Finite Group Databasebeta

← #20274023

Subgroups — #20274023

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#16811017C211yesmaximal, characteristic, complemented
3#16810983C211yesmaximal, characteristic, complemented
4#16810982C211yesmaximal, characteristic, complemented
5#16810362C211yesmaximal, characteristic, complemented
6#16810362C211yesmaximal, characteristic, complemented
7C53×#329C211yesmaximal, characteristic, complemented
8C2×Dic212—21no—
9C2×#11719913—21no—
10#13133217C2211yescharacteristic, complemented
11#13133206—21no—
12#13133205C2211yescharacteristic
13#13133205C2211yescharacteristic
14#13133203—21no—
15#13133203—21no—
16#13133187—21no—
17#13133187—21no—
18#13133185—21no—
19#13133142—21no—
20#13133141—21no—
21#13133139—21no—
22#13133138—21no—
23#13133128C2211yescharacteristic, complemented
24#124×C53C2211yescharacteristic
25#97×C53C2211yescharacteristic, complemented
26#97×C53C2211yescharacteristic, complemented
27C23×D53—21no—
28C2×Q8×C53D411yescharacteristic
29C2×D4×C53D411yescharacteristic
30C2×C22⋊D53—21no—
31C2×C22⋊D53—21no—
32C2×D212—21no—
33C2×D212D411yescharacteristic
34C2×D212D411yescharacteristic
35C2×Dic106—21no—
36C2×Dic106D411yescharacteristic
37C2×Dic106D411yescharacteristic
38C2×C8×C53—21no—
39C2×C8×C53—21no—
40C22⋊C4×C53—21no—
41C22⋊C4×C53—21no—
42C42×C53C2311yescharacteristic
43C22⋊Dic53—21no—
44#11719921—21no—
45#11719921—21no—
46#11719921—21no—
47#11719921—21no—
48#11719921—21no—
49#11719921—21no—
50C4⋊Dic53—21no—
51C4⋊Dic53—21no—
52C4×Dic53—21no—
53C2×#88322—21no—
54Dic212—42no—
55#11719913—42no—
56C23×C53—21no—
57C22×D53—21no—
58C22×D53—21no—
59C22×D53—81no—
60C22×D53—81no—
61C22×D53—81no—
62Q8×C53—41no—
63D4×C53—41no—
64C2×C4×C53—21no—
65C2×C4×C53C2×D411yescharacteristic
66C2×C4×C53C2×D411yescharacteristic
67C2×C4×C53C2×D411yescharacteristic
68C22⋊D53—81no—
69C22⋊D53—81no—
70C2×Dic53—21no—
71C2×Dic53—21no—
72C2×Dic53—41no—
73D212—41no—
74D212—41no—
75D212—81no—
76Dic106—41no—
77Dic106—41no—
78Dic106—41no—
79Dic106—41no—
80C8×C53—22no—
81C8×C53—41no—
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—
98Dic53—41no—
99Dic53—41no—
100Dic53—81no—
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#489—531no—
112#329—531no—
113#329—531no—
114#329D5311yescharacteristic, complemented
115C53#165911yescharacteristic, complemented, Sylow
116C2×Q16—1061no—
117C2×SD16—1061no—
118C4⋊Q8—531no—
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—1061no—
129#103—1061no—
130#102—1061no—
131#102—1061no—
132#97—531no—
133#97C2×D5311yescharacteristic, complemented
134#97C2×D5311yescharacteristic, complemented
135C24—1061no—
136C2×Q8—531no—
137C2×Q8—531no—
138C2×Q8—1061no—
139C2×Q8D21211yescharacteristic
140C2×D4—531no—
141C2×D4—531no—
142C2×D4—1061no—
143C2×D4—1061no—
144C2×D4—1061no—
145C2×D4D21211yescharacteristic
146Q16—2122no—
147SD16—2122no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—1061no—
151C4⋊C4—1061no—
152C4⋊C4—1061no—
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—2121no—
172Q8—2121no—
173Q8—2121no—
174Q8—2121no—
175D4—41no—
176D4—4241no—
177D4—4241no—
178D4—4241no—
179D4—2121no—
180D4—2121no—
181C2×C4—21no—
182C2×C4—1061no—
183C2×C4—1061no—
184C2×C4—2121no—
185C2×C4C2×D21211yescharacteristic
186C2×C4D4×D5311yescharacteristic
187C2×C4D4×D5311yescharacteristic
188C8—22no—
189C8—41no—
190C8—2121no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22—4244no—
195C22—2121no—
196C22—2121no—
197C22—2121no—
198C22—2121no—
199C22—2121no—
200C22—2121no—
201C22#1313318711yescharacteristic
202C4—21no—
203C4—21no—
204C4—21no—
205C4—41no—
206C4—4241no—
207C4—2121no—
208C4—2121no—
209C2—81no—
210C2—4241no—
211C2—4241no—
212C2#1681079211yescharacteristic
213C2#1681080211yescharacteristic
214C2#1681080311yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#329C2
socleC22×C53#120
hypercenter#329D53
hypocenterC53#1659
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #16811017C2semidirect
3. #16810983C2semidirect
4. #16810982C2semidirect
5. #16810362C2semidirect
6. #16810362C2semidirect
7. C53×#329C2semidirect
10. #13133217C22semidirect
23. #13133128C22semidirect
25. #97×C53C22semidirect
26. #97×C53C22semidirect
114. #329D53semidirect
115. C53#1659semidirect
133. #97C2×D53semidirect
134. #97C2×D53semidirect