Finite Group Databasebeta

← #20285110

Subgroups — #20285110

Order 6784.

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

normal subgroups only

This group has 424 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#16811174C211yesmaximal, characteristic
2C4⋊Q8×D53C211yesmaximal, characteristic, complemented
3#16810984C214yesmaximal, complemented
4#16810973C211yesmaximal, characteristic, complemented
5#16810971C211yesmaximal, characteristic, complemented
6#16810970C211yesmaximal, characteristic, complemented
7#16810623C214yesmaximal, complemented
8C4×#11719912C211yesmaximal, characteristic, complemented
9C53×#487C211yesmaximal, characteristic, complemented
10C2×#11719949C2212yes—
11C2×Q8×D53—21no—
12C2×#11719912C2212yes—
13#13133217C2211yescharacteristic
14#13133216—21no—
15#13133216C2212yes—
16#13133215—21no—
17#13133214C2211yescharacteristic
18#13133206C2212yes—
19#13133192—21no—
20#13133192—21no—
21#13133179—21no—
22#13133179—21no—
23C4⋊C4×D53—21no—
24C4⋊C4×D53C2212yes—
25C42×D53C2211yescharacteristic
26#13133141C2214yes—
27#13133139C2214yes—
28#13133137C2214yes—
29#13133127C2212yes—
30C4×#88322C2211yescharacteristic
31C4⋊Q8×C53C2211yescharacteristic
32C4⋊Q8×C53C2211yescharacteristic
33#103×C53C2214yes—
34C4×C8×C53C2211yescharacteristic
35C2×Q8×C53—21no—
36C2×Q8×C53C2312yes—
37#11719949—28no—
38Q8×D53—44no—
39C2×D212—22no—
40C2×D212—21no—
41C2×D212—21no—
42C2×D212C2312yes—
43C2×C4×D53—22no—
44C2×C4×D53—22no—
45C2×C4×D53D411yescharacteristic
46C2×C4×D53D411yescharacteristic
47C2×C4×D53C2311yescharacteristic
48C2×Dic106—21no—
49C2×Dic106—22no—
50C2×C8×C53C2312yes—
51C4⋊C4×C53—21no—
52C4⋊C4×C53—21no—
53C4⋊C4×C53—21no—
54C4⋊C4×C53C2312yes—
55C42×C53C2311yescharacteristic
56#11719921—22no—
57#11719921—22no—
58C4⋊Dic53—21no—
59C4⋊Dic53C2312yes—
60#11719919—22no—
61#11719919—22no—
62C4×Dic53D411yescharacteristic
63C4×Dic53D411yescharacteristic
64C4×Dic53C2311yescharacteristic
65C2×#88322C2312yes—
66#11719912—24no—
67C22×D53—22no—
68C22×D53C2×D411yescharacteristic
69Q8×C53—24no—
70Q8×C53—42no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53C2×D411yescharacteristic
74C2×C4×C53C2×D411yescharacteristic
75C2×C4×C53C2411yescharacteristic
76C2×Dic53—22no—
77C2×Dic53C2×D411yescharacteristic
78C2×Dic53C2×D411yescharacteristic
79C2×Dic53C2×D411yescharacteristic
80D212—24no—
81D212—44no—
82D212—42no—
83D212—42no—
84C4×D53—22no—
85C4×D53—22no—
86C4×D53—44no—
87C4×D53—44no—
88C4×D53#4114yes—
89Dic106—44no—
90Dic106—42no—
91C8×C53—22no—
92#88322—22no—
93C22×C53C22×D411yescharacteristic
94C2×D53—21no—
95C2×D53—21no—
96C2×D53—42no—
97C2×D53—42no—
98C2×D53—42no—
99C2×D53#12412yes—
100C4×C53—42no—
101C4×C53—42no—
102C4×C53AGL(1,ℤ/8)12yes—
103C4×C53#13712yes—
104C4×C53C2×#4112yes—
105Dic53—21no—
