Finite Group Databasebeta

← #20274631

Subgroups — #20274631

Order 6784.

612 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#11719947C211yesmaximal, characteristic
2C2×#13133132C211yesmaximal, characteristic, complemented
3C2×#13133131C212yesmaximal, complemented
4#16810484C212yesmaximal, complemented
5C2×#99×C53C211yesmaximal, characteristic, complemented
6C2×#11719947—22no—
7C2×#11719947—22no—
8C2×#11719947—24no—
9C2×#11719947—24no—
10C22×Dic106C411yescharacteristic, complemented
11C22×C22⋊D53C411yescharacteristic, complemented
12C22×C22⋊D53C2211yescharacteristic
13C23×Dic53C411yescharacteristic, complemented
14C23×Dic53C2211yescharacteristic
15C22×C4×D53C411yescharacteristic, complemented
16C2×C22⋊Dic53C2212yes—
17#13133132—22no—
18#13133131—24no—
19#13133124—24no—
20C22×D4×C53C2211yescharacteristic
21C2×C22⋊C4×C53C2212yes—
22#99×C53—22no—
23C24×C53C2×C411yescharacteristic
24C24×C53C2311yescharacteristic
25C23×D53C2×C411yescharacteristic
26C2×D4×C53—22no—
27C2×D4×C53—22no—
28C2×D4×C53D412yes—
29C2×D4×C53D412yes—
30C22×C4×C53—22no—
31C22×C4×C53C2×C411yescharacteristic
32C2×C22⋊D53—22no—
33C2×C22⋊D53—21no—
34C2×C22⋊D53—22no—
35C2×C22⋊D53—21no—
36C2×C22⋊D53—22no—
37C2×C22⋊D53—22no—
38C2×C22⋊D53D414yes—
39C22×Dic53—22no—
40C22×Dic53—21no—
41C22×Dic53—22no—
42C22×Dic53—21no—
43C22×Dic53—22no—
44C22×Dic53—22no—
45C22×Dic53—22no—
46C22×Dic53C2×C411yescharacteristic
47C22×Dic53C2×C411yescharacteristic
48C22×Dic53C2×C411yescharacteristic
49C22×Dic53D414yes—
50#11719947—216no—
51#11719947—216no—
52C2×C4×D53—22no—
53C2×C4×D53—22no—
54C2×C4×D53—22no—
55C2×Dic106—22no—
56C2×Dic106—22no—
57C2×Dic106—22no—
58C22⋊C4×C53—22no—
59C22⋊C4×C53—22no—
60C22⋊Dic53—22no—
61C22⋊Dic53—22no—
62C23×C53—21no—
63C23×C53—21no—
64C23×C53—42no—
65C23×C53—42no—
66C23×C53C22⋊C412yes—
67C23×C53C22×C411yescharacteristic
68C23×C53C2×D411yescharacteristic
69C23×C53C2×D412yes—
70C23×C53C2×D412yes—
71C23×C53C2×D411yescharacteristic
72C22×D53—21no—
73C22×D53—21no—
74C22×D53—42no—
75C22×D53C22⋊C412yes—
76D4×C53—24no—
77D4×C53—24no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—44no—
81C2×C4×C53—42no—
82C2×C4×C53C22⋊C412yes—
83C22⋊D53—28no—
84C22⋊D53—24no—
85C22⋊D53—24no—
86C2×Dic53—28no—
87C2×Dic53—21no—
88C2×Dic53—21no—
89C2×Dic53—21no—
90C2×Dic53—21no—
91C2×Dic53—21no—
92C2×Dic53—21no—
93C2×Dic53—24no—
94C2×Dic53—24no—
95C2×Dic53—44no—
96C2×Dic53—42no—
97C2×Dic53C22⋊C412yes—
98C2×Dic53C22⋊C412yes—
99C2×Dic53C22⋊C412yes—
100C4×D53—28no—
101Dic106—28no—
102C22×C53—24no—
103C22×C53—22no—
104C22×C53—22no—
105C22×C53—41no—
106C22×C53—42no—
107C22×C53—41no—
108C22×C53—42no—
109C22×C53—42no—
110C22×C53—42no—
111C22×C53C2×C22⋊C411yescharacteristic
112C22×C53C2×C22⋊C411yescharacteristic
113C22×C53C2×C22⋊C411yescharacteristic
114C22×C53#12012yes—
115C22×C53#12012yes—
116C2×D53—22no—
117C2×D53—42no—
118C2×D53—42no—
119C2×D53—42no—
120C4×C53—22no—
121C4×C53—82no—
122Dic53—22no—
123Dic53—22no—
124Dic53—22no—
125Dic53—82no—
126#1782—531nomaximal, Sylow
127C2×C53—22no—
128C2×C53—22no—
129C2×C53—21no—
130C2×C53—21no—
