Finite Group Databasebeta

← #20273790

Subgroups — #20273790

Order 6784.

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

normal subgroups only

This group has 236 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#16811333C211yesmaximal, characteristic, complemented
2#16810996C211yesmaximal, characteristic, complemented
3#16810823C211yesmaximal, characteristic, complemented
4#16810819C211yesmaximal, characteristic
5#16810371C211yesmaximal, characteristic, complemented
6#16810352C211yesmaximal, characteristic, complemented
7C53×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×#11719913—22no—
9C2×#11719924—21no—
10C2×D424—21no—
11#13133217C2211yescharacteristic
12#13133214C2211yescharacteristic
13#13133209—21no—
14#13133195—22no—
15#13133193—21no—
16C4×D212—21no—
17C4×D212C2211yescharacteristic
18C4⋊C4×D53—21no—
19C4⋊#88322C2211yescharacteristic
20#13133139—21no—
21#13133137—21no—
22C4×Q8×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic
24C4×C8×C53C2211yescharacteristic
25C2×Q8×C53D411yescharacteristic
26C2×D212—21no—
27C2×D212—21no—
28C2×D212—21no—
29C2×D212D411yescharacteristic
30C2×C4×D53—21no—
31C2×C4×D53—22no—
32C2×Dic106D412yes—
33C2×C8×C53—21no—
34C2×C8×C53—21no—
35C4⋊C4×C53—21no—
36C4⋊C4×C53D411yescharacteristic
37C42×C53—21no—
38C42×C53C2311yescharacteristic
39#11719924—42no—
40#11719921—22no—
41#11719921—21no—
42C4⋊Dic53—21no—
43C4⋊Dic53—21no—
44C4⋊Dic53D411yescharacteristic
45#11719919—22no—
46#11719919—22no—
47C2×#88322—21no—
48D424—42no—
49#11719913—44no—
50C22×D53—21no—
51C22×D53—41no—
52Q8×C53—21no—
53Q8×C53—21no—
54C2×C4×C53—21no—
55C2×C4×C53—21no—
56C2×C4×C53C2×D411yescharacteristic
57C2×C4×C53C2×D411yescharacteristic
58C2×C4×C53C2×D411yescharacteristic
59C2×Dic53—22no—
60C2×Dic53—21no—
61D212—21no—
62D212—42no—
63D212—41no—
64D212—41no—
65D212—41no—
66D212—41no—
67D212SD1612yescomplemented
68C4×D53—41no—
69C4×D53—41no—
70C4×D53—82no—
71Dic106—42no—
72C8×C53—22no—
73C8×C53—41no—
74#88322—41no—
75C22×C53#12011yescharacteristic
76C2×D53—22no—
77C2×D53—41no—
78C2×D53—41no—
79C2×D53—81no—
80C2×D53—81no—
81C2×D53—81no—
82C4×C53—21no—
83C4×C53—41no—
84C4×C53—41no—
85C4×C53C2×SD1611yescharacteristic
86C4×C53#13511yescharacteristic
87C4×C53AGL(1,ℤ/8)11yescharacteristic
88C4×C53#13711yescharacteristic
89Dic53—41no—
90Dic53—42no—
91#1622—531nomaximal, Sylow
92C2×C53#44811yescharacteristic
93C2×C53#44911yescharacteristic
94C2×C53#45311yescharacteristic
95D53—42no—
96D53—161no—
97#554—531no—
98#492—531no—
99#460—531no—
100#459—531no—
101#333—531no—
102Q8⋊C8D5311yescharacteristic, complemented
103D4⋊C8—531no—
104C53#162211yescharacteristic, complemented, Sylow
105C2×SD16—1062no—
106C2×SD16—1061no—
107C2×D8—1061no—
108C4⋊Q8—531no—
109#127—531no—
110#125—1061no—
111C22⋊Q8—1062no—
112C22⋊Q8—1061no—
113C4×Q8C2×D5311yescharacteristic
114C4×D4—531no—
115C4×D4—1061no—
116C2×C4⋊C4—1061no—
117C4⋊C8—531no—
118C4⋊C8C2×D5311yescharacteristic
119#102—1061no—
120#102—1061no—
121C4×C8C2×D5311yescharacteristic
122C2×Q8—532no—
123C2×Q8D21211yescharacteristic
124C2×D4—531no—
125C2×D4—1061no—
126C2×D4—1061no—
127C2×D4—1061no—
128C22×C4—1062no—
129C22×C4—1061no—
130SD16—2124no—
131SD16—2122no—
132D8—2122no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—1061no—
136C4⋊C4—21no—
137C4⋊C4—531no—
138C4⋊C4—1062no—
139C4⋊C4—1062no—
140C4⋊C4—1061no—
141C4⋊C4—1061no—
142C4⋊C4D21211yescharacteristic
143C22⋊C4—1062no—
144C22⋊C4—1061no—
145C42—21no—
146C42C22×D5311yescharacteristic
147C23—1061no—
148C23—2121no—
149Q8—21no—
150Q8—21no—
151Q8—2122no—
152D4—532no—
153D4—1061no—
154D4—2122no—
155D4—2121no—
156D4—2121no—
157D4—2121no—
158D4—2121no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—1061no—
162C2×C4—1062no—
163C2×C4—4242no—
164C2×C4—2121no—
165C2×C4—2121no—
166C2×C4C2×D21211yescharacteristic
167C2×C4D4×D5311yescharacteristic
168C2×C4D4×D5311yescharacteristic
169C8—22no—
170C8—41no—
171C8—2121no—
172C22—1062no—
173C22—4241no—
174C22—4241no—
175C22—4241no—
176C22—2121no—
177C22—2121no—
178C22#1313318711yescharacteristic
179C4—21no—
180C4—41no—
181C4—41no—
182C4—2121no—
183C4—2122no—
184C4SD16×D5311yescharacteristic
185C4#1313323911yescharacteristic
186C4#1313324511yescharacteristic
187C4#1313324911yescharacteristic
188C2—8481no—
189C2—2122no—
190C2#1681076311yescharacteristic
191C2#1681077011yescharacteristic
192C2#1681079211yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×Q8⋊C8C2
socleC22×C53#120
hypercenterQ8⋊C8D53
hypocenterC53#1622
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
1. #16811333C2semidirect
2. #16810996C2semidirect
3. #16810823C2semidirect
5. #16810371C2semidirect
6. #16810352C2semidirect
7. C53×Q8⋊C8C2semidirect
67. D212SD16semidirect
102. Q8⋊C8D53semidirect
104. C53#1622semidirect