Finite Group Databasebeta

← #20273788

Subgroups — #20273788

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#16811334C211yesmaximal, characteristic, complemented
2#16810997C211yesmaximal, characteristic, complemented
3#16810824C211yesmaximal, characteristic, complemented
4#16810819C211yesmaximal, characteristic
5#16810353C211yesmaximal, characteristic, complemented
6#16810350C211yesmaximal, characteristic, complemented
7C53×#333C211yesmaximal, characteristic, complemented
8C2×#11719913—21no—
9C2×#11719924—22no—
10C2×D424—21no—
11#13133214C2211yescharacteristic
12#13133208—21no—
13#13133197—21no—
14#13133193—22no—
15C4×Dic106C2211yescharacteristic
16C4×D212C2211yescharacteristic
17#13133179—21no—
18C4⋊C4×D53—21no—
19#13133139—21no—
20#13133139—21no—
21C4×#88322C2211yescharacteristic
22C4⋊Q8×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic
24C4⋊C8×C53C2211yescharacteristic
25C2×Q8×C53D412yes—
26C2×D212—21no—
27C2×D212—21no—
28C2×D212—21no—
29C2×D212D411yescharacteristic
30C2×C4×D53—21no—
31C2×C4×D53—22no—
32C2×Dic106D411yescharacteristic
33C2×C8×C53—21no—
34C2×C8×C53—21no—
35C4⋊C4×C53—21no—
36C4⋊C4×C53—21no—
37C42×C53C2311yescharacteristic
38#11719924—44no—
39#11719921—22no—
40#11719921—21no—
41C4⋊Dic53—22no—
42C4⋊Dic53D411yescharacteristic
43C4⋊Dic53D411yescharacteristic
44#11719919—22no—
45#11719919—21no—
46C4×Dic53—21no—
47C2×#88322—21no—
48D424—42no—
49#11719913—42no—
50C22×D53—21no—
51C22×D53—41no—
52Q8×C53—42no—
53C2×C4×C53—22no—
54C2×C4×C53C2×D411yescharacteristic
55C2×C4×C53C2×D411yescharacteristic
56C2×C4×C53C2×D411yescharacteristic
57C2×Dic53—21no—
58C2×Dic53—21no—
59C2×Dic53—21no—
60D212—21no—
61D212—42no—
62D212—41no—
63D212—41no—
64D212—41no—
65D212—41no—
66D212SD1612yescomplemented
67C4×D53—41no—
68C4×D53—41no—
69C4×D53—82no—
70Dic106—21no—
71Dic106—21no—
72C8×C53—41no—
73C8×C53—41no—
74#88322—22no—
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—42no—
84C4×C53C2×SD1611yescharacteristic
85C4×C53#13511yescharacteristic
86C4×C53AGL(1,ℤ/8)11yescharacteristic
87C4×C53#13711yescharacteristic
88Dic53—41no—
89Dic53—41no—
90Dic53—41no—
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#333D5311yescharacteristic, complemented
102Q8⋊C8—531no—
103D4⋊C8—531no—
104C53#162211yescharacteristic, complemented, Sylow
105C2×SD16—1061no—
106C2×SD16—1062no—
107C2×D8—1061no—
108C4⋊Q8C2×D5311yescharacteristic
109#127—531no—
110#125—1061no—
111C22⋊Q8—1061no—
112C22⋊Q8—1062no—
113C4×Q8—531no—
114C4×D4—531no—
115C4×D4—1061no—
116C2×C4⋊C4—1061no—
117C4⋊C8C2×D5311yescharacteristic
118C4⋊C8C2×D5311yescharacteristic
119#102—1061no—
120#102—1061no—
121C4×C8—531no—
122C2×Q8—531no—
123C2×Q8D21212yes—
124C2×D4—531no—
125C2×D4—1061no—
126C2×D4—1061no—
127C2×D4—1061no—
128C22×C4—1062no—
129C22×C4—1061no—
130SD16—2122no—
131SD16—2124no—
132D8—2122no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—1061no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—531no—
139C4⋊C4—531no—
140C4⋊C4—1062no—
141C4⋊C4—1062no—
142C4⋊C4—1061no—
143C22⋊C4—1062no—
144C22⋊C4—1061no—
145C42—1061no—
146C42C22×D5311yescharacteristic
147C23—1061no—
148C23—2121no—
149Q8—42no—
150Q8—1061no—
151Q8—1061no—
152D4—532no—
153D4—1061no—
154D4—2122no—
155D4—2121no—
156D4—2121no—
157D4—2121no—
158D4—2121no—
159C2×C4—22no—
160C2×C4—1061no—
161C2×C4—1061no—
162C2×C4—1061no—
163C2×C4—4242no—
164C2×C4—2121no—
165C2×C4—2121no—
166C2×C4C2×D21211yescharacteristic
167C2×C4D4×D5311yescharacteristic
168C2×C4D4×D5311yescharacteristic
169C8—41no—
170C8—41no—
171C8—1062no—
172C22—1062no—
173C22—4241no—
174C22—4241no—
175C22—4241no—
176C22—2121no—
177C22—2121no—
178C22#1313318711yescharacteristic
179C4—21no—
180C4—42no—
181C4—2121no—
182C4—2121no—
183C4—2121no—
184C4SD16×D5311yescharacteristic
185C4#1313324011yescharacteristic
186C4#1313324411yescharacteristic
187C4#1313324911yescharacteristic
188C2—8481no—
189C2—2122no—
190C2#1681076311yescharacteristic
191C2#1681077111yescharacteristic
192C2#1681079311yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#333C2
socleC22×C53#120
hypercenter#333D53
hypocenterC53#1622
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
1. #16811334C2semidirect
2. #16810997C2semidirect
3. #16810824C2semidirect
5. #16810353C2semidirect
6. #16810350C2semidirect
7. C53×#333C2semidirect
66. D212SD16semidirect
101. #333D53semidirect
104. C53#1622semidirect