Finite Group Databasebeta

← #20277525

Subgroups — #20277525

Order 6784.

540 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 177 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×#13133205C211yesmaximal, characteristic
2#16810527C212yesmaximal, complemented
3#16810515C212yesmaximal, complemented
4#16810512C212yesmaximal, complemented
5#16810501C212yesmaximal
6#16810497C212yesmaximal, complemented
7C4×C4⋊Dic53C211yesmaximal, characteristic, complemented
8#16810470C212yesmaximal, complemented
9C53×#383C211yesmaximal, characteristic, complemented
10C22×Dic106C2211yescharacteristic
11C22×D212C2211yescharacteristic
12C22×C4×D53C2212yes—
13#13133205C418yescomplemented
14C2×C4⋊Dic53C2212yes—
15C2×#11719919C2212yes—
16C2×#11719921C2212yes—
17C2×#11719921C2212yes—
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20C2×#11719921C2212yes—
21C2×C4×Dic53C2211yescharacteristic
22C2×C4×Dic53C2212yes—
23C2×C4×Dic53C2211yescharacteristic
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27C2×C42×C53C2211yescharacteristic
28#95×C53C2211yescharacteristic
29#95×C53C2212yes—
30#95×C53C2212yes—
31#95×C53C2211yescharacteristic
32C23×D53C2312yes—
33C22×C4×C53C2312yes—
34C22×C4×C53C2312yes—
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2311yescharacteristic
37C22×C4×C53C2311yescharacteristic
38C22×Dic53C2312yes—
39C22×Dic53C2312yes—
40C22×Dic53C2312yes—
41C2×D212—42no—
42C2×D212C2×C414yes—
43C2×C4×D53—24no—
44C2×C4×D53—42no—
45C2×C4×D53—42no—
46C2×Dic106—42no—
47C2×Dic106C2×C414yescomplemented
48C42×C53C2×C414yescomplemented
49#11719921—24no—
50#11719921—42no—
51#11719921—42no—
52#11719921C2×C418yescomplemented
53#11719921C2×C418yescomplemented
54C4⋊Dic53—24no—
55#11719919—42no—
56C4×Dic53—42no—
57C4×Dic53—41no—
58C4×Dic53—41no—
59C23×C53C2411yescharacteristic
60C22×D53—22no—
61C22×D53—22no—
62C22×D53—22no—
63C22×D53—22no—
64C22×D53—22no—
65C22×D53—22no—
66C22×D53C22×C414yes—
67C2×C4×C53—22no—
68C2×C4×C53—22no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53C22×C412yes—
78C2×C4×C53C22×C412yes—
79C2×C4×C53C22×C412yes—
80C2×Dic53—22no—
81C2×Dic53—22no—
82C2×Dic53—22no—
83C2×Dic53—22no—
84C2×Dic53—22no—
85C2×Dic53—22no—
86C2×Dic53—22no—
87C2×Dic53C22×C414yes—
88C2×Dic53#4114yes—
89D212—44no—
90C4×D53—48no—
91Dic106—44no—
92C22×C53C23×C411yescharacteristic
93C22×C53C2×#4112yes—
94C22×C53H(5,𝔽2)12yes—
95C22×C53H(5,𝔽2)11yescharacteristic
96C22×C532−1+411yescharacteristic
97C2×D53—24no—
98C2×D53—24no—
99C2×D53—24no—
100C2×D53—42no—
101C2×D53—42no—
102C2×D53—42no—
103C2×D53—42no—
104C4×C53—22no—
105C4×C53—22no—
106C4×C53—22no—
107C4×C53—42no—
108C4×C53—42no—
109Dic53—24no—
110Dic53—24no—
111Dic53—42no—
112#2337—531nomaximal, Sylow
113C2×C53C4×#4111yescharacteristic
114C2×C53#51811yescharacteristic
115C2×C53#52011yescharacteristic
116C2×C53#53812yes—
117C2×C53#54212yes—
