Finite Group Databasebeta

← #20281079

Subgroups — #20281079

Order 6784.

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

normal subgroups only

This group has 668 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×#13133201C212yesmaximal, complemented
2C2×#13133189C212yesmaximal, complemented
3#16810545C212yesmaximal, complemented
4#16810539C212yesmaximal, complemented
5#16810532C212yesmaximal, complemented
6#16810525C211yesmaximal, characteristic, complemented
7#16810521C211yesmaximal, characteristic, complemented
8#16810498C212yesmaximal, complemented
9C53×#390C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C22×C22⋊D53C2212yes—
12C22×C22⋊D53C2212yes—
13C23×Dic53C2212yes—
14#13133201—28no—
15#13133189—28no—
16C2×C4⋊Dic53C2212yes—
17C2×C4⋊Dic53C2211yescharacteristic
18C2×C4⋊Dic53C2211yescharacteristic
19C2×C4⋊Dic53C2211yescharacteristic
20C2×#11719919C2212yes—
21C2×C22⋊Dic53C2212yes—
22C2×#11719921C2212yes—
23C2×#11719921C2212yes—
24C2×#11719921C2212yes—
25C2×#11719921C2212yes—
26#13133124C2212yes—
27#13133124C2212yes—
28C22×D4×C53C2211yescharacteristic
29C2×C4⋊C4×C53C2211yescharacteristic
30C2×C22⋊C4×C53C2212yes—
31C2×C22⋊C4×C53C2212yes—
32C2×C42×C53C2211yescharacteristic
33C24×C53C2312yes—
34C23×D53C2312yes—
35C2×D4×C53—41no—
36C2×D4×C53—41no—
37C2×D4×C53D414yescomplemented
38C22×C4×C53C2312yes—
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2311yescharacteristic
42C2×C22⋊D53—24no—
43C2×C22⋊D53—24no—
44C2×C22⋊D53—24no—
45C2×C22⋊D53—42no—
46C2×C22⋊D53—42no—
47C2×C22⋊D53—42no—
48C22×Dic53—24no—
49C22×Dic53—24no—
50C22×Dic53—42no—
51C22×Dic53C2312yes—
52C22×Dic53C2312yes—
53C22×Dic53C2312yes—
54C2×D212—22no—
55C2×D212—41no—
56C2×D212—41no—
57C4⋊C4×C53—41no—
58C22⋊C4×C53—24no—
59C22⋊C4×C53—24no—
60C42×C53—41no—
61C22⋊Dic53—24no—
62#11719921—24no—
63#11719921—24no—
64#11719921—24no—
65#11719921—42no—
66C4⋊Dic53—24no—
67C4⋊Dic53—41no—
68C4⋊Dic53—41no—
69C4⋊Dic53D414yescomplemented
70#11719919—24no—
71C23×C53—22no—
72C23×C53—22no—
73C23×C53—22no—
74C23×C53—22no—
75C23×C53—22no—
76C23×C53—22no—
77C23×C53C2×D414yes—
78C23×C53C2411yescharacteristic
79C22×D53—22no—
80C22×D53—22no—
81C22×D53—22no—
82C22×D53—22no—
83C22×D53—22no—
84C22×D53—22no—
85C22×D53—22no—
86D4×C53—44no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53C2×D412yes—
97C2×C4×C53C2×D412yes—
98C2×C4×C53#4112yes—
99C22⋊D53—48no—
100C22⋊D53—48no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—48no—
110C2×Dic53C2×D414yes—
111D212—44no—
112C22×C53—24no—
113C22×C53—24no—
114C22×C53—24no—
115C22×C53—42no—
116C22×C53—42no—
117C22×C53—42no—
118C22×C53—42no—
119C22×C53C22×D411yescharacteristic
120C22×C53C22×D411yescharacteristic
121C22×C53C2×#4111yescharacteristic
122C22×C53H(5,𝔽2)11yescharacteristic
123C22×C53H(5,𝔽2)11yescharacteristic
124C22×C53H(5,𝔽2)11yescharacteristic
125C22×C532−1+411yescharacteristic
126C2×D53—42no—
127C2×D53—42no—
128C2×D53—42no—
129C2×D53—42no—
130C2×D53—42no—
131C2×D53—42no—
132C2×D53—42no—
133C4×C53—22no—
134C4×C53—22no—
135C4×C53—22no—
136C4×C53—42no—
137Dic53—24no—
138Dic53—42no—
139Dic53—42no—
140#2682—531nomaximal, Sylow
141C2×C53—44no—
142C2×C53#53411yescharacteristic
143C2×C53#53711yescharacteristic
144C2×C53#53811yescharacteristic
145C2×C53D4211yescharacteristic
146C2×C53#54711yescharacteristic
147C2×C53#55011yescharacteristic
