Finite Group Databasebeta

← #20275315

Subgroups — #20275315

Order 6784.

186 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 75 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#16810661C211yesmaximal, characteristic
2#16810649C211yesmaximal, characteristic
3#16810578C211yesmaximal, characteristic
4#16810388C211yesmaximal, characteristic
5#16810383C211yesmaximal, characteristic
6#16810383C211yesmaximal, characteristic
7C53×#336C211yesmaximal, characteristic
8C2×#11719917C2211yescharacteristic
9C2×#11719917C2211yescharacteristic
10#13133176C2211yescharacteristic
11#13133176C2211yescharacteristic
12#13133151C411yescharacteristic, complemented
13#13133151C411yescharacteristic, complemented
14#13133149C411yescharacteristic, complemented
15#13133149C411yescharacteristic, complemented
16#13133146—21no—
17#13133146—21no—
18#13133127—21no—
19C8×Dic53—21no—
20C22×C8×C53C2211yescharacteristic
21C22×C8×C53C2211yescharacteristic
22C2×C42×C53C2211yescharacteristic
23C22×C4×C53—21no—
24C22×C4×C53C2311yescharacteristic
25C2×C8×C53—21no—
26C2×C8×C53—21no—
27C2×C8×C53—21no—
28C2×C8×C53—21no—
29C2×C8×C53C2×C411yescharacteristic
30C2×C8×C53C2×C411yescharacteristic
31C2×C8×C53D411yescharacteristic
32C2×C8×C53D411yescharacteristic
33C42×C53—21no—
34C42×C53—21no—
35C22⋊Dic53—21no—
36C22⋊Dic53—21no—
37C4⋊Dic53C2×C411yescharacteristic
38C4⋊Dic53C2×C411yescharacteristic
39C4⋊Dic53C2×C411yescharacteristic
40C4⋊Dic53C2×C411yescharacteristic
41C4×Dic53—21no—
42C4×Dic53—21no—
43#11719917—21no—
44#11719917—21no—
45#11719917—21no—
46#11719917Q812yes—
47C2×#88322—21no—
48C2×#88322—21no—
49C23×C53#4111yescharacteristic
50C2×C4×C53—22no—
51C2×C4×C53—41no—
52C2×C4×C53—41no—
53C2×C4×C53C22×C411yescharacteristic
54C2×C4×C53C2×D411yescharacteristic
55C2×C4×C53C2×Q811yescharacteristic
56C2×C4×C53#4111yescharacteristic
57C2×C4×C53#4111yescharacteristic
58C2×C4×C53#4111yescharacteristic
59C2×Dic53—21no—
60C2×Dic53—21no—
61C2×Dic53—21no—
62C2×Dic53—21no—
63C8×C53—21no—
64C8×C53—21no—
65C8×C53—21no—
66C8×C53—21no—
67#88322—22no—
68#88322—41no—
69C22×C53—21no—
70C22×C53—21no—
71C22×C53C4×D411yescharacteristic
72C22×C53#12511yescharacteristic
73C22×C53#12611yescharacteristic
74C4×C53—42no—
75C4×C53#11711yescharacteristic
76C4×C53C4×Q811yescharacteristic
77C4×C53C22⋊Q811yescharacteristic
78C4×C53#12311yescharacteristic
79Dic53—41no—
80Dic53—41no—
81Dic53—41no—
82Dic53—41no—
83#1922—531nomaximal, Sylow
84C2×C53—21no—
85C2×C53—21no—
86C2×C53#38711yescharacteristic
87C2×C53#44311yescharacteristic
88C2×C53#44411yescharacteristic
89#427—531no—
90#424—531no—
91#405—531no—
92#341—531no—
93#339—531no—
94#339—531no—
95#336D5311yescharacteristic
96C53#192211yescharacteristic, complemented, Sylow
97C2×M16—531no—
98C2×M16—531no—
99C22×C8C2×D5311yescharacteristic
100C22×C8C2×D5311yescharacteristic
101#117—531no—
102#117—531no—
103C2×C42C2×D5311yescharacteristic
104#107—531no—
105#107—531no—
106#106—531no—
107#106—531no—
108C4⋊C8—1061no—
109C4⋊C8—1061no—
110#97—1061no—
111C4×C8—1061no—
112C22×C4—21no—
113C22×C4C22×D5311yescharacteristic
114M16—532no—
115M16—1061no—
116M16—1061no—
117M16—1061no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—1061no—
123C2×C8—1061no—
124C2×C8C4×D5311yescharacteristic
125C2×C8C4×D5311yescharacteristic
126C2×C8D21211yescharacteristic
127C2×C8D21211yescharacteristic
128C4⋊C4—531no—
129C4⋊C4—531no—
130C4⋊C4—531no—
131C4⋊C4—531no—
132C22⋊C4—1061no—
133C22⋊C4—1061no—
134C42—21no—
135C42—21no—
136C42—1061no—
137C42—1061no—
138C23#1171994511yescharacteristic
139C2×C4—22no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—1061no—
143C2×C4—1061no—
144C2×C4—1061no—
145C2×C4—1061no—
146C2×C4C2×C4×D5311yescharacteristic
147C2×C4C2×D21211yescharacteristic
148C2×C4#1171994711yescharacteristic
149C2×C4#1171994711yescharacteristic
150C2×C4#1171994711yescharacteristic
151C2×C4Q8×D5311yescharacteristic
152C8—21no—
153C8—21no—
154C8—21no—
155C8—21no—
156C8—1062no—
157C8—2121no—
158C22—21no—
159C22—21no—
160C22C4×D21211yescharacteristic
161C22#1313320711yescharacteristic
162C22#1313321011yescharacteristic
163C4—42no—
164C4—2121no—
165C4—2121no—
166C4—2121no—
167C4—2121no—
168C4#1313317311yescharacteristic
169C4#1313318211yescharacteristic
170C4#1313319711yescharacteristic
171C4#1313320111yescharacteristic
172C2—21no—
173C2—21no—
174C2#1681050811yescharacteristic
175C2#1681074111yescharacteristic
176C2#1681074611yescharacteristic
subgroupG/H
centerC2×C4C2×D212
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#336C2
socleC22×C53C4×D4
hypercenter#336D53
hypocenterC53#1922
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #336

Complements

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

NH ≅ G/Ntype
12. #13133151C4semidirect
13. #13133151C4semidirect
14. #13133149C4semidirect
15. #13133149C4semidirect
96. C53#1922semidirect