Finite Group Databasebeta

← #20280633

Subgroups — #20280633

Order 6784.

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

normal subgroups only

This group has 520 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×C4×C22⋊D53C211yesmaximal, characteristic
2#16810554C211yesmaximal, characteristic, complemented
3#16810551C212yesmaximal, complemented
4#16810549C211yesmaximal, characteristic, complemented
5#16810542C211yesmaximal, characteristic, complemented
6#16810538C212yesmaximal
7#16810534C211yesmaximal, characteristic, complemented
8#16810504C212yesmaximal, complemented
9#16810494C212yesmaximal, complemented
10#16810486C211yesmaximal, characteristic, complemented
11C2×C22⋊Q8×C53C211yesmaximal, characteristic, complemented
12C22×C22⋊D53C2211yescharacteristic
13C22×C4×D53C2211yescharacteristic
14C4×C22⋊D53—24no—
15C2×#11719919C2212yes—
16C2×#11719919C2211yescharacteristic
17C2×#11719919C2212yes—
18C2×C22⋊Dic53C2211yescharacteristic
19C2×C22⋊Dic53C2212yes—
20C2×#11719921C2211yescharacteristic
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23C2×C4×Dic53C2211yescharacteristic
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27#13133124C2212yes—
28#13133124C2211yescharacteristic
29#13133124C2211yescharacteristic
30#13133124C2212yes—
31C22×Q8×C53C2211yescharacteristic
32C23×C4×C53C2211yescharacteristic
33C22⋊Q8×C53—24no—
34C2×C4⋊C4×C53C2212yes—
35C2×C4⋊C4×C53C2211yescharacteristic
36C2×C22⋊C4×C53C2212yes—
37C24×C53C2311yescharacteristic
38C23×D53C2311yescharacteristic
39C2×Q8×C53—22no—
40C2×Q8×C53—42no—
41C22×C4×C53—42no—
42C22×C4×C53D414yes—
43C22×C4×C53C2312yes—
44C22×C4×C53C2312yes—
45C22×C4×C53C2311yescharacteristic
46C22×C4×C53C2311yescharacteristic
47C2×C22⋊D53—22no—
48C2×C22⋊D53—22no—
49C2×C22⋊D53—22no—
50C22×Dic53C2311yescharacteristic
51C22×Dic53C2312yes—
52C22×Dic53C2312yes—
53C22×Dic53C2311yescharacteristic
54C22×Dic53C2311yescharacteristic
55C2×C4×D53—22no—
56C2×C4×D53—42no—
57C4⋊C4×C53—24no—
58C4⋊C4×C53—22no—
59C22⋊C4×C53—24no—
60C22⋊Dic53—22no—
61C22⋊Dic53—42no—
62#11719921—24no—
63#11719921—22no—
64#11719921—42no—
65#11719919—22no—
66#11719919—24no—
67#11719919—42no—
68C4×Dic53—22no—
69C23×C53—21no—
70C23×C53—21no—
71C23×C53—22no—
72C23×C53—22no—
73C23×C53C2×D412yes—
74C23×C53C2411yescharacteristic
75C22×D53—21no—
76C22×D53—21no—
77C22×D53—22no—
78C22×D53—22no—
79C22×D53#4112yes—
80Q8×C53—44no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—44no—
90C2×C4×C53C2×D412yes—
91C2×C4×C53C2×D412yes—
92C22⋊D53—28no—
93C2×Dic53—21no—
94C2×Dic53—21no—
95C2×Dic53—21no—
96C2×Dic53—21no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53#4112yes—
106C4×D53—44no—
107C22×C53—22no—
108C22×C53—22no—
109C22×C53—22no—
110C22×C53—42no—
111C22×C53—42no—
112C22×C53C22×D411yescharacteristic
113C22×C53C2×#4111yescharacteristic
114C22×C53C2×#4111yescharacteristic
115C22×C53H(5,𝔽2)12yes—
116C22×C532−1+412yes—
117C2×D53—22no—
118C2×D53—24no—
119C2×D53—41no—
120C2×D53—41no—
121C2×D53—42no—
122C4×C53—22no—
123C4×C53—22no—
124C4×C53—42no—
125C4×C53—42no—
126Dic53—22no—
127Dic53—41no—
128Dic53—41no—
129Dic53—42no—
130Dic53—42no—
131#2638—531nomaximal, Sylow