148C2×C53#56211yescharacteristic
149D53—82no—
150C2×#123—532no—
151C2×#121—532no—
152#396—532no—
153#394—532no—
154#392—532no—
155#390—531no—
156#390D5311yescharacteristic, complemented
157#389—531no—
158#385—532no—
159C53#268211yescharacteristic, complemented, Sylow
160C22×D4—532no—
161C22×D4—532no—
162C22×D4—531no—
163C22×D4C2×D5311yescharacteristic
164C23×C4—532no—
165#123—1068no—
166#121—1068no—
167C2×C4⋊C4—531no—
168C2×C4⋊C4—532no—
169C2×C4⋊C4—532no—
170C2×C4⋊C4—531no—
171C2×C4⋊C4—531no—
172C2×C4⋊C4C2×D5311yescharacteristic
173C2×C22⋊C4—532no—
174C2×C22⋊C4—532no—
175C2×C22⋊C4—532no—
176C2×C22⋊C4—532no—
177C2×C22⋊C4—532no—
178C2×C22⋊C4C2×D5312yes—
179C2×C22⋊C4C2×D5312yes—
180C2×C42C2×D5311yescharacteristic
181#95—532no—
182#95—532no—
183C24—532no—
184C24C22×D5312yes—
185C2×D4—41no—
186C2×D4—41no—
187C2×D4—1064no—
188C2×D4—1064no—
189C2×D4—1064no—
190C2×D4—1062no—
191C2×D4—2121no—
192C2×D4—2121no—
193C2×D4—2122no—
194C2×D4—2122no—
195C2×D4—2122no—
196C2×D4D21214yescomplemented
197C22×C4—532no—
198C22×C4—532no—
199C22×C4—532no—
200C22×C4—1064no—
201C22×C4—1064no—
202C22×C4—2122no—
203C22×C4C22×D5312yes—
204C22×C4C22×D5311yescharacteristic
205C22×C4C22×D5311yescharacteristic
206C22×C4C22×D5311yescharacteristic
207C4⋊C4—41no—
208C4⋊C4—534no—
209C4⋊C4—1064no—
210C4⋊C4—1064no—
211C4⋊C4—2121no—
212C4⋊C4—2121no—
213C22⋊C4—24no—
214C22⋊C4—24no—
215C22⋊C4—1064no—
216C22⋊C4—1064no—
217C22⋊C4—1064no—
218C22⋊C4—1064no—
219C22⋊C4—2122no—
220C42—41no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—1062no—
228C23—1062no—
229C23—1062no—
230C23—1062no—
231C23—1062no—
232C23—1062no—
233C23—1062no—
234C23C2×D21214yes—
235C23C23×D5311yescharacteristic
236D4—44no—
237D4—2124no—
238D4—2128no—
239D4—2128no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—534no—
250C2×C4—1062no—
251C2×C4—1062no—
252C2×C4—1062no—
253C2×C4—1062no—
254C2×C4—1062no—
255C2×C4—1062no—
256C2×C4—1062no—
257C2×C4—1062no—
258C2×C4—2128no—
259C2×C4C2×D21212yes—
260C2×C4D4×D5312yes—
261C2×C4#1171994712yes—
262C22—24no—
263C22—24no—
264C22—24no—
265C22—42no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—2122no—
270C22—2122no—
271C22—2122no—
272C22—2122no—
273C22—2122no—
274C22—2122no—
275C22—2122no—
276C22C2×D4×D5311yescharacteristic
277C22C22×D21211yescharacteristic
278C22C2×#1171994711yescharacteristic
279C22#1313326211yescharacteristic
280C22#1313326211yescharacteristic
281C22#1313326311yescharacteristic
282C22#1313326411yescharacteristic
283C4—22no—
284C4—22no—
285C4—22no—
286C4—42no—
287C4—1064no—
288C4—2122no—
289C4—2122no—
290C2—44no—
291C2—4242no—
292C2#1681118511yescharacteristic
293C2#1681120011yescharacteristic
294C2#1681121011yescharacteristic
295C2D4×D21211yescharacteristic
296C2#1681127711yescharacteristic
297C2#1681129311yescharacteristic
298C2#1681136711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#390C2
hypercenter#390D53
hypocenterC53#2682
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#13133201C2semidirect
2. C2×#13133189C2semidirect
3. #16810545C2semidirect
4. #16810539C2semidirect
5. #16810532C2semidirect
6. #16810525C2semidirect
7. #16810521C2semidirect
8. #16810498C2semidirect
9. C53×#390C2semidirect
37. C2×D4×C53D4semidirect
69. C4⋊Dic53D4semidirect
156. #390D53semidirect
159. C53#2682semidirect
196. C2×D4D212semidirect