Finite Group Databasebeta

← #20285664

Subgroups — #20285664

Order 6784.

384 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 125 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
2C4⋊Q8×D53C211yesmaximal, characteristic
3#16811003C211yesmaximal, characteristic, complemented
4#16811002C211yesmaximal, characteristic, complemented
5#16810975C211yesmaximal, characteristic, complemented
6#16810832C214yesmaximal, complemented
7#16810780C214yesmaximal
8#16810674C211yesmaximal, characteristic
9C53×#487C211yesmaximal, characteristic, complemented
10C2×Q8×D53C2212yes—
11C2×Dic212C2214yes—
12C2×#11719925C2214yes—
13#13133217C2211yescharacteristic
14#13133217C2211yescharacteristic
15#13133216—21no—
16#13133216—21no—
17#13133215C2212yes—
18#13133208—21no—
19#13133197C2212yes—
20#13133193—21no—
21Q8×Dic53—21no—
22#13133182C2212yes—
23#13133175—21no—
24C4⋊C4×D53—21no—
25C4⋊C4×D53—21no—
26C42×D53C2211yescharacteristic
27#13133146C2212yes—
28#13133140C2214yes—
29#13133129C2212yes—
30C4×#88322C2211yescharacteristic
31C4⋊Q8×C53C2211yescharacteristic
32C4⋊Q8×C53C2211yescharacteristic
33#103×C53C2214yes—
34C4×C8×C53C2211yescharacteristic
35C2×Q8×C53—21no—
36C2×Q8×C53C2312yes—
37Q8×D53—44no—
38C2×C4×D53—22no—
39C2×C4×D53D411yescharacteristic
40C2×C4×D53D411yescharacteristic
41C2×C4×D53C2311yescharacteristic
42C2×Dic106—22no—
43C2×Dic106C2312yes—
44C2×Dic106C2312yes—
45C2×C8×C53C2312yes—
46C4⋊C4×C53—21no—
47C4⋊C4×C53—21no—
48C4⋊C4×C53—21no—
49C4⋊C4×C53C2312yes—
50C42×C53C2311yescharacteristic
51#11719925—44no—
52#11719921—22no—
53C4⋊Dic53—21no—
54C4⋊Dic53—21no—
55C4⋊Dic53—21no—
56C4⋊Dic53—21no—
57C4⋊Dic53—22no—
58#11719919—22no—
59#11719919—22no—
60#11719919—22no—
61C4×Dic53—22no—
62C4×Dic53D411yescharacteristic
63C4×Dic53D411yescharacteristic
64C4×Dic53C2311yescharacteristic
65C2×#88322C2312yes—
66Dic212—44no—
67C22×D53C2×D411yescharacteristic
68Q8×C53—24no—
69Q8×C53—42no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53C2×D411yescharacteristic
73C2×C4×C53C2×D411yescharacteristic
74C2×C4×C53C2411yescharacteristic
75C2×Dic53—22no—
76C2×Dic53—22no—
77C2×Dic53C2×D411yescharacteristic
78C2×Dic53C2×D411yescharacteristic
79C2×Dic53C2×D411yescharacteristic
80C4×D53—22no—
81C4×D53—22no—
82C4×D53—44no—
83C4×D53Q1614yescomplemented
84Dic106—24no—
85Dic106—24no—
86Dic106—44no—
87C8×C53—22no—
88#88322—22no—
89C22×C53C22×D411yescharacteristic
90C2×D53—21no—
91C2×D53—21no—
92C2×D53C2×Q1612yes—
93C4×C53—42no—
94C4×C53—42no—
95C4×C53C2×Q1612yes—
96C4×C53#13712yes—
97C4×C53H(5,𝔽2)12yes—
98Dic53—21no—
99Dic53—21no—
100Dic53—42no—
101Dic53—42no—
102Dic53C2×Q1612yes—
103#3209—531nomaximal, Sylow
104C2×C53#53511yescharacteristic
105C2×C53C22×Q1611yescharacteristic
106C2×C53C2×#13711yescharacteristic
107D53—22no—
108#533—531no—
109C2×C4⋊Q8—531no—
110#494—531no—
111#494—531no—
112#487—531no—
113#487D5311yescharacteristic, complemented
114#462—534no—
115#451—534no—
116#431—531no—
117C53#320911yescharacteristic, complemented, Sylow
118C22×Q8—532no—
119C2×Q16—534no—
120C2×Q16—534no—
121C4⋊Q8—532no—
122C4⋊Q8—531no—
123C4⋊Q8—531no—
124C4⋊Q8—1061no—
125C4⋊Q8—1061no—
126C4⋊Q8C2×D5311yescharacteristic
127C4⋊Q8C2×D5311yescharacteristic
128#125—1061no—
129C22⋊Q8—532no—
130C22⋊Q8—1061no—
131C4×Q8—532no—
132C4×Q8—1061no—
133#117—1061no—
134C2×C4⋊C4—1061no—
135C2×C4⋊C4—1061no—
136C2×C42—531no—
137C4⋊C8—532no—
138#103—534no—
139#103C2×D5314yes—
140C22⋊C8—532no—
141C4×C8—531no—
142C4×C8C2×D5311yescharacteristic
143C2×Q8—21no—
144C2×Q8—532no—
145C2×Q8—532no—
146C2×Q8—1062no—
147C2×Q8—2124no—
148C2×Q8C22×D5312yes—
149C22×C4—531no—
150C22×C4—531no—
151C22×C4—531no—
152C22×C4—1062no—
153Q16—2124no—
154Q16—2124no—
155C2×C8—532no—
156C2×C8C22×D5312yes—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—1062no—
161C4⋊C4—1062no—
162C4⋊C4—1062no—
163C4⋊C4—1062no—
164C4⋊C4—1061no—
165C4⋊C4—1061no—
166C4⋊C4—1061no—
167C4⋊C4—1061no—
168C4⋊C4C22×D5312yes—
169C22⋊C4—1062no—
170C42—531no—
171C42—531no—
172C42—531no—
173C42—1062no—
174C42C22×D5311yescharacteristic
175C23—531no—
176Q8—24no—
177Q8—42no—
178Q8—1064no—
179Q8—1064no—
180Q8—2124no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—531no—
184C2×C4—534no—
185C2×C4—531no—
186C2×C4—531no—
187C2×C4—1062no—
188C2×C4—1062no—
189C2×C4—1062no—
190C2×C4—1062no—
191C2×C4—2124no—
192C2×C4D4×D5311yescharacteristic
193C2×C4D4×D5311yescharacteristic
194C2×C4C23×D5311yescharacteristic
195C8—22no—
196C8—1062no—
197C22—532no—
198C22—1061no—
199C22—1061no—
200C22C2×D4×D5311yescharacteristic
201C4—42no—
202C4—42no—
203C4—532no—
204C4—1061no—
205C4—1061no—
206C4—2122no—
207C4—2122no—
208C4Q16×D5312yes—
209C4#1313324612yes—
210C4#1313326312yes—
211C2—1062no—
212C2#1681119111yescharacteristic
213C2C2×Q16×D5311yescharacteristic
214C2C2×#1313324611yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#487C2
socleC22×C53C22×D4
hypercenter#487D53
hypocenterC53#3209
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. #16811003C2semidirect
4. #16811002C2semidirect
5. #16810975C2semidirect
6. #16810832C2semidirect
9. C53×#487C2semidirect
83. C4×D53Q16semidirect
113. #487D53semidirect
117. C53#3209semidirect