Finite Group Databasebeta

← #20285617

Subgroups — #20285617

Order 6784.

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

normal subgroups only

This group has 402 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#16811201C212yesmaximal, complemented
2#16810959C212yesmaximal, complemented
3#16810937C212yesmaximal, complemented
4#16810869C212yesmaximal, complemented
5#16810864C211yesmaximal, characteristic, complemented
6#16810859C211yesmaximal, characteristic, complemented
7#16810787C212yesmaximal, complemented
8#16810770C212yesmaximal, complemented
9C53×#408C211yesmaximal, characteristic, complemented
10C2×#11719945C2212yes—
11C2×Dic212C2211yescharacteristic
12C2×#11719913C2212yes—
13C2×D424C2211yescharacteristic
14#13133196C2212yes—
15#13133196C2212yes—
16#13133193—22no—
17#13133191C2212yes—
18#13133191C2212yes—
19#13133189—22no—
20#13133189—22no—
21#13133188—22no—
22C2×C4⋊Dic53C2212yes—
23#13133151C2211yescharacteristic
24#13133151C2211yescharacteristic
25#13133151C2212yes—
26#13133142C2212yes—
27#13133142C2212yes—
28#13133139C2212yes—
29#13133139C2212yes—
30C2×#41×C53C2211yescharacteristic
31C2×M16×C53C2211yescharacteristic
32C22×C8×C53C2211yescharacteristic
33C22⋊C8×C53C2212yes—
34C22⋊C8×C53C2212yes—
35#41×C53—41no—
36#41×C53—41no—
37C2×Q8×C53D411yescharacteristic
38C2×D4×C53D412yes—
39C2×D4×C53D411yescharacteristic
40C22×C4×C53C2312yes—
41C22×C4×C53C2311yescharacteristic
42C2×C22⋊D53—22no—
43C2×C22⋊D53—22no—
44C2×C22⋊D53—22no—
45C22×Dic53—22no—
46#11719945—42no—
47#11719945—42no—
48C2×D212C2312yes—
49C2×C4×D53—22no—
50C2×Dic106C2312yes—
51M16×C53—41no—
52C2×C8×C53—41no—
53C2×C8×C53C2311yescharacteristic
54C2×C8×C53C2311yescharacteristic
55C2×C8×C53C2312yes—
56C22⋊Dic53—22no—
57C22⋊Dic53—22no—
58#11719921—22no—
59#11719921—22no—
60C4⋊Dic53—22no—
61C4⋊Dic53C2312yes—
62C4⋊Dic53C2312yes—
63#11719919—22no—
64#11719919—22no—
65Dic212—41no—
66D424—41no—
67#11719913—42no—
68C23×C53C2×D412yes—
69C23×C53C2×D411yescharacteristic
70C22×D53—22no—
71Q8×C53—41no—
72D4×C53—42no—
73D4×C53—41no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53C2×D412yes—
79C2×C4×C53C2×D411yescharacteristic
80C2×C4×C53C2411yescharacteristic
81C22⋊D53—44no—
82C22⋊D53—82no—
83C22⋊D53—82no—
84C2×Dic53—22no—
85C2×Dic53—22no—
86C2×Dic53—22no—
87C2×Dic53—82no—
88D212—22no—
89D212—22no—
90C4×D53—44no—
91Dic106—22no—
92Dic106—22no—
93C8×C53—22no—
94C8×C53—21no—
95C8×C53—21no—
96C22×C53—22no—
97C22×C53—21no—
98C22×C53—22no—
99C22×C53—21no—
100C22×C53—22no—
101C22×C53—21no—
102C22×C53C22×D411yescharacteristic
103C2×D53—42no—
104C2×D53—42no—
105C2×D53—42no—
106C4×C53—22no—
107C4×C53—21no—
108C4×C53H(5,𝔽2)11yescharacteristic
109C4×C53H(5,𝔽2)11yescharacteristic
110Dic53—42no—
111Dic53—42no—
112Dic53—42no—
113#3203—531nomaximal, Sylow
114C2×C53—42no—
