Finite Group Databasebeta

← #20285792

Subgroups — #20285792

Order 6784.

364 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 117 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#16811199C211yesmaximal, characteristic
2#16811015C211yesmaximal, characteristic, complemented
3#16811002C211yesmaximal, characteristic, complemented
4#16810993C211yesmaximal, characteristic, complemented
5#16810980C211yesmaximal, characteristic, complemented
6#16810845C212yesmaximal, complemented
7#16810833C212yesmaximal, complemented
8#16810783C212yesmaximal
9#16810778C212yesmaximal, complemented
10#16810684C211yesmaximal, characteristic, complemented
11C53×#541C211yesmaximal, characteristic, complemented
12C2×#11719945C2211yescharacteristic
13C22×Dic106C2211yescharacteristic
14C2×#11719925C2212yes—
15C2×#11719925C2212yes—
16C2×#11719925C2212yes—
17C2×#11719924C2212yes—
18#13133217C2211yescharacteristic
19#13133217C2211yescharacteristic
20#13133205C2211yescharacteristic
21#13133205C2211yescharacteristic
22#13133201—22no—
23#13133197—22no—
24C4⋊#88322C2212yes—
25#13133140C2212yes—
26#13133140C2212yes—
27#13133140C2212yes—
28#13133137C2212yes—
29C22⋊#88322C2212yes—
30#13133128C2211yescharacteristic
31C4×#88322C2211yescharacteristic
32C4⋊Q8×C53C2211yescharacteristic
33#126×C53—22no—
34#125×C53—22no—
35#125×C53C2211yescharacteristic
36C22⋊Q8×C53C2212yes—
37C4×Q8×C53C2212yes—
38#117×C53C2211yescharacteristic
39C2×Q8×C53C2312yes—
40C22×C4×C53C2311yescharacteristic
41C2×C22⋊D53—21no—
42C22×Dic53—21no—
43#11719945—42no—
44C2×D212C2311yescharacteristic
45C2×C4×D53—21no—
46C2×Dic106—21no—
47C2×Dic106—42no—
48C2×Dic106C2311yescharacteristic
49C2×Dic106C2311yescharacteristic
50C2×Dic106C2311yescharacteristic
51C4⋊C4×C53—22no—
52C4⋊C4×C53—22no—
53C4⋊C4×C53—22no—
54C4⋊C4×C53—22no—
55C4⋊C4×C53D412yes—
56C4⋊C4×C53C2312yes—
57C22⋊C4×C53—22no—
58C22⋊C4×C53D412yes—
59C42×C53—22no—
60C42×C53C2311yescharacteristic
61C42×C53C2311yescharacteristic
62#11719925—42no—
63#11719925—42no—
64#11719925—42no—
65#11719924—42no—
66#11719921—22no—
67#11719921—22no—
68C4⋊Dic53—22no—
69C4⋊Dic53—22no—
70C2×#88322C2312yes—
71C2×#88322C2312yes—
72C23×C53C2×D411yescharacteristic
73C22×D53—21no—
74Q8×C53—22no—
75Q8×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53C2×D411yescharacteristic
80C2×C4×C53C2×D412yes—
81C2×C4×C53C2×D412yes—
82C2×C4×C53C2411yescharacteristic
83C22⋊D53—42no—
84C2×Dic53—21no—
85C2×Dic53—21no—
86C2×Dic53—21no—
87C2×Dic53—42no—
88D212—22no—
89C4×D53—42no—
90Dic106—22no—
91Dic106—22no—
92Dic106—22no—
93Dic106—42no—
94#88322—22no—
95#88322—22no—
96C22×C53—21no—
97C22×C53—22no—
98C22×C53C22×D411yescharacteristic
99C2×D53—41no—
100C2×D53—42no—
101C4×C53—21no—
102C4×C53—22no—
103C4×C53—22no—
104C4×C53—42no—
105C4×C53—42no—
106C4×C53H(5,𝔽2)12yes—
107Dic53—41no—
108Dic53—41no—
109Dic53—41no—
110#3223—531nomaximal, Sylow
111C2×C53—41no—
112C2×C53#53511yescharacteristic
113C2×C53#57812yes—
114D53—81no—
115#541D5311yescharacteristic, complemented
116#536—531no—
117#497—531no—
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×Q16—532no—
132C2×SD16—532no—
133C4⋊Q8—531no—
134C4⋊Q8—531no—
135C4⋊Q8C2×D5311yescharacteristic
136#126—22no—
137#125—22no—
138#125C2×D5311yescharacteristic
139#124—531no—
140#124—531no—
141#123—1062no—
142C22⋊Q8—1062no—
143C22⋊Q8C2×D5312yes—
144C4×Q8C2×D5312yes—
145#117C2×D5311yescharacteristic
146C4⋊C8—532no—
147#103—532no—
148#103—532no—
149#103—532no—
150#102—532no—
151C22⋊C8—532no—
152#97—531no—
153C4×C8—531no—
154#41—2122no—
155C2×Q8—531no—
156C2×Q8—531no—
157C2×Q8—531no—
158C2×Q8—1061no—
159C2×Q8—2122no—
160C2×Q8C22×D5312yes—
161C2×D4—531no—
162C2×D4—1061no—
163C22×C4—1061no—
164C22×C4—1061no—
165C22×C4C22×D5311yescharacteristic
166Q16—2122no—
167Q16—2122no—
168Q16—2122no—
169SD16—2122no—
170C2×C8—532no—
171C2×C8—532no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—1062no—
177C4⋊C4—1062no—
178C4⋊C4C22⋊D5312yes—
179C4⋊C4C22×D5312yes—
180C22⋊C4—22no—
181C22⋊C4—1062no—
182C22⋊C4—1062no—
183C22⋊C4C22⋊D5312yes—
184C42—22no—
185C42C22×D5311yescharacteristic
186C42C22×D5311yescharacteristic
187C23—1061no—
188C23C2×C22⋊D5311yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—1062no—
192Q8—1062no—
193Q8—1062no—
194Q8—2122no—
195D4—1062no—
196D4—2122no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—1061no—
201C2×C4—1061no—
202C2×C4—1061no—
203C2×C4—2122no—
204C2×C4—2122no—
205C2×C4C2×C22⋊D5312yes—
206C2×C4C2×C22⋊D5312yes—
207C2×C4C2×C22⋊D5311yescharacteristic
208C2×C4C23×D5311yescharacteristic
209C8—1062no—
210C8—1062no—
211C22—22no—
212C22—21no—
213C22—2122no—
214C22—2121no—
215C22C22×C22⋊D5311yescharacteristic
216C4—22no—
217C4—22no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—2121no—
222C4—2121no—
223C4—2121no—
224C4#1313326312yes—
225C2—41no—
226C2—4241no—
227C2#1681118711yescharacteristic
228C2#1681143812yes—
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#541C2
socleC22×C53C22×D4
hypercenter#541D53
hypocenterC53#3223
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #16811015C2semidirect
3. #16811002C2semidirect
4. #16810993C2semidirect
5. #16810980C2semidirect
6. #16810845C2semidirect
7. #16810833C2semidirect
9. #16810778C2semidirect
10. #16810684C2semidirect
11. C53×#541C2semidirect
115. #541D53semidirect
126. C53#3223semidirect