Finite Group Databasebeta

← #20279202

Subgroups — #20279202

Order 6784.

596 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 141 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×#13133190C211yesmaximal, characteristic
2C2×C4⋊C4×D53C211yesmaximal, characteristic, complemented
3#16810550C211yesmaximal, characteristic, complemented
4#16810549C211yesmaximal, characteristic, complemented
5#16810548C211yesmaximal, characteristic, complemented
6#16810523C212yesmaximal, complemented
7#16810501C212yesmaximal, complemented
8#16810499C212yesmaximal, complemented
9#16810487C212yesmaximal, complemented
10#16810470C211yesmaximal, characteristic, complemented
11C53×#397C211yesmaximal, characteristic, complemented
12C22×D212C2211yescharacteristic
13C22×C22⋊D53C2211yescharacteristic
14C22×C22⋊D53C2211yescharacteristic
15C22×C4×D53C2211yescharacteristic
16C22×C4×D53C2212yes—
17#13133190—24no—
18C2×C4⋊Dic53C2211yescharacteristic
19C2×#11719919C2212yes—
20C2×#11719919C2211yescharacteristic
21C4⋊C4×D53—24no—
22C2×C22⋊Dic53C2212yes—
23C2×#11719921C2212yes—
24C2×#11719921C2211yescharacteristic
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27C2×C4×Dic53C2212yes—
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2211yescharacteristic
32C2×C4⋊C4×C53C2211yescharacteristic
33C2×C22⋊C4×C53C2211yescharacteristic
34C2×C22⋊C4×C53C2212yes—
35#95×C53C2212yes—
36#95×C53C2211yescharacteristic
37C24×C53C2311yescharacteristic
38C23×D53C2311yescharacteristic
39C23×D53C2311yescharacteristic
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2312yes—
42C22×C4×C53C2312yes—
43C22×C4×C53C2311yescharacteristic
44C2×C22⋊D53—22no—
45C2×C22⋊D53—24no—
46C2×C22⋊D53—22no—
47C2×C22⋊D53—24no—
48C22×Dic53C2311yescharacteristic
49C22×Dic53C2312yes—
50C22×Dic53C2312yes—
51C22×Dic53C2311yescharacteristic
52C2×D212—22no—
53C2×D212—24no—
54C2×C4×D53—24no—
55C2×C4×D53—24no—
56C2×C4×D53—42no—
57C2×C4×D53—42no—
58C2×C4×D53D414yes—
59C4⋊C4×C53—22no—
60C22⋊C4×C53—22no—
61C22⋊C4×C53—42no—
62C22⋊Dic53—42no—
63#11719921—24no—
64#11719921—22no—
65#11719921—42no—
66#11719921—42no—
67C4⋊Dic53—22no—
68#11719919—24no—
69#11719919—22no—
70C4×Dic53—24no—
71C23×C53—21no—
72C23×C53—22no—
73C23×C53—22no—
74C23×C53—21no—
75C23×C53—21no—
76C23×C53C2411yescharacteristic
77C22×D53—22no—
78C22×D53—21no—
79C22×D53—21no—
80C22×D53—21no—
81C22×D53—22no—
82C22×D53—22no—
83C22×D53—21no—
84C22×D53—21no—
85C22×D53C2×D412yes—
86C22×D53#4114yes—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—21no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53C2×D412yes—
97C22⋊D53—44no—
98C22⋊D53—44no—
99C2×Dic53—22no—
100C2×Dic53—21no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53C2×D412yes—
108C2×Dic53#4114yes—
109D212—44no—
110C4×D53—28no—
111C4×D53—48no—
112C22×C53—41no—
113C22×C53—41no—
114C22×C53—41no—
115C22×C53—42no—
116C22×C53—42no—
117C22×C53C22×D411yescharacteristic
118C22×C53C2×#4112yes—
119C22×C53C2×#4112yes—
120C22×C53H(5,𝔽2)11yescharacteristic
121C22×C53H(5,𝔽2)11yescharacteristic
122C2×D53—22no—
123C2×D53—22no—
124C2×D53—24no—
125C2×D53—24no—
126C2×D53—41no—
127C2×D53—41no—
128C2×D53—41no—
129C2×D53—42no—
130C2×D53—42no—
131C2×D53#12414yes—
132C4×C53—22no—
133C4×C53—41no—
134C4×C53—42no—
135C4×C53—42no—
136Dic53—41no—
137Dic53—42no—
138Dic53—42no—
