Finite Group Databasebeta

← #20285799

Subgroups — #20285799

Order 6784.

364 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 109 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#16811236C211yesmaximal, characteristic, complemented
2#16811198C211yesmaximal, characteristic
3#16811017C211yesmaximal, characteristic, complemented
4#16811002C211yesmaximal, characteristic, complemented
5#16810980C211yesmaximal, characteristic, complemented
6#16810848C212yesmaximal, complemented
7#16810832C212yesmaximal, complemented
8#16810788C212yesmaximal
9#16810780C212yesmaximal, complemented
10#16810685C211yesmaximal, characteristic, complemented
11C53×#491C211yesmaximal, characteristic, complemented
12C2×#11719949C2211yescharacteristic
13C2×Q8×D53C2211yescharacteristic
14C2×Dic212C2212yes—
15C2×#11719925C2212yes—
16C2×#11719925C2212yes—
17C2×#11719913C2212yes—
18#13133217C2211yescharacteristic
19#13133215C2211yescharacteristic
20#13133212—22no—
21#13133208—22no—
22#13133206C2211yescharacteristic
23#13133205C2211yescharacteristic
24#13133202—22no—
25#13133197C2212yes—
26#13133195—22no—
27#13133182C2212yes—
28#13133174C2211yescharacteristic
29#13133146C2212yes—
30#13133140C2212yes—
31#13133137C2212yes—
32#13133129C2212yes—
33C4×#88322C2211yescharacteristic
34C4⋊Q8×C53C2211yescharacteristic
35#125×C53C2211yescharacteristic
36#103×C53C2212yes—
37#103×C53C2212yes—
38#97×C53C2211yescharacteristic
39C2×Q8×C53C2311yescharacteristic
40C2×Q8×C53C2311yescharacteristic
41#11719949—42no—
42Q8×D53—42no—
43C2×D212—21no—
44C2×D212C2311yescharacteristic
45C2×C4×D53—21no—
46C2×C4×D53—21no—
47C2×C4×D53C2311yescharacteristic
48C2×Dic106—21no—
49C2×Dic106C2311yescharacteristic
50C2×Dic106C2312yes—
51C2×C8×C53C2312yes—
52C4⋊C4×C53—22no—
53C4⋊C4×C53—22no—
54C4⋊C4×C53C2312yes—
55C42×C53C2311yescharacteristic
56#11719925—42no—
57#11719925—42no—
58#11719921—22no—
59#11719921—22no—
60#11719921—22no—
61#11719921D412yes—
62C4⋊Dic53—22no—
63C4⋊Dic53—22no—
64#11719919—22no—
65#11719919—22no—
66#11719919D412yes—
67C4×Dic53—22no—
68C4×Dic53C2311yescharacteristic
69C2×#88322C2312yes—
70Dic212—42no—
71#11719913—42no—
72C22×D53—21no—
73C22×D53C2×D411yescharacteristic
74Q8×C53—22no—
75Q8×C53—22no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—22no—
79C2×C4×C53C2×D412yes—
80C2×C4×C53C2411yescharacteristic
81C2×Dic53—21no—
82C2×Dic53—22no—
83C2×Dic53C2×D411yescharacteristic
84C2×Dic53C2×D412yes—
85D212—22no—
86D212—42no—
87C4×D53—22no—
88C4×D53—42no—
89C4×D53—42no—
90Dic106—22no—
91Dic106—22no—
92Dic106—22no—
93Dic106—42no—
94C8×C53—22no—
95#88322—22no—
96C22×C53C22×D411yescharacteristic
97C2×D53—21no—
98C2×D53—22no—
99C2×D53—41no—
100C2×D53—42no—
101C4×C53—22no—
102C4×C53—41no—
103C4×C53—41no—
104C4×C53—42no—
105C4×C53H(5,𝔽2)12yes—
106Dic53—21no—
107Dic53—22no—
108Dic53—41no—
109Dic53—42no—
110#3223—531nomaximal, Sylow
111C2×C53#53511yescharacteristic
112C2×C53#57812yes—
113D53—41no—
114D53—81no—
115#541—531no—
116#536—531no—
117#497—531no—
118#494—531no—
119#491D5311yescharacteristic, complemented
120#488—531no—
121#464—532no—
122#462—532no—
123#452—532no—
124#451—532no—
125#433—531no—
126C53#322311yescharacteristic, complemented, Sylow
127C2×#41—531no—
128C22×Q8—531no—
129C2×Q16—532no—
130C2×Q16—532no—
131C2×Q16—532no—
132C2×SD16—532no—
133C4⋊Q8—531no—
134C4⋊Q8—531no—
135C4⋊Q8C2×D5311yescharacteristic
136#126—1062no—
137#125—1062no—
138#125C2×D5311yescharacteristic
139#124—531no—
140#124—531no—
141#123—1062no—
142C22⋊Q8—532no—
143C22⋊Q8—1062no—
144C4×Q8—532no—
145#117—531no—
146C4⋊C8—532no—
147#103—532no—
148#103C2×D5312yes—
149#103C2×D5312yes—
150#102—532no—
151C22⋊C8—532no—
152#97C2×D5311yescharacteristic
153C4×C8—531no—
154#41—2122no—
155C2×Q8—531no—
156C2×Q8—532no—
157C2×Q8—1061no—
158C2×Q8—2122no—
159C2×Q8C22×D5311yescharacteristic
160C2×Q8C22×D5311yescharacteristic
161C2×D4—531no—
162C2×D4—1061no—
163C22×C4—531no—
164C22×C4—1061no—
165C22×C4—1061no—
166Q16—2122no—
167Q16—2122no—
168Q16—2122no—
169SD16—2122no—
170C2×C8—532no—
171C2×C8C22×D5312yes—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—532no—
175C4⋊C4—1062no—
176C4⋊C4—1062no—
177C4⋊C4—1062no—
178C4⋊C4—1062no—
179C4⋊C4C22×D5312yes—
180C22⋊C4—532no—
181C22⋊C4—1062no—
182C22⋊C4—1062no—
183C22⋊C4—1062no—
184C42—531no—
185C42—1062no—
186C42C22×D5311yescharacteristic
187C23—531no—
188C23—1061no—
189Q8—22no—
190Q8—22no—
191Q8—1062no—
192Q8—1062no—
193Q8—1062no—
194Q8—2122no—
195D4—1062no—
196D4—2122no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—532no—
201C2×C4—531no—
202C2×C4—1062no—
203C2×C4—1062no—
204C2×C4—1061no—
205C2×C4—2122no—
206C2×C4—2122no—
207C2×C4D4×D5312yes—
208C2×C4C23×D5311yescharacteristic
209C8—22no—
210C8—1062no—
211C22—1062no—
212C22—1061no—
213C22—2122no—
214C22—2121no—
215C22C2×D4×D5311yescharacteristic
216C4—22no—
217C4—42no—
218C4—41no—
219C4—41no—
220C4—1062no—
221C4—1061no—
222C4—2121no—
223C4—2122no—
224C4#1313326312yes—
225C2—4241no—
226C2—2121no—
227C2#1681119111yescharacteristic
228C2#1681143912yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#491C2
socleC22×C53C22×D4
hypercenter#491D53
hypocenterC53#3223
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
1. #16811236C2semidirect
3. #16811017C2semidirect
4. #16811002C2semidirect
5. #16810980C2semidirect
6. #16810848C2semidirect
7. #16810832C2semidirect
9. #16810780C2semidirect
10. #16810685C2semidirect
11. C53×#491C2semidirect
119. #491D53semidirect
126. C53#3223semidirect