Finite Group Databasebeta

← #20291008

Subgroups — #20291008

Order 6784.

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

normal subgroups only

This group has 250 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
1#16811609C212yesmaximal, complemented
2#16811534C212yesmaximal, complemented
3#16811532C212yesmaximal, complemented
4#16811173C211yesmaximal, characteristic
5D53⋊M16—22no—
6#13133278C2214yes—
7#13133277—22no—
8C2×#11719945C411yescharacteristic
9C2×#11719945C2211yescharacteristic
10#13133217C2211yescharacteristic
11#13133214C411yescharacteristic
12#13133205C411yescharacteristic
13#13133205C2211yescharacteristic
14#13133191—22no—
15C4×D212—22no—
16C2×C42×C53C411yescharacteristic
17C22×C4×C53—21no—
18C22×C4×C53C2×C411yescharacteristic
19C2×C22⋊D53—21no—
20C2×C22⋊D53D412yes—
21#11719945—41no—
22#11719945—41no—
23#11719945—41no—
24#11719945—41no—
25C2×D212—22no—
26C2×D212C2×C411yescharacteristic
27C2×D212C2×C411yescharacteristic
28C2×C4×D53—21no—
29C2×C4×D53D412yes—
30C2×Dic106C2×C411yescharacteristic
31C2×Dic106C2311yescharacteristic
32#11719940—24no—
33#11719936—42no—
34D53⋊C8—42no—
35C42×C53—21no—
36C42×C53C2×C411yescharacteristic
37C42×C53C2×C411yescharacteristic
38#11719921—22no—
39#11719921—22no—
40C4⋊Dic53—22no—
41C23×C53C22⋊C411yescharacteristic
42C22×D53—21no—
43C22×D53C22⋊C412yes—
44C2×C4×C53—21no—
45C2×C4×C53—21no—
46C2×C4×C53—21no—
47C2×C4×C53—21no—
48C2×C4×C53—42no—
49C2×C4×C53C22⋊C411yescharacteristic
50C2×C4×C53C22×C411yescharacteristic
51C22⋊D53—42no—
52C22⋊D53—42no—
53C2×Dic53—21no—
54C2×Dic53C2×D412yes—
55D212—41no—
56D212—41no—
57D212—44no—
58C4×D53—22no—
59C4×D53—22no—
60C4×D53—41no—
61C4×D53—41no—
62Dic106—41no—
63Dic106—41no—
64#88324—44no—
65C22×C53—21no—
66C22×C53—41no—
67C22×C53C2×C22⋊C411yescharacteristic
68C2×D53—22no—
69C2×D53—42no—
70C2×D53—42no—
71C2×D53—41no—
72C4×C53—22no—
73C4×C53—21no—
74C4×C53—41no—
75C4×C53#9911yescharacteristic
76C4×C53#9911yescharacteristic
77Dic53—22no—
78Dic53—41no—
79#2132—531nomaximal, Sylow
80C2×C53—21no—
81C2×C53—41no—
82C2×C53C2×#9911yescharacteristic
83D53—42no—
84D53—81no—
85#532—531no—
86#413—532no—
87#356—532no—
88#355—532no—
89C53#213211yescharacteristic, complemented, Sylow
90C2×#41—531no—
91C2×#41—531no—
92C2×M16—1062no—
93C4⋊Q8—531no—
94#127—531no—
95#124—531no—
96#124—531no—
97#121—1062no—
98C4×D4—1062no—
99C2×C42#615711yescharacteristic
100#101—534no—
101#100—1062no—
102#41—2121no—
103#41—2121no—
104#41—2121no—
105#41—2121no—
106C2×Q8—531no—
107C2×Q8—531no—
108C2×D4—532no—
109C2×D4—531no—
110C2×D4—531no—
111C2×D4—1062no—
112C2×D4—1061no—
113C22×C4—21no—
114C22×C4—532no—
115C22×C4—1061no—
116C22×C4C2×#615711yescharacteristic
117M16—1064no—
118M16—2122no—
119C2×C8—2122no—
120C4⋊C4—1062no—
121C22⋊C4—1062no—
122C22⋊C4—1062no—
123C42—21no—
124C42C2×#615711yescharacteristic
125C42C2×#615711yescharacteristic
126C23—532no—
127C23—1061no—
128C23C22⋊#615711yescharacteristic
129Q8—2121no—
130Q8—2121no—
131D4—2121no—
132D4—2121no—
133D4—2124no—
134D4—2122no—
135D4—2122no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—42no—
141C2×C4—532no—
142C2×C4—1062no—
143C2×C4—1062no—
144C2×C4—1061no—
145C2×C4—2121no—
146C2×C4—2121no—
147C2×C4C22⋊#615711yescharacteristic
148C2×C4C22×#615711yescharacteristic
149C8—2124no—
150C22—21no—
151C22—41no—
152C22—1062no—
153C22—2122no—
154C22—2122no—
155C22—2121no—
156C22C2×C22⋊#615711yescharacteristic
157C4—22no—
158C4—21no—
159C4—41no—
160C4—1062no—
161C4—2121no—
162C4#1313327611yescharacteristic
163C4#1313327611yescharacteristic
164C2—21no—
165C2—41no—
166C2—4241no—
167C2—2122no—
168C2C2×#1313327611yescharacteristic
subgroupG/H
centerC4#13133276
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC2×C42×C53C4
socleC2×C53C2×#99
hypercenterC2×C42#6157
hypocenterC53#2132
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. #16811609C2semidirect
2. #16811534C2semidirect
3. #16811532C2semidirect
89. C53#2132semidirect