Finite Group Databasebeta

← #20277778

Subgroups — #20277778

Order 6784.

832 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 832 conjugacy classes of subgroups — too many to draw. Only its 209 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×#13133213C211yesmaximal, characteristic
2C2×C4×C22⋊D53C212yesmaximal, complemented
3C2×#13133177C212yesmaximal, complemented
4#16810523C212yesmaximal, complemented
5#16810523C212yesmaximal, complemented
6#16810503C211yesmaximal, characteristic, complemented
7#16810501C212yesmaximal, complemented
8#16810474C212yesmaximal, complemented
9C53×#386C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C22×C22⋊D53C2212yes—
12C22×C22⋊D53C2212yes—
13C23×Dic53C2211yescharacteristic
14C22×C4×D53C2212yes—
15#13133213C418yescomplemented
16C4×C22⋊D53—28no—
17#13133177—28no—
18C2×#11719919C2212yes—
19C2×#11719919C2212yes—
20C2×C22⋊Dic53C2212yes—
21C2×C22⋊Dic53C2211yescharacteristic
22C2×#11719921C2212yes—
23C2×#11719921C2212yes—
24C2×#11719921C2212yes—
25C2×C4×Dic53C2212yes—
26C2×C4×Dic53C2212yes—
27C2×C4×Dic53C2211yescharacteristic
28#13133124C2212yes—
29C22×D4×C53C2211yescharacteristic
30C23×C4×C53C2211yescharacteristic
31C2×C22⋊C4×C53C2212yes—
32C2×C22⋊C4×C53C2211yescharacteristic
33#95×C53C2212yes—
34C24×C53C2311yescharacteristic
35C24×C53C2311yescharacteristic
36C23×D53C2312yes—
37C2×D4×C53—41no—
38C2×D4×C53—41no—
39C2×D4×C53C2×C414yescomplemented
40C22×C4×C53—24no—
41C22×C4×C53—41no—
42C22×C4×C53C2312yes—
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2312yes—
45C2×C22⋊D53—24no—
46C2×C22⋊D53—24no—
47C2×C22⋊D53—24no—
48C2×C22⋊D53—24no—
49C2×C22⋊D53C2×C418yescomplemented
50C2×C22⋊D53C2×C418yescomplemented
51C22×Dic53—24no—
52C22×Dic53—41no—
53C22×Dic53C2312yes—
54C22×Dic53C2312yes—
55C22×Dic53C2312yes—
56C2×D212—41no—
57C2×D212—41no—
58C2×D212C2×C414yescomplemented
59C2×C4×D53—24no—
60C2×C4×D53—24no—
61C2×C4×D53—42no—
62C22⋊C4×C53—24no—
63C22⋊C4×C53—41no—
64C22⋊Dic53—24no—
65C22⋊Dic53—41no—
66#11719921—24no—
67#11719921—24no—
68#11719921—42no—
69#11719919—24no—
70#11719919D418yescomplemented
71C4×Dic53—24no—
72C4×Dic53—24no—
73C4×Dic53C2×C414yescomplemented
74C23×C53—22no—
75C23×C53—22no—
76C23×C53—22no—
77C23×C53—22no—
78C23×C53—21no—
79C23×C53—21no—
80C23×C53—21no—
81C23×C53—21no—
82C23×C53C22×C412yes—
83C23×C53C22×C412yes—
84C23×C53C2411yescharacteristic
85C22×D53—22no—
86C22×D53—22no—
87C22×D53—22no—
88C22×D53—22no—
89C22×D53—22no—
90C22×D53—22no—
91C22×D53C22×C414yes—
92D4×C53—44no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—44no—
101C2×C4×C53C22×C412yes—
102C2×C4×C53C2×D414yes—
103C22⋊D53—216no—
104C22⋊D53—216no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—44no—
111C2×Dic53C22×C414yes—
112C2×Dic53C2×D414yes—
113C2×Dic53C2×D414yes—
114C2×Dic53#4114yes—
115D212—44no—
116C4×D53—48no—
117C22×C53—24no—
118C22×C53—24no—
119C22×C53—22no—
120C22×C53—22no—
121C22×C53—41no—
122C22×C53—41no—
123C22×C53—42no—
124C22×C53—42no—
125C22×C53—41no—
126C22×C53—41no—
127C22×C53C23×C411yescharacteristic
128C22×C53C22×D412yes—
129C22×C53C2×#4112yes—
130C22×C53H(5,𝔽2)11yescharacteristic
131C22×C53H(5,𝔽2)11yescharacteristic
132C2×D53—24no—
133C2×D53—24no—
134C2×D53—24no—
135C2×D53—42no—
136C2×D53—42no—
137C2×D53—42no—
138C2×D53—42no—
139C4×C53—22no—
140C4×C53—24no—
141C4×C53—42no—
142Dic53—24no—
143Dic53—42no—
144Dic53C4×D418yescomplemented
145#2366—531nomaximal, Sylow
146C2×C53—42no—
147C2×C53—42no—
