Finite Group Databasebeta

← #20273703

Subgroups — #20273703

Order 6784.

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

normal subgroups only

This group has 280 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#16811185C211yesmaximal, characteristic
2#16810937C212yesmaximal, complemented
3#16810548C211yesmaximal, characteristic, complemented
4#16810358C212yesmaximal, complemented
5C53×#323C211yesmaximal, characteristic, complemented
6C22×C22⋊D53—21no—
7#13133201—22no—
8#13133191—21no—
9#13133191C2212yes—
10#13133187—22no—
11C2×C4⋊Dic53C2211yescharacteristic, complemented
12C2×C22⋊Dic53—21no—
13#13133151—22no—
14#13133139—22no—
15#13133124—21no—
16#13133124C2211yescharacteristic, complemented
17C2×C22⋊C4×C53C2211yescharacteristic
18C22⋊C8×C53C2212yescomplemented
19C24×C53D411yescharacteristic
20C23×D53—21no—
21C22×C4×C53D411yescharacteristic
22C22×C4×C53C2311yescharacteristic
23C2×C22⋊D53—22no—
24C2×C22⋊D53—22no—
25C2×C22⋊D53—42no—
26C2×C22⋊D53—42no—
27C22×Dic53—21no—
28C22×Dic53—21no—
29C22×Dic53—21no—
30C2×D212—21no—
31C2×D212D412yes—
32C2×C8×C53—22no—
33C22⋊C4×C53—22no—
34C22⋊Dic53—42no—
35#11719921—22no—
36#11719921—22no—
37C4⋊Dic53—21no—
38C4⋊Dic53—22no—
39C4⋊Dic53D412yes—
40C23×C53—21no—
41C23×C53—21no—
42C23×C53—41no—
43C23×C53—41no—
44C23×C53D812yes—
45C23×C53C2×D411yescharacteristic
46C22×D53—22no—
47C22×D53—42no—
48C22×D53—81no—
49C22×D53—81no—
50C2×C4×C53—21no—
51C2×C4×C53—41no—
52C2×C4×C53—41no—
53C2×C4×C53C2×D412yes—
54C22⋊D53—44no—
55C22⋊D53—42no—
56C22⋊D53—42no—
57C2×Dic53—22no—
58C2×Dic53—42no—
59C2×Dic53—41no—
60C2×Dic53—41no—
61C2×Dic53—41no—
62C2×Dic53—41no—
63C2×Dic53—41no—
64C2×Dic53—41no—
65D212—42no—
66D212—81no—
67C8×C53—42no—
68C22×C53—21no—
69C22×C53—21no—
70C22×C53—22no—
71C22×C53—42no—
72C22×C53—41no—
73C22×C53—41no—
74C22×C53—81no—
75C22×C53#12011yescharacteristic
76C22×C53C2×D811yescharacteristic
77C22×C53AGL(1,ℤ/8)11yescharacteristic
78C2×D53—42no—
79C2×D53—42no—
80C2×D53—42no—
81C2×D53—84no—
82C4×C53—22no—
83C4×C53—41no—
84Dic53—42no—
85Dic53—81no—
86Dic53—81no—
87#1607—531nomaximal, Sylow
88C2×C53—21no—
89C2×C53—21no—
90C2×C53—42no—
91C2×C53#44711yescharacteristic
92C2×C53#45511yescharacteristic
93C2×C53#45811yescharacteristic
94D53—82no—
95#534—531no—
96#480—532no—
97#397—531no—
98#327—532no—
99#323D5311yescharacteristic, complemented
100C53#160711yescharacteristic, complemented, Sylow
101C22×D4—1061no—
102#123—1062no—
103#121—532no—
104#121—1061no—
105#120—1062no—
106C2×C4⋊C4—531no—
107C2×C22⋊C4—1061no—
108C2×C22⋊C4C2×D5311yescharacteristic
109#107—1062no—
110#102—1062no—
111C22⋊C8C2×D5312yescomplemented
112#95—531no—
113#95—1061no—
114C24—1061no—
115C24D21211yescharacteristic
116C2×D4—532no—
117C2×D4—1061no—
118C2×D4—1062no—
119C2×D4—1062no—
120C2×D4—2122no—
121C2×D4—2122no—
122C22×C4—1061no—
123C22×C4—1061no—
124C22×C4—1061no—
125C22×C4D21211yescharacteristic
126C22×C4C22×D5311yescharacteristic
127C2×C8—22no—
128C4⋊C4—532no—
129C4⋊C4—1061no—
130C4⋊C4—1062no—
131C22⋊C4—22no—
132C22⋊C4—1062no—
133C22⋊C4—1062no—
134C22⋊C4—2122no—
135C23—21no—
136C23—21no—
137C23—41no—
138C23—41no—
139C23—1062no—
140C23—4241no—
141C23—4241no—
142C23—2122no—
143C23D42412yes—
144C23C2×D21211yescharacteristic
145D4—4241no—
146D4—2122no—
147D4—2124no—
148D4—2122no—
149D4—2122no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—1062no—
154C2×C4—2121no—
155C2×C4—2121no—
156C2×C4—2121no—
157C2×C4—2121no—
158C2×C4—2121no—
159C2×C4—2121no—
160C2×C4—2122no—
161C2×C4D4×D5312yes—
162C8—42no—
163C22—21no—
164C22—21no—
165C22—22no—
166C22—42no—
167C22—41no—
168C22—41no—
169C22—81no—
170C22—4244no—
171C22—2122no—
172C22—2122no—
173C22—2122no—
174C22#1313318711yescharacteristic
175C22C2×D42411yescharacteristic
176C22#1313324511yescharacteristic
177C4—22no—
178C4—41no—
179C4—4241no—
180C4—4241no—
181C4—2122no—
182C2—21no—
183C2—21no—
184C2—42no—
185C2—4242no—
186C2#1681075911yescharacteristic
187C2#1681080311yescharacteristic
188C2#1681081411yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#323C2
socleC22×C53#120
hypercenter#323D53
hypocenterC53#1607
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #16810937C2semidirect
3. #16810548C2semidirect
4. #16810358C2semidirect
5. C53×#323C2semidirect
11. C2×C4⋊Dic53C22semidirect
16. #13133124C22semidirect
18. C22⋊C8×C53C22semidirect
99. #323D53semidirect
100. C53#1607semidirect
111. C22⋊C8C2×D53semidirect