Finite Group Databasebeta

← #20274020

Subgroups — #20274020

Order 6784.

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

normal subgroups only

This group has 228 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#16811365C211yesmaximal, characteristic
2#16811015C211yesmaximal, characteristic, complemented
3#16810985C211yesmaximal, characteristic, complemented
4#16810363C211yesmaximal, characteristic, complemented
5#16810362C211yesmaximal, characteristic, complemented
6#16810362C211yesmaximal, characteristic, complemented
7C53×#489C211yesmaximal, characteristic, complemented
8C2×#11719925—21no—
9C2×#11719924—21no—
10#13133206—21no—
11#13133205C2211yescharacteristic
12#13133205C2211yescharacteristic
13#13133203—21no—
14#13133203—21no—
15#13133187—21no—
16#13133187—21no—
17#13133185—21no—
18#13133140—21no—
19#13133137—21no—
20#13133128C2211yescharacteristic, complemented
21#13133128C2211yescharacteristic, complemented
22C4⋊Q8×C53C2211yescharacteristic, complemented
23#124×C53C2211yescharacteristic
24#103×C53—21no—
25#102×C53—21no—
26#97×C53C2211yescharacteristic, complemented
27C23×D53—21no—
28C2×Q8×C53—21no—
29C2×Q8×C53D411yescharacteristic
30C2×D4×C53D411yescharacteristic
31C2×C22⋊D53—21no—
32C2×C22⋊D53—21no—
33C2×D212—21no—
34C2×D212D411yescharacteristic
35C2×D212D411yescharacteristic
36C2×Dic106D411yescharacteristic
37C2×Dic106D411yescharacteristic
38C2×C8×C53—21no—
39C4⋊C4×C53—21no—
40C4⋊C4×C53—21no—
41C22⋊C4×C53—21no—
42C22⋊C4×C53—21no—
43C42×C53C2311yescharacteristic
44C22⋊Dic53—21no—
45#11719925—42no—
46#11719924—42no—
47#11719921—21no—
48#11719921—21no—
49#11719921—21no—
50#11719921—21no—
51#11719921—21no—
52#11719921—21no—
53C4×Dic53—21no—
54C2×#88322—21no—
55C2×#88322—21no—
56C23×C53—21no—
57C22×D53—21no—
58C22×D53—21no—
59C22×D53—81no—
60C22×D53—81no—
61C22×D53—81no—
62Q8×C53—41no—
63Q8×C53—41no—
64Q8×C53—41no—
65D4×C53—41no—
66C2×C4×C53—21no—
67C2×C4×C53—41no—
68C2×C4×C53C2×D411yescharacteristic
69C2×C4×C53C2×D411yescharacteristic
70C2×C4×C53C2×D411yescharacteristic
71C22⋊D53—81no—
72C22⋊D53—81no—
73C2×Dic53—21no—
74C2×Dic53—21no—
75D212—41no—
76D212—41no—
77D212—81no—
78Dic106—41no—
79Dic106—41no—
80C8×C53—41no—
81#88322—22no—
82#88322—41no—
83C22×C53—41no—
84C22×C53—41no—
85C22×C53—41no—
86C22×C53#12011yescharacteristic
87C2×D53—41no—
88C2×D53—41no—
89C2×D53—41no—
90C2×D53—41no—
91C2×D53—41no—
92C2×D53—41no—
93C2×D53—84no—
94C4×C53—21no—
95C4×C53—21no—
96C4×C53—21no—
97C4×C53—41no—
98C4×C53—81no—
99Dic53—41no—
100Dic53—41no—
101#1659—531nomaximal, Sylow
102C2×C53—81no—
103C2×C53#45311yescharacteristic
104C2×C53#45511yescharacteristic
105C2×C53#45511yescharacteristic
106D53—81no—
107D53—81no—
108H(3,𝔽4)—531no—
109#497—531no—
110#489—531no—
111#489D5311yescharacteristic, complemented
112#329—531no—
113#329—531no—
114#329—531no—
115C53#165911yescharacteristic, complemented, Sylow
116C2×Q16—1061no—
117C2×SD16—1061no—
118C4⋊Q8C2×D5311yescharacteristic, complemented
119#124—531no—
120#124—531no—
121#124—1061no—
122#124—1061no—
123#124—1061no—
124#124C2×D5311yescharacteristic
125#120—1061no—
126#120—1061no—
127#120—1061no—
128#103—21no—
129#103—1061no—
130#102—21no—
131#102—1061no—
132#97—531no—
133#97—531no—
134#97C2×D5311yescharacteristic, complemented
135C24—1061no—
136C2×Q8—21no—
137C2×Q8—531no—
138C2×Q8—531no—
139C2×Q8C22⋊D5311yescharacteristic
140C2×D4—531no—
141C2×D4—531no—
142C2×D4—1061no—
143C2×D4—1061no—
144C2×D4—1061no—
145C2×D4C22⋊D5311yescharacteristic
146Q16—2122no—
147SD16—2122no—
148C2×C8—21no—
149C2×C8—1061no—
150C2×C8—1061no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C22⋊C4—21no—
154C22⋊C4—21no—
155C22⋊C4—1061no—
156C22⋊C4—1061no—
157C22⋊C4—1061no—
158C22⋊C4—1061no—
159C22⋊C4—1061no—
160C22⋊C4—1061no—
161C22⋊C4—1061no—
162C42—1061no—
163C42C22×D5311yescharacteristic
164C23—21no—
165C23—1061no—
166C23—1061no—
167C23—4241no—
168C23—4241no—
169C23—4241no—
170Q8—41no—
171Q8—41no—
172Q8—41no—
173Q8—2121no—
174Q8—2121no—
175D4—41no—
176D4—4241no—
177D4—4241no—
178D4—4241no—
179D4—2121no—
180D4—2121no—
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—1061no—
184C2×C4—1061no—
185C2×C4D4×D5311yescharacteristic
186C2×C4D4×D5311yescharacteristic
187C2×C4C2×C22⋊D5311yescharacteristic
188C8—41no—
189C8—1062no—
190C8—2121no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22—4244no—
195C22—2121no—
196C22—2121no—
197C22—2121no—
198C22—2121no—
199C22—2121no—
200C22—2121no—
201C22#1313318511yescharacteristic
202C4—21no—
203C4—21no—
204C4—21no—
205C4—41no—
206C4—81no—
207C4—2121no—
208C4—2121no—
209C2—81no—
210C2—4241no—
211C2—4241no—
212C2#1681078911yescharacteristic
213C2#1681079911yescharacteristic
214C2#1681080011yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#489C2
socleC22×C53#120
hypercenter#489D53
hypocenterC53#1659
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
2. #16811015C2semidirect
3. #16810985C2semidirect
4. #16810363C2semidirect
5. #16810362C2semidirect
6. #16810362C2semidirect
7. C53×#489C2semidirect
20. #13133128C22semidirect
21. #13133128C22semidirect
22. C4⋊Q8×C53C22semidirect
26. #97×C53C22semidirect
111. #489D53semidirect
115. C53#1659semidirect
118. C4⋊Q8C2×D53semidirect
134. #97C2×D53semidirect