Finite Group Databasebeta

← #20285744

Subgroups — #20285744

Order 6784.

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

normal subgroups only

This group has 386 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
1C2×#13133217C211yesmaximal, characteristic, complemented
2#16811157C211yesmaximal, characteristic
3#16811017C212yesmaximal, complemented
4#16810992C212yesmaximal, complemented
5#16810834C212yesmaximal, complemented
6#16810829C212yesmaximal, complemented
7#16810781C212yesmaximal, complemented
8#16810764C212yesmaximal
9C53×#432C211yesmaximal, characteristic, complemented
10C22×Dic106C2212yes—
11C2×Dic212C2214yes—
12C2×#11719913C2214yes—
13#13133217—21no—
14#13133217—21no—
15#13133217C2212yes—
16#13133217C2212yes—
17#13133211—21no—
18#13133209C2211yescharacteristic
19#13133205C2211yescharacteristic
20#13133200—21no—
21#13133196C2211yescharacteristic
22#13133191C2211yescharacteristic
23C4×Dic106C2211yescharacteristic
24C4×D212C2211yescharacteristic
25C4×C22⋊D53—21no—
26#13133174—21no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53—21no—
29#13133142C2214yes—
30#13133142C2214yes—
31C2×C42×C53C2211yescharacteristic
32C4⋊C8×C53C2212yes—
33C22⋊C8×C53C2212yes—
34#97×C53C2212yes—
35C22×C4×C53D411yescharacteristic
36C22×C4×C53D411yescharacteristic
37C22×C4×C53C2311yescharacteristic
38C2×C22⋊D53—21no—
39C22×Dic53—22no—
40C2×D212C2311yescharacteristic
41C2×C4×D53—21no—
42C2×Dic106—22no—
43C2×Dic106—44no—
44C2×Dic106C2314yes—
45C2×Dic106C2311yescharacteristic
46C2×C8×C53C2314yes—
47C42×C53D411yescharacteristic
48C42×C53D411yescharacteristic
49C42×C53C2311yescharacteristic
50C42×C53C2311yescharacteristic
51C22⋊Dic53—21no—
52#11719921—21no—
53#11719921—21no—
54#11719921—21no—
55#11719921—21no—
56C4⋊Dic53—22no—
57C4⋊Dic53—22no—
58C4⋊Dic53—22no—
59C4⋊Dic53—22no—
60C4⋊Dic53C2311yescharacteristic
61C4⋊Dic53C2311yescharacteristic
62#11719919—21no—
63#11719919—21no—
64#11719919—21no—
65#11719919—21no—
66C4×Dic53—21no—
67Dic212—44no—
68#11719913—44no—
69C23×C53C2×D411yescharacteristic
70C22×D53—21no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53C2×D411yescharacteristic
75C2×C4×C53C2×D411yescharacteristic
76C2×C4×C53C2×D411yescharacteristic
77C2×C4×C53C2×D411yescharacteristic
78C2×C4×C53C2×D411yescharacteristic
79C2×C4×C53C2411yescharacteristic
80C22⋊D53—42no—
81C2×Dic53—24no—
82C2×Dic53—21no—
83C2×Dic53—21no—
84C2×Dic53—21no—
85C2×Dic53—44no—
86D212—22no—
87C4×D53—42no—
88Dic106—24no—
89Dic106—24no—
90Dic106—22no—
91Dic106—44no—
92C8×C53—24no—
93C22×C53—21no—
94C22×C53—21no—
95C22×C53#13712yes—
96C22×C53C22×D411yescharacteristic
97C2×D53—41no—
98C2×D53—41no—
99C2×D53—41no—
100C4×C53—21no—
101C4×C53—21no—
102C4×C53—21no—
103C4×C53—21no—
104C4×C53#13712yes—
105C4×C53H(5,𝔽2)12yes—
106Dic53—44no—
107Dic53—41no—
108Dic53—41no—
109Dic53—41no—
