Finite Group Databasebeta

← #20279025

Subgroups — #20279025

Order 6784.

584 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 151 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×#13133177C211yesmaximal, characteristic
2C2×C4⋊C4×D53C211yesmaximal, characteristic, complemented
3#16810563C211yesmaximal, characteristic, complemented
4#16810561C211yesmaximal, characteristic, complemented
5#16810559C211yesmaximal, characteristic, complemented
6#16810518C212yesmaximal, complemented
7#16810499C212yesmaximal
8#16810494C212yesmaximal, complemented
9#16810482C211yesmaximal, characteristic, complemented
10#16810481C212yesmaximal, complemented
11C53×#399C211yesmaximal, characteristic, complemented
12C22×C22⋊D53C2211yescharacteristic
13C23×Dic53C2211yescharacteristic
14C22×C4×D53C2211yescharacteristic
15C22×C4×D53C2212yes—
16#13133177—24no—
17C2×C4⋊Dic53C2211yescharacteristic
18C2×#11719919C2212yes—
19C2×#11719919C2212yes—
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2212yes—
22C4⋊C4×D53—24no—
23C2×C22⋊Dic53C2212yes—
24C2×#11719921C2212yes—
25C2×#11719921C2211yescharacteristic
26C2×C4×Dic53C2211yescharacteristic
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2211yescharacteristic
30#13133124C2212yes—
31#13133124C2212yes—
32C2×C4⋊C4×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2212yes—
34C2×C22⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2212yes—
36#95×C53C2211yescharacteristic
37C24×C53C2311yescharacteristic
38C23×D53C2311yescharacteristic
39C22×C4×C53C2312yes—
40C22×C4×C53C2312yes—
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2311yescharacteristic
43C2×C22⋊D53—22no—
44C2×C22⋊D53—22no—
45C2×C22⋊D53Q814yes—
46C22×Dic53—22no—
47C22×Dic53—24no—
48C22×Dic53C2311yescharacteristic
49C22×Dic53C2312yes—
50C22×Dic53C2312yes—
51C22×Dic53C2311yescharacteristic
52C22×Dic53C2311yescharacteristic
53C2×C4×D53—24no—
54C2×C4×D53—24no—
55C2×C4×D53—42no—
56C2×C4×D53—42no—
57C2×C4×D53D414yes—
58C4⋊C4×C53—22no—
59C4⋊C4×C53—42no—
60C22⋊C4×C53—22no—
61C22⋊C4×C53—42no—
62C22⋊Dic53—42no—
63#11719921—22no—
64#11719921—24no—
65C4⋊Dic53—22no—
66#11719919—22no—
67#11719919—24no—
68#11719919—24no—
69#11719919—42no—
70C4×Dic53—22no—
71C23×C53—21no—
72C23×C53—22no—
73C23×C53—21no—
74C23×C53—22no—
75C23×C53C2×Q812yes—
76C23×C53C2411yescharacteristic
77C22×D53—22no—
78C22×D53—22no—
79C22×D53C2×D412yes—
80C22×D53C2×Q812yes—
81C22×D53#4112yes—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—21no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53C2×D412yes—
92C22⋊D53—28no—
93C2×Dic53—21no—
94C2×Dic53—21no—
95C2×Dic53—22no—
96C2×Dic53—21no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—44no—
104C2×Dic53C2×D412yes—
105C2×Dic53C2×Q812yes—
106C2×Dic53#4114yes—
107C4×D53—28no—
108C4×D53—48no—
109C22×C53—22no—
110C22×C53—24no—
111C22×C53—41no—
112C22×C53—41no—
113C22×C53—42no—
114C22×C53C22×D411yescharacteristic
115C22×C53C22×Q811yescharacteristic
116C22×C53C2×#4111yescharacteristic
117C22×C53C2×#4112yes—
118C22×C53H(5,𝔽2)12yes—
119C2×D53—22no—
120C2×D53—22no—
121C2×D53—24no—
122C2×D53—24no—
123C2×D53C22⋊Q814yes—
124C4×C53—22no—
125C4×C53—42no—
126C4×C53—41no—
127C4×C53—42no—
128Dic53—22no—
