Finite Group Databasebeta

← #20285844

Subgroups — #20285844

Order 6784.

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

normal subgroups only

This group has 404 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#16811019C211yesmaximal, characteristic, complemented
4#16810971C211yesmaximal, characteristic, complemented
5#16810970C211yesmaximal, characteristic, complemented
6#16810943C214yesmaximal, complemented
7#16810912C214yesmaximal, complemented
8#16810674C211yesmaximal, characteristic, complemented
9C53×#498C211yesmaximal, characteristic, complemented
10C2×#11719949—21no—
11C2×Q8×D53—21no—
12#13133217C2211yescharacteristic
13#13133216—21no—
14#13133216C2212yes—
15#13133215—21no—
16#13133214C2211yescharacteristic
17#13133206—21no—
18#13133192—21no—
19#13133192C2212yes—
20#13133179—21no—
21#13133179C2212yes—
22C4⋊C4×D53—21no—
23C4⋊C4×D53C2212yes—
24C42×D53C2211yescharacteristic
25#13133148C2214yes—
26#13133146C2212yes—
27#13133139C2214yes—
28#13133137C2214yes—
29#13133129C2212yes—
30C4×#88322C2211yescharacteristic
31C4⋊Q8×C53C2211yescharacteristic
32C4⋊Q8×C53C2211yescharacteristic
33#106×C53C2214yes—
34C4×C8×C53C2211yescharacteristic
35C2×Q8×C53—21no—
36C2×Q8×C53—21no—
37#11719949—44no—
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×C53C2312yes—
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—
66C22×D53—22no—
67C22×D53C2×D411yescharacteristic
68Q8×C53—42no—
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×Dic53C2×D411yescharacteristic
77C2×Dic53C2×D411yescharacteristic
78C2×Dic53C2×D411yescharacteristic
79D212—24no—
80D212—44no—
81D212—42no—
82D212—42no—
83C4×D53—22no—
84C4×D53—22no—
85C4×D53—44no—
86C4×D53—44no—
87C4×D53SD1614yes—
88Dic106—44no—
89Dic106—42no—
90C8×C53—22no—
91#88322—22no—
92C22×C53C22×D411yescharacteristic
93C2×D53—21no—
94C2×D53—21no—
95C2×D53—42no—
96C2×D53—42no—
97C2×D53—42no—
98C2×D53C2×SD1612yes—
99C4×C53—42no—
100C4×C53—42no—
101C4×C53C2×SD1612yes—
102C4×C53AGL(1,ℤ/8)12yes—
103C4×C532−1+412yes—
104Dic53—21no—
105Dic53—21no—
106Dic53—42no—
107Dic53C2×SD1612yes—
108#3227—531nomaximal, Sylow
109C2×C53#53611yescharacteristic
110C2×C53C22×SD1611yescharacteristic
111C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
112D53—22no—
113D53—82no—
114#532—531no—
115C2×C4⋊Q8—531no—
116#498—531no—
117#498D5311yescharacteristic, complemented
118#486—531no—
119#486—531no—
120#481—534no—
121#476—534no—
122#431—531no—
123C53#322711yescharacteristic, complemented, Sylow
124C2×#41—1061no—
125C22×Q8—1061no—
126C4⋊Q8—532no—
127C4⋊Q8—531no—
128C4⋊Q8—1061no—
129C4⋊Q8—1061no—
130C4⋊Q8C2×D5311yescharacteristic
131C4⋊Q8C2×D5311yescharacteristic
132#127—531no—
133#124—1061no—
134#121—532no—
135#121—1061no—
136C4×D4—532no—
137C4×D4—1061no—
138C2×C4⋊C4—532no—
139C2×C4⋊C4—1061no—
140C2×C42—531no—
141#106—534no—
142#106C2×D5314yes—
143C4⋊C8—532no—
144#102—534no—
145#102—534no—
146C22⋊C8—532no—
147C4×C8—531no—
148C4×C8C2×D5311yescharacteristic
149#41—2124no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—1062no—
153C2×Q8—1061no—
154C2×Q8—2124no—
155C2×D4—532no—
156C2×D4—1062no—
157C2×D4—1061no—
158C2×D4—1061no—
159C22×C4—531no—
160C22×C4—531no—
161C22×C4—531no—
162C22×C4—1062no—
163C22×C4—1062no—
164C2×C8—532no—
165C2×C8C22×D5312yes—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—532no—
169C4⋊C4—1062no—
170C4⋊C4—1062no—
171C4⋊C4—1061no—
172C4⋊C4C22×D5312yes—
173C4⋊C4C22×D5312yes—
174C22⋊C4—1062no—
175C22⋊C4—1062no—
176C42—531no—
177C42—531no—
178C42—531no—
179C42C22×D5311yescharacteristic
180C23—531no—
181C23—1062no—
182Q8—42no—
183Q8—42no—
184Q8—2124no—
185Q8—2122no—
186D4—1064no—
187D4—2124no—
188D4—2122no—
189D4—2122no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—531no—
193C2×C4—534no—
194C2×C4—531no—
195C2×C4—531no—
196C2×C4—1062no—
197C2×C4—1062no—
198C2×C4—1062no—
199C2×C4—2124no—
200C2×C4—2124no—
201C2×C4D4×D5311yescharacteristic
202C2×C4D4×D5311yescharacteristic
203C2×C4C23×D5311yescharacteristic
204C8—22no—
205C8—1062no—
206C22—532no—
207C22—1061no—
208C22—1061no—
209C22—2122no—
210C22—2122no—
211C22—2122no—
212C22C2×D4×D5311yescharacteristic
213C4—42no—
214C4—42no—
215C4—532no—
216C4—1061no—
217C4—1061no—
218C4—2122no—
219C4SD16×D5312yes—
220C4#1313324412yes—
221C4#1313326512yes—
222C2—1062no—
223C2—4242no—
224C2#1681119811yescharacteristic
225C2C2×SD16×D5311yescharacteristic
226C2C2×#1313324411yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#498C2
socleC22×C53C22×D4
hypercenter#498D53
hypocenterC53#3227
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D53C2semidirect
3. #16811019C2semidirect
4. #16810971C2semidirect
5. #16810970C2semidirect
6. #16810943C2semidirect
7. #16810912C2semidirect
8. #16810674C2semidirect
9. C53×#498C2semidirect
117. #498D53semidirect
123. C53#3227semidirect