131C2×C53—42no—
132C2×C53#37911yescharacteristic
133C2×C53#41012yes—
134D53—42no—
135C22×#41—531no—
136C2×#99—531no—
137C2×#99—532no—
138C2×#99D5311yescharacteristic, complemented
139#381—532no—
140C53#178211yescharacteristic, complemented, Sylow
141C2×#41—1064no—
142C2×#41—1064no—
143C2×#41—1062no—
144C2×#41—1062no—
145C22×Q8—531no—
146C22×D4—531no—
147C22×D4—531no—
148C22×D4C2×D5311yescharacteristic
149C23×C4—531no—
150C23×C4—531no—
151C23×C4—531no—
152C2×C22⋊C4—532no—
153C2×C22⋊C4C2×D5312yes—
154#99—22no—
155#99—1064no—
156#99—1062no—
157#95—1064no—
158C24—531no—
159C24C4×D5311yescharacteristic
160C24C22×D5311yescharacteristic
161#41—10616no—
162#41—10616no—
163C2×Q8—1062no—
164C2×Q8—1062no—
165C2×Q8—1062no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—534no—
169C2×D4—1062no—
170C2×D4—1062no—
171C2×D4—1061no—
172C2×D4—1062no—
173C2×D4—1061no—
174C2×D4—1062no—
175C2×D4D21212yes—
176C2×D4C22⋊D5312yes—
177C22×C4—22no—
178C22×C4—531no—
179C22×C4—531no—
180C22×C4—531no—
181C22×C4—534no—
182C22×C4—1062no—
183C22×C4—1062no—
184C22×C4—1062no—
185C22×C4—1062no—
186C22×C4—1061no—
187C22×C4—1062no—
188C22×C4—1061no—
189C22×C4—1062no—
190C22×C4—1062no—
191C22×C4—1062no—
192C22×C4C4×D5311yescharacteristic
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—1062no—
196C22⋊C4—1062no—
197C23—21no—
198C23—21no—
199C23—42no—
200C23—42no—
201C23—532no—
202C23—1061no—
203C23—1061no—
204C23—2122no—
205C23#1171992112yes—
206C23C2×C4×D5311yescharacteristic
207C23C2×D21211yescharacteristic
208C23D4×D5312yes—
209C23D4×D5312yes—
210C23C2×C22⋊D5311yescharacteristic
211Q8—1068no—
212D4—24no—
213D4—24no—
214D4—1068no—
215D4—1064no—
216D4—1064no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—44no—
220C2×C4—42no—
221C2×C4—532no—
222C2×C4—532no—
223C2×C4—532no—
224C2×C4—1068no—
225C2×C4—1068no—
226C2×C4—1061no—
227C2×C4—1061no—
228C2×C4—1061no—
229C2×C4—1061no—
230C2×C4—1061no—
231C2×C4—1061no—
232C2×C4—1064no—
233C2×C4—1064no—
234C2×C4—2124no—
235C2×C4—2122no—
236C2×C4#1171992112yes—
237C22—24no—
238C22—22no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—41no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—1062no—
247C22—2122no—
248C22—2122no—
249C22—2122no—
250C22C22⋊C4×D5311yescharacteristic
251C22C22⋊C4×D5311yescharacteristic
252C22C2×#1171992111yescharacteristic
253C22#1313318512yes—
254C22#1313318712yes—
255C4—22no—
256C4—82no—
257C4—1062no—
258C4—1062no—
259C4—1062no—
260C4—4242no—
261C2—22no—
262C2—22no—
263C2—21no—
264C2—21no—
265C2—42no—
266C2—2122no—
267C2#1681047811yescharacteristic
268C2#1681059512yes—
subgroupG/H
centerC22C22⋊C4×D53
commutatorC23×C53C22×C4
FrattiniC24C22×D53
FittingC2×#99×C53C2
socleC22×C53C2×C22⋊C4
hypercenterC2×#99D53
hypocenterC53#1782
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. C2×#13133132C2semidirect
3. C2×#13133131C2semidirect
4. #16810484C2semidirect
5. C2×#99×C53C2semidirect
10. C22×Dic106C4semidirect
11. C22×C22⋊D53C4semidirect
13. C23×Dic53C4semidirect
15. C22×C4×D53C4semidirect
138. C2×#99D53semidirect
140. C53#1782semidirect