Finite Group Databasebeta

← #20280197

Subgroups — #20280197

Order 6784.

520 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 141 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×#13133203C211yesmaximal, characteristic
2C2×Q8×Dic53C211yesmaximal, characteristic, complemented
3#16810550C211yesmaximal, characteristic, complemented
4#16810548C211yesmaximal, characteristic, complemented
5#16810534C211yesmaximal, characteristic, complemented
6#16810527C212yesmaximal, complemented
7#16810523C212yesmaximal
8#16810516C212yesmaximal, complemented
9#16810515C212yesmaximal, complemented
10#16810474C211yesmaximal, characteristic, complemented
11C53×#393C211yesmaximal, characteristic, complemented
12C22×Dic106C2211yescharacteristic
13C22×C22⋊D53C2211yescharacteristic
14#13133203—24no—
15Q8×Dic53—24no—
16C2×C4⋊Dic53C2212yes—
17C2×C4⋊Dic53C2211yescharacteristic
18C2×#11719919C2212yes—
19C2×C22⋊Dic53C2211yescharacteristic
20C2×C22⋊Dic53C2212yes—
21C2×#11719921C2211yescharacteristic
22C2×#11719921C2212yes—
23C2×#11719921C2211yescharacteristic
24C2×#11719921C2212yes—
25C2×C4×Dic53C2212yes—
26C2×C4×Dic53C2212yes—
27C2×C4×Dic53C2211yescharacteristic
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2211yescharacteristic
32C22×Q8×C53C2211yescharacteristic
33C2×C22⋊C4×C53C2212yes—
34C2×C22⋊C4×C53C2211yescharacteristic
35#95×C53C2212yes—
36#95×C53C2211yescharacteristic
37C24×C53C2311yescharacteristic
38C23×D53C2311yescharacteristic
39C2×Q8×C53—22no—
40C2×Q8×C53—24no—
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2312yes—
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2312yes—
45C2×C22⋊D53—22no—
46C2×C22⋊D53—24no—
47C22×Dic53C2311yescharacteristic
48C22×Dic53C2312yes—
49C22×Dic53C2312yes—
50C22×Dic53C2311yescharacteristic
51C22×Dic53C2311yescharacteristic
52C2×Dic106—22no—
53C2×Dic106—24no—
54C22⋊C4×C53—22no—
55C22⋊C4×C53—24no—
56C22⋊Dic53—22no—
57C22⋊Dic53—42no—
58#11719921—22no—
59#11719921—24no—
60#11719921—22no—
61#11719921—42no—
62C4⋊Dic53—22no—
63C4⋊Dic53—24no—
64#11719919—24no—
65C4×Dic53—24no—
66C4×Dic53—42no—
67C4×Dic53D414yes—
68C23×C53—21no—
69C23×C53—22no—
70C23×C53—22no—
71C23×C53—21no—
72C23×C53—21no—
73C23×C53C2411yescharacteristic
74C22×D53—21no—
75C22×D53—22no—
76C22×D53—22no—
77C22×D53—21no—
78C22×D53—21no—
79Q8×C53—28no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—21no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53C2×D412yes—
89C2×C4×C53#4114yes—
90C22⋊D53—44no—
91C2×Dic53—21no—
92C2×Dic53—21no—
93C2×Dic53—21no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53C2×D412yes—
102C2×Dic53C2×D412yes—
103C2×Dic53#4114yes—
104Dic106—44no—
105C22×C53—41no—
106C22×C53—41no—
107C22×C53—41no—
108C22×C53—42no—
109C22×C53—42no—
110C22×C53C22×D411yescharacteristic
111C22×C53C2×#4112yes—
112C22×C53C2×#4112yes—
113C22×C53H(5,𝔽2)11yescharacteristic
114C22×C532−1+411yescharacteristic
115C2×D53—41no—
116C2×D53—41no—
117C2×D53—41no—
118C2×D53—42no—
119C2×D53—42no—
120C4×C53—22no—
121C4×C53—24no—
122C4×C53—41no—
123C4×C53—42no—
124Dic53—22no—
125Dic53—41no—
126Dic53—42no—
127Dic53—42no—