106Dic53—21no—
107Dic53—42no—
108Dic53#12412yes—
109#3141—531nomaximal, Sylow
110C2×C53C2×#12411yescharacteristic
111C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
112C2×C53C2×#13711yescharacteristic
113D53—22no—
114D53—82no—
115#532—531no—
116C2×C4⋊Q8—531no—
117#489—534no—
118#487—531no—
119#487D5311yescharacteristic, complemented
120#486—531no—
121#486—531no—
122#417—534no—
123C4×M16—531no—
124C53#314111yescharacteristic, complemented, Sylow
125C2×#41—532no—
126C22×Q8—1061no—
127C2×M16—532no—
128C4⋊Q8—532no—
129C4⋊Q8—531no—
130C4⋊Q8—1061no—
131C4⋊Q8—1061no—
132C4⋊Q8C2×D5311yescharacteristic
133C4⋊Q8C2×D5311yescharacteristic
134#127—531no—
135#124—532no—
136#121—1061no—
137#121—1061no—
138C4×D4—1061no—
139C4×D4—1061no—
140C2×C4⋊C4—532no—
141C2×C4⋊C4—1061no—
142C2×C42—531no—
143#103—534no—
144#103C2×D5314yes—
145#102—534no—
146#102—534no—
147#97—532no—
148C4×C8—531no—
149C4×C8C2×D5311yescharacteristic
150#41—1068no—
151C2×Q8—21no—
152C2×Q8—1062no—
153C2×Q8—1061no—
154C2×Q8—2124no—
155C2×Q8C22×D5312yes—
156C2×D4—532no—
157C2×D4—1062no—
158C2×D4—1061no—
159C2×D4—1061no—
160C22×C4—531no—
161C22×C4—531no—
162C22×C4—531no—
163C22×C4—1062no—
164C22×C4—1062no—
165M16—1064no—
166C2×C8—532no—
167C2×C8C22×D5312yes—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—532no—
172C4⋊C4—1062no—
173C4⋊C4—1062no—
174C4⋊C4—1061no—
175C4⋊C4C22×D5312yes—
176C22⋊C4—1062no—
177C22⋊C4—1062no—
178C42—531no—
179C42—531no—
180C42—531no—
181C42C22×D5311yescharacteristic
182C23—531no—
183C23—1062no—
184Q8—24no—
185Q8—42no—
186Q8—2124no—
187Q8—2122no—
188D4—1064no—
189D4—2124no—
190D4—2122no—
191D4—2122no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—531no—
195C2×C4—534no—
196C2×C4—531no—
197C2×C4—531no—
198C2×C4—1062no—
199C2×C4—1062no—
200C2×C4—1062no—
201C2×C4—2124no—
202C2×C4—2124no—
203C2×C4D4×D5311yescharacteristic
204C2×C4D4×D5311yescharacteristic
205C2×C4C23×D5311yescharacteristic
206C8—22no—
207C8—1062no—
208C22—532no—
209C22—1061no—
210C22—1061no—
211C22—2122no—
212C22—2122no—
213C22—2122no—
214C22C2×D4×D5311yescharacteristic
215C4—42no—
216C4—42no—
217C4—532no—
218C4—1061no—
219C4—1061no—
220C4—2122no—
221C4#1313324412yes—
222C4#1313324912yes—
223C4#41×D5312yes—
224C2—1062no—
225C2—4242no—
226C2#124×D5311yescharacteristic
227C2C2×#1313324411yescharacteristic
228C2C2×#1313324911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#487C2
socleC22×C53C22×D4
hypercenter#487D53
hypocenterC53#3141
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
2. C4⋊Q8×D53C2semidirect
3. #16810984C2semidirect
4. #16810973C2semidirect
5. #16810971C2semidirect
6. #16810970C2semidirect
7. #16810623C2semidirect
8. C4×#11719912C2semidirect
9. C53×#487C2semidirect
119. #487D53semidirect
124. C53#3141semidirect