139Dic53#12314yes—
140#2504—531nomaximal, Sylow
141C2×C53—81no—
142C2×C53C2×#12311yescharacteristic
143C2×C53C2×#12411yescharacteristic
144C2×C53#53511yescharacteristic
145C2×C53#55312yes—
146C2×C53#55312yes—
147D53—24no—
148D53—81no—
149C2×#121—531no—
150C22×C4⋊C4—531no—
151#397—531no—
152#397—531no—
153#397—531no—
154#397D5311yescharacteristic, complemented
155#390—532no—
156#386—532no—
157#386—532no—
158#382—532no—
159C4×C22⋊C4—531no—
160C53#250411yescharacteristic, complemented, Sylow
161C22×D4—531no—
162C22×D4—531no—
163C22×D4—531no—
164C23×C4—531no—
165C23×C4—532no—
166#121—1064no—
167C2×C4⋊C4—532no—
168C2×C4⋊C4—531no—
169C2×C4⋊C4—531no—
170C2×C4⋊C4—1064no—
171C2×C4⋊C4C2×D5311yescharacteristic
172C2×C22⋊C4—531no—
173C2×C22⋊C4—532no—
174C2×C22⋊C4—532no—
175C2×C22⋊C4—532no—
176C2×C22⋊C4—532no—
177C2×C22⋊C4C2×D5311yescharacteristic
178C2×C22⋊C4C2×D5312yes—
179C2×C42—532no—
180#95—531no—
181#95—532no—
182#95—532no—
183#95—532no—
184#95C2×D5311yescharacteristic
185#95C2×D5312yes—
186C24—531no—
187C24—531no—
188C24C22×D5311yescharacteristic
189C2×D4—1062no—
190C2×D4—1064no—
191C2×D4—1062no—
192C2×D4—1064no—
193C2×D4—1062no—
194C2×D4—1064no—
195C22×C4—531no—
196C22×C4—534no—
197C22×C4—531no—
198C22×C4—532no—
199C22×C4—532no—
200C22×C4—1064no—
201C22×C4—1064no—
202C22×C4—2122no—
203C22×C4—2122no—
204C22×C4C22×D5311yescharacteristic
205C22×C4C22×D5312yes—
206C22×C4C22×D5312yes—
207C22×C4C22×D5311yescharacteristic
208C4⋊C4—22no—
209C4⋊C4—1064no—
210C4⋊C4—1062no—
211C4⋊C4—1062no—
212C22⋊C4—22no—
213C22⋊C4—42no—
214C22⋊C4—1064no—
215C22⋊C4—1062no—
216C22⋊C4—2122no—
217C22⋊C4—2122no—
218C22⋊C4—2122no—
219C42—1064no—
220C23—21no—
221C23—22no—
222C23—22no—
223C23—21no—
224C23—21no—
225C23—532no—
226C23—534no—
227C23—1062no—
228C23—1061no—
229C23—1061no—
230C23—1061no—
231C23—1062no—
232C23—1062no—
233C23—1061no—
234C23—1061no—
235C23C23×D5311yescharacteristic
236D4—2124no—
237D4—2124no—
238D4—2124no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—532no—
249C2×C4—534no—
250C2×C4—1068no—
251C2×C4—1061no—
252C2×C4—1062no—
253C2×C4—1062no—
254C2×C4—1062no—
255C2×C4—1062no—
256C2×C4—1062no—
257C2×C4—1062no—
258C2×C4—1062no—
259C2×C4—2128no—
260C2×C4D4×D5312yes—
261C22—42no—
262C22—42no—
263C22—41no—
264C22—41no—
265C22—41no—
266C22—534no—
267C22—1064no—
268C22—1064no—
269C22—1062no—
270C22—1062no—
271C22—2122no—
272C22—2122no—
273C22—2121no—
274C22—2121no—
275C22—2121no—
276C22C2×D4×D5311yescharacteristic
277C22#41×D5312yes—
278C22#41×D5312yes—
279C22#1313326211yescharacteristic
280C22#1313326311yescharacteristic
281C4—22no—
282C4—42no—
283C4—41no—
284C4—42no—
285C4—534no—
286C4—2122no—
287C4—2121no—
288C4—2122no—
289C2—81no—
290C2—1064no—
291C2—4241no—
292C2#123×D5311yescharacteristic
293C2#124×D5311yescharacteristic
294C2#1681118911yescharacteristic
295C2#1681131812yes—
296C2#1681132412yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#397C2
hypercenter#397D53
hypocenterC53#2504
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D53C2semidirect
3. #16810550C2semidirect
4. #16810549C2semidirect
5. #16810548C2semidirect
6. #16810523C2semidirect
7. #16810501C2semidirect
8. #16810499C2semidirect
9. #16810487C2semidirect
10. #16810470C2semidirect
11. C53×#397C2semidirect
154. #397D53semidirect
160. C53#2504semidirect