Finite Group Databasebeta

← #20285459

Subgroups — #20285459

Order 6784.

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

normal subgroups only

This group has 358 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#16811178C211yesmaximal, characteristic
2#16811178C211yesmaximal, characteristic
3#16811019C211yesmaximal, characteristic, complemented
4#16811002C211yesmaximal, characteristic, complemented
5#16810974C211yesmaximal, characteristic, complemented
6#16810880C214yesmaximal
7#16810727C214yesmaximal, complemented
8C4×#11719912C211yesmaximal, characteristic
9C53×#487C211yesmaximal, characteristic, complemented
10C2×#11719925C2214yes—
11C2×#11719912C2212yes—
12#13133217C2211yescharacteristic
13#13133217C2211yescharacteristic
14#13133208—21no—
15#13133208—21no—
16#13133197—21no—
17#13133197C2212yes—
18#13133193—21no—
19#13133193C2212yes—
20Q8×Dic53—21no—
21Q8×Dic53C2212yes—
22#13133182—21no—
23#13133182C2212yes—
24#13133175—21no—
25#13133175—21no—
26C42×D53C2211yescharacteristic
27#13133148C2214yes—
28#13133142C2214yes—
29#13133127C2212yes—
30C4×#88322C2211yescharacteristic
31C4⋊Q8×C53C2211yescharacteristic
32C4⋊Q8×C53C2211yescharacteristic
33#103×C53C2214yes—
34C4×C8×C53C2211yescharacteristic
35C2×Q8×C53—21no—
36C2×Q8×C53C2312yes—
37C2×C4×D53D411yescharacteristic
38C2×C4×D53D411yescharacteristic
39C2×C4×D53C2311yescharacteristic
40C2×Dic106—21no—
41C2×Dic106C2312yes—
42C2×C8×C53C2312yes—
43C4⋊C4×C53—21no—
44C4⋊C4×C53—21no—
45C4⋊C4×C53—21no—
46C4⋊C4×C53C2312yes—
47C42×C53C2311yescharacteristic
48#11719925—28no—
49#11719921—22no—
50#11719921—22no—
51C4⋊Dic53—21no—
52C4⋊Dic53—21no—
53C4⋊Dic53—21no—
54C4⋊Dic53—22no—
55C4⋊Dic53—22no—
56C4⋊Dic53C2312yes—
57#11719919—22no—
58#11719919—22no—
59C4×Dic53—22no—
60C4×Dic53—22no—
61C4×Dic53D411yescharacteristic
62C4×Dic53D411yescharacteristic
63C4×Dic53C2311yescharacteristic
64C2×#88322C2312yes—
65#11719912—24no—
66C22×D53C2×D411yescharacteristic
67Q8×C53—24no—
68Q8×C53—42no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53C2×D411yescharacteristic
72C2×C4×C53C2×D411yescharacteristic
73C2×C4×C53C2411yescharacteristic
74C2×Dic53—22no—
75C2×Dic53—22no—
76C2×Dic53C2×D411yescharacteristic
77C2×Dic53C2×D411yescharacteristic
78C2×Dic53C2×D411yescharacteristic
79C4×D53—22no—
80C4×D53—22no—
81C4×D53—22no—
82Dic106—24no—
83Dic106—42no—
84C8×C53—22no—
85#88322#4114yes—
86C22×C53C22×D411yescharacteristic
87C2×D53—21no—
88C2×D53—21no—
89C2×D53—21no—
90C4×C53—42no—
91C4×C53—42no—
92C4×C53#13712yes—
93C4×C53#13712yes—
94C4×C53C2×#4112yes—
95Dic53—21no—
96Dic53—21no—
97Dic53—21no—
98Dic53—42no—
99Dic53—42no—
100#3185—531nomaximal, Sylow
101C2×C53#53211yescharacteristic
102C2×C53C2×#13711yescharacteristic
103C2×C53C2×#13711yescharacteristic
104D53—41no—
105#533—531no—
106#533—531no—
107#498—531no—
108#494—531no—
109#487—531no—
110#487D5311yescharacteristic, complemented
111#470—534no—
112#441—534no—
113C4×M16—531no—
114C53#318511yescharacteristic, complemented, Sylow
115C2×Q16—534no—
116C2×M16—532no—
117C4⋊Q8—531no—
118C4⋊Q8—531no—
119C4⋊Q8C2×D5311yescharacteristic
120C4⋊Q8C2×D5311yescharacteristic
121#125—1061no—
122#125—1061no—
123C22⋊Q8—532no—
124C22⋊Q8—532no—
125C22⋊Q8—1061no—
126C22⋊Q8—1061no—
127C4×Q8—532no—
128C4×Q8—532no—
129C4×Q8—1061no—
130C4×Q8—1061no—
131#117—1061no—
132#117—1061no—
133C2×C42—531no—
134#106—534no—
135#103—534no—
136#103C2×D5314yes—
137#97—532no—
138C4×C8—531no—
139C4×C8C2×D5311yescharacteristic
140C2×Q8—21no—
141C2×Q8—532no—
142C2×Q8—1061no—
143C2×Q8C22×D5312yes—
144C22×C4—531no—
145C22×C4—531no—
146C22×C4—531no—
147Q16—1068no—
148M16—1064no—
149C2×C8—532no—
150C2×C8C22×D5312yes—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—532no—
155C4⋊C4—1062no—
156C4⋊C4—1062no—
157C4⋊C4—1062no—
158C4⋊C4—1062no—
159C4⋊C4—1061no—
160C4⋊C4—1061no—
161C4⋊C4—1061no—
162C4⋊C4C22×D5312yes—
163C22⋊C4—1062no—
164C22⋊C4—1062no—
165C42—531no—
166C42—531no—
167C42—531no—
168C42—1062no—
169C42—1062no—
170C42C22×D5311yescharacteristic
171C23—531no—
172Q8—24no—
173Q8—42no—
174Q8—1064no—
175Q8—2122no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—531no—
179C2×C4—531no—
180C2×C4—531no—
181C2×C4—1062no—
182C2×C4—1062no—
183C2×C4—1062no—
184C2×C4—1062no—
185C2×C4—1062no—
186C2×C4D4×D5311yescharacteristic
187C2×C4D4×D5311yescharacteristic
188C2×C4C23×D5311yescharacteristic
189C8—22no—
190C8—534no—
191C22—1061no—
192C22—1061no—
193C22—1061no—
194C22C2×D4×D5311yescharacteristic
195C4—42no—
196C4—42no—
197C4—1061no—
198C4—1061no—
199C4—1061no—
200C4—2122no—
201C4—2122no—
202C4#1313324612yes—
203C4#1313324912yes—
204C4#41×D5312yes—
205C2—2121no—
206C2#1681117211yescharacteristic
207C2C2×#1313324611yescharacteristic
208C2C2×#1313324911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#487C2
socleC22×C53C22×D4
hypercenter#487D53
hypocenterC53#3185
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
3. #16811019C2semidirect
4. #16811002C2semidirect
5. #16810974C2semidirect
7. #16810727C2semidirect
9. C53×#487C2semidirect
110. #487D53semidirect
114. C53#3185semidirect