Finite Group Databasebeta

← #20280508

Subgroups — #20280508

Order 6784.

548 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 133 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×#13133190C211yesmaximal, characteristic
2C2×#13133175C211yesmaximal, characteristic, complemented
3#16810558C211yesmaximal, characteristic, complemented
4#16810550C211yesmaximal, characteristic, complemented
5#16810548C211yesmaximal, characteristic, complemented
6#16810540C212yesmaximal, complemented
7#16810538C212yesmaximal, complemented
8#16810523C212yesmaximal, complemented
9#16810518C212yesmaximal, complemented
10#16810482C211yesmaximal, characteristic, complemented
11C53×#399C211yesmaximal, characteristic, complemented
12C22×D212C2211yescharacteristic
13C22×C22⋊D53C2211yescharacteristic
14C22×C22⋊D53C2211yescharacteristic
15C22×C4×D53C2211yescharacteristic
16#13133190—24no—
17#13133175—24no—
18C2×C4⋊Dic53C2211yescharacteristic
19C2×#11719919C2211yescharacteristic
20C2×#11719919C2212yes—
21C2×#11719919C2212yes—
22C2×C22⋊Dic53C2212yes—
23C2×#11719921C2212yes—
24C2×#11719921C2211yescharacteristic
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27C2×#11719921C2212yes—
28C2×C4×Dic53C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2211yescharacteristic
32C2×C4⋊C4×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2212yes—
34C2×C22⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2212yes—
36#95×C53C2211yescharacteristic
37C24×C53C2311yescharacteristic
38C23×D53C2311yescharacteristic
39C23×D53C2311yescharacteristic
40C22×C4×C53C2312yes—
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2312yes—
43C22×C4×C53C2311yescharacteristic
44C2×C22⋊D53—22no—
45C2×C22⋊D53—22no—
46C2×C22⋊D53—24no—
47C2×C22⋊D53—42no—
48C22×Dic53C2312yes—
49C22×Dic53C2312yes—
50C22×Dic53C2311yescharacteristic
51C22×Dic53C2311yescharacteristic
52C2×D212—22no—
53C2×D212—42no—
54C2×C4×D53—24no—
55C2×C4×D53D414yes—
56C4⋊C4×C53—22no—
57C4⋊C4×C53—42no—
58C22⋊C4×C53—22no—
59C22⋊C4×C53—42no—
60C22⋊Dic53—42no—
61#11719921—24no—
62#11719921—22no—
63#11719921—42no—
64#11719921—42no—
65#11719921—42no—
66C4⋊Dic53—22no—
67#11719919—22no—
68#11719919—24no—
69#11719919—42no—
70C4×Dic53—24no—
71C23×C53—21no—
72C23×C53—22no—
73C23×C53—22no—
74C23×C53—21no—
75C23×C53—21no—
76C23×C53C2411yescharacteristic
77C22×D53—22no—
78C22×D53—22no—
79C22×D53—21no—
80C22×D53—21no—
81C22×D53—21no—
82C22×D53—22no—
83C22×D53—22no—
84C22×D53—21no—
85C22×D53—21no—
86C22×D53C2×D412yes—
87C2×C4×C53—22no—
88C2×C4×C53—21no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53C2×D412yes—
97C22⋊D53—44no—
98C22⋊D53—44no—
99C2×Dic53—21no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53C2×D412yes—
108C2×Dic53#4114yes—
109D212—44no—
110C4×D53—28no—
111C22×C53—41no—
112C22×C53—41no—
113C22×C53—41no—
114C22×C53—42no—
115C22×C53—42no—
116C22×C53C22×D411yescharacteristic
117C22×C53C2×#4112yes—
118C22×C53H(5,𝔽2)12yes—
119C22×C53H(5,𝔽2)11yescharacteristic
120C22×C532−1+411yescharacteristic
121C2×D53—22no—
122C2×D53—22no—
123C2×D53—22no—
124C2×D53—42no—
125C2×D53—42no—
126C2×D53—41no—
127C2×D53—41no—
128C2×D53—41no—
129C2×D53—42no—
130C2×D53—42no—
131C4×C53—22no—
132C4×C53—41no—
133C4×C53—42no—
134C4×C53—42no—
135Dic53—41no—
136Dic53—42no—
137Dic53—42no—
138Dic53#12314yes—
