Finite Group Databasebeta

← #20279712

Subgroups — #20279712

Order 6784.

604 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 157 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×#13133215C211yesmaximal, characteristic, complemented
2C2×#13133206C211yesmaximal, characteristic
3C2×#13133205C211yesmaximal, characteristic
4C2×#13133179C211yesmaximal, characteristic
5#16810552C214yesmaximal, complemented
6#16810528C211yesmaximal, characteristic, complemented
7#16810482C214yesmaximal, complemented
8C4×#11719919C211yesmaximal, characteristic, complemented
9C53×#391C211yesmaximal, characteristic, complemented
10C22×Dic106C2211yescharacteristic
11C22×D212C2211yescharacteristic
12C22×C4×D53C2212yes—
13#13133215—24no—
14#13133206—24no—
15#13133205—24no—
16#13133179—24no—
17C2×#11719919C2212yes—
18C2×#11719919C2212yes—
19C2×#11719919C2212yes—
20C2×#11719921C2212yes—
21C2×#11719921C2214yes—
22C2×#11719921C2214yes—
23C2×C4×Dic53C2211yescharacteristic
24C2×C4×Dic53C2211yescharacteristic
25#13133124C2214yes—
26#13133124C2212yes—
27C22×Q8×C53C2211yescharacteristic
28C2×C4⋊C4×C53C2211yescharacteristic
29C2×C42×C53C2211yescharacteristic
30#95×C53C2214yes—
31C23×D53C2312yes—
32C2×Q8×C53—22no—
33C2×Q8×C53—22no—
34C2×Q8×C53—22no—
35C22×C4×C53C2312yes—
36C22×C4×C53C2312yes—
37C22×C4×C53C2312yes—
38C22×C4×C53C2311yescharacteristic
39C22×Dic53C2312yes—
40C22×Dic53C2312yes—
41C22×Dic53C2311yescharacteristic
42C22×Dic53C2311yescharacteristic
43C2×D212—22no—
44C2×D212—22no—
45C2×D212D414yes—
46C2×C4×D53—24no—
47C2×C4×D53—28no—
48C2×Dic106—22no—
49C2×Dic106—22no—
50C2×Dic106—22no—
51C4⋊C4×C53—22no—
52C42×C53—22no—
53#11719921—28no—
54#11719921—28no—
55#11719921—24no—
56#11719919—24no—
57#11719919—24no—
58#11719919—24no—
59C4×Dic53—22no—
60C4×Dic53D414yes—
61C23×C53C2411yescharacteristic
62C22×D53—22no—
63C22×D53—24no—
64C22×D53—22no—
65C22×D53—24no—
66C22×D53C2×D414yes—
67Q8×C53—44no—
68C2×C4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—24no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53C2×D412yes—
77C2×C4×C53C2×D412yes—
78C2×C4×C53#4112yes—
79C2×Dic53—22no—
80C2×Dic53—22no—
81C2×Dic53—22no—
82C2×Dic53—22no—
83C2×Dic53—22no—
84C2×Dic53—22no—
85C2×Dic53—22no—
86C2×Dic53C2×D412yes—
87C2×Dic53C2×D412yes—
88C2×Dic53#4114yes—
89D212—28no—
90C4×D53—48no—
91Dic106—44no—
92C22×C53C22×D411yescharacteristic
93C22×C53C22×D411yescharacteristic
94C22×C53C2×#4112yes—
95C22×C53C2×#4111yescharacteristic
96C22×C532−1+412yes—
97C2×D53—24no—
98C2×D53—28no—
99C2×D53—42no—
100C2×D53—42no—
101C2×D53—44no—
102C4×C53—42no—
103C4×C53—42no—
104C4×C53—42no—
105C4×C53#12114yes—
106Dic53—22no—
107Dic53—42no—
108Dic53—42no—
109Dic53#12414yescomplemented