110#3218—531nomaximal, Sylow
111C2×C53—22no—
112C2×C53#53511yescharacteristic
113C2×C53C2×#13711yescharacteristic
114C2×C53C2×#13711yescharacteristic
115D53—81no—
116C2×C4⋊Q8—531no—
117#529—531no—
118#497—532no—
119#491—532no—
120#462—532no—
121#461—532no—
122#451—532no—
123#448—532no—
124#432D5311yescharacteristic, complemented
125C53#321811yescharacteristic, complemented, Sylow
126C22×Q8—532no—
127C2×Q16—534no—
128C2×SD16—534no—
129C4⋊Q8—532no—
130C4⋊Q8—532no—
131C4⋊Q8—1061no—
132C4⋊Q8—1061no—
133#126—1061no—
134#125—531no—
135#124—531no—
136#123—1061no—
137C22⋊Q8—531no—
138#121—531no—
139C4×Q8—531no—
140C4×D4—531no—
141C4×D4—1061no—
142#117—1061no—
143C2×C4⋊C4—1061no—
144C2×C4⋊C4—1061no—
145C2×C42C2×D5311yescharacteristic
146C4⋊C8C2×D5312yes—
147#103—534no—
148#103—534no—
149C22⋊C8C2×D5312yes—
150#97C2×D5312yes—
151C2×Q8—534no—
152C2×Q8—531no—
153C2×Q8—1062no—
154C2×Q8—2124no—
155C2×D4—531no—
156C2×D4—1061no—
157C22×C4—1062no—
158C22×C4—1061no—
159C22×C4D21211yescharacteristic
160C22×C4D21211yescharacteristic
161C22×C4C22×D5311yescharacteristic
162Q16—2124no—
163SD16—2124no—
164C2×C8C22×D5314yes—
165C4⋊C4—531no—
166C4⋊C4—531no—
167C4⋊C4—1062no—
168C4⋊C4—1062no—
169C4⋊C4—1062no—
170C4⋊C4—1062no—
171C4⋊C4—1061no—
172C4⋊C4—1061no—
173C4⋊C4—1061no—
174C4⋊C4—1061no—
175C22⋊C4—1061no—
176C22⋊C4—1061no—
177C22⋊C4—1061no—
178C22⋊C4—1061no—
179C22⋊C4—1061no—
180C42—1061no—
181C42D21211yescharacteristic
182C42D21211yescharacteristic
183C42C22×D5311yescharacteristic
184C42C22×D5311yescharacteristic
185C23—1061no—
186C23C2×D21211yescharacteristic
187Q8—1064no—
188Q8—1062no—
189Q8—1064no—
190Q8—2124no—
191D4—1062no—
192D4—2122no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—1061no—
197C2×C4—1061no—
198C2×C4—1061no—
199C2×C4—1064no—
200C2×C4—2124no—
201C2×C4—2122no—
202C2×C4C2×D21211yescharacteristic
203C2×C4C2×D21211yescharacteristic
204C2×C4C2×D21211yescharacteristic
205C2×C4C2×D21211yescharacteristic
206C2×C4C2×D21211yescharacteristic
207C2×C4C23×D5311yescharacteristic
208C8—24no—
209C22—21no—
210C22—21no—
211C22—2121no—
212C22—2121no—
213C22—2121no—
214C22#1313325012yes—
215C22C22×D21211yescharacteristic
216C4—21no—
217C4—21no—
218C4—21no—
219C4—21no—
220C4—2121no—
221C4—2121no—
222C4—2124no—
223C4—2121no—
224C4#1313325012yes—
225C4#1313326312yes—
226C2—22no—
227C2—4241no—
228C2#1681119211yescharacteristic
229C2C2×#1313325011yescharacteristic
230C2C2×#1313325011yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#432C2
socleC22×C53C22×D4
hypercenter#432D53
hypocenterC53#3218
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
1. C2×#13133217C2semidirect
3. #16811017C2semidirect
4. #16810992C2semidirect
5. #16810834C2semidirect
6. #16810829C2semidirect
7. #16810781C2semidirect
9. C53×#432C2semidirect
124. #432D53semidirect
125. C53#3218semidirect