Finite Group Databasebeta

← #20280395

Subgroups — #20280395

Order 6784.

576 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 139 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×#13133204C211yesmaximal, characteristic, complemented
2C2×#13133173C211yesmaximal, characteristic, complemented
3#16810544C211yesmaximal, characteristic, complemented
4#16810537C211yesmaximal, characteristic, complemented
5#16810535C211yesmaximal, characteristic, complemented
6#16810526C212yesmaximal, complemented
7#16810516C212yesmaximal, complemented
8#16810511C212yesmaximal, complemented
9#16810503C212yesmaximal, complemented
10#16810468C211yesmaximal, characteristic, complemented
11C53×#392C211yesmaximal, characteristic
12C22×Dic106C2211yescharacteristic
13C23×Dic53C2211yescharacteristic
14#13133204—24no—
15#13133173—24no—
16C2×C4⋊Dic53C2212yes—
17C2×#11719919C2211yescharacteristic
18C2×#11719919C2211yescharacteristic
19C2×C22⋊Dic53C2212yes—
20C2×C22⋊Dic53C2212yes—
21C2×C22⋊Dic53C2212yes—
22C2×C22⋊Dic53C2211yescharacteristic
23C2×C22⋊Dic53C2212yes—
24C2×C22⋊Dic53C2212yes—
25C2×C4×Dic53C2212yes—
26C2×C4×Dic53C2211yescharacteristic
27C2×C4×Dic53C2211yescharacteristic
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2211yescharacteristic
32C22×D4×C53C2212yes—
33C22×D4×C53C2211yescharacteristic
34C2×C22⋊C4×C53C2212yes—
35C2×C22⋊C4×C53C2211yescharacteristic
36#95×C53C2211yescharacteristic
37C24×C53C2312yes—
38C24×C53C2311yescharacteristic
39C2×D4×C53—24no—
40C2×D4×C53—22no—
41C2×D4×C53—24no—
42C2×D4×C53—42no—
43C2×D4×C53—42no—
44C22×C4×C53C2311yescharacteristic
45C22×C4×C53C2311yescharacteristic
46C22×C4×C53C2312yes—
47C22×Dic53—22no—
48C22×Dic53—22no—
49C22×Dic53—22no—
50C22×Dic53C2312yes—
51C22×Dic53C2312yes—
52C22×Dic53C2311yescharacteristic
53C22×Dic53C2311yescharacteristic
54C22×Dic53C2311yescharacteristic
55C22×Dic53C2311yescharacteristic
56C2×Dic106—22no—
57C2×Dic106—42no—
58C22⋊C4×C53—22no—
59C22⋊C4×C53—24no—
60C22⋊Dic53—22no—
61C22⋊Dic53—24no—
62C22⋊Dic53—24no—
63C22⋊Dic53—24no—
64C22⋊Dic53—42no—
65C22⋊Dic53—42no—
66C4⋊Dic53—42no—
67#11719919—22no—
68#11719919—22no—
69C4×Dic53—24no—
70C4×Dic53—22no—
71C4×Dic53D414yescomplemented
72C23×C53—21no—
73C23×C53—21no—
74C23×C53—22no—
75C23×C53—22no—
76C23×C53—22no—
77C23×C53—22no—
78C23×C53—22no—
79C23×C53—22no—
80C23×C53—22no—
81C23×C53—22no—
82C23×C53—22no—
83C23×C53#4112yes—
84C23×C53C2411yescharacteristic
85D4×C53—48no—
86D4×C53—44no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—21no—
91C2×C4×C53—22no—
92C2×C4×C53C2×D412yes—
93C2×C4×C53#4114yes—
94C2×Dic53—28no—
95C2×Dic53—21no—
96C2×Dic53—21no—
97C2×Dic53—21no—
98C2×Dic53—21no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—21no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53C2×D412yes—
109C2×Dic53C2×D412yes—
110C2×Dic53#4112yes—
111Dic106—44no—
112C22×C53—22no—
113C22×C53—24no—
114C22×C53—41no—
115C22×C53—41no—
116C22×C53—42no—
117C22×C53—42no—
118C22×C53—42no—
119C22×C53—42no—
120C22×C53—42no—
121C22×C53—42no—
122C22×C53—42no—
123C22×C53—42no—
124C22×C53C22×D411yescharacteristic
125C22×C53C2×#4111yescharacteristic
126C22×C53C2×#4111yescharacteristic
127C22×C53C2×#4112yes—
128C22×C53H(5,𝔽2)12yes—
129C4×C53—22no—
130C4×C53—24no—
131C4×C53—41no—
