Finite Group Databasebeta

← #20285672

Subgroups — #20285672

Order 6784.

394 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 117 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, complemented
2#16811171C211yesmaximal, characteristic
3#16811003C211yesmaximal, characteristic, complemented
4#16810995C211yesmaximal, characteristic, complemented
5#16810975C211yesmaximal, characteristic, complemented
6#16810844C214yesmaximal, complemented
7#16810786C214yesmaximal, complemented
8#16810674C211yesmaximal, characteristic, complemented
9C53×#486C211yesmaximal, characteristic, complemented
10C2×#11719947C2212yes—
11C2×Dic212C2214yes—
12C2×#11719923C2214yes—
13#13133217C2211yescharacteristic
14#13133217C2211yescharacteristic
15#13133208—21no—
16#13133204C2212yes—
17#13133197C2212yes—
18#13133193—21no—
19#13133188—21no—
20#13133188—21no—
21Q8×Dic53—21no—
22#13133182C2212yes—
23D4×Dic53—21no—
24D4×Dic53—21no—
25#13133175—21no—
26C42×D53C2211yescharacteristic
27#13133146C2212yes—
28#13133140C2214yes—
29#13133129C2212yes—
30C4×#88322C2211yescharacteristic
31C4⋊Q8×C53C2211yescharacteristic
32#127×C53C2211yescharacteristic
33#102×C53C2214yes—
34C4×C8×C53C2211yescharacteristic
35C2×Q8×C53—21no—
36C2×D4×C53—21no—
37C2×D4×C53—21no—
38C2×D4×C53C2312yes—
39C2×C22⋊D53—22no—
40C22×Dic53—22no—
41#11719947—44no—
42C2×C4×D53D411yescharacteristic
43C2×C4×D53D411yescharacteristic
44C2×C4×D53C2311yescharacteristic
45C2×Dic106C2312yes—
46C2×Dic106C2312yes—
47C2×C8×C53C2312yes—
48C4⋊C4×C53—21no—
49C4⋊C4×C53C2312yes—
50C42×C53C2311yescharacteristic
51C22⋊Dic53—22no—
52C22⋊Dic53—22no—
53#11719923—44no—
54#11719921—22no—
55C4⋊Dic53—21no—
56C4⋊Dic53—21no—
57C4⋊Dic53—21no—
58C4⋊Dic53—21no—
59C4⋊Dic53—22no—
60#11719919—22no—
61C4×Dic53—22no—
62C4×Dic53D411yescharacteristic
63C4×Dic53D411yescharacteristic
64C4×Dic53C2311yescharacteristic
65C2×#88322C2312yes—
66Dic212—44no—
67C23×C53—22no—
68C22×D53C2×D411yescharacteristic
69Q8×C53—42no—
70D4×C53—24no—
71D4×C53—42no—
72D4×C53—42no—
73C2×C4×C53—22no—
74C2×C4×C53C2×D411yescharacteristic
75C2×C4×C53C2×D411yescharacteristic
76C2×C4×C53C2411yescharacteristic
77C22⋊D53—44no—
78C2×Dic53—22no—
79C2×Dic53—22no—
80C2×Dic53—44no—
81C2×Dic53C2×D411yescharacteristic
82C2×Dic53C2×D411yescharacteristic
83C2×Dic53C2×D411yescharacteristic
84C4×D53—22no—
85C4×D53—22no—
86C4×D53—22no—
87Dic106—24no—
88Dic106—24no—
89C8×C53—22no—
90#88322—22no—
91C22×C53—42no—
92C22×C53—42no—
93C22×C53—42no—
94C22×C53C22×D411yescharacteristic
95C2×D53—21no—
96C2×D53—21no—
97C2×D53—21no—
98C4×C53—42no—
99C4×C53#13512yes—
100C4×C53#13712yes—
101C4×C53H(5,𝔽2)12yes—
102Dic53—21no—
103Dic53—21no—
104Dic53—21no—
