Finite Group Databasebeta

← #20280854

Subgroups — #20280854

Order 6784.

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

normal subgroups only

This group has 534 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×#13133211C211yesmaximal, characteristic, complemented
2#16810546C212yesmaximal, complemented
3#16810542C211yesmaximal, characteristic, complemented
4#16810539C212yesmaximal, complemented
5#16810537C211yesmaximal, characteristic, complemented
6#16810534C211yesmaximal, characteristic, complemented
7#16810531C211yesmaximal, characteristic
8#16810523C212yesmaximal, complemented
9#16810517C212yesmaximal, complemented
10C4×C4⋊Dic53C211yesmaximal, characteristic, complemented
11C2×#124×C53C211yesmaximal, characteristic, complemented
12C22×C22⋊D53C2212yes—
13#13133211—24no—
14C2×C4⋊Dic53C2212yes—
15C2×#11719919C2211yescharacteristic
16C2×#11719919C2211yescharacteristic
17C2×#11719919C2211yescharacteristic
18C2×#11719919C2212yes—
19C2×C22⋊Dic53C2212yes—
20C2×C22⋊Dic53C2212yes—
21C2×C22⋊Dic53C2212yes—
22C2×#11719921C2211yescharacteristic
23C2×#11719921C2211yescharacteristic
24C2×#11719921C2211yescharacteristic
25C2×#11719921C2212yes—
26C2×C4×Dic53C2212yes—
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2211yescharacteristic
30#13133124C2211yescharacteristic
31C22×Q8×C53C2211yescharacteristic
32C22×D4×C53C2211yescharacteristic
33#124×C53—24no—
34C2×C22⋊C4×C53C2212yes—
35C2×C22⋊C4×C53C2212yes—
36C2×C42×C53C2211yescharacteristic
37C24×C53C2312yes—
38C23×D53C2311yescharacteristic
39C2×Q8×C53—22no—
40C2×Q8×C53—42no—
41C2×D4×C53—22no—
42C2×D4×C53—42no—
43C22×C4×C53C2312yes—
44C22×C4×C53C2311yescharacteristic
45C22×C4×C53C2311yescharacteristic
46C22×C4×C53C2311yescharacteristic
47C2×C22⋊D53—24no—
48C2×C22⋊D53—42no—
49C2×C22⋊D53—42no—
50C22×Dic53C2312yes—
51C22×Dic53C2311yescharacteristic
52C22×Dic53C2311yescharacteristic
53C22×Dic53C2311yescharacteristic
54C22×Dic53C2312yes—
55C22⋊C4×C53—24no—
56C22⋊C4×C53—24no—
57C42×C53D414yescomplemented
58C22⋊Dic53—24no—
59C22⋊Dic53—42no—
60C22⋊Dic53—42no—
61#11719921—22no—
62#11719921—22no—
63#11719921—22no—
64#11719921—42no—
65C4⋊Dic53—24no—
66#11719919—22no—
67#11719919—22no—
68#11719919—22no—
69#11719919—42no—
70C4×Dic53—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—21no—
80C22×D53—22no—
81C22×D53—21no—
82C22×D53—22no—
83C22×D53—21no—
84Q8×C53—44no—
85D4×C53—44no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53C2×D412yes—
94C2×C4×C53C2×D412yes—
95C2×C4×C53C2×D412yes—
96C22⋊D53—48no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—21no—
100C2×Dic53—21no—
101C2×Dic53—21no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—21no—
108C2×Dic53—21no—
109C2×Dic53#4114yes—
110C22×C53—42no—
111C22×C53—42no—
112C22×C53—42no—
113C22×C53—42no—
114C22×C53—42no—
115C22×C53—42no—
116C22×C53—42no—
117C22×C53C22×D411yescharacteristic
118C22×C53C2×#4112yes—
119C22×C53H(5,𝔽2)12yes—
120C22×C53H(5,𝔽2)11yescharacteristic
121C22×C532−1+411yescharacteristic
122C2×D53—41no—
123C2×D53—41no—
124C2×D53—41no—
125C2×D53—42no—
126C2×D53—42no—
127C4×C53—22no—
128C4×C53—22no—
129C4×C53—22no—
130C4×C53—42no—
131Dic53—24no—
132Dic53—41no—
133Dic53—41no—
134Dic53—41no—
135Dic53—42no—
136#2662—531nomaximal, Sylow