118D53—44no—
119C2×#124—531no—
120#391—532no—
121#388—532no—
122#388—532no—
123#386—532no—
124#385—532no—
125#383D5311yescharacteristic, complemented
126C4×C4⋊C4—531no—
127C4×C22⋊C4—532no—
128C53#233711yescharacteristic, complemented, Sylow
129C22×Q8—531no—
130C22×D4—531no—
131C23×C4—532no—
132#124—538no—
133C2×C4⋊C4—532no—
134C2×C4⋊C4—532no—
135C2×C22⋊C4—532no—
136C2×C22⋊C4—532no—
137C2×C22⋊C4—532no—
138C2×C22⋊C4—532no—
139C2×C22⋊C4—532no—
140C2×C42—531no—
141C2×C42—532no—
142C2×C42—531no—
143C2×C42C2×D5311yescharacteristic
144#95—532no—
145#95—532no—
146#95—532no—
147#95C2×D5312yes—
148#95C2×D5311yescharacteristic
149#95C2×D5312yes—
150#95C2×D5311yescharacteristic
151C24—532no—
152C2×Q8—534no—
153C2×Q8—2122no—
154C2×D4—534no—
155C2×D4—2122no—
156C22×C4—532no—
157C22×C4—532no—
158C22×C4—532no—
159C22×C4—1064no—
160C22×C4—2122no—
161C22×C4—2122no—
162C22×C4C22×D5312yes—
163C22×C4C22×D5312yes—
164C22×C4C22×D5311yescharacteristic
165C22×C4C22×D5311yescharacteristic
166C22×C4C22×D5311yescharacteristic
167C4⋊C4—1064no—
168C4⋊C4—2122no—
169C22⋊C4—538no—
170C22⋊C4—538no—
171C22⋊C4—1064no—
172C22⋊C4—2122no—
173C22⋊C4—2122no—
174C42—2121no—
175C42—2121no—
176C42—2122no—
177C42C4×D5314yescomplemented
178C23—534no—
179C23—1062no—
180C23—1062no—
181C23—1062no—
182C23—1062no—
183C23—1062no—
184C23—1062no—
185C23C23×D5311yescharacteristic
186Q8—2124no—
187D4—2124no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—534no—
199C2×C4—534no—
200C2×C4—1062no—
201C2×C4—1062no—
202C2×C4—1062no—
203C2×C4—1062no—
204C2×C4—1062no—
205C2×C4—1062no—
206C2×C4—1062no—
207C2×C4—2128no—
208C2×C4C2×C4×D5312yes—
209C2×C4C2×C4×D5312yes—
210C2×C4C2×C4×D5312yes—
211C22—1064no—
212C22—1064no—
213C22—1064no—
214C22—2122no—
215C22—2122no—
216C22—2122no—
217C22—2122no—
218C22C22×C4×D5311yescharacteristic
219C22#41×D5312yes—
220C22#1313326312yes—
221C22#1313326311yescharacteristic
222C22#1313326411yescharacteristic
223C4—22no—
224C4—22no—
225C4—22no—
226C4—42no—
227C4—42no—
228C4—1064no—
229C4—1064no—
230C4—2122no—
231C2—2124no—
232C2#1681109111yescharacteristic
233C2#1681109711yescharacteristic
234C2#1681110311yescharacteristic
235C2#1681121212yes—
236C2#1681123912yes—
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC53×#383C2
socleC23×C53C24
hypercenter#383D53
hypocenterC53#2337
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #383

Complements

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

NH ≅ G/Ntype
2. #16810527C2semidirect
3. #16810515C2semidirect
4. #16810512C2semidirect
6. #16810497C2semidirect
7. C4×C4⋊Dic53C2semidirect
8. #16810470C2semidirect
9. C53×#383C2semidirect
13. #13133205C4semidirect
47. C2×Dic106C2×C4semidirect
48. C42×C53C2×C4semidirect
52. #11719921C2×C4semidirect
53. #11719921C2×C4semidirect
125. #383D53semidirect
128. C53#2337semidirect
177. C42C4×D53semidirect