Finite Group Databasebeta

← #20285915

Subgroups — #20285915

Order 6784.

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

normal subgroups only

This group has 366 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#16811032C212yesmaximal, complemented
4#16810988C212yesmaximal, complemented
5#16810948C214yesmaximal, complemented
6#16810914C212yesmaximal, complemented
7#16810904C212yesmaximal, complemented
8C53×#432C211yesmaximal, characteristic, complemented
9C2×#11719945—21no—
10#13133217C2211yescharacteristic
11#13133217C2211yescharacteristic
12#13133217C2211yescharacteristic
13#13133214C2211yescharacteristic
14#13133209C2212yes—
15#13133205—21no—
16#13133196—21no—
17#13133196—21no—
18#13133191—21no—
19#13133191C2212yes—
20C4×Dic106—21no—
21C4×Dic106—21no—
22C4×D212—21no—
23C4×D212C2212yes—
24#13133176C2212yes—
25#13133151C2214yes—
26#13133149C2214yes—
27#13133139C2214yes—
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×Dic106—21no—
44C2×Dic106—21no—
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⋊Dic53C2312yes—
57C4⋊Dic53C2312yes—
58C4⋊Dic53C2312yes—
59#11719919—22no—
60#11719919—22no—
61C4×Dic53—22no—
62C23×C53C2×D411yescharacteristic
63C22×D53—22no—
64C2×C4×C53—22no—
65C2×C4×C53—22no—
66C2×C4×C53—41no—
67C2×C4×C53C2×D411yescharacteristic
68C2×C4×C53C2×D411yescharacteristic
69C2×C4×C53C2×D411yescharacteristic
70C2×C4×C53C2×D411yescharacteristic
71C2×C4×C53C2×D411yescharacteristic
72C2×C4×C53C2411yescharacteristic
73C22⋊D53—44no—
74C2×Dic53—22no—
75C2×Dic53—22no—
76C2×Dic53—22no—
77D212—24no—
78D212—42no—
79D212—42no—
80C4×D53—44no—
81Dic106—42no—
82Dic106—42no—
83Dic106—42no—
84C8×C53—24no—
85C22×C53—21no—
86C22×C53—21no—
87C22×C53—21no—
88C22×C53C22×D411yescharacteristic
89C2×D53—42no—
90C2×D53—42no—
91C2×D53—42no—
92C4×C53—21no—
93C4×C53—21no—
94C4×C53—21no—
95C4×C53—21no—
96C4×C53AGL(1,ℤ/8)12yes—
97C4×C532−1+412yes—
98Dic53—42no—
99Dic53—42no—
100Dic53—42no—
101#3235—531nomaximal, Sylow
102C2×C53—41no—
103C2×C53#53611yescharacteristic
104C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
105C2×C53#57511yescharacteristic
106D53—82no—
107#533—531no—
108#532—531no—
109#501—532no—
110#490—532no—
111#482—534no—
112#476—532no—
113#474—532no—
114#432D5311yescharacteristic, complemented
115C53#323511yescharacteristic, complemented, Sylow
116C2×#41—1061no—
117C4⋊Q8—531no—
118C4⋊Q8—531no—
119C4⋊Q8—531no—
120#127—531no—
121#125—532no—
122#124—1061no—
123C22⋊Q8—1061no—
124C22⋊Q8—1061no—
125#121—532no—
126#121—1061no—
127C4×Q8—1061no—
128C4×Q8—1061no—
129C4×D4—532no—
130C4×D4—1061no—
131#117—532no—
132C2×C42C2×D5311yescharacteristic
133#107—534no—
134#106—534no—
135C4⋊C8C2×D5312yes—
136#102—534no—
137#102—534no—
138C22⋊C8C2×D5312yes—
139#97C2×D5312yes—
140#41—2124no—
141C2×Q8—1061no—
142C2×Q8—1061no—
143C2×Q8—1061no—
144C2×D4—532no—
145C2×D4—1062no—
146C2×D4—1061no—
147C2×D4—1061no—
148C22×C4—1062no—
149C22×C4D21211yescharacteristic
150C22×C4D21211yescharacteristic
151C22×C4C22×D5311yescharacteristic
152C2×C8C22×D5314yes—
153C4⋊C4—532no—
154C4⋊C4—532no—
155C4⋊C4—532no—
156C4⋊C4—1061no—
157C4⋊C4—1062no—
158C4⋊C4—1062no—
159C4⋊C4—1061no—
160C4⋊C4—1061no—
161C22⋊C4—1062no—
162C22⋊C4—1062no—
163C22⋊C4—1062no—
164C42—1062no—
165C42D21211yescharacteristic
166C42D21211yescharacteristic
167C42C22×D5311yescharacteristic
168C42C22×D5311yescharacteristic
169C23—1062no—
170C23C2×D21211yescharacteristic
171Q8—2122no—
172Q8—2122no—
173Q8—2122no—
174D4—1064no—
175D4—2122no—
176D4—2122no—
177D4—2124no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—1062no—
182C2×C4—1062no—
183C2×C4—1062no—
184C2×C4—2124no—
185C2×C4C2×D21211yescharacteristic
186C2×C4C2×D21211yescharacteristic
187C2×C4C2×D21211yescharacteristic
188C2×C4C2×D21211yescharacteristic
189C2×C4C2×D21211yescharacteristic
190C2×C4C23×D5311yescharacteristic
191C8—24no—
192C22—21no—
193C22—21no—
194C22—21no—
195C22—2122no—
196C22—2122no—
197C22—2122no—
198C22C22×D21211yescharacteristic
199C4—21no—
200C4—21no—
201C4—21no—
202C4—21no—
203C4—2122no—
204C4—2122no—
205C4—2122no—
206C4#1313324512yes—
207C4#1313326412yes—
208C2—41no—
209C2—4242no—
210C2#1681119911yescharacteristic
211C2C2×#1313324511yescharacteristic
212C2#1681142311yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#432C2
socleC22×C53C22×D4
hypercenter#432D53
hypocenterC53#3235
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. #16811032C2semidirect
4. #16810988C2semidirect
5. #16810948C2semidirect
6. #16810914C2semidirect
7. #16810904C2semidirect
8. C53×#432C2semidirect
114. #432D53semidirect
115. C53#3235semidirect