Finite Group Databasebeta

← #20281733

Subgroups — #20281733

Order 6784.

528 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13133193C212yesmaximal, complemented
2#16810560C212yesmaximal, complemented
3#16810555C212yesmaximal, complemented
4#16810545C212yesmaximal, complemented
5#16810525C211yesmaximal, characteristic
6#16810521C211yesmaximal, characteristic, complemented
7#16810501C212yesmaximal
8#16810497C212yesmaximal, complemented
9C53×#391C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C22×C4×D53C2212yes—
12#13133193—28no—
13C2×C4⋊Dic53C2211yescharacteristic
14C2×C4⋊Dic53C2212yes—
15C2×C4⋊Dic53C2211yescharacteristic
16C2×C4⋊Dic53C2211yescharacteristic
17C2×C4⋊Dic53C2212yes—
18C2×#11719919C2212yes—
19C2×#11719919C2212yes—
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23C2×#11719921C2212yes—
24C2×#11719921C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27C22×Q8×C53C2211yescharacteristic
28C2×C4⋊C4×C53C2211yescharacteristic
29C2×C42×C53C2211yescharacteristic
30#95×C53C2212yes—
31#95×C53C2212yes—
32C23×D53C2312yes—
33C2×Q8×C53—41no—
34C2×Q8×C53—41no—
35C2×Q8×C53D414yescomplemented
36C22×C4×C53C2312yes—
37C22×C4×C53C2312yes—
38C22×C4×C53C2311yescharacteristic
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2311yescharacteristic
41C22×Dic53C2312yes—
42C22×Dic53C2312yes—
43C22×Dic53C2312yes—
44C2×D212—41no—
45C2×D212—41no—
46C2×D212Q814yescomplemented
47C2×C4×D53—24no—
48C2×C4×D53—42no—
49C2×C4×D53—42no—
50C4⋊C4×C53—41no—
51C42×C53—41no—
52#11719921—24no—
53#11719921—24no—
54#11719921—24no—
55#11719921—42no—
56#11719921—42no—
57C4⋊Dic53—24no—
58C4⋊Dic53—22no—
59C4⋊Dic53—41no—
60C4⋊Dic53—41no—
61C4⋊Dic53—42no—
62#11719919—24no—
63#11719919—24no—
64C23×C53C2411yescharacteristic
65C22×D53—22no—
66C22×D53—22no—
67C22×D53—22no—
68C22×D53—22no—
69C22×D53—22no—
70C22×D53—22no—
71C22×D53C2×Q814yes—
72Q8×C53—44no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53C2×D414yes—
85C2×C4×C53C2×D412yes—
86C2×C4×C53C2×Q812yes—
87C2×C4×C53#4112yes—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97D212—44no—
98C4×D53—48no—
99C22×C53C22×D411yescharacteristic
100C22×C53C22×Q811yescharacteristic
101C22×C53C2×#4111yescharacteristic
102C22×C53H(5,𝔽2)11yescharacteristic
103C22×C53H(5,𝔽2)11yescharacteristic
104C22×C53H(5,𝔽2)11yescharacteristic
105C22×C532−1+411yescharacteristic
106C2×D53—24no—
107C2×D53—24no—
108C2×D53—24no—
109C2×D53—42no—
110C2×D53—42no—
111C2×D53—42no—
112C2×D53—42no—
113C4×C53—24no—
114C4×C53—22no—
115C4×C53—22no—
116C4×C53—22no—
117C4×C53—42no—
118Dic53—42no—
119Dic53—42no—
120Dic53—42no—
121#2732—531nomaximal, Sylow
122C2×C53#53411yescharacteristic
123C2×C53#53911yescharacteristic
124C2×C53#54311yescharacteristic
125C2×C53D4×Q811yescharacteristic
126C2×C53#55011yescharacteristic
127C2×C53#55411yescharacteristic
128C2×C53#56211yescharacteristic
129D53—44no—
130C2×C22⋊Q8—532no—
131#399—532no—