132C2×C53—42no—
133C2×C53#52511yescharacteristic
134C2×C53#53411yescharacteristic
135C2×C53#53611yescharacteristic
136C2×C53#53912yes—
137C2×C53#54212yes—
138D53—42no—
139C2×C22⋊Q8D5311yescharacteristic, complemented
140C2×C4×D4—531no—
141#398—531no—
142#397—531no—
143#397—532no—
144#395—531no—
145#394—532no—
146#393—531no—
147#387—532no—
148#385—532no—
149#381—531no—
150C53#263811yescharacteristic, complemented, Sylow
151C22×Q8C2×D5311yescharacteristic
152C22×D4—531no—
153C23×C4—531no—
154C23×C4C2×D5311yescharacteristic
155C22⋊Q8—24no—
156C4×D4—1064no—
157C2×C4⋊C4—531no—
158C2×C4⋊C4—532no—
159C2×C4⋊C4—532no—
160C2×C4⋊C4C2×D5312yes—
161C2×C4⋊C4C2×D5311yescharacteristic
162C2×C22⋊C4—531no—
163C2×C22⋊C4—531no—
164C2×C22⋊C4—532no—
165C2×C22⋊C4—532no—
166C2×C22⋊C4—532no—
167C2×C22⋊C4C2×D5312yes—
168C2×C42—531no—
169#95—532no—
170#95—531no—
171#95—531no—
172#95—532no—
173#95—532no—
174#95—532no—
175#95—532no—
176C24—531no—
177C24C22×D5311yescharacteristic
178C2×Q8—22no—
179C2×Q8—42no—
180C2×D4—1062no—
181C2×D4—1062no—
182C2×D4—1062no—
183C22×C4—42no—
184C22×C4—531no—
185C22×C4—531no—
186C22×C4—532no—
187C22×C4—532no—
188C22×C4—531no—
189C22×C4—1062no—
190C22×C4—2122no—
191C22×C4C22⋊D5314yes—
192C22×C4C22×D5312yes—
193C22×C4C22×D5312yes—
194C22×C4C22×D5311yescharacteristic
195C22×C4C22×D5311yescharacteristic
196C4⋊C4—22no—
197C4⋊C4—24no—
198C4⋊C4—1064no—
199C4⋊C4—1062no—
200C4⋊C4—2122no—
201C22⋊C4—24no—
202C22⋊C4—1064no—
203C22⋊C4—1062no—
204C22⋊C4—1062no—
205C22⋊C4—2122no—
206C22⋊C4—2122no—
207C42—1062no—
208C23—21no—
209C23—21no—
210C23—22no—
211C23—22no—
212C23—532no—
213C23—1061no—
214C23—1061no—
215C23—1062no—
216C23—1062no—
217C23C2×C22⋊D5312yes—
218C23C23×D5311yescharacteristic
219Q8—44no—
220D4—1068no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—44no—
230C2×C4—532no—
231C2×C4—1061no—
232C2×C4—1061no—
233C2×C4—1061no—
234C2×C4—1061no—
235C2×C4—1062no—
236C2×C4—1062no—
237C2×C4—1062no—
238C2×C4—1062no—
239C2×C4—1062no—
240C2×C4—1062no—
241C2×C4—1062no—
242C2×C4—1062no—
243C2×C4—2124no—
244C2×C4C2×C22⋊D5312yes—
245C2×C4C2×C22⋊D5312yes—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—42no—
250C22—42no—
251C22—1062no—
252C22—1064no—
253C22—2121no—
254C22—2121no—
255C22—2122no—
256C22C22×C22⋊D5311yescharacteristic
257C22#41×D5311yescharacteristic
258C22#41×D5311yescharacteristic
259C22#1313326212yes—
260C22#1313326512yes—
261C4—22no—
262C4—22no—
263C4—42no—
264C4—42no—
265C4—1062no—
266C4—2121no—
267C4—2121no—
268C4—2122no—
269C4—2122no—
270C2—42no—
271C2—2122no—
272C2#1681113011yescharacteristic
273C2#1681118211yescharacteristic
274C2#1681119711yescharacteristic
275C2#1681121912yes—
276C2#1681124312yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×C22⋊Q8×C53C2
hypercenterC2×C22⋊Q8D53
hypocenterC53#2638
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #16810554C2semidirect
3. #16810551C2semidirect
4. #16810549C2semidirect
5. #16810542C2semidirect
7. #16810534C2semidirect
8. #16810504C2semidirect
9. #16810494C2semidirect
10. #16810486C2semidirect
11. C2×C22⋊Q8×C53C2semidirect
139. C2×C22⋊Q8D53semidirect
150. C53#2638semidirect