129Dic53—41no—
130Dic53—42no—
131Dic53—42no—
132Dic53#12314yes—
133#2487—531nomaximal, Sylow
134C2×C53—42no—
135C2×C53C2×C22⋊Q811yescharacteristic
136C2×C53C2×#12311yescharacteristic
137C2×C53#53511yescharacteristic
138C2×C53#55312yes—
139C2×C53#55412yes—
140D53—24no—
141C2×C4×D4—531no—
142C22×C4⋊C4—531no—
143#400—531no—
144#400—531no—
145#399—531no—
146#399D5311yescharacteristic, complemented
147#389—532no—
148#386—532no—
149#385—532no—
150#380—532no—
151#380—531no—
152C53#248711yescharacteristic, complemented, Sylow
153C22×D4—531no—
154C23×C4—531no—
155C23×C4—531no—
156C23×C4—532no—
157C4×D4—1064no—
158C2×C4⋊C4—532no—
159C2×C4⋊C4—531no—
160C2×C4⋊C4—531no—
161C2×C4⋊C4—532no—
162C2×C4⋊C4—532no—
163C2×C4⋊C4—1064no—
164C2×C4⋊C4C2×D5311yescharacteristic
165C2×C4⋊C4C2×D5312yes—
166C2×C22⋊C4—531no—
167C2×C22⋊C4—532no—
168C2×C22⋊C4—532no—
169C2×C22⋊C4C2×D5311yescharacteristic
170C2×C22⋊C4C2×D5312yes—
171C2×C42—531no—
172#95—532no—
173#95—532no—
174#95—531no—
175#95—532no—
176#95—532no—
177#95C2×D5311yescharacteristic
178C24—531no—
179C24C22×D5311yescharacteristic
180C2×D4—534no—
181C2×D4—1062no—
182C2×D4—1062no—
183C22×C4—531no—
184C22×C4—531no—
185C22×C4—534no—
186C22×C4—531no—
187C22×C4—532no—
188C22×C4—532no—
189C22×C4—1064no—
190C22×C4—1064no—
191C22×C4—1062no—
192C22×C4—1064no—
193C22×C4—2122no—
194C22×C4—2122no—
195C22×C4C22×D5312yes—
196C22×C4C22×D5312yes—
197C22×C4C22×D5311yescharacteristic
198C22×C4C22×D5311yescharacteristic
199C4⋊C4—22no—
200C4⋊C4—42no—
201C4⋊C4—1064no—
202C4⋊C4—1064no—
203C4⋊C4—1062no—
204C4⋊C4—1062no—
205C4⋊C4—2122no—
206C22⋊C4—22no—
207C22⋊C4—42no—
208C22⋊C4—1062no—
209C22⋊C4—1064no—
210C22⋊C4—2122no—
211C42—1062no—
212C23—21no—
213C23—22no—
214C23—21no—
215C23—22no—
216C23—532no—
217C23—532no—
218C23—532no—
219C23—1062no—
220C23—1062no—
221C23Q8×D5312yes—
222C23C23×D5311yescharacteristic
223D4—1068no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—532no—
234C2×C4—534no—
235C2×C4—532no—
236C2×C4—1068no—
237C2×C4—1061no—
238C2×C4—1061no—
239C2×C4—1061no—
240C2×C4—1062no—
241C2×C4—1062no—
242C2×C4—1062no—
243C2×C4—1062no—
244C2×C4—1062no—
245C2×C4—1062no—
246C2×C4—1062no—
247C2×C4—2128no—
248C2×C4—2124no—
249C2×C4D4×D5312yes—
250C22—24no—
251C22—22no—
252C22—42no—
253C22—41no—
254C22—41no—
255C22—534no—
256C22—1064no—
257C22—1064no—
258C22—1062no—
259C22—1062no—
260C22C2×D4×D5311yescharacteristic
261C22C2×Q8×D5311yescharacteristic
262C22#41×D5311yescharacteristic
263C22#41×D5312yes—
264C22#1313326212yes—
265C4—22no—
266C4—42no—
267C4—42no—
268C4—41no—
269C4—534no—
270C4—1062no—
271C4—2122no—
272C4—2121no—
273C4—2122no—
274C2—42no—
275C2—1064no—
276C2C22⋊Q8×D5311yescharacteristic
277C2#123×D5311yescharacteristic
278C2#1681118611yescharacteristic
279C2#1681131812yes—
280C2#1681132712yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#399C2
hypercenter#399D53
hypocenterC53#2487
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D53C2semidirect
3. #16810563C2semidirect
4. #16810561C2semidirect
5. #16810559C2semidirect
6. #16810518C2semidirect
8. #16810494C2semidirect
9. #16810482C2semidirect
10. #16810481C2semidirect
11. C53×#399C2semidirect
146. #399D53semidirect
152. C53#2487semidirect