Finite Group Databasebeta

← #20273945

Subgroups — #20273945

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#16810974C211yesmaximal, characteristic, complemented
3#16810904C211yesmaximal, characteristic, complemented
4#16810828C211yesmaximal, characteristic, complemented
5#16810371C211yesmaximal, characteristic, complemented
6#16810352C211yesmaximal, characteristic, complemented
7C53×D4⋊C8C211yesmaximal, characteristic, complemented
8C2×#11719923—21no—
9#13133217C2211yescharacteristic
10#13133217C2211yescharacteristic
11#13133203—22no—
12#13133191—21no—
13#13133188—21no—
14C4×D212C2211yescharacteristic
15#13133176—21no—
16#13133175—21no—
17#13133151—21no—
18C4⋊#88322C2211yescharacteristic
19#13133142—22no—
20#13133140—21no—
21#13133139—21no—
22C4×D4×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic
24C4×C8×C53C2211yescharacteristic
25C2×D4×C53D411yescharacteristic
26C22×C4×C53—21no—
27C2×C22⋊D53—22no—
28C2×D212D411yescharacteristic
29C2×C4×D53—21no—
30C2×Dic106—21no—
31C2×Dic106D412yes—
32C2×C8×C53—21no—
33C2×C8×C53—21no—
34C4⋊C4×C53D411yescharacteristic
35C22⋊C4×C53—21no—
36C42×C53C2311yescharacteristic
37C22⋊Dic53—22no—
38#11719923—42no—
39#11719921—22no—
40#11719921—21no—
41C4⋊Dic53—21no—
42C4⋊Dic53—21no—
43C4⋊Dic53—21no—
44C4⋊Dic53—21no—
45C4⋊Dic53D411yescharacteristic
46C4×Dic53—22no—
47C2×#88322—21no—
48C23×C53—21no—
49C22×D53—21no—
50D4×C53—21no—
51D4×C53—21no—
52C2×C4×C53—21no—
53C2×C4×C53—41no—
54C2×C4×C53—41no—
55C2×C4×C53C2×D411yescharacteristic
56C2×C4×C53C2×D411yescharacteristic
57C2×C4×C53C2×D411yescharacteristic
58C22⋊D53—82no—
59C2×Dic53—22no—
60C2×Dic53—21no—
61C2×Dic53—41no—
62D212—21no—
63D212—21no—
64C4×D53—41no—
65C4×D53—41no—
66Dic106—42no—
67Dic106—41no—
68Dic106—41no—
69C8×C53—41no—
70C8×C53—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#13511yescharacteristic
82C4×C53#13511yescharacteristic
83C4×C53AGL(1,ℤ/8)11yescharacteristic
84C4×C53#13711yescharacteristic
85Dic53—41no—
86Dic53—42no—
87Dic53—81no—
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#461—531no—
98#333—531no—
99D4⋊C8—531no—
100D4⋊C8D5311yescharacteristic, complemented
101C53#164611yescharacteristic, complemented, Sylow
102C2×SD16—1061no—
103C4⋊Q8—531no—
104C4⋊Q8—531no—
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—1062no—
116#103—1061no—
117#102—1061no—
118C4×C8C2×D5311yescharacteristic
119C2×Q8—532no—
120C2×Q8—1061no—
121C2×D4—531no—
122C2×D4—1062no—
123C2×D4D21211yescharacteristic
124C22×C4—21no—
125C22×C4—1061no—
126SD16—2122no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—1061no—
130C4⋊C4—531no—
131C4⋊C4—1061no—
132C4⋊C4—1061no—
133C4⋊C4—1061no—
134C4⋊C4—1061no—
135C4⋊C4D21211yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—1062no—
138C22⋊C4—1062no—
139C22⋊C4—1061no—
140C42—1062no—
141C42C22×D5311yescharacteristic
142C23—21no—
143C23—1061no—
144Q8—2122no—
145Q8—2121no—
146Q8—2121no—
147D4—21no—
148D4—21no—
149D4—1061no—
150D4—1061no—
151D4—4242no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—1061no—
156C2×C4—1062no—
157C2×C4—2121no—
158C2×C4—2121no—
159C2×C4—2121no—
160C2×C4C2×D21211yescharacteristic
161C2×C4D4×D5311yescharacteristic
162C2×C4D4×D5311yescharacteristic
163C8—41no—
164C8—41no—
165C8—2121no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—2121no—
170C22—2121no—
171C22—2121no—
172C22#1313318711yescharacteristic
173C4—21no—
174C4—41no—
175C4—4241no—
176C4—2121no—
177C4—2122no—
178C4#1313323611yescharacteristic
179C4#1313323911yescharacteristic
180C4#1313324511yescharacteristic
181C4#1313324611yescharacteristic
182C2—81no—
183C2—4241no—
184C2#1681077011yescharacteristic
185C2#1681078611yescharacteristic
186C2#1681080211yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×D4⋊C8C2
socleC22×C53#120
hypercenterD4⋊C8D53
hypocenterC53#1646
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. #16810974C2semidirect
3. #16810904C2semidirect
4. #16810828C2semidirect
5. #16810371C2semidirect
6. #16810352C2semidirect
7. C53×D4⋊C8C2semidirect
100. D4⋊C8D53semidirect
101. C53#1646semidirect