Finite Group Databasebeta

← #20280403

Subgroups — #20280403

Order 6784.

576 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 135 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×#13133206C211yesmaximal, characteristic, complemented
2C2×#13133173C211yesmaximal, characteristic, complemented
3#16810542C211yesmaximal, characteristic, complemented
4#16810541C211yesmaximal, characteristic, complemented
5#16810532C211yesmaximal, characteristic
6#16810523C212yesmaximal, complemented
7#16810512C212yesmaximal, complemented
8#16810512C212yesmaximal, complemented
9#16810500C212yesmaximal, complemented
10#16810468C211yesmaximal, characteristic, complemented
11C53×#393C211yesmaximal, characteristic, complemented
12C22×D212C2211yescharacteristic
13C22×C22⋊D53C2212yes—
14C23×Dic53C2211yescharacteristic
15#13133206—24no—
16#13133173—24no—
17C2×#11719919C2211yescharacteristic
18C2×#11719919C2211yescharacteristic
19C2×#11719919C2212yes—
20C2×C22⋊Dic53C2211yescharacteristic
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23C2×#11719921C2212yes—
24C2×#11719921C2212yes—
25C2×#11719921C2212yes—
26C2×C4×Dic53C2211yescharacteristic
27C2×C4×Dic53C2212yes—
28C2×C4×Dic53C2211yescharacteristic
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2211yescharacteristic
32C22×Q8×C53C2211yescharacteristic
33C2×C22⋊C4×C53C2211yescharacteristic
34C2×C22⋊C4×C53C2212yes—
35#95×C53C2212yes—
36#95×C53C2211yescharacteristic
37C24×C53C2311yescharacteristic
38C23×D53C2312yes—
39C2×Q8×C53—22no—
40C2×Q8×C53—42no—
41C22×C4×C53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2312yes—
44C22×C4×C53C2311yescharacteristic
45C2×C22⋊D53—24no—
46C2×C22⋊D53—24no—
47C2×C22⋊D53—42no—
48C22×Dic53—22no—
49C22×Dic53—22no—
50C22×Dic53—22no—
51C22×Dic53C2311yescharacteristic
52C22×Dic53C2311yescharacteristic
53C22×Dic53C2311yescharacteristic
54C22×Dic53C2312yes—
55C22×Dic53C2311yescharacteristic
56C2×D212—22no—
57C2×D212—42no—
58C22⋊C4×C53—22no—
59C22⋊C4×C53—42no—
60C22⋊Dic53—22no—
61#11719921—24no—
62#11719921—24no—
63#11719921—24no—
64#11719921—24no—
65#11719921—42no—
66#11719919—22no—
67#11719919—22no—
68#11719919—42no—
69C4×Dic53—22no—
70C4×Dic53—24no—
71C4×Dic53D414yescomplemented
72C23×C53—21no—
73C23×C53—21no—
74C23×C53—22no—
75C23×C53—22no—
76C23×C53#4112yes—
77C23×C53C2411yescharacteristic
78C22×D53—22no—
79C22×D53—22no—
80C22×D53—22no—
81C22×D53—22no—
82C22×D53—22no—
83C22×D53—22no—
84C22×D53—22no—
85Q8×C53—44no—
86C2×C4×C53—22no—
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×C53C2×D412yes—
96C22⋊D53—48no—
97C2×Dic53—28no—
98C2×Dic53—21no—
99C2×Dic53—21no—
100C2×Dic53—21no—
101C2×Dic53—21no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—21no—
106C2×Dic53—22no—
107C2×Dic53C2×D412yes—
108C2×Dic53C2×D412yes—
109C2×Dic53#4112yes—
110C2×Dic53#4114yes—
111D212—44no—
112C22×C53—22no—
113C22×C53—24no—
114C22×C53—41no—
115C22×C53—41no—
116C22×C53—42no—
117C22×C53C22×D411yescharacteristic
118C22×C53C2×#4111yescharacteristic
119C22×C53C2×#4112yes—
120C22×C53C2×#4111yescharacteristic
121C22×C53H(5,𝔽2)12yes—
122C2×D53—42no—
123C2×D53—42no—
124C2×D53—42no—