115C2×C53—41no—
116C2×C53#53411yescharacteristic
117C2×C53#57611yescharacteristic
118C2×C53#57811yescharacteristic
119D53—82no—
120#537—532no—
121#484—532no—
122#480—532no—
123#468—532no—
124#467—531no—
125#466—531no—
126#452—532no—
127#449—532no—
128#408D5311yescharacteristic, complemented
129C53#320311yescharacteristic, complemented, Sylow
130C2×#41—532no—
131C2×#41C2×D5311yescharacteristic
132C2×Q16—531no—
133C2×SD16—532no—
134C2×D8—531no—
135C2×M16C2×D5311yescharacteristic
136C22×C8C2×D5311yescharacteristic
137C22⋊Q8—532no—
138C22⋊Q8—532no—
139C22⋊Q8—1062no—
140#121—532no—
141#121—532no—
142#121—1062no—
143#121—1062no—
144#121—1062no—
145C2×C4⋊C4—532no—
146#107—531no—
147#107—531no—
148#107—532no—
149#103—532no—
150#103—532no—
151#102—532no—
152#102—532no—
153C22⋊C8C2×D5312yes—
154C22⋊C8C2×D5312yes—
155#41—41no—
156#41—41no—
157#41—2122no—
158#41—2122no—
159C2×Q8—532no—
160C2×Q8D21211yescharacteristic
161C2×D4—532no—
162C2×D4—1062no—
163C2×D4—1062no—
164C2×D4—1062no—
165C2×D4D21211yescharacteristic
166C2×D4D21212yes—
167C22×C4—1062no—
168C22×C4—1062no—
169C22×C4C22×D5312yes—
170C22×C4C22×D5311yescharacteristic
171Q16—2121no—
172SD16—2122no—
173D8—2121no—
174M16—41no—
175C2×C8—41no—
176C2×C8C22×D5311yescharacteristic
177C2×C8C22×D5312yes—
178C2×C8C22×D5311yescharacteristic
179C4⋊C4—532no—
180C4⋊C4—532no—
181C4⋊C4—1062no—
182C4⋊C4—1062no—
183C4⋊C4—1062no—
184C22⋊C4—1062no—
185C22⋊C4—1062no—
186C22⋊C4—1062no—
187C22⋊C4—1062no—
188C23—1062no—
189C23C2×D21211yescharacteristic
190C23C2×D21212yes—
191Q8—41no—
192Q8—1062no—
193Q8—1062no—
194D4—42no—
195D4—41no—
196D4—1062no—
197D4—1062no—
198D4—4242no—
199D4—4242no—
200D4—2124no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—1062no—
206C2×C4—1062no—
207C2×C4—1062no—
208C2×C4—4242no—
209C2×C4—2124no—
210C2×C4C2×D21211yescharacteristic
211C2×C4C2×D21212yes—
212C2×C4C23×D5311yescharacteristic
213C8—21no—
214C8—21no—
215C8—22no—
216C22—22no—
217C22—21no—
218C22—21no—
219C22—22no—
220C22—22no—
221C22—21no—
222C22—2122no—
223C22—2122no—
224C22—2122no—
225C22C22×D21211yescharacteristic
226C4—22no—
227C4—21no—
228C4—2122no—
229C4—2122no—
230C4—2122no—
231C4#1313326211yescharacteristic
232C4#1313326211yescharacteristic
233C2—42no—
234C2—41no—
235C2—4242no—
236C2#1681118511yescharacteristic
237C2#1681142811yescharacteristic
238C2#1681144011yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#408C2
socleC22×C53C22×D4
hypercenter#408D53
hypocenterC53#3203
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #16811201C2semidirect
2. #16810959C2semidirect
3. #16810937C2semidirect
4. #16810869C2semidirect
5. #16810864C2semidirect
6. #16810859C2semidirect
7. #16810787C2semidirect
8. #16810770C2semidirect
9. C53×#408C2semidirect
128. #408D53semidirect
129. C53#3203semidirect