Finite Group Databasebeta

← #20273941

Subgroups — #20273941

Order 6784.

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

normal subgroups only

This group has 212 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#16811338C211yesmaximal, characteristic
2#16810975C211yesmaximal, characteristic, complemented
3#16810900C211yesmaximal, characteristic, complemented
4#16810371C211yesmaximal, characteristic, complemented
5D4⋊#88322C211yesmaximal, characteristic, complemented
6#16810350C211yesmaximal, characteristic, complemented
7C53×#461C211yesmaximal, characteristic, complemented
8#13133217C2211yescharacteristic
9#13133203—22no—
10#13133191—21no—
11#13133188—21no—
12C4×D212C2211yescharacteristic
13#13133176—21no—
14#13133175—21no—
15#13133150—21no—
16C4⋊#88322C2211yescharacteristic
17#13133140—22no—
18#13133137—21no—
19C4×#88322C2211yescharacteristic
20C2×SD16×C53—21no—
21C4⋊Q8×C53C2211yescharacteristic
22C4×D4×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic
24#103×C53—21no—
25C2×Q8×C53—21no—
26C2×D4×C53D411yescharacteristic
27C22×C4×C53—21no—
28C2×C22⋊D53—22no—
29C2×D212D411yescharacteristic
30C2×C4×D53—21no—
31C2×Dic106D412yes—
32SD16×C53—42no—
33C2×C8×C53—21no—
34C4⋊C4×C53—21no—
35C4⋊C4×C53—21no—
36C4⋊C4×C53D411yescharacteristic
37C22⋊C4×C53—21no—
38C42×C53C2311yescharacteristic
39C22⋊Dic53—22no—
40#11719921—22no—
41#11719921—21no—
42C4⋊Dic53—21no—
43C4⋊Dic53—21no—
44C4⋊Dic53D411yescharacteristic
45C4×Dic53—22no—
46C2×#88322—21no—
47C2×#88322—21no—
48C23×C53—21no—
49C22×D53—21no—
50Q8×C53—41no—
51Q8×C53—41no—
52D4×C53—21no—
53D4×C53—21no—
54C2×C4×C53—21no—
55C2×C4×C53—41no—
56C2×C4×C53—41no—
57C2×C4×C53—41no—
58C2×C4×C53C2×D411yescharacteristic
59C2×C4×C53C2×D411yescharacteristic
60C2×C4×C53C2×D411yescharacteristic
61C22⋊D53—82no—
62C2×Dic53—22no—
63C2×Dic53—21no—
64D212—21no—
65D212—21no—
66C4×D53—41no—
67C4×D53—41no—
68Dic106—42no—
69C8×C53—41no—
70#88322—41no—
71#88322—41no—
72C22×C53—41no—
73C22×C53—41no—
74C22×C53—41no—
75C22×C53#12011yescharacteristic
76C2×D53—41no—
77C2×D53—41no—
78C2×D53—41no—
79C4×C53—21no—
80C4×C53—41no—
81C4×C53—81no—
82C4×C53#13511yescharacteristic
83C4×C53#13511yescharacteristic
84C4×C53AGL(1,ℤ/8)11yescharacteristic
85C4×C53#13711yescharacteristic
86Dic53—41no—
87Dic53—42no—
88#1646—531nomaximal, Sylow
89C2×C53—81no—
90C2×C53#44911yescharacteristic
91C2×C53#45211yescharacteristic
92C2×C53#45511yescharacteristic
93D53—81no—
94#555—531no—
95#487—531no—
96#474—531no—
97#461D5311yescharacteristic, complemented
98#333—531no—
99D4⋊C8—531no—
100D4⋊C8—531no—
101C53#164611yescharacteristic, complemented, Sylow
102C2×SD16—21no—
103C4⋊Q8—531no—
104C4⋊Q8C2×D5311yescharacteristic
105#124—1062no—
106#121—1061no—
107#121—1061no—
108C4×D4—531no—
109C4×D4C2×D5311yescharacteristic
110#117—1061no—
111#117—1061no—
112#107—1061no—
113C4⋊C8—531no—
114C4⋊C8C2×D5311yescharacteristic
115#103—21no—
116#103—1062no—
117#102—1061no—
118C4×C8—531no—
119C2×Q8—21no—
120C2×Q8—532no—
121C2×D4—531no—
122C2×D4—1062no—
123C2×D4C22⋊D5311yescharacteristic
124C22×C4—21no—
125C22×C4—1061no—
126SD16—42no—
127C2×C8—21no—
128C2×C8—1061no—
129C2×C8—1061no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—531no—
133C4⋊C4—1061no—
134C4⋊C4—1061no—
135C4⋊C4C22⋊D5311yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—1062no—
138C22⋊C4—1062no—
139C22⋊C4—1061no—
140C42—1062no—
141C42C22×D5311yescharacteristic
142C23—21no—
143C23—1061no—
144Q8—41no—
145Q8—41no—
146Q8—2122no—
147D4—21no—
148D4—21no—
149D4—1061no—
150D4—1061no—
151D4—4242no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—1061no—
157C2×C4—1062no—
158C2×C4—2121no—
159C2×C4—2121no—
160C2×C4D4×D5311yescharacteristic
161C2×C4D4×D5311yescharacteristic
162C2×C4C2×C22⋊D5311yescharacteristic
163C8—41no—
164C8—2121no—
165C8—2121no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—2121no—
170C22—2121no—
171C22—2121no—
172C22#1313318511yescharacteristic
173C4—21no—
174C4—41no—
175C4—81no—
176C4—2121no—
177C4—2122no—
178C4#1313323711yescharacteristic
179C4#1313323811yescharacteristic
180C4#1313324311yescharacteristic
181C4#1313324611yescharacteristic
182C2—81no—
183C2—4241no—
184C2#1681076611yescharacteristic
185C2#1681078211yescharacteristic
186C2#1681079911yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#461C2
socleC22×C53#120
hypercenter#461D53
hypocenterC53#1646
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #16810975C2semidirect
3. #16810900C2semidirect
4. #16810371C2semidirect
5. D4⋊#88322C2semidirect
6. #16810350C2semidirect
7. C53×#461C2semidirect
97. #461D53semidirect
101. C53#1646semidirect