Finite Group Databasebeta

← #20283063

Subgroups — #20283063

Order 6784.

644 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 201 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×#13133196C211yesmaximal, characteristic
3C22×C4×Dic53C211yesmaximal, characteristic
4#16810530C211yesmaximal, characteristic, complemented
5#16810528C211yesmaximal, characteristic, complemented
6#16810518C214yesmaximal, complemented
7#16810492C211yesmaximal, characteristic, complemented
8#16810481C214yesmaximal
9C2×C22⋊Q8×C53C211yesmaximal, characteristic
10C22×Dic106C2211yescharacteristic
11C23×Dic53C2212yes—
12#13133215—24no—
13#13133196—24no—
14C2×C4⋊Dic53C2211yescharacteristic
15C2×#11719919C2212yes—
16C2×#11719919C2214yes—
17C2×#11719919C2214yes—
18C2×C22⋊Dic53C2212yes—
19C2×C4×Dic53—24no—
20C2×C4×Dic53C2212yes—
21C2×C4×Dic53C2211yescharacteristic
22C2×C4×Dic53C2211yescharacteristic
23#13133124C2214yes—
24#13133124C2214yes—
25C22×Q8×C53C2211yescharacteristic
26C23×C4×C53C2211yescharacteristic
27C22⋊Q8×C53—24no—
28C2×C4⋊C4×C53C2212yes—
29C2×C4⋊C4×C53C2211yescharacteristic
30C2×C22⋊C4×C53C2212yes—
31C24×C53C2311yescharacteristic
32C2×Q8×C53—22no—
33C2×Q8×C53—22no—
34C2×Q8×C53—22no—
35C22×C4×C53—22no—
36C22×C4×C53—22no—
37C22×C4×C53D414yes—
38C22×C4×C53C2312yes—
39C22×C4×C53C2312yes—
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2311yescharacteristic
42C22×Dic53—24no—
43C22×Dic53—24no—
44C22×Dic53Q818yescomplemented
45C22×Dic53C2312yes—
46C22×Dic53C2312yes—
47C22×Dic53C2312yes—
48C22×Dic53C2312yes—
49C2×Dic106—22no—
50C2×Dic106—22no—
51C2×Dic106—22no—
52C4⋊C4×C53—24no—
53C4⋊C4×C53—22no—
54C22⋊C4×C53—24no—
55C22⋊Dic53—24no—
56C4⋊Dic53—22no—
57#11719919—24no—
58#11719919—28no—
59#11719919—28no—
60C4×Dic53—22no—
61C4×Dic53—24no—
62C4×Dic53D414yes—
63C23×C53—21no—
64C23×C53—22no—
65C23×C53—21no—
66C23×C53C2×D412yes—
67C23×C53C2×Q814yes—
68C23×C53C2411yescharacteristic
69Q8×C53—44no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—28no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—24no—
78C2×C4×C53C2×D412yes—
79C2×C4×C53C2×D412yes—
80C2×C4×C53C2×D412yes—
81C2×C4×C53#4112yes—
82C2×Dic53—22no—
83C2×Dic53—22no—
84C2×Dic53—216no—
85C2×Dic53—22no—
86C2×Dic53—22no—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53—24no—
90C2×Dic53C2×D414yes—
91C2×Dic53C2×Q814yes—
92C2×Dic53C2×Q814yes—
93C2×Dic53#4114yes—
94Dic106—44no—
95C22×C53—22no—
96C22×C53—24no—
97C22×C53—22no—
98C22×C53—24no—
99C22×C53C4⋊Q814yescomplemented
100C22×C53C22×D411yescharacteristic
101C22×C53C22×D411yescharacteristic
102C22×C53C22×Q812yes—
103C22×C53C2×#4112yes—
104C22×C53C2×#4111yescharacteristic
105C4×C53—22no—
106C4×C53—42no—
107C4×C53—42no—
108C4×C53#12114yes—
109Dic53—24no—
110Dic53—42no—
111Dic53—42no—
112Dic53C22⋊Q818yescomplemented
113#2857—531nomaximal, Sylow
114C2×C53—24no—
115C2×C53C2×#12111yescharacteristic
116C2×C53C2×C22⋊Q812yes—
