Finite Group Databasebeta

← #20274013

Subgroups — #20274013

Order 6784.

206 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 49 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#16811013C211yesmaximal, characteristic
2#16810989C212yesmaximal, complemented
3#16810365C212yesmaximal, complemented
4#16810363C211yesmaximal, characteristic, complemented
5C53×#563C211yesmaximal, characteristic, complemented
6C2×#11719924—21no—
7C2×#11719922—21no—
8#13133214C2211yescharacteristic
9#13133137—22no—
10#13133137—22no—
11#13133128C2211yescharacteristic
12#13133128C2212yes—
13C4⋊Q8×C53—21no—
14#126×C53—22no—
15#125×C53C2212yes—
16#124×C53C2211yescharacteristic
17#123×C53—21no—
18C22⋊Q8×C53—22no—
19C2×Q8×C53—21no—
20C2×Q8×C53D411yescharacteristic
21C2×D4×C53D411yescharacteristic
22C22×C4×C53—21no—
23C2×D212—22no—
24C2×D212—21no—
25C4⋊C4×C53—22no—
26C4⋊C4×C53—22no—
27C4⋊C4×C53—22no—
28C4⋊C4×C53D412yes—
29C4⋊C4×C53D412yes—
30C22⋊C4×C53—22no—
31C22⋊C4×C53—21no—
32C22⋊C4×C53—22no—
33C42×C53—21no—
34C42×C53C2311yescharacteristic
35#11719924—42no—
36#11719922—42no—
37C2×#88322—21no—
38C2×#88322—22no—
39C23×C53—21no—
40C22×D53—41no—
41Q8×C53—41no—
42Q8×C53—81no—
43D4×C53—41no—
44C2×C4×C53—22no—
45C2×C4×C53—21no—
46C2×C4×C53—22no—
47C2×C4×C53—81no—
48C2×C4×C53C2×D411yescharacteristic
49C2×C4×C53C2×D412yes—
50D212—44no—
51D212—41no—
52D212—41no—
53#88322—22no—
54#88322—42no—
55C22×C53—41no—
56C22×C53—42no—
57C22×C53#12011yescharacteristic
58C2×D53—81no—
59C2×D53—82no—
60C4×C53—21no—
61C4×C53—22no—
62C4×C53—42no—
63C4×C53—41no—
64C4×C53—42no—
65#1657—531nomaximal, Sylow
66C2×C53—81no—
67C2×C53#45311yescharacteristic
68C2×C53#45412yes—
69D53—161no—
70#563D5311yescharacteristic, complemented
71#496—531no—
72#490—532no—
73#330—532no—
74#329—531no—
75C53#165711yescharacteristic, complemented, Sylow
76C2×SD16—1061no—
77C2×D8—1061no—
78C4⋊Q8—21no—
79#127—531no—
80#126—22no—
81#125C2×D5312yes—
82#124C2×D5311yescharacteristic
83#123—21no—
84C22⋊Q8—22no—
85#102—1062no—
86#102—1062no—
87#97—532no—
88#97—531no—
89C2×Q8—21no—
90C2×Q8C22⋊D5311yescharacteristic
91C2×D4—1062no—
92C2×D4—1061no—
93C2×D4C22⋊D5311yescharacteristic
94C22×C4—21no—
95SD16—2122no—
96D8—2122no—
97C2×C8—1061no—
98C2×C8—1062no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4C22⋊D5312yes—
103C4⋊C4C22⋊D5312yes—
104C22⋊C4—22no—
105C22⋊C4—21no—
106C22⋊C4—22no—
107C42—21no—
108C42C22×D5311yescharacteristic
109C23—21no—
110C23—2121no—
111Q8—41no—
112Q8—81no—
113D4—41no—
114D4—2121no—
115D4—2121no—
116D4—2124no—
117C2×C4—22no—
118C2×C4—21no—
119C2×C4—22no—
120C2×C4—81no—
121C2×C4C2×C22⋊D5312yes—
122C2×C4C2×C22⋊D5311yescharacteristic
123C8—1062no—
124C8—2122no—
125C22—41no—
126C22—42no—
127C22—4241no—
128C22—4242no—
129C22#1313318611yescharacteristic
130C4—21no—
131C4—22no—
132C4—42no—
133C4—41no—
134C4—42no—
135C2—81no—
136C2—8481no—
137C2#1681079111yescharacteristic
138C2#1681079612yes—
subgroupG/H
centerC22#13133186
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#563C2
socleC22×C53#120
hypercenter#563D53
hypocenterC53#1657
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
2. #16810989C2semidirect
3. #16810365C2semidirect
4. #16810363C2semidirect
5. C53×#563C2semidirect
70. #563D53semidirect
75. C53#1657semidirect