Finite Group Databasebeta

← #20285906

Subgroups — #20285906

Order 6784.

368 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 111 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#16811174C211yesmaximal, characteristic
2#125×D53C211yesmaximal, characteristic, complemented
3#16811029C211yesmaximal, characteristic, complemented
4#16810989C211yesmaximal, characteristic, complemented
5#16810988C211yesmaximal, characteristic, complemented
6#16810943C212yesmaximal, complemented
7#16810938C212yesmaximal, complemented
8#16810926C212yesmaximal, complemented
9#16810922C212yesmaximal, complemented
10#16810678C211yesmaximal, characteristic, complemented
11C53×#501C211yesmaximal, characteristic, complemented
12C2×#11719949—21no—
13#13133214C2211yescharacteristic
14#13133209C2211yescharacteristic
15#13133208C2211yescharacteristic
16#13133208C2211yescharacteristic
17#13133207—21no—
18#13133207—21no—
19#13133206—21no—
20#13133192—21no—
21#13133192C2212yes—
22#13133179—21no—
23#13133179C2212yes—
24C4⋊C4×D53—21no—
25C4⋊C4×D53—21no—
26C4⋊C4×D53C2211yescharacteristic
27C4⋊C4×D53C2211yescharacteristic
28C42×D53C2211yescharacteristic
29#13133150C2212yes—
30#13133148C2212yes—
31#13133146C2212yes—
32#13133139C2212yes—
33#13133139C2212yes—
34#13133137C2212yes—
35#13133137C2212yes—
36#13133129C2212yes—
37#13133128C2211yescharacteristic
38C4⋊Q8×C53C2211yescharacteristic
39#125×C53C2211yescharacteristic
40#107×C53C2212yes—
41#106×C53C2212yes—
42#97×C53C2211yescharacteristic
43C2×Q8×C53—21no—
44#11719949—44no—
45C2×D212—22no—
46C2×D212—21no—
47C2×D212—21no—
48C2×D212C2312yes—
49C2×C4×D53—21no—
50C2×C4×D53—21no—
51C2×C4×D53—22no—
52C2×C4×D53D411yescharacteristic
53C2×C4×D53D411yescharacteristic
54C2×C4×D53C2311yescharacteristic
55C2×C8×C53C2312yes—
56C4⋊C4×C53—21no—
57C4⋊C4×C53—21no—
58C4⋊C4×C53—21no—
59C4⋊C4×C53C2311yescharacteristic
60C4⋊C4×C53C2312yes—
61C4⋊C4×C53C2311yescharacteristic
62C42×C53C2311yescharacteristic
63#11719921—22no—
64#11719921—22no—
65C4⋊Dic53—21no—
66C4⋊Dic53—21no—
67C4⋊Dic53C2311yescharacteristic
68C4⋊Dic53C2311yescharacteristic
69#11719919—21no—
70#11719919—21no—
71#11719919—21no—
72#11719919—21no—
73#11719919—21no—
74#11719919—21no—
75C4×Dic53D411yescharacteristic
76C4×Dic53D411yescharacteristic
77C4×Dic53C2311yescharacteristic
78C2×#88322C2312yes—
79C22×D53—22no—
80C22×D53C2×D411yescharacteristic
81Q8×C53—42no—
82C2×C4×C53—22no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53C2×D411yescharacteristic
86C2×C4×C53C2×D411yescharacteristic
87C2×C4×C53C2411yescharacteristic
88C2×Dic53—21no—
89C2×Dic53—21no—
90C2×Dic53C2×D411yescharacteristic
91C2×Dic53C2×D411yescharacteristic
92C2×Dic53C2×D411yescharacteristic
93D212—24no—
94D212—42no—
95D212—42no—
96D212—44no—
97C4×D53—22no—
98C4×D53—22no—
99C4×D53—41no—
100C4×D53—44no—
101C4×D53—42no—
102C4×D53—42no—
103C8×C53—22no—
104#88322—22no—
105C22×C53C22×D411yescharacteristic
106C2×D53—21no—
107C2×D53—21no—
108C2×D53—42no—
109C2×D53—42no—
110C2×D53—42no—
111C2×D53AGL(1,ℤ/8)12yes—
112C4×C53—21no—
113C4×C53—21no—
114C4×C53—42no—
115C4×C53—41no—
