Finite Group Databasebeta

← #20273846

Subgroups — #20273846

Order 6784.

214 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 55 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#16811334C211yesmaximal, characteristic
2#16810915C211yesmaximal, characteristic, complemented
3#16810910C211yesmaximal, characteristic
4#16810372C211yesmaximal, characteristic, complemented
5#16810355C211yesmaximal, characteristic, complemented
6#16810352C211yesmaximal, characteristic
7C53×#494C211yesmaximal, characteristic, complemented
8#13133208—21no—
9#13133197—21no—
10#13133193—22no—
11C4×Dic106C2211yescharacteristic
12C4×D212C2211yescharacteristic
13#13133179—21no—
14C4⋊C4×D53—21no—
15#13133150—21no—
16#13133148—21no—
17C4⋊#88322C2211yescharacteristic
18C4⋊#88322C2211yescharacteristic
19#13133140—21no—
20#13133137—21no—
21C2×Q16×C53—22no—
22C4⋊Q8×C53C2211yescharacteristic
23C4⋊Q8×C53C2211yescharacteristic
24C4×C8×C53C2211yescharacteristic
25C2×Q8×C53—21no—
26C2×Q8×C53D412yes—
27C2×D212D411yescharacteristic
28C2×C4×D53—21no—
29C2×C4×D53—22no—
30C2×Dic106D411yescharacteristic
31Q16×C53—44no—
32C2×C8×C53—21no—
33C4⋊C4×C53—21no—
34C4⋊C4×C53—21no—
35C4⋊C4×C53—21no—
36C4⋊C4×C53—21no—
37C42×C53C2311yescharacteristic
38#11719921—22no—
39#11719921—21no—
40C4⋊Dic53—22no—
41C4⋊Dic53D411yescharacteristic
42C4⋊Dic53D411yescharacteristic
43#11719919—22no—
44#11719919—21no—
45C4×Dic53—21no—
46C2×#88322—21no—
47C2×#88322—21no—
48C22×D53—21no—
49Q8×C53—42no—
50Q8×C53—42no—
51C2×C4×C53—22no—
52C2×C4×C53—41no—
53C2×C4×C53C2×D411yescharacteristic
54C2×C4×C53C2×D411yescharacteristic
55C2×C4×C53C2×D411yescharacteristic
56C2×Dic53—21no—
57C2×Dic53—21no—
58C2×Dic53—21no—
59D212—21no—
60D212Q1612yescomplemented
61C4×D53—41no—
62C4×D53—41no—
63C4×D53—82no—
64Dic106—21no—
65Dic106—21no—
66C8×C53—22no—
67#88322—41no—
68#88322—41no—
69C22×C53#12011yescharacteristic
70C2×D53—22no—
71C2×D53—41no—
72C2×D53—41no—
73C4×C53—21no—
74C4×C53—42no—
75C4×C53—81no—
76C4×C53C2×Q1611yescharacteristic
77C4×C53#13511yescharacteristic
78C4×C53#13711yescharacteristic
79C4×C53#13711yescharacteristic
80Dic53—41no—
81Dic53—41no—
82Dic53—41no—
83#1632—531nomaximal, Sylow
84C2×C53#45111yescharacteristic
85C2×C53#45211yescharacteristic
86C2×C53#45311yescharacteristic
87D53—42no—
88#554—531no—
89#494D5311yescharacteristic, complemented
90#477—531no—
91#476—531no—
92#333—531no—
93Q8⋊C8—531no—
94D4⋊C8—531no—
95C53#163211yescharacteristic, complemented, Sylow
96C2×Q16—22no—
97C4⋊Q8C2×D5311yescharacteristic
98C4⋊Q8C2×D5311yescharacteristic
99#125—1061no—
100C22⋊Q8—1061no—
101C22⋊Q8—1062no—
102C4×Q8—531no—
103C4×D4—531no—
104C4×D4—1061no—
105C2×C4⋊C4—1061no—
106#107—1061no—
107#106—1061no—
108C4⋊C8—531no—
109C4⋊C8—531no—
110#103—1061no—
111#102—1061no—
112C4×C8C2×D5311yescharacteristic
113C2×Q8—21no—
114C2×Q8—531no—
115C2×Q8C22⋊D5312yes—
116C2×D4—531no—
117C22×C4—1062no—
118C22×C4—1061no—
119Q16—44no—
120C2×C8—21no—
121C2×C8—1061no—
122C2×C8—1061no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—531no—
128C4⋊C4—531no—
129C4⋊C4—1062no—
130C4⋊C4—1062no—
131C4⋊C4—1061no—
132C22⋊C4—1062no—
133C22⋊C4—1061no—
134C42—1061no—
135C42C22×D5311yescharacteristic
136C23—1061no—
137Q8—42no—
138Q8—42no—
139Q8—1061no—
140Q8—1061no—
141D4—532no—
142D4—1061no—
143C2×C4—22no—
144C2×C4—41no—
145C2×C4—1061no—
146C2×C4—1061no—
147C2×C4—1061no—
148C2×C4—4242no—
149C2×C4—2121no—
150C2×C4—2121no—
151C2×C4D4×D5311yescharacteristic
152C2×C4D4×D5311yescharacteristic
153C2×C4C2×C22⋊D5311yescharacteristic
154C8—22no—
155C8—2121no—
156C8—2121no—
157C22—1062no—
158C22—2121no—
159C22—2121no—
160C22#1313318511yescharacteristic
161C4—21no—
162C4—42no—
163C4—81no—
164C4—2121no—
165C4—2121no—
166C4—2121no—
167C4Q16×D5311yescharacteristic
168C4#1313324011yescharacteristic
169C4#1313324911yescharacteristic
170C4#1313324911yescharacteristic
171C2—2122no—
172C2#1681077711yescharacteristic
173C2#1681078411yescharacteristic
174C2#1681079011yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#494C2
socleC22×C53#120
hypercenter#494D53
hypocenterC53#1632
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
2. #16810915C2semidirect
4. #16810372C2semidirect
5. #16810355C2semidirect
7. C53×#494C2semidirect
60. D212Q16semidirect
89. #494D53semidirect
95. C53#1632semidirect