Finite Group Databasebeta

← #20290146

Subgroups — #20290146

Order 6784.

264 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 66 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#16811641C211yesmaximal, characteristic, complemented
2#16811567C211yesmaximal, characteristic, complemented
3#16811475C214yesmaximal, complemented
4#16811174C211yesmaximal, characteristic
5C2×C4⋊#6157—21no—
6#13133284—21no—
7#13133274C2212yes—
8C4×#88324—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—42no—
23#11719949—42no—
24C2×D212—22no—
25C2×D212—21no—
26C2×D212—21no—
27C2×D212C2×C412yes—
28C2×C4×D53—22no—
29C2×C4×D53D411yescharacteristic
30C2×C4×D53D411yescharacteristic
31C2×C4×D53C2311yescharacteristic
32C2×#88324—22no—
33C4⋊#6157—42no—
34C4⋊C4×C53—21no—
35C4⋊C4×C53C2×C412yes—
36C42×C53C2×C411yescharacteristic
37#11719921—22no—
38#11719921—22no—
39C4×Dic53C2×C411yescharacteristic
40C4×Dic53D411yescharacteristic
41C4×Dic53D411yescharacteristic
42C22×D53—22no—
43C22×D53C2×D411yescharacteristic
44C2×#6157—42no—
45C2×#6157—42no—
46C2×#6157—42no—
47Q8×C53—42no—
48C2×C4×C53—22no—
49C2×C4×C53C22⋊C411yescharacteristic
50C2×C4×C53C22⋊C411yescharacteristic
51C2×C4×C53C22×C411yescharacteristic
52C2×Dic53C22⋊C411yescharacteristic
53C2×Dic53C22⋊C411yescharacteristic
54C2×Dic53C2×D411yescharacteristic
55D212—42no—
56D212—42no—
57D212—42no—
58D212—42no—
59D212—42no—
60C4×D53—41no—
61C4×D53—41no—
62C4×D53—44no—
63C4×D53D812yes—
64C4×D53SD1612yes—
65#88324—42no—
66C22×C53C2×C22⋊C411yescharacteristic
67C2×D53—21no—
68C2×D53—21no—
69C2×D53—42no—
70C2×D53—42no—
71C2×D53—42no—
72C2×D53C2×D811yescharacteristic
73C2×D53C2×SD1611yescharacteristic
74#6157—82no—
75C4×C53—21no—
76C4×C53—21no—
77C4×C53—42no—
78C4×C53#10212yes—
79Dic53—21no—
80Dic53—21no—
81Dic53#10212yes—
82#1517—531nomaximal, Sylow
83C2×C53#41011yescharacteristic
84C2×C53C2×#10211yescharacteristic
85C2×C53#42111yescharacteristic
86D53—21no—
87D53—21no—
88D53—82no—
89#532—531no—
90#431—531no—
91#382—531no—
92#327—534no—
93C53#151711yescharacteristic, complemented, Sylow
94C2×#41—1061no—
95C4⋊Q8#615711yescharacteristic, complemented
96#127—531no—
97#124—1061no—
98#121—532no—
99#121—1061no—
100C4×D4—532no—
101C4×D4—1061no—
102C2×C4⋊C4—1061no—
103C2×C42—531no—
104C4⋊C8—1061no—
105C22⋊C8—532no—
106C4×C8—1061no—
107#95—532no—
108#95—1061no—
109#41—2122no—
110#41—2122no—
111C2×Q8—21no—
112C2×D4—532no—
113C2×D4—1062no—
114C2×D4—1061no—
115C2×D4—1061no—
116C22×C4—531no—
117C22×C4—531no—
118C22×C4—531no—
119C22×C4—1062no—
120C22×C4—1062no—
121C2×C8—1062no—
122C4⋊C4—21no—
123C4⋊C4—2122no—
124C4⋊C4C2×#615712yes—
125C22⋊C4—1062no—
126C22⋊C4—1062no—
127C42—531no—
128C42—531no—
129C42—531no—
130C42C2×#615711yescharacteristic
131C23—531no—
132C23—1062no—
133Q8—42no—
134D4—2122no—
135D4—2122no—
136D4—2122no—
137D4—2122no—
138D4—2122no—
139C2×C4—22no—
140C2×C4—531no—
141C2×C4—532no—
142C2×C4—532no—
143C2×C4—531no—
144C2×C4—531no—
145C2×C4—2121no—
146C2×C4—2121no—
147C2×C4—2124no—
148C2×C4—2122no—
149C2×C4—2122no—
150C2×C4—2122no—
151C2×C4C22⋊#615711yescharacteristic
152C2×C4C22⋊#615711yescharacteristic
153C2×C4C22×#615711yescharacteristic
154C8—2122no—
155C22—531no—
156C22—531no—
157C22—1061no—
158C22—1061no—
159C22—2122no—
160C22—2122no—
161C22—2122no—
162C22C2×C22⋊#615711yescharacteristic
163C4—21no—
164C4—21no—
165C4—42no—
166C4—532no—
167C4—1061no—
168C4—1061no—
169C4—4242no—
170C4#1313327912yes—
171C2—1061no—
172C2—1061no—
173C2—4242no—
174C2#1681160411yescharacteristic
175C2C2×#1313327911yescharacteristic
176C2#1681162211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC4⋊Q8×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC4⋊Q8#6157
hypocenterC53#1517
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. #16811641C2semidirect
2. #16811567C2semidirect
3. #16811475C2semidirect
12. #13133214C4semidirect
17. #13133179C4semidirect
19. C4⋊Q8×C53C4semidirect
93. C53#1517semidirect
95. C4⋊Q8#6157semidirect