Finite Group Databasebeta

← #20284838

Subgroups — #20284838

Order 6784.

424 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 129 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#16811173C211yesmaximal, characteristic
2C2×C4×Dic106C211yesmaximal, characteristic, complemented
3#16810937C212yesmaximal, complemented
4#16810919C212yesmaximal, complemented
5#16810824C212yesmaximal, complemented
6#16810764C212yesmaximal, complemented
7#16810725C214yesmaximal, complemented
8C53×#432C211yesmaximal, characteristic, complemented
9C2×#11719945—21no—
10C22×Dic106C2211yescharacteristic
11C2×#11719913C2214yes—
12#13133217C2211yescharacteristic
13#13133214C2211yescharacteristic
14#13133205—21no—
15#13133191—21no—
16#13133191C2212yes—
17C4×Dic106—21no—
18C4×Dic106—21no—
19C4×Dic106C2212yes—
20C4×Dic106C2212yes—
21C4×D212—21no—
22C4×D212C2212yes—
23C2×C4⋊Dic53C2211yescharacteristic
24C2×#11719919—21no—
25C2×C4×Dic53—21no—
26#13133151C2214yes—
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—
37C22×Dic53—21no—
38C22×Dic53—21no—
39#11719945—44no—
40C2×D212—21no—
41C2×D212—21no—
42C2×D212C2312yes—
43C2×C4×D53—22no—
44C2×Dic106—21no—
45C2×Dic106—24no—
46C2×Dic106—21no—
47C2×Dic106C2312yes—
48C2×C8×C53C2314yes—
49C42×C53D411yescharacteristic
50C42×C53D411yescharacteristic
51C42×C53C2311yescharacteristic
52C42×C53C2311yescharacteristic
53#11719921—22no—
54#11719921—22no—
55C4⋊Dic53—21no—
56C4⋊Dic53—21no—
57C4⋊Dic53C2312yes—
58C4⋊Dic53C2312yes—
59#11719919—22no—
60#11719919—22no—
61C4×Dic53—22no—
62C4×Dic53—22no—
63#11719913—28no—
64C23×C53C2×D411yescharacteristic
65C22×D53—22no—
66C2×C4×C53—22no—
67C2×C4×C53—22no—
68C2×C4×C53—22no—
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—24no—
77C2×Dic53—22no—
78C2×Dic53—22no—
79C2×Dic53—22no—
80C2×Dic53—42no—
81D212—24no—
82D212—42no—
83D212—42no—
84C4×D53—44no—
85Dic106—24no—
86Dic106—22no—
87Dic106—42no—
88Dic106#4114yescomplemented
89C8×C53—24no—
90C22×C53—21no—
91C22×C53—21no—
92C22×C53AGL(1,ℤ/8)12yes—
93C22×C53C22×D411yescharacteristic
94C2×D53—42no—
95C2×D53—42no—
96C2×D53—42no—
97C4×C53—21no—
98C4×C53—21no—
99C4×C53—21no—
100C4×C53—21no—
101C4×C53#13712yes—
102C4×C53C2×#4112yes—
103Dic53—24no—
104Dic53—42no—
105Dic53—42no—
106#3112—531nomaximal, Sylow
107C2×C53—22no—
108C2×C53#52511yescharacteristic
109C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
110C2×C53C2×#13711yescharacteristic
111D53—82no—
112#532—531no—
113C2×C4×Q8—531no—
114#480—532no—
115#477—532no—
116#460—532no—
117#448—532no—
118#440—534no—
119#432D5311yescharacteristic, complemented
120C53#311211yescharacteristic, complemented, Sylow
121C2×#41—1061no—
122C22×Q8—531no—
123C2×SD16—534no—
124C4⋊Q8—531no—
125#127—531no—
126#124—1061no—
127#121—532no—
128#121—1061no—
129C4×Q8—532no—
130C4×Q8—532no—
131C4×Q8—1061no—
132C4×Q8—1061no—
133C4×D4—532no—
134C4×D4—1061no—
135C2×C4⋊C4—531no—
136C2×C4⋊C4—1061no—
137C2×C42—1061no—
138C2×C42C2×D5311yescharacteristic
139#107—534no—
140C4⋊C8C2×D5312yes—
141#103—534no—
142#102—534no—
143C22⋊C8C2×D5312yes—
144#97C2×D5312yes—
145#41—2124no—
146C2×Q8—532no—
147C2×Q8—1064no—
148C2×Q8—1061no—
149C2×Q8—1061no—
150C2×D4—532no—
151C2×D4—1062no—
152C2×D4—1061no—
153C2×D4—1061no—
154C22×C4—1061no—
155C22×C4—1061no—
156C22×C4—1062no—
157C22×C4D21211yescharacteristic
158C22×C4D21211yescharacteristic
159C22×C4C22×D5311yescharacteristic
160SD16—1068no—
161C2×C8C22×D5314yes—
162C4⋊C4—532no—
163C4⋊C4—532no—
164C4⋊C4—1061no—
165C4⋊C4—1062no—
166C4⋊C4—1062no—
167C4⋊C4—1061no—
168C22⋊C4—1062no—
169C22⋊C4—1062no—
170C42—1062no—
171C42—1062no—
172C42D21211yescharacteristic
173C42D21211yescharacteristic
174C42C22×D5311yescharacteristic
175C42C22×D5311yescharacteristic
176C23—1062no—
177C23C2×D21211yescharacteristic
178Q8—534no—
179Q8—1064no—
180Q8—1062no—
181Q8—2122no—
182D4—1064no—
183D4—2122no—
184D4—2124no—
185D4—2122no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—1064no—
190C2×C4—1062no—
191C2×C4—1062no—
192C2×C4—1062no—
193C2×C4—2124no—
194C2×C4—2122no—
195C2×C4C2×D21211yescharacteristic
196C2×C4C2×D21211yescharacteristic
197C2×C4C2×D21211yescharacteristic
198C2×C4C2×D21211yescharacteristic
199C2×C4C2×D21211yescharacteristic
200C2×C4C23×D5311yescharacteristic
201C8—24no—
202C22—21no—
203C22—21no—
204C22—2122no—
205C22—2122no—
206C22—2122no—
207C22#1313324512yes—
208C22C22×D21211yescharacteristic
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—1064no—
214C4—2122no—
215C4—2122no—
216C4#1313325012yes—
217C4#41×D5312yes—
218C2—22no—
219C2—4242no—
220C2#1681113511yescharacteristic
221C2C2×#1313324511yescharacteristic
222C2C2×#1313325011yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#432C2
socleC22×C53C22×D4
hypercenter#432D53
hypocenterC53#3112
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
2. C2×C4×Dic106C2semidirect
3. #16810937C2semidirect
4. #16810919C2semidirect
5. #16810824C2semidirect
6. #16810764C2semidirect
7. #16810725C2semidirect
8. C53×#432C2semidirect
88. Dic106#41semidirect
119. #432D53semidirect
120. C53#3112semidirect