Finite Group Databasebeta

← #20285728

Subgroups — #20285728

Order 6784.

390 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 121 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#16811180C211yesmaximal, characteristic, complemented
2#16811173C211yesmaximal, characteristic
3#16811017C212yesmaximal, complemented
4#16810983C212yesmaximal, complemented
5#16810834C212yesmaximal, complemented
6#16810824C212yesmaximal, complemented
7#16810787C214yesmaximal, complemented
8C53×#432C211yesmaximal, characteristic, complemented
9C2×#11719945C2212yes—
10C2×Dic212C2214yes—
11C2×#11719913C2214yes—
12#13133217C2211yescharacteristic
13#13133217C2211yescharacteristic
14#13133217C2211yescharacteristic
15#13133214C2211yescharacteristic
16#13133209—21no—
17#13133205C2212yes—
18#13133196—21no—
19#13133196C2212yes—
20#13133191—21no—
21#13133191—21no—
22C4×Dic106—21no—
23C4×Dic106C2212yes—
24C4×D212—21no—
25C4×D212—21no—
26#13133176—21no—
27#13133142C2214yes—
28#13133139C2214yes—
29C2×C42×C53C2211yescharacteristic
30C4⋊C8×C53C2212yes—
31C22⋊C8×C53C2212yes—
32#97×C53C2212yes—
33C22×C4×C53D411yescharacteristic
34C22×C4×C53D411yescharacteristic
35C22×C4×C53C2311yescharacteristic
36C2×C22⋊D53—22no—
37#11719945—44no—
38C2×D212—21no—
39C2×D212—21no—
40C2×D212C2312yes—
41C2×C4×D53—22no—
42C2×Dic106—21no—
43C2×Dic106C2312yes—
44C2×Dic106C2312yes—
45C2×C8×C53C2314yes—
46C42×C53D411yescharacteristic
47C42×C53D411yescharacteristic
48C42×C53C2311yescharacteristic
49C42×C53C2311yescharacteristic
50C22⋊Dic53—22no—
51#11719921—22no—
52#11719921—22no—
53C4⋊Dic53—21no—
54C4⋊Dic53—21no—
55C4⋊Dic53—21no—
56C4⋊Dic53—21no—
57C4⋊Dic53—21no—
58C4⋊Dic53C2312yes—
59#11719919—22no—
60#11719919—22no—
61C4×Dic53—22no—
62Dic212—44no—
63#11719913—44no—
64C23×C53C2×D411yescharacteristic
65C22×D53—22no—
66C2×C4×C53—22no—
67C2×C4×C53—22no—
68C2×C4×C53—41no—
69C2×C4×C53C2×D411yescharacteristic
70C2×C4×C53C2×D411yescharacteristic
71C2×C4×C53C2×D411yescharacteristic
72C2×C4×C53C2×D411yescharacteristic
73C2×C4×C53C2×D411yescharacteristic
74C2×C4×C53C2411yescharacteristic
75C22⋊D53—44no—
76C2×Dic53—22no—
77C2×Dic53—22no—
78C2×Dic53—22no—
79D212—24no—
80D212—42no—
81D212—42no—
82C4×D53—44no—
83Dic106—24no—
84Dic106—22no—
85Dic106—22no—
86Dic106—42no—
87C8×C53—24no—
88C22×C53—21no—
89C22×C53—21no—
90C22×C53—21no—
91C22×C53C22×D411yescharacteristic
92C2×D53—42no—
93C2×D53—42no—
94C2×D53—42no—
95C4×C53—21no—
96C4×C53—21no—
97C4×C53—21no—
98C4×C53—21no—
99C4×C53#13712yes—
100C4×C53H(5,𝔽2)12yes—
101Dic53—42no—
102Dic53—42no—
103Dic53—42no—
104#3216—531nomaximal, Sylow
105C2×C53—41no—
106C2×C53#53511yescharacteristic
107C2×C53C2×#13711yescharacteristic
108C2×C53#57511yescharacteristic
109D53—82no—
110#533—531no—
111#532—531no—
112#497—532no—
113#489—532no—
114#462—532no—
115#460—532no—
116#452—534no—
117#432D5311yescharacteristic, complemented
118C53#321611yescharacteristic, complemented, Sylow
119C2×#41—532no—
120C2×Q16—534no—
121C2×SD16—534no—
122C4⋊Q8—531no—
123C4⋊Q8—531no—
124C4⋊Q8—531no—
125#127—531no—
126#125—1061no—
127#124—532no—
128C22⋊Q8—532no—
129C22⋊Q8—1061no—
130#121—1061no—
131#121—1061no—
132C4×Q8—532no—
133C4×Q8—1061no—
134C4×D4—1061no—
135C4×D4—1061no—
136#117—1061no—
137C2×C42C2×D5311yescharacteristic
138C4⋊C8C2×D5312yes—
139#103—534no—
140#102—534no—
141C22⋊C8C2×D5312yes—
142#97C2×D5312yes—
143#41—2124no—
144C2×Q8—532no—
145C2×Q8—532no—
146C2×Q8—1061no—
147C2×D4—532no—
148C2×D4—1062no—
149C2×D4—1061no—
150C2×D4—1061no—
151C22×C4—1062no—
152C22×C4D21211yescharacteristic
153C22×C4D21211yescharacteristic
154C22×C4C22×D5311yescharacteristic
155Q16—2124no—
156SD16—2124no—
157C2×C8C22×D5314yes—
158C4⋊C4—532no—
159C4⋊C4—1061no—
160C4⋊C4—1061no—
161C4⋊C4—1061no—
162C4⋊C4—1061no—
163C4⋊C4—1062no—
164C4⋊C4—1062no—
165C4⋊C4—1061no—
166C22⋊C4—1062no—
167C22⋊C4—1062no—
168C22⋊C4—1062no—
169C42—1062no—
170C42D21211yescharacteristic
171C42D21211yescharacteristic
172C42C22×D5311yescharacteristic
173C42C22×D5311yescharacteristic
174C23—1062no—
175C23C2×D21211yescharacteristic
176Q8—1062no—
177Q8—1062no—
178Q8—1064no—
179Q8—2122no—
180D4—1064no—
181D4—2122no—
182D4—2124no—
183D4—2122no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—41no—
187C2×C4—1062no—
188C2×C4—1062no—
189C2×C4—1062no—
190C2×C4—2124no—
191C2×C4C2×D21211yescharacteristic
192C2×C4C2×D21211yescharacteristic
193C2×C4C2×D21211yescharacteristic
194C2×C4C2×D21211yescharacteristic
195C2×C4C2×D21211yescharacteristic
196C2×C4C23×D5311yescharacteristic
197C8—24no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—2122no—
202C22—2122no—
203C22—2122no—
204C22C22×D21211yescharacteristic
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—2122no—
210C4—2122no—
211C4—2122no—
212C4#1313325012yes—
213C4#1313326312yes—
214C2—41no—
215C2—4242no—
216C2#1681119211yescharacteristic
217C2C2×#1313325011yescharacteristic
218C2#1681142311yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#432C2
socleC22×C53C22×D4
hypercenter#432D53
hypocenterC53#3216
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
1. #16811180C2semidirect
3. #16811017C2semidirect
4. #16810983C2semidirect
5. #16810834C2semidirect
6. #16810824C2semidirect
7. #16810787C2semidirect
8. C53×#432C2semidirect
117. #432D53semidirect
118. C53#3216semidirect