Finite Group Databasebeta

← #20290136

Subgroups — #20290136

Order 6784.

252 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 56 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#16811646C211yesmaximal, characteristic, complemented
2#16811568C211yesmaximal, characteristic, complemented
3#16811475C214yesmaximal, complemented
4#16811174C211yesmaximal, characteristic
5#13133284—21no—
6#13133274C2212yes—
7#13133272—21no—
8(C2×#6157)⋊C4—21no—
9(C2×#6157)⋊C4—21no—
10(C2×#6157)⋊C4C2212yes—
11C2×#11719949—21no—
12#13133214C411yescharacteristic, complemented
13#13133206—21no—
14#13133192—21no—
15#13133192C2212yes—
16#13133179—21no—
17#13133179C412yescomplemented
18C42×D53C2211yescharacteristic
19C4⋊Q8×C53C411yescharacteristic, complemented
20C22×#6157—22no—
21C2×Q8×C53—21no—
22#11719949—44no—
23C2×D212—22no—
24C2×D212—21no—
25C2×D212—21no—
26C2×D212C2×C412yes—
27C2×C4×D53—22no—
28C2×C4×D53D411yescharacteristic
29C2×C4×D53D411yescharacteristic
30C2×C4×D53C2311yescharacteristic
31C2×#88324—22no—
32C4⋊C4×C53—21no—
33C4⋊C4×C53C2×C412yes—
34C42×C53C2×C411yescharacteristic
35#11719921—22no—
36#11719921—22no—
37C4×Dic53C2×C411yescharacteristic
38C4×Dic53D411yescharacteristic
39C4×Dic53D411yescharacteristic
40C22×D53—22no—
41C22×D53C2×D411yescharacteristic
42C2×#6157—42no—
43C2×#6157—42no—
44C2×#6157—42no—
45Q8×C53—42no—
46C2×C4×C53—22no—
47C2×C4×C53C22⋊C411yescharacteristic
48C2×C4×C53C22⋊C411yescharacteristic
49C2×C4×C53C22×C411yescharacteristic
50C2×Dic53C22⋊C411yescharacteristic
51C2×Dic53C22⋊C411yescharacteristic
52C2×Dic53C2×D411yescharacteristic
53D212—42no—
54D212—42no—
55D212—42no—
56D212—44no—
57C4×D53—22no—
58C4×D53—41no—
59C4×D53—41no—
60C4×D53—44no—
61#88324—42no—
62C22×C53C2×C22⋊C411yescharacteristic
63C2×D53—21no—
64C2×D53—21no—
65C2×D53—42no—
66C2×D53—42no—
67C2×D53—42no—
68C2×D53AGL(1,ℤ/8)12yes—
69#6157—82no—
70C4×C53—21no—
71C4×C53—21no—
72C4×C53—21no—
73C4×C53—42no—
74Dic53—21no—
75Dic53—21no—
76Dic53—21no—
77#1514—531nomaximal, Sylow
78C2×C53#41011yescharacteristic
79C2×C53#41811yescharacteristic
80C2×C53#42111yescharacteristic
81D53—22no—
82D53—82no—
83#532—531no—
84#432—531no—
85#383—531no—
86#327—534no—
87C53#151411yescharacteristic, complemented, Sylow
88C2×#41—1061no—
89C4⋊Q8#615711yescharacteristic, complemented
90#127—531no—
91#124—1061no—
92#121—532no—
93#121—1061no—
94C4×D4—532no—
95C4×D4—1061no—
96C2×C42—531no—
97C4⋊C8—1061no—
98C22⋊C8—532no—
99#97—1061no—
100#95—532no—
101#95—1061no—
102#95—1061no—
103#41—2124no—
104C2×Q8—21no—
105C2×D4—532no—
106C2×D4—1062no—
107C2×D4—1061no—
108C2×D4—1061no—
109C22×C4—531no—
110C22×C4—531no—
111C22×C4—531no—
112C22×C4—1062no—
113C22×C4—1062no—
114C2×C8—1062no—
115C4⋊C4—21no—
116C4⋊C4C2×#615712yes—
117C22⋊C4—1062no—
118C22⋊C4—1062no—
119C42—531no—
120C42—531no—
121C42—531no—
122C42C2×#615711yescharacteristic
123C23—531no—
124C23—1062no—
125Q8—42no—
126D4—2122no—
127D4—2122no—
128D4—2122no—
129D4—2124no—
130C2×C4—22no—
131C2×C4—531no—
132C2×C4—531no—
133C2×C4—531no—
134C2×C4—1062no—
135C2×C4—2121no—
136C2×C4—2121no—
137C2×C4—2124no—
138C2×C4—2122no—
139C2×C4—2122no—
140C2×C4—2122no—
141C2×C4C22⋊#615711yescharacteristic
142C2×C4C22⋊#615711yescharacteristic
143C2×C4C22×#615711yescharacteristic
144C8—2122no—
145C22—532no—
146C22—1061no—
147C22—1061no—
148C22—2122no—
149C22—2122no—
150C22—2122no—
151C22C2×C22⋊#615711yescharacteristic
152C4—21no—
153C4—21no—
154C4—21no—
155C4—42no—
156C4—1061no—
157C4—1061no—
158C4—1061no—
159C4—4242no—
160C2—1062no—
161C2—4242no—
162C2#1681160411yescharacteristic
163C2#1681161511yescharacteristic
164C2#1681162211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC4⋊Q8×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC4⋊Q8#6157
hypocenterC53#1514
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #16811646C2semidirect
2. #16811568C2semidirect
3. #16811475C2semidirect
12. #13133214C4semidirect
17. #13133179C4semidirect
19. C4⋊Q8×C53C4semidirect
87. C53#1514semidirect
89. C4⋊Q8#6157semidirect