Finite Group Databasebeta

← #20290108

Subgroups — #20290108

Order 6784.

228 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 58 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
2#16811559C211yesmaximal, characteristic
3#16811477C212yesmaximal
4#16811476C212yesmaximal
5#16811177C211yesmaximal, characteristic
6C2×C4×#6157—21no—
7#13133284—21no—
8#13133274C2212yes—
9#13133272—21no—
10(C2×#6157)⋊C4—21no—
11(C2×#6157)⋊C4C2212yes—
12#13133216C411yescharacteristic, complemented
13#13133215C411yescharacteristic, complemented
14#13133207—21no—
15#13133197C2211yescharacteristic
16#13133195—21no—
17#13133193C2211yescharacteristic
18C4×Dic106—21no—
19C4×Dic106C411yescharacteristic, complemented
20#13133174—21no—
21C42×D53C2211yescharacteristic
22C4×Q8×C53C411yescharacteristic, complemented
23C22×#6157—22no—
24C2×Q8×C53C2×C411yescharacteristic
25C2×C4×D53D411yescharacteristic
26C2×C4×D53D411yescharacteristic
27C2×C4×D53C2311yescharacteristic
28C2×Dic106—21no—
29C2×Dic106C2×C411yescharacteristic
30C2×#88324—22no—
31C4×#6157—24no—
32C4⋊C4×C53—21no—
33C4⋊C4×C53C2×C411yescharacteristic
34C42×C53—21no—
35C42×C53C2×C411yescharacteristic
36#11719921—21no—
37#11719921—21no—
38C4⋊Dic53—21no—
39C4⋊Dic53C2×C411yescharacteristic
40#11719919—21no—
41#11719919—21no—
42#11719919—21no—
43#11719919—21no—
44#11719919—21no—
45C4×Dic53—21no—
46C4×Dic53C2×C411yescharacteristic
47C4×Dic53D411yescharacteristic
48C4×Dic53D411yescharacteristic
49C22×D53C2×D411yescharacteristic
50C2×#6157—24no—
51C2×#6157—42no—
52C2×#6157—42no—
53Q8×C53—41no—
54C2×C4×C53—21no—
55C2×C4×C53—21no—
56C2×C4×C53C22⋊C411yescharacteristic
57C2×C4×C53C22⋊C411yescharacteristic
58C2×C4×C53C22×C411yescharacteristic
59C2×Dic53—21no—
60C2×Dic53—21no—
61C2×Dic53C22⋊C411yescharacteristic
62C2×Dic53C22⋊C411yescharacteristic
63C2×Dic53C2×D411yescharacteristic
64C4×D53—22no—
65C4×D53—22no—
66C4×D53—41no—
67Dic106—42no—
68Dic106—41no—
69#88324—42no—
70C22×C53C2×C22⋊C411yescharacteristic
71C2×D53—21no—
72C2×D53—21no—
73C2×D53#13712yes—
74#6157—44no—
75C4×C53—21no—
76C4×C53—21no—
77C4×C53—21no—
78C4×C53—41no—
79C4×C53—41no—
80Dic53—21no—
81Dic53—21no—
82Dic53—41no—
83Dic53—41no—
84Dic53#10412yes—
85#1506—531nomaximal, Sylow
86C2×C53#41011yescharacteristic
87C2×C53#41911yescharacteristic
88C2×C53C2×#10411yescharacteristic
89D53—22no—
90#533—531no—
91#432—531no—
92#376—531no—
93#328—532no—
94#328—532no—
95C53#150611yescharacteristic, complemented, Sylow
96C4⋊Q8—531no—
97C4⋊Q8—531no—
98#125—1061no—
99C22⋊Q8—531no—
100C22⋊Q8—531no—
101C22⋊Q8—1061no—
102C4×Q8—531no—
103C4×Q8—1061no—
104C4×Q8#615711yescharacteristic, complemented
105#117—1061no—
106C2×C42—531no—
107C2×C42—1061no—
108C4⋊C8—1061no—
109C22⋊C8—532no—
110#97—1061no—
111#95—532no—
112#95—1061no—
113C2×Q8—531no—
114C2×Q8—1061no—
115C2×Q8C2×#615711yescharacteristic
116C22×C4—531no—
117C22×C4—531no—
118C22×C4—531no—
119C22×C4—1062no—
120C2×C8—1062no—
121C4⋊C4—21no—
122C4⋊C4—531no—
123C4⋊C4—1061no—
124C4⋊C4—1061no—
125C4⋊C4—1061no—
126C4⋊C4—1061no—
127C4⋊C4—1061no—
128C4⋊C4—1061no—
129C4⋊C4C2×#615711yescharacteristic
130C22⋊C4—1061no—
131C22⋊C4—1061no—
132C42—21no—
133C42—531no—
134C42—531no—
135C42—531no—
136C42—1064no—
137C42—1061no—
138C42C2×#615711yescharacteristic
139C23—531no—
140Q8—41no—
141Q8—2121no—
142Q8—2122no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—531no—
146C2×C4—531no—
147C2×C4—531no—
148C2×C4—1062no—
149C2×C4—1062no—
150C2×C4—1064no—
151C2×C4—1061no—
152C2×C4—1061no—
153C2×C4—2121no—
154C2×C4—2122no—
155C2×C4—2122no—
156C2×C4C22⋊#615711yescharacteristic
157C2×C4C22⋊#615711yescharacteristic
158C2×C4C22×#615711yescharacteristic
159C8—2122no—
160C22—532no—
161C22—1061no—
162C22—1061no—
163C22C2×C22⋊#615711yescharacteristic
164C4—21no—
165C4—21no—
166C4—21no—
167C4—41no—
168C4—41no—
169C4—532no—
170C4—1061no—
171C4—1061no—
172C4—2124no—
173C4—2121no—
174C4—2121no—
175C2—1062no—
176C2#1681160211yescharacteristic
177C2#1681161611yescharacteristic
178C2D53⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC4×Q8×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC4×Q8#6157
hypocenterC53#1506
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
12. #13133216C4semidirect
13. #13133215C4semidirect
19. C4×Dic106C4semidirect
22. C4×Q8×C53C4semidirect
95. C53#1506semidirect
104. C4×Q8#6157semidirect