148C2×C53C2×C4×D412yes—
149C2×C53#51811yescharacteristic
150C2×C53D4211yescharacteristic
151C2×C53#55012yes—
152C2×C53#55911yescharacteristic
153D53—44no—
154C2×#127—531no—
155C2×C4×D4—532no—
156C2×C4×D4—532no—
157#390—532no—
158#390—532no—
159#386—531no—
160#386—532no—
161#386D5311yescharacteristic, complemented
162C4×C4⋊C4—532no—
163C53#236611yescharacteristic, complemented, Sylow
164C22×D4—532no—
165C22×D4—532no—
166C22×D4—531no—
167C22×D4C2×D5311yescharacteristic
168C23×C4—531no—
169C23×C4—532no—
170C23×C4C2×D5311yescharacteristic
171#127—538no—
172C4×D4—1068no—
173C4×D4—1068no—
174C2×C4⋊C4—532no—
175C2×C4⋊C4—532no—
176C2×C22⋊C4—532no—
177C2×C22⋊C4—532no—
178C2×C22⋊C4—532no—
179C2×C22⋊C4—531no—
180C2×C22⋊C4—532no—
181C2×C22⋊C4C2×D5312yes—
182C2×C22⋊C4C2×D5311yescharacteristic
183C2×C42—531no—
184C2×C42—532no—
185C2×C42—532no—
186#95—532no—
187#95C2×D5312yes—
188C24—532no—
189C24C22×D5311yescharacteristic
190C24C22×D5311yescharacteristic
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—538no—
194C2×D4—538no—
195C2×D4—534no—
196C2×D4—1064no—
197C2×D4—1064no—
198C2×D4—1064no—
199C2×D4—1064no—
200C2×D4—2121no—
201C2×D4—2121no—
202C2×D4C4×D5314yescomplemented
203C22×C4—24no—
204C22×C4—41no—
205C22×C4—532no—
206C22×C4—532no—
207C22×C4—532no—
208C22×C4—1064no—
209C22×C4—1064no—
210C22×C4—1064no—
211C22×C4—2122no—
212C22×C4—2121no—
213C22×C4C22×D5312yes—
214C22×C4C22×D5311yescharacteristic
215C22×C4C22×D5312yes—
216C4⋊C4—538no—
217C4⋊C4—1064no—
218C22⋊C4—24no—
219C22⋊C4—41no—
220C22⋊C4—1064no—
221C22⋊C4—1064no—
222C22⋊C4—1064no—
223C22⋊C4—2122no—
224C22⋊C4—2121no—
225C42—534no—
226C42—1064no—
227C42—1064no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—534no—
237C23—1062no—
238C23—1062no—
239C23—1062no—
240C23—1062no—
241C23—1062no—
242C23—1062no—
243C23C2×C4×D5312yes—
244C23C2×C4×D5312yes—
245C23C23×D5311yescharacteristic
246D4—44no—
247D4—10616no—
248D4—10616no—
249D4—2124no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—534no—
259C2×C4—534no—
260C2×C4—534no—
261C2×C4—534no—
262C2×C4—1062no—
263C2×C4—1062no—
264C2×C4—1062no—
265C2×C4—1062no—
266C2×C4—1062no—
267C2×C4—2128no—
268C2×C4—2124no—
269C2×C4C2×C4×D5312yes—
270C2×C4D4×D5314yes—
271C22—24no—
272C22—24no—
273C22—22no—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—1064no—
282C22—1064no—
283C22—1064no—
284C22—2122no—
285C22—2122no—
286C22—2122no—
287C22—2122no—
288C22C22×C4×D5311yescharacteristic
289C22C2×D4×D5312yes—
290C22#41×D5312yes—
291C22#1313326211yescharacteristic
292C22#1313326211yescharacteristic
293C4—22no—
294C4—24no—
295C4—42no—
296C4—538no—
297C4—1064no—
298C4—2122no—
299C2—42no—
300C2—42no—
301C2—2124no—
302C2C4×D4×D5312yes—
303C2#1681109611yescharacteristic
304C2#1681125511yescharacteristic
305C2#1681129112yes—
306C2#1681135711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC53×#386C2
socleC23×C53C24
hypercenter#386D53
hypocenterC53#2366
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D53C2semidirect
3. C2×#13133177C2semidirect
4. #16810523C2semidirect
5. #16810523C2semidirect
6. #16810503C2semidirect
7. #16810501C2semidirect
8. #16810474C2semidirect
9. C53×#386C2semidirect
15. #13133213C4semidirect
39. C2×D4×C53C2×C4semidirect
49. C2×C22⋊D53C2×C4semidirect
50. C2×C22⋊D53C2×C4semidirect
58. C2×D212C2×C4semidirect
70. #11719919D4semidirect
73. C4×Dic53C2×C4semidirect
144. Dic53C4×D4semidirect
161. #386D53semidirect
163. C53#2366semidirect
202. C2×D4C4×D53semidirect