116C4×C53—41no—
117C4×C532−1+412yes—
118Dic53—21no—
119Dic53—21no—
120Dic53—21no—
121Dic53—41no—
122Dic53—41no—
123#3234—531nomaximal, Sylow
124C2×C53#53611yescharacteristic
125C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
126C2×C53#57511yescharacteristic
127D53—22no—
128D53—82no—
129#532—531no—
130C2×#125—531no—
131#501—531no—
132#501D5311yescharacteristic, complemented
133#490—531no—
134#490—531no—
135#481—532no—
136#480—532no—
137#478—532no—
138#478—532no—
139#432—531no—
140C53#323411yescharacteristic, complemented, Sylow
141C2×#41—1061no—
142C4⋊Q8C2×D5311yescharacteristic
143#127—531no—
144#125—531no—
145#125—531no—
146#125—531no—
147#125—1061no—
148#125—1061no—
149#125C2×D5311yescharacteristic
150#124—1061no—
151#121—532no—
152#121—1061no—
153C4×D4—532no—
154C4×D4—1061no—
155C2×C4⋊C4—531no—
156C2×C4⋊C4—531no—
157C2×C4⋊C4—1061no—
158C2×C4⋊C4—1061no—
159C2×C42—531no—
160#107—532no—
161#107C2×D5312yes—
162#106—532no—
163#106C2×D5312yes—
164C4⋊C8—532no—
165#102—532no—
166#102—532no—
167#102—532no—
168#102—532no—
169C22⋊C8—532no—
170#97—531no—
171#97C2×D5311yescharacteristic
172#41—2124no—
173C2×Q8—21no—
174C2×D4—532no—
175C2×D4—1062no—
176C2×D4—1061no—
177C2×D4—1061no—
178C22×C4—531no—
179C22×C4—531no—
180C22×C4—531no—
181C22×C4—1062no—
182C22×C4—1061no—
183C22×C4—1061no—
184C2×C8—532no—
185C2×C8C22×D5312yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—531no—
190C4⋊C4—531no—
191C4⋊C4—1061no—
192C4⋊C4—1061no—
193C4⋊C4—1061no—
194C4⋊C4—1061no—
195C4⋊C4—1061no—
196C4⋊C4—1061no—
197C4⋊C4—1061no—
198C4⋊C4—1061no—
199C4⋊C4C22×D5311yescharacteristic
200C4⋊C4C22×D5311yescharacteristic
201C4⋊C4C22×D5312yes—
202C22⋊C4—1062no—
203C22⋊C4—1062no—
204C42—531no—
205C42—531no—
206C42—531no—
207C42C22×D5311yescharacteristic
208C23—531no—
209C23—1062no—
210Q8—42no—
211D4—1064no—
212D4—2122no—
213D4—2122no—
214D4—2124no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—531no—
219C2×C4—531no—
220C2×C4—531no—
221C2×C4—1062no—
222C2×C4—1062no—
223C2×C4—1061no—
224C2×C4—1061no—
225C2×C4—2121no—
226C2×C4—2124no—
227C2×C4—2122no—
228C2×C4—2122no—
229C2×C4D4×D5311yescharacteristic
230C2×C4D4×D5311yescharacteristic
231C2×C4C23×D5311yescharacteristic
232C8—22no—
233C8—1062no—
234C22—532no—
235C22—1061no—
236C22—1061no—
237C22—2122no—
238C22—2122no—
239C22—2122no—
240C22C2×D4×D5311yescharacteristic
241C4—21no—
242C4—21no—
243C4—42no—
244C4—41no—
245C4—41no—
246C4—1061no—
247C4—1061no—
248C4—1061no—
249C4—2121no—
250C4—2121no—
251C4#1313326512yes—
252C2—1062no—
253C2—4242no—
254C2#1681119811yescharacteristic
255C2AGL(1,ℤ/8)×D5311yescharacteristic
256C2#1681142211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#501C2
socleC22×C53C22×D4
hypercenter#501D53
hypocenterC53#3234
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
2. #125×D53C2semidirect
3. #16811029C2semidirect
4. #16810989C2semidirect
5. #16810988C2semidirect
6. #16810943C2semidirect
7. #16810938C2semidirect
8. #16810926C2semidirect
9. #16810922C2semidirect
10. #16810678C2semidirect
11. C53×#501C2semidirect
132. #501D53semidirect
140. C53#3234semidirect