Finite Group Databasebeta

← #20273851

Subgroups — #20273851

Order 6784.

214 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 57 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#16811330C211yesmaximal, characteristic
2#16811003C211yesmaximal, characteristic, complemented
3#16810919C211yesmaximal, characteristic, complemented
4#16810913C211yesmaximal, characteristic
5#16810371C211yesmaximal, characteristic, complemented
6#16810355C211yesmaximal, characteristic, complemented
7C53×D4⋊C8C211yesmaximal, characteristic
8C2×Dic212—22no—
9#13133217C2211yescharacteristic
10#13133217C2211yescharacteristic
11#13133208—21no—
12#13133196—21no—
13#13133194—22no—
14C4×Dic106C2211yescharacteristic
15D4×Dic53—21no—
16C2×C4⋊Dic53—21no—
17#13133151—21no—
18#13133148—21no—
19C4⋊#88322C2211yescharacteristic
20#13133142—21no—
21#13133138—21no—
22C4×D4×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic
24C4×C8×C53C2211yescharacteristic
25C2×D4×C53D411yescharacteristic
26C22×C4×C53—21no—
27C22×Dic53—22no—
28C2×Dic106—21no—
29C2×Dic106D411yescharacteristic
30C2×Dic106D412yes—
31C2×C8×C53—21no—
32C2×C8×C53—21no—
33C4⋊C4×C53D411yescharacteristic
34C22⋊C4×C53—21no—
35C42×C53C2311yescharacteristic
36C22⋊Dic53—22no—
37C4⋊Dic53—22no—
38C4⋊Dic53—21no—
39C4⋊Dic53—21no—
40C4⋊Dic53—21no—
41C4⋊Dic53—21no—
42C4⋊Dic53D411yescharacteristic
43#11719919—22no—
44#11719919—21no—
45C4×Dic53—21no—
46C2×#88322—21no—
47Dic212—44no—
48C23×C53—21no—
49D4×C53—21no—
50D4×C53Q1612yescomplemented
51C2×C4×C53—21no—
52C2×C4×C53—41no—
53C2×C4×C53—41no—
54C2×C4×C53C2×D411yescharacteristic
55C2×C4×C53C2×D411yescharacteristic
56C2×C4×C53C2×D411yescharacteristic
57C2×Dic53—22no—
58C2×Dic53—21no—
59C2×Dic53—21no—
60C2×Dic53—41no—
61C2×Dic53—82no—
62Dic106—21no—
63Dic106—21no—
64Dic106—42no—
65Dic106—42no—
66C8×C53—22no—
67C8×C53—41no—
68#88322—41no—
69C22×C53—22no—
70C22×C53—41no—
71C22×C53—41no—
72C22×C53#12011yescharacteristic
73C4×C53—21no—
74C4×C53—41no—
75C4×C53C2×Q1611yescharacteristic
76C4×C53#13511yescharacteristic
77C4×C53#13711yescharacteristic
78C4×C53#13711yescharacteristic
79Dic53—41no—
80Dic53—42no—
81Dic53—41no—
82Dic53—81no—
83#1632—531nomaximal, Sylow
84C2×C53—42no—
85C2×C53#45111yescharacteristic
86C2×C53#45211yescharacteristic
87C2×C53#45311yescharacteristic
88#554—531no—
89#494—531no—
90#477—531no—
91#476—531no—
92#333—531no—
93Q8⋊C8—531no—
94D4⋊C8D5311yescharacteristic
95C53#163211yescharacteristic, complemented, Sylow
96C2×Q16—1062no—
97C4⋊Q8—531no—
98C4⋊Q8—531no—
99#125—1061no—
100C22⋊Q8—1062no—
101C22⋊Q8—1061no—
102C4×Q8—531no—
103C4×D4—1061no—
104C4×D4C2×D5311yescharacteristic
105C2×C4⋊C4—1061no—
106#107—1061no—
107#106—1061no—
108C4⋊C8—531no—
109C4⋊C8C2×D5311yescharacteristic
110#103—1061no—
111#102—1061no—
112C4×C8C2×D5311yescharacteristic
113C2×Q8—531no—
114C2×Q8—532no—
115C2×Q8—1061no—
116C2×D4D21211yescharacteristic
117C22×C4—21no—
118C22×C4—1062no—
119Q16—2124no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—1061no—
123C4⋊C4—531no—
124C4⋊C4—1062no—
125C4⋊C4—1062no—
126C4⋊C4—1061no—
127C4⋊C4—1061no—
128C4⋊C4—1061no—
129C4⋊C4—1061no—
130C4⋊C4—1061no—
131C4⋊C4D21211yescharacteristic
132C22⋊C4—21no—
133C22⋊C4—1062no—
134C42—1061no—
135C42C22×D5311yescharacteristic
136C23—21no—
137Q8—1061no—
138Q8—1061no—
139Q8—2122no—
140Q8—2122no—
141D4—21no—
142D4Dic21212yescomplemented
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—1061no—
147C2×C4—1061no—
148C2×C4—1062no—
149C2×C4—4242no—
150C2×C4—2121no—
151C2×C4C2×D21211yescharacteristic
152C2×C4D4×D5311yescharacteristic
153C2×C4D4×D5311yescharacteristic
154C8—22no—
155C8—41no—
156C8—2121no—
157C22—22no—
158C22—41no—
159C22—41no—
160C22#1313318711yescharacteristic
161C4—21no—
162C4—41no—
163C4—4241no—
164C4—2121no—
165C4—2121no—
166C4—2122no—
167C4C2×Dic21211yescharacteristic
168C4#1313323611yescharacteristic
169C4#1313324611yescharacteristic
170C4#1313325011yescharacteristic
171C2—42no—
172C2#1681078111yescharacteristic
173C2#1681078611yescharacteristic
174C2#1681079211yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×D4⋊C8C2
socleC22×C53#120
hypercenterD4⋊C8D53
hypocenterC53#1632
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #16811003C2semidirect
3. #16810919C2semidirect
5. #16810371C2semidirect
6. #16810355C2semidirect
50. D4×C53Q16semidirect
95. C53#1632semidirect
142. D4Dic212semidirect