Finite Group Databasebeta

← #20285793

Subgroups — #20285793

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#16811234C211yesmaximal, characteristic, complemented
2#16811193C211yesmaximal, characteristic
3#16811002C211yesmaximal, characteristic, complemented
4#16810992C211yesmaximal, characteristic, complemented
5#16810977C211yesmaximal, characteristic, complemented
6#16810843C212yesmaximal, complemented
7#16810831C212yesmaximal, complemented
8#16810782C212yesmaximal
9#16810777C212yesmaximal, complemented
10#16810685C211yesmaximal, characteristic, complemented
11C53×#497C211yesmaximal, characteristic, complemented
12C2×#11719947C2211yescharacteristic
13C2×Q8×D53C2211yescharacteristic
14C2×#11719925C2212yes—
15C2×#11719925C2212yes—
16#13133217C2211yescharacteristic
17#13133215C2211yescharacteristic
18#13133212—22no—
19#13133209C2211yescharacteristic
20#13133207—22no—
21#13133204C2211yescharacteristic
22#13133199—22no—
23#13133194—22no—
24#13133193C2212yes—
25Q8×Dic53C2212yes—
26#13133174C2211yescharacteristic
27#13133146C2212yes—
28#13133142C2212yes—
29#13133142C2212yes—
30#13133141C2212yes—
31#13133138C2212yes—
32#13133129C2212yes—
33C4×#88322C2211yescharacteristic
34C2×Q16×C53C2212yes—
35C2×SD16×C53C2212yes—
36C4⋊Q8×C53C2211yescharacteristic
37#124×C53C2211yescharacteristic
38#97×C53C2211yescharacteristic
39C2×Q8×C53C2311yescharacteristic
40C2×Q8×C53C2312yes—
41C2×D4×C53C2311yescharacteristic
42C2×C22⋊D53—21no—
43C22×Dic53—21no—
44Q8×D53—42no—
45#11719947—42no—
46C2×C4×D53—21no—
47C2×C4×D53C2311yescharacteristic
48C2×Dic106—21no—
49C2×Dic106C2311yescharacteristic
50C2×Dic106C2311yescharacteristic
51Q16×C53—42no—
52SD16×C53—42no—
53C2×C8×C53C2312yes—
54C4⋊C4×C53—22no—
55C22⋊C4×C53—22no—
56C42×C53C2311yescharacteristic
57C22⋊Dic53—22no—
58#11719925—42no—
59#11719925—42no—
60#11719921—22no—
61#11719921D412yes—
62C4⋊Dic53—22no—
63C4⋊Dic53—22no—
64C4⋊Dic53C2312yes—
65#11719919—22no—
66#11719919—22no—
67#11719919—22no—
68#11719919D412yes—
69C4×Dic53—22no—
70C4×Dic53C2311yescharacteristic
71C2×#88322C2312yes—
72C23×C53—21no—
73C22×D53C2×D411yescharacteristic
74Q8×C53—22no—
75Q8×C53—22no—
76Q8×C53—22no—
77D4×C53—22no—
78C2×C4×C53—21no—
79C2×C4×C53—22no—
80C2×C4×C53C2×D412yes—
81C2×C4×C53C2411yescharacteristic
82C22⋊D53—42no—
83C2×Dic53—21no—
84C2×Dic53—21no—
85C2×Dic53—22no—
86C2×Dic53—42no—
87C2×Dic53C2×D411yescharacteristic
88C2×Dic53C2×D412yes—
89C4×D53—22no—
90C4×D53—42no—
91Dic106—22no—
92Dic106—22no—
93Dic106—42no—
94C8×C53—22no—
95#88322—22no—
96C22×C53—41no—
97C22×C53—42no—
98C22×C53C22×D411yescharacteristic
99C2×D53—21no—
100C2×D53—22no—
101C4×C53—22no—
102C4×C53—41no—
103C4×C53—42no—
104C4×C53H(5,𝔽2)12yes—
105Dic53—21no—
106Dic53—22no—
107Dic53—41no—
108Dic53—41no—
109Dic53—42no—
110#3223—531nomaximal, Sylow
111C2×C53—81no—
112C2×C53#53511yescharacteristic
113C2×C53#57812yes—
114D53—41no—
115#541—531no—
116#536—531no—
117#497D5311yescharacteristic, complemented
118#494—531no—
119#491—531no—
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×Q16C2×D5312yes—
132C2×SD16C2×D5312yes—
133C4⋊Q8—531no—
134C4⋊Q8—531no—
135C4⋊Q8C2×D5311yescharacteristic
136#126—1062no—
137#125—531no—
138#125—1062no—
139#124—531no—
140#124C2×D5311yescharacteristic
141#123—1062no—
142C22⋊Q8—532no—
143C22⋊Q8—1062no—
144C4×Q8—532no—
145#117—531no—
146C4⋊C8—532no—
147#103—532no—
148#103—532no—
149#103—532no—
150#102—532no—
151C22⋊C8—532no—
152#97C2×D5311yescharacteristic
153C4×C8—531no—
154#41—2122no—
155C2×Q8—531no—
156C2×Q8—531no—
157C2×Q8—1061no—
158C2×Q8—2122no—
159C2×Q8C22×D5311yescharacteristic
160C2×Q8C22×D5312yes—
161C2×D4—1061no—
162C2×D4C22×D5311yescharacteristic
163C22×C4—531no—
164C22×C4—1061no—
165C22×C4—1061no—
166Q16—42no—
167Q16—2122no—
168Q16—2122no—
169SD16—42no—
170C2×C8—532no—
171C2×C8C22×D5312yes—
172C4⋊C4—22no—
173C4⋊C4—532no—
174C4⋊C4—532no—
175C4⋊C4—1062no—
176C4⋊C4—1062no—
177C4⋊C4—1062no—
178C4⋊C4—1062no—
179C4⋊C4—1062no—
180C22⋊C4—22no—
181C22⋊C4—532no—
182C22⋊C4—1062no—
183C22⋊C4—1062no—
184C42—531no—
185C42—1062no—
186C42C22×D5311yescharacteristic
187C23—21no—
188C23—531no—
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—1062no—
193Q8—1062no—
194Q8—2122no—
195D4—22no—
196D4—2122no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—532no—
200C2×C4—531no—
201C2×C4—1062no—
202C2×C4—1062no—
203C2×C4—1061no—
204C2×C4—1061no—
205C2×C4—2122no—
206C2×C4—2122no—
207C2×C4D4×D5312yes—
208C2×C4C23×D5311yescharacteristic
209C8—22no—
210C8—1062no—
211C22—42no—
212C22—41no—
213C22—1062no—
214C22—1061no—
215C22C2×D4×D5311yescharacteristic
216C4—22no—
217C4—41no—
218C4—42no—
219C4—1062no—
220C4—1061no—
221C4—2122no—
222C4—2121no—
223C4—2121no—
224C4#1313326212yes—
225C2—81no—
226C2—2121no—
227C2#1681118611yescharacteristic
228C2#1681143912yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#497C2
socleC22×C53C22×D4
hypercenter#497D53
hypocenterC53#3223
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #16811234C2semidirect
3. #16811002C2semidirect
4. #16810992C2semidirect
5. #16810977C2semidirect
6. #16810843C2semidirect
7. #16810831C2semidirect
9. #16810777C2semidirect
10. #16810685C2semidirect
11. C53×#497C2semidirect
117. #497D53semidirect
126. C53#3223semidirect