Finite Group Databasebeta

← #20276666

Subgroups — #20276666

Order 6784.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 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#16811063C212yesmaximal, complemented
2#16810881C211yesmaximal, characteristic, complemented
3#16810433C212yesmaximal, complemented
4#16810405C211yesmaximal, characteristic, complemented
5C53×#575C211yesmaximal, characteristic
6C2×Dic212C2211yescharacteristic, complemented
7#13133196—21no—
8#13133163—22no—
9#13133161—22no—
10#13133157C2212yescomplemented
11#13133149C2211yescharacteristic, complemented
12#13133142—21no—
13C2×#41×C53—21no—
14#137×C53—22no—
15AGL(1,ℤ/8)×C53—22no—
16#135×C53—21no—
17#135×C53C2212yes—
18C2×M16×C53C2211yescharacteristic
19#41×C53—22no—
20#41×C53—22no—
21#41×C53—41no—
22#41×C53—41no—
23C2×Q8×C53—21no—
24C2×D4×C53—21no—
25C2×D4×C53—41no—
26C22×C4×C53—41no—
27C22×C4×C53D411yescharacteristic
28C2×Dic106—21no—
29Q16×C53—21no—
30Q16×C53D412yes—
31SD16×C53—22no—
32SD16×C53—22no—
33D8×C53—21no—
34D8×C53D412yes—
35M16×C53—22no—
36C2×C8×C53D411yescharacteristic
37C2×C8×C53C2311yescharacteristic
38C22⋊Dic53—41no—
39C4⋊Dic53—21no—
40#11719919—41no—
41Dic212—22no—
42#11719908—22no—
43C23×C53—41no—
44C23×C53D811yescharacteristic
45Q8×C53—22no—
46Q8×C53—41no—
47D4×C53—22no—
48D4×C53—41no—
49D4×C53—42no—
50D4×C53—42no—
51C2×C4×C53—21no—
52C2×C4×C53—21no—
53C2×C4×C53—42no—
54C2×C4×C53—41no—
55C2×C4×C53—42no—
56C2×C4×C53D811yescharacteristic
57C2×C4×C53C2×D411yescharacteristic
58C2×Dic53—41no—
59C2×Dic53—41no—
60Dic106—41no—
61Dic106—41no—
62C8×C53—21no—
63C8×C53C2×D412yes—
64C22×C53—21no—
65C22×C53—41no—
66C22×C53—42no—
67C22×C53—82no—
68C22×C53C2×D811yescharacteristic
69C4×C53—21no—
70C4×C53—42no—
71C4×C53#12011yescharacteristic
72C4×C53AGL(1,ℤ/8)11yescharacteristic
73Dic53—81no—
74Dic53—81no—
75#2191—531nomaximal, Sylow
76C2×C53—21no—
77C2×C53—41no—
78C2×C53—82no—
79C2×C53#44711yescharacteristic
80#575D5311yescharacteristic
81#510—532no—
82#470—531no—
83#360—532no—
84#349—531no—
85C53#219111yescharacteristic, complemented, Sylow
86C2×#41—21no—
87#137—22no—
88AGL(1,ℤ/8)—22no—
89#135—21no—
90#135C2×D5312yes—
91C2×Q16—531no—
92C2×M16C2×D5311yescharacteristic
93C22⋊Q8—1061no—
94Q32—1062no—
95SD32—1062no—
96M32—532no—
97#106—531no—
98#103—1061no—
99#41—22no—
100#41—22no—
101#41—41no—
102#41—41no—
103C2×Q8—21no—
104C2×Q8—1061no—
105C2×D4—21no—
106C2×D4—41no—
107C22×C4—41no—
108C22×C4C22⋊D5311yescharacteristic
109Q16—21no—
110Q16—1062no—
111Q16C22⋊D5312yes—
112SD16—22no—
113SD16—22no—
114D8—21no—
115D8C22⋊D5312yes—
116M16—22no—
117C2×C8C22⋊D5311yescharacteristic
118C2×C8C22×D5311yescharacteristic
119C4⋊C4—1061no—
120C4⋊C4—2121no—
121C22⋊C4—2121no—
122C16—1062no—
123C23—41no—
124C23#1171992211yescharacteristic
125Q8—22no—
126Q8—41no—
127Q8—2121no—
128Q8—2121no—
129D4—22no—
130D4—41no—
131D4—42no—
132D4—42no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—42no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—2121no—
139C2×C4—2121no—
140C2×C4#1171992211yescharacteristic
141C2×C4C2×C22⋊D5311yescharacteristic
142C8—21no—
143C8C2×C22⋊D5312yes—
144C22—21no—
145C22—41no—
146C22—42no—
147C22—82no—
148C22C2×#1171992211yescharacteristic
149C4—21no—
150C4—42no—
151C4—4241no—
152C4—4241no—
153C4#1313318611yescharacteristic
154C4#1313324311yescharacteristic
155C2—21no—
156C2—41no—
157C2—82no—
158C2#1681075711yescharacteristic
subgroupG/H
centerC2#16810757
commutatorC2×C8×C53C23
FrattiniC2×C8C22×D53
FittingC53×#575C2
socleC2×C53#447
hypercenter#575D53
hypocenterC53#2191
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #16811063C2semidirect
2. #16810881C2semidirect
3. #16810433C2semidirect
4. #16810405C2semidirect
6. C2×Dic212C22semidirect
10. #13133157C22semidirect
11. #13133149C22semidirect
85. C53#2191semidirect