Finite Group Databasebeta

← #20281739

Subgroups — #20281739

Order 6784.

592 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 137 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
2#16810539C212yesmaximal, complemented
3#16810532C212yesmaximal, complemented
4#16810531C212yesmaximal, complemented
5#16810530C211yesmaximal, characteristic, complemented
6#16810523C212yesmaximal, complemented
7#16810514C212yesmaximal, complemented
8#16810511C212yesmaximal, complemented
9C2×#124×C53C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C22×C22⋊D53C2212yes—
12C22×C22⋊D53C2212yes—
13#13133206—24no—
14C2×C4⋊Dic53C2211yescharacteristic
15C2×#11719919C2212yes—
16C2×C22⋊Dic53C2212yes—
17C2×C22⋊Dic53C2212yes—
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×C4×Dic53C2211yescharacteristic
23C2×C4×Dic53C2211yescharacteristic
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27C22×Q8×C53C2211yescharacteristic
28C22×D4×C53C2211yescharacteristic
29#124×C53—24no—
30C2×C22⋊C4×C53C2212yes—
31C2×C22⋊C4×C53C2212yes—
32C2×C42×C53C2211yescharacteristic
33C24×C53C2312yes—
34C23×D53C2312yes—
35C2×Q8×C53—41no—
36C2×Q8×C53—41no—
37C2×Q8×C53D414yescomplemented
38C2×D4×C53—22no—
39C2×D4×C53—41no—
40C2×D4×C53—41no—
41C22×C4×C53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2311yescharacteristic
45C2×C22⋊D53—24no—
46C2×C22⋊D53—42no—
47C2×C22⋊D53—42no—
48C2×C22⋊D53—42no—
49C2×C22⋊D53—42no—
50C2×C22⋊D53—42no—
51C22×Dic53C2312yes—
52C22×Dic53C2312yes—
53C22×Dic53C2312yes—
54C2×D212—22no—
55C2×D212—41no—
56C2×D212—41no—
57C22⋊C4×C53—24no—
58C22⋊C4×C53—24no—
59C42×C53—22no—
60C22⋊Dic53—24no—
61C22⋊Dic53—42no—
62#11719921—24no—
63#11719921—24no—
64#11719921—24no—
65#11719921—42no—
66C4⋊Dic53—41no—
67#11719919—42no—
68C4×Dic53—22no—
69C4×Dic53—41no—
70C23×C53—22no—
71C23×C53—22no—
72C23×C53—22no—
73C23×C53—22no—
74C23×C53—22no—
75C23×C53—22no—
76C23×C53—22no—
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—
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×C53C2×D412yes—
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#4114yes—
110D212—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×C53C2×#4112yes—
120C22×C53H(5,𝔽2)11yescharacteristic
121C22×C53H(5,𝔽2)11yescharacteristic
122C22×C53H(5,𝔽2)12yes—
123C2×D53—42no—
124C2×D53—42no—
125C2×D53—42no—
126C2×D53—42no—
127C2×D53—42no—
128C2×D53—42no—
129C2×D53—42no—
130C4×C53—22no—
131C4×C53—22no—
132C4×C53—22no—
133C4×C53—42no—
134Dic53—24no—
135Dic53—42no—
136Dic53—42no—
137#2733—531nomaximal, Sylow
138C2×C53—82no—
139C2×C53#53411yescharacteristic
140C2×C53#53812yes—
141C2×C53#55012yes—
142C2×C53#55511yescharacteristic
143C2×C53H(3,𝔽4)11yescharacteristic
144D53—82no—
145C2×#124—531no—
146C2×#124D5311yescharacteristic, complemented
147#394—532no—
148#392—532no—
149#392—532no—
150#391—531no—
151#390—532no—
152#388—532no—
153#388—532no—
154C53#273311yescharacteristic, complemented, Sylow
155C22×Q8C2×D5311yescharacteristic
156C22×D4—532no—
157C22×D4—532no—
158C22×D4—531no—
159C22×D4C2×D5311yescharacteristic
160#124—24no—
161#124—1064no—
162C2×C4⋊C4—532no—
163C2×C4⋊C4—531no—
164C2×C22⋊C4—532no—
165C2×C22⋊C4—532no—
166C2×C22⋊C4—532no—
167C2×C22⋊C4—532no—
168C2×C22⋊C4—532no—
169C2×C22⋊C4—532no—
170C2×C22⋊C4C2×D5312yes—
171C2×C22⋊C4C2×D5312yes—
172C2×C42—531no—
173C2×C42—531no—
174C2×C42C2×D5311yescharacteristic
175#95—532no—
176#95—532no—
177#95—532no—
178C24—532no—
179C24C22×D5312yes—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8C22⋊D5314yescomplemented
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—1064no—
187C2×D4—1062no—
188C2×D4—2121no—
189C2×D4—2121no—
190C2×D4—2122no—
191C2×D4—2122no—
192C2×D4—2122no—
193C2×D4—2122no—
194C2×D4—2122no—
195C22×C4—532no—
196C22×C4—532no—
197C22×C4—532no—
198C22×C4C22×D5312yes—
199C22×C4C22×D5311yescharacteristic
200C22×C4C22×D5311yescharacteristic
201C22×C4C22×D5311yescharacteristic
202C4⋊C4—2122no—
203C4⋊C4—2121no—
204C22⋊C4—24no—
205C22⋊C4—24no—
206C22⋊C4—1064no—
207C22⋊C4—1064no—
208C22⋊C4—1064no—
209C22⋊C4—1064no—
210C22⋊C4—2122no—
211C22⋊C4—2122no—
212C42—22no—
213C42—1062no—
214C42—2121no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—1062no—
223C23—1062no—
224C23—1062no—
225C23—1062no—
226C23—1062no—
227C23—1062no—
228C23—1062no—
229C23C23×D5311yescharacteristic
230Q8—44no—
231D4—44no—
232D4—2124no—
233D4—2128no—
234D4—2128no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—534no—
245C2×C4—1062no—
246C2×C4—1062no—
247C2×C4—1062no—
248C2×C4—1062no—
249C2×C4—1062no—
250C2×C4—1062no—
251C2×C4—1062no—
252C2×C4—1062no—
253C2×C4C2×C22⋊D5312yes—
254C2×C4C2×C22⋊D5312yes—
255C2×C4C2×C22⋊D5312yes—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—2122no—
264C22—2122no—
265C22—2122no—
266C22—2122no—
267C22—2122no—
268C22—2122no—
269C22—2122no—
270C22C22×C22⋊D5311yescharacteristic
271C22#41×D5312yes—
272C22#1313326211yescharacteristic
273C22#1313326211yescharacteristic
274C22#1313326312yes—
275C4—22no—
276C4—22no—
277C4—22no—
278C4—42no—
279C4—1064no—
280C4—2122no—
281C4—2122no—
282C2—82no—
283C2—4242no—
284C2#1681118211yescharacteristic
285C2#1681120712yes—
286C2#1681129212yes—
287C2#1681133511yescharacteristic
288C2#1681136511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#124×C53C2
hypercenterC2×#124D53
hypocenterC53#2733
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#13133206C2semidirect
2. #16810539C2semidirect
3. #16810532C2semidirect
4. #16810531C2semidirect
5. #16810530C2semidirect
6. #16810523C2semidirect
7. #16810514C2semidirect
8. #16810511C2semidirect
9. C2×#124×C53C2semidirect
37. C2×Q8×C53D4semidirect
146. C2×#124D53semidirect
154. C53#2733semidirect
182. C2×Q8C22⋊D53semidirect