137C2×C53—82no—
138C2×C53#53211yescharacteristic
139C2×C53#53511yescharacteristic
140C2×C53#53711yescharacteristic
141C2×C53#53812yes—
142C2×C53#54212yes—
143D53—81no—
144C2×#126—531no—
145C2×#124D5311yescharacteristic, complemented
146#396—532no—
147#395—531no—
148#394—531no—
149#394—532no—
150#393—531no—
151#392—531no—
152#390—532no—
153#388—532no—
154C4×C4⋊C4—531no—
155C53#266211yescharacteristic, complemented, Sylow
156C22×Q8C2×D5311yescharacteristic
157C22×D4—532no—
158C22×D4C2×D5311yescharacteristic
159#126—1064no—
160#124—24no—
161C2×C4⋊C4—531no—
162C2×C4⋊C4—531no—
163C2×C4⋊C4—531no—
164C2×C4⋊C4—532no—
165C2×C4⋊C4—532no—
166C2×C22⋊C4—532no—
167C2×C22⋊C4—531no—
168C2×C22⋊C4—531no—
169C2×C22⋊C4—531no—
170C2×C22⋊C4—532no—
171C2×C22⋊C4—532no—
172C2×C22⋊C4—532no—
173C2×C22⋊C4C2×D5312yes—
174C2×C22⋊C4C2×D5312yes—
175C2×C42—532no—
176C2×C42C2×D5311yescharacteristic
177#95—531no—
178#95—531no—
179#95—532no—
180#95—532no—
181C24—531no—
182C24C22×D5312yes—
183C2×Q8—22no—
184C2×Q8—42no—
185C2×D4—22no—
186C2×D4—42no—
187C2×D4—1064no—
188C2×D4—2122no—
189C2×D4—2122no—
190C22×C4—532no—
191C22×C4—532no—
192C22×C4—531no—
193C22×C4—531no—
194C22×C4—531no—
195C22×C4C22×D5312yes—
196C22×C4C22×D5311yescharacteristic
197C22×C4C22×D5311yescharacteristic
198C22×C4C22×D5311yescharacteristic
199C4⋊C4—1064no—
200C4⋊C4—1062no—
201C4⋊C4—1062no—
202C4⋊C4—1062no—
203C4⋊C4—2122no—
204C22⋊C4—24no—
205C22⋊C4—24no—
206C22⋊C4—1064no—
207C22⋊C4—1062no—
208C22⋊C4—1062no—
209C22⋊C4—1062no—
210C22⋊C4—2122no—
211C22⋊C4—2122no—
212C22⋊C4—2122no—
213C42—2122no—
214C42C22⋊D5314yescomplemented
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—1061no—
223C23—1062no—
224C23—1062no—
225C23—1061no—
226C23—1061no—
227C23C23×D5311yescharacteristic
228Q8—44no—
229D4—44no—
230D4—2128no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—534no—
239C2×C4—1062no—
240C2×C4—1062no—
241C2×C4—1061no—
242C2×C4—1061no—
243C2×C4—1061no—
244C2×C4—1062no—
245C2×C4—1062no—
246C2×C4—1062no—
247C2×C4—1062no—
248C2×C4—1062no—
249C2×C4—1061no—
250C2×C4—1061no—
251C2×C4C2×C22⋊D5312yes—
252C2×C4C2×C22⋊D5312yes—
253C2×C4C2×C22⋊D5312yes—
254C22—42no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—2121no—
262C22—2121no—
263C22—2121no—
264C22—2122no—
265C22—2122no—
266C22C22×C22⋊D5311yescharacteristic
267C22#41×D5312yes—
268C22#1313326212yes—
269C22#1313326311yescharacteristic
270C22#1313326411yescharacteristic
271C4—22no—
272C4—22no—
273C4—22no—
274C4—42no—
275C4—1064no—
276C4—2121no—
277C4—2121no—
278C4—2121no—
279C4—2122no—
280C2—82no—
281C2—4241no—
282C2#1681116911yescharacteristic
283C2#1681119011yescharacteristic
284C2#1681120111yescharacteristic
285C2#1681120712yes—
286C2#1681123712yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#124×C53C2
hypercenterC2×#124D53
hypocenterC53#2662
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#13133211C2semidirect
2. #16810546C2semidirect
3. #16810542C2semidirect
4. #16810539C2semidirect
5. #16810537C2semidirect
6. #16810534C2semidirect
8. #16810523C2semidirect
9. #16810517C2semidirect
10. C4×C4⋊Dic53C2semidirect
11. C2×#124×C53C2semidirect
57. C42×C53D4semidirect
145. C2×#124D53semidirect
155. C53#2662semidirect
214. C42C22⋊D53semidirect