Finite Group Databasebeta

← #20278330

Subgroups — #20278330

Order 6784.

952 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 185 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×#13133213C211yesmaximal, characteristic, complemented
2C2×#13133206C211yesmaximal, characteristic, complemented
3C2×#13133191C211yesmaximal, characteristic, complemented
4#16810531C214yesmaximal, complemented
5#16810530C211yesmaximal, characteristic, complemented
6#16810526C211yesmaximal, characteristic, complemented
7#16810502C214yesmaximal, complemented
8C4×C22⋊Dic53C211yesmaximal, characteristic, complemented
9C22×#41×C53C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C22×C22⋊D53C2214yes—
12C22×C22⋊D53C2212yes—
13#13133213—24no—
14#13133206—24no—
15#13133191—24no—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×C22⋊Dic53C2212yes—
18C2×C22⋊Dic53C2214yes—
19C2×#11719921C2212yes—
20C2×#11719921C2214yes—
21C2×C4×Dic53C2211yescharacteristic
22C2×C4×Dic53C2211yescharacteristic
23#13133124C2212yes—
24#13133124C2214yes—
25C2×#41×C53—24no—
26C2×#41×C53—24no—
27C2×#41×C53—24no—
28C22×Q8×C53C2211yescharacteristic
29C22×D4×C53C2212yes—
30C22×D4×C53C2211yescharacteristic
31C23×C4×C53C2212yes—
32C23×C4×C53C2211yescharacteristic
33C24×C53C2312yes—
34C24×C53C2311yescharacteristic
35C23×D53C2312yes—
36#41×C53—232no—
37C2×Q8×C53—22no—
38C2×Q8×C53—22no—
39C2×Q8×C53D414yescomplemented
40C2×D4×C53—24no—
41C2×D4×C53—24no—
42C2×D4×C53—24no—
43C2×D4×C53—22no—
44C2×D4×C53—22no—
45C2×D4×C53D414yescomplemented
46C22×C4×C53—24no—
47C22×C4×C53—24no—
48C22×C4×C53—24no—
49C22×C4×C53—22no—
50C22×C4×C53—22no—
51C22×C4×C53D414yescomplemented
52C22×C4×C53C2311yescharacteristic
53C22×C4×C53C2312yes—
54C22×C4×C53C2311yescharacteristic
55C2×C22⋊D53—24no—
56C2×C22⋊D53—28no—
57C2×C22⋊D53—28no—
58C2×C22⋊D53—44no—
59C2×C22⋊D53—44no—
60C22×Dic53C2312yes—
61C22×Dic53C2312yes—
62C22×Dic53C2312yes—
63C2×D212—22no—
64C2×D212—22no—
65C2×D212—22no—
66C22⋊Dic53—24no—
67C22⋊Dic53—44no—
68#11719921—24no—
69#11719921—28no—
70C4⋊Dic53—22no—
71C4×Dic53—22no—
72C4×Dic53—22no—
73C23×C53—22no—
74C23×C53—21no—
75C23×C53—22no—
76C23×C53—21no—
77C23×C53—24no—
78C23×C53—22no—
79C23×C53—22no—
80C23×C53—24no—
81C23×C53C2×D414yes—
82C23×C53C2×D412yes—
83C23×C53C2411yescharacteristic
84C22×D53—22no—
85C22×D53—22no—
86C22×D53—22no—
87C22×D53—24no—
88C22×D53—24no—
89Q8×C53—28no—
90D4×C53—216no—
91D4×C53—28no—
92C2×C4×C53—216no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—28no—
98C2×C4×C53C2×D412yes—
99C2×C4×C53C2×D414yes—
100C2×C4×C53C2×D412yes—
101C2×C4×C53C2×D412yes—
102C2×C4×C53C2×D412yes—
103C22⋊D53—416no—
104C22⋊D53—48no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—24no—
111C2×Dic53—22no—
112C2×Dic53#4114yescomplemented
113D212—44no—
114C22×C53—24no—
115C22×C53—22no—
116C22×C53—22no—
117C22×C53—22no—
118C22×C53—28no—
119C22×C53—42no—
120C22×C53—42no—
121C22×C53—42no—
122C22×C53—44no—
123C22×C53—42no—
124C22×C53C22×D411yescharacteristic
125C22×C53C22×D411yescharacteristic
126C22×C53C22×D411yescharacteristic
127C22×C53C2×#4112yes—
128C22×C53H(5,𝔽2)12yes—
129C2×D53—42no—
130C2×D53—42no—
131C2×D53—42no—
132C2×D53—44no—
133C2×D53—44no—
134C4×C53—22no—
135C4×C53—24no—
136C4×C53#12014yes—
137Dic53—24no—
138Dic53—42no—
139Dic53—42no—
140#2431—531nomaximal, Sylow
141C2×C53—42no—
142C2×C53—44no—
143C2×C53C2×#12011yescharacteristic
144C2×C53#53211yescharacteristic
145C2×C53#53511yescharacteristic
146C2×C53#54612yes—