128Dic53#12414yes—
129#2592—531nomaximal, Sylow
130C2×C53—81no—
131C2×C53#52511yescharacteristic
132C2×C53C2×#12411yescharacteristic
133C2×C53#53711yescharacteristic
134C2×C53#55612yes—
135C2×C53#55912yes—
136D53—81no—
137C2×#124—531no—
138C2×C4×Q8—531no—
139#397—531no—
140#397—531no—
141#393—531no—
142#393D5311yescharacteristic, complemented
143#391—532no—
144#390—532no—
145#388—532no—
146#388—532no—
147C4×C4⋊C4—531no—
148C53#259211yescharacteristic, complemented, Sylow
149C22×Q8—531no—
150C22×Q8C2×D5311yescharacteristic
151C22×D4—531no—
152#124—1064no—
153C4×Q8—1064no—
154C2×C4⋊C4—532no—
155C2×C4⋊C4—531no—
156C2×C4⋊C4—532no—
157C2×C22⋊C4—531no—
158C2×C22⋊C4—531no—
159C2×C22⋊C4—531no—
160C2×C22⋊C4—532no—
161C2×C22⋊C4—532no—
162C2×C22⋊C4—532no—
163C2×C22⋊C4C2×D5311yescharacteristic
164C2×C22⋊C4C2×D5312yes—
165C2×C42—531no—
166C2×C42—532no—
167C2×C42—532no—
168#95—531no—
169#95—532no—
170#95—532no—
171#95—532no—
172#95C2×D5311yescharacteristic
173#95C2×D5312yes—
174C24—531no—
175C24C22×D5311yescharacteristic
176C2×Q8—22no—
177C2×Q8—24no—
178C2×Q8—1062no—
179C2×Q8—1064no—
180C2×D4—1062no—
181C2×D4—1064no—
182C22×C4—531no—
183C22×C4—531no—
184C22×C4—531no—
185C22×C4—532no—
186C22×C4—532no—
187C22×C4C22×D5311yescharacteristic
188C22×C4C22×D5312yes—
189C22×C4C22×D5311yescharacteristic
190C22×C4C22×D5312yes—
191C4⋊C4—1064no—
192C4⋊C4—1064no—
193C4⋊C4—1062no—
194C22⋊C4—22no—
195C22⋊C4—24no—
196C22⋊C4—1062no—
197C22⋊C4—1062no—
198C22⋊C4—1062no—
199C22⋊C4—1064no—
200C22⋊C4—2122no—
201C22⋊C4—2122no—
202C42—534no—
203C42—1064no—
204C42—2122no—
205C23—21no—
206C23—22no—
207C23—22no—
208C23—21no—
209C23—21no—
210C23—1061no—
211C23—1062no—
212C23—1062no—
213C23—1061no—
214C23—1061no—
215C23C23×D5311yescharacteristic
216Q8—28no—
217Q8—2124no—
218D4—2124no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—532no—
228C2×C4—534no—
229C2×C4—532no—
230C2×C4—1061no—
231C2×C4—1061no—
232C2×C4—1062no—
233C2×C4—1061no—
234C2×C4—1062no—
235C2×C4—1062no—
236C2×C4—1062no—
237C2×C4—1062no—
238C2×C4—1062no—
239C2×C4—1062no—
240C2×C4D4×D5312yes—
241C2×C4#1171994714yes—
242C22—42no—
243C22—42no—
244C22—41no—
245C22—41no—
246C22—41no—
247C22—2122no—
248C22—2122no—
249C22—2121no—
250C22—2121no—
251C22—2121no—
252C22C2×D4×D5311yescharacteristic
253C22#41×D5312yes—
254C22C2×#1171994712yes—
255C22#1313326311yescharacteristic
256C22#1313326511yescharacteristic
257C4—22no—
258C4—24no—
259C4—42no—
260C4—41no—
261C4—534no—
262C4—1062no—
263C4—2122no—
264C4—2121no—
265C4—2122no—
266C2—81no—
267C2—4241no—
268C2#1681112911yescharacteristic
269C2#124×D5311yescharacteristic
270C2#1681120311yescharacteristic
271C2#1681134412yes—
272C2#1681135512yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#393C2
hypercenter#393D53
hypocenterC53#2592
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic53C2semidirect
3. #16810550C2semidirect
4. #16810548C2semidirect
5. #16810534C2semidirect
6. #16810527C2semidirect
8. #16810516C2semidirect
9. #16810515C2semidirect
10. #16810474C2semidirect
11. C53×#393C2semidirect
142. #393D53semidirect
148. C53#2592semidirect