125C2×D53—42no—
126C2×D53—42no—
127C2×D53—42no—
128C2×D53—42no—
129C4×C53—22no—
130C4×C53—41no—
131C4×C53—42no—
132C4×C53—42no—
133Dic53—22no—
134Dic53—22no—
135Dic53—22no—
136Dic53—24no—
137Dic53—41no—
138#2610—531nomaximal, Sylow
139C2×C53—42no—
140C2×C53#52511yescharacteristic
141C2×C53#53211yescharacteristic
142C2×C53#53511yescharacteristic
143C2×C53#55112yes—
144C2×C53#55512yes—
145D53—82no—
146C2×#124—531no—
147C2×#117—531no—
148#395—531no—
149#394—531no—
150#393D5311yescharacteristic, complemented
151#392—531no—
152#390—532no—
153#388—532no—
154#388—532no—
155#386—532no—
156#376—531no—
157C53#261011yescharacteristic, complemented, Sylow
158C22×Q8C2×D5311yescharacteristic
159C22×D4—532no—
160C22×D4—531no—
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#95C2×D5311yescharacteristic
182#95C2×D5312yes—
183C24—532no—
184C24C22×D5311yescharacteristic
185C2×Q8—22no—
186C2×Q8—42no—
187C2×D4—1064no—
188C2×D4—1064no—
189C2×D4—1062no—
190C2×D4—2122no—
191C2×D4—2122no—
192C22×C4—532no—
193C22×C4—531no—
194C22×C4—531no—
195C22×C4—531no—
196C22×C4—531no—
197C22×C4—1062no—
198C22×C4—1062no—
199C22×C4—1062no—
200C22×C4C22×D5312yes—
201C22×C4C22×D5312yes—
202C22×C4C22×D5311yescharacteristic
203C22×C4C22×D5311yescharacteristic
204C4⋊C4—1062no—
205C4⋊C4—1062no—
206C4⋊C4—2122no—
207C22⋊C4—22no—
208C22⋊C4—42no—
209C22⋊C4—1064no—
210C22⋊C4—1064no—
211C22⋊C4—1064no—
212C22⋊C4—1064no—
213C22⋊C4—1062no—
214C22⋊C4—2122no—
215C42—534no—
216C42—1062no—
217C42—1064no—
218C23—21no—
219C23—21no—
220C23—22no—
221C23—22no—
222C23—1062no—
223C23—1062no—
224C23—1062no—
225C23—1062no—
226C23—1062no—
227C23—1062no—
228C23—1062no—
229C23#1171994912yes—
230C23C23×D5311yescharacteristic
231Q8—44no—
232D4—2124no—
233D4—2128no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—532no—
244C2×C4—532no—
245C2×C4—534no—
246C2×C4—532no—
247C2×C4—1068no—
248C2×C4—1061no—
249C2×C4—1062no—
250C2×C4—1061no—
251C2×C4—1061no—
252C2×C4—1061no—
253C2×C4—1062no—
254C2×C4—1062no—
255C2×C4—1061no—
256C2×C4—1062no—
257C2×C4D4×D5312yes—
258C22—24no—
259C22—22no—
260C22—42no—
261C22—41no—
262C22—41no—
263C22—2122no—
264C22—2122no—
265C22—2122no—
266C22—2122no—
267C22—2122no—
268C22—2122no—
269C22—2122no—
270C22C2×D4×D5311yescharacteristic
271C22#41×D5312yes—
272C22#41×D5311yescharacteristic
273C22C2×#1171994911yescharacteristic
274C22#1313326212yes—
275C4—22no—
276C4—42no—
277C4—41no—
278C4—42no—
279C4—1064no—
280C4—1062no—
281C4—1062no—
282C4—1062no—
283C4—2121no—
284C2—42no—
285C2—4242no—
286C2#1681113611yescharacteristic
287C2#1681117211yescharacteristic
288C2#1681118811yescharacteristic
289C2#1681130012yes—
290C2#1681133512yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#393C2
hypercenter#393D53
hypocenterC53#2610
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#13133206C2semidirect
2. C2×#13133173C2semidirect
3. #16810542C2semidirect
4. #16810541C2semidirect
6. #16810523C2semidirect
7. #16810512C2semidirect
8. #16810512C2semidirect
9. #16810500C2semidirect
10. #16810468C2semidirect
11. C53×#393C2semidirect
71. C4×Dic53D4semidirect
150. #393D53semidirect
157. C53#2610semidirect