Finite Group Databasebeta

← #20282746

Subgroups — #20282746

Order 6784.

528 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 125 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×#13133195C211yesmaximal, characteristic, complemented
2C2×#13133188C211yesmaximal, characteristic
3#16810559C212yesmaximal, complemented
4#16810555C211yesmaximal, characteristic, complemented
5#16810553C211yesmaximal, characteristic, complemented
6#16810539C212yesmaximal, complemented
7#16810538C212yesmaximal, complemented
8#16810537C211yesmaximal, characteristic, complemented
9#16810535C212yesmaximal, complemented
10#16810485C211yesmaximal, characteristic, complemented
11C53×#394C211yesmaximal, characteristic, complemented
12C22×Dic106C2211yescharacteristic
13C22×C22⋊D53C2212yes—
14C22×C4×D53C2211yescharacteristic
15#13133195—24no—
16#13133188—24no—
17C2×C4⋊Dic53C2211yescharacteristic
18C2×C4⋊Dic53C2212yes—
19C2×#11719919C2211yescharacteristic
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2212yes—
22C2×C22⋊Dic53C2212yes—
23C2×C22⋊Dic53C2212yes—
24C2×C22⋊Dic53C2212yes—
25C2×#11719921C2211yescharacteristic
26C2×#11719921C2211yescharacteristic
27C2×#11719921C2212yes—
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2211yescharacteristic
32C22×D4×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C22⋊C4×C53C2212yes—
35C2×C22⋊C4×C53C2212yes—
36#95×C53C2211yescharacteristic
37C24×C53C2312yes—
38C23×D53C2311yescharacteristic
39C2×D4×C53—22no—
40C2×D4×C53—42no—
41C22×C4×C53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2311yescharacteristic
45C2×C22⋊D53—24no—
46C2×C22⋊D53—42no—
47C2×C22⋊D53—42no—
48C22×Dic53C2312yes—
49C22×Dic53C2311yescharacteristic
50C22×Dic53C2312yes—
51C22×Dic53C2311yescharacteristic
52C22×Dic53C2311yescharacteristic
53C2×C4×D53—42no—
54C2×C4×D53D414yes—
55C2×Dic106—22no—
56C2×Dic106—42no—
57C4⋊C4×C53—22no—
58C22⋊C4×C53—42no—
59C22⋊C4×C53—42no—
60C22⋊Dic53—24no—
61C22⋊Dic53—42no—
62C22⋊Dic53—42no—
63#11719921—22no—
64#11719921—22no—
65#11719921—42no—
66C4⋊Dic53—22no—
67C4⋊Dic53—42no—
68#11719919—22no—
69#11719919—22no—
70#11719919—42no—
71C23×C53—22no—
72C23×C53—22no—
73C23×C53—22no—
74C23×C53—22no—
75C23×C53—22no—
76C23×C53—22no—
77C23×C53—22no—
78C23×C53C2411yescharacteristic
79C22×D53—22no—
80C22×D53—21no—
81C22×D53—21no—
82C22×D53—22no—
83C22×D53C2×D412yes—
84D4×C53—44no—
85C2×C4×C53—22no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—22no—
89C2×C4×C53—21no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—21no—
93C2×C4×C53—22no—
94C2×C4×C53C2×D412yes—
95C22⋊D53—48no—
96C2×Dic53—22no—
97C2×Dic53—21no—
98C2×Dic53—22no—
99C2×Dic53—21no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—21no—
105C2×Dic53—22no—
106C2×Dic53—21no—
107C2×Dic53—22no—
108C2×Dic53C2×D412yes—
109C4×D53—44no—
110Dic106—44no—
111C22×C53—42no—
112C22×C53—42no—
113C22×C53—42no—
114C22×C53—42no—
115C22×C53—42no—
116C22×C53—42no—
117C22×C53—42no—
118C22×C53C22×D411yescharacteristic
119C22×C53H(5,𝔽2)12yes—
120C22×C53H(5,𝔽2)11yescharacteristic
121C22×C53H(5,𝔽2)11yescharacteristic
122C22×C532−1+412yes—
123C2×D53—22no—
124C2×D53—22no—
125C2×D53—22no—
126C2×D53—42no—
127C2×D53—42no—
128C4×C53—22no—
129C4×C53—41no—
130C4×C53—41no—