110#2547—531nomaximal, Sylow
111C2×C53C2×#12111yescharacteristic
112C2×C53C2×#12411yescharacteristic
113C2×C53#53611yescharacteristic
114C2×C53#54712yes—
115C2×C53#55612yes—
116D53—44no—
117C2×C4⋊Q8—531no—
118C2×#124—531no—
119C2×#124—531no—
120C2×C4×D4—531no—
121#397—534no—
122#391—531no—
123#391D5311yescharacteristic, complemented
124#380—534no—
125C4×C4⋊C4—531no—
126C53#254711yescharacteristic, complemented, Sylow
127C22×Q8—531no—
128C22×Q8C2×D5311yescharacteristic
129C22×D4—531no—
130C23×C4—532no—
131C4⋊Q8—1064no—
132#124—1064no—
133#124—1064no—
134C4×D4—1064no—
135C2×C4⋊C4—532no—
136C2×C4⋊C4—532no—
137C2×C4⋊C4—532no—
138C2×C4⋊C4C2×D5311yescharacteristic
139C2×C22⋊C4—532no—
140C2×C22⋊C4—534no—
141C2×C22⋊C4—534no—
142C2×C42—531no—
143C2×C42—531no—
144C2×C42C2×D5311yescharacteristic
145#95—532no—
146#95—534no—
147#95C2×D5314yes—
148C24—532no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—22no—
152C2×Q8—1062no—
153C2×Q8—1062no—
154C2×Q8—1062no—
155C2×D4—534no—
156C2×D4—1062no—
157C2×D4—1062no—
158C22×C4—531no—
159C22×C4—531no—
160C22×C4—532no—
161C22×C4—532no—
162C22×C4—1064no—
163C22×C4—1068no—
164C22×C4C22×D5312yes—
165C22×C4C22×D5312yes—
166C22×C4C22×D5312yes—
167C22×C4C22×D5311yescharacteristic
168C4⋊C4—22no—
169C4⋊C4—1064no—
170C4⋊C4—1064no—
171C4⋊C4—1064no—
172C22⋊C4—1064no—
173C22⋊C4—1068no—
174C22⋊C4—1068no—
175C42—22no—
176C42—534no—
177C42—1062no—
178C23—534no—
179C23—1062no—
180C23—1064no—
181C23—1062no—
182C23—1064no—
183C23C23×D5311yescharacteristic
184Q8—44no—
185Q8—2124no—
186D4—1068no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—532no—
196C2×C4—534no—
197C2×C4—532no—
198C2×C4—1062no—
199C2×C4—1062no—
200C2×C4—1062no—
201C2×C4—1062no—
202C2×C4—1062no—
203C2×C4—1062no—
204C2×C4—1062no—
205C2×C4—2128no—
206C2×C4#1171994512yes—
207C2×C4D4×D5312yes—
208C2×C4D4×D5312yes—
209C22—1068no—
210C22—1064no—
211C22—2124no—
212C22—2122no—
213C22—2122no—
214C22C2×D4×D5311yescharacteristic
215C22C2×D4×D5311yescharacteristic
216C22#41×D5312yes—
217C22C2×#1171994511yescharacteristic
218C22#1313326512yes—
219C4—42no—
220C4—42no—
221C4—42no—
222C4—534no—
223C4—1062no—
224C4—2122no—
225C4—2122no—
226C4#1313319014yes—
227C2—2124no—
228C2C2×#1313319011yescharacteristic
229C2#124×D5311yescharacteristic
230C2#1681119811yescharacteristic
231C2#1681127512yes—
232C2#1681134512yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#391C2
hypercenter#391D53
hypocenterC53#2547
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#13133215C2semidirect
5. #16810552C2semidirect
6. #16810528C2semidirect
7. #16810482C2semidirect
8. C4×#11719919C2semidirect
9. C53×#391C2semidirect
109. Dic53#124semidirect
123. #391D53semidirect
126. C53#2547semidirect