139#2624—531nomaximal, Sylow
140C2×C53—81no—
141C2×C53C2×#12311yescharacteristic
142C2×C53#53511yescharacteristic
143C2×C53#53711yescharacteristic
144C2×C53#54012yes—
145C2×C53#54212yes—
146D53—42no—
147D53—81no—
148C2×#121—531no—
149C2×#117—531no—
150#399—531no—
151#399D5311yescharacteristic, complemented
152#397—531no—
153#397—531no—
154#394—532no—
155#394—532no—
156#390—532no—
157#389—532no—
158#380—531no—
159C53#262411yescharacteristic, complemented, Sylow
160C22×D4—531no—
161C22×D4—531no—
162C22×D4—531no—
163C23×C4—531no—
164#121—1064no—
165#117—1064no—
166C2×C4⋊C4—531no—
167C2×C4⋊C4—531no—
168C2×C4⋊C4—532no—
169C2×C4⋊C4—532no—
170C2×C4⋊C4C2×D5311yescharacteristic
171C2×C4⋊C4C2×D5312yes—
172C2×C22⋊C4—532no—
173C2×C22⋊C4—531no—
174C2×C22⋊C4—532no—
175C2×C22⋊C4—532no—
176C2×C22⋊C4—532no—
177C2×C22⋊C4—532no—
178C2×C22⋊C4C2×D5311yescharacteristic
179C2×C22⋊C4C2×D5312yes—
180C2×C42—532no—
181#95—531no—
182#95—532no—
183#95—532no—
184#95C2×D5311yescharacteristic
185C24—531no—
186C24—531no—
187C24C22×D5311yescharacteristic
188C2×D4—1062no—
189C2×D4—1062no—
190C2×D4—1064no—
191C2×D4—1062no—
192C2×D4—2122no—
193C2×D4—2122no—
194C22×C4—531no—
195C22×C4—534no—
196C22×C4—532no—
197C22×C4—532no—
198C22×C4—531no—
199C22×C4—1064no—
200C22×C4C22×D5311yescharacteristic
201C22×C4C22×D5312yes—
202C22×C4C22×D5312yes—
203C22×C4C22×D5311yescharacteristic
204C4⋊C4—22no—
205C4⋊C4—42no—
206C4⋊C4—1062no—
207C4⋊C4—1064no—
208C4⋊C4—1062no—
209C4⋊C4—2122no—
210C22⋊C4—22no—
211C22⋊C4—42no—
212C22⋊C4—1064no—
213C22⋊C4—1062no—
214C22⋊C4—2122no—
215C22⋊C4—2122no—
216C22⋊C4—2122no—
217C22⋊C4—2122no—
218C42—1064no—
219C23—21no—
220C23—22no—
221C23—22no—
222C23—21no—
223C23—21no—
224C23—532no—
225C23—1062no—
226C23—1062no—
227C23—1061no—
228C23—1061no—
229C23—1061no—
230C23—1062no—
231C23—1062no—
232C23—1061no—
233C23—1061no—
234C23C23×D5311yescharacteristic
235D4—2124no—
236D4—2124no—
237D4—2124no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—532no—
248C2×C4—534no—
249C2×C4—1068no—
250C2×C4—1062no—
251C2×C4—1061no—
252C2×C4—1062no—
253C2×C4—1062no—
254C2×C4—1062no—
255C2×C4—1062no—
256C2×C4—1062no—
257C2×C4—1062no—
258C2×C4D4×D5312yes—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—42no—
263C22—42no—
264C22—1062no—
265C22—1062no—
266C22—1062no—
267C22—2121no—
268C22—2121no—
269C22—2121no—
270C22—2122no—
271C22—2122no—
272C22—2122no—
273C22—2122no—
274C22C2×D4×D5311yescharacteristic
275C22#41×D5312yes—
276C22#1313326212yes—
277C22#1313326311yescharacteristic
278C22#1313326511yescharacteristic
279C4—22no—
280C4—41no—
281C4—42no—
282C4—42no—
283C4—534no—
284C4—2121no—
285C4—2122no—
286C4—2122no—
287C2—81no—
288C2—4241no—
289C2—2122no—
290C2#123×D5311yescharacteristic
291C2#1681118611yescharacteristic
292C2#1681120311yescharacteristic
293C2#1681122512yes—
294C2#1681124312yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#399C2
hypercenter#399D53
hypocenterC53#2624
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
2. C2×#13133175C2semidirect
3. #16810558C2semidirect
4. #16810550C2semidirect
5. #16810548C2semidirect
6. #16810540C2semidirect
7. #16810538C2semidirect
8. #16810523C2semidirect
9. #16810518C2semidirect
10. #16810482C2semidirect
11. C53×#399C2semidirect
151. #399D53semidirect
159. C53#2624semidirect