132Dic53—22no—
133Dic53—22no—
134Dic53—22no—
135Dic53—42no—
136Dic53—41no—
137Dic53—42no—
138#2610—531nomaximal, Sylow
139C2×C53—42no—
140C2×C53—82no—
141C2×C53#52511yescharacteristic
142C2×C53#53211yescharacteristic
143C2×C53#53511yescharacteristic
144C2×C53#55112yes—
145C2×C53#55512yes—
146C2×#124—531no—
147C2×#117—531no—
148#395—531no—
149#394—531no—
150#393—531no—
151#392D5311yescharacteristic
152#390—532no—
153#388—532no—
154#388—532no—
155#386—532no—
156#376—531no—
157C53#261011yescharacteristic, complemented, Sylow
158C22×Q8—531no—
159C22×D4C2×D5312yes—
160C22×D4C2×D5311yescharacteristic
161C23×C4—531no—
162#124—1064no—
163#117—1064no—
164C2×C4⋊C4—531no—
165C2×C4⋊C4—531no—
166C2×C4⋊C4—532no—
167C2×C22⋊C4—531no—
168C2×C22⋊C4—532no—
169C2×C22⋊C4—532no—
170C2×C22⋊C4—532no—
171C2×C22⋊C4—532no—
172C2×C22⋊C4—532no—
173C2×C22⋊C4C2×D5311yescharacteristic
174C2×C22⋊C4C2×D5312yes—
175C2×C42—531no—
176C2×C42—531no—
177C2×C42—532no—
178#95—532no—
179#95—531no—
180#95—532no—
181#95—532no—
182#95C2×D5311yescharacteristic
183C24C22×D5312yes—
184C24C22×D5311yescharacteristic
185C2×Q8—1062no—
186C2×Q8—2122no—
187C2×D4—24no—
188C2×D4—22no—
189C2×D4—24no—
190C2×D4—42no—
191C2×D4—42no—
192C22×C4—531no—
193C22×C4—531no—
194C22×C4—531no—
195C22×C4—532no—
196C22×C4—532no—
197C22×C4—531no—
198C22×C4—1062no—
199C22×C4—1062no—
200C22×C4—1062no—
201C22×C4C22×D5311yescharacteristic
202C22×C4C22×D5311yescharacteristic
203C22×C4C22×D5312yes—
204C4⋊C4—1062no—
205C4⋊C4—1062no—
206C4⋊C4—2122no—
207C22⋊C4—22no—
208C22⋊C4—24no—
209C22⋊C4—1062no—
210C22⋊C4—1064no—
211C22⋊C4—1064no—
212C22⋊C4—1064no—
213C22⋊C4—2122no—
214C22⋊C4—2122no—
215C42—534no—
216C42—1062no—
217C42—1064no—
218C23—21no—
219C23—21no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23#1171994712yes—
230C23C23×D5311yescharacteristic
231Q8—2124no—
232D4—48no—
233D4—44no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—532no—
240C2×C4—532no—
241C2×C4—532no—
242C2×C4—1068no—
243C2×C4—1061no—
244C2×C4—1061no—
245C2×C4—1062no—
246C2×C4—1061no—
247C2×C4—1061no—
248C2×C4—1061no—
249C2×C4—1062no—
250C2×C4—1062no—
251C2×C4—1062no—
252C2×C4—1062no—
253C2×C4—1062no—
254C2×C4—1062no—
255C2×C4—1062no—
256C2×C4D4×D5312yes—
257C2×C4#1171994714yes—
258C22—24no—
259C22—22no—
260C22—42no—
261C22—41no—
262C22—41no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22C2×D4×D5311yescharacteristic
271C22#41×D5311yescharacteristic
272C22C2×#1171994711yescharacteristic
273C22C2×#1171994712yes—
274C22#1313326212yes—
275C4—24no—
276C4—22no—
277C4—41no—
278C4—1062no—
279C4—1062no—
280C4—1062no—
281C4—2122no—
282C4—2121no—
283C4—2122no—
284C2—42no—
285C2—82no—
286C2#1681113311yescharacteristic
287C2#1681117111yescharacteristic
288C2#1681118811yescharacteristic
289C2#1681130412yes—
290C2#1681133712yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#392C2
hypercenter#392D53
hypocenterC53#2610
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#13133204C2semidirect
2. C2×#13133173C2semidirect
3. #16810544C2semidirect
4. #16810537C2semidirect
5. #16810535C2semidirect
6. #16810526C2semidirect
7. #16810516C2semidirect
8. #16810511C2semidirect
9. #16810503C2semidirect
10. #16810468C2semidirect
71. C4×Dic53D4semidirect
157. C53#2610semidirect