125C2×D4—531no—
126C2×D4—1062no—
127C2×D4—1061no—
128C22×C4—24no—
129C22×C4—1062no—
130C22×C4—1061no—
131C22×C4D21211yescharacteristic
132C22×C4D21212yes—
133C22×C4C22×D5312yes—
134C22×C4C22×D5311yescharacteristic
135M16—24no—
136C2×C8C22×D5314yes—
137C4⋊C4—534no—
138C4⋊C4—532no—
139C4⋊C4—1062no—
140C4⋊C4—1062no—
141C22⋊C4—1064no—
142C22⋊C4—1061no—
143C22⋊C4—1062no—
144C22⋊C4—1062no—
145C42—1064no—
146C42—1061no—
147C23—22no—
148C23—42no—
149C23—1061no—
150C23C2×D21212yes—
151C23C2×D21211yescharacteristic
152Q8—1062no—
153D4—1062no—
154D4—2124no—
155D4—2122no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—1064no—
160C2×C4—1062no—
161C2×C4—1061no—
162C2×C4—2124no—
163C2×C4—2122no—
164C2×C4C2×D21212yes—
165C2×C4C2×D21211yescharacteristic
166C2×C4#1171994714yes—
167C2×C4C23×D5311yescharacteristic
168C8—24no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—44no—
174C22—2121no—
175C22—2122no—
176C22#1313320112yes—
177C22C22×D21211yescharacteristic
178C4—22no—
179C4—2121no—
180C4—2124no—
181C4—2122no—
182C4#1313320112yes—
183C4C2×#1171994712yes—
184C2—22no—
185C2—42no—
186C2—4241no—
187C2C2×#1313320111yescharacteristic
188C2#1681142312yes—
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#407C2
socleC22×C53C22×D4
hypercenter#407D53
hypocenterC53#3093
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#13133176C2semidirect
3. #16810965C2semidirect
4. #16810948C2semidirect
5. #16810665C2semidirect
6. #16810625C2semidirect
7. C53×#407C2semidirect
20. #13133151C22semidirect
21. #13133149C22semidirect
22. #13133142C22semidirect
23. #13133139C22semidirect
101. #407D53semidirect
102. C53#3093semidirect