147C2×C53#55012yes—
148D53—82no—
149C22×#41D5311yescharacteristic, complemented
150C2×#127—531no—
151C2×#124—531no—
152C2×#121—531no—
153#392—534no—
154#391—531no—
155#390—531no—
156#386—534no—
157C4×C22⋊C4—531no—
158C53#243111yescharacteristic, complemented, Sylow
159C2×#41—24no—
160C2×#41—24no—
161C2×#41—24no—
162C22×Q8C2×D5311yescharacteristic
163C22×D4—534no—
164C22×D4—532no—
165C22×D4—531no—
166C22×D4C2×D5312yes—
167C22×D4C2×D5311yescharacteristic
168C23×C4C2×D5312yes—
169C23×C4C2×D5311yescharacteristic
170#127—1064no—
171#124—1064no—
172#121—1064no—
173C2×C4⋊C4—531no—
174C2×C22⋊C4—532no—
175C2×C22⋊C4—532no—
176C2×C22⋊C4—534no—
177C2×C22⋊C4—534no—
178C2×C42—531no—
179C2×C42—531no—
180#95—532no—
181#95—534no—
182C24—532no—
183C24C22×D5311yescharacteristic
184C24C22×D5312yes—
185#41—232no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8C22⋊D5314yescomplemented
189C2×D4—24no—
190C2×D4—24no—
191C2×D4—24no—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—1068no—
195C2×D4—1064no—
196C2×D4—1068no—
197C2×D4—1062no—
198C2×D4—1062no—
199C2×D4—1062no—
200C2×D4—2124no—
201C2×D4—2124no—
202C2×D4C22⋊D5314yescomplemented
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—532no—
209C22×C4—532no—
210C22×C4—532no—
211C22×C4C22⋊D5314yescomplemented
212C22×C4C22×D5311yescharacteristic
213C22×C4C22×D5312yes—
214C22×C4C22×D5311yescharacteristic
215C4⋊C4—1062no—
216C22⋊C4—1064no—
217C22⋊C4—1064no—
218C22⋊C4—1068no—
219C22⋊C4—2124no—
220C42—1062no—
221C42—1062no—
222C23—22no—
223C23—21no—
224C23—22no—
225C23—21no—
226C23—22no—
227C23—24no—
228C23—24no—
229C23—22no—
230C23—1062no—
231C23—1062no—
232C23—1062no—
233C23—1064no—
234C23—1064no—
235C23C2×C22⋊D5314yes—
236C23C2×C22⋊D5312yes—
237C23C23×D5311yescharacteristic
238Q8—28no—
239D4—216no—
240D4—28no—
241D4—21216no—
242D4—2128no—
243D4—2124no—
244C2×C4—216no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—28no—
250C2×C4—534no—
251C2×C4—1062no—
252C2×C4—1062no—
253C2×C4—1062no—
254C2×C4—1062no—
255C2×C4—1062no—
256C2×C4—1062no—
257C2×C4—1064no—
258C2×C4C2×C22⋊D5312yes—
259C2×C4C2×C22⋊D5314yes—
260C2×C4C2×C22⋊D5312yes—
261C2×C4C2×C22⋊D5312yes—
262C2×C4C2×C22⋊D5312yes—
263C22—24no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—28no—
268C22—42no—
269C22—44no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—2124no—
274C22—2122no—
275C22—2122no—
276C22—2122no—
277C22—2124no—
278C22C22×C22⋊D5311yescharacteristic
279C22C22×C22⋊D5311yescharacteristic
280C22C22×C22⋊D5311yescharacteristic
281C22#41×D5312yes—
282C22#1313326312yes—
283C4—22no—
284C4—24no—
285C4—1064no—
286C4—2122no—
287C4—2122no—
288C4#1313318614yes—
289C2—42no—
290C2—44no—
291C2—4242no—
292C2C2×#1313318611yescharacteristic
293C2#1681116911yescharacteristic
294C2#1681118711yescharacteristic
295C2#1681126112yes—
296C2#1681129212yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC22×#41×C53C2
hypercenterC22×#41D53
hypocenterC53#2431
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#13133213C2semidirect
2. C2×#13133206C2semidirect
3. C2×#13133191C2semidirect
4. #16810531C2semidirect
5. #16810530C2semidirect
6. #16810526C2semidirect
7. #16810502C2semidirect
8. C4×C22⋊Dic53C2semidirect
9. C22×#41×C53C2semidirect
39. C2×Q8×C53D4semidirect
45. C2×D4×C53D4semidirect
51. C22×C4×C53D4semidirect
112. C2×Dic53#41semidirect
149. C22×#41D53semidirect
158. C53#2431semidirect
188. C2×Q8C22⋊D53semidirect
202. C2×D4C22⋊D53semidirect
211. C22×C4C22⋊D53semidirect