105Dic53—42no—
106Dic53—42no—
107#3210—531nomaximal, Sylow
108C2×C53—82no—
109C2×C53#53511yescharacteristic
110C2×C53C2×#13511yescharacteristic
111C2×C53C2×#13711yescharacteristic
112D53—41no—
113#533—531no—
114#532—531no—
115#494—531no—
116#492—531no—
117#487—531no—
118#486D5311yescharacteristic, complemented
119#464—534no—
120#452—534no—
121#431—531no—
122C53#321011yescharacteristic, complemented, Sylow
123C2×#41—532no—
124C2×Q16—534no—
125C2×SD16—534no—
126C4⋊Q8—531no—
127C4⋊Q8—531no—
128C4⋊Q8C2×D5311yescharacteristic
129#127C2×D5311yescharacteristic
130#125—1061no—
131#124—532no—
132C22⋊Q8—532no—
133C22⋊Q8—1061no—
134#121—1061no—
135#121—1061no—
136C4×Q8—532no—
137C4×Q8—1061no—
138C4×D4—1061no—
139C4×D4—1061no—
140#117—1061no—
141C2×C42—531no—
142C4⋊C8—532no—
143#103—534no—
144#102C2×D5314yes—
145C22⋊C8—532no—
146C4×C8—531no—
147C4×C8C2×D5311yescharacteristic
148#41—2124no—
149C2×Q8—21no—
150C2×Q8—532no—
151C2×Q8—532no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—1062no—
155C2×D4C22×D5312yes—
156C22×C4—531no—
157C22×C4—531no—
158C22×C4—531no—
159C22×C4—1062no—
160Q16—2124no—
161SD16—2124no—
162C2×C8—532no—
163C2×C8C22×D5312yes—
164C4⋊C4—21no—
165C4⋊C4—1062no—
166C4⋊C4—1062no—
167C4⋊C4—1061no—
168C4⋊C4—1061no—
169C4⋊C4—1061no—
170C4⋊C4—1061no—
171C4⋊C4C22×D5312yes—
172C22⋊C4—1062no—
173C22⋊C4—1062no—
174C22⋊C4—1062no—
175C42—531no—
176C42—531no—
177C42—531no—
178C42—1062no—
179C42C22×D5311yescharacteristic
180C23—22no—
181C23—531no—
182Q8—42no—
183Q8—1064no—
184Q8—1064no—
185D4—24no—
186D4—42no—
187D4—42no—
188D4—2124no—
189C2×C4—22no—
190C2×C4—531no—
191C2×C4—531no—
192C2×C4—531no—
193C2×C4—1062no—
194C2×C4—1062no—
195C2×C4—1062no—
196C2×C4—1062no—
197C2×C4—1062no—
198C2×C4—2124no—
199C2×C4D4×D5311yescharacteristic
200C2×C4D4×D5311yescharacteristic
201C2×C4C23×D5311yescharacteristic
202C8—22no—
203C8—1062no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22—1061no—
208C22—1061no—
209C22—1061no—
210C22C2×D4×D5311yescharacteristic
211C4—42no—
212C4—1061no—
213C4—1061no—
214C4—1061no—
215C4—2122no—
216C4—2122no—
217C4#1313323612yes—
218C4#1313324612yes—
219C4#1313326312yes—
220C2—82no—
221C2—2121no—
222C2#1681119111yescharacteristic
223C2C2×#1313323611yescharacteristic
224C2C2×#1313324611yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#486C2
socleC22×C53C22×D4
hypercenter#486D53
hypocenterC53#3210
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #16811178C2semidirect
3. #16811003C2semidirect
4. #16810995C2semidirect
5. #16810975C2semidirect
6. #16810844C2semidirect
7. #16810786C2semidirect
8. #16810674C2semidirect
9. C53×#486C2semidirect
118. #486D53semidirect
122. C53#3210semidirect