Finite Group Databasebeta

← #20278331

Subgroups — #20278331

Order 6784.

952 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C22×#11719947C211yesmaximal, characteristic, complemented
2C2×#13133204C211yesmaximal, characteristic, complemented
3C2×#13133188C211yesmaximal, characteristic, complemented
4#16810531C214yesmaximal, complemented
5#16810529C211yesmaximal, characteristic, complemented
6#16810526C211yesmaximal, characteristic, complemented
7#16810503C214yesmaximal, complemented
8C4×#11719921C211yesmaximal, characteristic, complemented
9C2×#127×C53C211yesmaximal, characteristic, complemented
10C2×#11719947—24no—
11C2×#11719947—24no—
12C2×#11719947—24no—
13C22×Dic106C2211yescharacteristic
14C22×C22⋊D53C2212yes—
15C22×C22⋊D53C2212yes—
16C23×Dic53C2212yes—
17C22×C4×D53C2211yescharacteristic
18#13133204—24no—
19#13133188—24no—
20C2×C4⋊Dic53C2211yescharacteristic
21C2×C22⋊Dic53C2212yes—
22C2×C22⋊Dic53C2214yes—
23C2×C22⋊Dic53C2214yes—
24C2×#11719921C2212yes—
25C2×C4×Dic53C2211yescharacteristic
26#13133124C2212yes—
27#13133124C2214yes—
28C22×D4×C53C2211yescharacteristic
29C22×D4×C53C2214yes—
30C22×D4×C53C2211yescharacteristic
31#127×C53—24no—
32C2×C42×C53C2211yescharacteristic
33C24×C53C2312yes—
34C24×C53C2312yes—
35C23×D53C2311yescharacteristic
36C2×D4×C53—28no—
37C2×D4×C53—22no—
38C2×D4×C53—22no—
39C2×D4×C53—22no—
40C2×D4×C53—22no—
41C2×D4×C53—22no—
42C2×D4×C53—28no—
43C2×D4×C53—44no—
44C2×D4×C53D414yescomplemented
45C22×C4×C53C2312yes—
46C22×C4×C53C2311yescharacteristic
47C2×C22⋊D53—24no—
48C2×C22⋊D53—24no—
49C2×C22⋊D53—24no—
50C2×C22⋊D53—24no—
51C2×C22⋊D53—44no—
52C22×Dic53—24no—
53C22×Dic53—24no—
54C22×Dic53—24no—
55C22×Dic53C2312yes—
56C22×Dic53C2312yes—
57C22×Dic53C2311yescharacteristic
58C22×Dic53C2312yes—
59#11719947—232no—
60C2×C4×D53—22no—
61C2×C4×D53—22no—
62C2×C4×D53D414yescomplemented
63C2×Dic106—22no—
64C2×Dic106—22no—
65C2×Dic106D414yescomplemented
66C42×C53—22no—
67C22⋊Dic53—24no—
68C22⋊Dic53—28no—
69C22⋊Dic53—44no—
70#11719921—24no—
71C4⋊Dic53—22no—
72C4×Dic53—22no—
73C23×C53—22no—
74C23×C53—22no—
75C23×C53—24no—
76C23×C53—24no—
77C23×C53—22no—
78C23×C53—22no—
79C23×C53—22no—
80C23×C53—24no—
81C23×C53—24no—
82C23×C53C2×D414yes—
83C23×C53C2411yescharacteristic
84C22×D53—21no—
85C22×D53—21no—
86C22×D53—22no—
87C22×D53—22no—
88C22×D53C2×D412yes—
89D4×C53—28no—
90D4×C53—416no—
91D4×C53—44no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53C2×D412yes—
95C2×C4×C53C2×D412yes—
96C2×C4×C53C2×D412yes—
97C2×C4×C53#4114yescomplemented
98C22⋊D53—216no—
99C22⋊D53—48no—
100C2×Dic53—216no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—21no—
104C2×Dic53—21no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53—24no—
109C2×Dic53—22no—
110C2×Dic53C2×D414yes—
111C2×Dic53C2×D412yes—
112C4×D53—28no—
113Dic106—28no—
114C22×C53—24no—
115C22×C53—28no—
116C22×C53—42no—
117C22×C53—42no—
118C22×C53—44no—
119C22×C53—42no—
120C22×C53—42no—
121C22×C53—42no—
122C22×C53—44no—
123C22×C53—44no—
124C22×C53C22×D411yescharacteristic
125C22×C53C22×D411yescharacteristic
126C22×C53C22×D411yescharacteristic
127C22×C53C2×#4112yes—
128C22×C53H(5,𝔽2)12yes—
129C2×D53—22no—
130C2×D53—22no—
131C2×D53—22no—
132C2×D53—42no—
133C2×D53—42no—
134C4×C53—24no—
135C4×C53#12014yes—
136Dic53—24no—
137Dic53—22no—
138Dic53—42no—
139Dic53—42no—
140#2431—531nomaximal, Sylow
141C2×C53—44no—
142C2×C53—82no—