132#398—532no—
133#396—532no—
134#391D5311yescharacteristic, complemented
135#390—531no—
136#389—531no—
137#386—532no—
138#385—532no—
139C53#273211yescharacteristic, complemented, Sylow
140C22×Q8C2×D5311yescharacteristic
141C22×D4—531no—
142C23×C4—532no—
143C22⋊Q8—1068no—
144C2×C4⋊C4—532no—
145C2×C4⋊C4—532no—
146C2×C4⋊C4—532no—
147C2×C4⋊C4—531no—
148C2×C4⋊C4—531no—
149C2×C4⋊C4—531no—
150C2×C4⋊C4—532no—
151C2×C4⋊C4C2×D5311yescharacteristic
152C2×C22⋊C4—532no—
153C2×C22⋊C4—532no—
154C2×C22⋊C4—532no—
155C2×C22⋊C4—532no—
156C2×C22⋊C4—532no—
157C2×C42C2×D5311yescharacteristic
158#95—532no—
159#95—532no—
160#95C2×D5312yes—
161#95C2×D5312yes—
162C24—532no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8D21214yescomplemented
166C2×D4—534no—
167C2×D4—2121no—
168C2×D4—2121no—
169C22×C4—532no—
170C22×C4—532no—
171C22×C4—532no—
172C22×C4—1064no—
173C22×C4—2122no—
174C22×C4—2122no—
175C22×C4C22×D5312yes—
176C22×C4C22×D5312yes—
177C22×C4C22×D5311yescharacteristic
178C22×C4C22×D5311yescharacteristic
179C22×C4C22×D5311yescharacteristic
180C4⋊C4—41no—
181C4⋊C4—1064no—
182C4⋊C4—1064no—
183C4⋊C4—1064no—
184C4⋊C4—1062no—
185C4⋊C4—2121no—
186C4⋊C4—2121no—
187C4⋊C4—2122no—
188C22⋊C4—1064no—
189C22⋊C4—1064no—
190C22⋊C4—1064no—
191C22⋊C4—2122no—
192C22⋊C4—2122no—
193C42—41no—
194C23—534no—
195C23—1062no—
196C23—1062no—
197C23—1062no—
198C23—1062no—
199C23—1062no—
200C23—1062no—
201C23C23×D5311yescharacteristic
202Q8—44no—
203D4—2124no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—1062no—
216C2×C4—1062no—
217C2×C4—1062no—
218C2×C4—1062no—
219C2×C4—1062no—
220C2×C4—1062no—
221C2×C4—1062no—
222C2×C4—1062no—
223C2×C4—1062no—
224C2×C4—2128no—
225C2×C4C2×D21212yes—
226C2×C4C2×D21214yes—
227C2×C4Q8×D5312yes—
228C2×C4#1171994912yes—
229C22—1064no—
230C22—1064no—
231C22—1064no—
232C22—2122no—
233C22—2122no—
234C22—2122no—
235C22—2122no—
236C22C22×D21211yescharacteristic
237C22C2×Q8×D5311yescharacteristic
238C22C2×#1171994911yescharacteristic
239C22#1313326211yescharacteristic
240C22#1313326211yescharacteristic
241C22#1313326311yescharacteristic
242C22#1313326411yescharacteristic
243C4—24no—
244C4—22no—
245C4—22no—
246C4—22no—
247C4—42no—
248C4—2122no—
249C4—2122no—
250C4—2122no—
251C2—2124no—
252C2#1681118511yescharacteristic
253C2#1681122211yescharacteristic
254C2#1681124811yescharacteristic
255C2Q8×D21211yescharacteristic
256C2#1681129511yescharacteristic
257C2#1681133411yescharacteristic
258C2#1681136711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#391C2
hypercenter#391D53
hypocenterC53#2732
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#13133193C2semidirect
2. #16810560C2semidirect
3. #16810555C2semidirect
4. #16810545C2semidirect
6. #16810521C2semidirect
8. #16810497C2semidirect
9. C53×#391C2semidirect
35. C2×Q8×C53D4semidirect
46. C2×D212Q8semidirect
134. #391D53semidirect
139. C53#2732semidirect
165. C2×Q8D212semidirect