Finite Group Databasebeta

← #20283952

Subgroups — #20283952

Order 6784.

738 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C23×Dic106C211yesmaximal, characteristic, complemented
2C2×#13133250C212yesmaximal, complemented
3#16811134C211yesmaximal, characteristic
4#16810781C214yesmaximal, complemented
5#16810764C214yesmaximal, complemented
6#16810633C212yesmaximal, complemented
7C53×#407C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic
9C22×Dic106—22no—
10C22×Dic106—21no—
11C22×Dic106—28no—
12C22×Dic106C2214yes—
13C22×Dic106C2212yes—
14C23×Dic53—21no—
15#13133250—28no—
16C2×Dic212C2214yescomplemented
17C2×#11719913C2214yescomplemented
18#13133200—21no—
19#13133196C2212yes—
20#13133191C2212yes—
21#13133186—21no—
22C4×C22⋊D53—22no—
23#13133176C2211yescharacteristic
24#13133142C2218yescomplemented
25C23×C4×C53C2211yescharacteristic
26C2×M16×C53C2212yes—
27C22⋊C8×C53C2214yes—
28C24×C53D411yescharacteristic
29C22×C4×C53—24no—
30C22×C4×C53D412yes—
31C22×C4×C53D411yescharacteristic
32C22×C4×C53C2312yes—
33C22×C4×C53C2311yescharacteristic
34C2×C22⋊D53—22no—
35C2×C22⋊D53—21no—
36C22×Dic53—28no—
37C22×Dic53—24no—
38C22×Dic53—22no—
39#11719945—24no—
40C2×D212C2311yescharacteristic
41C2×C4×D53—21no—
42C2×Dic106—28no—
43C2×Dic106—216no—
44C2×Dic106—24no—
45C2×Dic106—22no—
46C2×Dic106—24no—
47C2×Dic106—216no—
48C2×Dic106D418yescomplemented
49C2×Dic106C2311yescharacteristic
50C2×Dic106C2314yes—
51M16×C53—24no—
52C2×C8×C53C2314yes—
53C22⋊Dic53—21no—
54C22⋊Dic53—22no—
55#11719921—22no—
56C4⋊Dic53C2312yes—
57#11719919—22no—
58C4×Dic53—21no—
59Dic212—28no—
60#11719913—28no—
61C23×C53—22no—
62C23×C53—24no—
63C23×C53C2×D411yescharacteristic
64C23×C53C2×D412yes—
65C22×D53—21no—
66C2×C4×C53—22no—
67C2×C4×C53—24no—
68C2×C4×C53—24no—
69C2×C4×C53C2×D412yes—
70C2×C4×C53C2×D411yescharacteristic
71C2×C4×C53C2×D414yes—
72C2×C4×C53C2411yescharacteristic
73C22⋊D53—42no—
74C22⋊D53—44no—
75C2×Dic53—216no—
76C2×Dic53—28no—
77C2×Dic53—24no—
78C2×Dic53—21no—
79C2×Dic53—22no—
80D212—22no—
81C4×D53—42no—
82Dic106—216no—
83Dic106—28no—
84Dic106—24no—
85Dic106—22no—
86Dic106C2×D418yes—
87C8×C53—24no—
88C22×C53—28no—
89C22×C53—22no—
90C22×C53—22no—
91C22×C53—22no—
92C22×C53#12012yes—
93C22×C53#13714yescomplemented
94C22×C53C22×D411yescharacteristic
95C2×D53—41no—
96C2×D53—42no—
97C4×C53—22no—
98C4×C53#12012yes—
99C4×C53C22×D412yes—
100Dic53—28no—
101Dic53—41no—
102Dic53—42no—
103#3004—531nomaximal, Sylow
104C2×C53—22no—
105C2×C53—24no—
106C2×C53C2×#12011yescharacteristic
107C2×C53C2×#13712yes—
108D53—81no—
109C23×Q8—531no—
110C2×#137—532no—
111#525—531no—
112#451—534no—
113#448—534no—
114#419—532no—
115#407D5311yescharacteristic, complemented
116C53#300411yescharacteristic, complemented, Sylow
117C2×#41—531no—
118C22×Q8—532no—
119C22×Q8—534no—
120C22×Q8—1068no—
121C22×Q8—1062no—
122C22×Q8—1061no—
123C23×C4—1061no—
124C23×C4C2×D5311yescharacteristic
125#137—1068no—
126C2×Q16—534no—
127C2×SD16—534no—
128C2×M16C2×D5312yes—
129#123—1061no—
130C22⋊Q8—532no—
131#121—532no—
132#120—1061no—
133C4×D4—1062no—
134#117—531no—
135#103—538no—
136C22⋊C8C2×D5314yes—
137C24D21211yescharacteristic
138#41—1064no—
139C2×Q8—538no—
140C2×Q8—534no—
141C2×Q8—531no—
142C2×Q8—10616no—
143C2×Q8—1068no—
144C2×Q8—10616no—
145C2×Q8—1064no—
146C2×Q8—1062no—
147C2×Q8—1064no—
148C2×D4—531no—
149C2×D4—1062no—
150C2×D4—1061no—
151C22×C4—24no—
152C22×C4—1068no—
153C22×C4—1064no—
154C22×C4—1062no—
155C22×C4—1061no—
156C22×C4D21212yes—
157C22×C4D21211yescharacteristic
158C22×C4C22×D5312yes—
159C22×C4C22×D5311yescharacteristic
160Q16—1068no—
161SD16—1068no—
162M16—24no—
163C2×C8C22×D5314yes—
164C4⋊C4—532no—
165C4⋊C4—1062no—
166C22⋊C4—1061no—
167C22⋊C4—1062no—
168C22⋊C4—1062no—
169C42—1061no—
170C23—22no—
171C23—24no—
172C23—1061no—
173C23C2×D21211yescharacteristic
174C23C2×D21212yes—
175Q8—538no—
176Q8—10616no—
177Q8—1068no—
178Q8—1064no—
179Q8—1062no—
180D4—1062no—
181D4—2122no—
182D4—2124no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—10616no—
187C2×C4—1068no—
188C2×C4—1064no—
189C2×C4—1062no—
190C2×C4—1061no—
191C2×C4—2122no—
192C2×C4C2×D21212yes—
193C2×C4C2×D21211yescharacteristic
194C2×C4D4×D5314yes—
195C2×C4C23×D5311yescharacteristic
196C8—24no—
197C22—28no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—2121no—
202C22—2122no—
203C22#1313318712yes—
204C22#1313325014yescomplemented
205C22C22×D21211yescharacteristic
206C4—22no—
207C4—1068no—
208C4—2121no—
209C4—2122no—
210C4#1313318712yes—
211C4C2×D4×D5312yes—
212C2—22no—
213C2—24no—
214C2—4241no—
215C2C2×#1313318711yescharacteristic
216C2C2×#1313325012yes—
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#407C2
socleC22×C53C22×D4
hypercenter#407D53
hypocenterC53#3004
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C23×Dic106C2semidirect
2. C2×#13133250C2semidirect
4. #16810781C2semidirect
5. #16810764C2semidirect
6. #16810633C2semidirect
7. C53×#407C2semidirect
16. C2×Dic212C22semidirect
17. C2×#11719913C22semidirect
24. #13133142C22semidirect
48. C2×Dic106D4semidirect
93. C22×C53#137semidirect
115. #407D53semidirect
116. C53#3004semidirect
204. C22#13133250semidirect