143C2×C53C2×#12011yescharacteristic
144C2×C53#53211yescharacteristic
145C2×C53#53511yescharacteristic
146C2×C53#54612yes—
147C2×C53#55012yes—
148D53—42no—
149C22×#41—531no—
150C2×#127D5311yescharacteristic, complemented
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—1064no—
160C2×#41—1064no—
161C2×#41—1064no—
162C22×Q8—531no—
163C22×D4—532no—
164C22×D4—532no—
165C22×D4C2×D5311yescharacteristic
166C22×D4C2×D5314yes—
167C22×D4C2×D5311yescharacteristic
168C23×C4—531no—
169C23×C4—532no—
170#127—24no—
171#124—1064no—
172#121—1064no—
173C2×C4⋊C4—531no—
174C2×C22⋊C4—532no—
175C2×C22⋊C4—534no—
176C2×C22⋊C4—532no—
177C2×C22⋊C4—534no—
178C2×C42—531no—
179C2×C42C2×D5311yescharacteristic
180#95—532no—
181#95—534no—
182C24—531no—
183C24C22×D5312yes—
184C24C22×D5312yes—
185#41—10632no—
186C2×Q8—534no—
187C2×Q8—1062no—
188C2×Q8—1062no—
189C2×D4—28no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—28no—
195C2×D4—22no—
196C2×D4—44no—
197C2×D4—1064no—
198C2×D4—1064no—
199C2×D4—1064no—
200C2×D4—1064no—
201C2×D4—2124no—
202C2×D4C22⋊D5314yescomplemented
203C22×C4—531no—
204C22×C4—532no—
205C22×C4—534no—
206C22×C4—532no—
207C22×C4—532no—
208C22×C4—1064no—
209C22×C4—1064no—
210C22×C4—1064no—
211C22×C4—1062no—
212C22×C4—1062no—
213C22×C4C22×D5312yes—
214C22×C4C22×D5311yescharacteristic
215C4⋊C4—1062no—
216C22⋊C4—1064no—
217C22⋊C4—1068no—
218C22⋊C4—1064no—
219C22⋊C4—2124no—
220C42—22no—
221C42—1062no—
222C23—22no—
223C23—22no—
224C23—24no—
225C23—24no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—24no—
230C23—24no—
231C23—532no—
232C23—1061no—
233C23—1061no—
234C23—1062no—
235C23—1062no—
236C23C2×C22⋊D5314yes—
237C23C23×D5311yescharacteristic
238Q8—1068no—
239D4—28no—
240D4—416no—
241D4—44no—
242D4—10616no—
243D4—2128no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—532no—
247C2×C4—534no—
248C2×C4—10616no—
249C2×C4—1061no—
250C2×C4—1062no—
251C2×C4—1061no—
252C2×C4—1062no—
253C2×C4—1068no—
254C2×C4—1062no—
255C2×C4—1062no—
256C2×C4—1062no—
257C2×C4—1062no—
258C2×C4—1064no—
259C2×C4D4×D5312yes—
260C2×C4D4×D5312yes—
261C2×C4#1171994714yescomplemented
262C2×C4C2×C22⋊D5312yes—
263C22—28no—
264C22—24no—
265C22—44no—
266C22—42no—
267C22—42no—
268C22—44no—
269C22—44no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—1062no—
274C22—1062no—
275C22—1062no—
276C22—2122no—
277C22—2122no—
278C22C2×D4×D5311yescharacteristic
279C22C2×D4×D5311yescharacteristic
280C22C22×C22⋊D5311yescharacteristic
281C22C2×#1171994712yes—
282C22#1313326212yes—
283C4—24no—
284C4—1062no—
285C4—1064no—
286C4—2122no—
287C4—2122no—
288C4#1313318514yes—
289C2—44no—
290C2—82no—
291C2—2122no—
292C2C2×#1313318511yescharacteristic
293C2#1681117111yescharacteristic
294C2#1681118611yescharacteristic
295C2#1681126712yes—
296C2#1681129312yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#127×C53C2
hypercenterC2×#127D53
hypocenterC53#2431
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C22×#11719947C2semidirect
2. C2×#13133204C2semidirect
3. C2×#13133188C2semidirect
4. #16810531C2semidirect
5. #16810529C2semidirect
6. #16810526C2semidirect
7. #16810503C2semidirect
8. C4×#11719921C2semidirect
9. C2×#127×C53C2semidirect
44. C2×D4×C53D4semidirect
62. C2×C4×D53D4semidirect
65. C2×Dic106D4semidirect
97. C2×C4×C53#41semidirect
150. C2×#127D53semidirect
158. C53#2431semidirect
202. C2×D4C22⋊D53semidirect
261. C2×C4#11719947semidirect