117C2×C53C2×C4⋊Q811yescharacteristic
118C2×C53#53211yescharacteristic
119C2×C53#53312yes—
120C2×C4⋊Q8—531no—
121C2×C22⋊Q8—531no—
122C2×C22⋊Q8D5311yescharacteristic
123C22×C42—531no—
124#391—531no—
125#391—531no—
126#389—534no—
127#384—531no—
128#380—534no—
129C53#285711yescharacteristic, complemented, Sylow
130C22×Q8—531no—
131C22×Q8C2×D5311yescharacteristic
132C23×C4—532no—
133C23×C4C2×D5311yescharacteristic
134C4⋊Q8—1064no—
135C22⋊Q8—24no—
136C22⋊Q8—1064no—
137C2×C4⋊C4—532no—
138C2×C4⋊C4—531no—
139C2×C4⋊C4—534no—
140C2×C4⋊C4—534no—
141C2×C4⋊C4C2×D5311yescharacteristic
142C2×C4⋊C4C2×D5312yes—
143C2×C22⋊C4—532no—
144C2×C22⋊C4C2×D5312yes—
145C2×C42—531no—
146C2×C42—532no—
147C2×C42—531no—
148C2×C42—1064no—
149#95—534no—
150#95—534no—
151C24C22×D5311yescharacteristic
152C2×Q8—22no—
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—1062no—
156C2×Q8—1062no—
157C2×Q8—1062no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—532no—
161C22×C4—538no—
162C22×C4—532no—
163C22×C4—532no—
164C22×C4—532no—
165C22×C4—1064no—
166C22×C4—1064no—
167C22×C4C22⋊D5314yes—
168C22×C4C22×D5312yes—
169C22×C4C22×D5312yes—
170C22×C4C22×D5311yescharacteristic
171C22×C4C22×D5311yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—24no—
174C4⋊C4—1068no—
175C4⋊C4—1068no—
176C4⋊C4—1064no—
177C4⋊C4—1062no—
178C22⋊C4—24no—
179C22⋊C4—1064no—
180C42—534no—
181C42—1064no—
182C42—1062no—
183C23—21no—
184C23—22no—
185C23—21no—
186C23Q8×D5314yes—
187C23C2×C22⋊D5312yes—
188C23C23×D5311yescharacteristic
189Q8—44no—
190Q8—2124no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—28no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—534no—
200C2×C4—534no—
201C2×C4—534no—
202C2×C4—534no—
203C2×C4—10616no—
204C2×C4—1062no—
205C2×C4—1062no—
206C2×C4—1062no—
207C2×C4—1062no—
208C2×C4—1062no—
209C2×C4—1064no—
210C2×C4—1062no—
211C2×C4D4×D5312yes—
212C2×C4#1171994712yes—
213C2×C4C2×C22⋊D5312yes—
214C2×C4C2×C22⋊D5312yes—
215C22—22no—
216C22—24no—
217C22—22no—
218C22—24no—
219C22#1313321514yescomplemented
220C22C2×D4×D5311yescharacteristic
221C22C22×C22⋊D5311yescharacteristic
222C22C2×Q8×D5312yes—
223C22#41×D5312yes—
224C22C2×#1171994711yescharacteristic
225C4—22no—
226C4—42no—
227C4—42no—
228C4—538no—
229C4—1064no—
230C4—2122no—
231C4—2122no—
232C4#1313318914yes—
233C2—24no—
234C2C2×#1313318911yescharacteristic
235C2C22⋊Q8×D5312yes—
236C2C2×#1313321511yescharacteristic
237C2#1681116911yescharacteristic
238C2#1681117512yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×C22⋊Q8×C53C2
hypercenterC2×C22⋊Q8D53
hypocenterC53#2857
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#13133215C2semidirect
4. #16810530C2semidirect
5. #16810528C2semidirect
6. #16810518C2semidirect
7. #16810492C2semidirect
44. C22×Dic53Q8semidirect
99. C22×C53C4⋊Q8semidirect
112. Dic53C22⋊Q8semidirect
129. C53#2857semidirect
219. C22#13133215semidirect