Finite Group Databasebeta

← #20274012

Subgroups — #20274012

Order 6784.

206 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 45 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#16811377C211yesmaximal, characteristic, complemented
2#16811010C211yesmaximal, characteristic
3#16810987C211yesmaximal, characteristic, complemented
4#16810365C211yesmaximal, characteristic, complemented
5#16810364C211yesmaximal, characteristic, complemented
6#16810362C211yesmaximal, characteristic, complemented
7C53×#490C211yesmaximal, characteristic, complemented
8C2×#11719923—21no—
9C2×#11719922—21no—
10#13133216—21no—
11#13133212—21no—
12#13133212—21no—
13#13133209C2211yescharacteristic
14#13133205C2211yescharacteristic
15#13133202—21no—
16#13133197—21no—
17#13133195—21no—
18#13133138—21no—
19#13133138—21no—
20#13133128C2211yescharacteristic
21#13133128C2211yescharacteristic
22#127×C53C2211yescharacteristic
23#125×C53C2211yescharacteristic
24#102×C53—21no—
25#102×C53—21no—
26#97×C53C2211yescharacteristic
27C2×D4×C53—21no—
28C2×D4×C53—21no—
29C2×D4×C53—21no—
30C2×D212D411yescharacteristic
31C2×C4×D53—21no—
32C2×Dic106—21no—
33C2×Dic106D411yescharacteristic
34C2×C8×C53—21no—
35C4⋊C4×C53—21no—
36C4⋊C4×C53—21no—
37C4⋊C4×C53D411yescharacteristic
38C4⋊C4×C53D411yescharacteristic
39C42×C53C2311yescharacteristic
40#11719923—42no—
41#11719922—42no—
42#11719921—21no—
43#11719921—21no—
44#11719921—21no—
45#11719921—21no—
46#11719921—21no—
47C4⋊Dic53—21no—
48C4⋊Dic53D411yescharacteristic
49C4⋊Dic53D411yescharacteristic
50#11719919—21no—
51#11719919—21no—
52#11719919—21no—
53C4×Dic53—21no—
54C2×#88322—21no—
55C2×#88322—21no—
56C23×C53—41no—
57C22×D53—21no—
58D4×C53—42no—
59D4×C53—42no—
60D4×C53—41no—
61D4×C53—41no—
62C2×C4×C53—21no—
63C2×C4×C53—21no—
64C2×C4×C53C2×D411yescharacteristic
65C2×C4×C53C2×D411yescharacteristic
66C2×C4×C53C2×D411yescharacteristic
67C2×Dic53—21no—
68C2×Dic53—21no—
69C2×Dic53—21no—
70D212—41no—
71C4×D53—81no—
72Dic106—41no—
73Dic106—81no—
74C8×C53—41no—
75#88322—22no—
76#88322—41no—
77C22×C53—81no—
78C22×C53—81no—
79C22×C53—81no—
80C22×C53#12011yescharacteristic
81C2×D53—41no—
82C2×D53—41no—
83C2×D53—41no—
84C4×C53—21no—
85C4×C53—21no—
86C4×C53—21no—
87C4×C53—41no—
88C4×C53—41no—
89Dic53—41no—
90Dic53—41no—
91Dic53—41no—
92#1657—531nomaximal, Sylow
93C2×C53—161no—
94C2×C53#45311yescharacteristic
95C2×C53#45411yescharacteristic
96C2×C53#45411yescharacteristic
97D53—81no—
98#563—531no—
99#496—531no—
100#490—531no—
101#490D5311yescharacteristic, complemented
102#330—531no—
103#330—531no—
104#329—531no—
105C53#165711yescharacteristic, complemented, Sylow
106C2×SD16—1061no—
107C2×D8—1061no—
108C4⋊Q8—1061no—
109#127C2×D5311yescharacteristic
110#126—1061no—
111#126—1061no—
112#125—531no—
113#125C2×D5311yescharacteristic
114#124—531no—
115#123—1061no—
116C22⋊Q8—1061no—
117C22⋊Q8—1061no—
118#102—21no—
119#102—21no—
120#102—1061no—
121#102—1061no—
122#97—531no—
123#97—531no—
124#97C2×D5311yescharacteristic
125C2×Q8—531no—
126C2×Q8—1061no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—531no—
131C22×C4—1061no—
132SD16—2122no—
133D8—2122no—
134C2×C8—21no—
135C2×C8—1061no—
136C2×C8—1061no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—531no—
140C4⋊C4—531no—
141C4⋊C4—1061no—
142C4⋊C4—1061no—
143C4⋊C4—1061no—
144C4⋊C4—1061no—
145C4⋊C4C22⋊D5311yescharacteristic
146C4⋊C4C22⋊D5311yescharacteristic
147C22⋊C4—1061no—
148C22⋊C4—1061no—
149C22⋊C4—1061no—
150C22⋊C4—1061no—
151C22⋊C4—1061no—
152C42—1061no—
153C42C22×D5311yescharacteristic
154C23—41no—
155C23—1061no—
156Q8—4241no—
157Q8—2121no—
158D4—41no—
159D4—41no—
160D4—42no—
161D4—42no—
162D4—2121no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—1061no—
166C2×C4—1061no—
167C2×C4—1061no—
168C2×C4—4241no—
169C2×C4D4×D5311yescharacteristic
170C2×C4D4×D5311yescharacteristic
171C2×C4C2×C22⋊D5311yescharacteristic
172C8—41no—
173C8—1062no—
174C8—2121no—
175C22—81no—
176C22—81no—
177C22—81no—
178C22—2121no—
179C22—2121no—
180C22—2121no—
181C22#1313318511yescharacteristic
182C4—21no—
183C4—21no—
184C4—21no—
185C4—41no—
186C4—41no—
187C4—2121no—
188C4—2121no—
189C4—2121no—
190C2—161no—
191C2—4241no—
192C2#1681078911yescharacteristic
193C2#1681079411yescharacteristic
194C2#1681079511yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#490C2
socleC22×C53#120
hypercenter#490D53
hypocenterC53#1657
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #16811377C2semidirect
3. #16810987C2semidirect
4. #16810365C2semidirect
5. #16810364C2semidirect
6. #16810362C2semidirect
7. C53×#490C2semidirect
101. #490D53semidirect
105. C53#1657semidirect