Finite Group Databasebeta

← #20290990

Subgroups — #20290990

Order 6784.

286 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 54 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
1C2×#13133276C212yesmaximal, complemented
2#16811529C212yesmaximal, complemented
3#16811527C212yesmaximal, complemented
4#16811187C211yesmaximal, characteristic
5C2×C22⋊#6157—22no—
6#13133276—22no—
7#13133276C2214yes—
8C22×D212C2211yescharacteristic
9#13133213C411yescharacteristic, complemented
10#13133213C2211yescharacteristic
11#13133206C411yescharacteristic, complemented
12#13133206C2211yescharacteristic
13#13133191—22no—
14#13133185—22no—
15#13133176C411yescharacteristic, complemented
16C2×#41×C53C411yescharacteristic, complemented
17C23×D53D412yes—
18C22×#6157—42no—
19#41×C53—42no—
20C2×Q8×C53C2×C411yescharacteristic, complemented
21C2×D4×C53—21no—
22C2×D4×C53C2×C411yescharacteristic, complemented
23C22×C4×C53—21no—
24C22×C4×C53C2×C411yescharacteristic
25C2×C22⋊D53—22no—
26C2×C22⋊D53—22no—
27C2×D212—42no—
28C2×D212C2×C411yescharacteristic
29C2×D212D412yes—
30C2×D212C2311yescharacteristic
31C22⋊#6157—24no—
32C22⋊#6157—42no—
33C22⋊Dic53—21no—
34#11719921—22no—
35#11719921—22no—
36C4⋊Dic53—21no—
37C4×Dic53C2×C411yescharacteristic, complemented
38C4×Dic53C2×C411yescharacteristic, complemented
39C23×C53—21no—
40C23×C53C22⋊C411yescharacteristic
41C22×D53—22no—
42C22×D53—22no—
43C22×D53—82no—
44C22×D53C22⋊C412yes—
45C22×D53C2×D412yes—
46C2×#6157—44no—
47C2×#6157—84no—
48Q8×C53—41no—
49D4×C53—41no—
50D4×C53—42no—
51C2×C4×C53—22no—
52C2×C4×C53—21no—
53C2×C4×C53—42no—
54C2×C4×C53C22⋊C411yescharacteristic
55C2×C4×C53C22×C411yescharacteristic
56C22⋊D53—82no—
57C22⋊D53—82no—
58C2×Dic53—21no—
59C2×Dic53—21no—
60D212—41no—
61D212—42no—
62D212—41no—
63C22×C53—41no—
64C22×C53—42no—
65C22×C53—41no—
66C22×C53#9912yes—
67C22×C53C2×C22⋊C411yescharacteristic
68C2×D53—22no—
69C2×D53—22no—
70C2×D53—44no—
71C2×D53—42no—
72C2×D53—42no—
73C2×D53—82no—
74#6157—84no—
75C4×C53—21no—
76C4×C53—21no—
77C4×C53—41no—
78Dic53—41no—
79Dic53—41no—
80#2127—531nomaximal, Sylow
81C2×C53—21no—
82C2×C53—22no—
83C2×C53—81no—
84C2×C53C2×#9911yescharacteristic
85D53—42no—
86D53—42no—
87#535—531no—
88C2×#99—532no—
89#354—532no—
90#353—532no—
91C53#212711yescharacteristic, complemented, Sylow
92C2×#41#615711yescharacteristic, complemented
93C22×D4—531no—
94#127—531no—
95#127—531no—
96#124—531no—
97#124—531no—
98#121—1062no—
99#120—1062no—
100#117—531no—
101C2×C22⋊C4—1062no—
102#99—534no—
103#99—1062no—
104C24—532no—
105#41—42no—
106C2×Q8C2×#615711yescharacteristic, complemented
107C2×D4—21no—
108C2×D4—532no—
109C2×D4—531no—
110C2×D4—531no—
111C2×D4—1062no—
112C2×D4—1062no—
113C2×D4—2122no—
114C2×D4C2×#615711yescharacteristic, complemented
115C22×C4—21no—
116C22×C4—2122no—
117C22×C4C2×#615711yescharacteristic
118C4⋊C4—1061no—
119C22⋊C4—1062no—
120C22⋊C4—1062no—
121C22⋊C4—1061no—
122C22⋊C4—1064no—
123C22⋊C4—2122no—
124C42—531no—
125C42—531no—
126C23—21no—
127C23—532no—
128C23—532no—
129C23—1062no—
130C23—1062no—
131C23—4242no—
132C23C22⋊#615711yescharacteristic
133Q8—41no—
134D4—41no—
135D4—42no—
136D4—4242no—
137D4—4242no—
138D4—2122no—
139D4—2121no—
140D4—2121no—
141C2×C4—22no—
142C2×C4—21no—
143C2×C4—42no—
144C2×C4—1061no—
145C2×C4—1061no—
146C2×C4—4244no—
147C2×C4—2124no—
148C2×C4C22⋊#615711yescharacteristic
149C2×C4C22×#615711yescharacteristic
150C22—41no—
151C22—42no—
152C22—41no—
153C22—1062no—
154C22—1062no—
155C22—4242no—
156C22—2122no—
157C22—2122no—
158C22—2124no—
159C22#1313327512yes—
160C22C2×C22⋊#615711yescharacteristic
161C4—21no—
162C4—21no—
163C4—41no—
164C4—4244no—
165C4—2121no—
166C4—2121no—
167C2—21no—
168C2—22no—
169C2—81no—
170C2—2122no—
171C2—2122no—
172C2C2×#1313327511yescharacteristic
subgroupG/H
centerC2C2×#13133275
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC2×#41×C53C4
socleC2×C53C2×#99
hypercenterC2×#41#6157
hypocenterC53#2127
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×#13133276C2semidirect
2. #16811529C2semidirect
3. #16811527C2semidirect
9. #13133213C4semidirect
11. #13133206C4semidirect
15. #13133176C4semidirect
16. C2×#41×C53C4semidirect
20. C2×Q8×C53C2×C4semidirect
22. C2×D4×C53C2×C4semidirect
37. C4×Dic53C2×C4semidirect
38. C4×Dic53C2×C4semidirect
91. C53#2127semidirect
92. C2×#41#6157semidirect
106. C2×Q8C2×#6157semidirect
114. C2×D4C2×#6157semidirect