131C4×C53—42no—
132Dic53—22no—
133Dic53—41no—
134Dic53—41no—
135Dic53—42no—
136Dic53—42no—
137#2816—531nomaximal, Sylow
138C2×C53—82no—
139C2×C53#53411yescharacteristic
140C2×C53#53511yescharacteristic
141C2×C53#53611yescharacteristic
142C2×C53#56212yes—
143C2×C53#56212yes—
144D53—42no—
145C2×C22⋊Q8—531no—
146C2×#121—531no—
147#399—532no—
148#398—531no—
149#398—531no—
150#394—531no—
151#394—532no—
152#394—532no—
153#394D5311yescharacteristic, complemented
154#393—532no—
155#381—531no—
156C53#281611yescharacteristic, complemented, Sylow
157C22×Q8—531no—
158C22×D4—532no—
159C22×D4C2×D5311yescharacteristic
160C23×C4—531no—
161C22⋊Q8—1064no—
162#121—1064no—
163C2×C4⋊C4—531no—
164C2×C4⋊C4—531no—
165C2×C4⋊C4—531no—
166C2×C4⋊C4—532no—
167C2×C4⋊C4—532no—
168C2×C4⋊C4C2×D5311yescharacteristic
169C2×C22⋊C4—532no—
170C2×C22⋊C4—531no—
171C2×C22⋊C4—531no—
172C2×C22⋊C4—532no—
173C2×C22⋊C4—532no—
174C2×C22⋊C4—532no—
175C2×C22⋊C4C2×D5312yes—
176C2×C22⋊C4C2×D5312yes—
177#95—532no—
178#95—531no—
179#95—532no—
180#95—532no—
181#95C2×D5311yescharacteristic
182C24—531no—
183C24C22×D5312yes—
184C2×Q8—1062no—
185C2×Q8—2122no—
186C2×D4—22no—
187C2×D4—42no—
188C2×D4—1064no—
189C2×D4—2122no—
190C2×D4—2122no—
191C22×C4—531no—
192C22×C4—534no—
193C22×C4—532no—
194C22×C4—531no—
195C22×C4—532no—
196C22×C4—531no—
197C22×C4—2122no—
198C22×C4C22×D5312yes—
199C22×C4C22×D5311yescharacteristic
200C22×C4C22×D5311yescharacteristic
201C22×C4C22×D5311yescharacteristic
202C4⋊C4—22no—
203C4⋊C4—1062no—
204C4⋊C4—1062no—
205C4⋊C4—1062no—
206C4⋊C4—2122no—
207C4⋊C4—2122no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—1062no—
211C22⋊C4—1062no—
212C22⋊C4—1064no—
213C22⋊C4—2122no—
214C22⋊C4—2122no—
215C22⋊C4—2122no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—532no—
224C23—1062no—
225C23—1061no—
226C23—1061no—
227C23—1062no—
228C23C23×D5311yescharacteristic
229Q8—2124no—
230D4—44no—
231D4—2128no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—532no—
242C2×C4—1062no—
243C2×C4—1061no—
244C2×C4—1062no—
245C2×C4—1061no—
246C2×C4—1062no—
247C2×C4—1062no—
248C2×C4—1061no—
249C2×C4—1062no—
250C2×C4—1061no—
251C2×C4—1062no—
252C2×C4—1062no—
253C2×C4—1062no—
254C2×C4—2124no—
255C2×C4D4×D5312yes—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—1062no—
264C22—1062no—
265C22—1062no—
266C22—2122no—
267C22—2122no—
268C22C2×D4×D5311yescharacteristic
269C22#1313326211yescharacteristic
270C22#1313326212yes—
271C22#1313326311yescharacteristic
272C22#1313326412yes—
273C4—22no—
274C4—42no—
275C4—41no—
276C4—41no—
277C4—1062no—
278C4—2122no—
279C4—2121no—
280C4—2121no—
281C4—2122no—
282C2—82no—
283C2—2122no—
284C2#1681118311yescharacteristic
285C2#1681118911yescharacteristic
286C2#1681119311yescharacteristic
287C2#1681136812yes—
288C2#1681137012yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#394C2
hypercenter#394D53
hypocenterC53#2816
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#13133195C2semidirect
3. #16810559C2semidirect
4. #16810555C2semidirect
5. #16810553C2semidirect
6. #16810539C2semidirect
7. #16810538C2semidirect
8. #16810537C2semidirect
9. #16810535C2semidirect
10. #16810485C2semidirect
11. C53×#394C2semidirect
153. #394D53semidirect
156. C53#2816semidirect